Research ArticleThermomechanical Characterisation of Copper Diamond andBenchmarking with the MultiMat Experiment

Marcus Portelli 12 Michele Pasquali 3 Federico Carra 2 Alessandro Bertarelli 2

Pierluigi Mollicone 1 Nicholas Sammut 1 Oscar Sacristan de Frutos 2

Jorge Guardia Valenzuela 2 Erich Neubauer 4 Michael Kitzmantel 4

and David Grech 4

1University of Malta Msida 2080 Malta2CERN Esplanade des Particules 1 1211 Geneva 23 Switzerland3Sapienza University of Rome Via Eudossiana 18 Rome 00184 Italy4RHP-Technology GmbH Seibersdorf 2444 Austria

Correspondence should be addressed to Marcus Portelli marcusportellioutlookcom

Received 13 July 2020 Revised 25 November 2020 Accepted 12 December 2020 Published 5 January 2021

Academic Editor Trupti Ranjan Lenka

Copyright copy 2021Marcus Portelli et al+is is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

+e High-Luminosity Large Hadron Collider upgrade at CERN will result in an increase in the energy stored in the circulatingparticle beams making it necessary to assess the thermomechanical performance of currently used and newly developed materialsfor use in beam intercepting devices such as collimators and absorbers +is study describes the thermomechanical charac-terisation of a novel copper diamond grade selected for use in tertiary collimators of the HL-LHC +e data obtained are used tobuild an elastoplastic material model and implemented in numerical simulations performed to benchmark experimental dataobtained from the recently completed MultiMat experiment conducted at CERNrsquos HiRadMat facility where various materialsshaped as slender rods were tested under particle beam impact +e analyses focus on the dynamic longitudinal and flexuralresponse of the material with results showing that the material model is capable of replicating the material behaviour to asatisfactory level in both thermal and structural domains accurately matching experimental measurements in terms of tem-perature frequency content and amplitude

1 Introduction

+is work presents the material characterisation of thecopper diamond (CuCD) RHP3434 grade conducted atCERNrsquos facilities which includes a combination oftemperature-dependent measurements for density spe-cific heat capacity conductivity and coefficient ofthermal expansion along with impulse excitation tech-nique (IET) tests for the derivation of the dynamic elasticproperties and a four-point bending test for the stress-strain behaviour of the material in the plastic regime +ematerial model derived from the characterisation wasimplemented in transient thermomechanical simulationsperformed in the finite element analysis software ANSYS

modelling various types of particle beam impacts onCuCD samples tested at CERN

+e study is organised as follows Following the intro-duction Section 2 details information about copper diamondand the grade studied Section 3 details the material charac-terisation campaign carried out on the CuCD RHP3434 gradeconsisting of measurements of temperature-dependent prop-erties elastic constants and the plasticity curve Section 4presents the benchmarking of the material model by imple-menting it in a simulation compared with data obtained in theMultiMat experiment+is is followed by a discussion of resultsand future analyses were proposed for an extensivemodelling ofthe material behaviour in a wide variety of applications andscenarios

HindawiShock and VibrationVolume 2021 Article ID 8879400 18 pageshttpsdoiorg10115520218879400

2 Copper Diamond

Copper diamond is a novel composite material developed asa baseline material for use in collimators in the upcomingHL-LHC upgrade of the CERNrsquos LHC particle acceleratorspecifically as a replacement of the tungsten heavy alloyInermet180 +e material grade tested in this study de-veloped by RHP Technology [1] consists of diamond par-ticles dispersed in a copper matrix and hot pressed in a sparkplasma sintering process with various binding materials attemperatures slightly lower than the melting temperature ofcopper [2 3]+e binding elements form carbides during thesintering process aiding in the bonding of diamond andcopper particles which otherwise have a lack of affinity [4]

+e copper diamond grade tested in the MultiMat ex-periment CuCD RHP3434 developed by RHP Technologyconsists of a 50-50 volume percentage distribution of copperand diamond with a measured porosity of approximately8 determined by calculating the expected density of themixture and measuring the actual density of the compositematerial Microscopy images of the CuCD RHP3434 testedin this study with diamond particles dispersed in a coppermatrix are shown in Figure 1

CuCD is of particular interest in the field of beamintercepting devices due to the combination of propertiesprovided by the two main material constituentsmdashcoppercontributes to the thermal and electrical conductivity of thematerial while the diamond particles further improve ther-mal conductivity and aid in reducing the density and thecoefficient of thermal expansion [5] When compared withInermet180 CuCD has been identified as a more robustmaterial which can withstand a higher beam intensity withoutsevere damage and need for replacement [6] One of the mainlimitations of the material is the degradation of the diamond-copper boundary at temperatures above 250ndash300degC a result ofthe mismatch in the coefficient of thermal expansion betweenthe twomaterials causing the detaching of diamonds from thecopper matrix +is results in an irreversible change in ma-terial properties such as a reduction in thermal diffusivitywhich decreases with each thermal cycle above 250degC [7]

+e material is one of several materials tested in theMultiMat experiment conducted at CERNrsquos HiRadMat(high radiation to materials) facility in 2017 [8ndash11] whichuses beams from the super proton synchrotron (SPS) ac-celerator to test components and materials exposed to ex-treme conditions in CERNrsquos accelerator facility +eMultiMat experiment or HRMT36 was conducted in Oc-tober 2017 and aimed to offer a reusable platform for testingmaterials under high-intensity beam impacts [6 7] +etestbench shown in Figure 2 hosted a number of targetstations each having a length of 1m setup on a rotatablebarrel equipped with a Geneva mechanism allowing themovement of each target station into the shooting positionto be impacted by the incoming particle beam Testedspecimens were equipped with thermal probes and straingauges and placed on graphitic supports Other materialstested in the MultiMat experiment included heavy alloyssuch as titanium zirconium molybdenum Inermet180 andtantalum tungsten carbide-graphite composites silicon

carbide and carbon foams [13ndash16] +e pulse intensityranged from 1 to 288 bunches with a typical bunch intensityof 13times1011 protons Different types of beam impact weretested in the experiment namely axially centred impacts(resulting in a dynamic longitudinal response) offset im-pacts (resulting in an additional flexural response) andgrazing impacts (for the testing of material coatings)

3 Material Characterisation

A material characterisation campaign was carried out at theCERN laboratories to build an elasto-plastic material modelfor the CuCD RHP3434 grade comprising of the following

(i) A thermomechanical characterisation consisting oftemperature-dependent measurements for densityspecific heat capacity conductivity and coefficientof thermal expansion

(ii) An impulse excitation technique test for thecomputation of elastic properties (namely Youngrsquosmodulus shear modulus and Poissonrsquos ratio) atroom temperature

(iii) A four-point bending test at room temperature forthe analysis of the stress-strain behaviour of thematerial including plasticity

31 ermomechanical Characterisation A thermo-mechanical characterisation campaign was conductedmeasuring material properties as a function of temperatureMeasurements were made for the coefficient of thermalexpansion density specific heat capacity and conductivityof the material +e measurements were limited up to atemperature of approximately 300degC above which theboundary between the diamond particles and the coppermatrix was found to degrade irreversibly resulting in aregression in thermal properties [7] Note that the error barsshown in Figures 3ndash6 refer to the standard deviation be-tween measurements for different samples

For the measurement of the linear coefficient of thermalexpansion α a Netzsch DIL 402E dilatometer [17] wasutilised to measure the expansion (in the heating phase)and shrinkage (in the cooling phase) of CuCD specimensfollowing the ASTM E228-17 standard ldquoStandard Test

Figure 1 Microscopy image of CuCD RHP3434 with diamondparticles dispersed in a copper matrix

2 Shock and Vibration

Method for Linear ermal Expansion of Solid Materialswith a Push-Rod Dilatometerrdquo [18] Measured values for thelinear coefficient of thermal expansion are shown inFigure 3

Initial tests for the calculation of density ρ were carriedout using the Archimedes principle where the specimen isimmersed in a liquid and its volume is determined by thedisplacement of liquid+emass of the specimen can then bemeasured to calculate the density [19] Due to the highporosity of the CuCD grade being tested the alcohol used inthe experiment infiltrates the pores resulting in an over-estimation of the density calculated It was therefore deemedpreferable to calculate the density frommeasurements of thespecimensrsquo dimensions and weight Five specimens ofvarying dimensions were measured at three different pointsalong their length +e dimensions were averaged and thedensity was calculated for each specimen +e results for thefive measured specimens were similarly averaged to achievea final value for density at room temperature (ρ0) equal to5700 kgmiddotmminus3 Taking into consideration the volumetric co-efficient of thermal expansion β 3α the density can beexpressed as a function of temperature by consideringtemperature-dependent measurements for the coefficient ofthermal expansion as shown in equation (1) where ρ0 is theinitial density and T0 is the initial temperature ie roomtemperature +e temperature-dependent measurements forthe density of the CuCD RHP3434 grade are shown inFigure 4

ρ kgmminus31113960 1113961

ρ01 + β T minus T0( 1113857

(1)

For the measurement of the specific heat capacity (Cp) adifferential scanning calorimeter (DSC) was used With sucha setup the specific heat capacity was determined via theratio method [20] For each test sample three tests must beperformed the first test being a ldquobaselinerdquo which recordssystem characteristics and allows removal of the system biasfrom experimental data After a repeatable baseline has beenrecorded a reference material must be tested followingwhich the experimental sample is tested in exactly the same

way as reference [21 22] +e specific heat capacity mea-surements for the CuCD RHP3434 grade are shown inFigure 5

+e thermal diffusivity was measured by using a laserflash apparatus (LFA) which shoots a short laser pulse ontoone of the specimenrsquos faces andmeasures the temperature onthe opposite face with the use of an optical pyrometerKnowing the specimenrsquos thickness the thermal diffusivityacross a range of temperatures can be computed from thesereadings+e conductivity k was then found by multiplyingthe diffusivity (in m2middotsminus1) the density and the specific heatcapacity [23]+e calculated thermal conductivity results areshown in Figure 6

5

6

7

8

9

10

0 50 100 150 200 250 300 350

CTE

(10ndash6

K)

Temperature (degC)

Figure 3 Linear coefficient of thermal expansion of CuCDRHP3434

5580

5620

5660

5700

5740

0 50 100 150 200 250 300 350

Den

sity

(kg

m3 )

Temperature (degC)

Figure 4 Density of CuCD RHP3434

Figure 2 Rotatable barrel and mounted target stations in theHRMT36 experiment testbench

Shock and Vibration 3

Measured values for the CTE density specific heatcapacity and thermal conductivity are summarised at roomtemperature in Table 1

32 Impulse Excitation Technique Test An impulse excita-tion technique test was conducted to determine the elasticproperties of the CuCDRHP3434 grade As will be discussedin this section the expected flexural and torsional fre-quencies to be measured in the test were first calculated inaccordance with industry standards following which the testwas performed and the excited modes of interest identified+is was followed by a preliminary modal finite elementanalysis conducted with estimated elastic constants frommeasurements on other CuCD grades +e elastic properties

of the material were then optimised to the frequenciesmeasured in the IET test

+e impulse excitation technique test is a dynamicnondestructive material characterisation technique whichmeasures the resonant frequencies of a material which aresubsequently used to calculate the materialrsquos elastic con-stants at room temperature or at elevated temperatures [24]+e measurement consists of hitting the sample with ahammer and recording the induced vibration modes bymeans of a microphone +e vibrational signal measured inthe time domain is then converted to the frequency domainby a fast Fourier transform (FFT) following which theresonant frequencies can be determined allowing for thecalculation of the elastic properties of the material takinginto consideration the dimensions at hand

Different frequency modes (resonant frequencies) can beexcited in the specimen and are mainly dependent on theposition of the support wires the mechanical impulse andthe measuring deviceWith regard to the elastic properties ofthe material the two most important resonant frequenciesare the flexural and torsional frequencies which are mostlydependent on Youngrsquos and shear moduli for isotropicmaterials+e specimenrsquos elastic properties can be calculatedby considering the dimensions and geometry weight andthe measured resonant frequencies in accordance with theASTM E1876-15 ldquoStandard Test Method for DynamicYoungrsquos Modulus Shear Modulus and Poissonrsquos Ratio byImpulse Excitation of Vibrationrdquo [25 26]

+e natural flexural frequency denoted as ff is char-acteristic of Youngrsquos modulus E For the measurement ofthis frequency the specimen is supported at the nodes(where the amplitude of vibration is zero) and is excited atone of the antinodes (ie one of the ends of the rod) in orderto provoke maximum vibration as shown in Figure 7(a)Following the flexural frequency measurement Youngrsquosmodulus can be calculated by

E 09465m middot f

2f

b⎛⎝ ⎞⎠

L3

h31113888 1113889Rf (2)

wherem is the mass b h and L are the width thickness andlength of the specimen ff is the measured flexural frequencyand Rf is a geometrical correction factor which represents theaspect ratio of the specimen depending on the thickness andlength of the specimen

As can be seen in equation (2) the mass and dimensionsof the sample need to be measured accurately for reliablecalculation of Youngrsquos modulus Similarly the naturaltorsional frequency denoted as ft is characteristic of the

350

400

450

500

550

600

0 50 100 150 200 250

Ther

mal

cond

uctiv

ity (W

mK)

Temperature (degC)

Figure 6 +ermal conductivity of CuCD RHP3434

Table 1 CuCD RHP3434material properties at room temperature

Property Symbol Value Standarduncertainty Units

Density ρ 5700 plusmn33 kgmiddotmminus3

Specific heat capacity Cp 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity k 537 plusmn15 Wmiddotmminus1 Kminus1

Linear coefficient ofthermal expansion α 619 plusmn02 10minus6 Kminus1

350

400

450

500

550

600

650

0 50 100 150 200 250

Spec

ific h

eat c

apac

ity (J

kgK

)

Temperature (degC)

Figure 5 Specific heat capacity of CuCD RHP3434

4 Shock and Vibration

shear modulus In this case the specimen is supported alongboth longitudinal and transverse axes as can be seen inFigure 7(b) while the mechanical excitation is performed atone corner inducing a twisting motion in the beam ratherthan flexing it Following the torsional frequency mea-surement the shear modulus can be calculated by thefollowing

G 4 middot L middot m middot f

2t

b middot hRt (3)

where ft is the measured torsional frequency and Rt is ageometrical correction factor Poissonrsquos ratio ] is calculatedby ] [(E2G) minus 1] and the material damping can also becharacterised by considering the logarithmic decrement ofthe decaying amplitude of vibration

While the solutions defined by equations (2) and (3)provide adequate results for elastic properties a numericalapproach encompasses the triaxiality of the problem and canallow for the optimisation of the elastic constants allowingfor the reverse engineering of these values from the mea-sured frequencies with great accuracy In addition a nu-merical approach has the benefit of allowing the calculationof elastic constants for anisotropic bodies +e followingsection describes the numerical approach adopted in thisstudy to determine the elastic properties of the tested CuCDgrade For a specimen with a square cross section and madeof an isotropic material measurements of resonant fre-quencies in different transverse planes (referred to as in-plane and through-plane) are identical since the orientationof the specimen does not have an effect on the materialproperties Whilst the method described in the followingsection refers to isotropic bodiesmdashas is the case forCuCDmdashit can easily be extended to anisotropic bodies byconsidering flexural and torsional mode measurements forboth in-plane and through-plane conditions

+e finite element analysis optimisation was implementedin the software ANSYS Workbench [27] using a modal FEManalysis which for a defined geometry calculates the resonantfrequencies for the given elastic properties and density Ar-bitrary elastic properties are initially input which are thenoptimised with the resonant frequencies measured from theexperiment During preprocessing the geometry is dividedinto sub-blocks allowing the definition of several lines and

surfaces which can be constrained to force the excitation of aparticular resonance mode +e density value input in thesimulation is also of particular importance given that thefrequency response is directly dependent on this property (fora given set of properties and dimensions a higher densityresults in lower frequencies and vice versa) +e experi-mentally derived density values detailed in the previoussections of this study were used for the CuCD grade inquestion (ie 5700 kgm3 at room temperature)

If the specimen tested has a rectangular cross section twopossible flexural planes are present During the simulationsetup the flexural mode in the direction of interest (ie thedirection excited in the experiment) is forced by defining asymmetrical plane in this direction +is ensures movement inthe direction perpendicular to the symmetrical plane is re-stricted For the excitation of the torsional mode the longi-tudinal and transverse axes are fixed (ie displacement equal tozero) blocking movement of these lines in all directions Aconstraint equation is applied to the two faces at the extremitiesof the specimen promoting a natural torsional behaviour wherethe two faces rotate in opposite directions +e relevantboundary conditions for the two analyses are shown in Figure 8

Once the initial simulations are setup and completed(utilising approximate values for the elastic constants) theresulting first flexural and torsional frequencies are para-metrised and optimised to the experimental measurements+e variables influencing the two frequencies (flexural andtorsional) are Youngrsquos modulus and Poissonrsquos ratio (theshear modulus is calculated from the other properties) Arange of acceptable results is defined for each parameter

+e IET test was carried out on a specimen with di-mensions 247times1241times 1035mm3 (lengthtimeswidthtimes

thickness) A spectrogram showing the intensity of theexcited frequencies with respect to time in the testedspecimen setup in the configuration shown in Figure 7(a) isshown in Figure 9 As shown in the spectrogram there arevarious excited frequencies in the measured range In orderto identify the modes of interest using Youngrsquos modulus of208GPa and Poissonrsquos ratio of 022 as a first approximationthe first flexural and torsional modes were calculated byequations (2) and (3) as shown in Table 2+is allows for theidentification of the first torsional and flexural frequenciesfrom the various unwanted frequencies excited in theexperiment

(a) (b)

Figure 7 IET setup for flexural (a) and torsional (b) mode excitation

Shock and Vibration 5

From the experimental results the first flexural fre-quency can be easily identified with a peak at 1016 kHz+eslender nature of the specimen made it difficult to excite thetorsional vibration however two possible frequencies cor-responding to the torsional mode were identified in the

flexural measurement at 6794 kHz and 7238 kHz (two faintlines between the 5583 kHz and 9082 kHz readings)

Following the identification of the first flexural mode andthe two possible torsional modes the initial estimates forYoungrsquos modulus and Poissonrsquos ratiomdash208GPa and 022respectivelymdashwere used as a primary input along with thedensity of the material in the optimisation procedure A firstestimation of the flexural and torsional modes was com-puted for the given material properties +e first flexural andtorsional modes determined by the modal analyses areshown in Figure 10 As can be seen additional lsquoundesiredrsquomodes such as the first longitudinal harmonic can also beexcited since movement in this direction is unrestricted

(a) (b)

Figure 8 Boundary conditions forcing mode of interest (a) symmetry plane promoting flexural vibration and (b) fixed axes promotingtorsional vibration and a constraint equation limiting motion of the two extremities of the specimen to rotate in opposite directions

14558319

0036231110

[Hz][Pa][s]

1328430009099575

[Hz][Pa][s]

908203001697770

[Hz][Pa][s]

286255000570350

[Hz][Pa][s]

1016240064722175

[Hz][Pa][s]

7238520001971475

[Hz][Pa][s]

679401000260141

[Hz][Pa][s]

15325930003508175

[Hz][Pa][s]

18077090006504325

[Hz][Pa][s]

13

12

11

10

9

8

7

6

5

4

3

2

1

0

Tim

e (s)

Inte

nsity

(Pa)

Frequency (Hz)10000 20000 30000

1μ

10μ

100μ

1m

10m

Figure 9 Spectrogram showing the IET test flexural frequency measurement as a function of time

Table 2 Calculated flexural and torsional modes for positions Aand B with initial estimates of elastic properties (E 208GPa and] 022)

Specimen dimensionslengthtimeswidthtimes thickness

Flexuralmode (kHz)

Torsionalmode (kHz)

247times1241times 1035mm3 1047 7054

6 Shock and Vibration

47277 Max

42056

36835

31614

26393

21172

15951

1073

55088

28774 Min

000

2500 7500

5000 10000 (mm)

Y

XZ

(a)

Y

XZ

57916 Max

51481

45046

38611

32176

2574

19305

1287

64351

0 Min

000

2500 7500

5000 10000 (mm)

(b)

Y

XZ

33437 Max

29722

26007

22292

18577

14862

11147

7432

37171

02117 Min

000

2500 7500

5000 10000 (mm)

(c)

Figure 10 First flexural and torsional modes simulated with estimated material properties (a b) and additional modes such as the firstlongitudinal one (c)

Shock and Vibration 7

Following the initial modal analysis with estimatedproperties the computed frequencies were parametrised andoptimised with the frequencies measured in the experimentYoungrsquos modulus was constricted to a range between187GPa and 229GPa while Poissonrsquos ratio was restricted tovalues ranging from 01 to 03 +e optimisation procedurevaries these two values attempting to match the computedfrequencies with the measured values in the experiment

+e relatively minor difference (11) in the two mea-sured frequencies considered for the torsional mode resultedin a significant effect on the computed Poissonrsquos ratio Forff 1016 kHz and ft 6794 kHz Youngrsquos modulus iscomputed at 1937GPa with Poissonrsquos ratio of 0248 whilefor ff 1016 kHz and ft 7238 kHz Youngrsquos modulus iscalculated at 1936GPa with Poissonrsquos ratio of 0104Considering previous data on similar CuCD grades thevalue of ] 0248 was deemed to be the physically valid oneand retained for the material model along with a roundedYoungrsquos modulus value of 194GPa [13] +e final values forYoungrsquos modulus and Poissonrsquos ratio are shown in Table 3

33 Plasticity-Four-Point Bending Experiment To model thematerial behaviour beyond the elastic regime a four-pointbending test was performed allowing for the implementa-tion of a multilinear hardening model in the numericalanalyses A ZwickRoell Z400 universal testing machine wasused for the four-point flexural test in accordance with theASTM C651 standard ldquoFlexural Strength of Carbon andGraphite by Four-Point Loadingrdquo [28] With this experi-mental setup as shown in the schematic in Figure 11 thebending force is applied at two points along the length of thespecimen which is supported at its extremities Specimenswith an average length of 30mm width of 53mm andthickness of 51mm were tested with such a setup

In the experiment the bending force is progressivelyincreased with the corresponding induced axial strain in thespecimen measured with a strain gauge glued to its surface+e maximum bending moment M is deduced from theforce F and the distance between the load application and thesupports L1 byM (L1F2) In the linear elastic domainthe stress distribution through the sample thickness can beassumed to be linear and the maximum can be calculated byσ (Mh2I) where I is the flexural moment of inertia and his the sample thickness However the force-strain relationmeasured in the experiment is nonlinear indicating thatthere is a nonlinear stress distribution across the thickness ofthe specimen as shown in Figure 12 +e maximum stressthus has to be calculated taking into consideration thisnonlinearity For small curvatures the axial strain can beconsidered to be linearly distributed through the thickness[7]

+e stress must therefore be calculated out of the generalmoment-strain relationship +e moment can be expressedas an integration of the stress distribution through thethickness by

M b 1113946h2

minush2σz(x)xdx (4)

where x is the vertical coordinate from the neutral axis b isthe specimenrsquos width and h is its thickness (height ie alongx) +e moment can be expressed in terms of the measuredstrain as follows

M εtotmax1113872 1113873 2b 1113946h2

0σz

2x

hεtotmax1113874 1113875xdx (5)

where σz((2xh)εtotmax) indicates the stress-strain curve whichcan be numerically derived given the applied momentM(εtotmax) and the measured strain εtotmax +e problem can besolved for each measured value of force and strain resultingin a piecewise stress-strain relationship ie a multilinearhardening curve describing the plastic behaviour of thematerial shown in Figure 13 +e material can be seen toquickly lose linearity and undergoes a significant amount ofplastic deformation as a result of the highly ductile coppermatrix

Table 3 Youngrsquos modulus and Poissonrsquos ratio results determinedby the IET optimisation procedure

Property Symbol Value Uncertainty UnitsYoungrsquos modulus E 194 plusmn32 GPaPoissonrsquos ratio ] 0248 plusmn0002 mdash

F2F2

LL1 L1

Figure 11 Four-point bending experimental setup

StrainStress

Figure 12 Axial strain (linear) and stress (nonlinear) distributionthrough the sample thickness

8 Shock and Vibration

One aspect which is not considered in the model is thecompression behaviour of the material which is assumed tobe identical to that in tension +is assumption may not bevalid at high compressive strains at which point the dia-mond particles may come in contact In contrast when thematerial is loaded in tension the bond between the diamondparticles and the copper matrix may be lost

4 Benchmarking of the Material Model

41 HRMT36 e MultiMat Experiment CERNrsquos HiR-adMat facility is used to test materials used for componentsexposed to extreme conditions in CERNrsquos acceleratorcomplex [10] Upcoming LHC upgrades such as the high-luminosity LHC (HL-LHC) upgrade and the proposedFuture Circular Collider (FCC) require an increase in thebeam energy and thus bring the need for high-performingmaterials+is is especially true for use in components at riskof exposure to accidental beam impacts such as collimatorsand other beam intercepting devices [29]

Experiments conducted in the HiRadMat facility includetests on full-scale collimator jaws [29ndash31] as well as specimenswith simple geometries allowing the benchmarking of nu-merical results obtained through numerical analyses [9 14]+e HRMT36 ldquoMultiMatrdquo experiment conducted in October2017 was designed to offer a reusable platform to test a widerange of novel materials developed for beam interceptingdevices impacted by particle beams with energy densities inthe range of those experienced in the HL-LHC [32 33]

+e test bench hosted 16 target stations mounted on arotatable barrel with each station having a total length of1m+ematerial specimens were mounted in series and hadlengths of 120 or 247mm with cross sections varying from8times 8 to 12times115mm2 18 different materials were tested intotal+e acquisition system included electrical strain gaugesand temperature probes placed on the material specimensalong with remote instrumentation including a laser

Doppler vibrometer (LDV) oriented towards the top face ofthe specimens and a side-mounted radiation-hard high-definition camera [12]+e dimensions for CuCD specimenstested and typical positions of strain gauges and thermalprobes are shown in Figure 14+e target station hosting theCuCD RHP3434 samples consisted of 4 specimens in serieseach having a length of 247mm (with a 3mm space in-between specimens) and a square cross section with an edgelength of 10mm All four specimens were equipped withlongitudinal strain gauges to measure dynamic phenomenainduced by the particle beam impacts

Other than copper diamond (of which two grades from twodifferent suppliers were tested) various materials and materialgrades were tested in the MultiMat experiment including 4grades of MoGr 3 different coatings (Cu Mo and TiN) andnovel carbon-based materials such as highly ordered pyrolyticgraphite (HOPG) and titanium-graphite (TG-1100) +e pulseintensity to which specimens were subjected to ranged from 1to 288 bunches with a typical bunch intensity of 13times1011protons along with nominal beam RMS sizes of 025times 02505times 05 and 2times 2mm2 +e three different types of impacttested were axially centred impacts offset impacts and grazingimpacts (where a coating was present)

42 Modelling and Benchmarking with Experimental Results+e material model for the CuCD RHP3434 grade con-sisting of the measured thermomechanical temperature-dependent properties (Figures 2ndash5 summarised in Table 1)dynamic elastic constants from the IET test (Table 3) andmultilinear hardening curve from the four-point bendingtest (Figure 13) as shown in the previous sections wasincluded in a simply-coupled implicit dynamic thermo-mechanical simulation performed in the FEA softwareANSYS +e material is assumed to be isotropic in naturebased on previous study of the material [6] and is assumedto behave in an elasto-plastic manner based on the four-point bending test presented in this study Similar analyseshave been performed on other materials tested in theMultiMat experiment and information on the thermo-mechanical modelling aspects in such scenarios is detailed inprevious studies [13 15] Similarly analytical solutions forthe dynamic response of slender beams subjected to thermalshock have also been studied extensively [34ndash39]

Figure 15 details the flow of data of the simply-coupledthermomechanical analysis starting with the FLUKA energydeposition maps which volumetrically map the transfer ofenergy in the tested material specimens as a result of theparticle beam impact [40] +is is used as an input in transientthermal analyses which compute the temperature in thespecimens as a result of the beam impact+e thermal analyseswere modelled adiabatically given that the thermal evolutionwithin the specimen is in the order of seconds compared withthe dynamic phenomena observed which are in the order ofmicroseconds [35] +e mesh size used in thermal analysesconsidered the energy density map bin sizemdashin this case themesh consisted of 50 divisions along the longitudinal coor-dinate and 60 divisions across the cross section resulting in amodel with 379181 elements 3D 20-node high-order thermal

0 0005 001 0015 002 0025Strain (mm)

0

50

100

150

200

250

300

350

400St

ress

(MPa

)

Figure 13 Stress-strain diagram for CuCD RHP3434 obtainedfrom the four-point bending test

Shock and Vibration 9

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

Method for Linear ermal Expansion of Solid Materialswith a Push-Rod Dilatometerrdquo [18] Measured values for thelinear coefficient of thermal expansion are shown inFigure 3

Initial tests for the calculation of density ρ were carriedout using the Archimedes principle where the specimen isimmersed in a liquid and its volume is determined by thedisplacement of liquid+emass of the specimen can then bemeasured to calculate the density [19] Due to the highporosity of the CuCD grade being tested the alcohol used inthe experiment infiltrates the pores resulting in an over-estimation of the density calculated It was therefore deemedpreferable to calculate the density frommeasurements of thespecimensrsquo dimensions and weight Five specimens ofvarying dimensions were measured at three different pointsalong their length +e dimensions were averaged and thedensity was calculated for each specimen +e results for thefive measured specimens were similarly averaged to achievea final value for density at room temperature (ρ0) equal to5700 kgmiddotmminus3 Taking into consideration the volumetric co-efficient of thermal expansion β 3α the density can beexpressed as a function of temperature by consideringtemperature-dependent measurements for the coefficient ofthermal expansion as shown in equation (1) where ρ0 is theinitial density and T0 is the initial temperature ie roomtemperature +e temperature-dependent measurements forthe density of the CuCD RHP3434 grade are shown inFigure 4

ρ kgmminus31113960 1113961

ρ01 + β T minus T0( 1113857

(1)

For the measurement of the specific heat capacity (Cp) adifferential scanning calorimeter (DSC) was used With sucha setup the specific heat capacity was determined via theratio method [20] For each test sample three tests must beperformed the first test being a ldquobaselinerdquo which recordssystem characteristics and allows removal of the system biasfrom experimental data After a repeatable baseline has beenrecorded a reference material must be tested followingwhich the experimental sample is tested in exactly the same

way as reference [21 22] +e specific heat capacity mea-surements for the CuCD RHP3434 grade are shown inFigure 5

+e thermal diffusivity was measured by using a laserflash apparatus (LFA) which shoots a short laser pulse ontoone of the specimenrsquos faces andmeasures the temperature onthe opposite face with the use of an optical pyrometerKnowing the specimenrsquos thickness the thermal diffusivityacross a range of temperatures can be computed from thesereadings+e conductivity k was then found by multiplyingthe diffusivity (in m2middotsminus1) the density and the specific heatcapacity [23]+e calculated thermal conductivity results areshown in Figure 6

5

6

7

8

9

10

0 50 100 150 200 250 300 350

CTE

(10ndash6

K)

Temperature (degC)

Figure 3 Linear coefficient of thermal expansion of CuCDRHP3434

5580

5620

5660

5700

5740

0 50 100 150 200 250 300 350

Den

sity

(kg

m3 )

Temperature (degC)

Figure 4 Density of CuCD RHP3434

Figure 2 Rotatable barrel and mounted target stations in theHRMT36 experiment testbench

Shock and Vibration 3

Measured values for the CTE density specific heatcapacity and thermal conductivity are summarised at roomtemperature in Table 1

32 Impulse Excitation Technique Test An impulse excita-tion technique test was conducted to determine the elasticproperties of the CuCDRHP3434 grade As will be discussedin this section the expected flexural and torsional fre-quencies to be measured in the test were first calculated inaccordance with industry standards following which the testwas performed and the excited modes of interest identified+is was followed by a preliminary modal finite elementanalysis conducted with estimated elastic constants frommeasurements on other CuCD grades +e elastic properties

of the material were then optimised to the frequenciesmeasured in the IET test

+e impulse excitation technique test is a dynamicnondestructive material characterisation technique whichmeasures the resonant frequencies of a material which aresubsequently used to calculate the materialrsquos elastic con-stants at room temperature or at elevated temperatures [24]+e measurement consists of hitting the sample with ahammer and recording the induced vibration modes bymeans of a microphone +e vibrational signal measured inthe time domain is then converted to the frequency domainby a fast Fourier transform (FFT) following which theresonant frequencies can be determined allowing for thecalculation of the elastic properties of the material takinginto consideration the dimensions at hand

Different frequency modes (resonant frequencies) can beexcited in the specimen and are mainly dependent on theposition of the support wires the mechanical impulse andthe measuring deviceWith regard to the elastic properties ofthe material the two most important resonant frequenciesare the flexural and torsional frequencies which are mostlydependent on Youngrsquos and shear moduli for isotropicmaterials+e specimenrsquos elastic properties can be calculatedby considering the dimensions and geometry weight andthe measured resonant frequencies in accordance with theASTM E1876-15 ldquoStandard Test Method for DynamicYoungrsquos Modulus Shear Modulus and Poissonrsquos Ratio byImpulse Excitation of Vibrationrdquo [25 26]

+e natural flexural frequency denoted as ff is char-acteristic of Youngrsquos modulus E For the measurement ofthis frequency the specimen is supported at the nodes(where the amplitude of vibration is zero) and is excited atone of the antinodes (ie one of the ends of the rod) in orderto provoke maximum vibration as shown in Figure 7(a)Following the flexural frequency measurement Youngrsquosmodulus can be calculated by

E 09465m middot f

2f

b⎛⎝ ⎞⎠

L3

h31113888 1113889Rf (2)

wherem is the mass b h and L are the width thickness andlength of the specimen ff is the measured flexural frequencyand Rf is a geometrical correction factor which represents theaspect ratio of the specimen depending on the thickness andlength of the specimen

As can be seen in equation (2) the mass and dimensionsof the sample need to be measured accurately for reliablecalculation of Youngrsquos modulus Similarly the naturaltorsional frequency denoted as ft is characteristic of the

350

400

450

500

550

600

0 50 100 150 200 250

Ther

mal

cond

uctiv

ity (W

mK)

Temperature (degC)

Figure 6 +ermal conductivity of CuCD RHP3434

Table 1 CuCD RHP3434material properties at room temperature

Property Symbol Value Standarduncertainty Units

Density ρ 5700 plusmn33 kgmiddotmminus3

Specific heat capacity Cp 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity k 537 plusmn15 Wmiddotmminus1 Kminus1

Linear coefficient ofthermal expansion α 619 plusmn02 10minus6 Kminus1

350

400

450

500

550

600

650

0 50 100 150 200 250

Spec

ific h

eat c

apac

ity (J

kgK

)

Temperature (degC)

Figure 5 Specific heat capacity of CuCD RHP3434

4 Shock and Vibration

shear modulus In this case the specimen is supported alongboth longitudinal and transverse axes as can be seen inFigure 7(b) while the mechanical excitation is performed atone corner inducing a twisting motion in the beam ratherthan flexing it Following the torsional frequency mea-surement the shear modulus can be calculated by thefollowing

G 4 middot L middot m middot f

2t

b middot hRt (3)

where ft is the measured torsional frequency and Rt is ageometrical correction factor Poissonrsquos ratio ] is calculatedby ] [(E2G) minus 1] and the material damping can also becharacterised by considering the logarithmic decrement ofthe decaying amplitude of vibration

While the solutions defined by equations (2) and (3)provide adequate results for elastic properties a numericalapproach encompasses the triaxiality of the problem and canallow for the optimisation of the elastic constants allowingfor the reverse engineering of these values from the mea-sured frequencies with great accuracy In addition a nu-merical approach has the benefit of allowing the calculationof elastic constants for anisotropic bodies +e followingsection describes the numerical approach adopted in thisstudy to determine the elastic properties of the tested CuCDgrade For a specimen with a square cross section and madeof an isotropic material measurements of resonant fre-quencies in different transverse planes (referred to as in-plane and through-plane) are identical since the orientationof the specimen does not have an effect on the materialproperties Whilst the method described in the followingsection refers to isotropic bodiesmdashas is the case forCuCDmdashit can easily be extended to anisotropic bodies byconsidering flexural and torsional mode measurements forboth in-plane and through-plane conditions

+e finite element analysis optimisation was implementedin the software ANSYS Workbench [27] using a modal FEManalysis which for a defined geometry calculates the resonantfrequencies for the given elastic properties and density Ar-bitrary elastic properties are initially input which are thenoptimised with the resonant frequencies measured from theexperiment During preprocessing the geometry is dividedinto sub-blocks allowing the definition of several lines and

surfaces which can be constrained to force the excitation of aparticular resonance mode +e density value input in thesimulation is also of particular importance given that thefrequency response is directly dependent on this property (fora given set of properties and dimensions a higher densityresults in lower frequencies and vice versa) +e experi-mentally derived density values detailed in the previoussections of this study were used for the CuCD grade inquestion (ie 5700 kgm3 at room temperature)

If the specimen tested has a rectangular cross section twopossible flexural planes are present During the simulationsetup the flexural mode in the direction of interest (ie thedirection excited in the experiment) is forced by defining asymmetrical plane in this direction +is ensures movement inthe direction perpendicular to the symmetrical plane is re-stricted For the excitation of the torsional mode the longi-tudinal and transverse axes are fixed (ie displacement equal tozero) blocking movement of these lines in all directions Aconstraint equation is applied to the two faces at the extremitiesof the specimen promoting a natural torsional behaviour wherethe two faces rotate in opposite directions +e relevantboundary conditions for the two analyses are shown in Figure 8

Once the initial simulations are setup and completed(utilising approximate values for the elastic constants) theresulting first flexural and torsional frequencies are para-metrised and optimised to the experimental measurements+e variables influencing the two frequencies (flexural andtorsional) are Youngrsquos modulus and Poissonrsquos ratio (theshear modulus is calculated from the other properties) Arange of acceptable results is defined for each parameter

+e IET test was carried out on a specimen with di-mensions 247times1241times 1035mm3 (lengthtimeswidthtimes

thickness) A spectrogram showing the intensity of theexcited frequencies with respect to time in the testedspecimen setup in the configuration shown in Figure 7(a) isshown in Figure 9 As shown in the spectrogram there arevarious excited frequencies in the measured range In orderto identify the modes of interest using Youngrsquos modulus of208GPa and Poissonrsquos ratio of 022 as a first approximationthe first flexural and torsional modes were calculated byequations (2) and (3) as shown in Table 2+is allows for theidentification of the first torsional and flexural frequenciesfrom the various unwanted frequencies excited in theexperiment

(a) (b)

Figure 7 IET setup for flexural (a) and torsional (b) mode excitation

Shock and Vibration 5

From the experimental results the first flexural fre-quency can be easily identified with a peak at 1016 kHz+eslender nature of the specimen made it difficult to excite thetorsional vibration however two possible frequencies cor-responding to the torsional mode were identified in the

flexural measurement at 6794 kHz and 7238 kHz (two faintlines between the 5583 kHz and 9082 kHz readings)

Following the identification of the first flexural mode andthe two possible torsional modes the initial estimates forYoungrsquos modulus and Poissonrsquos ratiomdash208GPa and 022respectivelymdashwere used as a primary input along with thedensity of the material in the optimisation procedure A firstestimation of the flexural and torsional modes was com-puted for the given material properties +e first flexural andtorsional modes determined by the modal analyses areshown in Figure 10 As can be seen additional lsquoundesiredrsquomodes such as the first longitudinal harmonic can also beexcited since movement in this direction is unrestricted

(a) (b)

Figure 8 Boundary conditions forcing mode of interest (a) symmetry plane promoting flexural vibration and (b) fixed axes promotingtorsional vibration and a constraint equation limiting motion of the two extremities of the specimen to rotate in opposite directions

14558319

0036231110

[Hz][Pa][s]

1328430009099575

[Hz][Pa][s]

908203001697770

[Hz][Pa][s]

286255000570350

[Hz][Pa][s]

1016240064722175

[Hz][Pa][s]

7238520001971475

[Hz][Pa][s]

679401000260141

[Hz][Pa][s]

15325930003508175

[Hz][Pa][s]

18077090006504325

[Hz][Pa][s]

13

12

11

10

9

8

7

6

5

4

3

2

1

0

Tim

e (s)

Inte

nsity

(Pa)

Frequency (Hz)10000 20000 30000

1μ

10μ

100μ

1m

10m

Figure 9 Spectrogram showing the IET test flexural frequency measurement as a function of time

Table 2 Calculated flexural and torsional modes for positions Aand B with initial estimates of elastic properties (E 208GPa and] 022)

Specimen dimensionslengthtimeswidthtimes thickness

Flexuralmode (kHz)

Torsionalmode (kHz)

247times1241times 1035mm3 1047 7054

6 Shock and Vibration

47277 Max

42056

36835

31614

26393

21172

15951

1073

55088

28774 Min

000

2500 7500

5000 10000 (mm)

Y

XZ

(a)

Y

XZ

57916 Max

51481

45046

38611

32176

2574

19305

1287

64351

0 Min

000

2500 7500

5000 10000 (mm)

(b)

Y

XZ

33437 Max

29722

26007

22292

18577

14862

11147

7432

37171

02117 Min

000

2500 7500

5000 10000 (mm)

(c)

Figure 10 First flexural and torsional modes simulated with estimated material properties (a b) and additional modes such as the firstlongitudinal one (c)

Shock and Vibration 7

Following the initial modal analysis with estimatedproperties the computed frequencies were parametrised andoptimised with the frequencies measured in the experimentYoungrsquos modulus was constricted to a range between187GPa and 229GPa while Poissonrsquos ratio was restricted tovalues ranging from 01 to 03 +e optimisation procedurevaries these two values attempting to match the computedfrequencies with the measured values in the experiment

+e relatively minor difference (11) in the two mea-sured frequencies considered for the torsional mode resultedin a significant effect on the computed Poissonrsquos ratio Forff 1016 kHz and ft 6794 kHz Youngrsquos modulus iscomputed at 1937GPa with Poissonrsquos ratio of 0248 whilefor ff 1016 kHz and ft 7238 kHz Youngrsquos modulus iscalculated at 1936GPa with Poissonrsquos ratio of 0104Considering previous data on similar CuCD grades thevalue of ] 0248 was deemed to be the physically valid oneand retained for the material model along with a roundedYoungrsquos modulus value of 194GPa [13] +e final values forYoungrsquos modulus and Poissonrsquos ratio are shown in Table 3

33 Plasticity-Four-Point Bending Experiment To model thematerial behaviour beyond the elastic regime a four-pointbending test was performed allowing for the implementa-tion of a multilinear hardening model in the numericalanalyses A ZwickRoell Z400 universal testing machine wasused for the four-point flexural test in accordance with theASTM C651 standard ldquoFlexural Strength of Carbon andGraphite by Four-Point Loadingrdquo [28] With this experi-mental setup as shown in the schematic in Figure 11 thebending force is applied at two points along the length of thespecimen which is supported at its extremities Specimenswith an average length of 30mm width of 53mm andthickness of 51mm were tested with such a setup

In the experiment the bending force is progressivelyincreased with the corresponding induced axial strain in thespecimen measured with a strain gauge glued to its surface+e maximum bending moment M is deduced from theforce F and the distance between the load application and thesupports L1 byM (L1F2) In the linear elastic domainthe stress distribution through the sample thickness can beassumed to be linear and the maximum can be calculated byσ (Mh2I) where I is the flexural moment of inertia and his the sample thickness However the force-strain relationmeasured in the experiment is nonlinear indicating thatthere is a nonlinear stress distribution across the thickness ofthe specimen as shown in Figure 12 +e maximum stressthus has to be calculated taking into consideration thisnonlinearity For small curvatures the axial strain can beconsidered to be linearly distributed through the thickness[7]

+e stress must therefore be calculated out of the generalmoment-strain relationship +e moment can be expressedas an integration of the stress distribution through thethickness by

M b 1113946h2

minush2σz(x)xdx (4)

where x is the vertical coordinate from the neutral axis b isthe specimenrsquos width and h is its thickness (height ie alongx) +e moment can be expressed in terms of the measuredstrain as follows

M εtotmax1113872 1113873 2b 1113946h2

0σz

2x

hεtotmax1113874 1113875xdx (5)

where σz((2xh)εtotmax) indicates the stress-strain curve whichcan be numerically derived given the applied momentM(εtotmax) and the measured strain εtotmax +e problem can besolved for each measured value of force and strain resultingin a piecewise stress-strain relationship ie a multilinearhardening curve describing the plastic behaviour of thematerial shown in Figure 13 +e material can be seen toquickly lose linearity and undergoes a significant amount ofplastic deformation as a result of the highly ductile coppermatrix

Table 3 Youngrsquos modulus and Poissonrsquos ratio results determinedby the IET optimisation procedure

Property Symbol Value Uncertainty UnitsYoungrsquos modulus E 194 plusmn32 GPaPoissonrsquos ratio ] 0248 plusmn0002 mdash

F2F2

LL1 L1

Figure 11 Four-point bending experimental setup

StrainStress

Figure 12 Axial strain (linear) and stress (nonlinear) distributionthrough the sample thickness

8 Shock and Vibration

One aspect which is not considered in the model is thecompression behaviour of the material which is assumed tobe identical to that in tension +is assumption may not bevalid at high compressive strains at which point the dia-mond particles may come in contact In contrast when thematerial is loaded in tension the bond between the diamondparticles and the copper matrix may be lost

4 Benchmarking of the Material Model

41 HRMT36 e MultiMat Experiment CERNrsquos HiR-adMat facility is used to test materials used for componentsexposed to extreme conditions in CERNrsquos acceleratorcomplex [10] Upcoming LHC upgrades such as the high-luminosity LHC (HL-LHC) upgrade and the proposedFuture Circular Collider (FCC) require an increase in thebeam energy and thus bring the need for high-performingmaterials+is is especially true for use in components at riskof exposure to accidental beam impacts such as collimatorsand other beam intercepting devices [29]

Experiments conducted in the HiRadMat facility includetests on full-scale collimator jaws [29ndash31] as well as specimenswith simple geometries allowing the benchmarking of nu-merical results obtained through numerical analyses [9 14]+e HRMT36 ldquoMultiMatrdquo experiment conducted in October2017 was designed to offer a reusable platform to test a widerange of novel materials developed for beam interceptingdevices impacted by particle beams with energy densities inthe range of those experienced in the HL-LHC [32 33]

+e test bench hosted 16 target stations mounted on arotatable barrel with each station having a total length of1m+ematerial specimens were mounted in series and hadlengths of 120 or 247mm with cross sections varying from8times 8 to 12times115mm2 18 different materials were tested intotal+e acquisition system included electrical strain gaugesand temperature probes placed on the material specimensalong with remote instrumentation including a laser

Doppler vibrometer (LDV) oriented towards the top face ofthe specimens and a side-mounted radiation-hard high-definition camera [12]+e dimensions for CuCD specimenstested and typical positions of strain gauges and thermalprobes are shown in Figure 14+e target station hosting theCuCD RHP3434 samples consisted of 4 specimens in serieseach having a length of 247mm (with a 3mm space in-between specimens) and a square cross section with an edgelength of 10mm All four specimens were equipped withlongitudinal strain gauges to measure dynamic phenomenainduced by the particle beam impacts

Other than copper diamond (of which two grades from twodifferent suppliers were tested) various materials and materialgrades were tested in the MultiMat experiment including 4grades of MoGr 3 different coatings (Cu Mo and TiN) andnovel carbon-based materials such as highly ordered pyrolyticgraphite (HOPG) and titanium-graphite (TG-1100) +e pulseintensity to which specimens were subjected to ranged from 1to 288 bunches with a typical bunch intensity of 13times1011protons along with nominal beam RMS sizes of 025times 02505times 05 and 2times 2mm2 +e three different types of impacttested were axially centred impacts offset impacts and grazingimpacts (where a coating was present)

42 Modelling and Benchmarking with Experimental Results+e material model for the CuCD RHP3434 grade con-sisting of the measured thermomechanical temperature-dependent properties (Figures 2ndash5 summarised in Table 1)dynamic elastic constants from the IET test (Table 3) andmultilinear hardening curve from the four-point bendingtest (Figure 13) as shown in the previous sections wasincluded in a simply-coupled implicit dynamic thermo-mechanical simulation performed in the FEA softwareANSYS +e material is assumed to be isotropic in naturebased on previous study of the material [6] and is assumedto behave in an elasto-plastic manner based on the four-point bending test presented in this study Similar analyseshave been performed on other materials tested in theMultiMat experiment and information on the thermo-mechanical modelling aspects in such scenarios is detailed inprevious studies [13 15] Similarly analytical solutions forthe dynamic response of slender beams subjected to thermalshock have also been studied extensively [34ndash39]

Figure 15 details the flow of data of the simply-coupledthermomechanical analysis starting with the FLUKA energydeposition maps which volumetrically map the transfer ofenergy in the tested material specimens as a result of theparticle beam impact [40] +is is used as an input in transientthermal analyses which compute the temperature in thespecimens as a result of the beam impact+e thermal analyseswere modelled adiabatically given that the thermal evolutionwithin the specimen is in the order of seconds compared withthe dynamic phenomena observed which are in the order ofmicroseconds [35] +e mesh size used in thermal analysesconsidered the energy density map bin sizemdashin this case themesh consisted of 50 divisions along the longitudinal coor-dinate and 60 divisions across the cross section resulting in amodel with 379181 elements 3D 20-node high-order thermal

0 0005 001 0015 002 0025Strain (mm)

0

50

100

150

200

250

300

350

400St

ress

(MPa

)

Figure 13 Stress-strain diagram for CuCD RHP3434 obtainedfrom the four-point bending test

Shock and Vibration 9

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

Measured values for the CTE density specific heatcapacity and thermal conductivity are summarised at roomtemperature in Table 1

32 Impulse Excitation Technique Test An impulse excita-tion technique test was conducted to determine the elasticproperties of the CuCDRHP3434 grade As will be discussedin this section the expected flexural and torsional fre-quencies to be measured in the test were first calculated inaccordance with industry standards following which the testwas performed and the excited modes of interest identified+is was followed by a preliminary modal finite elementanalysis conducted with estimated elastic constants frommeasurements on other CuCD grades +e elastic properties

of the material were then optimised to the frequenciesmeasured in the IET test

+e impulse excitation technique test is a dynamicnondestructive material characterisation technique whichmeasures the resonant frequencies of a material which aresubsequently used to calculate the materialrsquos elastic con-stants at room temperature or at elevated temperatures [24]+e measurement consists of hitting the sample with ahammer and recording the induced vibration modes bymeans of a microphone +e vibrational signal measured inthe time domain is then converted to the frequency domainby a fast Fourier transform (FFT) following which theresonant frequencies can be determined allowing for thecalculation of the elastic properties of the material takinginto consideration the dimensions at hand

Different frequency modes (resonant frequencies) can beexcited in the specimen and are mainly dependent on theposition of the support wires the mechanical impulse andthe measuring deviceWith regard to the elastic properties ofthe material the two most important resonant frequenciesare the flexural and torsional frequencies which are mostlydependent on Youngrsquos and shear moduli for isotropicmaterials+e specimenrsquos elastic properties can be calculatedby considering the dimensions and geometry weight andthe measured resonant frequencies in accordance with theASTM E1876-15 ldquoStandard Test Method for DynamicYoungrsquos Modulus Shear Modulus and Poissonrsquos Ratio byImpulse Excitation of Vibrationrdquo [25 26]

+e natural flexural frequency denoted as ff is char-acteristic of Youngrsquos modulus E For the measurement ofthis frequency the specimen is supported at the nodes(where the amplitude of vibration is zero) and is excited atone of the antinodes (ie one of the ends of the rod) in orderto provoke maximum vibration as shown in Figure 7(a)Following the flexural frequency measurement Youngrsquosmodulus can be calculated by

E 09465m middot f

2f

b⎛⎝ ⎞⎠

L3

h31113888 1113889Rf (2)

wherem is the mass b h and L are the width thickness andlength of the specimen ff is the measured flexural frequencyand Rf is a geometrical correction factor which represents theaspect ratio of the specimen depending on the thickness andlength of the specimen

As can be seen in equation (2) the mass and dimensionsof the sample need to be measured accurately for reliablecalculation of Youngrsquos modulus Similarly the naturaltorsional frequency denoted as ft is characteristic of the

350

400

450

500

550

600

0 50 100 150 200 250

Ther

mal

cond

uctiv

ity (W

mK)

Temperature (degC)

Figure 6 +ermal conductivity of CuCD RHP3434

Table 1 CuCD RHP3434material properties at room temperature

Property Symbol Value Standarduncertainty Units

Density ρ 5700 plusmn33 kgmiddotmminus3

Specific heat capacity Cp 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity k 537 plusmn15 Wmiddotmminus1 Kminus1

Linear coefficient ofthermal expansion α 619 plusmn02 10minus6 Kminus1

350

400

450

500

550

600

650

0 50 100 150 200 250

Spec

ific h

eat c

apac

ity (J

kgK

)

Temperature (degC)

Figure 5 Specific heat capacity of CuCD RHP3434

4 Shock and Vibration

shear modulus In this case the specimen is supported alongboth longitudinal and transverse axes as can be seen inFigure 7(b) while the mechanical excitation is performed atone corner inducing a twisting motion in the beam ratherthan flexing it Following the torsional frequency mea-surement the shear modulus can be calculated by thefollowing

G 4 middot L middot m middot f

2t

b middot hRt (3)

where ft is the measured torsional frequency and Rt is ageometrical correction factor Poissonrsquos ratio ] is calculatedby ] [(E2G) minus 1] and the material damping can also becharacterised by considering the logarithmic decrement ofthe decaying amplitude of vibration

While the solutions defined by equations (2) and (3)provide adequate results for elastic properties a numericalapproach encompasses the triaxiality of the problem and canallow for the optimisation of the elastic constants allowingfor the reverse engineering of these values from the mea-sured frequencies with great accuracy In addition a nu-merical approach has the benefit of allowing the calculationof elastic constants for anisotropic bodies +e followingsection describes the numerical approach adopted in thisstudy to determine the elastic properties of the tested CuCDgrade For a specimen with a square cross section and madeof an isotropic material measurements of resonant fre-quencies in different transverse planes (referred to as in-plane and through-plane) are identical since the orientationof the specimen does not have an effect on the materialproperties Whilst the method described in the followingsection refers to isotropic bodiesmdashas is the case forCuCDmdashit can easily be extended to anisotropic bodies byconsidering flexural and torsional mode measurements forboth in-plane and through-plane conditions

+e finite element analysis optimisation was implementedin the software ANSYS Workbench [27] using a modal FEManalysis which for a defined geometry calculates the resonantfrequencies for the given elastic properties and density Ar-bitrary elastic properties are initially input which are thenoptimised with the resonant frequencies measured from theexperiment During preprocessing the geometry is dividedinto sub-blocks allowing the definition of several lines and

surfaces which can be constrained to force the excitation of aparticular resonance mode +e density value input in thesimulation is also of particular importance given that thefrequency response is directly dependent on this property (fora given set of properties and dimensions a higher densityresults in lower frequencies and vice versa) +e experi-mentally derived density values detailed in the previoussections of this study were used for the CuCD grade inquestion (ie 5700 kgm3 at room temperature)

If the specimen tested has a rectangular cross section twopossible flexural planes are present During the simulationsetup the flexural mode in the direction of interest (ie thedirection excited in the experiment) is forced by defining asymmetrical plane in this direction +is ensures movement inthe direction perpendicular to the symmetrical plane is re-stricted For the excitation of the torsional mode the longi-tudinal and transverse axes are fixed (ie displacement equal tozero) blocking movement of these lines in all directions Aconstraint equation is applied to the two faces at the extremitiesof the specimen promoting a natural torsional behaviour wherethe two faces rotate in opposite directions +e relevantboundary conditions for the two analyses are shown in Figure 8

Once the initial simulations are setup and completed(utilising approximate values for the elastic constants) theresulting first flexural and torsional frequencies are para-metrised and optimised to the experimental measurements+e variables influencing the two frequencies (flexural andtorsional) are Youngrsquos modulus and Poissonrsquos ratio (theshear modulus is calculated from the other properties) Arange of acceptable results is defined for each parameter

+e IET test was carried out on a specimen with di-mensions 247times1241times 1035mm3 (lengthtimeswidthtimes

thickness) A spectrogram showing the intensity of theexcited frequencies with respect to time in the testedspecimen setup in the configuration shown in Figure 7(a) isshown in Figure 9 As shown in the spectrogram there arevarious excited frequencies in the measured range In orderto identify the modes of interest using Youngrsquos modulus of208GPa and Poissonrsquos ratio of 022 as a first approximationthe first flexural and torsional modes were calculated byequations (2) and (3) as shown in Table 2+is allows for theidentification of the first torsional and flexural frequenciesfrom the various unwanted frequencies excited in theexperiment

(a) (b)

Figure 7 IET setup for flexural (a) and torsional (b) mode excitation

Shock and Vibration 5

From the experimental results the first flexural fre-quency can be easily identified with a peak at 1016 kHz+eslender nature of the specimen made it difficult to excite thetorsional vibration however two possible frequencies cor-responding to the torsional mode were identified in the

flexural measurement at 6794 kHz and 7238 kHz (two faintlines between the 5583 kHz and 9082 kHz readings)

Following the identification of the first flexural mode andthe two possible torsional modes the initial estimates forYoungrsquos modulus and Poissonrsquos ratiomdash208GPa and 022respectivelymdashwere used as a primary input along with thedensity of the material in the optimisation procedure A firstestimation of the flexural and torsional modes was com-puted for the given material properties +e first flexural andtorsional modes determined by the modal analyses areshown in Figure 10 As can be seen additional lsquoundesiredrsquomodes such as the first longitudinal harmonic can also beexcited since movement in this direction is unrestricted

(a) (b)

Figure 8 Boundary conditions forcing mode of interest (a) symmetry plane promoting flexural vibration and (b) fixed axes promotingtorsional vibration and a constraint equation limiting motion of the two extremities of the specimen to rotate in opposite directions

14558319

0036231110

[Hz][Pa][s]

1328430009099575

[Hz][Pa][s]

908203001697770

[Hz][Pa][s]

286255000570350

[Hz][Pa][s]

1016240064722175

[Hz][Pa][s]

7238520001971475

[Hz][Pa][s]

679401000260141

[Hz][Pa][s]

15325930003508175

[Hz][Pa][s]

18077090006504325

[Hz][Pa][s]

13

12

11

10

9

8

7

6

5

4

3

2

1

0

Tim

e (s)

Inte

nsity

(Pa)

Frequency (Hz)10000 20000 30000

1μ

10μ

100μ

1m

10m

Figure 9 Spectrogram showing the IET test flexural frequency measurement as a function of time

Table 2 Calculated flexural and torsional modes for positions Aand B with initial estimates of elastic properties (E 208GPa and] 022)

Specimen dimensionslengthtimeswidthtimes thickness

Flexuralmode (kHz)

Torsionalmode (kHz)

247times1241times 1035mm3 1047 7054

6 Shock and Vibration

47277 Max

42056

36835

31614

26393

21172

15951

1073

55088

28774 Min

000

2500 7500

5000 10000 (mm)

Y

XZ

(a)

Y

XZ

57916 Max

51481

45046

38611

32176

2574

19305

1287

64351

0 Min

000

2500 7500

5000 10000 (mm)

(b)

Y

XZ

33437 Max

29722

26007

22292

18577

14862

11147

7432

37171

02117 Min

000

2500 7500

5000 10000 (mm)

(c)

Figure 10 First flexural and torsional modes simulated with estimated material properties (a b) and additional modes such as the firstlongitudinal one (c)

Shock and Vibration 7

Following the initial modal analysis with estimatedproperties the computed frequencies were parametrised andoptimised with the frequencies measured in the experimentYoungrsquos modulus was constricted to a range between187GPa and 229GPa while Poissonrsquos ratio was restricted tovalues ranging from 01 to 03 +e optimisation procedurevaries these two values attempting to match the computedfrequencies with the measured values in the experiment

+e relatively minor difference (11) in the two mea-sured frequencies considered for the torsional mode resultedin a significant effect on the computed Poissonrsquos ratio Forff 1016 kHz and ft 6794 kHz Youngrsquos modulus iscomputed at 1937GPa with Poissonrsquos ratio of 0248 whilefor ff 1016 kHz and ft 7238 kHz Youngrsquos modulus iscalculated at 1936GPa with Poissonrsquos ratio of 0104Considering previous data on similar CuCD grades thevalue of ] 0248 was deemed to be the physically valid oneand retained for the material model along with a roundedYoungrsquos modulus value of 194GPa [13] +e final values forYoungrsquos modulus and Poissonrsquos ratio are shown in Table 3

33 Plasticity-Four-Point Bending Experiment To model thematerial behaviour beyond the elastic regime a four-pointbending test was performed allowing for the implementa-tion of a multilinear hardening model in the numericalanalyses A ZwickRoell Z400 universal testing machine wasused for the four-point flexural test in accordance with theASTM C651 standard ldquoFlexural Strength of Carbon andGraphite by Four-Point Loadingrdquo [28] With this experi-mental setup as shown in the schematic in Figure 11 thebending force is applied at two points along the length of thespecimen which is supported at its extremities Specimenswith an average length of 30mm width of 53mm andthickness of 51mm were tested with such a setup

In the experiment the bending force is progressivelyincreased with the corresponding induced axial strain in thespecimen measured with a strain gauge glued to its surface+e maximum bending moment M is deduced from theforce F and the distance between the load application and thesupports L1 byM (L1F2) In the linear elastic domainthe stress distribution through the sample thickness can beassumed to be linear and the maximum can be calculated byσ (Mh2I) where I is the flexural moment of inertia and his the sample thickness However the force-strain relationmeasured in the experiment is nonlinear indicating thatthere is a nonlinear stress distribution across the thickness ofthe specimen as shown in Figure 12 +e maximum stressthus has to be calculated taking into consideration thisnonlinearity For small curvatures the axial strain can beconsidered to be linearly distributed through the thickness[7]

+e stress must therefore be calculated out of the generalmoment-strain relationship +e moment can be expressedas an integration of the stress distribution through thethickness by

M b 1113946h2

minush2σz(x)xdx (4)

where x is the vertical coordinate from the neutral axis b isthe specimenrsquos width and h is its thickness (height ie alongx) +e moment can be expressed in terms of the measuredstrain as follows

M εtotmax1113872 1113873 2b 1113946h2

0σz

2x

hεtotmax1113874 1113875xdx (5)

where σz((2xh)εtotmax) indicates the stress-strain curve whichcan be numerically derived given the applied momentM(εtotmax) and the measured strain εtotmax +e problem can besolved for each measured value of force and strain resultingin a piecewise stress-strain relationship ie a multilinearhardening curve describing the plastic behaviour of thematerial shown in Figure 13 +e material can be seen toquickly lose linearity and undergoes a significant amount ofplastic deformation as a result of the highly ductile coppermatrix

Table 3 Youngrsquos modulus and Poissonrsquos ratio results determinedby the IET optimisation procedure

Property Symbol Value Uncertainty UnitsYoungrsquos modulus E 194 plusmn32 GPaPoissonrsquos ratio ] 0248 plusmn0002 mdash

F2F2

LL1 L1

Figure 11 Four-point bending experimental setup

StrainStress

Figure 12 Axial strain (linear) and stress (nonlinear) distributionthrough the sample thickness

8 Shock and Vibration

One aspect which is not considered in the model is thecompression behaviour of the material which is assumed tobe identical to that in tension +is assumption may not bevalid at high compressive strains at which point the dia-mond particles may come in contact In contrast when thematerial is loaded in tension the bond between the diamondparticles and the copper matrix may be lost

4 Benchmarking of the Material Model

41 HRMT36 e MultiMat Experiment CERNrsquos HiR-adMat facility is used to test materials used for componentsexposed to extreme conditions in CERNrsquos acceleratorcomplex [10] Upcoming LHC upgrades such as the high-luminosity LHC (HL-LHC) upgrade and the proposedFuture Circular Collider (FCC) require an increase in thebeam energy and thus bring the need for high-performingmaterials+is is especially true for use in components at riskof exposure to accidental beam impacts such as collimatorsand other beam intercepting devices [29]

Experiments conducted in the HiRadMat facility includetests on full-scale collimator jaws [29ndash31] as well as specimenswith simple geometries allowing the benchmarking of nu-merical results obtained through numerical analyses [9 14]+e HRMT36 ldquoMultiMatrdquo experiment conducted in October2017 was designed to offer a reusable platform to test a widerange of novel materials developed for beam interceptingdevices impacted by particle beams with energy densities inthe range of those experienced in the HL-LHC [32 33]

+e test bench hosted 16 target stations mounted on arotatable barrel with each station having a total length of1m+ematerial specimens were mounted in series and hadlengths of 120 or 247mm with cross sections varying from8times 8 to 12times115mm2 18 different materials were tested intotal+e acquisition system included electrical strain gaugesand temperature probes placed on the material specimensalong with remote instrumentation including a laser

Doppler vibrometer (LDV) oriented towards the top face ofthe specimens and a side-mounted radiation-hard high-definition camera [12]+e dimensions for CuCD specimenstested and typical positions of strain gauges and thermalprobes are shown in Figure 14+e target station hosting theCuCD RHP3434 samples consisted of 4 specimens in serieseach having a length of 247mm (with a 3mm space in-between specimens) and a square cross section with an edgelength of 10mm All four specimens were equipped withlongitudinal strain gauges to measure dynamic phenomenainduced by the particle beam impacts

Other than copper diamond (of which two grades from twodifferent suppliers were tested) various materials and materialgrades were tested in the MultiMat experiment including 4grades of MoGr 3 different coatings (Cu Mo and TiN) andnovel carbon-based materials such as highly ordered pyrolyticgraphite (HOPG) and titanium-graphite (TG-1100) +e pulseintensity to which specimens were subjected to ranged from 1to 288 bunches with a typical bunch intensity of 13times1011protons along with nominal beam RMS sizes of 025times 02505times 05 and 2times 2mm2 +e three different types of impacttested were axially centred impacts offset impacts and grazingimpacts (where a coating was present)

42 Modelling and Benchmarking with Experimental Results+e material model for the CuCD RHP3434 grade con-sisting of the measured thermomechanical temperature-dependent properties (Figures 2ndash5 summarised in Table 1)dynamic elastic constants from the IET test (Table 3) andmultilinear hardening curve from the four-point bendingtest (Figure 13) as shown in the previous sections wasincluded in a simply-coupled implicit dynamic thermo-mechanical simulation performed in the FEA softwareANSYS +e material is assumed to be isotropic in naturebased on previous study of the material [6] and is assumedto behave in an elasto-plastic manner based on the four-point bending test presented in this study Similar analyseshave been performed on other materials tested in theMultiMat experiment and information on the thermo-mechanical modelling aspects in such scenarios is detailed inprevious studies [13 15] Similarly analytical solutions forthe dynamic response of slender beams subjected to thermalshock have also been studied extensively [34ndash39]

Figure 15 details the flow of data of the simply-coupledthermomechanical analysis starting with the FLUKA energydeposition maps which volumetrically map the transfer ofenergy in the tested material specimens as a result of theparticle beam impact [40] +is is used as an input in transientthermal analyses which compute the temperature in thespecimens as a result of the beam impact+e thermal analyseswere modelled adiabatically given that the thermal evolutionwithin the specimen is in the order of seconds compared withthe dynamic phenomena observed which are in the order ofmicroseconds [35] +e mesh size used in thermal analysesconsidered the energy density map bin sizemdashin this case themesh consisted of 50 divisions along the longitudinal coor-dinate and 60 divisions across the cross section resulting in amodel with 379181 elements 3D 20-node high-order thermal

0 0005 001 0015 002 0025Strain (mm)

0

50

100

150

200

250

300

350

400St

ress

(MPa

)

Figure 13 Stress-strain diagram for CuCD RHP3434 obtainedfrom the four-point bending test

Shock and Vibration 9

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

shear modulus In this case the specimen is supported alongboth longitudinal and transverse axes as can be seen inFigure 7(b) while the mechanical excitation is performed atone corner inducing a twisting motion in the beam ratherthan flexing it Following the torsional frequency mea-surement the shear modulus can be calculated by thefollowing

G 4 middot L middot m middot f

2t

b middot hRt (3)

where ft is the measured torsional frequency and Rt is ageometrical correction factor Poissonrsquos ratio ] is calculatedby ] [(E2G) minus 1] and the material damping can also becharacterised by considering the logarithmic decrement ofthe decaying amplitude of vibration

While the solutions defined by equations (2) and (3)provide adequate results for elastic properties a numericalapproach encompasses the triaxiality of the problem and canallow for the optimisation of the elastic constants allowingfor the reverse engineering of these values from the mea-sured frequencies with great accuracy In addition a nu-merical approach has the benefit of allowing the calculationof elastic constants for anisotropic bodies +e followingsection describes the numerical approach adopted in thisstudy to determine the elastic properties of the tested CuCDgrade For a specimen with a square cross section and madeof an isotropic material measurements of resonant fre-quencies in different transverse planes (referred to as in-plane and through-plane) are identical since the orientationof the specimen does not have an effect on the materialproperties Whilst the method described in the followingsection refers to isotropic bodiesmdashas is the case forCuCDmdashit can easily be extended to anisotropic bodies byconsidering flexural and torsional mode measurements forboth in-plane and through-plane conditions

+e finite element analysis optimisation was implementedin the software ANSYS Workbench [27] using a modal FEManalysis which for a defined geometry calculates the resonantfrequencies for the given elastic properties and density Ar-bitrary elastic properties are initially input which are thenoptimised with the resonant frequencies measured from theexperiment During preprocessing the geometry is dividedinto sub-blocks allowing the definition of several lines and

surfaces which can be constrained to force the excitation of aparticular resonance mode +e density value input in thesimulation is also of particular importance given that thefrequency response is directly dependent on this property (fora given set of properties and dimensions a higher densityresults in lower frequencies and vice versa) +e experi-mentally derived density values detailed in the previoussections of this study were used for the CuCD grade inquestion (ie 5700 kgm3 at room temperature)

If the specimen tested has a rectangular cross section twopossible flexural planes are present During the simulationsetup the flexural mode in the direction of interest (ie thedirection excited in the experiment) is forced by defining asymmetrical plane in this direction +is ensures movement inthe direction perpendicular to the symmetrical plane is re-stricted For the excitation of the torsional mode the longi-tudinal and transverse axes are fixed (ie displacement equal tozero) blocking movement of these lines in all directions Aconstraint equation is applied to the two faces at the extremitiesof the specimen promoting a natural torsional behaviour wherethe two faces rotate in opposite directions +e relevantboundary conditions for the two analyses are shown in Figure 8

Once the initial simulations are setup and completed(utilising approximate values for the elastic constants) theresulting first flexural and torsional frequencies are para-metrised and optimised to the experimental measurements+e variables influencing the two frequencies (flexural andtorsional) are Youngrsquos modulus and Poissonrsquos ratio (theshear modulus is calculated from the other properties) Arange of acceptable results is defined for each parameter

+e IET test was carried out on a specimen with di-mensions 247times1241times 1035mm3 (lengthtimeswidthtimes

thickness) A spectrogram showing the intensity of theexcited frequencies with respect to time in the testedspecimen setup in the configuration shown in Figure 7(a) isshown in Figure 9 As shown in the spectrogram there arevarious excited frequencies in the measured range In orderto identify the modes of interest using Youngrsquos modulus of208GPa and Poissonrsquos ratio of 022 as a first approximationthe first flexural and torsional modes were calculated byequations (2) and (3) as shown in Table 2+is allows for theidentification of the first torsional and flexural frequenciesfrom the various unwanted frequencies excited in theexperiment

(a) (b)

Figure 7 IET setup for flexural (a) and torsional (b) mode excitation

Shock and Vibration 5

From the experimental results the first flexural fre-quency can be easily identified with a peak at 1016 kHz+eslender nature of the specimen made it difficult to excite thetorsional vibration however two possible frequencies cor-responding to the torsional mode were identified in the

flexural measurement at 6794 kHz and 7238 kHz (two faintlines between the 5583 kHz and 9082 kHz readings)

Following the identification of the first flexural mode andthe two possible torsional modes the initial estimates forYoungrsquos modulus and Poissonrsquos ratiomdash208GPa and 022respectivelymdashwere used as a primary input along with thedensity of the material in the optimisation procedure A firstestimation of the flexural and torsional modes was com-puted for the given material properties +e first flexural andtorsional modes determined by the modal analyses areshown in Figure 10 As can be seen additional lsquoundesiredrsquomodes such as the first longitudinal harmonic can also beexcited since movement in this direction is unrestricted

(a) (b)

Figure 8 Boundary conditions forcing mode of interest (a) symmetry plane promoting flexural vibration and (b) fixed axes promotingtorsional vibration and a constraint equation limiting motion of the two extremities of the specimen to rotate in opposite directions

14558319

0036231110

[Hz][Pa][s]

1328430009099575

[Hz][Pa][s]

908203001697770

[Hz][Pa][s]

286255000570350

[Hz][Pa][s]

1016240064722175

[Hz][Pa][s]

7238520001971475

[Hz][Pa][s]

679401000260141

[Hz][Pa][s]

15325930003508175

[Hz][Pa][s]

18077090006504325

[Hz][Pa][s]

13

12

11

10

9

8

7

6

5

4

3

2

1

0

Tim

e (s)

Inte

nsity

(Pa)

Frequency (Hz)10000 20000 30000

1μ

10μ

100μ

1m

10m

Figure 9 Spectrogram showing the IET test flexural frequency measurement as a function of time

Table 2 Calculated flexural and torsional modes for positions Aand B with initial estimates of elastic properties (E 208GPa and] 022)

Specimen dimensionslengthtimeswidthtimes thickness

Flexuralmode (kHz)

Torsionalmode (kHz)

247times1241times 1035mm3 1047 7054

6 Shock and Vibration

47277 Max

42056

36835

31614

26393

21172

15951

1073

55088

28774 Min

000

2500 7500

5000 10000 (mm)

Y

XZ

(a)

Y

XZ

57916 Max

51481

45046

38611

32176

2574

19305

1287

64351

0 Min

000

2500 7500

5000 10000 (mm)

(b)

Y

XZ

33437 Max

29722

26007

22292

18577

14862

11147

7432

37171

02117 Min

000

2500 7500

5000 10000 (mm)

(c)

Figure 10 First flexural and torsional modes simulated with estimated material properties (a b) and additional modes such as the firstlongitudinal one (c)

Shock and Vibration 7

Following the initial modal analysis with estimatedproperties the computed frequencies were parametrised andoptimised with the frequencies measured in the experimentYoungrsquos modulus was constricted to a range between187GPa and 229GPa while Poissonrsquos ratio was restricted tovalues ranging from 01 to 03 +e optimisation procedurevaries these two values attempting to match the computedfrequencies with the measured values in the experiment

+e relatively minor difference (11) in the two mea-sured frequencies considered for the torsional mode resultedin a significant effect on the computed Poissonrsquos ratio Forff 1016 kHz and ft 6794 kHz Youngrsquos modulus iscomputed at 1937GPa with Poissonrsquos ratio of 0248 whilefor ff 1016 kHz and ft 7238 kHz Youngrsquos modulus iscalculated at 1936GPa with Poissonrsquos ratio of 0104Considering previous data on similar CuCD grades thevalue of ] 0248 was deemed to be the physically valid oneand retained for the material model along with a roundedYoungrsquos modulus value of 194GPa [13] +e final values forYoungrsquos modulus and Poissonrsquos ratio are shown in Table 3

33 Plasticity-Four-Point Bending Experiment To model thematerial behaviour beyond the elastic regime a four-pointbending test was performed allowing for the implementa-tion of a multilinear hardening model in the numericalanalyses A ZwickRoell Z400 universal testing machine wasused for the four-point flexural test in accordance with theASTM C651 standard ldquoFlexural Strength of Carbon andGraphite by Four-Point Loadingrdquo [28] With this experi-mental setup as shown in the schematic in Figure 11 thebending force is applied at two points along the length of thespecimen which is supported at its extremities Specimenswith an average length of 30mm width of 53mm andthickness of 51mm were tested with such a setup

In the experiment the bending force is progressivelyincreased with the corresponding induced axial strain in thespecimen measured with a strain gauge glued to its surface+e maximum bending moment M is deduced from theforce F and the distance between the load application and thesupports L1 byM (L1F2) In the linear elastic domainthe stress distribution through the sample thickness can beassumed to be linear and the maximum can be calculated byσ (Mh2I) where I is the flexural moment of inertia and his the sample thickness However the force-strain relationmeasured in the experiment is nonlinear indicating thatthere is a nonlinear stress distribution across the thickness ofthe specimen as shown in Figure 12 +e maximum stressthus has to be calculated taking into consideration thisnonlinearity For small curvatures the axial strain can beconsidered to be linearly distributed through the thickness[7]

+e stress must therefore be calculated out of the generalmoment-strain relationship +e moment can be expressedas an integration of the stress distribution through thethickness by

M b 1113946h2

minush2σz(x)xdx (4)

where x is the vertical coordinate from the neutral axis b isthe specimenrsquos width and h is its thickness (height ie alongx) +e moment can be expressed in terms of the measuredstrain as follows

M εtotmax1113872 1113873 2b 1113946h2

0σz

2x

hεtotmax1113874 1113875xdx (5)

where σz((2xh)εtotmax) indicates the stress-strain curve whichcan be numerically derived given the applied momentM(εtotmax) and the measured strain εtotmax +e problem can besolved for each measured value of force and strain resultingin a piecewise stress-strain relationship ie a multilinearhardening curve describing the plastic behaviour of thematerial shown in Figure 13 +e material can be seen toquickly lose linearity and undergoes a significant amount ofplastic deformation as a result of the highly ductile coppermatrix

Table 3 Youngrsquos modulus and Poissonrsquos ratio results determinedby the IET optimisation procedure

Property Symbol Value Uncertainty UnitsYoungrsquos modulus E 194 plusmn32 GPaPoissonrsquos ratio ] 0248 plusmn0002 mdash

F2F2

LL1 L1

Figure 11 Four-point bending experimental setup

StrainStress

Figure 12 Axial strain (linear) and stress (nonlinear) distributionthrough the sample thickness

8 Shock and Vibration

One aspect which is not considered in the model is thecompression behaviour of the material which is assumed tobe identical to that in tension +is assumption may not bevalid at high compressive strains at which point the dia-mond particles may come in contact In contrast when thematerial is loaded in tension the bond between the diamondparticles and the copper matrix may be lost

4 Benchmarking of the Material Model

41 HRMT36 e MultiMat Experiment CERNrsquos HiR-adMat facility is used to test materials used for componentsexposed to extreme conditions in CERNrsquos acceleratorcomplex [10] Upcoming LHC upgrades such as the high-luminosity LHC (HL-LHC) upgrade and the proposedFuture Circular Collider (FCC) require an increase in thebeam energy and thus bring the need for high-performingmaterials+is is especially true for use in components at riskof exposure to accidental beam impacts such as collimatorsand other beam intercepting devices [29]

Experiments conducted in the HiRadMat facility includetests on full-scale collimator jaws [29ndash31] as well as specimenswith simple geometries allowing the benchmarking of nu-merical results obtained through numerical analyses [9 14]+e HRMT36 ldquoMultiMatrdquo experiment conducted in October2017 was designed to offer a reusable platform to test a widerange of novel materials developed for beam interceptingdevices impacted by particle beams with energy densities inthe range of those experienced in the HL-LHC [32 33]

+e test bench hosted 16 target stations mounted on arotatable barrel with each station having a total length of1m+ematerial specimens were mounted in series and hadlengths of 120 or 247mm with cross sections varying from8times 8 to 12times115mm2 18 different materials were tested intotal+e acquisition system included electrical strain gaugesand temperature probes placed on the material specimensalong with remote instrumentation including a laser

Doppler vibrometer (LDV) oriented towards the top face ofthe specimens and a side-mounted radiation-hard high-definition camera [12]+e dimensions for CuCD specimenstested and typical positions of strain gauges and thermalprobes are shown in Figure 14+e target station hosting theCuCD RHP3434 samples consisted of 4 specimens in serieseach having a length of 247mm (with a 3mm space in-between specimens) and a square cross section with an edgelength of 10mm All four specimens were equipped withlongitudinal strain gauges to measure dynamic phenomenainduced by the particle beam impacts

Other than copper diamond (of which two grades from twodifferent suppliers were tested) various materials and materialgrades were tested in the MultiMat experiment including 4grades of MoGr 3 different coatings (Cu Mo and TiN) andnovel carbon-based materials such as highly ordered pyrolyticgraphite (HOPG) and titanium-graphite (TG-1100) +e pulseintensity to which specimens were subjected to ranged from 1to 288 bunches with a typical bunch intensity of 13times1011protons along with nominal beam RMS sizes of 025times 02505times 05 and 2times 2mm2 +e three different types of impacttested were axially centred impacts offset impacts and grazingimpacts (where a coating was present)

42 Modelling and Benchmarking with Experimental Results+e material model for the CuCD RHP3434 grade con-sisting of the measured thermomechanical temperature-dependent properties (Figures 2ndash5 summarised in Table 1)dynamic elastic constants from the IET test (Table 3) andmultilinear hardening curve from the four-point bendingtest (Figure 13) as shown in the previous sections wasincluded in a simply-coupled implicit dynamic thermo-mechanical simulation performed in the FEA softwareANSYS +e material is assumed to be isotropic in naturebased on previous study of the material [6] and is assumedto behave in an elasto-plastic manner based on the four-point bending test presented in this study Similar analyseshave been performed on other materials tested in theMultiMat experiment and information on the thermo-mechanical modelling aspects in such scenarios is detailed inprevious studies [13 15] Similarly analytical solutions forthe dynamic response of slender beams subjected to thermalshock have also been studied extensively [34ndash39]

Figure 15 details the flow of data of the simply-coupledthermomechanical analysis starting with the FLUKA energydeposition maps which volumetrically map the transfer ofenergy in the tested material specimens as a result of theparticle beam impact [40] +is is used as an input in transientthermal analyses which compute the temperature in thespecimens as a result of the beam impact+e thermal analyseswere modelled adiabatically given that the thermal evolutionwithin the specimen is in the order of seconds compared withthe dynamic phenomena observed which are in the order ofmicroseconds [35] +e mesh size used in thermal analysesconsidered the energy density map bin sizemdashin this case themesh consisted of 50 divisions along the longitudinal coor-dinate and 60 divisions across the cross section resulting in amodel with 379181 elements 3D 20-node high-order thermal

0 0005 001 0015 002 0025Strain (mm)

0

50

100

150

200

250

300

350

400St

ress

(MPa

)

Figure 13 Stress-strain diagram for CuCD RHP3434 obtainedfrom the four-point bending test

Shock and Vibration 9

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

From the experimental results the first flexural fre-quency can be easily identified with a peak at 1016 kHz+eslender nature of the specimen made it difficult to excite thetorsional vibration however two possible frequencies cor-responding to the torsional mode were identified in the

flexural measurement at 6794 kHz and 7238 kHz (two faintlines between the 5583 kHz and 9082 kHz readings)

Following the identification of the first flexural mode andthe two possible torsional modes the initial estimates forYoungrsquos modulus and Poissonrsquos ratiomdash208GPa and 022respectivelymdashwere used as a primary input along with thedensity of the material in the optimisation procedure A firstestimation of the flexural and torsional modes was com-puted for the given material properties +e first flexural andtorsional modes determined by the modal analyses areshown in Figure 10 As can be seen additional lsquoundesiredrsquomodes such as the first longitudinal harmonic can also beexcited since movement in this direction is unrestricted

(a) (b)

Figure 8 Boundary conditions forcing mode of interest (a) symmetry plane promoting flexural vibration and (b) fixed axes promotingtorsional vibration and a constraint equation limiting motion of the two extremities of the specimen to rotate in opposite directions

14558319

0036231110

[Hz][Pa][s]

1328430009099575

[Hz][Pa][s]

908203001697770

[Hz][Pa][s]

286255000570350

[Hz][Pa][s]

1016240064722175

[Hz][Pa][s]

7238520001971475

[Hz][Pa][s]

679401000260141

[Hz][Pa][s]

15325930003508175

[Hz][Pa][s]

18077090006504325

[Hz][Pa][s]

13

12

11

10

9

8

7

6

5

4

3

2

1

0

Tim

e (s)

Inte

nsity

(Pa)

Frequency (Hz)10000 20000 30000

1μ

10μ

100μ

1m

10m

Figure 9 Spectrogram showing the IET test flexural frequency measurement as a function of time

Table 2 Calculated flexural and torsional modes for positions Aand B with initial estimates of elastic properties (E 208GPa and] 022)

Specimen dimensionslengthtimeswidthtimes thickness

Flexuralmode (kHz)

Torsionalmode (kHz)

247times1241times 1035mm3 1047 7054

6 Shock and Vibration

47277 Max

42056

36835

31614

26393

21172

15951

1073

55088

28774 Min

000

2500 7500

5000 10000 (mm)

Y

XZ

(a)

Y

XZ

57916 Max

51481

45046

38611

32176

2574

19305

1287

64351

0 Min

000

2500 7500

5000 10000 (mm)

(b)

Y

XZ

33437 Max

29722

26007

22292

18577

14862

11147

7432

37171

02117 Min

000

2500 7500

5000 10000 (mm)

(c)

Figure 10 First flexural and torsional modes simulated with estimated material properties (a b) and additional modes such as the firstlongitudinal one (c)

Shock and Vibration 7

Following the initial modal analysis with estimatedproperties the computed frequencies were parametrised andoptimised with the frequencies measured in the experimentYoungrsquos modulus was constricted to a range between187GPa and 229GPa while Poissonrsquos ratio was restricted tovalues ranging from 01 to 03 +e optimisation procedurevaries these two values attempting to match the computedfrequencies with the measured values in the experiment

+e relatively minor difference (11) in the two mea-sured frequencies considered for the torsional mode resultedin a significant effect on the computed Poissonrsquos ratio Forff 1016 kHz and ft 6794 kHz Youngrsquos modulus iscomputed at 1937GPa with Poissonrsquos ratio of 0248 whilefor ff 1016 kHz and ft 7238 kHz Youngrsquos modulus iscalculated at 1936GPa with Poissonrsquos ratio of 0104Considering previous data on similar CuCD grades thevalue of ] 0248 was deemed to be the physically valid oneand retained for the material model along with a roundedYoungrsquos modulus value of 194GPa [13] +e final values forYoungrsquos modulus and Poissonrsquos ratio are shown in Table 3

33 Plasticity-Four-Point Bending Experiment To model thematerial behaviour beyond the elastic regime a four-pointbending test was performed allowing for the implementa-tion of a multilinear hardening model in the numericalanalyses A ZwickRoell Z400 universal testing machine wasused for the four-point flexural test in accordance with theASTM C651 standard ldquoFlexural Strength of Carbon andGraphite by Four-Point Loadingrdquo [28] With this experi-mental setup as shown in the schematic in Figure 11 thebending force is applied at two points along the length of thespecimen which is supported at its extremities Specimenswith an average length of 30mm width of 53mm andthickness of 51mm were tested with such a setup

In the experiment the bending force is progressivelyincreased with the corresponding induced axial strain in thespecimen measured with a strain gauge glued to its surface+e maximum bending moment M is deduced from theforce F and the distance between the load application and thesupports L1 byM (L1F2) In the linear elastic domainthe stress distribution through the sample thickness can beassumed to be linear and the maximum can be calculated byσ (Mh2I) where I is the flexural moment of inertia and his the sample thickness However the force-strain relationmeasured in the experiment is nonlinear indicating thatthere is a nonlinear stress distribution across the thickness ofthe specimen as shown in Figure 12 +e maximum stressthus has to be calculated taking into consideration thisnonlinearity For small curvatures the axial strain can beconsidered to be linearly distributed through the thickness[7]

+e stress must therefore be calculated out of the generalmoment-strain relationship +e moment can be expressedas an integration of the stress distribution through thethickness by

M b 1113946h2

minush2σz(x)xdx (4)

where x is the vertical coordinate from the neutral axis b isthe specimenrsquos width and h is its thickness (height ie alongx) +e moment can be expressed in terms of the measuredstrain as follows

M εtotmax1113872 1113873 2b 1113946h2

0σz

2x

hεtotmax1113874 1113875xdx (5)

where σz((2xh)εtotmax) indicates the stress-strain curve whichcan be numerically derived given the applied momentM(εtotmax) and the measured strain εtotmax +e problem can besolved for each measured value of force and strain resultingin a piecewise stress-strain relationship ie a multilinearhardening curve describing the plastic behaviour of thematerial shown in Figure 13 +e material can be seen toquickly lose linearity and undergoes a significant amount ofplastic deformation as a result of the highly ductile coppermatrix

Table 3 Youngrsquos modulus and Poissonrsquos ratio results determinedby the IET optimisation procedure

Property Symbol Value Uncertainty UnitsYoungrsquos modulus E 194 plusmn32 GPaPoissonrsquos ratio ] 0248 plusmn0002 mdash

F2F2

LL1 L1

Figure 11 Four-point bending experimental setup

StrainStress

Figure 12 Axial strain (linear) and stress (nonlinear) distributionthrough the sample thickness

8 Shock and Vibration

One aspect which is not considered in the model is thecompression behaviour of the material which is assumed tobe identical to that in tension +is assumption may not bevalid at high compressive strains at which point the dia-mond particles may come in contact In contrast when thematerial is loaded in tension the bond between the diamondparticles and the copper matrix may be lost

4 Benchmarking of the Material Model

41 HRMT36 e MultiMat Experiment CERNrsquos HiR-adMat facility is used to test materials used for componentsexposed to extreme conditions in CERNrsquos acceleratorcomplex [10] Upcoming LHC upgrades such as the high-luminosity LHC (HL-LHC) upgrade and the proposedFuture Circular Collider (FCC) require an increase in thebeam energy and thus bring the need for high-performingmaterials+is is especially true for use in components at riskof exposure to accidental beam impacts such as collimatorsand other beam intercepting devices [29]

Experiments conducted in the HiRadMat facility includetests on full-scale collimator jaws [29ndash31] as well as specimenswith simple geometries allowing the benchmarking of nu-merical results obtained through numerical analyses [9 14]+e HRMT36 ldquoMultiMatrdquo experiment conducted in October2017 was designed to offer a reusable platform to test a widerange of novel materials developed for beam interceptingdevices impacted by particle beams with energy densities inthe range of those experienced in the HL-LHC [32 33]

+e test bench hosted 16 target stations mounted on arotatable barrel with each station having a total length of1m+ematerial specimens were mounted in series and hadlengths of 120 or 247mm with cross sections varying from8times 8 to 12times115mm2 18 different materials were tested intotal+e acquisition system included electrical strain gaugesand temperature probes placed on the material specimensalong with remote instrumentation including a laser

Doppler vibrometer (LDV) oriented towards the top face ofthe specimens and a side-mounted radiation-hard high-definition camera [12]+e dimensions for CuCD specimenstested and typical positions of strain gauges and thermalprobes are shown in Figure 14+e target station hosting theCuCD RHP3434 samples consisted of 4 specimens in serieseach having a length of 247mm (with a 3mm space in-between specimens) and a square cross section with an edgelength of 10mm All four specimens were equipped withlongitudinal strain gauges to measure dynamic phenomenainduced by the particle beam impacts

Other than copper diamond (of which two grades from twodifferent suppliers were tested) various materials and materialgrades were tested in the MultiMat experiment including 4grades of MoGr 3 different coatings (Cu Mo and TiN) andnovel carbon-based materials such as highly ordered pyrolyticgraphite (HOPG) and titanium-graphite (TG-1100) +e pulseintensity to which specimens were subjected to ranged from 1to 288 bunches with a typical bunch intensity of 13times1011protons along with nominal beam RMS sizes of 025times 02505times 05 and 2times 2mm2 +e three different types of impacttested were axially centred impacts offset impacts and grazingimpacts (where a coating was present)

42 Modelling and Benchmarking with Experimental Results+e material model for the CuCD RHP3434 grade con-sisting of the measured thermomechanical temperature-dependent properties (Figures 2ndash5 summarised in Table 1)dynamic elastic constants from the IET test (Table 3) andmultilinear hardening curve from the four-point bendingtest (Figure 13) as shown in the previous sections wasincluded in a simply-coupled implicit dynamic thermo-mechanical simulation performed in the FEA softwareANSYS +e material is assumed to be isotropic in naturebased on previous study of the material [6] and is assumedto behave in an elasto-plastic manner based on the four-point bending test presented in this study Similar analyseshave been performed on other materials tested in theMultiMat experiment and information on the thermo-mechanical modelling aspects in such scenarios is detailed inprevious studies [13 15] Similarly analytical solutions forthe dynamic response of slender beams subjected to thermalshock have also been studied extensively [34ndash39]

Figure 15 details the flow of data of the simply-coupledthermomechanical analysis starting with the FLUKA energydeposition maps which volumetrically map the transfer ofenergy in the tested material specimens as a result of theparticle beam impact [40] +is is used as an input in transientthermal analyses which compute the temperature in thespecimens as a result of the beam impact+e thermal analyseswere modelled adiabatically given that the thermal evolutionwithin the specimen is in the order of seconds compared withthe dynamic phenomena observed which are in the order ofmicroseconds [35] +e mesh size used in thermal analysesconsidered the energy density map bin sizemdashin this case themesh consisted of 50 divisions along the longitudinal coor-dinate and 60 divisions across the cross section resulting in amodel with 379181 elements 3D 20-node high-order thermal

0 0005 001 0015 002 0025Strain (mm)

0

50

100

150

200

250

300

350

400St

ress

(MPa

)

Figure 13 Stress-strain diagram for CuCD RHP3434 obtainedfrom the four-point bending test

Shock and Vibration 9

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

47277 Max

42056

36835

31614

26393

21172

15951

1073

55088

28774 Min

000

2500 7500

5000 10000 (mm)

Y

XZ

(a)

Y

XZ

57916 Max

51481

45046

38611

32176

2574

19305

1287

64351

0 Min

000

2500 7500

5000 10000 (mm)

(b)

Y

XZ

33437 Max

29722

26007

22292

18577

14862

11147

7432

37171

02117 Min

000

2500 7500

5000 10000 (mm)

(c)

Figure 10 First flexural and torsional modes simulated with estimated material properties (a b) and additional modes such as the firstlongitudinal one (c)

Shock and Vibration 7

Following the initial modal analysis with estimatedproperties the computed frequencies were parametrised andoptimised with the frequencies measured in the experimentYoungrsquos modulus was constricted to a range between187GPa and 229GPa while Poissonrsquos ratio was restricted tovalues ranging from 01 to 03 +e optimisation procedurevaries these two values attempting to match the computedfrequencies with the measured values in the experiment

+e relatively minor difference (11) in the two mea-sured frequencies considered for the torsional mode resultedin a significant effect on the computed Poissonrsquos ratio Forff 1016 kHz and ft 6794 kHz Youngrsquos modulus iscomputed at 1937GPa with Poissonrsquos ratio of 0248 whilefor ff 1016 kHz and ft 7238 kHz Youngrsquos modulus iscalculated at 1936GPa with Poissonrsquos ratio of 0104Considering previous data on similar CuCD grades thevalue of ] 0248 was deemed to be the physically valid oneand retained for the material model along with a roundedYoungrsquos modulus value of 194GPa [13] +e final values forYoungrsquos modulus and Poissonrsquos ratio are shown in Table 3

33 Plasticity-Four-Point Bending Experiment To model thematerial behaviour beyond the elastic regime a four-pointbending test was performed allowing for the implementa-tion of a multilinear hardening model in the numericalanalyses A ZwickRoell Z400 universal testing machine wasused for the four-point flexural test in accordance with theASTM C651 standard ldquoFlexural Strength of Carbon andGraphite by Four-Point Loadingrdquo [28] With this experi-mental setup as shown in the schematic in Figure 11 thebending force is applied at two points along the length of thespecimen which is supported at its extremities Specimenswith an average length of 30mm width of 53mm andthickness of 51mm were tested with such a setup

In the experiment the bending force is progressivelyincreased with the corresponding induced axial strain in thespecimen measured with a strain gauge glued to its surface+e maximum bending moment M is deduced from theforce F and the distance between the load application and thesupports L1 byM (L1F2) In the linear elastic domainthe stress distribution through the sample thickness can beassumed to be linear and the maximum can be calculated byσ (Mh2I) where I is the flexural moment of inertia and his the sample thickness However the force-strain relationmeasured in the experiment is nonlinear indicating thatthere is a nonlinear stress distribution across the thickness ofthe specimen as shown in Figure 12 +e maximum stressthus has to be calculated taking into consideration thisnonlinearity For small curvatures the axial strain can beconsidered to be linearly distributed through the thickness[7]

+e stress must therefore be calculated out of the generalmoment-strain relationship +e moment can be expressedas an integration of the stress distribution through thethickness by

M b 1113946h2

minush2σz(x)xdx (4)

where x is the vertical coordinate from the neutral axis b isthe specimenrsquos width and h is its thickness (height ie alongx) +e moment can be expressed in terms of the measuredstrain as follows

M εtotmax1113872 1113873 2b 1113946h2

0σz

2x

hεtotmax1113874 1113875xdx (5)

where σz((2xh)εtotmax) indicates the stress-strain curve whichcan be numerically derived given the applied momentM(εtotmax) and the measured strain εtotmax +e problem can besolved for each measured value of force and strain resultingin a piecewise stress-strain relationship ie a multilinearhardening curve describing the plastic behaviour of thematerial shown in Figure 13 +e material can be seen toquickly lose linearity and undergoes a significant amount ofplastic deformation as a result of the highly ductile coppermatrix

Table 3 Youngrsquos modulus and Poissonrsquos ratio results determinedby the IET optimisation procedure

Property Symbol Value Uncertainty UnitsYoungrsquos modulus E 194 plusmn32 GPaPoissonrsquos ratio ] 0248 plusmn0002 mdash

F2F2

LL1 L1

Figure 11 Four-point bending experimental setup

StrainStress

Figure 12 Axial strain (linear) and stress (nonlinear) distributionthrough the sample thickness

8 Shock and Vibration

One aspect which is not considered in the model is thecompression behaviour of the material which is assumed tobe identical to that in tension +is assumption may not bevalid at high compressive strains at which point the dia-mond particles may come in contact In contrast when thematerial is loaded in tension the bond between the diamondparticles and the copper matrix may be lost

4 Benchmarking of the Material Model

41 HRMT36 e MultiMat Experiment CERNrsquos HiR-adMat facility is used to test materials used for componentsexposed to extreme conditions in CERNrsquos acceleratorcomplex [10] Upcoming LHC upgrades such as the high-luminosity LHC (HL-LHC) upgrade and the proposedFuture Circular Collider (FCC) require an increase in thebeam energy and thus bring the need for high-performingmaterials+is is especially true for use in components at riskof exposure to accidental beam impacts such as collimatorsand other beam intercepting devices [29]

Experiments conducted in the HiRadMat facility includetests on full-scale collimator jaws [29ndash31] as well as specimenswith simple geometries allowing the benchmarking of nu-merical results obtained through numerical analyses [9 14]+e HRMT36 ldquoMultiMatrdquo experiment conducted in October2017 was designed to offer a reusable platform to test a widerange of novel materials developed for beam interceptingdevices impacted by particle beams with energy densities inthe range of those experienced in the HL-LHC [32 33]

+e test bench hosted 16 target stations mounted on arotatable barrel with each station having a total length of1m+ematerial specimens were mounted in series and hadlengths of 120 or 247mm with cross sections varying from8times 8 to 12times115mm2 18 different materials were tested intotal+e acquisition system included electrical strain gaugesand temperature probes placed on the material specimensalong with remote instrumentation including a laser

Doppler vibrometer (LDV) oriented towards the top face ofthe specimens and a side-mounted radiation-hard high-definition camera [12]+e dimensions for CuCD specimenstested and typical positions of strain gauges and thermalprobes are shown in Figure 14+e target station hosting theCuCD RHP3434 samples consisted of 4 specimens in serieseach having a length of 247mm (with a 3mm space in-between specimens) and a square cross section with an edgelength of 10mm All four specimens were equipped withlongitudinal strain gauges to measure dynamic phenomenainduced by the particle beam impacts

Other than copper diamond (of which two grades from twodifferent suppliers were tested) various materials and materialgrades were tested in the MultiMat experiment including 4grades of MoGr 3 different coatings (Cu Mo and TiN) andnovel carbon-based materials such as highly ordered pyrolyticgraphite (HOPG) and titanium-graphite (TG-1100) +e pulseintensity to which specimens were subjected to ranged from 1to 288 bunches with a typical bunch intensity of 13times1011protons along with nominal beam RMS sizes of 025times 02505times 05 and 2times 2mm2 +e three different types of impacttested were axially centred impacts offset impacts and grazingimpacts (where a coating was present)

42 Modelling and Benchmarking with Experimental Results+e material model for the CuCD RHP3434 grade con-sisting of the measured thermomechanical temperature-dependent properties (Figures 2ndash5 summarised in Table 1)dynamic elastic constants from the IET test (Table 3) andmultilinear hardening curve from the four-point bendingtest (Figure 13) as shown in the previous sections wasincluded in a simply-coupled implicit dynamic thermo-mechanical simulation performed in the FEA softwareANSYS +e material is assumed to be isotropic in naturebased on previous study of the material [6] and is assumedto behave in an elasto-plastic manner based on the four-point bending test presented in this study Similar analyseshave been performed on other materials tested in theMultiMat experiment and information on the thermo-mechanical modelling aspects in such scenarios is detailed inprevious studies [13 15] Similarly analytical solutions forthe dynamic response of slender beams subjected to thermalshock have also been studied extensively [34ndash39]

Figure 15 details the flow of data of the simply-coupledthermomechanical analysis starting with the FLUKA energydeposition maps which volumetrically map the transfer ofenergy in the tested material specimens as a result of theparticle beam impact [40] +is is used as an input in transientthermal analyses which compute the temperature in thespecimens as a result of the beam impact+e thermal analyseswere modelled adiabatically given that the thermal evolutionwithin the specimen is in the order of seconds compared withthe dynamic phenomena observed which are in the order ofmicroseconds [35] +e mesh size used in thermal analysesconsidered the energy density map bin sizemdashin this case themesh consisted of 50 divisions along the longitudinal coor-dinate and 60 divisions across the cross section resulting in amodel with 379181 elements 3D 20-node high-order thermal

0 0005 001 0015 002 0025Strain (mm)

0

50

100

150

200

250

300

350

400St

ress

(MPa

)

Figure 13 Stress-strain diagram for CuCD RHP3434 obtainedfrom the four-point bending test

Shock and Vibration 9

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

Following the initial modal analysis with estimatedproperties the computed frequencies were parametrised andoptimised with the frequencies measured in the experimentYoungrsquos modulus was constricted to a range between187GPa and 229GPa while Poissonrsquos ratio was restricted tovalues ranging from 01 to 03 +e optimisation procedurevaries these two values attempting to match the computedfrequencies with the measured values in the experiment

+e relatively minor difference (11) in the two mea-sured frequencies considered for the torsional mode resultedin a significant effect on the computed Poissonrsquos ratio Forff 1016 kHz and ft 6794 kHz Youngrsquos modulus iscomputed at 1937GPa with Poissonrsquos ratio of 0248 whilefor ff 1016 kHz and ft 7238 kHz Youngrsquos modulus iscalculated at 1936GPa with Poissonrsquos ratio of 0104Considering previous data on similar CuCD grades thevalue of ] 0248 was deemed to be the physically valid oneand retained for the material model along with a roundedYoungrsquos modulus value of 194GPa [13] +e final values forYoungrsquos modulus and Poissonrsquos ratio are shown in Table 3

33 Plasticity-Four-Point Bending Experiment To model thematerial behaviour beyond the elastic regime a four-pointbending test was performed allowing for the implementa-tion of a multilinear hardening model in the numericalanalyses A ZwickRoell Z400 universal testing machine wasused for the four-point flexural test in accordance with theASTM C651 standard ldquoFlexural Strength of Carbon andGraphite by Four-Point Loadingrdquo [28] With this experi-mental setup as shown in the schematic in Figure 11 thebending force is applied at two points along the length of thespecimen which is supported at its extremities Specimenswith an average length of 30mm width of 53mm andthickness of 51mm were tested with such a setup

In the experiment the bending force is progressivelyincreased with the corresponding induced axial strain in thespecimen measured with a strain gauge glued to its surface+e maximum bending moment M is deduced from theforce F and the distance between the load application and thesupports L1 byM (L1F2) In the linear elastic domainthe stress distribution through the sample thickness can beassumed to be linear and the maximum can be calculated byσ (Mh2I) where I is the flexural moment of inertia and his the sample thickness However the force-strain relationmeasured in the experiment is nonlinear indicating thatthere is a nonlinear stress distribution across the thickness ofthe specimen as shown in Figure 12 +e maximum stressthus has to be calculated taking into consideration thisnonlinearity For small curvatures the axial strain can beconsidered to be linearly distributed through the thickness[7]

+e stress must therefore be calculated out of the generalmoment-strain relationship +e moment can be expressedas an integration of the stress distribution through thethickness by

M b 1113946h2

minush2σz(x)xdx (4)

where x is the vertical coordinate from the neutral axis b isthe specimenrsquos width and h is its thickness (height ie alongx) +e moment can be expressed in terms of the measuredstrain as follows

M εtotmax1113872 1113873 2b 1113946h2

0σz

2x

hεtotmax1113874 1113875xdx (5)

where σz((2xh)εtotmax) indicates the stress-strain curve whichcan be numerically derived given the applied momentM(εtotmax) and the measured strain εtotmax +e problem can besolved for each measured value of force and strain resultingin a piecewise stress-strain relationship ie a multilinearhardening curve describing the plastic behaviour of thematerial shown in Figure 13 +e material can be seen toquickly lose linearity and undergoes a significant amount ofplastic deformation as a result of the highly ductile coppermatrix

Table 3 Youngrsquos modulus and Poissonrsquos ratio results determinedby the IET optimisation procedure

Property Symbol Value Uncertainty UnitsYoungrsquos modulus E 194 plusmn32 GPaPoissonrsquos ratio ] 0248 plusmn0002 mdash

F2F2

LL1 L1

Figure 11 Four-point bending experimental setup

StrainStress

Figure 12 Axial strain (linear) and stress (nonlinear) distributionthrough the sample thickness

8 Shock and Vibration

One aspect which is not considered in the model is thecompression behaviour of the material which is assumed tobe identical to that in tension +is assumption may not bevalid at high compressive strains at which point the dia-mond particles may come in contact In contrast when thematerial is loaded in tension the bond between the diamondparticles and the copper matrix may be lost

4 Benchmarking of the Material Model

41 HRMT36 e MultiMat Experiment CERNrsquos HiR-adMat facility is used to test materials used for componentsexposed to extreme conditions in CERNrsquos acceleratorcomplex [10] Upcoming LHC upgrades such as the high-luminosity LHC (HL-LHC) upgrade and the proposedFuture Circular Collider (FCC) require an increase in thebeam energy and thus bring the need for high-performingmaterials+is is especially true for use in components at riskof exposure to accidental beam impacts such as collimatorsand other beam intercepting devices [29]

Experiments conducted in the HiRadMat facility includetests on full-scale collimator jaws [29ndash31] as well as specimenswith simple geometries allowing the benchmarking of nu-merical results obtained through numerical analyses [9 14]+e HRMT36 ldquoMultiMatrdquo experiment conducted in October2017 was designed to offer a reusable platform to test a widerange of novel materials developed for beam interceptingdevices impacted by particle beams with energy densities inthe range of those experienced in the HL-LHC [32 33]

+e test bench hosted 16 target stations mounted on arotatable barrel with each station having a total length of1m+ematerial specimens were mounted in series and hadlengths of 120 or 247mm with cross sections varying from8times 8 to 12times115mm2 18 different materials were tested intotal+e acquisition system included electrical strain gaugesand temperature probes placed on the material specimensalong with remote instrumentation including a laser

Doppler vibrometer (LDV) oriented towards the top face ofthe specimens and a side-mounted radiation-hard high-definition camera [12]+e dimensions for CuCD specimenstested and typical positions of strain gauges and thermalprobes are shown in Figure 14+e target station hosting theCuCD RHP3434 samples consisted of 4 specimens in serieseach having a length of 247mm (with a 3mm space in-between specimens) and a square cross section with an edgelength of 10mm All four specimens were equipped withlongitudinal strain gauges to measure dynamic phenomenainduced by the particle beam impacts

Other than copper diamond (of which two grades from twodifferent suppliers were tested) various materials and materialgrades were tested in the MultiMat experiment including 4grades of MoGr 3 different coatings (Cu Mo and TiN) andnovel carbon-based materials such as highly ordered pyrolyticgraphite (HOPG) and titanium-graphite (TG-1100) +e pulseintensity to which specimens were subjected to ranged from 1to 288 bunches with a typical bunch intensity of 13times1011protons along with nominal beam RMS sizes of 025times 02505times 05 and 2times 2mm2 +e three different types of impacttested were axially centred impacts offset impacts and grazingimpacts (where a coating was present)

42 Modelling and Benchmarking with Experimental Results+e material model for the CuCD RHP3434 grade con-sisting of the measured thermomechanical temperature-dependent properties (Figures 2ndash5 summarised in Table 1)dynamic elastic constants from the IET test (Table 3) andmultilinear hardening curve from the four-point bendingtest (Figure 13) as shown in the previous sections wasincluded in a simply-coupled implicit dynamic thermo-mechanical simulation performed in the FEA softwareANSYS +e material is assumed to be isotropic in naturebased on previous study of the material [6] and is assumedto behave in an elasto-plastic manner based on the four-point bending test presented in this study Similar analyseshave been performed on other materials tested in theMultiMat experiment and information on the thermo-mechanical modelling aspects in such scenarios is detailed inprevious studies [13 15] Similarly analytical solutions forthe dynamic response of slender beams subjected to thermalshock have also been studied extensively [34ndash39]

Figure 15 details the flow of data of the simply-coupledthermomechanical analysis starting with the FLUKA energydeposition maps which volumetrically map the transfer ofenergy in the tested material specimens as a result of theparticle beam impact [40] +is is used as an input in transientthermal analyses which compute the temperature in thespecimens as a result of the beam impact+e thermal analyseswere modelled adiabatically given that the thermal evolutionwithin the specimen is in the order of seconds compared withthe dynamic phenomena observed which are in the order ofmicroseconds [35] +e mesh size used in thermal analysesconsidered the energy density map bin sizemdashin this case themesh consisted of 50 divisions along the longitudinal coor-dinate and 60 divisions across the cross section resulting in amodel with 379181 elements 3D 20-node high-order thermal

0 0005 001 0015 002 0025Strain (mm)

0

50

100

150

200

250

300

350

400St

ress

(MPa

)

Figure 13 Stress-strain diagram for CuCD RHP3434 obtainedfrom the four-point bending test

Shock and Vibration 9

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

One aspect which is not considered in the model is thecompression behaviour of the material which is assumed tobe identical to that in tension +is assumption may not bevalid at high compressive strains at which point the dia-mond particles may come in contact In contrast when thematerial is loaded in tension the bond between the diamondparticles and the copper matrix may be lost

4 Benchmarking of the Material Model

41 HRMT36 e MultiMat Experiment CERNrsquos HiR-adMat facility is used to test materials used for componentsexposed to extreme conditions in CERNrsquos acceleratorcomplex [10] Upcoming LHC upgrades such as the high-luminosity LHC (HL-LHC) upgrade and the proposedFuture Circular Collider (FCC) require an increase in thebeam energy and thus bring the need for high-performingmaterials+is is especially true for use in components at riskof exposure to accidental beam impacts such as collimatorsand other beam intercepting devices [29]

Experiments conducted in the HiRadMat facility includetests on full-scale collimator jaws [29ndash31] as well as specimenswith simple geometries allowing the benchmarking of nu-merical results obtained through numerical analyses [9 14]+e HRMT36 ldquoMultiMatrdquo experiment conducted in October2017 was designed to offer a reusable platform to test a widerange of novel materials developed for beam interceptingdevices impacted by particle beams with energy densities inthe range of those experienced in the HL-LHC [32 33]

+e test bench hosted 16 target stations mounted on arotatable barrel with each station having a total length of1m+ematerial specimens were mounted in series and hadlengths of 120 or 247mm with cross sections varying from8times 8 to 12times115mm2 18 different materials were tested intotal+e acquisition system included electrical strain gaugesand temperature probes placed on the material specimensalong with remote instrumentation including a laser

Doppler vibrometer (LDV) oriented towards the top face ofthe specimens and a side-mounted radiation-hard high-definition camera [12]+e dimensions for CuCD specimenstested and typical positions of strain gauges and thermalprobes are shown in Figure 14+e target station hosting theCuCD RHP3434 samples consisted of 4 specimens in serieseach having a length of 247mm (with a 3mm space in-between specimens) and a square cross section with an edgelength of 10mm All four specimens were equipped withlongitudinal strain gauges to measure dynamic phenomenainduced by the particle beam impacts

Other than copper diamond (of which two grades from twodifferent suppliers were tested) various materials and materialgrades were tested in the MultiMat experiment including 4grades of MoGr 3 different coatings (Cu Mo and TiN) andnovel carbon-based materials such as highly ordered pyrolyticgraphite (HOPG) and titanium-graphite (TG-1100) +e pulseintensity to which specimens were subjected to ranged from 1to 288 bunches with a typical bunch intensity of 13times1011protons along with nominal beam RMS sizes of 025times 02505times 05 and 2times 2mm2 +e three different types of impacttested were axially centred impacts offset impacts and grazingimpacts (where a coating was present)

42 Modelling and Benchmarking with Experimental Results+e material model for the CuCD RHP3434 grade con-sisting of the measured thermomechanical temperature-dependent properties (Figures 2ndash5 summarised in Table 1)dynamic elastic constants from the IET test (Table 3) andmultilinear hardening curve from the four-point bendingtest (Figure 13) as shown in the previous sections wasincluded in a simply-coupled implicit dynamic thermo-mechanical simulation performed in the FEA softwareANSYS +e material is assumed to be isotropic in naturebased on previous study of the material [6] and is assumedto behave in an elasto-plastic manner based on the four-point bending test presented in this study Similar analyseshave been performed on other materials tested in theMultiMat experiment and information on the thermo-mechanical modelling aspects in such scenarios is detailed inprevious studies [13 15] Similarly analytical solutions forthe dynamic response of slender beams subjected to thermalshock have also been studied extensively [34ndash39]

Figure 15 details the flow of data of the simply-coupledthermomechanical analysis starting with the FLUKA energydeposition maps which volumetrically map the transfer ofenergy in the tested material specimens as a result of theparticle beam impact [40] +is is used as an input in transientthermal analyses which compute the temperature in thespecimens as a result of the beam impact+e thermal analyseswere modelled adiabatically given that the thermal evolutionwithin the specimen is in the order of seconds compared withthe dynamic phenomena observed which are in the order ofmicroseconds [35] +e mesh size used in thermal analysesconsidered the energy density map bin sizemdashin this case themesh consisted of 50 divisions along the longitudinal coor-dinate and 60 divisions across the cross section resulting in amodel with 379181 elements 3D 20-node high-order thermal

0 0005 001 0015 002 0025Strain (mm)

0

50

100

150

200

250

300

350

400St

ress

(MPa

)

Figure 13 Stress-strain diagram for CuCD RHP3434 obtainedfrom the four-point bending test

Shock and Vibration 9

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

elements were adopted for this analysis +is provided a goodbalance between accuracy in thermal gradient computed andcomputation time+e thermal analysis consisted of two stepsone modelling the energy deposition and the other spanningover a longer time period in order to compute the thermalevolution across the material

+e temperature field generated from the thermalanalysis was imported as a thermal load in the transientstructural analysis which models the dynamic phenomenameasured in the MultiMat experimental campaign +estructural analysis consists of three stepsmdashthe first coveringthe energy deposition period and consisting of 10 substepsthe second considering the dynamic longitudinal responseand covering five longitudinal wave oscillations and thethird considering the flexural response covering threebending oscillations +e substep for each phase was setaccordingly depending on the phenomenon of interest andthe CourantndashFriedrichsndashLewy (CFL) condition was con-sidered to determine the appropriate element size consid-ering the speed of sound of the material and the respectivetime step +is resulted in a 247times10times10mm3 model with15350 elements and 70753 nodes Higher order 3D 20-nodesolid elements with quadratic displacement behaviour wereused for the structural analysis With regards to boundaryconditions a simply supported configuration was modelledrestricting vertical movement of the bottom edges at theextremities of the specimen as shown in Figure 16 [13]

+e thermomechanical material model adopted in theanalysis was built from measurements discussed in theMaterial Characterisation section Temperature-dependentvalues for the linear coefficient of thermal expansiondensity specific heat capacity and thermal conductivitywere adopted as shown in Figures 3ndash6 in the +ermo-mechanical Characterisation section Youngrsquos modulus andPoissonrsquos ratio were obtained as described in the ImpulseExcitation Technique Test section while the multilinearhardening curve was found as shown in the Plasticity-Four-Point Bending Experiment section +e material propertiesat room temperature are summarised in Table 4

Beam parameters for the two pulses modelled in nu-merical simulations such as pulse duration intensitytransverse beam size σ and vertical offset ηy are shown inTable 5 +e first shot had a total intensity of 1433times1011protons (1 bunch) and a vertical offset of 3mm while thesecond shot consisted of a longer central 12-bunch shotwith a total intensity of 6880times1012 protons Modelling twodistinctly different pulses allows one to test the ability of theformulated material model to simulate different scenarios+e calculated energy deposited along the length of thespecimens along with the temperature along the length ofthe specimens following beam impact (ie at 25 ns for shot128 and at 300 ns for shot 132) can be seen in Figure 17Note that the temperature line was probed along the beamimpact position ie at a vertical offset of 3mm for shot 128and along the longitudinal axis of the specimen for shot 132

ANSYS transient structuralFLUKA ANSYS transient thermal

Energydeposition

mapTemperature

field

Figure 15 Flowchart detailing flow of data for the weakly coupled thermomechanical analysis

X

YZ

10m

m

247mm

LF1 LF2

BF1

10mmBF2 BF3 BF4

RF1

Figure 14 Typical dimensions and position of longitudinal strain gauges (black) and temperature sensors (red) on the specimensLongitudinal strain gauges are referenced by a BF (bottom face) or RF (right face) notation and thermal probes are referenced by an LF (leftface) notation

Z

X

Y

Figure 16 Structural model ldquosimply-supportedrdquo boundary con-dition setup (bold edges constrained in X-direction)

Table 4 Summary of the material model for CuCD RHP3434 atroom temperature

Property Value Uncertainty UnitsDensity 5700 plusmn33 kgmiddotmminus3

Specific heat capacity 407 plusmn12 Jmiddotkgminus1 Kminus1

+ermal conductivity 537 plusmn15 Wmiddotmminus1 Kminus1

Coefficient of thermalexpansion 619 plusmn02 10minus6 Kminus1

Youngrsquos modulus 194 plusmn32 GPaPoissonrsquos ratio 0248 0002 mdash

10 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

+e most loaded CuCD specimen can be seen to be thesecond in the target stationmdashthis specimen was chosen tomodel the dynamic phenomena in the structural analysisdue to the lower signal-to-noise ratio achieved as a result ofthe higher deposited energy and subsequent larger

amplitudes in the propagating waves Figure 18 shows ex-perimental measurements obtained from a thermal probepositioned on the outer surface of the second specimen at alength of 823mm LF1 in Figure 14 compared with thermalanalysis results for the modelled specimen probed at the

Table 5 Beam parameters for the studied impacts (pulse 128 and pulse 132)

Pulse no Pulse duration (ns) No of bunches Bunch intensity (protonsbunch) Total intensity (protons) σ (mm) ηy (mm)128 25 1 1433times1011 1433times1011 05 3132 300 12 5733times1011 6880times1012 05 0

0 200 400 600 800 1000Length (mm)

0

2

4

6

8

10

12

14

16

18

20

Ener

gy p

er cm

(Jc

m)

Specimen 1Specimen 2

Specimen 3Specimen 4

(a)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

0

10

20

30

40

50

60

70

80

90

Ener

gy p

er cm

(Jc

m)

(b)

0 200 400 600 800 1000Length (mm)

20

30

40

50

60

70

80

Tem

pera

ture

(degC)

Specimen 1Specimen 2

Specimen 3Specimen 4

(c)

0 200 400 600 800 1000Length (mm)

Specimen 1Specimen 2

Specimen 3Specimen 4

20

40

60

80

100

120

140

160

180

200

Tem

pera

ture

(degC)

(d)

Figure 17 Energy deposited along the length of the specimens in the CuCD RHP target station for shots 128 (a) and 132 (b) andtemperature along the length of specimens for each respective shot (c d)

Shock and Vibration 11

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

same location +e thermal probe has a reaction time in theorder of seconds and is not capable of capturing variations oftemperature in the order of microseconds +is coupledwith issues related to contact resistance between the thermalprobe and the tested specimens results in a delay in ex-perimental measurements to reach the maximum temper-ature when compared with the numerical analysis +esimulation can otherwise be seen to accurately predict themaximum temperature at the surface of the specimen

+e temperature field obtained from the thermal analyseswas imported in a structural analysis modelling the materialrsquosdynamic behaviour upon beam impact +e dynamic longi-tudinal and flexural response was modelled by considering asimply supported specimen (modelling the experimentalsetup) Again the second specimen in the target station (beingthe most loaded) was modelled and results were probed at alength of 154mm at a point at the centre of the specimenrsquosbottom face +ese results were compared with experimentalresults obtained from a longitudinal strain gauge attached tothe specimen at the same position +e dynamic longitudinalresponse for shot 128 probed at a length of 154mm on thebottom face of the specimen obtained with the application ofthe material model described in the study is compared withexperimental results in Figure 19 Along with a minor butmeasurable difference in amplitude the main discrepancybetween the two signals can be seen to be in the longitudinalfrequency with the numerical result having a frequency of118 kHz compared with an experimental result of 130 kHz(10 difference) +e speed of sound of the materialmdashonwhich the frequency is dependentmdashis a function of thedensity of the specimen and the materialrsquos stiffness (Youngrsquosmodulus in the elastic domain or δσδε in general)

+e relatively high variability inherent with this type ofmaterial (a result of the large amount of porosity) may causea variation of properties between different specimen batcheseg between the ones tested in the characterisation cam-paign and those tested in the MultiMat experiment +e

10 15 20 25 30Time (s)

20

22

24

26

28

30

32Te

mpe

ratu

re (deg

C)

SimulationExperiment

0 5

(a)

SimulationExperiment

0 5 10 15 20 25 30Time (s)

20

25

30

35

40

45

50

55

60

Tem

pera

ture

(degC)

(b)

Figure 18 +ermal analysis results compared with experimental results for shot 128 and shot 132 probed on the surface of the secondspecimen at a length of 823mm

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 19 Longitudinal strain for shot 128 probed at a length of154mm on the bottom face of the second specimen+e numericalmodel with Youngrsquos modulus of 194GPa and the multilinearhardening model

12 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

density measured in the characterisation campaign is as-sumed to be valid since this takes into account the materialporosity suggesting that the variations are results of anunderestimation of the materialrsquos stiffness in the model +ematerial model was thus updated with a higher Youngrsquosmodulus of 220GPa a value that provides the best fit forlongitudinal and flexural frequencies between experimentaland numerical data retaining all other properties Simula-tion results compared with the experimental result for thisscenario are shown in Figure 20 +e increase in Youngrsquosmodulus results in a lsquocompressionrsquo of the waveform (ie anincrease in frequency) which leads to a better fit with ex-perimental data with an improved longitudinal frequency of126 kHz +e benefits of using a multilinear hardeningmodel rather than an elastic model can be clearly seen inFigure 21 which shows the result for a perfectly elastic (ieno plasticity and not multilinear) model with Youngrsquosmodulus of 220GPa It can be seen that in this case thelongitudinal frequency is retained to a certain extent but themodel fails to correctly simulate the high frequency contentof the signal which can be seen to have a much higheramplitude than the one observed in the experiment

With regard to the flexural response shown in Figure 22for the same beam impact (shot 128) the numerical result (withmultilinear hardening and a Youngrsquos Modulus of 220 GPa) canbe seen to overestimate the amplitude of oscillation +isdiscrepancy is addressed by improving thematerial model withthe introduction of Rayleigh damping to the analysis with aRayleigh damping ratio of ζ 861 at 450Hz (the flexuralfrequency) for a stiffness coefficient β 6088times10minus5 +eresulting waveform can be seen to closely follow the experi-mentally measured signal With the implementation of Ray-leigh damping at the flexural frequency it can be also seen thatinformation at higher frequencies is lost

+e implementation of the multilinear hardening modelalready produced a decay in amplitude of numerical results asa result of dissipation in the longitudinal response as could beobserved by comparing the numerical results in Figure 21 withthose shown in Figure 20 Similarly for the flexural responsethe numerical results without the use of the Rayleigh dampingcoefficient (also shown in Figure 22) can also be seen to exhibita reduction in amplitude in both the lower frequency flexuraloscillations as well as in the higher frequency content+erefore a hardening model with a lower yield stress wouldinduce an earlier plasticisation which would allow for a degreeof control of dissipation of the signal without requiring theneed to resort to numerical damping

A summary of the longitudinal and flexural frequenciesobtained through the different models implemented isshown in Table 6

+e repeatability of the multilinear kinematic hardeningmodel adopted was tested by simulating a higher intensitycentral shot (pulse no 132 in Table 5) +e longitudinal re-sponses experimental and numerical for shot 132 are shown inFigure 23 Similar to the previously modelled pulse the sim-ulation follows the experimental result with a good degree ofaccuracy in terms of wave shape amplitude and frequency forthe longitudinal wave +e model can be seen to underestimatethe amplitude of high frequency content in the signalmdashthis can

indicate that the hardening curve adopted is overstating thedissipation of energy at higher intensities An additional factorwhich needs to be considered when looking at this discrepancyis the inhomogeneity of the materialrsquos microstructure which iscomposed of diamond particles dispersed in a copper matrix+e diamond particle size is an important factor especiallywhen comparedwith thewavelength of the travellingwave sinceinternal reflections in the material can become significant andhave an effect on the high frequency content observed in themeasured signal

5 Discussion

+emultilinear hardeningmaterial model obtained through thethermomechanical characterisation campaign described in thisstudy proved accurate in replicating the shape of the experi-mentally obtained signal improving on a purely elasticmodel interms of longitudinal wave shape and amplitude relative to theexperimentally obtained signal Despite this one can see thatYoungrsquos modulus computed from the impulse excitationtechnique testmeasurements underestimates the value observedin the MultiMat experiment measurements +is is believed tobe a result of the high variability of the material due to the highlevel of porosity present as well as to possible errors inmodelling the plastic part of the hardening curve

Differences between the longitudinal wave results obtainedexperimentally and those generated numerically can be ob-served in terms of high frequency dissipative effects as observedin Figure 23 +is discrepancy suggests that the hardeningmodel adopted does not encompass the full range of behaviourof thematerial which is represented as a homogeneous entity inthe numerical analyses conducted On a mesoscale level thematerial is actually composed of diamond particles dispersed ina copper matrix +is results in stress waves propagatingthrough the material having to travel from one medium toanother which results in the wave splitting into respectivetransmitted and reflective portions in accordancewith the shockimpedance of the two constituents of the material +e lon-gitudinal signal in experimental results can be seen to reduce inamplitude significantly following the first oscillation whichcould similarly be related to this phenomenon As mentionedin this study the effects mentioned were treated in a homo-geneous model through the application of damping and themodification of the yield stress of thematerial which results in aloss of energy and thus a dissipation of the propagating waves+is is fundamentally different from the wave-particle inter-action discussed since in this case themechanism in play is wavedispersion at the particle-matrix interface

+e following considerations are hence proposed forfuture study of the material

(i) +e modelling of internal material effects

Longitudinal strain signals measured experimentally canbe seen to decay as the wave propagates through the ma-terial +is phenomenon is believed to be a result of theinhomogeneous mesoscale structure of CuCD which iscomposed of diamond particles (with particle size varyingbetween 40 and 200 μm) in the copper matrix resulting inthe stress waves propagating through the medium being split

Shock and Vibration 13

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

into transmitted and reflected portions depending on theshock impedance of the two materials resulting in theperceived decay of the signal While such a phenomenonconserves the total energy of the system dissipative effectsrelated to internal friction are also believed to be contrib-uting to this decay +e effects related to the microstructureof the material can be studied by modelling the diamondparticles on a mesoscale level continuing on the workproposed by Carra [5] Figure 24 shows a schematic of a

homogeneous and a mesoscale model the former assumes amaterial behaviour which is constant across the geometrywhile the mesoscale model attempts to model the real-lifescenario where in the case of CuCD diamond particles aredispersed in a copper matrix

(ii) Strain rate and temperature-dependent testing

+e presented thermomechanical characterisation cam-paign (excluding the IET test) tested the material in quasi-static

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 20 Longitudinal strain for shot 128 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

0 50 100 150 200 250 300 350 400Time (micros)

ndash100

ndash50

0

50

100

150

Stra

in (micro

mm

)

ExperimentSimulation

Figure 21 Longitudinal strain for shot 128 with a linear elastic model (E 220GPa) plotted against experimental data measured at a lengthof 154mm on the bottom face of the second specimen

14 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

0 2 3 4 5 6 71Time (ms)

ndash50

0

50

100

Stra

in (micro

mm

)

ExperimentSimulationDamped

Figure 22 Flexural response for the multilinear model with E 220GPa with and without Rayleigh damping for shot 128 probed at a lengthof 154mm on the bottom face of the second specimen

Table 6 Comparison of measured experimental frequencies with numerical results from perfectly elastic andmultilinear hardeningmodels

ExperimentSimulation multilinear hardening

modelSimulation elastic

modelSimulation multilinear hardening

modelE 194GPa E 220GPa E 220GPa

Longitudinal frequency(kHz) 1303 118 122 126

Flexural frequency (Hz) 452 434 444 455

ndash100

ndash50

0

50

100

150

200

250

300

350

400

450

Stra

in (micro

mm

)

0 50 100 150 200 250 300 350 400Time (micros)

ExperimentSimulation

Figure 23 Longitudinal strain for shot 132 probed at a length of 154mm on the bottom face of the second specimen+e numerical modelwith Youngrsquos modulus of 220GPa and the multilinear hardening model

Shock and Vibration 15

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

conditions at room temperature which can result in a dif-ference in material behaviour compared with the dynamicconditions induced by sudden particle beam impacts+e split-Hopkinson bar test is designed to characterise material be-haviour at elevated strain rates and temperatures [41] allowingfor the formulation of strength models such as the John-sonndashCook and ZerillindashArmstrong models which describe thematerial behaviour at extreme conditions A split-Hopkinsonbar test on CuCD RHP3434 samples will be conducted

Additionally future wave propagation problems andanalyses could benefit from deviating from the finite ele-ment method Alternatives based on machine learningexist such as isogeometric analysis (IGA) and mesh-freemethods In such analyses approximations for partialdifferential equation solutions can be found with an energyapproach via the application of deep neural networks(DNNs) [42 43] An IGA approach provides some ad-vantages with respect to FEA namely eliminating thegeometric approximation error and reducing dissipationerrors and numerical dispersion [44] Finally a sensitivityanalysis quantifying the influence of all uncertain inputparameters on the modelrsquos outputs would contribute indetermining the impact of experimental uncertainty onresults [45]

6 Conclusions

Copper diamond is a novel composite material currentlybeing developed for possible implementation in the colli-mation system for use in future upgrades of the LHC andnew particle accelerators such as the high-luminosity LHCupgrade and the Future Circular Collider

+is study presents a material characterisation cam-paign for the material conducted at CERN mainlycomposed of measurements of temperature-dependentmaterial properties testing for the elastic properties of thematerial and testing of the plastic behaviour of the ma-terial +e material model was then implemented in asimply-coupled thermomechanical analysis performed inANSYS the results of which were benchmarked withexperimentally measured data from the MultiMat ex-periment conducted in CERNrsquos HiRadMat facility +ehomogeneous model implemented was able to replicatethe longitudinal wave shape and amplitude with some

discrepancies observed in terms of the high frequencycontent in the longitudinal signal +is is believed to be aresult of a combination of factors namely an overesti-mation of the plasticity induced in the material as well asmesoscale effects related to the interaction of stress waveswith diamond particles having a diameter which is close tothe wavelength With this in mind a number of con-siderations for future study of the material are proposednamely the effect of the yield stress and plasticity on thedynamic material response the modelling of internalmaterial effects by considering the mesoscale structure ofthe material as well as further testing of the material athigh strain rates and temperatures

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is project has received funding from the EuropeanUnionrsquos Horizon 2020 Research and Innovation Programmeunder grant no 730871

References

[1] lsquoRHP-Technology GmbHrsquo 2019 httpwwwrhp-technologycom

[2] J Guardia-Valenzuela ldquoDevelopment and characterisation ofnovel graphite-matrix composite material for thermal man-agement applicationsrdquo Escuela de Ingenieria Y ArquitecturaMSc thesis Universidad de Zaragoza Zaragoza Spain 2015

[3] J Guardia-Valenzuela A Bertarelli F Carra N MarianiS Bizzaro and R Arenal ldquoDevelopment and properties ofhigh thermal conductivity molybdenum carbide-graphitecompositesrdquo Carbon vol 135 pp 72ndash84 2018

[4] T Weiszliggarber CopperDiamond Composites for Heat SinkApplications httpswwwnanoanalytikfraunhoferdeenlibraryCS1html Fraunhofer IFAM 2019

HomogeneousCuCD

(a)

Diamondparticle

Coppermatrix

(b)

Figure 24 (a) Homogeneous and (b) mesoscale models

16 Shock and Vibration

[5] F Carra ermomechanical Response of Advanced Materialsunder Quasi Instantaneous Heating PhD thesis Politecnico diTorino Italy 2017

[6] R Bruce and S Redaelli CuCD-Based Tertiary CollimatorPresented at the HL-LHC TCC Meeting CERN GenevaSwitzerland 2019

[7] F Carra J G Valenzuela P Gradassi and L K MettlerResults on Simulations of New Materials and CompositesCERN Geneva Switzerlandhttpsedmscernchdocument13252523

[8] A Bertarelli E Berthome V Boccone et al ldquoAn experimentto test advanced materials impacted by intense proton pulsesat CERN HiRadMat facilityrdquo Nuclear Instruments andMethods in Physics Research Section B Beam Interactions withMaterials and Atoms vol 308 pp 88ndash99 2013

[9] A Bertarelli High Energy Tests of Advanced Materials forBeam Intercepting Devices at CERN HiRadMat FacilityCERN San Sebastian Spain 2012

[10] I Efthymiopoulos HiRadMat A New Irradiation Facility forMaterial Testing at CERN CERN San Sebastian Spain 2011

[11] C Torregrosa ldquo+e Hiradmat 27 experiment exploring high-density materials response at extreme conditions for anti-proton productionrdquo in Proceedings of the 7th InternationalParticle Accelerator Conference (IPAC 2016) Busan KoreaJune 2016

[12] F Carra A Bertarelli E Berthome and C Fichera ldquolsquo+eldquoMultimatrdquo experiment at CERNHiRadMat facility advancedtesting of novel materials and instrumentation for HL-LHCcollimatorsrdquo Journal of Physics Conference Series vol 8742017

[13] M Pasquali A Bertarelli C Accettura et al ldquoDynamic re-sponse of advanced materials impacted by particle beams theMultiMat experimentrdquo Journal of Dynamic Behaviour ofMaterials vol 5 pp 266ndash295 2019

[14] A Bertarelli G Arnau Izquierdo F Carra A DallocchioM Gil Costa and N Mariani ldquoResearch and development ofnovel advanced materials for next-generation collimatorsrdquo inProceedings of the 2nd International Particle AcceleratorConference San Sebastian Spain 2011

[15] M Portelli A Bertarelli F Carra M Pasquali N Sammutand P Mollicone ldquoNumerical and experimental bench-marking of the dynamic response of SiC and TZM specimensin theMultiMat experimentrdquoMechanics ofMaterials vol 138Article ID 103169 2019

[16] A Bertarelli F Carra M Garlasche et al ldquoInnovativeMoCndashgraphite composite for thermal management andthermal shock applicationsrdquo in Proceedings of the 2015 31stermal Measurement Modeling amp Management Symposium(SEMI-THERM) pp 56ndash59 San Jose CA USA 2015

[17] lsquoDIL 402 Expedis Classicrsquo httpswwwnetzsch-thermal-analysiscomenproducts-solutionsdilatometerdil-402-expedis-classic

[18] ASTM International Standard Test Method for Linear er-mal Expansion of Solid Materials with a Push-Rod Dilatom-eter ASTM International Standard ASTM E228-17 2017

[19] lsquoSartorius Quintix Analytical Balance 224-1xrsquo httpswwwnorthernbalancecoukproductsartorius-quintix-analytical-balance-224-1x

[20] DIN ldquolsquo+ermal analysis-differential thermal analysis (DTA)and differential scanning calorimetry (DSC)-general Princi-plesrdquo 2019 DIN Standard DIN 510072019-04

[21] J Guardia-Valenzuela Mathematical Model and Extrapola-tion for ermal Diffusivity and Specific Heat EDMS No1540215 2015

[22] T D Burchell ldquoGraphite properties and characteristicsrdquoComprehensive Nuclear Materials no 2 pp 285ndash305 2012

[23] D R Lide CRC Handbook of Chemistry and Physics CRCPress Boca Raton FL USA 90th edition 2009

[24] M Borg A Bertarelli F Carra et al ldquo+ermostructuralcharacterization and structural elastic property optimizationof novel high luminosity LHC collimation materials atCERNrdquo Physical Review Accelerators and Beams vol 21 no 32018

[25] ASTM International lsquoStandard Test Method for DynamicYoungrsquos Modulus Shear Modulus and Poissonrsquos Ratio byImpulse Excitation of Vibrationrsquo ASTM International Stan-dard ASTM E1876-15 2015

[26] ASTM International lsquoStandard Test Method for DynamicYoungrsquos Modulus Shear Modulus and Poissonrsquos Ratio forAdvanced Ceramics by Impulse Excitation of VibrationrsquoASTM International Standard ASTM C1259-15 2015

[27] ANSYS Inc ANSYS Mechanical Userrsquos Guide SoftwareVersion 150 ANSYS Inc Canonsburg PA USA 2013

[28] ASTM International Standard Test Method for FlexuralStrength of Manufactured Carbon and Graphite Articles UsingFour-Point Loading at Room Temperature ASTM Interna-tional Standard ASTM C651-15 2015

[29] L Marrodan M P Fiol M I Gaitan et al ldquoAbsence oflatitudinal gradient in oligoclonal bands prevalence inArgentinardquo Multiple Sclerosis and Related Disorders vol 462020 102582

[30] A Bertarelli ldquoPermanent deformation of the LHC collimatorjaws induced by shock beam impact an analytical and nu-merical interpretationrdquo in Proceedings of the European Par-ticle Accelerator Conference (EPAC 2006) Edinburgh UKJune 2006

[31] M Cauchi O Aberle R W Assmann et al ldquoHigh energybeam impact tests on a LHC tertiary collimator at the CERNhigh-radiation to materials facilityrdquo Physical Review SpecialTopics-Accelerators and Beams vol 17 no 2 Article ID021004 2014

[32] E Quaranta Investigation of Collimator Materials for the HighLuminosity Large Hadron Collider Politecnico di MilanoMilano Italy 2017

[33] A Bertarelli F Carra N Mariani and S Bizzaro Develop-ment and Testing of Novel Advanced Materials with Very Highermal Shock Resistance EuCARD-2 Scientific ReportCERN-ACC-2014-0306 2014 httpscdscernchrecord1973365filesCERN-ACC- 2014-0306pdf

[34] A Bertarelli Beam-Induced Damage Mechanisms and theirCalculation CERN 0007ndash8328 2016 httpse-publishingcernchindexphpCYRarticleview234

[35] M Portelli A Bertarelli F Carra L K Mettler P Molliconeand N Sammut ldquoNumerical simulation of long rods impactedby particle beamsrdquo Physical Review Accelerators and Beamsvol 21 no 6 2018

[36] A Bertarelli A Dallocchio and T Kurtyka ldquoDynamic re-sponse of rapidly heated cylindrical rods longitudinal andflexural behaviorrdquo Journal of AppliedMechanics vol 75 no 32008

[37] P Sievers ldquoElastic stress waves in matter due to rapid heatingby an intense high-energy particle beamrdquo CERN TechnicalNote No LAB IIBT74-2 1974

[38] H Bargmann ldquolsquoDynamic response of external targets underthermal shockrsquordquo CERN Technical Note No LAB IIBTInt73-3 1973

Shock and Vibration 17

[39] B A Boley ldquoApproximate analyses of thermally inducedvibrations of beams and platesrdquo Journal of Applied Mechanicsvol 24 no 3 pp 413ndash416

[40] A Ferrari P R Sala A Fasso and J Ranft FLUKA A Multi-Particle Transport Code (Program Version 2005) CERNGeneva Switzerland 2005

[41] G T Gray ldquoClassic split-Hopkinson pressure bar testingrdquo inASM Handbook Volume 8 Mechanical Testing and Evalua-tion Scientific Research Publishing Inc Wuhan China 2000

[42] E Samaniego C Anitescu S Goswami et al ldquoAn energyapproach to the solution of partial differential equations incomputational mechanics via machine learning conceptsimplementation and applicationsrdquo Computer Methods inApplied Mechanics and Engineering vol 362 Article ID112790 2020

[43] T Rabczuk and T Belytschko ldquoA three-dimensional largedeformation meshfree method for arbitrary evolving cracksrdquoComputer Methods in Applied Mechanics and Engineeringvol 196 no 29-30 pp 2777ndash2799 2007

[44] L Pegolotti L Dede and A Quarteroni ldquoIsogeometricAnalysis of the electrophysiology in the human heart nu-merical simulation of the bidomain equations on the atriardquoComputer Methods in Applied Mechanics and Engineeringvol 343 pp 52ndash73 2019

[45] N Vu-Bac T Lahmer X Zhuang T Nguyen-+oi andT Rabczuk ldquoA software framework for probabilistic sensi-tivity analysis for computationally expensive modelsrdquo Ad-vances in Engineering Software vol 100 pp 19ndash31 2016

18 Shock and Vibration