ceg_simone_20031013

Continuous-DiscontinuousModelling of Failure

Continuous-DiscontinuousModelling of Failure

Proefschrift

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J. T. Fokkema,voorzitter van het College van Promoties,

in het openbaar te verdedigen op maandag 13 oktober 2003 om 10.30 uurdoor Angelo SIMONE

ingegnere civile laureato al Politecnico di Milanogeboren te Taranto, Italie

Dit proefschrift is goedgekeurd door de promotor:Prof. dr. ir. J. Blaauwendraad

Toegevoegd promotor:Dr. ir. L. J. Sluys

Samenstelling promotiecommissie:

Rector Magnificus Voorzitter

Prof. dr. ir. J. Blaauwendraad Technische Universiteit Delft, promotor

Dr. ir. L. J. Sluys Technische Universiteit Delft, toegevoegd promotor

Prof. dr. ir. M. G. D. Geers Technische Universiteit Eindhoven

Prof. dr. ir. E. van der Giessen Rijksuniversiteit Groningen

Dr. M. Jirasek Ecole Polytechnique Federale de Lausanne, Zwitserland

Prof. dr. R. Larsson Chalmers Tekniska Hogskola, Zweden

Prof. dr. ir. F. Molenkamp Technische Universiteit Delft, reservelid

Prof. dr. ir. J. G. Rots Technische Universiteit Delft

Published and distributed by DUP Science

DUP Science is an imprint ofDelft University PressP.O. Box 982600 MG Delftthe NetherlandsTelephone: +31 15 27 85 678Telefax: + 31 15 27 85 706E-mail: [email protected]

ISBN 90-407-2434-2

Keywords: continuous-discontinuous failure, finite-element method, damage

Copyright c© 2003 by A. Simone

All rights reserved. No part of the material protected by this copyright notice may be repro-duced or utilised in any form or by any means, electronic or mechanical, including photocopy-ing, recording or by any information storage and retrieval system, without written permissionfrom the publisher: Delft University Press

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Printed in the Netherlands

Contents

Preface vii

List of symbols and abbreviations ix

1 From continuous to continuous-discontinuous failure representation 11.1 The need for discontinuous failure descriptions . . . . . . . . . . . . . . . . . . 11.2 Failure characterisation and numerical strategies . . . . . . . . . . . . . . . . . 41.3 Requirements for distributed failure models . . . . . . . . . . . . . . . . . . . . 61.4 Models for inelastic bulk deformation . . . . . . . . . . . . . . . . . . . . . . . 61.5 A model for separation across cohesive surfaces . . . . . . . . . . . . . . . . . 12

2 Continuous-discontinuous failure in standard media 132.1 Problem fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Discretisation and linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.5 Element technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Continuous-discontinuous failure in gradient-enhanced media 413.1 Problem fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Discretisation and linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5 Element technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.7 Stress-strain relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.8 Damage initiation and discontinuities . . . . . . . . . . . . . . . . . . . . . . . 653.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Continuous-discontinuous failure in rate-dependent media 714.1 Rate-dependent elastoplastic-damage models . . . . . . . . . . . . . . . . . . . 724.2 Element technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.4 Traction-free discontinuities in rate-dependent and non-local media . . . . . . 964.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Conclusions 101

vi Contents

A Conventional interface and PU-based discontinuous elements 105A.1 Conventional continuous interface element . . . . . . . . . . . . . . . . . . . . 105A.2 Partition of unity-based discontinuous elements . . . . . . . . . . . . . . . . . 108A.3 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B Some essentials of generalised functions 115

C Constitutive models for softening materials 117C.1 Considerations on numerical modelling of concrete . . . . . . . . . . . . . . . 119C.2 Non-local versus viscous regularisation . . . . . . . . . . . . . . . . . . . . . . 121

D Interpolation requirements for a class of gradient-enhanced media 125D.1 Governing equations and spatial discretisation . . . . . . . . . . . . . . . . . . 125D.2 Element performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127D.3 About terminology: mixed method versus coupled problem . . . . . . . . . . 130D.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

E Incorrect failure characterisation in non-local media 135E.1 Some basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135E.2 Damage characterisation in mode I problems . . . . . . . . . . . . . . . . . . . 137E.3 Damage characterisation in shear band problems . . . . . . . . . . . . . . . . . 145E.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

References 153

Author/editor index 167

Subject index 171

Summary 173

Samenvatting 175

Curriculum vitae 177

Preface

Failure of most engineering materials is a phenomenon which is charac-terised by the development of a process zone in which microcracks arise,deformations localise and the accumulation of damage eventually leads totraction-free macrocracks with eventual total loss of load-carrying capacityat the final stage. This thesis explores applications of a method combiningnovel techniques of continuous and discontinuous numerical failure anal-ysis. Continuous and discontinuous approaches to failure have been ex-tensively studied in the past and have now reached a reasonable degreeof maturity. Various strategies are known to cope with the problems origi-nating from the use of continuous softening stress-strain relationships, andtechniques are available which allow displacement discontinuities to crossthrough solid elements when the discontinuous stress-crack opening ap-proach is considered. It is desirable that a numerical approach to failuredescription is performed in a unified fashion, in all its stages.

The basic idea behind this study is the combination of continuous anddiscontinuous failure descriptions to achieve a better characterisation of thewhole failure process. Aim of this thesis is to illustrate that a discontinuousinterpolation of the problem fields

- enables a more realistic description of failure,- adds flexibility to continuous modelling,- solves some problems inherent to some regularised models, and- allows a different interpretation of model parameters.

Outline. The motivations behind a continuous-discontinuous approach tofailure are put forward in Chapter 1. The chapter is completed by some basicnotions regarding models for degradation in the bulk volume and across thediscontinuity surface used in the remainder of this thesis.

The basic tool of the continuous-discontinuous approach to failure, dis-placement discontinuities based on the partition of unity property of finite-element shape functions, is described in Chapter 2. A similarity is drawnbetween conventional interface elements and partition of unity-based dis-continuous elements; some applications to elastic and strain-hardening me-dia conclude the chapter.

viii Preface

When strain-softening media are considered, regularisation techniquesmust be employed to avoid a dependence of the solution on the discretisa-tion. The applicability of discontinuities in regularised media is discussed inChapters 3 and 4 where discontinuities are considered in gradient-enhancedand rate-dependent media respectively, and where it is shown that the un-derlying regularised continuum determines the quality of the discontinuousenhancement.

Notation. In most of this thesis tensor notation is used. Details on tensorialalgebra can be found in References [77, 84]. Matrix notation (referred to as‘engineering notation’ in the remainder of the thesis) is considered in the lin-earisation and discretisation of the governing equations in Chapters 2 and3 and in Appendix A. First-order tensors (representing vectors) are denotedby bold lower-case Latin letters, second-order tensors (vectors in engineer-ing notation) by bold lower-case Latin or Greek letters, and fourth-ordertensors (matrices in engineering notation) by bold capital Latin letters. In

a = b c = d, (1)

(1)2 indicates the second relation (c = d), and in

a = b = c, (2)

(2)2 relates to a = c. A list of symbols and abbreviations is given on page ix.

Acknowledgements. The research presented in this thesis was carried outat the Faculty of Civil Engineering and Geosciences at Delft University ofTechnology under the supervision of L.J. Sluys. Support for this researchwas provided by the BEO programme (special fund from Delft Universityof Technology for excellent research).

I am greatly indebted to J. Alfaiate, H. Askes, J. Blaauwendraad,G.L. Chiusa, K. De Proft, F.P.X. Everdij, M.G.D. Geers, M. Jirasek, P. Ka-bele, E. Kuhl, P. Lura, A. Meda, A.V. Metrikine, J. Pamin, T. Pannachet,R.H.J. Peerlings, J.J.C. Remmers, L.J. Sluys and G.N. Wells for their adviceand suggestions.

A.S.

Delft, the NetherlandsJune 2003

List of symbols and abbreviations

Latin symbols

a softening parameter in softening evolution lawa vector of regular nodal displacement degrees of freedomA cross-sectional areab softening parameter in softening evolution law

interface width (Appendix A)b vector of extra nodal displacement degrees of freedomBe matrix containing derivatives of Ne

Bu matrix containing derivatives of Nu

c gradient parameterC interface constitutive matrixCe fourth-order compliance tensord spring stiffness (Chapter 2)

notch size (Chapter 3)di vector in the direction of the Gauss point iD tangent matrix for the bulk materialDe fourth-order linear-elastic constitutive tensor

linear-elastic constitutive tangent matrixDp fourth-order elastoplastic consistent tangent tensorDpd fourth-order elastoplastic-damage consistent tangent tensore non-local equivalent straine component of e when Ω is crossed by Γd

e+/− positive/negative part of ee component of e when Ω is crossed by Γd

el local equivalent strainE Young’s modulusf yield functionfcd stress level corresponding to εcd

ft tensile strengthft,ω stress level corresponding to a specific damage value ωfκ ∂ f /∂κfλ ∂ f /∂λfσ ∂ f /∂σfσσ ∂ fσ/∂σfext/int,i external/internal force vector for dof iG f fracture energyH hardening parameterH1

0 , H1 Sobolev spacesHΓd Heaviside function centred at Γd

x List of symbols and abbreviations

Iε1 first invariant of the strain tensorI complementary energyI fourth-order identity tensorJε2 second invariant of the deviatoric strain tensorJσ2 second invariant of the deviatoric stress tensorJ total potential energyk ratio of the compressive and tensile strength for concreteKI mode I stress intensity factorK global stiffness matrixKi j partition of K related to i j degrees of freedomK partition of K in interface and PU-based discontinuous elementsl length scalel f fibre lengthL lengthL2 Sobolev spaceL Lagrangian functionalL differential operatorm shape parameter for cohesive law at the discontinuity (Chapter 2)

nodes in interface and PU-based discontinuous elements (Appendix A)m inward unit normal to Ω+

n outward unit normal to ΩN parameter in the overstress functionφN displacement shape function matrix (Appendix A)Ne non-local equivalent strain shape function matrixNu displacement shape function matrixp vector of regular nodal non-local equivalent strain degrees of freedomP applied loadq vector of extra nodal non-local equivalent strain degrees of freedomr interaction radiusr special second-order tensor (Chapter 4)

De :∇su (Appendix D)R distance from the crack tip along the crack lineR special fourth-order tensor (Chapter 4)

rotation matrix (Appendix A)s special second-order tensor (Chapter 4)

test function (Appendix D)t timet tractions at the discontinuityt prescribed tractionsti traction in the tangential s or normal n directiontmax tensile strengthT tangent matrix for the discontinuityJuK uniaxial displacement jumpuy vertical displacementu displacement fieldJuK displacement jumpu prescribed displacements

List of symbols and abbreviations xi

u component of u when Ω is crossed by Γd

u component of u when Ω is crossed by Γd

Uu space of trial displacementsv deflectionv generic unit vector (Chapter 3 and Appendix B)

test function (Appendix D)V f fibre volume fractionVi volume related to Gauss point ix Cartesian line coordinatex, y Cartesian spatial coordinatesw crack opening displacementwe weight function for the non-local equivalent strain ewe component of we when Ω is crossed by Γd

we component of we when Ω is crossed by Γd

wi weight associated to the Gauss point iwmax crack opening displacement at zero loadwu weight function for the displacement uwu component of wu when Ω is crossed by Γd

wu component of wu when Ω is crossed by Γd

Wu,0 space of admissible displacement variations

Greek symbols

α softening parameter in damage evolution lawβ softening parameter in damage evolution lawΓ boundary surface of ΩΓd discontinuity surfaceΓ

+/−d positive/negative part of Γd

Γt boundary surface where prescribed tractions are appliedΓ

+/−t positive/negative part of Γt

Γu boundary surface where prescribed displacement are appliedΓ

+/−u positive/negative part of Γu

δΓd Dirac-delta function∆t time incrementε uniaxial strainεi principal strain in the i directionεcd strain level for continuous to discontinuous transitionε strain tensor

strain array in engineering notationεe elastic component of the strain tensorεp plastic component of the strain tensorεvp viscoplastic component of the strain tensorθ angleκ equivalent plastic strain (plasticity)

deformation history parameter (damage)

xii List of symbols and abbreviations

κ0 threshold of damage initiationκc ultimate equivalent strain at whichω = 1κ equivalent plastic strain for rate-dependent effective stress spaceλ plastic multiplierν Poisson’s ratioρ distance between points x and yσ uniaxial stressσ uniaxial yield stressσ0 yield stress for perfect plasticityσ0 initial yield stress (cohesion) for hardening/softening plasticityσe effective stress defined by the yield criterion in plasticityσi principal stress in the i directionσ stress tensor

stress array in engineering notationσd deviatoric part of the stress tensorστ relaxation timeφ overstress function (Chapter 4)

regular function (Appendix B)ψ weight function in non-local averagingω scalar damage variableωcrit critical scalar damage valueω criticalω for the introduction of a cohesive discontinuityΩ body volume or surface or length (boundary excluded)Ω+/− positive/negative part of ΩΩ body volume or surface or length (boundary included)Ω+/− positive/negative part of Ω

Meaning of indices

Subscripts

a related to a dofsb related to b dofsd related to discontinuity surfacee related to non-local equivalent strain ee related to the elastic zoneh discretised quantityn related to the normal direction to Γd

p related to p dofsp related to the process zoneq related to q dofss related to the tangential direction to Γd

u related to displacement u

List of symbols and abbreviations xiii

Superscripts

′ admissible variatione elastic quantityp plastic quantitys symmetric partT transposevp viscoplastic quantity

Abbreviations

C continuous modellingcf confer = Latin for ‘compare’cmod crack mouth opening displacementcmsd crack mouth sliding displacementCOD crack opening displacementD discontinuous modellingdof(s) degree(s) of freedomECC engineered cementitious compositeet al. et alia = Latin for ‘and others’FEM finite-element methodFRC fibre-reinforced cement/concreteHPFRCC high performance fibre-reinforced cement compositei.e. id est = Latin for ‘that is’LVDT linear variable differential transformerPU partition of unityRHS right-hand sideSFR steel-fibre reinforcedSIFCON slurry infiltrated fibre concrete

Chapter 1

From continuous to continuous-discontinuousfailure representation

A numerical strategy for a realistic characterisation of failure should con-sider the representation of nucleation of defects into microcracks, the evolu-tion of microcracks into macrocracks and the correct macrocrack-microcrackinteraction [87]. This is an extremely complicated task which can be pursuedin a phenomenological approach to failure, although the value of such a pro-cedure is debatable (see Appendix C).

When modelling failure phenomena, the use of displacement disconti-nuities is advocated to achieve a better representation of the entire fail-ure process. Compared to a continuous failure analysis, a continuous-discontinuous failure analysis, in which discontinuities in the problem fieldsarise as a natural consequence of strain localisation, may lead to a morerealistic description of the entire failure process, from diffuse microcrack-ing to macroscopic traction-free cracks as depicted in Figure 1.1. The useof a continuous-discontinuous approach in which displacement discontinu-ities are considered as the natural evolution of strain localisation has beenconsidered e.g. by Grootenboer [75], Rots [152], Ren and Bicanic [145], Ka-bele and Horii [89], Jirasek and Zimmermann [86], Oliver et al. [123, 124]and Wells et al. [194]. An indication of other continuous-discontinuous ap-proaches can be found in References [24, 26–28, 57, 81, 90, 106, 111, 112] andreferences herein.

In the rest of this chapter, the motivations for a continuous-discontinuousapproach to failure are briefly recalled and a parallel is drawn betweenfailure characterisations and numerical strategies for failure description. Asummary of the models for failure description used in the remainder of thisthesis is given in the last part of this chapter.

1.1 The need for discontinuous failure descriptions

In a smeared approach to failure in quasi-brittle materials [59, 60, 151, 153,154, 180, 181], the cracked material is treated as a continuum and displace-

2 Chapter 1 From continuous to continuous-discontinuous failure representation

(a) (b) (c)

Figure 1.1 Three different stages of failure in a compact-tension specimen: (a) failure ini-tiation, (b) failure propagation and (c) close to complete failure (the shaded part indicatesmicrocracking while the thick white line represents a macrocrack).

ment continuity is assumed across the cracked region. To describe the lossof load-carrying capacity, specific material properties [19, 138] and strain-softening constitutive relationships with a residual stress are usually em-ployed (see Figure 1.2a).

Reasons of practical and of theoretical nature are behind the use of strain-softening constitutive relationships with residual stress. From the practicalside, this setting is, obviously, a very convenient one since it allows numer-ical analyses to be performed in a continuous framework. Using e.g. con-tinuum damage softening constitutive relationships with full stress relax-ation at significant strain values, such as the one depicted in Figure 1.2b(solid line), poses the problem of dealing with damage values equal tounity, i.e. with a singular stiffness matrix. Conversely, considerations of the-oretical nature lead to the conclusion that the asymptote of such constitu-tive relationships might be useful in reproducing the long tail observed inload-displacement diagrams of concrete specimens which can be related tocrack bridging [78, 108]. Unfortunately, due to the inability of describing akinematic discontinuity in the primal field in a continuous setting, the as-sumption of a smeared reproduction of displacement discontinuity across acracked region usually results in a too stiff mechanical response [48, 152] andin spurious damage growth [64]. Smeared degradation constitutive models,such as continuum damage or plasticity models, are best suited for mod-elling diffuse microcracking, in strain-softening or -hardening materials, be-fore macrocracks become dominant. A better approximation of failure pro-cesses can be achieved by using numerical techniques in which a disconti-nuity is naturally endowed in the model itself and is activated at some stageduring localisation (by using a cohesive discontinuity) or after the locali-

1.1 The need for discontinuous failure descriptions 3

strainstrain(a) (b)

stre

ss

stre

ss

Figure 1.2 Softening stress-strain curves employed in a continuous failure description:(a) with residual stress and (b) with full stress relaxation (solid line) or negligible residualstress (dashed line).

sation process has been completed (by using a traction-free discontinuity).Failure can then be realistically described as progressive material degrada-tion which develops into a discrete crack, for which a discontinuity in thedisplacement field is a suitable representation [110, 193].

To illustrate one of the incongruities due to an erroneous use of strain-softening relationships in a continuous setting, the global response in termsof load-deflection curve for a beam loaded in four-point bending [78, 130]is reported in Figure 1.3. The analysis has been performed in the frame-work of a regularised damage model with a strain-softening constitutiverelationship of the type depicted in Figure 1.2a (see Reference [130] andSection 3.6.1). In this regularised model (implicit gradient-enhanced dam-age continuum model [131]), damage evolution is governed by a non-localmeasure (non-local equivalent strain e) of the strain tensor. The compari-son with the experimental curve reported in Figure 1.3a can be consideredsatisfactory but it must be realised that it is the result of an unrealistic repre-sentation of the stress-strain situation in the points surrounding the notch.From the first principal stress–non-local equivalent strain curve depicted inFigure 1.3b, it is evident that part of the load-deflection curve in Figure 1.3ais to be attributed to the contribution of the integration points surroundingthe notch along the crack line. In a more realistic failure representation ofquasi-brittle materials, the contribution of the points surrounding the notchshould vanish close to local failure. The relevance of strain-softening lawswith residual stress (of the type shown in Figure 1.2a) is questionable sincethere is no physical rationale behind their definition and their use alters the


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

b

a

v [mm]

experimentsimulation

(a)

0

0.5

1

1.5

2

2.5

3

P [k

N]

0 0.01 0.02 0.03 0.04 0.05

b

e [−]

a

(b)

0

1

2

3

4

σ 1[M

Pa]

Figure 1.3 Notched beam in four-point bending (30 mm notch depth [78, 130]): (a) load-deflection curve and (b) first principal stress (σ1) –non-local equivalent strain (e) curve fora point ahead the notch (the points a and b in (b) indicate the integration point contributionto the global response depicted in (a) for an integration point ahead the notch; also, a andb indicate the value of the deflection v in (a) and of the corresponding strain level e for theintegration point ahead the notch in (b) at different stages of the loading process—note thatthe contribution b in (b) is pathological since it is related to a value of σ1 which remainsconstant, and different than zero, for increasing deformation values e).

understanding of the whole failure process. The use of constitutive lawswith full stress relaxation at significant strain, in conjuction with a numeri-cal technique in which the problem fields are allowed to develop a disconti-nuity, may enable a more realistic characterisation of failure processes.

1.2 Failure characterisation and numerical strategies

Considerations of different nature from the ones in the previous section sug-gest that the use of a discontinuous interpolation of problem field couldbe beneficial, in terms of added flexibility with respect to continuous mod-elling, to the description of the failure behaviour of cementitious compos-ites.

Failure in cementitious composite materials ranges from quasi-brittlefailure of plain concrete or conventional fibre-reinforced cement or con-crete (FRC) to ductile failure observed in some high performance fibre-reinforced cement composite (HPFRCC) [3, 158]. In plain concrete andconventional FRC, an initial linear response is followed by a softeningbranch which is characterised by the widening of a single crack. Most ofthe deformation relates to crack splitting and can be coherently described,

1.2 Failure characterisation and numerical strategies 5

uu

(b)(a)

stre

ss

C

ε

b

D

a

stre

ss

C

ε

b

Da

Figure 1.4 Strategies for failure analysis of cementitious composites: idealised tensile be-haviour for (a) quasi-brittle composites and (b) ductile composites (C = continuous mod-elling, D = discontinuous modelling, ε = strain, JuK = displacement jump across the discon-tinuity surface).

in a computational framework, by softening cohesive surfaces where allthe non-linearity is concentrated—these formulations make use of stress-displacement relations across the discontinuity surface. An alternative re-sides in the use of regularised continuous approaches to failure in whichsoftening stress-strain relations are employed. In some HPFRCC, an initiallinear response is followed by an extended hardening branch characterisedby a uniform deformation stage due to the multiple cracking process. Themultiple cracking phase is followed by a softening branch which stems fromthe widening of a single crack. This characterisation is typical of variouscomposites such as SIFCON [117] (randomly distributed short steel fibresinfiltrated with cement slurry with fibre volume fraction V f = 5− 20%) orECC (cementitious matrix with e.g. V f = 1− 2% of short random syntheticfibres [100]). The behaviour prior to softening can be described by a stress-strain relationship which can be deduced after direct translation of uniax-ial tensile tests due to the relatively homogeneous distribution of cracks inthe hardening regime. The softening part of the global response can be de-scribed by means of cohesive surfaces.

The numerical strategy just described is purely phenomenological and isbased on the direct translation of the above failure characterisations intoa computational framework. A schematic representation of the strategiesfor failure analyses of cementitious composites is reported in Figure 1.4,where C and D indicates the continuous and the discontinuous range re-spectively. In the representation of quasi-brittle failure (see Figure 1.4a),analyses can be conducted in a continuous framework or in a combinedcontinuous-discontinuous setting. Traditionally, this is done, in the latter ap-proach, by plugging a discontinuity at the end of the elastic regime, just be-


fore the softening phase (at point a), thus avoiding the use of regularised me-dia. However, the introduction of a discontinuity close to failure (at point b)could be beneficial to a class of regularised models, improving failure rep-resentation as will be discussed in Chapter 3. A discontinuity can also beintroduced between points a and b. This may be useful when only someinelasticity is allowed in the continuum. This approach requires the use ofregularised media and could be considered representative of the behaviourof some steel fibre-reinforced composites. When failure in ductile compos-ite is considered (see Figure 1.4b), it is necessary to allow some degree ofinelasticity to the bulk to represent diffuse degradation. A discontinuity isthen introduced at the onset of localisation (at point a).

1.3 Requirements for distributed failure models

When introducing discontinuities in the problems fields, some requirementson the underlying continuum description must be satisfied. Obvious re-quirements are related to the nature of the continuum model which shouldbe regularised—numerical results must be independent of mesh type, size andorientation—and should produce a failure mode which is physically reasonable(the location of failure initiation and the evolution of failure should be cor-rectly predicted). Less obvious requirements, but equally important, residein the ability of the model to allow the formation of a localised strain profilewith full stress relaxation at significant deformation and without spurious damagegrowth close to failure for a correct description of a stress-free crack—in otherwords, the problem fields of the continuous model should converge to a re-alistic discontinuous state. The above requirements should also be taken intoaccount when a conventional continuous failure analysis is considered.

1.4 Models for inelastic bulk deformation

Inelastic bulk behaviour can be effectively described by continuum damageor plasticity theories which can be opportunely tailored to account for therepresentation of the behaviour of various materials. A summary of quasi-static damage and plasticity models used in this thesis is given next. Thesummary is limited to the geometrically linear case. A more detailed de-scription can be found in Crisfield [46, 47] and Jirasek and Bazant [84].

1.4 Models for inelastic bulk deformation 7

Continuum damage theories

Continuum damage theories are usually considered to describe materialdegradation as a consequence of the growth of microstructural defects. Thebasic assumption characterising isotropic elasticity-based damage modelsresides in the progressive reduction of the elastic stiffness via a scalar dam-age parameterω, ranging from 0 (virgin material) to 1 (completely damagedmaterial). Consequently, the basic relation between total stress σ and totalstrain ε reads

σ = (1−ω) De:ε, (1.1)

where De is the fourth-order linear-elastic constitutive tensor. It is also as-sumed that the damage history of the material is considered a function ofthe monotonically increasing deformation history parameter κ, whose evo-lution is governed by the Kuhn-Tucker relations

κ ≥ 0, el −κ ≤ 0, κ (el −κ) = 0, (1.2)

which are related to the definition of a scalar measure el of the strain tensor.Failure characterisation is defined next through a damage evolution law andan equivalent strain definition.

Equivalent strain definition. Different weight can be given to the compo-nents of the strain tensor in the construction of an equivalent strain measure.When only tensile strains are relevant, use can be made of the expression

el =

√√√√ 3

∑i=1〈εi〉2, (1.3)

due to Mazars [104, 105], with 〈εi〉 = (εi + |εi|)/2 andεi the principal strains,based on the positive principal strain components 〈εi〉. Conversely, when allthe components are considered, the modified von Mises definition [63, 188]

el =k− 1

2k (1− 2ν)Iε1 +

12k

√√√√ (k− 1)2

(1− 2ν)2 I2ε1 +

6k

(1 + ν)2 Jε2, (1.4)

with

Iε1 = tr (ε) , Jε2 = tr (ε ·ε)− 13 tr2 (ε) , (1.5)

k the ratio of the compressive and tensile strength and ν the Poisson’s ratio,can be used.


Damage evolution laws. Some characteristics of global behaviour of aspecimen can be directly related to the shape of the damage evolution lawas briefly discussed in Section 1.1. There are two classes of damage evolu-tion laws which are diversified by the strain value at which a fully damagedstate is reached. In the first category there are laws for which a critical dam-age value is reached at a significant strain level such as the linear softeningdamage evolution law

ω =

0 if κ < κ01− κ0

κκc−κκc−κ0

if κ0 ≤ κ ≤ κc

1 if κ > κc,(1.6)

which is characterised by a linear decrease of the stress until a zero-stresslevel is reached at ultimate strain κc with κ0 the threshold of damage initia-tion, or the modified power softening law [64]

ω =

0 if κ < κ0

1−(κ0κ

)β ( κc−κκc−κ0

)αif κ0 ≤ κ ≤ κc

1 if κ > κc,

(1.7)

which is a modified version of the linear softening law (1.6) in which themodel parametersα andβ influence the slope and the shape of the softeningcurve, respectively.

In the second category of damage evolution laws, critical damage is neverreached—it is possible to reach unit damage only asymptotically, at infinitestrain values. One such law is the exponential softening law [134]

ω =

0 if κ < κ01− κ0

κ (1−α +αexp (−β (κ −κ0))) if κ ≥ κ0,(1.8)

withα and βmodel parameters influencing the residual stress level and theslope of the softening curve, respectively.

Regularisation in presence of softening constitutive behaviour. Whenstrain-softening constitutive equations are exploited to describe the progres-sive loss of load-carrying capacity in a continuous setting, regularisationtechniques must be considered to preserve well-posedness of the govern-ing equations. A widely used technique relies on the inclusion of non-local


terms in the constitutive equations. In the non-local damage model pro-posed by Pijaudier-Cabot and Bazant [139], non-locality enters the consti-tutive equations through the definition of a non-local scalar measure e ofthe strain tensor defined in the body Ω as

e(x) =

∫

Ωψ (y; x) el (y) dΩ (y)∫

Ωψ (y; x) dΩ (y)

, (1.9)

whereψ is a homogeneous and isotropic weight function which determinesthe influence of point y on x through a decaying weight which is a functionof a length scale l. The normalised Gaussian function

ψ (ρ) =1

2π l2 exp(− ρ

2

2l2

)in R2, (1.10)

where ρ is defined as the distance between the points y and x, is usuallytaken as the weight function in integral non-local models. The non-localstate variable e replaces its local counterpart el in the Kuhn-Tucker rela-tions (1.2) which now read

κ ≥ 0, e−κ ≤ 0, κ (e−κ) = 0. (1.11)

An approximate differential format of (1.9)-(1.10) has been derived by Peer-lings et al. [134] and reads

e− c∇2e = el in Ω, (1.12)

where c = 12 l2 and ∇2 is the Laplacian operator. Equation (1.12) is a modi-

fied Helmholtz equation and is combined with the boundary condition

∇e · n = 0 on Γ , (1.13)

where n is the outward unit normal at the boundary Γ of Ω and ∇e is thegradient of e. In a finite-element context, e represents an additional degree offreedom to the standard ones. The equivalence of the two formats has beendiscussed by Peerlings et al. [135].

Continuum plasticity theories

The flow theory of plasticity is usually considered to describe elastoplasticmaterial behaviour. In the geometrically linear case, the basic assumption is


the decomposition of the strain tensor into an elastic and a plastic compo-nent:

ε = εe +εp. (1.14)

Consequently, the relation between total stress and elastic strain yields

σ = De:εe = De: (ε−εp) . (1.15)

The plastic strain rate εp is postulated, in the case of an associated flow ruleto which this study is restricted, as

εp = λ fσ , (1.16)

where the rate λ of the plastic strain multiplier λ determines the magnitudeof the plastic flow while the gradient fσ = ∂ f /∂σ to the yield surface f isa tensor indicating the direction of the plastic strain-rate. Similar to damagetheories, loading-unloading conditions are expressed by the Kuhn-Tuckerrelations

λ ≥ 0, f ≤ 0, λ f = 0. (1.17)

In ideal plasticity, the yield function is expressed as

f (σ) = σe (σ)−σ0, (1.18)

which is a function of the stress tensor only and where σe is the effectivestress, as defined by the yield criterion, and σ0 is the yield stress. Soften-ing/hardening constitutive behaviour is introduced by making the yieldstress a function of a scalar measure κ of the plastic strain tensor. The equiv-alent plastic strain κ is defined through the strain-hardening hypothesiswhich defines the relation of proportionality between the equivalent plas-tic strain-rate κ and the plastic strain-rate multiplier λ. The characterisationof elastoplastic material behaviour is based on the definition of the yieldfunction, limited here to isotropic hardening, through the effective and yieldstress definitions.

Yield function. Two yield functions have been considered. The first is theclassical von Mises yield function

f (σ ,κ) =√

3Jσ2 − σ (κ) , (1.19)


σ1

σ2

σ−

σ−

Figure 1.5 Smoothed Rankine yield function [128].

where σ (κ) is the uniaxial yield stress whose expression will be specifiedlater and Jσ2 = 1

2σd:σd is the second invariant of the deviatoric stress ten-

sor σd. The second yield function is the Rankine yield function which iswidely used to describe cracking in quasi-brittle materials. It is a principalstress criterion and is characterised by a vertex in the principal stress space.Its smoothed version, proposed by Pamin [128] and depicted in Figure 1.5,reads

f (σ ,κ) =

σ1 − σ (κ) if σ2 ≤ 0√σ2

1 +σ22 − σ (κ) if σ2 > 0,

(1.20)

where σi are the principal stresses (σ1 > σ2), and has been considered insome of the analyses reported in Chapter 4 and Appendices C and E. Forboth yield criteria, under the assumption of strain hardening, κ = λ. Planestress and plane strain plasticity have been considered (plane stress plas-ticity has been formulated in the framework proposed by Simo and Tay-lor [167]).

In describing softening constitutive behaviour, the rule governing σ (κ)has been given an exponential form [69] according to

σ (κ) = σ0 ((1 + a) exp (−bκ)− a exp (−2bκ)) , (1.21)

with a and b model parameters and σ0 the initial cohesion (or yield stress).Conversely, in case of hardening material behaviour, a linear hardeningmodel of the type

σ (κ) = σ0 + Hκ, (1.22)


where H is the hardening parameter, is considered.

Regularisation in presence of softening constitutive behaviour. Theintroduction of rate terms in the constitutive equations has been consid-ered to preserve the regularity of the governing equations in case of strain-softening constitutive equations. Rate-dependent regularisation is consid-ered in a viscoplasticity framework specified by the models of Perzyna [137]and Duvaut-Lions [54, 165]. Similar to rate-independent plasticity, in smallstrain viscoplasticity the strain tensor is decomposed into an elastic and aviscoplastic component according to

ε = εe +εvp (1.23)

which is combined with the elastic stress-strain law (1.15)1. The two vis-coplasticity formats will be described in Chapter 4 in the context of theircoupling to damage.

1.5 A model for separation across cohesive surfaces

As an approximation of the behaviour of quasi-brittle materials, materialdegradation can be considered as lumped at cohesive surfaces [185, 197].Depending on the modelling characterisation of the failure process, degra-dation can be initiated on cohesive surfaces or can be considered as the nat-ural evolution of diffuse bulk degradation into a localised crack with someresidual stress. In general terms, the traction at a discontinuity surface Γdreads

t = f (JuK, history) on Γd. (1.24)

A simple law can be formulated in terms of tractions tn and displacementjumps (Crack Opening Displacement, COD) w normal to the discontinuitysurface:

tn(w) =

tmax

(1− w

wmax

)mif 0 ≤ w ≤ wmax

0 if w > wmax,(1.25)

where tmax is the tensile strength, wmax is the COD at zero stress and m > 1is a shape parameter. In the analysis of fibre composites, wmax relates to thefibre length and m depends on the fibre type [82].

Chapter 2

Continuous-discontinuous failure in standard media∗

Displacement discontinuities are incorporated in standard finite elementsfor the numerical treatment of geometrical discontinuities, e.g. rockjoints [72], or for the description of problems with evolving boundaries, e.g.cracks in quasi-brittle materials [159] or delamination in composite mate-rials [6]. The incorporation of a discontinuity in the displacement field isconventionally pursued by using interface elements or embedded disconti-nuity elements. With interface elements, a discontinuous displacement fieldis described by the relative displacement of a double set of nodes at inter-element boundaries. This requires special mesh generators and, in the caseof evolving boundaries, remeshing procedures [25]. Alternatively, the effectof a discontinuity in the displacement field can be described by the use ofembedded discontinuity elements in which the displacement discontinuity,described by using the Heaviside function, is incorporated into the finite-element formulation as an incompatible strain mode and can pass throughsolids elements [7, 122, 166].

A different approach employs the partition of unity property of finite-element shape functions (the sum of the shape functions must equal unityat each spatial point) [52, 107]. Within this approach, the standard approxi-mation basis is enriched locally with special functions. This enrichment re-sults in extra degrees of freedom for the nodes in the domain subjected tothe enrichment, without modification of the mesh topology. The effect ofa displacement discontinuity is described by enriching the standard finite-element polynomial basis with the Heaviside function [110]. The outcome ofthis approach is a class of elements, in the following referred to as ‘partitionof unity-based discontinuous elements’ (in short ‘PU-based discontinuouselements’), which is kinematically equivalent to the class of conventionalinterface elements, the key difference being the possibility of arbitrarily lo-cating the discontinuity within the domain of an element. When PU-baseddiscontinuous elements are used, the interface response is described by anadditional set of global degrees of freedom and by a constitutive law at the∗ Based on References [168, 171, 174]

14 Chapter 2 Continuous-discontinuous failure in standard media

Γu

Γt

+Γd d−Γ

+Ω−Ω

−Ω

+Ω

dΓ

Γu

n

m

t−

Figure 2.1 Body Ω crossed by a discontinuity Γd.

discontinuity.Next, the characterisation of the problem fields for a body crossed by a

discontinuity is recalled. The variational formulation and its linearised dis-crete format are derived following standard procedures. Some issues relatedto the element technology are discussed and some applications to stationaryand propagating discontinuities in elastic and inelastic bulk materials con-clude the chapter.

2.1 Problem fields

When a body is crossed by a discontinuity, it is necessary to characterise theproblem fields in a proper way. Here, the displacement jump is described bythe Heaviside function operating on a smooth and continuous function. Dif-ferent from Wells and Sluys [193], the strain field is defined everywhere inthe body except at the discontinuity surface. This format leads to a straight-forward derivation of a variational statement in local and non-local (e.g.gradient-enhanced) continua as will be illustrated later in this chapter andin Chapter 3, respectively.

The body Ω, depicted in Figure 2.1, is bounded by Γ and is crossed bya discontinuity surface Γd which divides the body into two sub-domains,Ω+ and Ω− (Ω = Ω+ ∪Ω−). The boundary surface of the body Ω consistsof three mutually disjoint boundary surfaces, Γu, Γt and Γd (Γ = Γu ∪ Γt).Displacements u are prescribed on Γu, while tractions t are prescribed on Γt.

2.1 Problem fields 15

Figure 2.2 Discretised body Ω crossed by a dis-continuity Γd (the white circles indicates the nodeswith extra degrees of freedom b).

In the body Ω, the displacement field can be decomposed as

u(x, t) = u(x, t) +HΓd (x) u(x, t), (2.1)

whereHΓd (x) is the Heaviside function centred at the discontinuity surfaceΓd (HΓd = 1 if x ∈ Ω+,HΓd = 0 if x ∈ Ω−) and u and u are continuous func-tions on Ω. In the geometrically linear case, the strain field inΩ is computedas the symmetric part of the gradient of the displacement field and reads

ε (x, t) = ∇su (x, t) +HΓd (x)∇su (x, t) if x /∈ Γd, (2.2)

where (·)s refers to the symmetric part of (·).

2.1.1 Problem field interpolation

The discrete representation of a discontinuity can be rigorously achievedthrough a discontinuous interpolation of the problem fields [110, 193]. Tobegin with, it is useful to consider the discretised version of the body Ωdepicted in Figure 2.2. In standard finite elements, each node of the discre-tised medium is given a standard set a of degrees of freedom representingthe displacements in the Cartesian directions. With PU-based discontinu-ous elements, a node is given an extra set b of degrees of freedom whena discontinuity crosses the support of that node. Following standard pro-cedures [110, 193], the discretised format of (2.1) reads, for nodes whosesupport is crossed by a discontinuity,

uh = Nua +HΓd Nub, (2.3)

where Nu is an array containing standard finite-element shape functionsand the global nodal degrees of freedom a and b represent, in the arrange-ment of (2.3), the total displacement field. The displacement jump across the


discontinuity Γd is given by

JuhK = NubΓd

. (2.4)

The strain field in (2.2) can be discretised in a similar fashion and reads,using engineering notation for the stress tensor,

εh = Bua +HΓd Bub if x /∈ Γd, (2.5)

where Bu = LNu and L is the differential operator

L =

∂∂x 0 0 ∂

∂y 0 ∂∂z

0 ∂∂y 0 ∂

∂x∂∂z 0

0 0 ∂∂z 0 ∂

∂y∂

∂x

T

. (2.6)

For nodes whose support is not crossed by a discontinuity, the Heavisidefunction is a constant function over their supports and therefore it is not con-sidered. Consequently, since there is no enhancement, the standard finite-element interpolation uh = Nua is retrieved. Note that the inclusion of in-ternal discontinuity surfaces in a finite element is equivalent to the applica-tion of natural boundary conditions, without modifications of the originalfinite-element mesh, at the discontinuity surface.

The interpolation in (2.3) can be understood as an enrichment of the stan-dard polynomial finite-element spaces by a special function (the HeavisidefunctionHΓd ) which reflects known information (the presence of a displace-ment jump) about the boundary value problem. The special function HΓd ismultiplied with the shape functions Nu (which are also partitions of unity)and then pasted to the existing finite-element basis to construct a conform-ing approximation [52, 107, 121].

The PU-based discontinuous element approach can be termed as a‘smeared discontinuity approach’ since the displacement jump is describedat the nodes whose support is crossed by a discontinuity by the b degrees offreedom. Indeed, there are no modifications of the existing mesh topology,i.e. no extra nodes are added where a discontinuity intersects an element(see e.g. the application in Section 2.6.1). However, contrary to embeddeddiscontinuity elements [7, 122, 166], in PU-based discontinuous elements thediscontinuity contribution to the element stiffness matrix is properly takeninto account (see Section 2.5 under Numerical integration). A formulation ofembedded discontinuity elements with extra nodes where a discontinuityintersects an element has been derived by Alfaiate et al. [4, 5].

2.2 Governing equations 17

2.2 Governing equations

The equilibrium equations and boundary conditions for the body Ωwithoutbody forces can be summarised by

∇·σ = 0 in Ω, σn = t on Γt, σm = t on Γd, (2.7)

where σ is the Cauchy stress tensor, n is the outward unit normal to thebody and m is the inward unit normal to Ω+ on Γd (see Figure 2.1). Equa-tion (2.7)3 represents traction continuity at the discontinuity surface Γd. Thestrong form is completed by the boundary conditions

u = u on Γu, u = 0 on Γu, (2.8)

where u is a prescribed displacement. Equation (2.8)1 is a standard essentialboundary condition while (2.8)2 has been imposed to simplify the finite-element implementation [193]. The constitutive relationships for bulk anddiscontinuity degradation will be specified in Section 2.6.

2.3 Variational formulation

Following standard procedures, the governing equations (2.7)1 to (2.7)3 willbe cast in a weak form. The space of trial displacements is defined by thefunction u = u +HΓd u with u and u ∈ Uu where

Uu =

ui and ui | ui and ui ∈ H1(Ω) and ui|Γu = ui, ui|Γu = 0

(2.9)

with H1 a Sobolev space; ui and ui indicate the ith component of u and u,respectively. The space of admissible displacement variations is defined bythe weight function wu = wu +HΓd wu with wu and wu ∈ Wu,0 where

Wu,0 =

wu,i and wu,i | wu,i and wu,i ∈ H1(Ω)

and wu,i|Γu = wu,i|Γu = 0

, (2.10)

where wu,i and wu,i indicate the ith component of wu and wu, respectively.The equilibrium equations (2.7)1 are multiplied by the weight function wu ∈Wu,0, which is decomposed into wu and wu consistent with the displacementdecomposition in (2.1), and integrated over the domainΩ to obtain the weakequilibrium statement

∫

Ω(wu +HΓd wu) · (∇·σ) dΩ = 0. (2.11)


The term related to the continuous part of the displacement field (wu) can beexpanded using integration by parts, Gauss’ theorem and the relationshipσn = t on Γt to yield

∫

Ωwu· (∇·σ) dΩ =

∫

Ω∇· (σwu) dΩ−

∫

Ω∇swu:σ dΩ

=∫

Γt

wu·t dΓ −∫

Ω∇swu:σ dΩ. (2.12)

Similarly, the term related to the discontinuous part of the displacementfield (HΓd wu) is expanded using integration by parts:

∫

ΩHΓd wu· (∇·σ) dΩ =

∫

Ω+wu· (∇·σ) dΩ

=∫

Ω+∇· (σwu) dΩ−

∫

Ω+∇swu:σ dΩ. (2.13)

Using Gauss’ theorem and the relationships σm = t and σn = t, the firstterm of the RHS of (2.13) reads

∫

Ω+∇· (σwu) dΩ =

∫

Γ+t

wu· (σn) dΓ −∫

Γ+d

wu· (σm) dΓ

=∫

Γ+t

wu·t dΓ −∫

Γ+d

wu·t dΓ , (2.14)

where Γ+t and Γ+

d are parts of the boundary ∂Ω+. The weak form thereforereads

∫

Ω∇swu:σ dΩ+

∫

Ω+∇swu:σ dΩ+

∫

Γd

wu·t dΓ

=∫

Γt

(wu +HΓd w) ·t dΓ , (2.15)

in which the terms related to Γt and Γ+t have been collected under the same

integral using the Heaviside function. Since the function wu is continuousacross the discontinuity and since the notation Γ+

d has been introduced toindicate which part of the discontinuity is under analysis, the domain Γ+

d ofthe integral of the traction t has been changed into Γd.

From the decomposition of the displacement field it follows that any ad-missible variation w of u can be regarded as admissible variations wu andwu, thus leading to two variational statements. Taking first variation wu

2.4 Discretisation and linearisation 19

(wu = 0) and then wu (wu = 0) yields the final form of the variationalstatements:∫

Ω∇swu:σ dΩ =

∫

Γt

wu·t dΓ ∀ wu ∈ Wu,0 (2.16a)∫

Ω+∇swu:σ dΩ+

∫

Γd

wu·t dΓ =∫

Γ+t

wu·t dΓ ∀ wu ∈ Wu,0. (2.16b)

The second variational statement ensures that traction continuity is satisfiedin a weak sense across the discontinuity Γd. The two variational statementsin (2.16) resemble a coupled problem in which the fields u and u are coupledin the continuum through the expression of the stress field and the effect ofthe discontinuity is taken into account by the integral over Γd. It is also worthnoting that the variational statements in (2.16), compared to the variationalstatement related to a body under a continuous kinematic field,

∫

Ω∇swu:σ dΩ =

∫

Γt

wu·t dΓ ∀ wu ∈ H10(Ω), (2.17)

could have also been derived writing the principle of virtual work appliedto the body Ω under the continuum displacement field u and applied tothe body Ω+ under the displacement fieldHΓd u related to the displacementjump. Note also that the principle of virtual work for a body crossed by adiscontinuity is recovered by summing (2.16a) and (2.16b) [4, 103].

2.4 Discretisation and linearisation

Next, the linearised format of the equilibrium equation is derived. In thefollowing, the subscript h is dropped from discretised quantities and engi-neering notation forσ and ε is used.

2.4.1 Problem field description

Following a Bubnov-Galerkin approach, (2.1) and (2.2) can be discretised ineach element with extra degrees of freedom b using a format similar to (2.3)and (2.5):

u = Nua u = Nub ∇su = Bua ∇su = Bub (2.18)wu = Nua′ wu = Nub′ ∇swu = Bua′ ∇swu = Bub′. (2.19)

In the above relations, the primes refer to admissible variations. For ele-ments with standard degrees of freedom a only, the problem fields can bediscretised in a standard fashion.


2.4.2 Discretised and linearised weak governing equations

The discrete format of the problem fields leads to the two discrete weakgoverning equations

∫

ΩBT

uσ dΩ =∫

Γt

NTu t dΓ (2.20a)

∫

Ω+BT

uσ dΩ+∫

Γd

NTu t dΓ =

∫

Γ+t

NTu t dΓ , (2.20b)

from which the equivalent nodal force vector related to admissible varia-tions of a and b result in

fint,a =∫

ΩBT

uσ dΩ fext,a =∫

Γt

NTu t dΓ (2.21a)

fint,b =∫

Ω+BT

uσ dΩ+∫

Γd

NTu t dΓ fext,b =

∫

Γ+t

NTu t dΓ . (2.21b)

The linearised form of the discretised weak governing equation is obtainedby substituting the stress rate in the bulk

σ = Dε = D(Bua +HΓd Bub

), (2.22)

with D the tangent matrix for the bulk material, and the traction rate at adiscontinuity

t = TJuK = T(Nub

) Γd

, (2.23)

where T relates traction rate t and displacement jump rate JuK, in (2.20).After standard manipulations, the linearised weak form of the governingequations at iteration i within a time step n reads

[K n,i−1

aa K n,i−1ab

K n,i−1ba K n,i−1

bb

] [δa n,i

δb n,i

]=

[f n

ext,a

f next,b

]−[

f n,i−1int,a

f n,i−1int,b

], (2.24)

where

Kaa =∫

ΩBT

u DBu dΩ (2.25a)

Kab =∫

Ω+BT

u DBu dΩ (2.25b)

2.5 Element technology 21

Kba = KTab =

∫

Ω+BT

u DBu dΩ (2.25c)

Kbb =∫

Ω+BT

u DBu dΩ+∫

Γd

NTu TNu dΓ (2.25d)

f int,a =∫

ΩBT

uσ dΓ (2.25e)

f int,b =∫

Ω+BT

uσ dΩ+∫

Γd

NTu t dΓ (2.25f)

f ext,a =∫

Γt

NTu t dΓ (2.25g)

f ext,b =∫

Γ+t

NTu t dΓ . (2.25h)

Symmetry of the stiffness matrix is assured if the material tangent matricesD and T are symmetric. The derivation of the variational formulation pre-sented here differs from the one reported by Wells and Sluys [193] but yieldsthe same weak and linearised weak form of the governing equations for abody crossed by a discontinuity.

Note that the arrays Nu multiplying a and b in (2.3) are normally not thesame since only part of the extra degrees of freedom in the array b mightbe activated. Consequently, the matrices Bu multiplying a and b in (2.5) arealso normally not the same. However, since the system of equations in (2.24)is assembled only for the active degrees of freedom, it is still possible to usethe standard Nu and Bu matrices.

2.5 Element technology

In the following, some issues pertinent to the implementation of PU-baseddiscontinuous elements are discussed. Other issues, such as the choiceand the activation of the extra degrees of freedom, are discussed in Ref-erences [51, 110, 193].

Introducing a discontinuity. As soon as a critical situation is detected inthe element ahead of a discontinuity tip the discontinuity is extended. Inthe examples reported in Sections 2.6.3 and 2.6.4, a cohesive discontinuityis extended from a pre-existing traction-free discontinuity; this correspondsto the introduction of a discontinuity at point a as depicted in Figure 1.4. Aprincipal stress criterion is used. This criterion is applied by sampling the


discontinuity tip

Figure 2.3 Extension of a discontinuity withinthe body Ω (the black circles indicates nodes withstandard degrees of freedom a while the whitecircles indicates nodes with extra degrees of free-dom b for the nodes whose support is crossed bya discontinuity).

maximum principal tensile stress at all the integration points in the elementahead the discontinuity tip at the end of a load increment. Mesh refinementstudies suggest that the total energy dissipated during crack propagation inan elastic medium is a constant material parameter [193]. This gives an in-dication of the validity of the above criterion. From a mathematical point ofview this is not correct since, for very high mesh densities, the elastic solu-tion is recovered, and criteria based on stress or strain quantities are mean-ingless. In that perspective, the use of criteria based on energy considera-tions should be considered [109]. However, since the finite-element solutionis known to converge very slowly to the elastic solution, the above maxi-mum stress criterion can be accepted with some confidence. For the meshdensities considered in this chapter, this criterion can be considered equiva-lent to the sampling of the maximum principal stress at some distance fromthe crack tip as suggested by Williams and Ewing [195]. In inelastic localmedia, this issue is of no concern since the stress at the discontinuity tip isbounded in the inelastic regime.

To preserve the robustness of the Newton-Raphson solution procedure, adiscontinuity is introduced as a straight segment at the end of a load incre-ment in the element ahead of a discontinuity tip if the initiation criterion isfulfilled. This procedure is repeated in the elements ahead of this elementwith the extended discontinuity until the initiation criterion is no longersatisfied.

To reproduce a crack tip, the displacement jump at the discontinuity tipis set to zero. This is achieved by considering only standard a dofs for thenodes on an element boundary touched by a discontinuity (see Figure 2.3).When a discontinuity is extended in a neighbouring element, the nodes be-hind the discontinuity tip are enhanced.


Figure 2.4 Composite integration scheme for a quadrilateralelement crossed by a discontinuity (the crossed circles are in-tegration points on the discontinuity and the black dots areintegration points for the continuum). discontinuity

Orienting a discontinuity. For the example in Section 2.6.4, the directionof discontinuity extension is computed using the principal directions of anon-local stress tensor which is calculated as a weighted average of stressesusing a Gaussian weight function [86, 193]. The weight wi associated withintegration point i are computed from

wi =1

(2π)3/2r3exp

(−‖di‖2

2r2

), (2.26)

where di is the vector in the direction of the integration point i and the in-teraction radius r is equal to three times the average element size ahead thediscontinuity tip. The discontinuity propagates in the direction normal tothe direction of the maximum non-local principal stress. When the disconti-nuity is close to a boundary, the discontinuity extension direction is alignedwith the previous discontinuity segment to avoid an incorrect direction de-termination due to the bias produced by an unsymmetric domain in (2.26).

Numerical integration. When dealing with integration of the element ma-trices only on a part of an element domain, it is necessary to consider al-ternative integration rules [110]. When an element is intersected by a dis-continuity, the two resulting domains are triangulated and each triangularsub-domain is mapped to a parent unit triangle over which a three-pointsymmetric quadrature rule, with interior points within the triangular sub-domain, is considered (see Figure 2.4). The discontinuity contribution to thestiffness matrix and the internal force vector are integrated on the discon-tinuity through a 2–point Gauss integration scheme unless otherwise indi-cated. For elements not crossed by a discontinuity, a standard 2× 2 Gaussintegration scheme is used for numerical convenience.

It is worth noting that the resulting composite integration scheme for thecontinuum reported in Figure 2.4, which consists of 24 integration points ineight triangular sub-domains, integrates correctly a quadratic function over


spring

L L

x

P

Figure 2.5 Geometry and boundary conditions for the tension test.

a parent quadrilateral element. In the numerical integration of the elementmatrices, this composite integration scheme results, for the terms containingthe strain energy density, in a full integration rule for bilinear quadrilateralelements and in a reduced integration rule for a quadratic quadrilateral ele-ments. The actual construction of the composite integration rule is based onan exact algebraic inverse iso-parametric mapping for bilinear quadrilateralelements [201]. This technique is based on the use of quadrilateral elementswith straight sides and evenly spaced side nodes.

Transfer of history data. The transfer of history data is performed withinan element only for those elements crossed by a discontinuity in which thebulk experiences inelastic deformations [194]. The maximum value of theequivalent plastic strain κ within the element not yet crossed by a disconti-nuity is considered as representative of the state of the entire element. Oncethis value is identified, the transfer of the history data to it related is per-formed by copying this history array to the integration points related to thecomposite integration scheme as defined in the above paragraph. After thetransfer is performed, the increment which led to the discontinuity exten-sion is recomputed to allow a consistent dissipation in the elements aroundthe discontinuity tip.

2.6 Applications

Some applications of PU-based discontinuous elements are presented. Tobegin with, a simple one-dimensional example is given to get acquaintedwith this class of elements. Application to quasi-brittle and to ductile fail-ure of cementitious composites follows a study of the performance of themethod with respect to a problem in which conventional interface elementsdo not perform well.

2.6 Applications 25

21 3 4

L L

1

LL/2

2 3 4

L/2

(a) (b)

discontinuitydiscontinuity

Figure 2.6 Discretisation for the tension test with (a) conventional interface elements and(b) PU-based discontinuous elements (the black circles indicates the standard degrees offreedom a while the white circles indicates the extra degrees of freedom b; the discontinuityis indicated by the vertical dotted line).

2.6.1 Setting the scene: a simple one-dimensional example

A simple one-dimensional example is given next to illustrate some of theissues involved in the element technology for PU-based discontinuous ele-ments. The example deals with the elastic solution of a one-dimensional barin tension with a spring in the cross section as depicted in Figure 2.5. The so-lution will be given for standard and for PU-based discontinuous elements.The bar is modelled by means of the two discretisations depicted in Fig-ure 2.6. The spring is first represented by a conventional interface element(see Figure 2.6a) while the discontinuous interpolation of (2.3) is exploitedfor the discretisation depicted in Figure 2.6b. The domain Ω+ is given by0 < x < L in Figure 2.5. In the following, E is the Young’s modulus, A is thecross-sectional area, and d is the spring stiffness. The element types neededfor the finite-element solution are depicted in Figure 2.7.

Stiffness matrix computation. For completeness, the stiffness matricesof truss and conventional interface elements are reported. For the one-dimensional truss element of length L depicted in Figure 2.8a the stiffnessmatrix reads

Ktruss =EAL

[1 −1−1 1

]. (2.27)

A one-dimensional conventional interface element can be conceived as atranslational spring element with stiffness d = (EA) /L in (2.27).

The sub-matrices in (2.25) are expanded for the truss element with a dis-continuity in the middle section depicted in Figure 2.8b. Note that the pos-


3

2 3

(c)

1 42 3(a) (b)

1 2 4

Figure 2.7 Element assembly for the discretisation with PU-based discontinuous elements(note the effect of the choice of the domain Ω+ on the activation of the extra degrees offreedom b).

itive part Ω+ of the domain goes from x = 0 to x = L/2 (HΓd = 1 forx < L/2; see Figure 2.8b) and the presence of extra degrees of freedom bfor both nodes. Sub-matrix Kaa is the same as Ktruss. The remaining sub-matrices are expanded as

Kab = Kba =∫

Ω+BT

u DBu dΩ =

L/2∫

0

BTu DBu d x =

EA2L

[1 −1−1 1

]. (2.28)

For Kbb, the volume integral equals Kab while the surface integral is evalu-ated on x = L/2 and yields

∫

Γd

NTu TNu dΓ =

(NT

u TNu

) x=L/2

=d4

[1 11 1

]. (2.29)

Assembly of the sub-matrices into the element stiffness matrix gives

K =

[K aa K ab

K ba K bb

]=

EAL − EA

LEA2L − EA

2L

− EAL

EAL − EA

2LEA2L

EA2L − EA

2LEA2L + d

4 − EA2L + d

4

− EA2L

EA2L − EA

2L + d4

EA2L + d

4

. (2.30)

If the discontinuity does not cross the element but one of the element nodeshas got extra degrees of freedom b (see e.g. Figure 2.7a) the element stiffnessmatrix can be derived following the procedure described above and reads

K =2EA

L

1 −1 −1−1 1 1−1 1 1

, (2.31)

2.6 Applications 27

bi bj

j

L

i

a ax

i j

L

xa ai j i j

(a) (b)

−Ω+Ω

dΓ

Figure 2.8 One-dimensional (a) truss element and (b) PU-based discontinuous truss ele-ment.

where the degrees of freedom have been ordered in the sequence[a1 a2 b2

](note that the truss length is L/2).

Assembly and solution. For the bar with the spring modelled by a con-ventional interface element (see Figure 2.6a), assembly of local stiffness ma-trices into the global stiffness matrix for the active degrees of freedom resultsin the system of equations

EAL + d −d 0

−d EAL + d − EA

L

0 − EAL

EAL

a2

a3

a4

=

0

0

P

, (2.32)

which yields

[a2 a3 a4

]=[ PL

EAPLEA + P

d2PLEA + P

d

]. (2.33)

For the discretisation with the PU-based discontinuous element (see Fig-ure 2.6b), the global system of equations reads

3EAL − EA

L52

EAL − 1

2EAL 0

− EAL

3EAL − 1

2EAL

12

EAL − 2EA

L52

EAL − 1

2EAL

52

EAL + d

4 − 12

EAL + d

4 0

− 12

EAL

12

EAL − 1

2EAL + d

412

EAL + d

4 0

0 − 2EAL 0 0 2EA

L

a2

a3

b2

b3

a4

=

0

0

0

0

P

(2.34)


1 2 43

+

=

H

(a)

(b)

(c)

u

u~

u

Figure 2.9 Displacement field for the one-dimensional tension test: (a) total displace-ment field given by the sum of (b) the u fieldand (c) the HΓd u field (see (2.35) for the nu-merical values).

and its solution yields

[a2 a3 b2 b3 a4

]

=[ 1

2PLEA + P

d32

PLEA + P

d − Pd − P

d2PLEA + P

d

]. (2.35)

For interface elements, the displacement jump across the discontinuity Γdis expressed as the difference JuhK = a3 − a2 = P/d of the displacement ofthe doubled nodes. In PU-based discontinuous elements, the displacementjump is expressed through (2.4) as JuhK = Nub|Γd = 0.5× (b2 + b3) = −P/d(see Figure 2.9). Note that the minus sign in the displacement jump valueindicates that the local frame of the discontinuity is in the opposite directionwith respect to the global frame x—in other words, the negative sign is dueto the choice of the positive part of the domain.

2.6.2 Linear-elastic analysis of a notched beam: the problem ofoscillations in the traction profile

As partition of unity-based discontinuous elements become more and moreused [110, 143, 193], it is important to assess their limitations. The perfor-mance of PU-based discontinuous elements in a more realistic setting is ex-amined through the analysis of the two-dimensional notched beam depictedin Figure 2.10 in which the discontinuity is allowed only along the centralline of the beam, later referred to as the ‘crack line’. This linear elastic testwas used by Rots [151] to test conventional interface elements. Four-nodeand eight-node quadrilateral elements are used under plane stress condi-tions. A Young’s modulus equal to 2× 104 N/mm2 and a Poisson’s ratio ν

2.6 Applications 29

20

450

100

P

Figure 2.10 Notched beam (depth=100 mm; all dimensions in millimetres).

equal to 0.2 have been used for the continuum. The traction-separation rela-tion in (1.24) is formulated in a local s,n coordinate system. A simple elasticlaw of the type

[tstn

]= TsnJuKsn =

[ds 00 dn

] [usun

](2.36)

is used, where ds and dn are constant, us and un are the displacement jumpsin the local (discontinuity) reference system and ts and tn are the tangen-tial and normal interface tractions, respectively. To reproduce pure mode Iopening, only displacement jumps in the horizontal direction are activated.The notch is described as a traction-free discontinuity (ds = dn = 0 N/mm3)and to simulate perfect contact prior to the loss of material coherence alongthe crack line, a high value of the interface stiffness is there considered. Thisapproach is usually called the ‘dummy stiffness’ approach and it is oftenused in combination with conventional interface elements.

The analyses performed with conventional interface elements reportedby Rots [151] show that the normal traction profile along the crack line ishighly dependent on the stiffness of the interface and on the chosen numer-ical integration scheme. In particular, it was shown that high values of thenormal stiffness in combination with exact Gauss integration (2–point rulefor linear element and 3–point rule for quadratic element) lead to significantoscillations in the normal tractions. The results of the analyses performedwith PU-based discontinuous elements are reported in Figures 2.12 and 2.13,where the normal tractions have been sampled at the integration points onthe discontinuity (similar results have been reported by Audi et al. [11] andby Remmers et al. [144]). The stiffness dn at the discontinuity ranges from2× 103 to 2× 105 N/mm3. Results depicted in the upper part of Figures 2.12


Figure 2.11 Unstructured mesh for the notched beam (943 elements;the discontinuity is indicated by the heavy line).

and 2.13 have been obtained using structured meshes with the discontinu-ity lying within elements (element size = 3.33 mm)—the discontinuity inter-sected undistorted quadrilateral elements through opposite midsides. Themesh in Figure 2.11 has been used for the results depicted in the lower partof Figures 2.12 and 2.13. In structured meshes, PU-based discontinuous ele-ments show the same spurious traction oscillations of conventional interfaceelements when an exact Gauss quadrature scheme is used for the integrationof the traction forces at the discontinuity (upper part of Figures 2.12 and2.13). Only a nodal integration scheme gives a smooth traction profile for allthe values of the dummy stiffness (in the computations, the trapezoidal rule,also called 2–point closed Newton-Cotes formula, for linear element and theSimpson’s rule, also called 3–point Newton-Cotes formula, for quadratic el-ement were used). These results are similar to those reported by Rots [151]and by Schellekens and de Borst [160] for conventional interface elements.However, in unstructured meshes with a high value of the dummy stiffness,the oscillations in the traction profile are always present (lower part of Fig-ures 2.12 and 2.13). Results for the limit case of a discontinuity lying on anelement side for a structured mesh are reported in Figure 2.14. It is stressedthat the oscillations are not due to a poorly conditioned system of equationsbut stem from the numerical integration of the contribution of the disconti-nuity to the stiffness matrix. Indeed, the oscillations are present with Gaussintegration schemes (see Figures 2.12a and 2.12b) and disappear with nodalintegration schemes (see Figures 2.13a and 2.13b) when a structured mesh isconsidered.

Similar to conventional interface elements, the reasons behind the oscil-lations in the traction profile for PU-based discontinuous elements are notwell understood. A heuristic remedy is the use of nodal integration schemes.However, this approach is successful only for special mesh configurations.

2.6 Applications 31

−2 −2

−2 −2

−1 −1

−1 −1

0 0

0 0

1 1

1 1

2 2

2 2

3 3

3 3

4 4

4 4

5

4

3

5

4

3

5

4

3

5

4

3

normal traction [MPa] normal traction [MPa]

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm


d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

20 20

20 20

40 40

40 40

60 60

60 60

80 80

80 80

100 100

100 100

posi

tion

[mm

]

posi

tion

[mm

]

posi

tion

[mm

]

posi

tion

[mm

]

(a) (b)

(c) (d)

Figure 2.12 Traction profiles along the crack line with Gauss integration scheme fortwo mesh structures: structured mesh with (a) linear and (b) quadratic elements andunstructured mesh with (c) linear and (d) quadratic elements (the position is measuredfrom the bottom of the beam).

In Appendix A, an attempt to link the behaviour of standard interface ele-ments and PU-based discontinuous elements is presented along the lines ofthe derivations pursued by Schellekens and de Borst [160]. Indeed, an anal-ysis of the stiffness matrices of the elements showed that the two method-ologies share the structure of the terms governing the interface response.The problem of the oscillations in the traction profile can be circumventedby activating the degrees of freedom responsible for the displacement jumpwhen they are required [110, 193], thus avoiding the initial elastic branch,as in the following examples. However, this problem may still be present iffavourable conditions exist, e.g. high stress gradient and high discontinuitystiffness in elastic regime.


−2 −2

−2 −2

−1 −1

−1 −1

0 0

0 0

1 1

1 1

2 2

2 2

3 3

3 3

4 4

4 4

5

4

3

5

4

3

5

4

3

5

4

3


d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm


d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

20 20

20 20

40 40

40 40

60 60

60 60

80 80

80 80

100 100

100 100

posi

tion

[mm

]

posi

tion

[mm

]

posi

tion

[mm

]

posi

tion

[mm

]

(a) (b)

(d)(c)

Figure 2.13 Traction profiles along the crack line with nodal integration scheme(trapezoidal rule for linear element and Simpson’s rule for quadratic element) for twomesh structures: structured mesh with (a) linear and (b) quadratic elements and un-structured mesh with (c) linear and (d) quadratic elements (the position is measuredfrom the bottom of the beam).

2.6.3 Quasi-brittle failure in a FRC beam in bending

Similar to quasi-brittle materials such as plain concrete, some fibre-reinforced materials fail as a consequence of the development of a majorcrack. Their behaviour up to failure can then be described by the same tech-niques used for plain concrete by employing discrete softening constitutivelaws [193].

The flexural behaviour of a series of mortar beams reinforced witharamide fibres depicted in Figure 2.15 has been analysed by Ward andLi [191]. The beam was tested with different fibre types and volume frac-tions. Aramide (kevlar) fibres with l f = 6.4 mm and fibre volume fractionV f = 0.5, 1.0, 1.5% were used in the beams considered for the numerical

2.6 Applications 33

−2 −2

−2 −2

−1 −1

−1 −1

0 0

0 0

1 1

1 1

2 2

2 2

3 3

3 3

4 4

4 4

5

4

3

5

4

3

5

4

3

5

4

3


d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm


d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

d=2 x 10 MPa/mm

20 20

20 20

40 40

40 40

60 60

60 60

80 80

80 80

100 100

100 100

posi

tion

[mm

]

posi

tion

[mm

]

posi

tion

[mm

]

posi

tion

[mm

]

(a) (b)

(d)(c)

Figure 2.14 Traction profiles along the crack line with (a, b) Gauss and (c, d) nodal in-tegration scheme for structured mesh with discontinuity along element side: (a, c) lin-ear and (b, d) quadratic elements (the position is measured from the bottom of thebeam).

analyses. In the experiment, the load was applied via displacement controland the midspan deflection was measured by a linear variable differentialtransformer (LVDT) placed at the centre of the beam. The beam has beenanalysed using a structured mesh with 8–node quadrilateral elements inplane stress with element size in the central part of the beam equal to 3 mm;a Young’s modulus equal to 30 GPa and a Poisson’s ratio equal to 0.2 wereused. Inelasticity was introduced through the cohesive law in (1.25) withm = 2 and composite tensile strength tmax = 1.96, 2.60, 3.25 MPa for a fibrecontent equal to 0.5, 1.0, 1.5%, respectively—these model parameters havebeen derived by Kullaa [94] ‘from micromechanical properties using a mi-cromechanical statistical tensile model.’ This analysis falls under the casedepicted in Figure 1.4a, with the discontinuity activated at point a (a co-


d

d

d d

thickness=0.55dd = 114 mm

Figure 2.15 Geometry and boundary con-ditions for the bending test.

0

simulationexperiment

increasing Vf

0.2 0.4 0.6 0.8 1 1.2Deflection [mm]

0

5

10

15

Load

[kN

]

Figure 2.16 Load-deflection curves for fi-bre volume fractions V f = 0.5, 1.0, 1.5%.

hesive discontinuity is extended along the central line of the beam from apre-existing vertical traction-free discontinuity placed at the bottom of thebeam).

The numerical and experimental load-deflection curves are reported inFigure 2.16 in which the applied load is plotted against the midspan de-flection. A good agreement was found only for the beam with the highestfibre volume fraction (V f = 1.5%); for the other fibre volume fractions thepeak load of the beams was predicted lower and the post-peak responsewas more ductile—these results are very similar to those reported by Kul-laa [94] who used conventional interface elements. As suggested by Wellsand Sluys [193], a possible remedy to modify the post peak regime couldbe the introduction of a more complicated dissipation mechanism throughthe introduction of displacement contributions parallel to the crack (slid-ing). Such an approach may lead to a better fit between numerical and ex-perimental curves but is rather difficult to justify on physical grounds (seeAppendix C for a brief discussion on this issue).

2.6.4 Ductile failure in a composite compact-tension specimen

In a class of engineered cementitious composites (ECC) [100], inelasticitydevelops as a consequence of multiple cracking which is initiated on planesnormal to the maximum principal stress. In tensile specimens, cracking isdiffuse over a large part of the specimen and the portion of total displace-ment associated to it is not recoverable when the specimen is unloaded to

2.6 Applications 35

63.5

127

63.5

25.4

360

310

117

thickness=61

300

Figure 2.17 Compact-tension specimen (all dimensions in mm).

zero stress state. The composite can then be idealised as homogeneous andcontinuous in the multiple cracking stage and the Rankine yield criterioncan be used to characterise inelastic behaviour in the hardening phase be-fore localisation [89]. It is noted that the use of anisotropic plasticity is bettersuited to describe diffuse cracking with a preferential directional pattern.However, since in this study the failure mode is controlled by uniaxial ex-tension, the use of anisotropic plasticity is not required [61]. Localisation ofdeformation is accounted for by means of the cohesive law in (1.25).

A compact-tension specimen, tested by Li and Hashida [98], was anal-ysed using the procedure depicted in Figure 1.4b with a cohesive discon-tinuity activated at point a. The material was an ECC with 2% by vol-ume of 12 mm long polyethylene fibres. The geometry of the specimenis depicted in Figure 2.17. In the experiment, the load was applied at theloading pins via deformation control using a LVDT placed near the notchtip. The following model parameters have been used for the analysis [89]:Young’s modulus 22 GPa, Poisson’s ratio 0.2, matrix cracking strengthft = σ0 = 2.2 MPa (see (1.22)), and continuous-discontinuous transitionstrain value εcd = 5.78% with corresponding stress level fcd = 4.32 MPaas depicted in Figure 2.18a. The above model parameters could have beendetermined on the basis of the composite microstructure following the pro-cedure described in Kanda et al. [92]. The cohesive discontinuity is extended


ε

σσ

(b)

εcd

f t

f cd f cd

wmax u(a)

Figure 2.18 Stress-deformation relations for ECC: (a) contin-uum and (b) discrete constitutive laws (dashed lines indicateunloading behaviour).

0 10 20 30 40 50Load−line displacement [mm]

0

experimentsimulation

2

4

6

8

10

Load

[kN

]

Figure 2.19 Load-displacement curves for ECC compact-tension specimen.

from a pre-existing horizontal traction-free discontinuity which representspart of the notch and its direction is computed according to the proceduredescribed in Section 2.5 under Orienting a discontinuity.

Inelasticity in the continuum bulk was introduced by Rankine plasticity(see (1.20) and Figure 2.18a), while the cohesive law in (1.25) with tmax = fcd,wmax = 0.5l f , l f = 12 mm and m = 1 was used to describe the inelas-tic behaviour across the macrocrack (see Figure 2.18b). In the finite-elementsimulations, an unstructured mesh of 2964 4–node quadrilateral elements inplane stress with an average element size in the central part of the specimenequal to 4 mm was used.

The simulated behaviour compares reasonably well with the experimen-tal data as depicted in Figure 2.19 and the softening effect due to the activa-

2.6 Applications 37

5 mm

50 mm20 mm

10 mm

κ0.3 % 6 %

Figure 2.20 Multiple and localised cracking at different load-line displacements (the thickline represents the discontinuity; traction-free and cohesive segments are represented in whiteand black, respectively).

Figure 2.21 Close-up of the failure zone close to the notch.


(a) (b)

Figure 2.22 Experimental results for the failure of the ECC compact tension specimen:(a) close-up of the cracked zone near the notch for load-line displacement equal to21.12 mm and (b) detail of a crack bridged by fibres (from Li and Wu [100] and Kabeleand Horii [88]).

tion of the discrete crack is fully captured by the model. Note that the highvalue of the total deformation is due to the high density of multiple cracks(visual inspection of the specimen indicated the presence of several cracksper square centimetre [98]). The extent of multiple cracking can be estimatedfrom the equivalent plastic strain contour plots depicted in Figure 2.20 alongwith cohesive and traction-free discontinuity segments. The reproduced lo-cal behaviour is similar to the experimental one, reported in Figures 2.22aand 2.23 [88, 100], with a large portion of the specimen undergoing multiplecracking. The discontinuity propagates through solids elements as depictedin Figure 2.21, where a close-up of the failure zone close to the notch is rep-resented.

Figures 2.22a and 2.23 clearly motivate a continuous-discontinuous ap-proach to failure according to the strategies described in Section 1.2 as aproper approach to the description of failure processes in which a localisedcrack stems from diffuse degradation.

2.7 Concluding remarks

PU-based discontinuous elements represent a promising alternative to con-ventional interface elements in problems with evolving boundaries. Com-pared to conventional interface elements, the major advantage is that meshadjustment along the propagating boundary is avoided since a discontinu-

2.7 Concluding remarks 39

Figure 2.23 Experimental results for the failure of the ECC compact tension speci-men: multiple cracking and localised crack at failure (from Kabele and Horii [88]).

ity can pass arbitrarily through solid elements.It is worth noting that the performance of PU-based discontinuous ele-

ments deteriorates when this class of elements is used with a ‘dummy stiff-ness’ approach, a key ingredient in many cohesive-zone models [185, 197](see also Appendix A where PU-based discontinuous elements are con-trasted with conventional interface elements).

Chapter 3

Continuous-discontinuous failure ingradient-enhanced media∗

The performance of some non-local models, in either integral [139] or dif-ferential format [134], deteriorates in the final stage of failure [64]. At com-plete failure of a material point, which is understood as a discontinuity i.e. aphysical crack, numerical interaction between the two physically separatedparts of the body persists and causes an exchange of information betweenthe two sides of the discontinuity. This information, in the form of locally ex-tremely high strain values, stimulates damage growth away from the pro-cess zone, i.e. a spurious extension of the continuum damaged zone. Thisphenomenon, most evident in problems with large crack opening, such ase.g. in the failure of composite specimens [64], manifests itself also in prob-lems with small crack opening [130].

In this chapter, the introduction of traction-free discontinuities in a differ-ential version of a non-local damage model, the so-called implicit gradient-enhanced continuum damage model [134], is presented. Discontinuities areintroduced when the material reaches a fully damaged state. The inclusionof internal discontinuity surfaces, where boundary conditions are appliedwithout modifications of the original finite-element mesh, avoids non-localinteractions across the crack. The unbounded strain on the surface has noinfluence on the non-local field and the unrealistic damage growth typi-cal of this class of regularised continuum models is avoided. After the in-troduction of a discontinuity in the problem fields, coupling between fullydamaged locations and the surrounding material ceases and damage awayfrom the discontinuity tip is frozen; additional degradation is described bythe damaging material in the process zone, around the discontinuity tip.As a discontinuity propagates, the non-local equivalent strain increases atthe discontinuity tip and ceases behind it since the non-local interaction be-tween opposite sides of the discontinuity is avoided. This allows a realisticdescription of macroscopic cracks evolving from a material with microc-

∗ Based on References [175, 176]

42 Chapter 3 Continuous-discontinuous failure in gradient-enhanced media

racks. Numerical simulations illustrate the performance of the continuous-discontinuous gradient-enhanced damage model in preventing the spuriousresponse normally observed prior to complete failure with the continuousmodel alone.

3.1 Problem fields

In the gradient-enhanced damage continuum model proposed by Peerlingset al. [134], the problem is characterised by the displacement field u and bythe scalar non-local equivalent strain field e (e is an extra degree of freedomwhich is added to the standard displacement degrees of freedom). The de-composition of the displacement field u and of the strain field ε into a con-tinuous and a discontinuous part has been described in Section 2.1 (see (2.1)and (2.2)). Similar to the strain field in (2.2), the non-local equivalent strainfield for the body Ω crossed by a discontinuity (depicted in Figure 2.1) canbe decomposed as

e(x, t) = e (x, t) +HΓd (x) e (x, t) if x /∈ Γd, (3.1)

where e and e are continuous functions on Ω. In the domain of an elementwhere extra degrees of freedom are active, an approximation of the non-local equivalent strain field in (3.1) is given by

eh = Nep +HΓd Neq if x /∈ Γd, (3.2)

where Ne is a matrix containing usual finite-element shape functions andthe global nodal degrees of freedom p and q represent, in the arrangementof (3.2), the total non-local equivalent strain field. When the support of anode is not crossed by a discontinuity, the standard finite-element interpola-tion is retrieved for that node. Note that (3.1) can be derived by substitutionof the local strain field from (2.2) into the non-local averaging in (1.9). Thedecomposition in (3.1) is a direct consequence of the definition of the strainfield (2.2).

3.2 Governing equations

The equilibrium equations and boundary conditions for the elastic body Ω,recalled in (2.7), can be used in a damage continuum model by adopting theconstitutive relation

σ = (1−ω) De:ε in Ω (3.3)

3.2 Governing equations 43

for an elasticity-based isotropic description of continuum damage. Asbriefly recalled in Section 1.4, in presence of softening constitutive be-haviour, an approximate gradient-enhanced version of the above dam-age continuum model can be conveniently employed to preserve well-posedness of the governing equations. The approximate differential formatof the non-local averaging of the local equivalent strain el in (1.9)-(1.10) re-sults in the modified Helmholtz equation [134]

e− c∇2e = el in Ω, (3.4)

for e , where c = 12 l2 is the gradient parameter with l the length scale. Equa-

tion (3.4) is an extra equation which is added to the equilibrium equation.The homogeneous natural boundary conditions

∇e · n = 0 on Γ (3.5)

complete the coupled system of equations.From the decomposition of the non-local equivalent strain e in (3.1) and

using (3.5), the boundary conditions at the discontinuity surface Γd (see Fig-ure 2.1) can be written as

∇e ·m = 0 on Γ−d (3.6)∇ (e + e) ·m = 0 on Γ+

d . (3.7)

Since the function e is a continuous function, e+ = e−, where e+/− indicatesthe value of e on Γ+/−

d . Therefore,

(∇e ·m)Γ+

d

= (∇e ·m)Γ−d

= 0. (3.8)

From (3.7) and (3.8), it follows that ∇e · m = 0 on Γ+d . In summary, the

boundary conditions for the non-local equivalent strain e at the discontinu-ity surface are

∇e ·m = 0 on Γ+/−d (3.9)

∇e ·m = 0 on Γ+d . (3.10)

3.2.1 Propagating discontinuities and boundary conditions

Together with the boundary conditions for the displacement fields, theabove boundary conditions allow the development of the process zone into


Ωp

Ωe

Ωd : ω = 1

: < ω < 0 1

0: ω =

Figure 3.1 Damage distribution in a continuum (adapted from Peerlings [131]).

a macroscopic crack. Indeed, (3.4) is only valid on the domain Ωe ∪Ωp ofthe body where 0 ≤ ω < 1 (see Figure 3.1). Since the process zone Ωp andthe fully damaged domain Ωd will increase during a computation, the nu-merical procedure must consider the evolution of the internal boundary Γd.

The evolution of the fully damaged zone can be dealt with by modifyingthe domain of the problem (using remeshing and adaptive techniques [8, 39]or removing fully damaged elements as done by Peerlings [131, 133]). Toavoid singularities in the system due to ω = 1, exponential-like soften-ing laws, such as e.g. (1.8), can be used. In this way, the damaging bodywill never experience complete damage. This may sound appealing fromthe numerical point of view in the sense that it is then possible to performcomputations in a continuous framework and consider as fully damagedzones those parts of the body where damage is close to unity. However,since in those parts of the body the local equivalent strain reaches extremelyhigh values, due to the non-local averaging either integral or differentialdamage models transfer information from physically fully damaged zonesto partly damaged zones, resulting in spurious damage growth away fromthe discontinuity tip [64]. With the continuous-discontinuous description offailure, the incorporation of the standard boundary conditions for the prob-lem fields where damage has reached its critical value is automatic and al-lows the development of boundary surfaces. The application of the standardboundary conditions to u and e on the internal boundary surfaces alters theaveraging procedure and implies that the non-local averaging for point A inFigure 3.2b is limited to the shaded part only. Classical non-local models are

3.3 Variational formulation 45

Figure 3.2 Non-local averagingfor point A close to a numericalmacrocrack (highly damaged lo-cations) in the darker area: (a) astandard non-local model aver-ages on and across the numeri-cal macrocrack while (b) the non-local averaging is limited to theshaded part when a true discon-tinuity is present (adapted fromWells [192]). (a) (b)

discontinuity

highly damaged zone

A A

unable to detect a numerical macrocrack (highly damaged locations) andnon-local averaging for point A is performed across and on the numericalmacrocrack (see Figure 3.2a) with consequent spurious damage growth.

3.3 Variational formulation

In this section, following standard procedures, the modified Helmholtzequation will be cast in a weak form. The weak form of the equilibriumequations has been derived in Section 2.3. The weak statements of the gov-erning equations of the continuous-discontinuous gradient-enhanced dam-age model will be coupled through the definition of the stress field in Sec-tion 3.4. To begin with, the space of trial non-local equivalent strains is de-fined by the function e(x, t) = e(x, t) +HΓd e(x, t) with e and e ∈ H1(Ω)while the space of admissible non-local equivalent strain variations is de-fined by the weight function we(x) = we(x) + HΓd we(x) with we and we∈ H1(Ω). Equation (3.4) can be cast in a variational form by multiplicationwith a scalar weight function we ∈ H1(Ω) (split into we and we) and byintegration over the domain Ω. The weak form of the modified Helmholtzequation reads

∫

Ω(we +HΓd we) (e +HΓd e) dΩ− c

∫

Ω(we +HΓd we)∇2 (e +HΓd e) dΩ

=∫

Ω(we +HΓd we) el dΩ ∀ we ∈ H1(Ω). (3.11)

From the decomposition of the problem fields it follows that any admissiblevariation we of e can be regarded as admissible variations we and we, thusleading to two variational statements. Taking first variations we (we = 0),


and then we (we = 0), leads to:

∫

Ωwe (e +HΓd e) dΩ− c

∫

Ωwe∇2 (e +HΓd e) dΩ

=∫

Ωweel dΩ ∀ we ∈ H1(Ω) (3.12a)

∫

ΩHΓd we (e +HΓd e) dΩ− c

∫

ΩHΓd we∇2 (e +HΓd e) dΩ

=∫

ΩHΓd weel dΩ ∀ we ∈ H1(Ω). (3.12b)

Using the product rule for the Laplacian of a discontinuous scalar field (seeAppendix B), the term ∇2 (HΓdφ) in the previous equations is expressed as

∇2 (HΓdφ) = HΓd∇2φ+φ∇δΓd ·m + 2δΓd∇φ·m, (3.13)

where δΓd is the Dirac-delta function centred at the discontinuity surfaceΓd. Substitution of the above relation into (3.12) leads to terms of the type∫Ωφ∇δΓd ·m dΩ which can be expanded using the directional derivative of

a functionφ in the direction of a generic unit vector v (Dvφ = ∇φ·v):

∫

Ω(∇δΓd ·v)φ dΩ =

∫

ΩDvδΓdφ dΩ = −

∫

Γd

Dvφ dΓ

= −∫

Γd

∇φ·v dΓ . (3.14)

Note that (3.14) has been derived using the relation∫

Ω(∇δΓd )φ dΩ = −

∫

Γd

∇φ dΓ (3.15)

for the Dirac-delta function δΓd (see Appendix B). Using Green’s theoremand after the application of the boundary conditions (3.5), (3.9) and (3.10),the variational statements in (3.12) can be written as

∫

Ωwe e dΩ+

∫

Ω+we e dΩ+ c

∫

Ω∇we·∇e dΩ+ c

∫

Ω+∇we·∇e dΩ

=∫

Ωweel dΩ ∀ we ∈ H1(Ω) (3.16a)

3.4 Discretisation and linearisation 47∫

Ω+we e dΩ+

∫

Ω+we e dΩ+ c

∫

Ω+∇we·∇e dΩ+ c

∫

Ω+∇we·∇e dΩ

=∫

Ω+weel dΩ ∀ we ∈ H1(Ω). (3.16b)

It is interesting to recall the weak statement of the continuous modelalone, as derived by Peerlings et al. [134]:

∫

Ωwee dΩ+ c

∫

Ω∇we·∇e dΩ =

∫

Ωweel dΩ ∀ we ∈ H1(Ω). (3.17)

Unlike the variational statement of the equilibrium counterpart of the cou-pled problem (see Section 2.3), the coupling is already present in the varia-tional statements in (3.16a) and (3.16b). However, collecting the continuousand discontinuous part of the interpolation under the expression in (3.1)allows for a comparison with the standard variational statement alongthe lines of the previous comparison for the equilibrium equations in Sec-tion 2.3.

3.4 Discretisation and linearisation

In this section the linearised form of the governing equations for a bodycrossed by a discontinuity is developed. In what follows, the subscript hwill be dropped from discretised quantities and engineering notation for σand ε will be used.

3.4.1 Problem field description

Using a Bubnov-Galerkin approach, (3.1) can be discretised in each elementaffected by the enhancement using

e = Nep e = Neq ∇e = Bep ∇e = Beq (3.18)we = Nep′ we = Neq′ ∇we = Bep′ ∇we = Beq′ (3.19)

for the non-local equivalent strain field where p and q are standard andextra dofs, respectively. A similar discretisation has been considered for thedisplacement and strain fields in Section 2.4.


3.4.2 Discretised and linearised weak governing equations

Substitution of the discretised non-local equivalent strain into (3.16), leadsto two discrete weak governing equations:

∫

ΩNT

e Ne p dΩ+∫

Ω+NT

e Ne q dΩ+∫

ΩBT

e cBe p dΩ

+∫

Ω+BT

e cBe q dΩ =∫

ΩNT

e el dΩ (3.20a)

∫

Ω+NT

e Ne p dΩ+∫

Ω+NT

e Ne q dΩ+∫

Ω+BT

e cBe p dΩ

+∫

Ω+BT

e cBe q dΩ =∫

Ω+NT

e el dΩ, (3.20b)

from which the ‘equivalent nodal force’ vectors related to admissible varia-tions of p and q result in

fint,p =∫

Ω

(NT

e Ne p + BTe cBe p−NT

e el

)dΩ

+∫

Ω+

(NT

e Ne + BTe cBe

)q dΩ (3.21a)

fint,q =∫

Ω+

(NT

e Ne p + BTe cBe p−NT

e el

)dΩ

+∫

Ω+

(NT

e Ne + BTe cBe

)q dΩ. (3.21b)

It is interesting to note that the equivalent nodal force vectors take care of thediffusive nature of the modified Helmholtz equation since they are not self-equilibrated vectors within a finite element. The external part of the RHS isempty due to the applied boundary conditions (fext,p = fext,q = 0). The equi-librium counterpart of the above equations has been derived in Section 2.4.To develop a consistent incremental-iterative full Newton-Raphson proce-dure, the governing equations are linearised following standard procedures(see e.g. References [64, 84]). At iteration i within a time step n, the discre-

3.4 Discretisation and linearisation 49

tised coupled boundary value problem can be written in matrix format as:

K n,i−1aa K n,i−1

ab K n,i−1ap K n,i−1

aq

K n,i−1ba K n,i−1

bb K n,i−1bp K n,i−1

bq

K n,i−1pa K n,i−1

pb K n,i−1pp K n,i−1

pq

K n,i−1qa K n,i−1

qb K n,i−1qp K n,i−1

qq

δ a n,i

δ b n,i

δ p n,i

δ q n,i

=

f next,a

f next,b

0

0

−

f n,i−1int,a

f n,i−1int,b

f n,i−1vint,p

f n,i−1int,q

, (3.22)

with the symmetries Kba = Kab, Kbp = Kbq = Kaq, Kqa = Kqb = Kpb,Kqp = Kqq = Kpq, and

Kaa =∫

ΩBT

u (1−ω) DeBu dΩ (3.23a)

Kab =∫

Ω+BT

u (1−ω) DeBu dΩ (3.23b)

Kap = −∫

ΩBT

u

[∂ω∂κ

] [∂κ∂e

]DeεNe dΩ (3.23c)

Kaq = −∫

Ω+BT

u

[∂ω∂κ

] [∂κ∂e

]DeεNe dΩ (3.23d)

Kbb =∫

Ω+BT

u (1−ω) DeBu dΩ+∫

Γd

NTu TNu dΓ (3.23e)

Kpa = −∫

ΩNT

e

[∂el

∂ε

]T

Bu dΩ (3.23f)

Kpb = −∫

Ω+NT

e

[∂el

∂ε

]T

Bu dΩ (3.23g)

Kpp =∫

Ω

(NT

e Ne + BTe cBe

)dΩ (3.23h)

Kpq =∫

Ω+

(NT

e Ne + BTe cBe

)dΩ, (3.23i)


u

ε

discontinuouscontinuous

δ

Figure 3.3 From continuous to discontinuous displace-ment/strain profiles as a consequence of strain localisation(adapted from Jirasek [83]).

where De is the linear-elastic constitutive matrix. As in the standardgradient-enhanced damage formulation, the stiffness matrix is not symmet-ric. The terms in the RHS of the discretised boundary value problem aredefined in (2.21) and (3.21).


The finite-element implementation for the continuum response mainlyfollows the one proposed by Peerlings et al. [134] for the gradient-enhanced model. The proposed model has been implemented with linearand quadratic quadrilateral elements. A discussion of the performance oflinear and quadratic elements in the gradient-enhanced damage model canbe found in Appendix D. Next, some issues pertinent to the element tech-nology for the continuous-discontinuous gradient-enhanced model will bediscussed. The integration scheme has been described in Section 2.5.

Introducing a discontinuity. In a damaging continuum a critical situationcan be defined as one which corresponds to damage values close, or equal,to unity in the element ahead of a discontinuity tip. The introduction of adiscontinuity at (almost) total loss of load-carrying capacity is in line withthe narrowing of the strain profile as a consequence of strain localisation


d i

2θ/

r i

discontinuity tip

Figure 3.4 Determination of the propagation direction.

(see Figure 3.3). Introducing a discontinuity before complete localisation,e.g. when damage is around 90%, poses the problem of defining the cor-rect energy dissipation for the regularised continuous-discontinuous modelwhich is problematic, even in a one-dimensional setting. In the applicationsin Sections 3.6.1 and 3.6.2, a traction-free discontinuity is inserted into anelement when the damage at all its integration points is larger than a crit-ical value ωcrit. The insertion of a discontinuity with damage larger thanωcrit at one integration point is considered in Section 3.6.3 and in Chapter 4.Simulations performed with both criteria (ω in one integration point versusω in all integration points larger thanωcrit) on the same test gave identicalresults upon mesh refinement.

Orienting a discontinuity. In a regularised continuum, the direction of adiscontinuity cannot be analytically derived from bifurcation analyses andphenomenologically based criteria must be used. In this context, since cri-teria based on the stress tensor cannot be reliably used [193], the directionof maximum accumulation of the non-local equivalent strain in a V-shapedwindow ahead of a discontinuity tip is used (the V-shaped window spans acircular sector withθ = 90, see Figure 3.4). This criterion can be justified bythe observation of numerically computed non-local equivalent strain pro-files and by the experimental strain fields shown by Geers et al. [68]. Thevector in the direction of the discontinuity propagation is computed as

dΓd = ∑i∈S

eiViwidi

‖di‖, (3.24)

where S is the set of integration points i in the V-shaped window ahead of adiscontinuity tip, ei is the non-local equivalent strain at integration point i,


Vi and wi are the volume and the weight associated with integration pointi, and di is the vector in the direction of the integration point i [194]. Theweights wi are computed using the Gaussian weight function in (2.26) withinteraction radius r equal to four times the length scale of the gradient-enhanced damage model. When the discontinuity is close to a boundary, thediscontinuity extension direction is aligned with the previous discontinuitysegment.

Transfer of history data. The transfer of history data within an elementis performed considering the maximum value of κ within the element asrepresentative of the state of the entire element (see also Section 2.5 underTransfer of history data). Smoothing techniques, as e.g. the one employed byWells et al. [194], could produce smaller values for the maximum value κ ofthe non-local equivalent strain e driving damage evolution, violating thusthe Kuhn-Tucker conditions in (1.11). The error introduced diminishes uponmesh refinement. The increment which led to the critical damage value isrecomputed after the transfer is performed.

3.6 Applications

In this section, the continuous-discontinuous approach to failure is ap-plied to quasi-brittle failure in concrete and composite specimens. A fullNewton-Raphson procedure has been used to trace the response in the non-linear regime. Although the algorithm was consistently derived to obtainquadratic convergence, the performance of the model close to failure andbefore the extension of a discontinuity in a critically damaged area deterio-rated (asω→ 1). However, after the extension of a discontinuity, quadraticconvergence was retrieved. All the applications are analysed following theprocedure described in Figure 1.4a on page 5 with a traction-free disconti-nuity (i.e. with t = 0 in (1.24) and (2.25f) and T = 0 in (3.23e)) activatedat point b—the use of cohesive discontinuities is not feasible because of theproblems discussed in Section 3.8.

The first application (concrete beam in four-point bending) illustrates thatthe problem of spurious damage growth close to failure can be avoidedby using the proposed continuous-discontinuous strategy. This applicationalso illustrates that softening laws with high residual stress induce an in-correct energy dissipation. In the second application (composite compact-tension test), the continuous and the continuous-discontinuous approaches

3.6 Applications 53

150 150

500

150

d

100

1/2 P 1/2 P

Figure 3.5 Four-point bending test (thickness = 50 mm; all dimensions in mm).

are properly compared and the results highlight the effect of a constantlength scale (see also Section 3.8). The last application (single-edge notchedbeam in anti-symmetric four-point-shear loading) illustrates the quality ofthe criterion for the determination of the direction of discontinuity propa-gation.

3.6.1 Concrete beam in four-point bending

A four-point bending test of a concrete beam with different notch sizes dis analysed (see Figure 3.5). To enable comparison with the experiments re-ported in Reference [78], the vertical displacement of a point placed at thebottom of the beam and with an offset of 7.5 mm from the centreline of thebeam is used for the measurement of the deflection v. The following modelparameters, fitted for the continuous problem, are adopted for the simu-lation [130]: Young’s modulus E = 40000 MPa; Poisson’s ratio ν = 0.2;exponential damage evolution law (1.8) with κ0 = 0.000075, α = 0.92and β = 300; modified von Mises definition of the local equivalent strain(1.4) with k = 10 and gradient parameter c = 4 mm2. The simulation isperformed under plane stress conditions. The load is applied via an im-posed displacement and quadrilateral elements with quadratic interpola-tion for the displacement and bilinear interpolation for the non-local equiv-alent strain have been used (see Appendix D). The notch is simulated as atraction-free discontinuity (t = 0 in (1.24)). The traction-free discontinuityis extended, starting from the tip at the notch, when the damage at all in-


0 0.1 0.2 0.3 0.4 00.5 0.10.6 0.2

50

30

10

0.7 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

0

1

2

3

4

5

P [k

N]

P [k

N]

continuous−discontinuouscontinuous experiment

continuouscontinuous−discontinuous

v [mm]

(a)

v [mm]

(b)

h=10, 5, 2.5 mm

Figure 3.6 Load-deflection curves for simulations with (a) different meshes for a 10 mmdeep notch (h is the element size in the central part of the beam) and (b) different notch sized for the medium element size mesh (h = 5 mm) and experimental results [78].

notch

continuous failure

continuous−discontinuous failure

Figure 3.7 Four-point bending test: non-local equivalent strain evolution (plotted in thesame scale for the 10 mm deep notch beam with medium element size mesh; close-up of thecentral part of the beam).

tegration points in the element ahead of the discontinuity tip is larger thana critical value, set to ωcrit = 0.999; the discontinuity is prescribed to bevertical (consequently, (3.24) is not used here).

3.6 Applications 55

Results of the analyses for the 10 mm deep notch beam are reported in Fig-ure 3.6a for different meshes. In the central part of the mesh, the coarse,medium and fine mesh element size h is 10 mm, 5 mm and 2.5 mm, respec-tively. Due to the small difference in the response between the medium andthe fine discretisation (see Figure 3.6a), the former, with a 10 mm deep notch,has been used for the results depicted in Figures 3.7 and 3.8. From the anal-ysis, it is evident that the introduction of a discontinuity during the com-putation influences the global (load-displacement curve) and local (dam-age and non-local equivalent strain profiles) behaviour. In particular, Fig-ure 3.7 shows that the activity of the non-local equivalent strain is mobilisedonly around the discontinuity tip for the continuum-discontinuous model.This translates into the more realistic damage profiles depicted in Figure 3.8where, in contrast to the continuous model alone, the width of the damagezone at the bottom of the specimen does not increase with the continuous-discontinuous failure model. However, the use of an exponential softeningrelationship with a high residual stress at the moment of the enhancementcauses the marked drops reported in Figure 3.6a and makes the comparison

ω1

0.1

continuous−discontinuous failure

continuous failure

Figure 3.8 Four-point bending test: damage evolution (the discontinuity is represented bythe white thick line; 10 mm deep notch beam with medium element size mesh; close-up ofthe central part of the beam).


thickness 3.8 mm

W/50

0.25 W

0.275 W

0.325 W

0.25 W

1.20 W

1.25 W

d

Figure 3.9 Compact-tension specimen (W = 50 mm, d = 10 mm).

with the experiment, reported in Figure 3.6b, difficult (this issue is discussedin Section 3.7). Responses of computations with higher values ofωcrit showa better agreement at the global level with experimental results but this isdue to the delayed or, in some cases, precluded extension of a discontinu-ity. The use of different damage law parameters to achieve lower residualstress values in the softening relationship produced an unsatisfactory com-parison in the post-peak response (note that this consideration relates to thecontinuum model alone since the discontinuity is extended at a later stageas depicted in Figure 3.6). Introduction of displacement discontinuities re-quires a re-assessment of the continuum model parameters governing thepost-peak response, which has not been done.

3.6.2 Composite compact-tension specimen

The compact-tension specimen depicted in Figure 3.9 [10], experimentallytested by Geers et al. [68] and numerically analysed by Geers et al. [65],Peerlings [131] and de Borst et al. [34], has been numerically investigatedwith the continuous-discontinuous approach. The specimen is placed ontwo loading pins whose action has been modelled by applying two verti-

3.6 Applications 57

Figure 3.10 Load-cmod diagrams for thecompact-tension specimen for different meshresolutions (h indicates the average elementsize in the central part of the specimen).

0 1 2 3 4 5cmod [mm]

0

500

1000

1500

2000

F [N

]

experimenth=2, 1, 0.5 mm

cal forces at the uppermost and lowermost node of the pinholes via defor-mation control. In the simulations, indirect displacement control has beenused, with the displacement (crack mouth opening displacement, cmod)measured between two markers placed 25 mm from the left edge and 14 mmfrom the symmetry axis of the specimen [34]. The vertical displacement ofthe mid-side point on the right central part of the specimen has been re-strained, as well as the horizontal displacement of the right lowermost anduppermost corners. A small part of the notch is simulated as a traction-freediscontinuity. The traction-free discontinuity is extended when the dam-age at all integration points in the element ahead of the discontinuity tipis larger than ωcrit = 0.99995, and its direction is computed according tothe direction of maximum non-local equivalent strain as described in Sec-tion 3.5 under Orienting a discontinuity. The simulations are performed usingunstructured meshes of bilinear quadrilateral elements under plane stressconditions with average element sizes h in the central part of the specimenequal to 2, 1 and 0.5 mm with 443, 1209 and 3955 elements, respectively.The model parameters have been adopted from Geers et al. [65]: damagegrowth is expressed via the power law in (1.7) with κ0 = 0.011, κc = 0.5,α = 5 and β = 0.75; the equivalent strain definition is based on the positiveprincipal strain components (see (1.3)); Young’s modulus E = 3200 MPa;Poisson’s ratio ν = 0.28 and gradient parameter c = 2 mm2. To avoid dam-age growth in the elements around the pinholes, a higher value of κ0 hasbeen given to elements in these areas. Furthermore, an almost horizontalbranch was added to the softening law at κ = 0.385κc, which corresponds,in an ideal one-dimensional uniform tension test, to an almost nil residualstress (0.175%κ0E; see Section 3.8).

The load-cmod response is shown in Figure 3.10. The agreement betweenthe experimental response and the results of the simulations is excellent.


cmod = 2.00 mm

cmod = 2.75 mm

cmod = 5.00 mm(a) (b)

ω e

0.1 1 0.1 0.5

Figure 3.11 Evolution of the failure process: (a) damage ω and (b) non-local equivalentstrain e contour plots.

3.6 Applications 59

cmod = 5 mm(a) (b)

ω e

0.1 1 0.1 0.5

Figure 3.12 Final failure pattern with the continuous model (no propagating discontinuity):(a) damageω and (b) non-local equivalent strain e contour plots.

Figure 3.13 Load-cmod diagramsfor the compact-tension specimen(h = 0.5 mm): comparison betweenthe continuous and the continuous-discontinuous model.

0 1 2 3 4 5cmod [mm]

0

500

1000

1500

2000

F [N

]


However, the response computed with the finer mesh (h = 0.5 mm) showsbumps after the extension of the discontinuity (this issue is discussed indetail in Section 3.8). The evolution of the fracture process is shown in Fig-ure 3.11 for the simulation related to the finest of the meshes used (h =0.5 mm). For a comparison, the local and global response of the standardcontinuous model (no propagating discontinuity) and of the continuous-discontinuous (with a propagating discontinuity) model are shown in Fig-ure 3.12 and Figure 3.13, respectively. Although the differences in the globalresponse are not significant (see Figure 3.13), the continuous-discontinuous


100

20

520 20

20 20

10/11P1/11P

440

Figure 3.14 Single-edge notched beam [161] (depth = 100 mm; all dimensions in mm).

model provides a better representation of the failure process (compare Fig-ure 3.11 to Figure 3.12).

3.6.3 Single-edge notched beam in anti-symmetric four-point-shearloading

A single-edge notched beam, depicted in Figure 3.14, is subjected to an anti-symmetric four-point-shear loading [161]. Figure. 3.14 shows the appliedboundary conditions which result in a curved crack path, from the lowerright part of the notch towards a point to the right of the lower-right sup-port as depicted in Figure 3.15. The boundary conditions are specified byconstraining the displacement at the upper-right support in both directionsand by constraining the vertical displacement at the upper-left support. Thesupports have widths of 20 mm with the centre located 20 mm out of themidspan of the beam.

The beam is analysed in a plane stress situation with bilinear quadrilateralelements; the simulations are performed using unstructured meshes withaverage element size h in the central part of beam equal to 4, 2 and 1 mmwith 674, 2148 and 7308 elements, respectively. The model parameters are

Figure 3.15 Experimental crack patterns from three single-edgenotched beams (adapted from Schlangen [161]).

3.6 Applications 61

0 0.05 0.1 0.15 0.2cmsd [mm]

0

10

20

30

40

50

P [k

N]


h=4, 2, 1 mm

(a)

0 0.05 0.1 0.15 0.2cmsd [mm]

0

10

20

30

40

50

P [k

N]


experiment

(b)

Figure 3.16 Applied load against crack mouth sliding displacement (cmsd): (a) convergencestudies and (b) comparison with the experimental response [161] for the h = 1 mm mesh (hindicates the average element size in the central part of the beam).

adopted unaltered from Peerlings et al. [132] who performed analyses ina purely continuous setting: Young’s modulus E = 35000 MPa, Poisson’sratio ν = 0.2, gradient parameter c = 1 mm2, modified von Mises equiv-alent strain definition (1.4) with k = 10, exponential softening law (1.8)with κ0 = 0.00006, α = 0.96 and β = 100. The loading platens have aYoung’s modulus one order of magnitude larger than the concrete. The loadis applied by means of an indirect displacement control procedure with thecrack mouth sliding displacement (cmsd), defined as the relative verticaldisplacement of the opposite faces of the notch, taken as control parame-ter [131]. The traction-free discontinuity is extended when damage is largerthatωcrit = 0.99 at one integration point in the element ahead of the discon-tinuity tip. It starts from an initial horizontal traction-free discontinuity (thehorizontal traction-free discontinuity goes from (222.5; 80.1) to (223; 80.1),with the origin of the coordinate system placed at the lowermost left cornerof the beam). As in the previous example, the direction of the discontinu-ity is aligned with the direction of the maximum accumulation of non-localequivalent strain.

Similar to the case of the four-point bending test reported in Section 3.6.1,the global response of the continuous-discontinuous model, reported in Fig-ure 3.16 in terms of load-crack mouth sliding displacement, is more brit-tle than that of the continuous model. This makes the comparison withthe experiment difficult (see Figure 3.16b). This issue is discussed in the


(a)

(b)

(c)

0.1 1

ω

Figure 3.17 Final damage distribution and traction-free discontinuity for the (a) h = 4 mm,(b) h = 2 mm and (c) h = 1 mm meshes (close-up of the central part of the beam).

3.6 Applications 63

Figure 3.18 Close-up of the final damage distribution and traction-free discontinuity for the h = 1 mm mesh near the notch.

(a)

(b)

0.1 1

ω

Figure 3.19 Final failure state for (a) the continuous-discontinuousmodel and (b) the continuous model (no propagating discontinuity)for the h = 1 mm mesh (close-up of the central part of the beam).


next section. However, it is worth noting that the convergence of the iter-ative scheme for the continuous-discontinuous approach is faster (see Fig-ure 3.16a). The final failure pattern is reported in Figures 3.17 to 3.19 to-gether with a comparison of failure patterns between the continuous andthe continuous-discontinuous model. In particular, Figure 3.17 shows thatthe evolution of damage is in good agreement with the observed crack pat-tern reported in Figure 3.15 and that the direction of the discontinuity iscorrectly determined even with the coarsest of the discretisations (see Fig-ure 3.17a). A close-up of the final damage distribution and traction-freediscontinuity for the h = 1 mm mesh is shown in Figure 3.18, and thecontinuous-discontinuous model and the continuous model are contrastedin Figure 3.19 where it is clear that the introduction of the discontinuityavoids the unrealistic damage growth close to the notch (see Figure 3.19a).

3.7 Stress-strain relationships

Enhancing an element with a discontinuous interpolation of the problemfields when the damage at all its integration points is larger than ωcrit =0.999 or larger than ωcrit = 0.9999 may result in significant differencesin terms of the global response. To understand the differences originat-ing from the choice of stress-strain relationships, the normalised softeningstress-strain paths for a one-dimensional uniform loading state field arecontrasted. The model parameters of the four-point bending test and thecompact-tension test previously analysed are adopted. The results of thisanalysis are shown in Figure 3.20. For the exponential law (1.8) (see Fig-ure 3.20a), an increment of less than 1% in damage requires an incrementof approximately 512% in deformation which corresponds to a drop in thenormalised stress of about 39%; for the power law (1.7) (see Figure 3.20b),only an increase of 23% in the deformation is necessary to drop the stress ofabout 88% for the same increment of damage.

The drops in the global response reported in Figures 3.6 and 3.16 are dueto an incorrect energy dissipation related to the high residual stress at themoment of the enhancement of the displacement field—despite high dam-age (ω u 1), high stresses can still develop. Although the asymptote of theexponential softening curve in Figure 3.20a might be useful in reproducingthe long tail observed in load-displacement diagrams of concrete specimenswhich is related to crack bridging [78, 108], the physical relevance of theselaws is questionable since an analogous and more pronounced phenomenon

3.8 Damage initiation and discontinuities 65

0 0.1 0.2 0.3 0.40 0.50.01deformation

0.02 0.03 0.04 0.05 0.06deformation

(a) (b)

0

0.2

0.4

0.6

0.8

1

norm

alis

ed s

tress

0

0.2

0.4

0.6

0.8

1no

rmal

ised

stre

ss

ωσε

=0.9999=0.08=0.06

ωσε

=0.999=0.02551=0.2985

ωσε

=0.9999=0.0031=0.36875

εσω =0.999

=0.12974=0.0098

Figure 3.20 Normalised stress-strain softening curves for (a) exponential and (b) power lawof damage growth.

occurs in fibre-reinforced composite polymers where softening laws withfull stress relaxation at finite strain—or softening laws with a small residualstress—suffice.

3.8 Damage initiation and discontinuities

A different phenomenon can be observed in the global response of the com-pact tension test depicted in Figure 3.10. In the simulations of the compact-tension test with the finest of the meshes, the load-displacement curveshowed a bump after the extension of a discontinuity. To better understandthe cause of this problem, it is important to realise that, as a result of non-local averaging, the non-local equivalent strain differs quantitatively andqualitatively from its local counterpart. With a strongly concave boundary,as in the case of a sharp notch, non-local regularisation results in a finitevalue of the damage driving quantity at the crack tip (see Appendix E),unlike in the case of the standard continuum where the equivalent strainmimics the singularity of the strain field. However, in contrast to a standardcontinuum, the maximum of the damage driving quantity is not located atthe tip, but shifted away from it. Simple analytical considerations reportedin Appendix E show that, for a planar crack, the position of the maximumof the non-local equivalent strain is a function of the length scale (i.e. themaximum of the non-local equivalent strain is at the crack tip for a lengthscale equal to zero). As a consequence, damage reaches its maximum insidethe specimen, away from the tip. Note that although the condition for the


x

y

(a) (b)

60 mm

4 mm

12 mm

60 mm

Figure 3.21 Compact-tension test: (a) geometry (thickness = 1 mm) and (b) discreti-sation (traction-free discontinuity from x = 12 mm to x = 16 mm at y = 30 mm).

propagation of a discontinuity might be satisfied in the specimen, it is notnecessarily at the crack tip. Eventually, due to increasing loading, the por-tion of the specimen where damage has its maximum enlarges and reachesthe crack tip, thus allowing for the propagation of the discontinuity. Further,for computational reasons, since damage can reach values larger than ωcritaway from the discontinuity tip, the use of exponential-like softening laws(or any other law with full stress relaxation at infinite strain) is necessary toavoid singularity in the global stiffness matrix.

The discontinuity propagates through more that one element at a timeand, as a consequence, is accompanied by strong variations of the localequivalent strain which in turn affect its non-local counterpart. It is shownlater that this influence is proportional to the length scale. This affects theloading condition at points ahead of the new discontinuity tip which nowexperience unloading and, due to an increased loading level, generates astructural pseudo-elastic loading, resulting in an incorrect global response.

Numerical considerations. To gain more insight into the nature of theproblem, a simpler compact-tension test geometry, depicted in Figure 3.21a,has been considered. To reproduce the effect of the discretisation on non-local averaging, different values of c have been used, keeping the mesh un-changed (structured mesh with 4 mm×4 mm linear quadrilateral elements;element size esize = 4×

√2 mm; see Figure 3.21b). The load is applied at the

uppermost and lowermost left corners (x = 0 mm, y = 0 mm / y = 60 mm)via an applied displacement. The uppermost and lowermost right corners

3.8 Damage initiation and discontinuities 67

Figure 3.22 Load-displacement curves for in-creasing non-locality parameter l (the dots rep-resent the extension of the discontinuity).

0 5 10displacement [mm]

0

100

200

300

400

500

reac

tion

[N]

0 5 100

100

200

300

400

500

l=4, 6, 8 mm

are restrained in the horizontal direction while the nodes at x = 60 mm,y = ±2 mm are restrained in the vertical direction. To avoid damage wherethe load is applied, a higher value of κ0 (see (1.7)) has been given to thematerial in the area around the uppermost and lowermost left corners. Themodel parameters used in the previous simulation of the compact-tensiontest (see Section 3.6.2) with ν = 0 and β = 1 have been considered. A hor-izontal traction-free discontinuity is placed in the shaded element depictedin Figure 3.21b. The traction-free discontinuity is extended horizontally inthe neighbouring element(s) if at all its(their) integration points damage islarger than ωcrit = 0.99999. The influence of the non-local averaging is il-lustrated in Figure 3.22, where load-displacement curves related to the ver-tical uppermost restrained node for different values of the length scale havebeen plotted. The values used for c are 8 mm2, 18 mm2 and 32 mm2 whichcorrespond to a length scale l equal to 4 mm, 6 mm and 8 mm. Comparedto the element size esize, only the last one permits sufficient non-local in-teraction. However, no estimate of the necessary/sufficient number of el-ements in the process zone is available for differential non-local modelsand the ratio esize/l is often taken very close to unity and sometimes evenlarger [34, 66, 93, 130, 132]. From the curves in Figure 3.22 it is evident thatan increasing non-local interaction leads to bumps in the load-displacementresponse after the extension of the discontinuity (the extension is marked bythe dots). The bumps in Figure 3.22 originate from the loading situation afterthe extension of the discontinuity in the elements ahead of the discontinuitytip. This phenomenon can be understood as the combination of two effects:(i) the sudden extension of the discontinuity through several elements dueto the shift of the maximum of the non-local equivalent strain away from thecrack tip and (ii) the influence of non-local interaction on the unloading ofpoints adjacent to the discontinuity, with the influence proportional to the


0

tip new tip tip new tip

0 1010 2020 3030 4040 5050 6060

e

(b)

xy=+32

(a)

xy=+32

e

Figure 3.23 Non-local equivalent strain profile before (solid line) and after (dotted line) theextension of the discontinuity for (a) l = 4 mm and (b) l = 8 mm (not to scale).

degree of non-local interaction.The non-local equivalent strain profiles depicted in Figure 3.23, for differ-

ent degrees of non-local interaction before and after the first extension of thediscontinuity, illustrate the problem with the loading condition discussedabove. Due to the definition of the loading function (see (1.11)), when non-local interaction is strong (see Figure 3.23b) all the integration points aheadof the discontinuity tip will temporarily unload after its extension. Despitethe high damage value in these integration points, they contribute to theRHS through residual stresses resulting in spurious loading at the globallevel. If the non-local interaction is weak (see Figure 3.23a), the difference inthe profiles of the non-local equivalent strain ahead of the discontinuity tipis small. The integration points between the new discontinuity tip and thepoint where the non-local equivalent strain exhibits its maximum value willunload and, since the unloading is limited, there is no evidence at the globallevel. The influence of the length scale on the evolution of the solution ar-ray (u and e) is depicted in Figure 3.24. The extension of the discontinuity,marked by the dots, is accompanied by strong variations of the non-localequivalent strain field (see section 4.4 for similar analyses performed with arate-dependent model). Note that this phenomenon is related to the resolu-tion of the mesh with respect to the length scale as depicted in Figure 3.24:if a coarse mesh is used it is likely that bumps in the load-displacementresponse will not appear (see Figure 3.6). It is also stressed that the use ofcohesive discontinuities do not solve nor alleviate the problem of the bumpswhich is only related to the non-local nature of the underlying continuum.Finally, the use of a model with a variable length scale, for which the length


pseudo−time

(a) (b)

0

0.2

0.4

0.6

0.8

uy

uy

|u | x

|u | x

1e e

scal

ed s

olut

ion

arra

y

pseudo−time0

0.2

0.4

0.6

0.8

1

scal

ed s

olut

ion

arra

y

Figure 3.24 Evolution of u and e for a point located at x = 32 mm, y = 32 mm for (a) l =4 mm and (b) l = 8 mm (the dots represent the extension of the discontinuity and the dottedline indicates the first moment at which the point is behind the discontinuity tip).

scale goes to zero close to failure (see e.g. Reference [64]), might solve theproblem of the unloading after the extension of a discontinuity, thus reduc-ing or eliminating the bumps. However, the continuous model will still beunable to predict damage initiation correctly as illustrated in Appendix E.


When discontinuities are allowed in an implicit gradient-enhanced contin-uum damage model, spurious growth of damage can be prevented. Themodel successfully eliminates the interaction between fully and partiallydamaged material locations. A continuous model with the same goal wasintroduced by Geers et al. [64] but, unlike that approach, in which a variablelength scale was used for the gradual transition from a regularised to a stan-dard continuum, in the continuous-discontinuous model considered hereonly minimal assumptions (i.e. (2.1) and (3.1)) are necessary to introduce atrue discontinuity. Although in a different context, the transition from a reg-ularised continuum model to discrete cracking was analysed by Jirasek andZimmermann [86] using embedded discontinuities.

The model presented in this chapter eliminates the problem of spuriousdamage growth but some properties of the underlying non-local continuummodel, illustrated in Appendix E, makes the transition from continuous tocontinuous-discontinuous failure problematic. In the next chapter, a differ-ent regularisation technique, based on rate-dependence of the governingequations, will be employed.

Chapter 4

Continuous-discontinuous failure in rate-dependentmedia∗

A displacement discontinuity can be considered as originating either fromcomplete material failure in quasi-brittle materials, representing genuinematerial separation, or from failure of the cementitious matrix in somesteel fibre-reinforced concrete (SFR-concrete), representing crack bridging.In both cases, prior to the development of a displacement discontinuity, thematerial undergoes a phase of degradation which can be described by a con-tinuous approach. In this chapter, degradation in the continuum is describedby means of a rate-dependent elastoplastic-damage model in which rate-dependency is considered in the framework of Perzyna viscoplasticity. Thecoupling to damage is crucial to the subsequent introduction of a disconti-nuity as it (i) allows the necessary narrowing of the width of the localisationzone (converging to a discrete surface) which preludes the discontinuity inthe displacement field and (ii) allows full stress relaxation at the integrationpoint level response (see Section 1.3). This is an important aspect of the cou-pling to damage since in standard viscoplastic models [54, 137] full stressrelaxation is difficult to obtain due to the viscous contribution to the stressfield [179]. In describing the behaviour of SFR-concrete, the discontinuityis given cohesive forces to describe the crack bridging effect of the fibres.Further, the use of a coupled elastoplastic-damage model ensures a morerealistic representation of the behaviour of quasi-brittle materials since theassumption of unloading to the origin in a damage model is simplistic [17]and does not reflect experimental evidence [108].

An approach similar to the one presented in this chapter has been con-sidered by Wells et al. [194] who used traction-free discontinuities in a stan-dard Perzyna viscoplastic model. However, as already mentioned and aswill be illustrated next, the use of such a model does not represent a real-istic physical situation since a traction-free discontinuity is considered at astage in which the material (i.e. the integration point in a finite-element con-

∗ Based on References [172, 173, 177]

72 Chapter 4 Continuous-discontinuous failure in rate-dependent media

text) still has some residual stress. Note that in standard viscoplasticity it isproblematic to control the residual stress in the strain-softening constitutiverelationship when the localisation process is completed. If the simulation iscontrolled in displacements, a constant residual stress is usually obtainedwith constant stepping; conversely, in load control situations, it is even pos-sible to obtain hardening effects. With the coupling to damage, as proposednext, it is always possible to control the residual stress level.

After the detailed description of the continuum constitutive model, someaspects of the finite-element technology will be discussed (the kinematicsof the continuous-discontinuous model used in this chapter has been de-scribed in Chapter 2). Three applications in failure mechanics problems il-lustrate the performance of the approach. The case of inserting discontinu-ities in a differential version of a non-local continuum description [134] (asdescribed in Chapter 3) is qualitatively compared to the present approach.In Section 4.4 it is illustrated that the nature of the regularisation techniqueplays a major role in a continuous-discontinuous approach.

4.1 Rate-dependent elastoplastic-damage models

In small strain viscoplasticity, the strain tensorε is decomposed into an elas-tic εe and a viscoplastic εvp component according to (1.23). Consequently,the elastic stress-strain relationship (1.15)1 yields

σ = De: (ε−εvp) . (4.1)

The viscoplastic flow rule will be specified later for Perzyna and Duvaut-Lions viscoplasticity models—a general overview of the two viscoplasticityformats can be found in Jirasek and Bazant [84] while a detailed comparisoncan be found in Sluys [179] and Runesson et al. [157]. The coupling of iso-tropic damage and plasticity is introduced by adopting the effective stressconcept and the strain equivalence principle [96]. In such a framework, asimple and quite general algorithmic formulation, based on the operatorsplitting technique [42, 163, 164], can be derived. An alternative algorithmicformulation is based on the direct linearisation of all the relevant quantitiesas will be specified next. The two formulations are equivalent [42] and thelatter approach will be used for a Duvaut-Lions–type model while the for-mer for a Perzyna model. In the following derivations, the quantity refersto a rate-dependent homogenised quantity (like the quantity with the super-script vp), to a rate-dependent effective quantity, ˜ to a quasi-static (rate-

4.1 Rate-dependent elastoplastic-damage models 73

independent) effective quantity and to a quasi-static (rate-independent)homogenised quantity.

The algorithmic procedure for the coupled model hinges on the knowl-edge of the stress tensor, the algorithmic tangent moduli and the equivalentplastic strain, all defined in the effective space. Key point for the followingderivations is the expression of the rate-dependent effective stress tensor σas a function of (i) the rate dependent homogenised stress tensorσ , (ii) of theelastic strain tensor εe, or (iii) of the total strain tensor ε and the viscoplasticstrain tensor εvp:

σ =σ

1−ω = De:εe = De: (ε−εvp) , (4.2)

whereω is a scalar valued damage variable (0 ≤ ω ≤ 1). To preserve well-posedness of the governing equations when softening constitutive relation-ships are used, damage evolution must be postulated as some function ofa regularised monotonically increasing deformation history invariant κ. Inthis chapter, damage evolution is postulated as

ω (κ) =

0 if κ ≤ κ0α (1− exp (−βκ)) if κ > κ0,

(4.3)

with α and β model parameters and κ0 the threshold of damage initiation(note that damage is plastically induced).

4.1.1 Duvaut-Lions–type model

In the viscoplastic model proposed by Simo et al. [165]—this model isa modified version of the model originally proposed by Duvaut and Li-ons [54]—the viscoplastic flow rule is written in the form

εvp =1τ

Ce: (σ − σ) , (4.4)

where τ is the relaxation time, Ce = (De)−1 is the fourth-order elastic com-pliance tensor and σ indicates the projection ofσ on the static yield surface,i.e. the rate-independent stress. The rate of the hardening parameter κ isdefined in a similar fashion as

κ =1τ

(κ −κ) . (4.5)


A backward Euler time integration scheme gives

κn+1 =τ

τ + ∆tκn +

∆tτ + ∆t

κn+1. (4.6)

For the next derivations, it is useful to recall the expressions [116, 198],for the rate-independent case, of the differential of the incremental plasticmultiplier

d (∆λ) =fσ :R:d (∆ε)

fσ :R: fσ − fκκλ, (4.7)

with d (∆λ) = dλ, and of the fourth-order consistent tangent tensor

Dp = R− R: fσ ⊗ fσ :Rfσ :R: fσ − fκκλ

(4.8)

where

R =(I + ∆λDe fσσ

)−1 De, (4.9)

fσ = ∂ f /∂σ , fσσ = ∂ fσ/∂σ , fκ = ∂ f /∂κ, κλ = ∂κ/∂λ and I is the fourth-order identity tensor.

Stress update and algorithmic tangent

Since the following operations are performed in a rate-independent effec-tive stress space while the previous relations ((4.4) to (4.9)) were defined ina rate-independent homogenised stress space, it is necessary to replace thequantities with and with ˜.

The stress update is obtained by combining relations (4.2)1 and (4.2)3 toexpress the stress-strain relationship as [69]

σ = (1−ω) De: (ε−εvp) . (4.10)

For the following derivations, the viscoplastic strain rate

εvp =1τ

Ce: (σ − ˜σ) , (4.11)

formulated in the rate-independent effective stress space, can be expressedmore conveniently as a function ofσ and ˜σ :

εvp =1τ

Ce:(

σ

1−ω −˜σ)

. (4.12)


Time differentiation of (4.10), with substitution of εvp from the above ex-pression, gives the stress rate

σ +

(1τ

+ω

1−ω

)σ =

1−ωτ

˜σ + (1−ω) De:ε (4.13)

which can be integrated, replacing time derivatives by their discrete coun-terparts, using a backward Euler algorithm:

∆σ

∆t+

(1τ

+∆ω

∆t (1−ωn+1)

)σn+1 =

1−ωn+1

τ˜σn+1

+ (1−ωn+1) De:∆ε

∆t. (4.14)

Rearranging terms yields the incremental stress update relation

σn+1 =(σn + (1−ωn+1) De:∆ε) + ∆t

τ (1−ωn+1) ˜σn+1

1 + ∆tτ

+ ωn+1−ωn1−ωn+1

(4.15)

as a function of quantities from the previous step and from the rate-independent effective stress space. After the update of the effective rate-independent stress ˜σn+1 with a standard return mapping scheme in the ef-fective stress space, and the update of κn+1 in (4.6) and ofωn+1 in (4.3), theupdate ofσn+1 follows as mere function evaluations.

The algorithmic tangent operator Dpd is obtained from the linearised for-mat of the constitutive relationships at the end of the time step—note thatd (n) = 0 → d (n+1) = d (∆). To this end, (4.14) is expressed in aformat which is more convenient for the next derivations as

∆σ +∆tτσn+1 +

∆ω

1−ωn+1σn+1 −

∆tτ

(1−ωn+1) ˜σn+1

− (1−ωn+1) De:∆ε = 0. (4.16)

Differentiation of the above stress update relation requires the evaluation ofd (∆ω) which can be expressed, using (4.6), as

d (∆ω) =∂ωn+1

∂κn+1d (κn+1) =

∂ωn+1

∂κn+1

∆tτ + ∆t

d ( ˜κn+1) . (4.17)

From the relation between the plastic multiplier and the equivalent plasticstrain (d ˜κ = d ˜λ for the specific choice of the yield function, see Section 1.4),


after using the discrete form of (4.7) and employing the symmetry of ˜R,the variation of the damage increment can be related to the variation of thestrain tensor:

d (∆ω) =∂ωn+1

∂κn+1

∆tτ + ∆t

˜Rn+1: ˜fσ ,n+1˜fσ ,n+1: ˜Rn+1: ˜fσ ,n+1 − ˜fκ,n+1 ˜κλ,n+1

:d (∆ε)

= rn+1:d (∆ε) , (4.18)

where d (∆ω) has been expressed as a function of quantities in the effectiverate-independent space. After noting that

d(

∆ω

1−ωn+1σn+1

)=

1−ωn

1−ωn+1De:εe

n+1d (∆ω)

+ωn+1 −ωn

1−ωn+1d (∆σ) , (4.19)

where use has been made ofσn+1/(1−ωn+1) = De:εen+1, and that

d ((1−ωn+1) De:∆ε) = (1−ωn+1) De:d (∆ε)− (De:∆ε) d (∆ω) , (4.20)

the algorithmic tangent can be written as

Dpdn+1 =

∆tτ (1−ωn+1) ˜Dp

n+1 + (1−ωn+1) De − sn+1 ⊗ rn+1

1 + ∆tτ

+ ωn+1−ωn1−ωn+1

(4.21)

with

sn+1 =∆tτ

˜σn+1 +1−ωn

1−ωn+1De:εe

n+1 + De:∆ε. (4.22)

Note that the algorithmic tangent is not symmetric. The integration proce-dure requires only the evaluation of Dpd

n+1 after the computation of ˜Dpn+1 in

the effective stress space for the rate independent problem [46, 162].

4.1.2 Perzyna model

The rate-dependent isotropic elastoplastic-damage model described next isderived from the class of models proposed by Ju [87], where the notion ofoperator split [42] allows a very simple algorithmic treatment [87, 163, 164].


Stress update and algorithmic tangent

The stress update relation at the end of the time step follows from rela-tions (4.2)1:

σn+1 = (1−ωn+1) σn+1 (4.23)

where the damage value is updated through

ωn+1 = α (1− exp (−βκn+1)) , (4.24)

with the equivalent plastic strain in the effective space for the rate-dependent elastoplastic problem.

The fourth-order algorithmic tangent stiffness tensor Dpdn+1 is defined by

d (∆σ) = Dpdn+1:d (∆ε) for variations d (∆ε) of the current strain increment

∆ε. To derive the consistent tangent operator, (4.23) is differentiated at tn+1to obtain (dropping the subscript n + 1):

d (∆σ) = (1−ω) d (∆σ)− d (∆ω) σ . (4.25)

The variation d (∆ω) of the damage increment can be related to d (∆ε) by

d (∆ω) =∂ω∂κ

dκ =∂ω∂κ

dλ =∂ω∂κ

r:d (∆ε) , (4.26)

where r is a second-order tensor which depends on the plasticity modelin the effective stress space and which will be specified later. Substitutingd (∆ω) from the above expression in (4.25), and recalling that d (∆σ) =Dp:d (∆ε), yields the consistent tangent operator for the elastoplastic-damage model:

Dpd = (1−ω) Dp − ∂ω∂κσ ⊗ r. (4.27)

The rate-dependent response of the plasticity model in the effective stressspace is governed by the Perzyna viscoplastic model [137]. In presence ofplastic flow ( f ≥ 0, where f is the yield function in the effective stress space),the viscoplastic strain rate for the Perzyna model is expressed in the asso-ciative form according to

εvp =1τφ fσ , (4.28)


where the overstress function is given by the following power-law form

φ(

f)

=

(fσ0

)N

, (4.29)

with σ0 the initial yield stress and N (N ≥ 1) a real number. After stan-dard manipulations [47, 116], the algorithmic treatment of the constitutiveequations for Perzyna viscoplasticity in the effective stress space yields therelation for d (∆λ) as

d (∆λ) =fσ :R:d (∆ε)

fσ :R: fσ − fκκλ + τ/(∆tφ f

) , (4.30)

with d (∆λ) = dλ, the increment of the plastic multiplier, and the consistenttangent

Dp = R− R: fσ ⊗ fσ :R


) (4.31)

with R defined in (4.9). After employing the symmetry of R, the second-order tensor r (see (4.26)), required for the evaluation of the consistent tan-gent operator (4.27) for the elastoplastic-damage model, reads

r =R: fσ


) . (4.32)

The consistent tangent operator for the elastoplastic-damage model is read-ily available by direct substitution of the above expressions into (4.27). As forthe rate-dependent elastoplastic-damage model of the Duvaut-Lions–type,the consistent tangent operator in (4.27) is not symmetric. The step-by-stepintegration procedure is very similar to that of standard plasticity, the dif-ference being the presence of the damage update which requires only theevaluation of (4.23) and (4.27).

4.1.3 Influence of model parameters

The influence of the model parameters on the two rate-dependentelastoplastic-damage models is studied by considering the integration pointlevel response for the von Mises yield function by means of a one-element


0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

stre

ss [M

Pa]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

2.5

3

stre

ss [M

Pa]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

2.5

3

stre

ss [M

Pa]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

2.5

3

stre

ss [M

Pa]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

2.5

3

stre

ss [M

Pa]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

stre

ss [M

Pa]

α =1

α =0 β =0

β =100

τ =1 s

τ =0.0001 s

τ =1 s

τ =0.0001 s

a=2

a=−1

(e)

b=100

b=1

(f)

influence of softening law parameters

(a) (b)

influence of damage law parameters

(c) (d)

influence of relaxation time

Figure 4.1 Duvaut-Lions–type model, influence of the model parameters on the constitutiveresponse: effect of the damage law parameters (a)α withα = 0, 0.5, 1 and β = 100 and (b) βwith β =0, 1, 10, 100 and α = 1 on the rate-dependent elastoplastic-damage model (a = 1,b = 1, τ = 0.01 s); effect of the relaxation time τ with τ = 0.0001, 0.01, 1 s on the rate-dependent (c) elastoplastic model with a = α = β = 0, b = 1 and (d) elastoplastic-damagemodel with a = 0, b = 1, α = 1, β = 100; effect of softening law parameters (e) a witha = −1, 0, 1, 2 and b = 10 and (f) b with b = 1, 10, 100 and a = 0 on the rate-independent(τ = 0 s) elastoplastic model.


0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

st

ress

[MP

a]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

st

ress

[MP

a]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

st

ress

[MP

a]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

st

ress

[MP

a]

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

st

ress

[MP

a]

α =0

α =1 β =100

β =0

τ =0.001 s

τ =1000 s

τ =0.001 s

τ =1000 s

0 0.025 0.05 0.075 0.1displacement [mm]

0

0.5

1

1.5

2

st

ress

[MP

a]

influence of softening law parameters

a=2

a=−1

(e) (f)

b=100

b=1

(a) (b)

influence of damage law parameters

influence of relaxation time

(c) (d)

Figure 4.2 Perzyna model, influence of the model parameters on the constitutive response:effect of the damage law parameters (a) α with α = 0, 0.5, 1 and β = 100 and (b) β withβ =0, 1, 10, 100 andα = 1 on the rate-dependent elastoplastic-damage model (a = 1, b = 1,τ = 1 s); effect of the relaxation time τ with τ = 0.001, 1, 1000 s on the rate-dependent(c) elastoplastic model with a = α = β = 0, b = 1 and (d) elastoplastic-damage model witha = 0, b = 1,α = 1, β = 100; effect of softening law parameters (e) a with a = −1, 0, 1, 2 andb = 10 and (f) b with b = 1, 10, 100 and a = 0 on the rate-independent (τ = 0 s) elastoplasticmodel.


00 0.0250.025 0.050.05 0.0750.075 0.10.1displacement [mm]displacement [mm]

00

0.50.5

11

1.51.5

22

stre

ss [M

Pa]

stre

ss [M

Pa]

(a) (b)

Figure 4.3 Response of the model to a series of loading-unloading-reloading cycles com-pared to a monotonic load response for (a) Duvaut-Lions–type model with τ = 0.01 s and(b) Perzyna model with N = 1 and τ = 0.5 s.

test in displacement control on a 8-node quadrilateral element (element size1 mm × 1 mm). The element is subjected to monotonic linearly increas-ing uniaxial loading at constant strain rate (∆t = 0.0001 s until the finaldisplacement of 0.1 mm is reached in 1000 steps). The model parametersadopted are: Young’s modulus E = 100 MPa and Poisson’s ratio ν = 0. Thesoftening rule governing the cohesion capacity of the material is given anexponential form according to

σ (κ) = σ0 ((1 + a) exp (−bκ)− a exp (−2bκ)) , (4.33)

where a and b are model parameters, with the initial cohesion σ0 = 1 MPa.Note that the softening behaviour is represented both by damage and plas-ticity effects through (4.3) and (4.33), respectively. The results of the analy-ses are shown in Figures 4.1 and 4.2. The effective reduction of the residualstress due to damage as in Figures 4.1a and 4.1b and the effect of the relax-ation time τ reported in Figures 4.1c and 4.1d are worth noting. The effect ofthe softening rule parameters is depicted in Figures 4.1e and 4.1f. Analogousconsiderations hold for Figure 4.2. The constitutive response to a series ofloading-unloading-reloading cycles is reported in Figure 4.3 along with theresponse of the model to monotonic loading (model parameters are: a = 0,b = 1,α = 1, β = 100; relaxation time τ = 0.01 s and τ = 0.5 s with N = 1for the Duvaut-Lions–type model and the Perzyna model, respectively).


4.1.4 Regularisation properties

The regularisation properties of the models are demonstrated consideringa bar of length L = 100 mm, thickness t = 1 mm and width increas-ing from 8 mm at the restrained end to 10 mm at the free end (see Fig-ure 4.4). This geometrical configuration was chosen to avoid dependence ofthe width of the localisation zone on the finite-element discretisation [189].In the finite-element discretisations, the boundary conditions are prescribedrestraining both bottom node directions and vertical top node direction; themesh consists of one row of equally spaced elements. The bar is subjected tomonotonic tensile loading with constant average strain rate obtained by in-creasing the displacement at the free end linearly in time in 1000 steps with∆t = 0.0001 s until the final displacement of 0.1 mm. Von Mises yield func-tion with yield stress σ0 = 2 MPa is adopted. Other model parameters are:Young’s modulus E = 24000 MPa, Poisson’s ratio ν = 0, a = 0 and b = 100in the exponential softening law (4.33) andα = 1 and β = 300 for the expo-nential damage evolution law (4.3). Relaxation time is set to τ = 0.005 s forthe Duvaut-Lions–type model and to τ = 3 s with N = 1 for the Perzynamodel. These model parameters have been chosen for numerical conve-nience. Linear quadrilateral elements have been used. The results of thesimulations for different discretisations (20, 40, 80 and 160 equally spaced el-ements) have been reported in Figure 4.5 for the Duvaut-Lions–type modeland in Figure 4.6 for the Perzyna model together with the results for therate-dependent elastoplastic model (α = β = 0 in the exponential dam-age evolution law (4.3)). Close to failure, computations with the Duvaut-Lions–type model did not converge in the local Newton-Raphson iterationscheme for the computation of the plastic multiplier (a maximum of 50 localiteration was allowed) and the analyses were stopped (see Figure 4.5b). Ananalysis of the problem showed that the evaluation of the derivatives of theyield function was problematic due to an almost zero value of the yield func-tion for some integration points in the localisation zone (this situation wasfound to be related to the trial stress being of the same order of magnitudeas the stress increment De:∆ε). Similar problems were encountered for stan-

u_

100

8 10Figure 4.4 Geometry and boundaryconditions for the tensile test (all dimen-sions in mm).


0 0.01 0.02 0.03elongation [mm]

0

5

10

15

20re

actio

n fo

rce

[N]

20 el.

160 el.

20 − 160 el.

0.02 0.0225 0.025 0.0275 0.03elongation [mm]

0

1

2

3

4

reac

tion

forc

e [N

]

160 el.

20 el.

(a) (b)

Figure 4.5 Duvaut-Lions–type model: (a, b) load-displacement curves for 20, 40, 80 and160 element discretisations for the rate-dependent elastoplastic (dashed line) and the rate-dependent elastoplastic-damage (solid line) model; (b) close-up.

Figure 4.6 Perzyna model: load-displacement curves for 20, 40, 80 and160 element discretisations for the rate-dependent elastoplastic (dashed line) andthe rate-dependent elastoplastic-damage(solid line) model.

0 0.025 0.05 0.075 0.1elongation [mm]

0

5

10

15

20

reac

tion

forc

e [N

]

20 el.

160 el.

160 el.

20 el.

dard Duvaut-Lions viscoplasticity, i.e. without the coupling to damage. Inthe following, only the Perzyna model will be considered. The curves for thePerzyna model reported in Figure 4.6 show convergence to a unique solu-tion and the curves obtained with the rate-dependent elastoplastic-damagemodel clearly show mesh-dependence close to failure due to strain localisa-tion in one element (of course, this does not mean that there is a patholog-ical mesh dependence since the model is regularised, but indicates that themodel is able to reproduce a highly localised strain field). This is clearer fromthe stroboscopic evolution plot of the equivalent plastic strain κ reported inFigure 4.7 where strain localisation due to damage is evident. The effect ofthe viscous regularisation is evident from Figures 4.8–4.9 where a higherrelaxation time corresponds to a higher energy dissipation and a wider lo-calisation zone. In particular, it is worth noting the influence of the coupling


0 25 50 75 100distance from clamped end [mm]

0

0.5

1.0

1.5


0

0.5

1.0

1.5


0

0.5

0.1

1.5


0

0.5

1.0

1.5

κ [−

]x

100

κ [−

]x

100

κ [−

]x

100

κ [−

]x

100

(a)

20 el.

(c)

20 el.

elastoplastic model

elastoplastic−damage model(b)

160 el.

(d)

160 el.

Figure 4.7 Perzyna model: stroboscopic evolution of the equivalent plastic strain for therate-dependent (a, b) elastoplastic and (c, d) elastoplastic-damage model for (a, c) 20 and (b,d) 160 element discretisations.

between damage and plasticity on the equivalent plastic strain reported inFigure 4.9: it is evident that the coupling allows a more localised equivalentstrain profile compared to the plasticity model alone while leaving almostunchanged the width of the localisation zone. The load-displacement curvesand the damage profiles depicted in Figure 4.8 indicate that this model canproperly represent strain localisation at the global and local level throughfull stress relaxation and localised damage profile, respectively (see also Sec-tion 1.3 and Figure 3.3).


In this section some aspects of the finite-element implementation arediscussed—details on the integration scheme and on the transfer of historydata can be found in Section 2.5.

4.3 Applications 85

0 0.05 0.1 0.15 0.2 0 20elongation [mm]

40 60 80 100distance from clamped end [mm]

0

5

10

15

20re

actio

n fo

rce

[N]

0

0.2

0.4

0.6

0.8

1

dam

age

[−]

τ =3 s

τ =9 s

τ =3 s

τ =9 s

(a) (b)

Figure 4.8 Perzyna model: effect of the relaxation time on (a) the global response and on(b) damage profile at the end of the computation.

Introducing a discontinuity. The inclusion of displacement discontinu-ities represents, in this context, either genuine separation of material (i.e. astress-free discontinuity) or the formation of cohesive surfaces. In the formercase, a discontinuity is extended when damage at one or more integrationpoints in the element ahead of the discontinuity tip is above a critical value;in the latter case this condition has to be fulfilled at all the integration pointsin the element ahead of the discontinuity tip. The critical damage value,close to unity for quasi-brittle materials, will be specified in the applicationto SFR-concrete.

Orienting a discontinuity. The discontinuity direction is aligned with thedirection of maximum dissipation. This is achieved performing a non-localaveraging of the equivalent plastic strain, the quantity driving damage evo-lution, ahead of the discontinuity tip. The vector in the direction of the dis-continuity propagation is computed as in Section 3.5, replacing ei with κiin (3.24). The interaction radius for the Gaussian weight function in (2.26)is taken equal to three times the average element size ahead of the discon-tinuity tip. When the discontinuity is close to a boundary, the discontinuityextension direction is aligned with the previous discontinuity segment.


0 200 40 6020 8040 10060 80distance from clamped end [mm]

100distance from clamped end [mm]

0

0.005

0

0.01

0.005 κ [−

]

0.01

κ [−

]

τ =3 s

τ =9 s τ =9 s

τ =3 s

(b)(a)

Figure 4.9 Perzyna model: effect of the relaxation time on the equivalent plastic strain forthe rate-dependent (a) elastoplastic and (b) elastoplastic-damage model.

0

500

1000

1500

2000

0 0.5 1 1.5 2 2.5 3

appl

ied

load

[N]

u [mm]

Figure 4.10 Inclusion of discontinuities ina rate-dependent elastoplastic model: load-displacement response for a biaxial speci-men under tensile loading (adapted fromWells [192]).

4.3 Applications

The model for bulk degradation described in Section 4.1.2 is endowed withsome properties which make it a suitable tool for failure analyses. In contrastto standard rate-dependent elastoplastic models, characterised by a constantwidth of the localisation zone (see Figure 4.7b) and by a high residual stressdue to the viscous contribution, this rate-dependent elastoplastic-damagemodel allows the progressive narrowing of the localisation zone (see Fig-ure 4.7d) and full stress relaxation, which can be interpreted as a stress-freecrack in a continuous setting. If the regularised model is unable to describethe narrowing of the degraded zone, the inclusion of a traction-free discon-tinuity is problematic and its use should be avoided. An example of sucha model is the gradient elastoplasticity model proposed by Muhlhaus andAifantis [114] and Pamin [128] which is not able to ‘properly model com-

4.3 Applications 87

Figure 4.11 Load-cmod diagrams for thecompact-tension specimen for different meshresolutions (h indicates the average elementsize in the central part of the specimen).

0 1 2 3 4 5cmod [mm]

0

500

1000

1500

2000

F [N

]

experiment

h=2, 1, 0.5 mm

plete failure’ since gradient contributions make it ‘impossible to reach zerostress values’ (quoted from Engelen et al. [55]). A similar problem is to befound in classical viscoplasticity formulations (like e.g in Duvaut-Lions orPerzyna viscoplasticity) in which the stress-strain relationship presents ahorizontal plateau or even an increasing stress due to the viscous stresscontribution [179]. When traction-free discontinuities are considered in arate-dependent viscoplastic model, load-displacement curves exhibit a saw-tooth–like shape caused by high residual stresses which have not been dis-sipated [192, 194] (see Figure 4.10 and also Figures 3.6 and 3.16). In the fol-lowing, some applications to problems involving strain localisation will beanalysed.

4.3.1 Composite compact-tension specimen

The compact-tension specimen studied in Section 3.6.2 is now analysed con-sidering the model for bulk degradation described in Section 4.1.2. Thesimulations are performed using the same unstructured meshes of bilinearquadrilateral elements used in Section 3.6.2. The total cmod (5 mm) has beenapplied in 200 steps with ∆cmod = 0.025 mm. Plastic flow is described bya smoothed Rankine yield criterion [128] with σ0 = 35.2 MPa under theassumption of a plane stress situation. Softening parameters are: a = 1.5and b = 13 in (4.33) and α = 1 and β = 20 in (4.3). Other model parame-ters have been chosen as follows: N = 1 and relaxation time τ = 20 s forthe viscous regularisation of (4.28), Young’s modulus E = 3200 MPa andPoisson’s ratio ν = 0.28. Of the above model parameters, E, ν and σ0 havebeen adopted from Geers et al. [65]. The remaining parameters have beenchosen to ensure a qualitative fit at the global and local levels in terms ofload-cmod curve and damage profile, respectively, with the data reported


(a) (b)

0.1 1

ω

Figure 4.12 Failure state at cmod = 5 mm: damage contour plot for (a) coarse (h =2 mm) and (b) fine (h = 0.5 mm) mesh (the thick white line indicates the traction-freediscontinuity).

by Geers et al. [65]. To avoid damage growth in the elements around thepinholes, a higher value of σ0 has been given to elements in these areas. Thecritical damage value for discontinuity extension (traction-free) has been setto 0.99. An initial horizontal traction-free discontinuity has been placed inthe notch zone as starting point for the discontinuity extension. This hasno effect on the global/local response since the traction-free discontinuitymimics the real notch.

The load-cmod response is shown in Figure 4.11. The marked drop in theload-cmod curve for the coarsest of the discretisations (h = 2 mm) relates tothe high residual stress in the element at the moment of the first extensionof the discontinuity—only one integration points had damage values largerthan 0.99 while in the remaining integration points damage was around 0.7.The following discontinuity extensions are characterised by a smooth globalresponse as a consequence of comparable damage values at all integrationpoints of one element. The failure pattern at cmod = 5 mm is shown in Fig-ures 4.12 and 4.13. The continuous-discontinuous failure strategy provides aclear indication of the extent of the discontinuity and of the zone of continu-ous degradation. However, it is worth noting that no significant differenceshave been observed with respect to computations performed in a continu-ous framework with the finest of the discretisations in terms of local and

4.3 Applications 89

(a) (b)

Figure 4.13 Failure state at cmod = 5 mm: discontinuity extension for (a) coarse(h = 2 mm) and (b) fine (h = 0.5 mm) mesh.

global responses.A comparison of results obtained with the continuous-discontinuous im-

plicit gradient-enhanced continuum damage model described in Chapter 3and the continuous-discontinuous rate-dependent elastoplastic-damagemodel is reported in Figures 4.14 and 4.15. Both continuous-discontinuousregularised models are able to reproduce the experimental results with rea-sonable accuracy, as depicted in Figure 4.14. However, the comparison interms of failure evolution in Figure 4.15 clearly indicates that the gradient-enhancement produces a delayed discontinuity extension caused by theshift of the maximum of the quantity driving damage evolution (see Sec-tions 3.8 and 4.4 and Appendix E for more details on this issue).

4.3.2 Strip footing near a slope

A more interesting failure pattern can be observed from the numerical anal-ysis of the rigid and rough strip footing resting on the crest of a slope de-picted in Figure 4.16. A 0.6 m deep initial traction-free discontinuity is lo-cated on the right side of the footing. The domain has been discretised byusing 8-node quadrilateral elements with reduced integration scheme—930elements for the coarse mesh and 3720 elements for the fine mesh. Planestrain J2 softening plasticity with cohesive capacity according to (4.33) withinitial cohesion σ0 = 0.00005 MPa, a = 1 and b = 100 and with damagegrowth according to (4.3), with α = 0.9999 and β = 200, has been used


0 1 2 3 4 5cmod [mm]

0 1 2 3 4 5cmod [mm]

0

500

1000

1500

2000

F [N

]

0

500

1000

1500

2000

F [N

]

experiment

h=2, 1, 0.5 mm

experimenth=2, 1, 0.5 mm

(b)(a)

Figure 4.14 Comparison of load-cmod diagrams for the compact-tension specimen for dif-ferent mesh resolutions h for (a) implicit gradient-enhanced continuum damage model (seeChapter 3) and (b) rate-dependent elastoplastic-damage model.

to describe the material behaviour. The remaining model parameters havebeen chosen as follows: N = 1 and relaxation time τ = 1000 s, Young’smodulus E = 0.1 MPa (the strip is assumed to be elastic and one hundredtimes stiffer) and Poisson’s ratio ν = 0.2. The above model parameters havebeen chosen for numerical convenience but may be thought of as represent-ing the undrained behaviour of a heavily overconsolidated soil. The highvalue of the relaxation time allows a meaningful width of the localisationband. The rigid footing has been loaded in displacement control and thetotal vertical displacement u = 100 mm has been applied in 50 steps with∆u = 2 mm. The critical damage value for discontinuity extension (traction-free) has been set to 0.99.

The failure of the slope develops in the typical curved shape depictedin Figures 4.17b and 4.17d and is accompanied by localised deformationswhich are well described by the continuous model and by the followingdiscontinuity. The direction of the discontinuity is properly described evenwith the coarse mesh due to the significant extension of the equivalent plas-tic strain beyond the discontinuity tip. Note however that, due to the un-symmetric geometry of the slope stability problem, the alignment of thediscontinuity with the previous discontinuity segments close to the leftboundary is necessary in order to obtain a realistic failure pattern. Similar tothe simulations of the composite compact-tension specimen, no significantdifferences have been observed between a continuous and a continuous-discontinuous analysis in terms of local and global responses. The use of

4.3 Applications 91

cmod = 2.00 mm

cmod = 2.75 mm

cmod = 5.00 mm(a) (b)

0.1 1

ω

Figure 4.15 Evolution of the failure process for (a) implicit gradient-enhanced continuumdamage model (see Chapter 3) and (b) rate-dependent elastoplastic-damage model.


u =0y

u =

0x

5 5

12

10

u

0.6

Figure 4.16 Geometry and boundary conditions forthe slope (all dimensions in m).

traction-free discontinuities does not improve failure representation whenbulk degradation obeys the rate-dependent elastoplastic-damage model de-scribed in Section 4.1.2.

4.3.3 Steel fibre-reinforced concrete beam

Discontinuities can be better employed for the description of SFR-concrete.Numerical modelling of SFR-concrete is based on the use of stress-crackopening relationships [76]. Within the stress-crack opening approach, a rea-sonable approximation of the tensile behaviour of SFR-concrete can be ob-tained by using a bilinear relationship [99] of the type depicted in Fig-ure 4.18a. According to Stang and Olesen [184], ‘the first part of the bilin-ear relationship reflects a combination of the concrete contribution and theinitial fibre bridging action, while the second part reflects the fibre bridgingaction only.’ For normal strength concrete and low fibre content, the firstbranch of the bilinear relationship can be taken equal to that of plain con-crete, so that the effect of the fibres is reflected only by the second branch.The numerical strategy proposed for the modelling of SFR-concrete (see Fig-ure 1.4a) consists in the use of the model for bulk degradation, presented in

4.3 Applications 93

(a)

(c) (d)

(b)

Figure 4.17 Slope failure: (a, c) damage contour plot and discontinuity at failure and (b,d) deformed state (magnified 100 times) for different mesh resolutions (the thick white lineindicates the discontinuity).

Section 4.1.2, to describe, in a continuous fashion, the initial degradation ofthe cementitious matrix, represented by the first part of the curve in Fig-ure 4.18a. A cohesive discontinuity is introduced at a specified level ω ofdamage of the cementitious matrix, corresponding to a residual stress ftω(see Figures 4.18b and 4.19), and represents the second part of the curve inFigure 4.18a. The constitutive model for the cohesive discontinuity is de-fined by a linear decreasing relationship between the stress σ and the crack


uwmax

σf

(a)

σ

(b)

f t

κ

t

1

−

0.70.8

0.9ω =−

ω =

ω =−

Figure 4.18 Constitutive models: (a) bilinear constitutive relationship for SFR-concrete and (b) qualitative stress-equivalent plastic strain plot with critical dam-age values ω for the activation of the cohesive discontinuity.

uwmax

f t ω

σσf t

κcohesive discontinuitysoftening continuum

ω−

Figure 4.19 From continuous to discontinuous failure description in SRF-concrete(the dashed line indicates that the continuum close to the discontinuity experiencesincreasing dissipation after the introduction of a cohesive discontinuity).

opening w with σ = ftω at w = 0 and σ = 0 for w ≥ wmax where wmax isrelated to the fibre length.

A notched beam in four point bending with a 150 mm× 150 mm cross sec-tion, a 550 mm span and a 25 mm notch is analysed. The load is applied viadisplacement control at a distance of 75 mm from the midspan and the totaldeflection v (0.4 mm) has been applied in 160 steps with ∆v = 0.0025 mm.The beam was discretised with bilinear quadrilateral elements with elementsize, in the central part of the beam, of 2.73 mm × 3.125 mm. Bulk degra-dation is described by plane stress smoothed Rankine plasticity [128] withft = σ0 = 2 MPa and with softening parameters a = 0, b = 200, α = 0.99and β = 400. The other model parameters have been chosen as follows:N = 1 and relaxation time τ = 20 s, Young’s modulus E = 30 GPa andPoisson’s ratio ν = 0.2. The notch has been described by a traction-freediscontinuity. SFR-concrete is characterised by the extension of a cohesivediscontinuity related to the fibre action at three different damage values,ω = 0.7, 0.8 and 0.9, from the traction-free notch—note that the cohesive

4.3 Applications 95

0 0.1 0.2 0.3 0.4deflection [mm]

plain concrete

SFR−concrete

0

5

10

15

20

load

[KN

]

0.70.8

0.9ω =−

ω =−ω =−

Figure 4.20 Load-deflection curves forplain and SFR-concrete.

ω

(a) (b)

0.1 1

Figure 4.21 Damage contour plot for(a) plain and (b) SFR-concrete (ω = 0.8) atdeflection = 0.4 mm.

discontinuity is extended if all integration points reach the critical damagevalue ω. For plain concrete, a traction-free discontinuity was extended atωcrit = ω = 0.989 from the traction-free notch.

The values of ftω at which damage reaches the above values are depictedin Figure 4.18b in the qualitative plot of the first principal stress σ1 versusthe equivalent plastic strain κ for an integration point in the element aheadof the notch. The value of wmax was chosen equal to 25 mm for all the sim-ulations. Figures 4.20 and 4.21 clearly indicate the effect of the cohesive dis-continuity on the global and local response, respectively. Figure 4.21 indi-cates that the use of a cohesive discontinuity promotes dissipation acrossthe discontinuity surface as well as in the continuum. A more ductile re-sponse, which corresponds to a wider zone of high damage concentration,is obtained with a cohesive discontinuity (see Figure 4.21b) compared tothe response obtained with a traction-free discontinuity (see Figure 4.21a),where high damage values are concentrated only close to the discontinuity.

The damage values for the activation of the fibre and the shape of the co-hesive relationship have been chosen for numerical convenience—they canbe related e.g. to the strength of the cementitious matrix. The aim of this aca-demic example was to illustrate how changes at integration point level (un-derstood as changes at the material level) influence the structural responsein a regularised continuous-discontinuous model. Note that this approachis not feasible in a non-local continuum because of the severe oscillations inthe quantity governing damage growth (see Section 3.8).


4.4 Traction-free discontinuities in rate-dependent andnon-local media

It is interesting to further compare the performance of this rate-dependentcontinuous-discontinuous model to the one obtained when discontinuitiesare considered in a non-local medium (see Chapter 3). To this end, numeri-cal analyses similar to those performed with the gradient-enhanced damagemodel in Section 3.8 have been performed with the rate-dependent model.Although some of the following issues have already been touched uponin Section 3.8, they are recalled here to draw a parallel between the useof traction-free discontinuities in gradient-enhanced and in rate-dependentmedia. In the examples, the model parameters for the viscous regularisationare similar to those used in Section 4.3.1. A finer discretisation, compared tothe one used in Section 3.8, has been employed in the numerical simulations.

An important conclusion can be drawn from the results of the simulationsof the compact-tension test, shown in Figures 4.22 and 4.23, and it is relatedto the actual meaning of the inclusion of discontinuities (of any nature, co-hesive or traction-free), which are local in space, in problem fields whichare, by definition, non-local. As already discussed in Section 3.8 and as de-picted in Figure 4.22a, for the analyses performed with the implicit gradient-enhanced continuum damage model [134], of all the basic problem fields(vertical displacement field uy and non-local equivalent strain field e), onlythe non-local strain field suffered from severe oscillations upon a traction-free discontinuity extension. Analogous analyses depicted in Figure 4.22band performed with the rate-dependent model did not show oscillations inthe equivalent plastic strain κ. This can be considered as a consequence ofthe nature of the regularisation employed which is non-local in space for theimplicit gradient-enhanced continuum damage model and local in space forthis rate-dependent elastoplastic-damage.

The examples reported in Section 3.8 showed that the use of discontinu-ities in a gradient-enhanced medium can be problematic due to the signifi-cant changes in failure initiation and characterisation (see also Appendix E).When a sharp crack is considered in a non-local medium, damage initia-tion is predicted ahead of the crack tip and not at the crack tip itself. Thisshift in the location of damage initiation is proportional to the length scaleparameter. Numerical evidence showed that the shift of the maximum ofthe non-local equivalent strain away from the crack tip is even more pro-nounced in the non-linear regime than it is in the elastic regime. The con-

4.4 Traction-free discontinuities in rate-dependent and non-local media 97

pseudo−time0

0.2

0.4

0.6

0.8

uy

uy

a

1e κ

scal

ed s

olut

ion

arra

y

pseudo−time0

0.2

0.4

0.6

0.8

1

scal

ed s

olut

ion

arra

y

(a) (b)

Figure 4.22 Pseudo-time evolution of (a) the vertical displacement field and thedissipation-driving quantity e for implicit gradient-enhanced continuum damage model(see Chapter 3) and of (b) the vertical displacement field and κ for the rate-dependentelastoplastic-damage model in a compact tension test for point a (the discontinuity prop-agates from the notch along the dotted line; the dots represent the extension of the dis-continuity and the dotted line indicates the first moment at which the point is behind thediscontinuity tip).

0 1 2 3 4displacement [mm]

0

100

200

300

reac

tion

forc

e [N

]

0 1 2 3 4displacement [mm]

0

100

200

300

reac

tion

forc

e [N

]

(a) (b)

Figure 4.23 Load-displacement curves for (a) implicit gradient-enhanced continuumdamage model (see Chapter 3) and (b) rate-dependent elastoplastic-damage model with apropagating discontinuity (the dots represent the extension of the discontinuity; note thatthe extension is performed through several elements at a time for the gradient-enhancedmodel and through one element at a time for the rate dependent model).

sequence is that, for a reasonably fine discretisation, several elements arecrossed at a time and the discontinuity extension can be delayed as reportedin Figure 4.15a. The oscillations of the dissipation driving quantity are morepronounced by the release of more than one element at a time. This causes a


00 10 2010 3020 4030 5040 6050 60position [mm]position [mm]

(b)(a)

0

0.05

0.1

0.15

0

κ [−

]

0.5

1

1.5

2κ

[−]

crack tipinitial initialcrack tip

propagating disc.fixed disc.

propagating disc.

Figure 4.24 Profile of the equivalent plastic strain κ for the rate-dependent elastoplastic-damage model: (a) comparison of profiles with fixed and propagating discontinuity and(b) close-up of the profile in case of propagating discontinuity (profiles plotted close tofailure).

situation of artificial shock-wise crack propagation and of spurious unload-ing in the points ahead of the newly extended discontinuity tip resultingin the bumps in the load-displacement curve as depicted in Figure 4.23a.For the rate-dependent elastoplastic-damage model (plane stress Rankine),some numerical evidence suggests that the dissipation-driving field quan-tity is maximum, at the beginning of the dissipation process, at the crack tip.Mesh refinement studies also indicated that the energy dissipated during aload process converges to a non-zero value. From here it can be concludedthat the dissipation-driving field quantity converges to a finite value at thecrack tip (these conclusions are based on numerical considerations and, assuch, must be validated by analytical studies). When the rate-dependentmodel is considered, usually one element at a time is crossed by the dis-continuity and the load-displacement curves do not show bumps (see Fig-ure 4.23b).


A discontinuous problem field interpolation is intended to give a better fail-ure representation. The justification for the use of displacement discontinu-ities in the rate-dependent elastoplastic-damage model stems from a betterinterpretation of some field quantities which gain a clearer physical mean-ing. With reference to the example reported in Section 4.4, when consider-


ing the evolution of the equivalent plastic strain κ ahead of the disconti-nuity tip as depicted in Figure 4.24, the propagating discontinuity avoidsthe artificial growth of the equivalent plastic strain which, in a continuoussetting, is the response of the model to strain localisation. When a traction-free discontinuity is considered, this is the only observable difference whenthe responses of the continuous and the continuous-discontinuous modelsare compared (load-displacement curves and damage profiles are identical).Conversely, when a cohesive discontinuity is considered, as was presentedin Section 4.3.3, the continuous-discontinuous strategy proved to be effec-tive in the qualitative description of SFR-concrete.

Unlike standard regularised models in which failure description can beproblematic close to complete failure (e.g. difficulties in reaching full stressrelaxation in rate- [179] and gradient-dependent [55] media and spuriousdamage growth in gradient-dependent media [64]), the continuum modelalone is a valuable tool for the description of strain localisation phenomena.Further, its combination to a technique in which displacement discontinu-ities are considered can enhance failure representations in a wide range ofapplications. It is stressed that the continuous-discontinuous strategy de-scribed in this chapter is intended, at this stage, as a tool to the description ofphenomena in which a displacement discontinuity arises as a consequenceof strain localisation. More insight into the determination of model param-eters, using the technique described e.g. in References [44, 91], is needed totransform this approach from a descriptive tool into a predictive tool.

Chapter 5

Conclusions

The design of safe structures is intimately linked to the understand-ing of failure processes of engineering materials. Only in exceptionalcircumstances—e.g. in building demolition—failure is a desirable event; inall other cases, failure must be delayed or, if possible, prevented. The roleof the analyst is that of providing practitioners with reliable tools for fail-ure analysis so that failure processes can be prevented or controlled. In re-cent times, a host of constitutive models and computational tools have beendeveloped to assist practitioners in the analysis of (quasi-)brittle and duc-tile fracture. Amongst all these models and tools, damage and plasticitytheories and continuous and discontinuous analyses have emerged, respec-tively. Damage and plasticity theories as well as continuous and discontinu-ous analyses indicate relative situations; whether a particular failure processis better represented within a damage or a plasticity framework through acontinuous or a discontinuous analysis depends on the situation. In mostcases, a ‘realistic’ failure representation should consider a coupled damage-plasticity model in a combined continuous-discontinuous failure represen-tation, rather than a specific constitutive model in a continuous or discon-tinuous framework.

The development of computational strategies to combine techniques ofcontinuous and discontinuous numerical failure analysis was the objectiveof this thesis. A continuous-discontinuous approach to failure is advocatedas a way to improve classical continuous failure representation by (i) allow-ing a different interpretation of model parameters (see Chapters 1 and 3),(ii) enabling a more realistic failure representation by solving some prob-lems inherent to some regularised models (see Chapter 3), and (iii) addingflexibility to continuous modelling alone (see Chapters 2 and 4). Besidesthe applications presented in this thesis, the superiority of a continuous-discontinuous approach to failure, compared to a conventional continuousapproach, is evident. The particular choice of partition of unity-based dis-continuous elements as the tool for the discontinuous enhancement in thecontinuous-discontinuous failure approach was instrumental in (i) provid-

102 Chapter 5 Conclusions

ing a mechanically sound approach when compared to embedded disconti-nuity element techniques, and (ii) in providing a clear framework in whichthe continuous-discontinuous analysis is cast when compared to the use ofconventional interface elements coupled to remeshing and adaptive tech-niques. The only assumption behind the application of this method is thedecomposition of the displacement field into the sum of a continuous and adiscontinuous field.

However, the simplicity of the partition of unity-based discontinuous el-ement concept is counterbalanced by the difficulties related to its imple-mentation which requires heavy modifications to element routines and ac-cess to the whole code (which is not an option in commercial finite-elementmethod packages). Partition of unity-based discontinuous elements are notthe panacea for all the problems involving discrete cracking. Indeed, parti-tion of unity-based discontinuous elements are best suited to describe majorcracks. When the focus is on the description of distributed failure in a dis-continuous setting, conventional interface elements or embedded discon-tinuity elements, within a cohesive-zone approach, should be used. Whenpartition of unity-discontinuous elements are used, there should be an ex-tra set of degrees of freedom defined for each crack; although possible inprinciple, the use of many extra set of degrees of freedoms is problematic tohandle, expecially in the case of intersecting discontinuities, and should beavoided.

The main conclusions of this study can be summarised as follows:

1. model parameters should be reassessed when continuous(-discontin-uous) failure descriptions are considered (see Sections 1.1 and 3.6.1and Appendix C);

2. partition of unity-based discontinuous elements represent no generalremedy against the shortcomings of conventional interface elements(only in terms of stress oscillations) and their use should not be consid-ered in combination to a ‘dummy stiffness’ approach (see Section 2.6.2and Appendix A);

3. although the use of discontinuities helps in avoiding spurious damagegrowth in gradient-enhanced media (see Chapter 3), discontinuitiesand non-locality do not get along well (see Sections 3.8 and 4.4);

4. Duvaut-Lions viscoplasticity is poorly suited for the description ofsoftening media close to failure (see Section 4.1.4);

Conclusions 103

5. compared to the gradient-enhanced damage model used described inChapter 3, the rate-dependent elastoplastic-damage model describedin Chapter 4 can be used with coarser meshes to adequately representstrain localisation (see Section 4.3);

6. the continuous-discontinuous rate-dependent elastoplastic-damagemodel described in Chapter 4 results in the most realistic and com-putationally effective description of failure;

7. the underlying regularised continuum determines the quality of thediscontinuous enhancement (see Section 4.3.1);

8. gradient models derived from non-local formulations can be classifiedas coupled problems, not requiring mixed finite-element formulationsin the classical sense (see Appendix D), and

9. constitutive models based on a non-local dissipation-driving variablemay lead to incorrect failure initiation and propagation in arbitraryloading scenarios due to a fundamental flaw in damage characterisa-tion (see Appendix E).

Appendix A

Conventional interface and PU-based discontinuouselements∗

The performance of PU-based discontinuous elements deteriorates whenthese elements are used with a ‘dummy stiffness’ approach because of theincorrect representation of the traction profile along the crack line (see Sec-tion 2.6.2). When oscillations in the traction profile are present, the non-linear interface response cannot be activated correctly.

An examination of the interface contribution to the stiffness matrix of PU-based discontinuous elements through the conventional interface elementperspective [160] may provide clarity on their limitation and cast some lighttowards possible improvements. In what follows, it is argued that the causeof the problem could be traced back to the contribution of the displacementdiscontinuity to the element stiffness matrix. A comparison between planeconventional and PU-based discontinuous elements is elaborated, and it isshown that the two methodologies share the same structure of the part ofthe stiffness matrix responsible for interface behaviour. The effect of differ-ent integration schemes on conventional continuous interface elements andon PU-based discontinuous elements is investigated and their influence onthe structure of the stiffness matrix of PU-based discontinuous elements isillustrated.

A.1 Conventional continuous interface element

The stiffness matrix of a conventional isoparametric m-node line interfaceelement is expressed as

K = bξ=+1∫

ξ=−1

BTCB∂x∂ξ

dξ , (A.1)

∗ Based on References [168, 171]

106 Appendix A Conventional interface and PU-based discontinuous elements

where b is the width of the interface and B is the 2× 2m matrix

B =

[ −N N 0 00 0 −N N

](A.2)

containing isoparametric shape functions Ni with

N =[

N1, ..., Nm]

. (A.3)

The interface constitutive matrix C is obtained by the constitutive matrixin the local s,n system after the transformation C = RTsnRT, with R therotation matrix from the the local frame into the global frame of reference,

R =

[cosαxs −sinαxssinαxs cosαxs

], (A.4)

and Tsn a traction-separation relation of the type

Tsn =

[ds 00 dn

], (A.5)

where ds and dn are constant. For the element depicted in Figure A.1, the lo-cal and global frames of reference coincides, therefore C = Tsn. The degreesof freedom have been ordered in the sequence

[u 1

n , ..., u mn u 1

s , ..., u ms]

according to the node numbering of Figure A.1. For a conventional linearline interface element the N matrix is equal to

N =[ 1

2 (1−ξ) 12 (1 +ξ)

](A.6)

while for a conventional quadratic line interface elements it reads

N =[ 1

2

(−ξ +ξ2) 1

2

(ξ +ξ2) (

1−ξ2) ] . (A.7)

Expansion of the term BTCB results in the block diagonal matrix

BTCB =

[Kn 00 Ks

], (A.8)

where Ki is given by

Ki = di

[K −K−K K

](A.9)

A.1 Conventional continuous interface element 107

ξ

x

y

5

23

6

1

4s

n

Figure A.1 Conventional 6-node line interface element.

and di is the stiffness at the interface in the direction i. The structure of thesub-matrix K depends on the interpolation along the conventional interfaceelement. For line elements it results in

K =

[N2

1 N1N2

N1N2 N22

]. (A.10)

Analytical integration (and a 2–point Gauss integration scheme) of the termsNiN j in (A.10) results in a full matrix while nodal integration (2–point closedNewton-Cotes formula i.e. trapezoidal rule) removes the coupling termsNiN j with i 6= j. For a quadratic line interface element the submatrix Kreads

K =

N21 N1N2 N1N3

N2N1 N22 N2N3

N3N1 N3N2 N23

, (A.11)

and the use of analytical integration (and a 3–point Gauss integrationscheme) results in a full matrix while the coupling terms NiN j are deletedfor i 6= j using again nodal integration (3–point Newton-Cotes formula i.e.Simpson’s rule). In other words, the terms responsible for the coupling be-tween the degrees of freedom are deleted if a nodal integration scheme isused.


(a) (b) (c) (d) (e)

34

1

4

1

2

3 3

2

4

1 1

4

3

2

1

43

2

2

Figure A.2 Possible configurations for a horizontal discontinuity crossing a PU-baseddiscontinuous linear quadrilateral element.

A.2 Partition of unity-based discontinuous elements

For the two-dimensional m-node PU-based discontinuous element depictedin Figure A.2, where m = 4, the contribution of Kbb (see (2.24)) on Γd reads

Kbb,Γd =∫

Γd

NTRTsnRTN dΓ = bξ=+1∫

ξ=−1

NTTsnN∂x∂ξ

dξ , (A.12)

where N is the 2× 2m matrix

N =

[N 00 N

](A.13)

containing the isoparametric shape functions of the element with

N =[

N1, ..., Nm]

. (A.14)

The sequence[

u 1x , ..., u m

x u 1y , ..., u m

y

], with the corner nodes in the first

m positions and the interior nodes in the remaining positions, has been usedfor the ordering of the degrees of freedom in (A.13). Expansion of the inte-grand in (A.12) results in the block diagonal matrix

NTTsnN =

[Kn 00 Ks

], (A.15)

where Ki = diK and K is given by

K =

N21 N1N2 . . . N1Nm

N2N1 N22 . . . N2Nm

......

. . ....

NmN1 NmN2 . . . N2m

. (A.16)

A.2 Partition of unity-based discontinuous elements 109

0.5000 0.5000

0.1667 0.1667

(a)

0.5000 0.5000

0.50000.5000

(b)

Figure A.3 Eigenmodes and corresponding eigenvalues for a linear PU-based discontin-uous element with discontinuity placed along an element side for 2–point (a) Gauss and(b) trapezoidal integration rule.

It is worth to stress the similarity between matrix K in (A.16) and matrix Kin (A.9) defined for conventional line interface elements.

To illustrate the significance of the coupling between degrees of freedomin a PU-based discontinuous element, a linear quadrilateral element (m = 4)is analysed next. In the case of Figure A.2a and with reference to matrix Kin (A.16), the use of a trapezoidal rule will activate the interaction betweennodes 1 and 4 and between nodes 2 and 3 only (‘natural’ coupling), pre-venting, for instance, the coupling between node 1 and node 2 (‘pathologi-cal’ coupling); more specifically, the terms N1N2, N1N3, N2N1, N2N4, N3N1,N3N4, N4N2 and N4N3 cancel. When a discontinuity is placed similar to theone depicted in Figure A.2b and a 2-point Gauss integration scheme is con-sidered, the coupling is understood to be between node 2 and the remainingnodes. By using a trapezoidal rule, the pathological coupling is only par-tially reduced (see the lower part of Figure 2.13 where the oscillations areless pronounced than in the lower part of Figure 2.12) but it is still presentsince the terms N1N2, N2N1, N2N3 and N3N2 do not cancel. The cases ofFigures A.2c and A.2d fall under the the same category of Figure A.2a (i.e.pathological coupling disappears with a trapezoidal rule). In the limit case


of a discontinuity placed along an element side (see Figure A.2e), matrix Nin (A.12) reduces to a 2× 4 matrix containing the isoparametric shape func-tions of the element related to the side on which the discontinuity lies (i.e.N1 and N2 for side 12 in Figure A.2e) and matrix K in (A.16) is identical tomatrix K of a conventional line interface element as defined in (A.9). Thisidentity links the two approaches since conventional interface elements andPU-based discontinuous elements retain the same block diagonal structurefor the terms related to the discontinuity.

The previous observations on the coupling of degrees of freedom are con-firmed by an eigenvalue analysis performed on the part

∫Γd

NTTN dΓ of thesub-matrix Kbb (see (2.24)) which directly contributes to the stiffness of thediscontinuity. The interpretation of the eigenvalue analysis is straightfor-ward and clearly indicates that the reason of the oscillating behaviour couldbe indeed rooted in the pathological coupling between degrees of freedom.Unit values for the length, the surface area and the dummy stiffnesses ds anddn have been assumed. The results of the eigenvalue analyses are shownin Figures A.3 and A.4 for a PU-based discontinuous element with a dis-continuity placed along the right vertical side. The pathological couplingof the nodal displacements is evident when a Gauss integration scheme isused; on the contrary, the pathological coupling disappears with nodal in-tegration scheme. These results are similar to those reported by Rots andSchellekens [155]. Consequently, with two-dimensional linear and quadraticPU-based discontinuous elements and in the particular situation of a dis-continuity placed along an element side, only the use of nodal integrationscheme for the integration of the terms related to the discontinuity Γd guar-antees a smooth traction profile for all the values of the dummy stiffness.Analogous analyses have been performed on a wide range of element anddiscontinuity configurations and the results are reported in Table A.1. Theanalysis for the 8–node quadrilateral element (marked with an asterisk intable A.1) shows pathological coupling but the results reported in the up-per part of Figure 2.13 do not show stress oscillations. This can be explainedby further decomposition of the matrix K (see (A.16)) into a matrix whichcontains the contribution to the stiffness matrix of the integration points lo-cated along the element sides and a matrix which contains the contributionto the stiffness matrix of the integration point located in the centre of thediscontinuity. Along the line of reasoning pursued by Schellekens and deBorst [160], it can be shown that the pathological coupling is introduced bythe integration point located in the centre of the discontinuity and that, only

A.3 Concluding remarks 111

Table A.1 Appearance of pathological coupling between degrees of freedom for pos-sible configurations between a discontinuity and a plane element (YES indicates theappearance of coupling; the asterisk indicates a particular situation which is explainedon page 110).

Gauss (3 pts)

Simpson

Gauss (2 pts)

Trapezoidal

Yes Yes Yes Yes

Yes Yes NoYes *

Yes Yes Yes Yes

Yes NoNo No

Gauss (2 pts)

Trapezoidal

Gauss (3 pts)

Simpson

Yes Yes No

YesYesYes

Yes Yes Yes

Yes Yes No

for this special configuration, the element behaves like if no pathologicalcoupling exists. To summarise, the use of nodal integration is effective indeleting the pathological coupling between degrees of freedom and in pre-venting the appearance of oscillations in the stress profile only in particularsituations.


(b)

1.6667

1.6667

1.6667 1.6667

6.6667

(a)

0.8037

1.6667

0.8037

5.5296

1.6667

5.5296

6.6667

Figure A.4 Eigenmodes and corresponding eigenvalues for a quadratic PU-based discontinuous element with discontinuity placed along an element sidefor 3–point (a) Gauss and (b) Simpson’s integration rule.

A.3 Concluding remarks 113

A.3 Concluding remarks

When partition of unity-based discontinuous elements are used with a‘dummy stiffness’ approach, similar to conventional interface elements,their performance is severely affected by the position of the discontinuity.Spurious oscillations in the traction profile that appear when a ‘dummystiffness’ approach is considered (see Chapter 2), are linked to the appear-ance of pathological coupling between degrees of freedom in the part of thestiffness matrix responsible for the ‘interface’ behaviour. It was illustratedthat the pathological coupling between degrees of freedom can be elimi-nated only in few circumstances. Therefore, partition of unity-based discon-tinuous elements represent no general remedy against the shortcomings ofconventional interface elements and their use should not be considered incombination to a ‘dummy stiffness’ approach.

Appendix B

Some essentials of generalised functions

Some relations involving the Dirac-delta function δΓd (x) and the HeavisidefunctionHΓd (x) are recalled [73, 74, 183] and derived. The body Ω, depictedin Figure B.1, is crossed by a discontinuity surface Γd which divides the bodyinto two sub-domains, Ω+ and Ω− (Ω = Ω+ ∪Ω−). The Dirac-delta func-tion is centred at the discontinuity surface Γd and the Heaviside functionis discontinuous across the discontinuity surface Γd (HΓd = 1 if x ∈ Ω+,HΓd = 0 if x ∈ Ω−). A regular functionφ(x) is defined in the domainΩ andm is the inward unit normal to Ω+. The following relations hold betweenthe continuous functionφ and the Dirac-delta function δΓd :

∫

ΩδΓdφ dΩ =

∫

Γd

φ dΓ , (B.1)

∫

Ω(∇δΓd )φ dΩ = −

∫

Γd

∇φ dΓ . (B.2)

The directional derivative of the function φ in the direction of a genericunit vector v is defined as Dvφ = ∇φ·v. All the properties of standard

−Ω

+Ω

dΓ

m

Figure B.1 Body Ω crossed by a discontinuity Γd: definition of Ω+ and Ω−.

116 Appendix B Some essentials of generalised functions

derivation hold and∫

Ω(∇δΓd ·v)φ dΩ =

∫

ΩDvδΓdφ dΩ = −

∫

Γd

Dvφ dΓ

= −∫

Γd

∇φ·v dΓ . (B.3)

The last term in (B.3) is non zero only if the vector v is not perpendicular to∇φ. The gradient of the Heaviside function, with reference to Figure B.1, isdefined as [73]

∇HΓd = δΓd m (B.4)

while the Laplacian is derived as

∇2HΓd = ∇· (∇HΓd ) = ∇· (δΓd m) = δΓd∇·m +∇δΓd ·m = ∇δΓd ·m. (B.5)

The Laplacian of the discontinuous scalar functionHΓdφ then yields

∇2 (HΓdφ) = HΓd∇2φ+φ∇δΓd ·m + 2δΓd∇φ·m. (B.6)

Note that the function φ must be continuously differentiable in (B.2) andtwice continuously differentiable in (B.6)

Appendix C

Constitutive models for softening materials

The phenomenological behaviour of many engineering materials is charac-terised by a phenomenon known as softening: when a specimen is loadedbeyond its elastic limit, an increase of load-carrying capacity is still possi-ble up to a certain level of increasing deformation; upon further loadingthe load-carrying capacity is progressively lost until failure (this last phe-nomenon is referred to as softening [108]; see Figure C.1a).

In a continuous setting, softening behaviour can be reproduced by con-sidering strain-softening constitutive relationships such as the one depictedin Figure C.1b. However, when a standard inviscid continuum modelis equipped with strain-softening constitutive relationships its numeri-cal characterisation is not unique as numerical solutions of the boundaryvalue problem suffer from a severe dependence on the spatial discretisa-tion [14, 16, 139]. Mesh-dependent results are caused by a change in type ofthe governing boundary value problem in the process zone—the problembecomes ill-posed: elliptic equations for quasi-static problems become hy-perbolic [36] and hyperbolic equations in wave propagation problems withclassical continuum models become elliptic [95]. Nevertheless, in the frame-work of the existing phenomenological description of material behaviour,it is possible to objectively characterise localisation phenomena through theinclusion, implicitly or explicitly, of a length scale into the field equationsor the material description (models making use of a length scale are usu-ally known as ‘regularised’ models or as ‘localisation limiters’). Commonapproaches are indicated below.

– Fracture energy-based approaches: the softening portion of the stress-strain curve is adjusted according to the element size to ensure properinternal energy dissipation during the failure process [19, 138, 140, 151,153, 154].

– Rate-dependent models: these models are based on the inclusion of rate-dependence in the constitutive equations [53, 87, 101, 118, 163, 164,179, 180, 189, 190]—it could be artificial viscosity or material rate-

118 Appendix C Constitutive models for softening materials

strain

stre

ss

(b)

0 0.05 0.1 0.15 0.2cmsd [mm]

0

10

20

30

40

50

P [k

N]

experiment

(a)

Figure C.1 Modelling structural softening: (a) envelop of load-displacement curves(adapted from Schlangen [161]) and (b) softening stress-strain constitutive relationship.

dependence such as e.g. Perzyna [136, 137] or Duvaut-Lions [54] vis-coplasticity.

– Higher-order continuum models:

- micro-polar continuum models: additional degrees of freedom,micro-rotations [45], are added to the continuum description al-lowing the strains to localise in a band of finite size, dictated byan internal length scale [31, 35, 50, 113, 115, 179];

- gradient methods: the stresses depend on strain gradients [1, 2, 33,95, 114, 128, 179, 182];

- non-local continuum models: the stresses at a point depend on thestrains in a non-vanishing region around the point itself (math-ematically this is similar to gradient methods above) [15, 17, 18,21, 134, 139, 141].

Although there is no commonly accepted way of treating strain localisa-tion problems, their correct characterisation is of prime interest since strainlocalisation can be usefully interpreted as a failed region. It is therefore im-portant to understand the limitations of current approaches for the descrip-tion of failure. To this end, some considerations on numerical modelling ofconcrete structures are reported next while a non-local and a rate-dependentmodel are contrasted in Section C.2.

C.1 Considerations on numerical modelling of concrete 119

Figure C.2 Schematic representation of materialknowledge when defining a micromechanical consti-tutive model for fibre-reinforced concrete (note thatgeometrical and mechanical properties can be definedfor each of the material constituents).

fibre

fibre

aggregateaggregate

cement paste

cement paste

xx

x?

?

C.1 Considerations on numerical modelling of concrete

There are different opinions regarding the strategy to choose when a con-stitutive model is used. Basically one could consider micromechanical ormacromechanical approaches. Both procedures require a certain degree ofhomogenisation which is considered at the end of the analysis in the for-mer case and at the beginning in the latter, assuming a priori the knowl-edge of the mutual action of each single material constituents. In both cases,the level of knowledge regarding material constituents and their mutual in-teraction is imperfect. Considering e.g. micromechanical modelling, mate-rial knowledge can be represented in a matrix form (see Figure C.2 for themicromechanical modelling of fibre-reinforced concrete) in which only thediagonal terms are more or less known. The knowledge regarding someof the off-diagonal terms is incomplete or neglected because of the diffi-culties in determining the mechanical responses of the single constituentsand their coupling in independent experiments (see e.g. References [92, 97]where fibre-fibre interaction and crack-closure effect in fibre-reinforced con-crete is not considered). Similar considerations hold for macromechanicalapproaches.

When defining a constitutive model, missing information could be re-trieved through inverse methods for parameter determination. These meth-ods make use of specific information derived e.g. from the local behaviourin terms of stress, strain or displacement fields [49, 63, 67, 68, 80, 102] or byusing the size effect method [40]. Considering only global information interms of a load-displacement curve [150] gives rise to an ill-posed problemand parameters cannot be uniquely identified (see References [40, 63])—as


a side remark, it is worth noting that an inverse method which makes useof the global behaviour is suggested in the FRC design guidelines in Refer-ences [146, 147] to determine the stress-crack opening relationship for FRCmaterials from load-deflection test data [184]. Further, when inverse meth-ods are considered, it is fundamental that the parameters identified by theinverse procedure are the most basic parameters and do not represent a col-lection of other factors. Also, it has to be made sure that all the mechanismsinvolved in the failure process are clearly identified and that their interac-tion is well defined. Equally important is the use of error-estimation-drivenadaptive procedure in connection with one of the above inverse methods forparameter determination to avoid the influence of finite-element discreti-sation errors [8, 149]. Unfortunately, inverse methods are prone to fail (i.e.convergence is difficult to achieve) when many parameters are consideredin the parameter identification procedure. This circumstance poses seriouslimitations to their application.

To conclude, there are strong arguments against the use of many of themodels and procedures currently considered in computational mechanicsof quasi-brittle materials (see e.g. References [20, 23, 63, 70, 71, 85, 126, 127,178, 184, 196, 200] and Sections 1.1 and 3.6.1 and Appendix E). As an exam-ple, the fracture energy G f , a key ingredient in many cohesive-zone models,seems to be size-dependent [187, 196, 200]. Further, in Reference [126] it isdemonstrated that the length scale l of the non-local damage model pro-posed by Pijaudier-Cabot and Bazant [139] is not a material property de-pending on the maximum aggregate size [22]; rather, it is a ‘material func-tion depending on the strain and stress field in the neighbourhood of apoint, especially for points in the fracture process zone’ [126] (an analogousconclusion can be found in the review paper by Bazant et al. [20]). It is alsorather obvious that a model for material characterisation should be valid un-der complex stress states and loading histories. Although the abundance offormulations to describe failure in concrete, and their widespread use, noneof the constitutive models derived so far seems to be general enough to deal

Figure C.3 Experimental crack patterns from three single-edgenotched beams (adapted from Schlangen [161]).

C.2 Non-local versus viscous regularisation 121

Figure C.4 Applied load against crackmouth sliding displacement (cmsd) for therate-dependent elastoplastic-damage model(with Rankine plasticity) and for the im-plicit gradient-enhanced continuum damagemodel (with modified von Mises and Mazarsequivalent strain definitions).

0 0.05 0.1 0.15 0.2cmsd [mm]

0

10

20

30

40

50

P [k

N]

Rankinemodified von Mises

Mazars (curved crack)

experimentMazars (straight curve)

with real-life situations. Even the obvious requirement of obtaining similarpredictions with different constitutive models for the same test is not war-ranted [71]. This suggests that the mechanics governing failure in concreteis far from clearly understood.

C.2 Non-local versus viscous regularisation

The implicit gradient-enhanced continuum damage model [134] used inChapter 3 and the rate-dependent elastoplastic-damage model described inChapter 4 are contrasted through the analysis of a single-edge notched beamin anti-symmetric four-point-shear loading [161]. This specimen has alreadybeen analysed in Chapter 3 with the continuous-discontinuous approach.The single-edge notched beam has been often considered in the literature totest constitutive models with respect to their ability to reproduce the experi-mental failure pattern [9, 32, 59, 60, 109, 128, 132, 149, 151, 153, 154, 161, 193].Here this test is considered to highlight some of the characteristics of theregularised models used in Chapter 3 and 4. In particular, attention will bedrawn to the ability of the models to reproduce the experimental crack pat-tern reported in Figure C.3.

The simulations are performed considering the finest of the discretisationused in Section 3.6.3. For the analyses with the gradient-enhanced damagemodel, the parameters reported on page 61 are used in combination withthe Mazars (1.3) and modified von Mises (1.4) definitions of the equivalentstrain. For the rate-dependent elastoplastic-damage model, plastic flow isdescribed by a smoothed Rankine yield criterion [128] with a yield stressequal to 1.4 MPa under the assumption of a plane stress situation. Softeningparameters are: a = −1 and b = 300 in (4.33) and α = 0.9999 and β = 170in (4.3). Other model parameters are taken as follows: N = 1 and relaxation


(a) (b)

(c) (d)

0.1 1

ω

Figure C.5 Final failure pattern with (a) the rate-dependent elastoplastic damage model(Rankine plasticity) and the gradient-enhanced damage model with (b) modified von Misesand (c, d) Mazars equivalent strain (close-up of the central part of the beam; the failure profilein (d) relates to softening parameters different from the ones used in (b) and (c)).

time τ = 10 s for the viscous regularisation of (4.28), Young’s modulus E =35000 MPa and Poisson’s ratio ν = 0.2. The total cmsd (0.2 mm) has beenapplied in 200 steps with ∆cmsd = 0.001 mm. The above model parametershave been chosen to ensure a qualitative fit with the experimental data atthe global level in terms of the initial slope of the load-cmsd curve and thepeak load.

Results of the simulations are reported in Figure C.4 and C.5. The re-sults reported with the dotted line in Figure C.4 and in Figure C.5d havebeen obtained with a modified set of softening parameters as suggestedby Rodrıguez-Ferran and Huerta [149] (the softening parameter β in (1.8)is taken equal to 7000 instead of 100; this results in a steeper stress-strainrelationships). Although the global response in terms of applied load ver-sus cmsd, reported in Figure C.4 for the gradient-enhanced model with the

C.2 Non-local versus viscous regularisation 123

modified von Mises equivalent strain and the rate-dependent elastoplastic-damage model with Rankine plasticity, is satisfactory, the local response, interms of damage profiles reported in Figures C.5a-b, indicates that a dif-ferent failure mode has taken place. In particular, Figures C.5a indicate thepresence of two curved cracks of which the one on the right is the domi-nant one. All the cracks in Figure C.5a develop simultaneously while of allthe cracks reported in Figure C.5b, only the major curved crack is active un-til failure; the lateral cracks are active only in the early stages of the failureprocess. This latter fact is in agreement with the experimental observationby Schlangen [161]. However, the damage level of those secondary cracks isnot confirmed by the experimental crack pattern in Figure C.3.

Figure C.5d indicates that the correct failure pattern, reported in Fig-ure C.3, can be reproduced with a mode I failure criterion (i.e. by usingthe Mazars equivalent strain with the gradient-enhanced damage model).However, softening parameters must be properly chosen to obtain an ade-quate fit of the global response (see Figure C.4 and Rodrıguez-Ferran andHuerta [149] for a similar discussion).

Appendix D

Interpolation requirements for a class ofgradient-enhanced media∗

Some properties of the gradient-enhanced continuum damage model pro-posed by Peerlings et al. [134] are discussed by means of a numerical andtheoretical study. In particular, it is shown that gradient-enhanced damagemodels [32, 134] derived from integral non-local models [21, 139] do not re-quire the use of mixed finite-element formulation, as suggested by de Borstet al. [32], but rather belong to the class of coupled problems. Hence, theBabuska-Brezzi condition does not apply to this type of models. Conse-quently, the use of linear interpolation functions for displacements as wellfor non-local equivalent strains yields excellent performance in terms of rateof convergence and of convergence of the numerical procedure. Stress oscil-lations may arise for this particular interpolation, but these do not originatefrom the variational defects of the discretisation technique. On the contrary,they can be handled by simple post-processing techniques.

D.1 Governing equations and spatial discretisation

Following standard procedures [134], the governing equations of the im-plicit gradient-enhanced continuum damage model (see Chapter 3) are castin a weighted residual weak form. For completeness, the equilibrium equa-tions (2.7) (without body forces) and the modified Helmholtz equation (3.4)are recalled:

∇·σ = 0 in Ω, e− c∇2e = el in Ω. (D.1)

The weak format of the previous governing equations results in∫

Ω∇swu:σ dΩ =

∫

Γt

wu·t dΓ (D.2)

∫

Ω(c∇we·∇e + we (e− el)) dΩ = 0, (D.3)

∗ Based on Reference [169]

126 Appendix D Interpolation requirements for a class of gradient-enhanced media

p, u

u

p

x

x x

xx

ω

ε

σ

e

Figure D.1 Origin of the stress oscillations with linear interpolation functionsfor Nu and Ne in a one-dimensional tension test.

where wu and we are test functions for the displacement and the non-localequivalent strain field. The weak governing equations are discretised, at el-ement level, using a Bubnov-Galerkin approach. The displacement and thenon-local equivalent strain fields are discretised in each element by

uh = Nua, eh = Nep, (D.4)

where the interpolation functions Nu and Ne link the continuous-valued dis-placement and non-local equivalent strain fields to their discretised nodalcomponents in the vectors a and p. Details on the numerical formulationcan be found in Reference [134].

A question now arises with respect to the choice of the order of the inter-polation functions Nu and Ne. Since the non-local equivalent strain field isa scalar function of the strain tensor, an obvious choice would be to take theorder of the interpolation for the displacement one order higher than for thenon-local equivalent strain field (i.e. Nu quadratic and Ne linear or Nu cubicand Ne quadratic, etc.). Using the same order for the interpolation functions(linear for instance) could lead to some misunderstandings in the interpre-tation of the results of the numerical simulation. The issue was illustratedwith reference to a one-dimensional tension test with piece-wise linear in-

D.2 Element performance 127

Figure D.2 Linear-linear (left) and quadratic-linear(right) bar element (white and black circles indi-cate displacement and non-local equivalent straindegrees of freedom, respectively).

terpolation of u and e by Peerlings [131]. Due to the linearity of the displace-ment field within each element, the local strain field (equal to the equivalentstrain field in this case) is piecewise constant within each element. However,the damage, depending on the non-local equivalent strain, is close to linearwithin the element. The combination of the linearly varying damage and ofthe constant strain within each element throughσ = (1−ω)ε yields the os-cillatory behaviour in the stress depicted in Figure D.1. However, it must berealised that the stress field depicted in Figure D.1 is a valid solution of theweak equilibrium problem and it is accompanied by correct displacement,non-local equivalent strain and damage profiles and by quadratic conver-gence of the numerical algorithm.

When a strain-based stress update algorithm is used [134], the oscillationsin the stress profile do not constitute a problem and this apparent anomalycan be easily solved at the post-processor level, e.g. sampling the stress ata point located in the middle of the element in case of a one-dimensionalsetting. A heuristic way to avoid oscillations has been suggested by Peer-lings [131], in which the use of linear interpolation functions for Nu and Neand taking the damage variable uniform in each element ensures a smoothstress profile. Note that this technique can also be used for stress-based up-date algorithms.

D.2 Element performance

The effect of the interpolation function on the results is analysed here byusing a one-dimensional bar which is clamped at one end and with a pre-scribed displacement incrementally applied at the free end. The bar has alength of 100 mm, a cross sectional area of 10 mm2 and a geometrical imper-fection in the middle section (10% reduction of the cross sectional area for10% of the length of the bar). Young’s modulus is equal to 20000 MPa. Thelinear softening damage evolution law (1.6) withκ0 = 0.001 andκc = 0.0125has been used. The gradient parameter is set to c = 1 mm2. In the following,the use of linear interpolation functions for Nu and Ne is indicated with thewording ‘linear-linear’ while the use of quadratic interpolation functions for


0 0.01 0.02 0.03 0.04 0.05displacement [mm]

0

5

10

15

20

load

[N]

linear−linear

quadratic−linear

962 dofs3842 dofs

Figure D.3 Load-displacement curve for different interpolation schemes.

10

10

10

10

10

−6

−5

−4

−3

−2

102 103 104

linear−linear

quadratic−linear

p p −

| | a a −| |ref

ref

number of degrees of freedom

erro

r in

solu

tion

norm

non−local equivalent strain norm

displacement norm

Figure D.4 Rate of convergence for different interpolation schemes.

D.2 Element performance 129

020

4060

80100

00.005

0.0100.015

0.0200.025

0.0300

0.5

1

1.5

2

2.5

stre

ss [M

Pa]

end displacement [mm] x [mm]

Figure D.5 Evolution of the stress profile for the linear-linear discretisation.

020

4060

80100

00.005

0.0100.015

0.0200.025

0.0300

0.5

1

1.5

2

2.5

x [mm]

stre

ss [M

Pa]

end displacement [mm]

Figure D.6 Evolution of the stress profile for the quadratic-linear discretisation.


Nu and of linear interpolation functions for Ne is indicated with ‘quadratic-linear’. The comparison has been done with the same number of degreesof freedom and for two finite-element discretisations. In the first discretisa-tion, the total number of dofs is 962 (480 linear-linear elements versus 320quadratic-linear elements) and in the second it is 3842 (1920 linear-linearelements versus 1280 quadratic-linear elements). As a measure of the accu-racy of the approximation (see Figure D.3) the difference in solution norms|are f − a| and |pre f − p| between a reference solution (the solution obtainedwith a quadratic-linear bar element with 7682 dofs) and the solutions ob-tained in various computations have been compared. Figure D.4 shows thatthe use of a higher-order interpolation for the displacement field with re-spect to the non-local equivalent strain leads to a smaller computational ef-fort (at equivalent error level). However, the rate of convergence is the samefor the two interpolations.

The curves in Figure D.4 have been obtained imposing a final displace-ment of 0.425 mm at the free end of the bar in displacement control andevaluating the difference in solution norm in 81 equally distributed samplepoints. The stress profile for different interpolation orders is shown in Fig-ures D.5 and D.6. Figure D.5 clearly shows the stress oscillations with linear-linear interpolation functions while the use of a quadratic-linear interpola-tion function restores the correct stress profile (see Figure D.6). The compar-ison has been done with 120 linear-linear bar elements and 80 quadratic-linear bar elements for a given displacement of 0.03 mm at the free end ofthe bar. Note that the profiles were constructed by connecting the valuessampled at Gauss points. The real profiles resemble the one in Figure D.1. Itis worthwhile noting that oscillations remain local (they only occur wherethe damage varies quickly) and that the amplitude of oscillations is andremains limited. It is therefore questionable how serious these oscillationsshould be taken. In the next section some theoretical arguments will begiven to support the idea of using interpolation functions of the same or-der for Nu and Ne.

D.3 About terminology: mixed method versus coupled problem

In order to justify the use of different orders of interpolation related to thediscretisation of the displacement and the non-local equivalent strain fields,it has been argued by de Borst et al. [32] that gradient-enhanced damagemodels [32, 134] derived from integral non-local models [21, 139] requires

D.3 About terminology: mixed method versus coupled problem 131

the use of mixed finite-element formulations. Broadly speaking, a mixedmethod for a second-order elliptic boundary value problem is a methodin which one or more auxiliary variables are introduced, the second-orderproblem is rewritten as a first-order system of equations which is cast intoa variational form and then discretised within a Bubnov-Galerkin method,thus obtaining direct approximations to both the original and the auxiliaryvariables.

For instance, considering the field equations of the equilibrium problemwith a homogeneous Dirichlet boundary condition [142]

∇·σ + f = ∇· (De:ε) + f = ∇· (De:∇su) + f = 0 in Ωu = 0 on Γ ,

(D.5)

where f are the body forces and De is the fourth-order linear-elastic consti-tutive tensor, its solution u is given by the minimisation, with respect to asuitable test function v belonging to the Sobolev space H1

0 (Ω), of the totalpotential energy J (v):

J (v) =12

∫

Ωε:De:ε dΩ−

∫

Ωf·v dΩ

=12

∫

Ω∇sv:De:∇sv dΩ−

∫

Ωf·v dΩ. (D.6)

By introduction of the quantity r = De:∇su, where u is the solution to theequilibrium problem stated above, the original problem can be restated withrespect to the new couple of unknowns u and r. The problem has been re-stated into the first-order system

r = De:∇su in Ω∇·r + f = 0 in Ωu = 0 on Γ

(D.7)

and its solution r is given by the minimisation of the complementary energy

I(s) =12

∫

Ωs: (De)−1 :s dΩ, (D.8)

where s is a suitable test function which belongs to the Sobolev spaceL2 (Ω) and satisfies ∇·s + f = 0 in Ω and whose divergence belongs toL2 (Ω). The solution r is related to the solution u of (D.5) satisfying (D.7)3through (D.7)1 [142].


The restated second-order equilibrium problem in (D.7) is more appeal-ing than the original one with respect to those situations in which an ap-proximation of r has to be found which is at least as accurate as that of theapproximation v. Starting from the minimisation of the total complemen-tary energy, it is possible to build a discretised finite-element solution to theproblem. However, this is a difficult task because of the fulfilment of the con-straint∇·r + f = 0. These problems can be solved by making use of a mixedmethod. These methods are based on the introduction of a Lagrangian mul-tiplier to relax the constraint∇·r + f = 0. The Lagrangian functional can beformulated as

L(s, v) = I(s) +∫

Ω(∇·s + f) ·v dΩ. (D.9)

The solution to the problem is then to minimise over s and maximise overv, that is to search for the couple s and v which is the saddle-point ofL(·, ·). The existence and uniqueness of a solution to the discretised formof the saddle-point problem of L(·, ·) is assured by a compatibility condi-tion, which links the order of interpolation for the unknowns (this condi-tion is known in literature as ‘inf-sup condition’ or ‘Babuska-Brezzi condi-tion’ [12, 37, 79]), and by the ellipticity condition [38, 142]. Moreover, con-vergence of the numerical implementation cannot be assured if the Babuska-Brezzi condition is not satisfied [13].

From the above definition, it is clear that the problem defined by thetwo equations (D.1)1 and (D.1)2 of the gradient-enhanced description of thecontinuum is not of the mixed type, since the diffusion equation can notbe retrieved from the equilibrium equation: (D.1)1 and (D.1)2 form a cou-pled problem in which neither domain or set of dependent variables can besolved if separated from the other [202, vol 1, p 542]. Therefore, the compati-bility condition is not necessary and, as a consequence, the order of interpo-lation of the enhanced strain field does not have to be related to the one ofthe displacement field. Stress oscillations that appear in a linear-linear inter-polated element can be solved by means of post-processing techniques. Forthe class of gradient-enhanced damage models derived from integral non-local models [21, 139], it is always possible to formulate the weak form insuch a way that it resembles a coupled formulation. The use of mixed for-mulations is also possible but leads to unnecessary complications (see e.g.Comi [43] where, although in a different gradient-enhanced damage for-mulation, a mixed formulation is used). Further, according to Zienkiewiczand Taylor [202, vol 1, p 542], ‘in coupled systems the solution of any single

D.4 Concluding remarks 133

system is a well-posed problem and is possible when the variables corre-sponding to the other system are prescribed. This is not always the case inmixed formulations:’ due to the well-posed nature of each of the two sys-tems in the gradient-enhanced damage models proposed by Peerlings etal. [134], namely a set of equilibrium equations (see (D.1)1) and a modifiedHelmholtz equation (see (D.1)2), well-posedness for gradient-enhanced con-tinuum damage formulations is guaranteed.

D.4 Concluding remarks

Gradient models derived from non-local formulations can be classified ascoupled problems, not requiring mixed finite-element formulations in theclassical sense. Consequently, the inf-sup or Babuska-Brezzi condition is notrelevant for this class of models. In practice, this means that linear-linear in-terpolation functions can be used, instead of quadratic-linear interpolationfunctions. This can be advantageous when quadratic finite-element meshesare not available or difficult to construct. The use of linear-linear elementsgives the same convergence rate as the use of quadratic-linear elements.Stress oscillations that appear in a linear-linear interpolated element can besolved by means of post-processing techniques. It is emphasised that the useof higher order functions is not related to the well-posedness of the problembut rather to the occurrence of stress oscillations: stress oscillations are notrelated to the Babuska-Brezzi condition.

Appendix E

Incorrect failure characterisation in non-local media∗

Correct failure characterisation, in terms of damage initiation and propa-gation, is a fundamental property of any sound model. Considering failureinitiation, a quantitative discrepancy between the response of a regularisedand a standard continuum is accepted but a qualitative resemblance mustbe maintained. It is obvious that a wrong prediction of the correct locationor moment of initiation may lead to a misrepresentation of the failure modeand therefore of the failure load. Failure propagation is as important as fail-ure initiation and, in a continuous failure representation, gives an indicationof the failure mechanism. A realistic failure propagation is of paramountimportance when the assessment of a constitutive model is concerned. Inthe class of non-local models analysed here [55, 129, 134, 139], only thedissipation-driving variable is given a non-local character.

In this appendix it is shown that the choice of a non-local quantity asdamage-driving quantity produces non-physical damage initiation awayfrom the crack tip in mode I problems and a wrong failure pattern in shearband problems. Damage initiation in a class of non-local damage models,within an integral [139] as well as a differential [134] formulation, is anal-ysed. The findings of this study are not limited to damage mechanics butextend easily to other dissipation mechanisms, e.g. plasticity [55, 129], if asimilar form of regularisation is employed.

E.1 Some basic notions

In the remainder of this appendix, use is made of some notions from Chap-ter 1 which are briefly recalled for convenience. In particular, the non-localequivalent strain e is defined as [139]

e(x) =

∫

Ωψ (y; x) el (y) dΩ (y)∫

Ωψ (y; x) dΩ (y)

, (E.1)

∗ Based on Reference [170]

136 Appendix E Incorrect failure characterisation in non-local media

u

u

4h

h

ba p

2h

ψ

y

x

o

ba

e

e

(a) (b)

l

Figure E.1 Compact tension specimen: (a) geometry and bound-ary conditions and (b) local (el) and non-local (e) equivalent strainfield along line ab.

with the homogeneous and isotropic weight function

ψ (ρ) =1

2π l2 exp(− ρ

2

2l2

)in R2 (E.2)

which depends on the length scale l and on the distance ρ between thesource point y and the receiving point x. With these definitions, the de-nominator in (E.1) sums to unity for an infinite and regular domain. In anapproximate differential version of the non-local model (implicit gradient-enhanced damage model [134]), (E.1) is expanded in a Taylor’s series aroundx which yields, after some manipulations and with the use of (E.2), the mod-

E.2 Damage characterisation in mode I problems 137

ified Helmholtz equation

e− 12

l2∇2e = el in Ω. (E.3)

Equation (E.3) is complemented by the homogeneous natural boundary con-ditions

∇e · n = 0 on Γ , (E.4)

where n is the outward unit normal at the boundary Γ ofΩ. The equivalenceof (E.1)-(E.2) and (E.3)-(E.4) has been discussed by Peerlings et al. [135].

E.2 Damage characterisation in mode I problems

Proper failure characterisation relies on correct failure initiation. In quasi-brittle failure analyses of notched specimens, experimental evidence showsthat cracks propagate from the notch [108]. Proper modelling of quasi-brittlematerial behaviour must reproduce this phenomenon.

Damage initiation in mode I is analysed by means of the compact ten-sion specimen with a pre-existing crack of length h depicted in Figure E.1a.Numerical analyses showed that the elastic contour plots of the non-localdamage-driving quantity e is maximum at some distance from the cracktip. More specifically, the maximum of the non-local equivalent strain e wasfound along the line ab, as qualitatively depicted in Figure E.1b (analyticalresult) and in Figure E.2b (numerical result), and not at the crack tip. Thus,damage initiation is predicted inside the specimen, rather than at the cracktip. In Figure E.1b, the profiles of the local equivalent strain el and of thenon-local equivalent strain e—the latter obtained through (E.1)—are plottedalong the line ab. First, consider the profile of el. The local equivalent strain,as predicted by Griffith’s theory, equals 0 from point a to the crack tip. Atthe crack tip it is infinite, after which it decays monotonically to its finitevalue at point b. The profile of the non-local counterpart e differs from theprofile of el in that (i) no singularity is present (as already noted by Peer-lings et al. [135]), and (ii) the maximum occurs not at the crack tip but atsome point between the crack tip and point b along the crack line. Note thatthe crack is discretised as a set of zero measure in this example and, as such,along line ab, it does not influence the integral in the denominator in (E.1)—the denominator in (E.1) is the normalising factor in the non-local averaging


(b)(a)

min max

Figure E.2 Qualitative contour plot of (a) the local elastic equiv-alent strain el and (b) the non-local elastic equivalent strain e: dueto the non-local averaging the maximum of the non-local equiv-alent strain shifts away from the tip.

near free boundaries. In other words, for all points along line ab that are rea-sonably far from the edges of the specimen, the denominator of (E.1) yieldsthe same value, therefore the shift of maximum from el to e is not the resultof a varying averaging volume. The shift of the maximum of the non-localequivalent strain e from the crack tip is a phenomenon which is independentof the stress situation (plane stress/plane strain) and of the choice of the lo-cal equivalent strain definition as will be illustrated in Section E.2.2. Indeed,this can be explained by considering that non-local averaging of the unsym-metrical local strain field el is performed through a symmetric function ψ.The shift of the maximum away from the crack tip will be proven analyt-ically and illustrated numerically in Sections E.2.1 and E.2.2, respectively.

E.2.1 Analytical considerations

In the ideal situation of a planar crack in an infinite plate loaded in mode I,such as the one depicted in Figure E.3, the linear elastic stress field is singu-lar at the crack tip [58]. Analytical manipulations under the assumption of a


x2

θ

r

o R, x1

Figure E.3 Linear elastic crack problem in an infinite domain.

plane stress situation yield the Cartesian strains at a distance r and an angleθ from the crack tip o:

ε11 (r,θ) =KI

E√

2πrcos

12θ

(1− ν − (1 + ν) sin

12θ sin

32θ

)(E.5)

ε22 (r,θ) =KI

E√

2πrcos

12θ

(1− ν + (1 + ν) sin

12θ sin

32θ

)(E.6)

ε12 (r,θ) =KI

E√

2πr(1 + ν) cos

12θ sin

12θ cos

32θ (E.7)

ε33 (r,θ) =ν

ν − 1(ε11 +ε22) = − 2νKI

E√

2πrcos

12θ, (E.8)

from which the local equivalent strain according to the von Mises equivalentstrain

el =1

1 + ν

√−3Jε2 (E.9)

reads

el (r,θ) =

√2KI

4E√πr

√(1 + cosθ) (5− 3 cosθ). (E.10)

In the above relations, E is the Young’s modulus, ν the Poisson’s ratio, Jε2 isthe second invariant of the deviatoric strain tensor and KI the mode I stressintensity factor.

Next, the non-local equivalent strain e is investigated. First, its value at thecrack tip is considered. Second, it will be shown that the crack tip value eoof e is not maximum since larger values of e occur at locations away fromthe crack tip. To demonstrate that damage initiation is incorrectly predictedwith the class of non-local models analysed here, it is necessary and suf-ficient to demonstrate that the absolute maximum of the damage-drivingquantity does not occur at the crack tip.


Crack tip value of non-local equivalent strain. Following Peerlings etal. [135], it can be demonstrated that the non-local equivalent strain e has afinite value eo at the crack tip. To this end, the two-dimensional normalisedisotropic Gaussian weight function in (E.2) along with the local equivalentstrain expression in (E.10) is substituted in the non-local averaging of (E.1)to give

eo =KI

8π3/2El2

∞∫

0

1√r

exp(− r2

2l2

)r dr

+π∫

−π

√2 (1 + cosθ) (5− 3 cosθ) dθ. (E.11)

The first integral in (E.11) can be rewritten as

∞∫

0

√r exp

(− r2

2l2

)dr =

l3/2 Γ( 3

4

)4√

2, (E.12)

where Γ (·) indicates the gamma function; the second integral in (E.11)yields:

+π∫

−π

√2 (1 + cosθ) (5− 3 cosθ) dθ = 8

√2− 2

3

√6 ln

(7− 4

√3)

. (E.13)

Thus,

eo =KI

E√

l

4√

18 Γ( 3

4

)

12π3/2

[4√

3− ln(

7− 4√

3)]≈ 0.3612

KI

E√

l, (E.14)

which proves that the non-local equivalent strain is not singular at the cracktip (for l 6= 0). An analogous result was presented in References [131, 135].One of the conclusions reported by Peerlings [131] is that the absolute max-imum of the non-local equivalent strain is at the crack tip. This issue is dis-cussed next.

Non-local equivalent strain along the crack line. The search for largervalues of the non-local equivalent strain is restricted to the points along thecrack line ab (see Figure E.1b). A point p is considered along line ab whereby


the distance from the crack tip to p is denoted by R. The weight function in(E.2) is written for a point p as

ψp (r,θ, R) =1

2π l2 exp

(− (rcosθ− R)2 + (rsinθ)2

2l2

), (E.15)

and with this expression the non-local equivalent strain at a distance R fromthe crack tip along the crack line reads

eR =

∞∫

0

+π∫

−πψp (r,θ, R) el (r,θ) r dθ dr (E.16)

for which a closed form solution could not be obtained. Numerical evalua-tion of the integral in (E.16), for a given R, indicates that the maximum ofthe non-local equivalent strain is not positioned at the crack tip as depictedin Figure E.4a. The linear dependence of the position of the maximum of thenon-local equivalent strain on the length scale l is depicted in Figure E.4bwhich shows that the non-local equivalent strain is maximum at the cracktip only for l = 0 mm, i.e. only for a local damage model. These results ex-tend to a finite specimen width if the effect of a finite geometry is reflectedin the stress intensity factor KI.

Note that, in general, the use of non-local averaging of field quantitieswith isotropic weight functions results in a modification of failure charac-terisation. In the class of non-local elasticity models proposed by Eringenet al. [56], the stress field value at the crack tip is finite but, similar to thenon-local damage model considered here, its maximum occurs at some dis-tance from the crack tip along the crack line. The modification of the weightfunction ψ in order to preserve the qualitative character of the field to beaveraged (the local field) is problematic and leads to tailor-made solutionswhich are not recommended when a differential version of the non-localdamage model is considered.

E.2.2 Numerical analysis

The compact tension specimen depicted in Figure E.1a has been numericallyanalysed by using an integral and a differential non-local damage modelwith the finite-element method. In the numerical simulations only the upperpart of the specimen has been discretised due to symmetry and the load has


0 5 10R [mm]

0 1 2 3 4l [mm]

[mm]

0

0.1

0.2

0.3

0.4

0

0.5

eR

1

2

3

4

Rmax

Rmax

(a) (b)

Figure E.4 Profile of the non-local equivalent strain at distance R from the tip forl = 1 mm (a) and (b) linear dependence of the position of the maximum of the non-local equivalent strain Rmax on the length scale l for unit values of E and KI.

been applied via an imposed displacement. The following model parame-ters have been adopted for the simulation: Young’s modulus E = 1000 MPa;Poisson’s ratio ν = 0; exponential damage evolution law

ω =

0 if κ ≤ κ0

1− κ0κ

(1−α +αe−β(κ−κ0)

)if κ > κ0,

(E.17)

with threshold of damage initiation κ0 = 0.0003; softening parameters α =0.99 andβ = 1000; length scale l = 0.2 mm. The equivalent strain definition

el =

√√√√ 3

∑i=1〈εi〉2, (E.18)

based on the positive principal strain components 〈εi〉 = (εi + |εi|)/2 whereεi are the principal strains, has been used. The height of the specimen hasbeen taken as 4h = 2 mm. The mesh used for the simulations has beenchosen such that a sufficient resolution of the non-local field is obtained.

Contour plots of the non-local equivalent strain at the onset of damageinitiation are reported in Figure E.5. Clearly, the maximum of the non-localequivalent strain has shifted, both for the integral model and the differen-tial model. Due to the shifting, damage is expected to initiate, wrongly, awayfrom the crack tip. However, as depicted in Figure E.6, damage profiles closeto global failure of the specimen give no indication of the incorrect damage


(a) (b)

x0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1

x0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1

y y

0.0003

2E−05

e

Figure E.5 Contour plot of the non-local equivalent strain for (a) the integral non-localdamage and (b) the differential non-local damage model at the onset of damage initiation,i. e. in the elastic stage (measures in mm; crack tip at 0.5 mm).

(a) (b)

x

y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

y

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

x

ω1

0

Figure E.6 Contour plot of the damage field for (a) the integral non-local damage and(b) the differential non-local damage model close to failure of the specimen (measures inmm; crack tip at 0.5 mm).


P, v

(a)

P, v

2h

h

(b)

P, v

Figure E.7 Geometry and boundary conditions for the specimen in biaxialcompression: (a) full specimen and (b) half specimen (h = 60 mm; the shadedpart indicates the imperfection; imperfection size in (a) is h/10× h/10).

initiation. Experience with the differential version of the non-local damagemodel indicates that this is a common situation in mode I problems andthat failure characterisation close to failure is quite similar to the ones ob-tained with other constitutive models. Consequently, the shift of the maxi-mum of the non-local equivalent strain away from the crack tip can be con-sidered ‘harmless’ as long as the final failure characterisation is concerned.However, it must be realised that the use of a non-local dissipation-drivingvariable leads to a non-physical damage initiation. The shift of the maxi-mum of the non-local equivalent strain away from the crack tip is present,although less evident, also in case of cracks or notches modelled as stronglynon-convex entities with non-zero volume, i.e. when there are no strain sin-gularities.

E.3 Damage characterisation in shear band problems 145

(a) (b) (c)

p p p

Figure E.8 Schematic formation of a shear band: (a) initiation and (b) expansion of theplastic zone and (c) further localisation/expansion within/of the plastic zone (strainlocalisation is triggered by a weak spot in the left bottom corner).

E.3 Damage characterisation in shear band problems

The correct determination of shear bands is of prime interest and it is di-rectly linked to the occurrence of possible collapse mechanisms in manyengineering problems. Specimens under compressive loading are usuallycharacterised by the formation of shear bands whose inclination can be de-termined analytically. Results obtained within the flow theory of plastic-ity [156] have been extended to scalar damage models by Rizzi et al. [148]and Carol and Willam [41] and apply to an infinite geometry for a standard(i.e. not regularised) medium. Their results have been derived for a specificchoice of the equivalent strain definition and their generalisation to otherequivalent strain definitions, although possible in principle, is not withinthe scope of this study and therefore is not considered here. In what fol-lows, it is illustrated that non-local regularisation techniques significantlyalter failure propagation during strain localisation.

To illustrate the problem, shear band simulations under a plane stress anda plane strain condition have been performed with the gradient-enhancedcontinuum damage model for the two-dimensional specimen depicted inFigure E.7. Both geometries were considered to assess the validity of theboundary condition (E.4) which was indeed verified: both geometries gaveidentical results and in the following only the geometry in Figure E.7bis considered. A detailed analysis regarding the treatment of the non-


0 0.02 0.04 0.06 0.08v [mm]

0

50

100

150

200

P [N

]

Figure E.9 Load-displacement curve forthe shear band problem (relevant fieldscorresponding to the white circles are de-picted in Figures E.10 and E.11).

local equivalent strain at the boundaries for a mode I problem was givenby Peerlings et al. [135]. In numerical simulations of quasi-static shearband formation under compressive loading, shear bands are usually trig-gered by an imperfection (placed on the left bottom corner for the spec-imen in Figure E.7b). After the shear band has been initiated, expansionof the plastic zone and further localisation within the plastic zone is ob-served [199] (for a schematic initiation and evolution of the shear band seeFigure E.8). Shear bands are characterised by their stationary nature in thesense that their position is determined after their formation (see e.g. Ref-erence [120] and references herein for experimental shear bands and Ref-erences [29, 30, 62, 119, 125, 179, 182, 186, 194, 199] for some numerical re-sults). The inclination angle that the shear band forms with the horizon-tal axis is determined mainly by assumptions related to the constitutivemodel, to the Poisson’s ratio and to the plane stress or plane strain con-dition [41, 148, 156, 179, 182] while the width of the shear band, in a contin-uous description of failure, is dictated by the length scale (i.e. the larger thelength scale, the wider the band width).

In the numerical analyses, the material has been characterised by aYoung’s modulus E = 20000, a Poisson’s ration ν = 0.2, the exponen-tial softening law (E.17) with κ0 = 0.0001, α = 0.99 and β = 300 andthe von Mises equivalent strain (E.9). The load has been applied via dis-placement control. The imperfection has been given a reduced value of κ0(κ0 = 0.00005) and the mesh density has been chosen to ensure a sufficientresolution of the non-local field. To begin with, the evolution of the shearband, in terms of non-local equivalent strain and damage fields has beenanalysed for the specimen in Figure E.7b with a length scale l = 2 mm un-der a plane strain condition. Results depicted in Figures E.10 and E.11 relate


v=0.009 mm v=0.013 mm v=0.016 mm

v=0.021 mm v=0.030 mm v=0.038 mm v=0.080 mm

v=0.006 mm

0.030.0003e

Figure E.10 Shear band evolution: contour plots of the non-local equivalent strain field (seeFigure E.9).

v=0.006 mm v=0.009 mm v=0.013 mm v=0.016 mm

v=0.021 mm v=0.030 mm v=0.038 mm v=0.080 mm

ω1.00.5

Figure E.11 Shear band evolution: contour plots of the damage field (see Figure E.9).


ω1.00.5

0.030.0003e

Figure E.12 Shear band close to failure: contour plots of the non-local equivalent strain field(top) and of the damage field (bottom) for l = 1 mm (left) and l = 2 mm (right) for planestress (odd columns) and plane strain (even columns).

to the load-displacement diagram in Figure E.9, where the applied load Pis plotted against the absolute value v of the vertical displacement. In thecontour plots in Figures E.10 and E.11, only values larger than the thresholdin the respective legends have been plotted. It is clear that the shear band‘migrates’ from the weak spot, where it was initiated, to the opposite side ofthe specimen along the horizontal boundary as depicted in Figures E.10 andE.11. Similar results have been reported by Engelen et al. [55] and Pamin etal. [129]. It is stressed that the ‘migrating’ shear band is not the product ofan improper treatment of the boundary conditions since, as already statedbefore, the geometries reported in Figure E.7 give identical results. This par-ticular appearance of the shear band is simply due to a wrong prediction ofthe positioning of localised shearing and has the same nature of the shift ofthe maximum of the non-local equivalent strain in mode I problems.

The effect of a larger length scale is reported in Figure E.12 together with acomparison between plane stress and plane strain conditions close to failure.Similar to shear bands in a plasticity context [156] and as reported by Caroland Willam [41], the only noticeable difference between plane stress and


plane strain resides in a different inclination of the shear band with respectto the horizonal axis which does not correspond to the numerical results inFigure E.12. Further, with an increasing non-local effect a wider shear bandwidth is expected, while it is also noted that a more pronounced shift of theshear band takes place. In one case (plane stress situation with l = 2 mm)the non-local equivalent strain field mimics a mode I situation and almosthalf of the specimen is damaged, which is not realistic.

0 0.02 0.04 0.06 0.08v [mm]

0

50

100

150

200

P [N

]

(a)

0 0.05 0.1 0.15 0.2v [mm]

0

50

100

150

200

P [N

]

(b)

Figure E.13 Load-displacement curve for the shear band problem with the rate-dependentelastoplastic-damage model (see Chapter 4): relaxation time (a) τ = 4 s and (b) τ = 8 s(relevant fields corresponding to the white circles are depicted in Figures E.14 and E.15 forτ = 4 s and τ = 8 s, respectively).

The shear band problem was also analysed with the viscous model de-scribed in Chapter 4. The simulations were performed with a von Misesplane stress rate-dependent damage-elastoplasticity for two values of therelaxation time τ in (4.28). Other model parameters are: Young’s modulusE = 20000, Poisson’s ratio ν = 0.2, N = 1 in (4.29), softening parametersa = −1 and b = 200 in (1.21) and α = 0.99 and β = 300 in (4.3) andyield stress equal to 2 MPa with a reduction of 10% in the weak spot. Thetotal displacement was applied at constant strain rate (for τ = 4 s the fi-nal displacement (0.08 mm) was applied in 160 steps while for τ = 8 s thefinal displacement (0.2 mm) was applied in 400 steps). Results reported inFigures E.14 and E.15 relate to the load-displacement diagram reported inFigure E.13. The shear band evolves correctly, is stationary and its width isset by the relaxation time [179]. The slight shift of the shear band is due tothe rectangular geometry of the imperfection and is not influenced by therelaxation time.


ω1.00.5

v=0.028 mm v=0.043 mm v=0.062 mm v=0.080 mm

Figure E.14 Shear band evolution: contour plots of the damage field with the rate-dependent elastoplastic-damage model (see Chapter 4) for τ = 4 s (see Figure E.13a).

ω1.00.5

v=0.050 mm v=0.075 mm v=0.099 mm v=0.200 mm

Figure E.15 Shear band evolution: contour plots of the damage field with the rate-dependent elastoplastic-damage model (see Chapter 4) for τ = 8 s (see Figure E.13b).

E.4 Concluding remarks

When non-local averaging is considered as a regularisation technique, theuse of a non-local variable as degradation-driving variable induces incorrectfailure initiation and propagation—this issue has been investigated throughanalytical and numerical analyses for mode I and shear band problems. Theparticular choice of the examples in mode I failure was aimed at empha-sising that the standard formulation of non-local damage model, either inintegral [139] or in differential [134] format, is unsuited to correctly describedamage initiation when degradation stems from a strong inhomogeneityof the strain field. It is stressed that this phenomenon is not caused, in thepresent case, by boundary effects on the non-local averaging. Analogousresults can be found in cases with strong gradients of the quantity to beaveraged due to e.g. concave boundaries, concentrated loads and material

E.4 Concluding remarks 151

inhomogeneities. Although the shift of the maximum of the dissipation-driving variable may not alter the final failure representation in mode I dom-inated problems, it does affect the transition from continuous to continuous-discontinuous failure in a gradient-enhanced damage model as discussed inChapters 3 and 4. The numerical study of a shear band problem illustratedthat the non-local averaging is responsible for the non-stationary shear bandwhich results in an unrealistic failure pattern. This has been studied for var-ious length scale values under plane stress and plane strain conditions. Theshear band problem was also analysed with a rate-dependent elastoplastic-damage model which gave correct results.

To summarise, constitutive models based on a non-local dissipation-driving variable may lead to incorrect failure initiation and propagation inarbitrary loading scenarios due to a fundamental flaw in damage character-isation. Their use is acceptable with some reserve only in mode I problems.

References

The number printed in italics indicates the page where the publication iscited.

[1] E. C. Aifantis. On the microstructural origin of certain inelastic models. Journal of EngineeringMaterials Technology, 106:326–330, 1984. 118

[2] E. C. Aifantis. The physics of plastic deformation. International Journal of Plasticity, 3:211–247,1987. 118

[3] P. C. Aıtcin. High Performance Concrete. E & FN Spon, London, 1998. 4

[4] J. Alfaiate, A. Simone, and L. J. Sluys. Non-homogeneous displacement jumps in strong embed-ded discontinuities. International Journal of Solids and Structures, 2002. In press. 16, 19

[5] J. Alfaiate, A. Simone, and L. J. Sluys. A new approach to strong embedded discontinuities. InN. Bicanic, R. de Borst, H. A. Mang, and G. Meschke, editors, Computational Modelling of ConcreteStructures – EURO-C 2003, pages 33–41. Swets & Zeitlinger, Lisse, 17-20 March 2003. 16

[6] G. Alfano and M. A. Crisfield. Finite element interface models for the delamination analysis oflaminated composites: mechanical and computational issues. International Journal for NumericalMethods in Engineering, 50(7):1701–1736, 2001. 13

[7] F. Armero and K. Garikipati. An analysis of strong discontinuities in multiplicative finite strainplasticity and their relation with the numerical simulation of strain localization. InternationalJournal of Solids and Structures, 33(20–22):2863–2885, 1996. 13, 16

[8] H. Askes. Advanced Spatial Discretisation Strategies for Localised Failure. PhD thesis, Delft Universityof Technology, 2000. 44, 120

[9] H. Askes and L. J. Sluys. Remeshing strategies for adaptive ALE analysis of strain localisation.European Journal of Mechanics A/Solids, 19:447–467, 1999. 121

[10] ASTM. Metals Test Methods and Analytical Procedures, Designation: E 399–90; E 647–99. InAnnual Book of ASTM Standards, volume 03.01. American Society for Testing and Materials, WestConshohocken, Pennsylvania, 2000. 56

[11] M. Audi, J. Krcek, J. Zeman, and M. Sejnoha. Constant strain triangular element with embeddeddiscontinuity based on partition of unity. Building Research Journal, 2003. Submitted. 29

[12] I. Babuska. The finite element method with lagrangian multipliers. Numerische Mathematik,20:179–192, 1973. 132

[13] K. J. Bathe. Finite Element Procedures. Prentice-Hall, New Jersey, 1996. 132

[14] Z. P. Bazant. Instability, ductility and size-effect in strain-softening concrete. Journal of EngineeringMechanics, 102:331–344, 1976. 117

154 References

[15] Z. P. Bazant. Mechanics of distributed cracking. Applied Mechanics Reviews, 39:675–705, 1986.118

[16] Z. P. Bazant and L. Cedolin. Stability of Structures: Elastic, Inelastic, Fracture and Damage Theories.Oxford University Press, New York, 1991. 117

[17] Z. P. Bazant and M Jirasek. Nonlocal integral formulations of plasticity and damage: survey ofprogress. Journal of Engineering Mechanics, 128(11):1119–1149, 2002. 71, 118

[18] Z. P. Bazant and F. B. Lin. Non-local yield limit degradation. International Journal for NumericalMethods in Engineering, 26:1805–1823, 1985. 118

[19] Z. P. Bazant and B. Oh. Crack band theory for fracture of concrete. Materials and Struc-tures/Materiaux et Constructions, 16:155–177, 1983. 2, 117

[20] Z. P. Bazant, J. Ozbolt, and R. Eligehausen. Fracture size effect: review of evidence for concretestructures. Journal of Structural Engineering, 120(8):2377–2398, 1994. 120

[21] Z. P. Bazant and G. Pijaudier-Cabot. Nonlocal continuum damage, localization instability andconvergence. Journal of Applied Mechanics, 55:287–293, 1988. 118, 125, 130, 132

[22] Z. P. Bazant and G. Pijaudier-Cabot. Measurement of characteristic length of nonlocal continuum.Journal of Engineering Mechanics, 115:755–767, 1989. 120

[23] T. Belytschko and K. Mish. Computability in non-linear solid mechanics. International Journal forNumerical Methods in Engineering, 52(1–2):3–21, 2001. 120

[24] N. Bicanic. From continua to discontinua—modelling of progressive discontinuities. In R. deBorst, N. Bicanic, H. A. Mang, and G. Meschke, editors, Computational Modelling of Concrete Struc-tures, pages 921–930, Rotterdam, 1998. Balkema. 1

[25] T. N. Bittencourt, P. A. Wawrzynek, A. R. Ingraffea, and J. L. Souza. Quasi-automatic simulationof crack propagation for 2D LEFM problems,. Engineering Fracture Mechanics, 52(2):321–334, 1996.13

[26] L. Bode, J. L. Tailhan, G. Pijaudier-Cabot, C. La Borderie, and J. L. Clement. Failure analysis ofinitially cracked concrete structures. Journal of Engineering Mechanics, 123:1153–1160, 1997. 1

[27] G. Bolzon. Formulation of a triangular element with an embedded interface via isoparametricmapping. Computational Mechanics, 27:463–473, 2001. 1

[28] G. Bolzon and A. Corigliano. Finite elements with embedded displacement discontinuity. a gen-eralised variable formulation. International Journal for Numerical Methods in Engineering, 49:1227–1266, 2000. 1

[29] R. I. Borja. A finite element model for strain localization analysis of strongly discontinuous fieldsbased on standard Galerkin approximation. Computer Methods in Applied Mechanics and Engineer-ing, 190:1529–1549, 2000. 146

[30] R. de Borst. Simulation of strain localisation: a reappraisal of the Cosserat continuum. EngineeringComputations, 8:317–332, 1991. 146

[31] R. de Borst. A generalisation of J2-flow theory for polar continua. Computer Methods in AppliedMechanics and Engineering, 103:347–362, 1993. 118

[32] R. de Borst, A. Benallal, and O. M. Heeres. A gradient-enhanced damage approach to fracture.Journal de Physique IV, 6:491–502, 1996. 121, 125, 130

References 155

[33] R. de Borst and H.-B. Muhlhaus. Gradient-dependent plasticity - formulation and algorithmicaspects. International Journal for Numerical Methods in Engineering, 35(3):521–539, 1992. 118

[34] R. de Borst, J. Pamin, and M. G. D. Geers. On coupled gradient-dependent plasticity and damagetheories with a view to localization analysis. European Journal of Mechanics A/Solids, 18(6):939–962,1999. 56, 57, 67

[35] R. de Borst and L. J. Sluys. Localisation in a Cosserat continuum under static and dynamic loadingconditions. Computer Methods in Applied Mechanics and Engineering, 90:805–827, 1991. 118

[36] R. de Borst, L. J. Sluys, H.-B. Muhlhaus, and J. Pamin. Fundamental issues in finite elementanalyses of localization of deformation. Engineering Computations, 10:99–121, 1993. 117

[37] F. Brezzi. On the existence, uniqueness and approximation of saddle-point problems arising fromlagrangian multipliers. Revue Francaise d’Automatique, d’Informatique et de Recherche Operationnelle.Analyse Numerique, 8:129–151, 1974. 132

[38] F. Brezzi and M. Fortin. Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York,1991. 132

[39] D. Brokken. Numerical Modelling of Ductile Fracture in Blanking. PhD thesis, Eindhoven Universityof Technology, 1999. 44

[40] J. Carmeliet. Optimal estimation of gradient damage parameters from localization phenomena inquasi-brittle materials. Mechanics of Cohesive-Frictional Materials, 4(1):1–16, 1999. 119

[41] I. Carol and K. Willam. Application of analytical solutions in elasto-plasticity to localization anal-ysis of damage models. In D. R. J. Owen, E. Onate, and E. Hinton, editors, Computational Plasticity,Fundamentals and Applications, pages 714–719, Barcelona, Spain, 1997. CIMNE. 145, 146, 148

[42] A. Chorin, T. J. R. Hughes, M. F. McCracken, and J. E. Marsden. Product formulas and numericalalgorithms. Communications in Pure and Applied Mathematics, 31:205–256, 1978. 72, 76

[43] C. Comi. Computational modelling of gradient-enhanced damage in quasi-brittle materials. Me-chanics of Cohesive-Frictional Materials, 4:17–36, 1999. 132

[44] A. Corigliano and S. Mariani. Parameter identification of a time-dependent elastic-damage in-terface model for the simulation of debonding in composites. Composites Science and Technology,61:191–203, 2000. 99

[45] E. Cosserat and F. Cosserat. Theorie des Corps Deformables. Librairie Scientifique A. Herman et Fils,Paris, 1909. 118

[46] M. A. Crisfield. Non-Linear Finite Element Analysis of Solids and Structures. John Wiley & Sons Ltd,Chichester, England, 1996. 6, 76

[47] M. A. Crisfield. Non-Linear Finite Element Analysis of Solids and Structures: Advanced Topics. JohnWiley & Sons Ltd, Chichester, England, 1997. 6, 78

[48] M. A. Crisfield and J. Wills. Numerical comparison involving different ‘concrete-models’. InComputational Mechanics of Concrete Structures – Advances and Applications, IABSE Colloquium –Delft 1987, Report, volume 54, pages 177–187. Delft University Press, Delft, the Netherlands, 1987.2

156 References

[49] K. De Proft, G. N. Wells, L. J. Sluys, and W. De Wilde. A combined experimental-numerical studyto cyclic behaviour of limestone. In D. R. J. Owen, E. Onate, and B. Suarez, editors, Proceedingsof the 7th International Conference on Computational Plasticity – Complas 2003, page 120. CIMNE,Barcelona, Spain, 7-10 April 2003. 119

[50] A. Dietsche, P. Steinmann, and K. Willam. Micropolar elasto-plasticity and its role in localisationanalysis. International Journal of Plasticity, 9:813–831, 1992. 118

[51] J. Dolbow, N. Moes, and T. Belytschko. Discontinuous enrichment in finite elements with a parti-tion of unity method. Finite Elements in Analysis and Design, 36:235–260, 2000. 21

[52] C. A. M. Duarte and J. T. Oden. H-p clouds – an h-p meshless method. Numerical Methods forPartial Differential Equations, 12(6):673–705, 1996. 13, 16

[53] J. F. Dube, G. Pijaudier-Cabot, and C. La Borderie. Rate dependent damage model for concrete indynamics. Journal of Engineering Mechanics, 122(10):939–947, 1996. 117

[54] G. Duvaut and J. L. Lions. Les Inequations en Mechanique et en Physique. Dunod, Paris, France, 1972.12, 71, 73, 118

[55] R. A. B. Engelen, M. G. D. Geers, and F. P. T. Baaijens. Nonlocal implicit gradient-enhanced elasto-plasticity for the modelling of softening behaviour. International Journal of Plasticity, 19(4):403–433,2003. 87, 99, 135, 148

[56] A. C. Eringen, C. G. Speziale, and B. S. Kim. Crack-tip problem in non-local elasticity. Journal ofthe Mechanics and Physics of Solids, 25:339–255, 1977. 141

[57] H. Espinosa, P. D. Zavattieri, and S. K. Dwivedi. A finite deformation continuum/discrete modelfor the description of fragmentation and damage in brittle materials. Journal of the Mechanics andPhysics of Solids, 46(10):1909–1942, 1998. 1

[58] H. L. Ewalds and R. J. H. Wanhill. Fracture Mechanics. Edward Arnold, London, England, 1984.138

[59] P. H. Feenstra. Computational Aspects of Biaxial Stress in Plain and Reinforced Concrete. PhD thesis,Delft University of Technology, 1993. 1, 121

[60] P. H. Feenstra and R. de Borst. Aspects of robust computational modeling for plain and reinforcedconcrete. HERON, 38(4):3–76, 1993. 1, 121

[61] S. Fichant, C. La Borderie, and G. Pijaudier-Cabot. Isotropic and anisotropic descriptions of dam-age in concrete structures. Mechanics of Cohesive-Frictional Materials, 4:339–359, 1999. 35

[62] F. X. Garaizar and J. Trangenstein. Adaptive mesh refinement and front-tracking for shear bandsin an antiplane shear model. SIAM Journal on Scientific Computing, 20(2):750–779, 1998. 146

[63] M. G. D. Geers. Experimental Analysis and Computational Modelling of Damage and Fracture. PhDthesis, Eindhoven University of Technology, 1997. 7, 119, 120

[64] M. G. D. Geers, R. de Borst, W. A. M. Brekelmans, and R. H. J. Peerlings. Strain-based transient-gradient damage model for failure analyses. Computer Methods in Applied Mechanics and Engineer-ing, 160:133–153, 1998. 2, 8, 41, 44, 48, 69, 99

[65] M. G. D. Geers, R. de Borst, W. A. M. Brekelmans, and R. H. J. Peerlings. Validation and in-ternal length scale determination for a gradient damage model: application to short glass-fibre-reinforced polypropylene. International Journal of Solids and Structures, 36:2557–2583, 1999. 56, 57,87, 88

References 157

[66] M. G. D. Geers, R. de Borst, and R. H. J. Peerlings. Damage and crack modeling in single-edgeand double-edge notched concrete beams. Engineering Fracture Mechanics, 65:247–261, 2000. 67

[67] M. G. D. Geers, R. de Borst, and A. A. J. M. Peijs. Mixed numerical-experimental identification ofnonlocal characteristics of random fibre reinforced composites. Composite Science and Technology,59:1569–1578, 1999. 119

[68] M. G. D. Geers, T. Peijs, W. A. M. Brekelmans, and R. de Borst. Experimental monitoring ofstrain localization and failure behaviour of composite materials. Composite Science and Technology,56:1283–1290, 1996. 51, 56, 119

[69] J. F. Georgin, W. Nechnech, L. J. Sluys, and J. F. Reynouard. A coupled damage-viscoplasticitymodel for localization problems. In H. A. Mang, F. G. Rammerstorfer, and J. Eberhardsteiner,editors, Proceedings of the Fifth World Congress on Computational Mechanics – WCCM V, volume I,page 428. Vienna University of Technology, Austria, July 7-12 2002. 11, 74

[70] W. H. Gerstle and A. R. Ingraffea. Does bond-slip exist? In S. P. Shah, S. E. Swartz, and M. L.Wang, editors, Proceedings of Conference on Micromechanics of Quasi-brittle Materials, pages 407–416.Elsevier Science Publishing LTD, June 6-8 1990. 120

[71] S. Ghavamian and I. Carol. Benchmarking of concrete cracking constitutive laws: MECA project.In N. Bicanic, R. de Borst, H. A. Mang, and G. Meschke, editors, Computational Modelling of ConcreteStructures – EURO-C 2003, pages 179–187. Swets & Zeitlinger, Lisse, 17-20 March 2003. 120, 121

[72] R. E. Goodman, R. L. Taylor, and T. L. Brekke. A model for the mechanics of jointed rocks. Journalof the Soil Mechanics and Foundations Division, 94:637–659, 1968. 13

[73] W. G. Gray and P. C. Y. Lee. On the theorems for local volume averaging of multiphase systems.International Journal of Multiphase Flow, 3:333–340, 1977. 115, 116

[74] W. G. Gray, A. Leijnse, R. L. Kolar, and C. A. Blain. Mathematical Tools for Changing Scales in theAnalysis of Physical Systems. CRC, Boca Raton, Florida, 1993. 115

[75] H. J. Grootenboer. Finite Element Analysis of Two-Dimensional Reinforced Concrete Structures, Tak-ing Account of Non-Linear Physical Behaviour and the Development of Discrete Cracks. PhD thesis,Technische Hogeschool Delft, 1979. 1

[76] A. Hillerborg. Analysis of fracture by means of the fictitious crack model, particularly for fibrereinforced concrete. The International Journal of Cement Composites, 2(4):177–184, 1980. 92

[77] G. A. Holzapfel. Nonlinear Solid Mechanics. John Wiley & Sons, New York, 2000. viii

[78] D. A. Hordijk. Local Approach to Fatigue of Concrete. PhD thesis, Delft University of Technology,1991. Also published as Tensile and tensile fatigue behaviour of concrete: experiments, modellingand analyses. HERON, 37(1):3–79, 1992. 2–4, 53, 54, 64

[79] T. J. R. Hughes. The Finite Element Method - Linear Static and Dynamic Finite Element Analysis.Prentice-Hall, London, England, 1987. 132

[80] C. Iacono, L. J. Sluys, and J. G. M. van Mier. Development of an inverse procedure for parameterestimates of numerical models. In N. Bicanic, R. de Borst, H. A. Mang, and G. Meschke, editors,Computational Modelling of Concrete Structures – EURO-C 2003, pages 259–268. Swets & Zeitlinger,Lisse, 17-20 March 2003. 119

[81] J. Janson and J. Hult. Fracture mechanics and damage mechanics - a combined approach. Journalde Mechanique Appliquee, 1(1):69–83, 1977. 1

158 References

[82] Y. S. Jenq and S. P. Shah. Crack propagation in fiber-reinforced concrete. Journal of StructuralEngineering, 112(1):19–34, 1986. 12

[83] M. Jirasek. Modelling of localized damage and fracture in quasibrittle materials. In P. A. Vermeer,S. Diebels, W. Ehlers, H. J. Herrmann, S. Luding, and E. Ramm, editors, Continuous and Discon-tinuous Modelling of Cohesive-Frictional Materials, volume 568, pages 17–19. Springer, Berlin, 2001.50

[84] M. Jirasek and Z. P. Bazant. Inelastic Analysis of Structures. John Wiley & Sons, New York, 2002.viii, 6, 48, 72

[85] M. Jirasek and B. Patzak. Models for quasibrittle failure: theoretical and computational aspects.In Z. Waszczyszyn and J. Pamin, editors, Solids, Structures and Coupled Problems in Engineering,Proceedings of the Second European Conference on Computational Mechanics, volume 1, pages 70–71,Cracow, Poland, 2001. Fundacja Zdrowia Publicznego, ‘Vesalius’ Publisher. On CD-ROM. 120

[86] M. Jirasek and T. Zimmermann. Embedded crack model. Part II: Combination with smearedcracks. International Journal for Numerical Methods in Engineering, 50:1291–1305, 2001. 1, 23, 69

[87] J. W. Ju. On energy-based coupled elastoplastic damage theories: constitutive modeling and com-putational aspects. International Journal of Solids and Structures, 25(7):803–833, 1989. 1, 76, 117

[88] P. Kabele. Assessment of structural performance of engineered cementitious composites by com-puter simulation. Habilitation thesis, Czech Technical University, Prague, November 2000. 38,39

[89] P. Kabele and H. Horii. Analytical model for fracture behaviour of pseudo strain-hardening ce-mentitious composites. Concrete Library International (proc. of JSCE), 29:105–120, 1997. 1, 35

[90] P. Kabele, E. Yamaguchi, and H. Horii. FEM-BEM superposition method for analysis of quasi-brittle structures. International Journal of Fracture, 100:249–274, 1999. 1

[91] R. E. Kalman. A new approach to linear filtering and prediction problems. Journal of Basic Engi-neering, 82:35–45, 1960. 99

[92] T. Kanda, Z. Lin, and V. C. Li. Modeling of tensile stress-strain relation of pseudo strain-hardeningcementitious composites. Journal of Materials in Civil Engineering, 12(2):147–156, 2000. 35, 119

[93] E. Kuhl, E. Ramm, and R. de Borst. An anisotropic gradient damage model for quasi-brittle ma-terials. Computer Methods in Applied Mechanics and Engineering, 183:87–103, 2000. 67

[94] J. Kullaa. Finite element modelling of fibre-reinforced brittle materials. HERON, 42(2):75–95, 1997.33, 34

[95] D. Lasry and T. Belytschko. Localization limiters in transient problems. International Journal ofSolids and Structures, 24:581–597, 1988. 117, 118

[96] J. Lemaitre. A Course on Damage Mechanics. Springer-Verlag, Berlin, second edition, 1996. 72

[97] F. Li and Z. Li. Continuum damage mechanics based modeling of fiber reinforced concrete intension. International Journal of Solids and Structures, 38:777–793, 2001. 119

[98] V. C. Li and T. Hashida. Ductile fracture in cementitious materials? In Z. P. Bazant, editor, FractureMechanics of Concrete Structures. Proceedings of the First International Conference on Fracture Mechanicsof Concrete and Concrete Structures – FramCoS 1, pages 526–535, London, 1992. Elsevier. 35, 38

References 159

[99] V. C. Li, H. Stang, and H. Krenchel. Micromechanics of crack bridging in fiber reinforced concrete.Materials and Structures/Materiaux et Constructions, 26:486–494, 1993. 92

[100] V. C. Li and H. C. Wu. Conditions for pseudo strain-hardening in fiber reinforced brittle matrixcomposites. Applied Mechanics Reviews, 45(8):390–398, 1992. 5, 34, 38

[101] B. Loret and J. H. Prevost. Dynamic strain localization in elasto-(visco)plastic solids. Part I. Gen-eral formulation and one-dimensional example. Computer Methods in Applied Mechanics and Engi-neering, 83:247–273, 1990. 117

[102] R. Mahnken and E. Kuhl. Parameter identification of gradient enhanced damage models with thefinite element method. European Journal of Mechanics A/Solids, 18:819–835, 1999. 119

[103] L. E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall International,Englewood Cliffs, New Jersey, 1969. 19

[104] J. Mazars. Application de la Mecanique de l’Endommagement au Comportement Non-Lineaire et a laRupture du Beton de Structure. PhD thesis, Universite Paris VI, France, 1984. 7

[105] J. Mazars and G. Pijaudier-Cabot. Continuum damage theory - application to concrete. Journal ofEngineering Mechanics, 115:345–365, 1989. 7

[106] J. Mazars and G. Pijaudier-Cabot. From damage to fracture mechanics and conversely: a com-bined approach. International Journal of Solids and Structures, 33(20–22):3327–3342, 1996. 1

[107] J. M. Melenk and I. Babuska. The partition of unity finite element method: basic theory andapplications. Computer Methods in Applied Mechanics and Engineering, 139(1–4):289–314, 1996. 13,16

[108] J. G. M. van Mier. Fracture Processes of Concrete. CRC Press, Inc., Boca Raton, Florida, 1997. 2, 64,71, 117, 137

[109] N. Moes and T. Belytschko. Extended finite element method for cohesive crack growth. Engineer-ing Fracture Mechanics, 69:813–833, 2002. 22, 121

[110] N. Moes, J. Dolbow, and T. Belytschko. A finite element method for crack growth without remesh-ing. International Journal for Numerical Methods in Engineering, 46(1):131–150, 1999. 3, 13, 15, 21,23, 28, 31

[111] S. Mohammadi, D. R. J. Owen, and D. Peric. A combined finite/discrete element algorithm fordelamination analysis of composites. Finite Elements in Analysis and Design, 28:321–336, 1998. 1

[112] S. Mohammadi, D. R. J. Owen, and D. Peric. 3D progressive damage analysis of composites by fi-nite/discrete element approach. In E. Onate, G. Bugeda, and B. Suarez, editors, European Congresson Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, 2000. CIMNE. OnCD-ROM. 1

[113] H.-B. Muhlhaus. Application of Cosserat theory in numerical solutions of limit load problems.Archive of Applied Mechanics (Ingenieur Archiv), 59:124–137, 1989. 118

[114] H.-B. Muhlhaus and E. C. Aifantis. A variational principle for gradient plasticity. InternationalJournal of Solids and Structures, 28:845–857, 1991. 86, 118

[115] H.-B. Muhlhaus and I. Vardoulakis. The thickness of shear bands in granular materials.Geotechnique, 37:271–283, 1987. 118

160 References

[116] T. Munz, K. Willam, and K. Runesson. Viscoplastic algorithmic operators and their localizationproperties. In R. de Borst, N. Bicanic, H. A. Mang, and G. Meschke, editors, Computational Mod-elling of Concrete Structures, pages 219–228, Rotterdam, 1998. Balkema. 74, 78

[117] A. E. Naaman. SIFCON: tailored properties for structural performance. In A. E. Naaman andH. W. Reinhardt, editors, High Performance Fibre Reinforced Cement Composites – HPFRCC-91, pages18–38, London, 1991. E & FN Spon. 5

[118] A. Needleman. Material rate dependence and mesh sensitivity in localization problems. ComputerMethods in Applied Mechanics and Engineering, 67:69–85, 1988. 117

[119] A. Needleman and V. Tvergaard. Finite element analysis of localization in plasticity. In J. T.Oden and G. F. Carey, editors, Finite Elements: Special Problems in Solid Mechanics, volume V, pages94–157. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1984. 146

[120] S. Nemat-Nasser and N. Okada. Radiographic and microscopic observation of shear bands ingranular materials. Geotechnique, 51(9):753–765, 2001. 146

[121] J. T. Oden, C. A. M. Duarte, and O. C. Zienkiewicz. A new cloud-based hp finite element method.Computer Methods in Applied Mechanics and Engineering, 153(1–2):117–126, 1998. 16

[122] J. Oliver. Modelling strong discontinuities in solid mechanics via strain softening constitutiveequations. Part 2: numerical simulation. International Journal for Numerical Methods in Engineering,39(21):3601–3623, 1996. 13, 16

[123] J. Oliver, M. Cervera, and O. Manzoli. Strong discontinuities and continuum plasticity models:the strong discontinuity approach. International Journal of Plasticity, 15:319–351, 1999. 1

[124] J. Oliver, A. E. Huespe, M. D. G. Pulido, and E. Chaves. From continuum mechanics to fracturemechanics: the strong discontinuity approach. Engineering Fracture Mechanics, 69:113–136, 2002.1

[125] M. Ortiz, Y. Leroy, and A. Needleman. A finite element method for localized failure analysis.Computer Methods in Applied Mechanics and Engineering, 61:189–214, 1987. 146

[126] J. Ozbolt. General microplane model for concrete. In F. H. Wittmann, editor, Numerical Models inFracture Mechanics of Concrete, pages 173–187, Rotterdam, Brookfield, 1993. A.A. Balkema. 120

[127] J. Ozbolt and Z. P. Bazant. Numerical smeared fracture analysis: nonlocal microcrack interactionapproach. International Journal for Numerical Methods in Engineering, 39:635–661, 1996. 120

[128] J. Pamin. Gradient-Dependent Plasticity in Numerical Simulation of Localisations Phenomena. PhDthesis, Delft University of Technology, 1994. 11, 86, 87, 94, 118, 121

[129] J. Pamin, H. Askes, and R. de Borst. Two gradient plasticity theories discretized with the Element-free Galerkin Method. Computer Methods in Applied Mechanics and Engineering, 192:2377–2403,2003. 135, 148

[130] J. Pamin and R. de Borst. Gradient-enhanced damage and plasticity models for plain and rein-forced concrete. In W. Wunderlich, editor, Proceedings of the European Conference on ComputationalMechanics – ECCM’99, pages 482–483, paper no. 636, Munich, 1999. Technical University of Mu-nich. 3, 4, 41, 53, 67

[131] R. H. J. Peerlings. Enhanced Damage Modelling for Fracture and Fatigue. PhD thesis, EindhovenUniversity of Technology, 1999. 3, 44, 56, 61, 127, 140

References 161

[132] R. H. J. Peerlings, R. de Borst, W. A. M. Brekelmans, and M. G. D. Geers. Gradient-enhanceddamage modelling of concrete fracture. Mechanics of Cohesive-Frictional Materials, 3:323–342, 1998.61, 67, 121

[133] R. H. J. Peerlings, R. de Borst, W. A. M. Brekelmans, and M. G. D. Geers. Gradient-enhanceddamage modelling of high-cycle fatigue. International Journal for Numerical Methods in Engineering,49:1547–1569, 2000. 44

[134] R. H. J. Peerlings, R. de Borst, W. A. M. Brekelmans, and J. H. P. de Vree. Gradient-enhanced dam-age for quasi-brittle materials. International Journal for Numerical Methods in Engineering, 39:3391–3403, 1996. 8, 9, 41–43, 47, 50, 72, 96, 118, 121, 125–127, 130, 133, 135, 136, 150

[135] R. H. J. Peerlings, M. G. D. Geers, R. de Borst, and W. A. M. Brekelmans. A critical comparison ofnonlocal and gradient-enhanced softening continua. International Journal of Solids and Structures,38:7723–7746, 2001. 9, 137, 140, 146

[136] P. Perzyna. The constitutive equations for rate sensitive plastic materials. Quarterly of AppliedMathematics, 20(4):321–332, 1962. 118

[137] P. Perzyna. Fundamental problems in viscoplasticity. In Advances in Applied Mechanics, volume 9,pages 243–377. Academic Press, New York, 1966. 12, 71, 77, 118

[138] S. Pietruszczak and Z. Mroz. Finite element analysis of deformation of strain softening materials.International Journal for Numerical Methods in Engineering, 17:327–334, 1981. 2, 117

[139] G. Pijaudier-Cabot and Z. P. Bazant. Nonlocal damage theory. Journal of Engineering Mechanics,113:1512–1533, 1987. 9, 41, 117, 118, 120, 125, 130, 132, 135, 150

[140] G. Pijaudier-Cabot, Z. P. Bazant, and M. Tabbara. Comparison of various models for strain-softening. Engineering Computations, 5:141–150, 1988. 117

[141] G. Pijaudier-Cabot and A. Benallal. Strain localization and bifurcation in a nonlocal continuum.International Journal of Solids and Structures, 30:1761–1775, 1993. 118

[142] A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin, 1994. 131, 132

[143] J. J. C. Remmers, R. de Borst, and A. Needleman. A cohesive segments method for the simulationof crack growth. Computational Mechanics, 31(1–2):69–77, 2003. 28

[144] J. J. C. Remmers, G. N. Wells, and R. de Borst. Analysis of delamination growth with discon-tinuous finite elements. In Z. Waszczyszyn and J. Pamin, editors, Solids, Structures and CoupledProblems in Engineering, Proceedings of the Second European Conference on Computational Mechanics,volume 2, pages 906–907, Cracow, Poland, 2001. Fundacja Zdrowia Publicznego, ‘Vesalius’ Pub-lisher. On CD-ROM. 29

[145] Z. Ren and N. Bicanic. Simulation of progressive fracturing under dynamic loading conditions.Communications in Numerical Methods in Engineering, 13:127–138, 1997. 1

[146] RILEM. TC 162-TDF: Test and design methods for steel fibre reinforced concrete. Materials andStructures/Materiaux et Constructions, 33(225):3–5, 2000. 120

[147] RILEM. TC 162-TDF: Test and design methods for steel fibre reinforced concrete. Materials andStructures/Materiaux et Constructions, 33(226):75–81, 2000. 120

[148] E. Rizzi, I. Carol, and K. Willam. Localization analysis of elastic degradation with application toscalar damage. Journal of Engineering Mechanics, 121(4):541–554, 1995. 145, 146

162 References

[149] A. Rodrıguez-Ferran and A. Huerta. Error estimation and adaptivity for nonlocal damage models.International Journal of Solids and Structures, 37:7501–7528, 2000. 120–123

[150] P. E. Roelfstra and F. H. Wittmann. Numerical method to link strain softening with failure ofconcrete. In F. H. Wittmann, editor, Fracture Toughness and Fracture Energy, pages 163–175, Ams-terdam, 1986. Elsevier. 119

[151] J. G. Rots. Computational Modeling of Concrete Fracture. PhD thesis, Delft University of Technology,1988. 1, 28–30, 117, 121

[152] J. G. Rots. Removal of finite elements in strain-softening analysis of tensile fracture. In Z. P. Bazant,editor, Fracture Mechanics of Concrete Structures. Proceedings of the First International Conference onFracture Mechanics of Concrete and Concrete Structures – FramCoS 1, pages 330–338, London, 1992.Elsevier. 1, 2

[153] J. G. Rots and J. Blaauwendraad. Crack models for concrete: Discrete or smeared? Fixed, multi-directional or rotating? HERON, 34(1):3–59, 1989. 1, 117, 121

[154] J. G. Rots, P. Nauta, G. M. A. Kusters, and J. Blaauwendraad. Smeared crack approach and fracturelocalization in concrete. HERON, 30(1):3–48, 1985. 1, 117, 121

[155] J. G. Rots and J. C. J. Schellekens. Interface elements in concrete mechanics. In N. Bicanic and H. A.Mang, editors, Computer Aided Analysis and Design of Concrete Structures, pages 909–918, Swansea,U.K., 1990. Pinerigde Press. 110

[156] K. Runesson, N. S. Ottosen, and D. Peric. Discontinuous bifurcation of elasto-plastic solutions inplane stress and plane strain. International Journal of Plasticity, 7:99–121, 1991. 145, 146, 148

[157] K. Runesson, M. Ristinmaa, and L. Mahler. A comparison of viscoplasticity formats and algo-rithms. Mechanics of Cohesive-Frictional Materials, 4(1):75–98, 1999. 72

[158] H. G. Russel. ACI defines High Performance Concrete. Concrete International, 21(2):56–57, 1999.4

[159] V. E. Saouma, E. Bruhwiler, and H. L. Boggs. A review of fracture mechanics applied to concretedams. International Journal of Dam Engineering, 1(1):41–57, 1990. 13

[160] J. C. J. Schellekens and R. de Borst. On the numerical integration of interface elements. Interna-tional Journal for Numerical Methods in Engineering, 36:43–66, 1993. 30, 31, 105, 110

[161] E. Schlangen. Experimental and Numerical Analysis of Fracture Processes in Concrete. PhD thesis,Delft University of Technology, 1993. Also published in HERON, 38(2):3–117, 1993. 60, 61, 118,120, 121, 123

[162] J. C. Simo and T. J. R. Hughes. Computational Inelasticity. Springer, London, 1998. 76

[163] J. C. Simo and J. W. Ju. Strain- and stress-based continuum damage models–I. Formulation. Inter-national Journal of Solids and Structures, 23(7):821–840, 1987. 72, 76, 117

[164] J. C. Simo and J. W. Ju. Strain- and stress-based continuum damage models–II. Computationalaspects. International Journal of Solids and Structures, 23(7):841–869, 1987. 72, 76, 117

[165] J. C. Simo, J. G. Kennedy, and S. Govindjee. Non-smooth multisurface plasticity and viscoplas-ticity. loading/unloading conditions and numerical algorithms. International Journal for NumericalMethods in Engineering, 26:2161–2185, 1988. 12, 73

References 163

[166] J. C. Simo, J. Oliver, and F. Armero. An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. Computational Mechanics, 12:277–296, 1993. 13, 16

[167] J. C. Simo and R. L. Taylor. A return mapping algorithm for plane stress elastoplasticity. Interna-tional Journal for Numerical Methods in Engineering, 22:649–670, 1986. 11

[168] A. Simone. Partition of unity-based discontinuous elements for interface phenomena: computa-tional issues. 2003. Submitted. 13, 105

[169] A. Simone, H. Askes, R. H. J. Peerlings, and L. J. Sluys. Interpolation requirements for implicitgradient-enhanced continuum damage models. Communications in Numerical Methods in Engineer-ing, 19(7):563–572, 2003. See also the corrigenda which will appear in Communications in NumericalMethods in Engineering. 125

[170] A. Simone, H. Askes, and L. J. Sluys. Incorrect initiation and propagation of failure in non-localand gradient-enhanced media. International Journal of Solids and Structures, 2003. In press. 135

[171] A. Simone, J. J. C. Remmers, and G. N. Wells. An interface element based on the partition of unity.Technical Report CM2001-007, Delft University of Technology, Delft, the Netherlands, November2001. 13, 105

[172] A. Simone and L. J. Sluys. Continuous-discontinuous failure analysis in a rate-dependent elasto-plastic damage model. In F. Bontempi, editor, Proceedings of the 2nd International Structural Engi-neering and Construction Conference – ISEC-02, 23-26 September 2003. 71

[173] A. Simone and L. J. Sluys. The use of displacement discontinuities in a rate-dependent medium.Computer Methods in Applied Mechanics and Engineering, 2003. In press. 71

[174] A. Simone, L. J. Sluys, and P. Kabele. Combined continuous/discontinuous failure of cementi-tious composites. In N. Bicanic, R. de Borst, H. A. Mang, and G. Meschke, editors, ComputationalModelling of Concrete Structures – EURO-C 2003, pages 133–137. Swets & Zeitlinger, Lisse, 17-20March 2003. 13

[175] A. Simone, G. N. Wells, and L. J. Sluys. Discontinuous modelling of crack propagation in agradient-enhanced continuum. In H. A. Mang, F. G. Rammerstorfer, and J. Eberhardsteiner, edi-tors, Proceedings of the Fifth World Congress on Computational Mechanics – WCCM V, volume I, page130. Vienna University of Technology, Austria, 7–12 July 2002. 41

[176] A. Simone, G. N. Wells, and L. J. Sluys. From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Computer Methods in Applied Mechanics and Engineering,2002. In press. 41

[177] A. Simone, G. N. Wells, and L. J. Sluys. Discontinuities in regularised media. In D. R. J. Owen,E. Onate, and B. Suarez, editors, Proceedings of the 7th International Conference on ComputationalPlasticity – Complas 2003, page 90. CIMNE, Barcelona, Spain, 7-10 April 2003. On CD-ROM. 71

[178] D. Simons. Grid convergence in several simple problems. In G. H. Hulbert, editor, Sixth U.S. Na-tional Congress on Computational Mechanics, page 279. Mechanical Engineering Department, Uni-versity of Michigan, 1–3 August 2001. 120

[179] L. J. Sluys. Wave Propagation, Localisation and Dispersion in Softening Solids. PhD thesis, DelftUniversity of Technology, 1992. 71, 72, 87, 99, 117, 118, 146, 149

[180] L. J. Sluys and R. de Borst. Wave propagation and localization in a rate-dependent crackedmedium—model formulation and one-dimensional examples. International Journal of Solids andStructures, 29:2945–2958, 1992. 1, 117

164 References

[181] L. J. Sluys and R. de Borst. Failure in plain and reinforced concrete—an analysis of crack widthand crack spacing. International Journal of Solids and Structures, 33(20–22):3257–3276, 1996. 1

[182] L. J. Sluys, R. de Borst, and H.-B. Muhlhaus. Wave propagation, localization and dispersion in agradient-dependent medium. International Journal of Solids and Structures, 30(9):1153–1171, 1993.118, 146

[183] I. Stakgold. Green’s Functions and Boundary Value Problems. John Wiley & Sons, New York, 1979.115

[184] H. Stang and J. F. Olesen. On the interpretation of bending tests on frc-materials. In H. Mihashiand K. Rokugo, editors, Fracture Mechanics of Concrete Structures. Proceedings of the Third Interna-tional Conference on Fracture Mechanics of Concrete and Concrete Structures – FramCoS 3, volume 1,pages 511–520, Freiburg, Germany, 1998. Aedificatio Publishers. 92, 120

[185] M. G. A. Tijssens, L. J. Sluys, and E. van der Giessen. Numerical simulation of quasi-brittle fractureusing damaging cohesive surfaces. European Journal of Mechanics A/Solids, 19(5):761–779, 2000. 12,39

[186] V. Tvergaard, A. Needleman, and K. K. Lo. Flow localization in the plane strain tensile test. Journalof the Mechanics and Physics of Solids, 29(2):115–142, 1981. 146

[187] M. R. A. van Vliet. Size Effect in Tensile Fracture of Concrete and Rock. PhD thesis, Delft Universityof Technology, 2000. 120

[188] J. H. P. de Vree, W. A. M. Brekelmans, and M. A. J. van Gils. Comparison of nonlocal approachesin continuum damage mechanics. Computers and Structures, 55:581–588, 1995. 7

[189] W. M. Wang. Stationary and Propagative Instabilities in Metals – A Computational Point of View. PhDthesis, Delft University of Technology, 1997. 82, 117

[190] W. M. Wang, L. J. Sluys, and R. de Borst. Viscoplasticity for instabilities due to strain softeningand strain-rate softening. International Journal for Numerical Methods in Engineering, 40:3839–3864,1997. 117

[191] R. J. Ward and V. C. Li. Dependence of flexural behavior of fiber reinforced mortar on materialfracture resistance and beam size. ACI Materials Journal, 87(6):627–637, 1990. 32

[192] G. N. Wells. Discontinuous Modelling of Strain Localisation and Failure. PhD thesis, Delft Universityof Technology, 2001. 45, 86, 87

[193] G. N. Wells and L. J. Sluys. A new method for modelling cohesive cracks using finite elements.International Journal for Numerical Methods in Engineering, 50(12):2667–2682, 2001. 3, 14, 15, 17,21–23, 28, 31, 32, 34, 51, 121

[194] G. N. Wells, L. J. Sluys, and R. de Borst. Simulating the propagation of displacement disconti-nuities in a regularised strain-softening medium. International Journal for Numerical Methods inEngineering, 53(5):1235–1256, 2002. 1, 24, 52, 71, 87, 146

[195] J. G. Williams and P. D. Ewing. Fracture under complex stress – the angled crack problem. Inter-national Journal of Fracture Mechanics, 8(4):441–446, 1972. 22

[196] F. H. Wittmann, H. Mihashi, and N. Nomura. Size effect on fracture energy of concrete. Engineer-ing Fracture Mechanics, 35:107–115, 1990. 120

[197] X.-P. Xu and A. Needleman. Numerical simulations of fast crack growth in brittle solids. Journalof the Mechanics and Physics of Solids, 42:1397–1434, 1994. 12, 39

References 165

[198] L. F. Zeng, G. Horrigmoe, and R. Andersen. Numerical implementation of constitutive integrationfor rate-independent elastoplasticity. Computational Mechanics, 18:387–396, 1996. 74

[199] A. Zervos, P. Papanastasiou, and I. Vardoulakis. A finite element displacement formulation forgradient elastoplasticity. International Journal for Numerical Methods in Engineering, 50:1369–1388,2001. 146

[200] D. Zhang and K. Wu. Fracture process zone of notched three-point-bending concrete beams.Cement and Concrete Research, 29:1887–1892, 1999. 120

[201] C. Zhao, B. E. Hobbs, H.-B. Muhlhaus, and A. Ord. A consistent point-searching algorithm for so-lution interpolation in unstructured meshes consisting of 4-node bilinear quadrilateral elements.International Journal for Numerical Methods in Engineering, 45(10):1509–1526, 1999. 24

[202] O. C. Zienkiewicz and R. L. Taylor. The Finite Element Method. Butterworth-Heinemann, Oxford,UK, fifth edition, 2000. 132

Author/editor index

Aifantis, E. C., 118Aifantis, E. C., see Muhlhaus, H.-B., 86, 118Aıtcin, P. C., 4Alfaiate, J., 16, 19Alfano, G., 13Andersen, R., see Zeng, L. F., 74Armero, F., 13, 16Armero, F., see Simo, J. C., 13, 16Askes, H., 44, 120, 121Askes, H., see Pamin, J., see Simone, A., 125,

135, 148ASTM, 56Audi, M., 29

Baaijens, F. P. T., see Engelen, R. A. B., 87, 99,135, 148

Babuska, I., 132Babuska, I., see Melenk, J. M., 13, 16Bathe, K. J., 132Bazant, Z. P., 1, 2, 35, 38, 71, 117, 118, 120, 125,

130, 132Bazant, Z. P., viii, see Jirasek, M., see Ozbolt, J.,

see Pijaudier-Cabot, G., 6, 9, 41, 48,72, 117, 118, 120, 125, 130, 132, 135,150

Belytschko, T., 120Belytschko, T., see Dolbow, J., see Lasry, D., see

Moes, N., 3, 13, 15, 21–23, 28, 31,117, 118, 121

Benallal, A., see Borst, R. de, seePijaudier-Cabot, G., 118, 121, 125,130

Bicanic, N., 1, 13, 16, 110, 119–121Bicanic, N., see Borst, R. de, see Ren, Z., 1, 74, 78Bittencourt, T. N., 13Blaauwendraad, J., see Rots, J. G., 1, 117, 121Blain, C. A., see Gray, W. G., 115Bode, L., 1Boggs, H. L., see Saouma, V. E., 13Bolzon, G., 1Bontempi, F., 71Borja, R. I., 146Borst, R. de, 1, 56, 57, 67, 74, 78, 117, 118, 121,

125, 130, 146Borst, R. de, see Bicanic, N., see Feenstra, P. H.,

see Geers, M. G. D., see Kuhl, E., seePamin, J., see Peerlings, R. H. J., seeRemmers, J. J. C., see Schellekens, J.C. J., see Sluys, L. J., see Wang, W.

M., see Wells, G. N., 1–4, 8, 9, 13, 16,24, 28–31, 41–44, 47, 48, 50–53, 56,57, 61, 67, 69, 71, 72, 87, 88, 96, 99,105, 110, 117–121, 125–127, 130,133, 135–137, 140, 146, 148, 150

Brekelmans, W. A. M., see Geers, M. G. D., seePeerlings, R. H. J., see Vree, J. H. P.de, 2, 7–9, 41–44, 47, 48, 50, 51, 56,57, 61, 67, 69, 72, 87, 88, 96, 99, 118,119, 121, 125–127, 130, 133,135–137, 140, 146, 150

Brekke, T. L., see Goodman, R. E., 13Brezzi, F., 132Brokken, D., 44Bruhwiler, E., see Saouma, V. E., 13Bugeda, G., see Onate, E., 1

Carey, G. F., see Oden, J. T., 146Carmeliet, J., 119Carol, I., 145, 146, 148Carol, I., see Ghavamian, S., see Rizzi, E., 120,

121, 145, 146Cedolin, L., see Bazant, Z. P., 117Cervera, M., see Oliver, J., 1Chaves, E., see Oliver, J., 1Chorin, A., 72, 76Clement, J. L., see Bode, L., 1Comi, C., 132Corigliano, A., 99Corigliano, A., see Bolzon, G., 1Cosserat, E., 118Cosserat, F., see Cosserat, E., 118Crisfield, M. A., 2, 6, 76, 78Crisfield, M. A., see Alfano, G., 13

De Proft, K., 119De Wilde, W., see De Proft, K., 119Diebels, S., see Vermeer, P. A., 50Dietsche, A., 118Dolbow, J., 21Dolbow, J., see Moes, N., 3, 13, 15, 21, 23, 28, 31Duarte, C. A. M., 13, 16Duarte, C. A. M., see Oden, J. T., 16Dube, J. F., 117Duvaut, G., 12, 71, 73, 118Dwivedi, S. K., see Espinosa, H., 1

Eberhardsteiner, J., see Mang, H. A., 11, 41, 74Ehlers, W., see Vermeer, P. A., 50

168 Author/editor index

Eligehausen, R., see Bazant, Z. P., 120Engelen, R. A. B., 87, 99, 135, 148Eringen, A. C., 141Espinosa, H., 1Ewalds, H. L., 138Ewing, P. D., see Williams, J. G., 22

Feenstra, P. H., 1, 121Fichant, S., 35Fortin, M., see Brezzi, F., 132

Garaizar, F. X., 146Garikipati, K., see Armero, F., 13, 16Geers, M. G. D., 2, 7, 8, 41, 44, 48, 51, 56, 57, 67,

69, 87, 88, 99, 119, 120Geers, M. G. D., see Borst, R. de, see Engelen, R.

A. B., see Peerlings, R. H. J., 9, 44,56, 57, 61, 67, 87, 99, 121, 135, 137,140, 146, 148

Georgin, J. F., 11, 74Gerstle, W. H., 120Ghavamian, S., 120, 121Giessen, E. van der, see Tijssens, M. G. A., 12,

39Gils, M. A. J. van, see Vree, J. H. P. de, 7Goodman, R. E., 13Govindjee, S., see Simo, J. C., 12, 73Gray, W. G., 115, 116Grootenboer, H. J., 1

Hashida, T., see Li, V. C., 35, 38Heeres, O. M., see Borst, R. de, 121, 125, 130Herrmann, H. J., see Vermeer, P. A., 50Hillerborg, A., 92Hinton, E., see Owen, D. R. J., 145, 146, 148Hobbs, B. E., see Zhao, C., 24Holzapfel, G. A., viiiHordijk, D. A., 2–4, 53, 54, 64Horii, H., see Kabele, P., 1, 35Horrigmoe, G., see Zeng, L. F., 74Huerta, A., see Rodrıguez-Ferran, A., 120–123Huespe, A. E., see Oliver, J., 1Hughes, T. J. R., 132Hughes, T. J. R., see Chorin, A., see Simo, J. C.,

72, 76Hulbert, G. H., 120Hult, J., see Janson, J., 1

Iacono, C., 119Ingraffea, A. R., see Bittencourt, T. N., see

Gerstle, W. H., 13, 120

Janson, J., 1Jenq, Y. S., 12Jirasek, M, see Bazant, Z. P., 71, 118

Jirasek, M., viii, 1, 6, 23, 48, 50, 69, 72, 120Ju, J. W., 1, 76, 117Ju, J. W., see Simo, J. C., 72, 76, 117

Kabele, P., 1, 35, 38, 39Kabele, P., see Simone, A., 13Kalman, R. E., 99Kanda, T., 35, 119Kennedy, J. G., see Simo, J. C., 12, 73Kim, B. S., see Eringen, A. C., 141Kolar, R. L., see Gray, W. G., 115Krcek, J., see Audi, M., 29Krenchel, H., see Li, V. C., 92Kuhl, E., 67Kuhl, E., see Mahnken, R., 119Kullaa, J., 33, 34Kusters, G. M. A., see Rots, J. G., 1, 117, 121

La Borderie, C., see Bode, L., see Dube, J. F., seeFichant, S., 1, 35, 117

Lasry, D., 117, 118Lee, P. C. Y., see Gray, W. G., 115, 116Leijnse, A., see Gray, W. G., 115Lemaitre, J., 72Leroy, Y., see Ortiz, M., 146Li, F., 119Li, V. C., 5, 34, 35, 38, 92Li, V. C., see Kanda, T., see Ward, R. J., 32, 35,

119Li, Z., see Li, F., 119Lin, F. B., see Bazant, Z. P., 118Lin, Z., see Kanda, T., 35, 119Lions, J. L., see Duvaut, G., 12, 71, 73, 118Lo, K. K., see Tvergaard, V., 146Loret, B., 117Luding, S., see Vermeer, P. A., 50

Mahler, L., see Runesson, K., 72Mahnken, R., 119Malvern, L. E., 19Mang, H. A., 11, 41, 74Mang, H. A., see Bicanic, N., see Borst, R. de, 1,

13, 16, 74, 78, 110, 119–121Manzoli, O., see Oliver, J., 1Mariani, S., see Corigliano, A., 99Marsden, J. E., see Chorin, A., 72, 76Mazars, J., 1, 7McCracken, M. F., see Chorin, A., 72, 76Melenk, J. M., 13, 16Meschke, G., see Bicanic, N., see Borst, R. de, 1,

13, 16, 74, 78, 119–121Mier, J. G. M. van, 2, 64, 71, 117, 137Mier, J. G. M. van, see Iacono, C., 119Mihashi, H., 92, 120Mihashi, H., see Wittmann, F. H., 120

Author/editor index 169

Mish, K., see Belytschko, T., 120Moes, N., 3, 13, 15, 21–23, 28, 31, 121Moes, N., see Dolbow, J., 21Mohammadi, S., 1Mroz, Z., see Pietruszczak, S., 2, 117Muhlhaus, H.-B., 86, 118Muhlhaus, H.-B., see Borst, R. de, see Sluys, L.

J., see Zhao, C., 24, 117, 118, 146Munz, T., 74, 78

Naaman, A. E., 5Nasser, S. Nemat-, see Nemat-Nasser, S.Nauta, P., see Rots, J. G., 1, 117, 121Nechnech, W., see Georgin, J. F., 11, 74Needleman, A., 117, 146Needleman, A., see Ortiz, M., see Remmers, J. J.

C., see Tvergaard, V., see Xu, X.-P.,12, 28, 39, 146

Nemat-Nasser, S., 146Nomura, N., see Wittmann, F. H., 120

Oden, J. T., 16, 146Oden, J. T., see Duarte, C. A. M., 13, 16Oh, B., see Bazant, Z. P., 2, 117Okada, N., see Nemat-Nasser, S., 146Olesen, J. F., see Stang, H., 92, 120Oliver, J., 1, 13, 16Oliver, J., see Simo, J. C., 13, 16Onate, E., 1Onate, E., see Owen, D. R. J., 71, 119, 145, 146,

148Ord, A., see Zhao, C., 24Ortiz, M., 146Ottosen, N. S., see Runesson, K., 145, 146, 148Owen, D. R. J., 71, 119, 145, 146, 148Owen, D. R. J., see Mohammadi, S., 1Ozbolt, J., 120Ozbolt, J., see Bazant, Z. P., 120

Pamin, J., 3, 4, 11, 41, 53, 67, 86, 87, 94, 118, 121,135, 148

Pamin, J., see Borst, R. de, see Waszczyszyn, Z.,29, 56, 57, 67, 117, 120

Papanastasiou, P., see Zervos, A., 146Patzak, B., see Jirasek, M., 120Peerlings, R. H. J., 3, 8, 9, 41–44, 47, 50, 56, 61,

67, 72, 96, 118, 121, 125–127, 130,133, 135–137, 140, 146, 150

Peerlings, R. H. J., see Geers, M. G. D., seeSimone, A., 2, 8, 41, 44, 48, 56, 57,67, 69, 87, 88, 99, 125

Peijs, A. A. J. M., see Geers, M. G. D., 119Peijs, T., see Geers, M. G. D., 51, 56, 119Peric, D., see Mohammadi, S., see Runesson, K.,

1, 145, 146, 148

Perzyna, P., 12, 71, 77, 118Pietruszczak, S., 2, 117Pijaudier-Cabot, G., 9, 41, 117, 118, 120, 125,

130, 132, 135, 150Pijaudier-Cabot, G., see Bazant, Z. P., see Bode,

L., see Dube, J. F., see Fichant, S., seeMazars, J., 1, 7, 35, 117, 118, 120,125, 130, 132

Prevost, J. H., see Loret, B., 117Pulido, M. D. G., see Oliver, J., 1

Quarteroni, A., 131, 132

Ramm, E., see Kuhl, E., see Vermeer, P. A., 50,67

Rammerstorfer, F. G., see Mang, H. A., 11, 41,74

Reinhardt, H. W., see Naaman, A. E., 5Remmers, J. J. C., 28, 29Remmers, J. J. C., see Simone, A., 13, 105Ren, Z., 1Reynouard, J. F., see Georgin, J. F., 11, 74RILEM, 120Ristinmaa, M., see Runesson, K., 72Rizzi, E., 145, 146Rodrıguez-Ferran, A., 120–123Roelfstra, P. E., 119Rokugo, K., see Mihashi, H., 92, 120Rots, J. G., 1, 2, 28–30, 110, 117, 121Runesson, K., 72, 145, 146, 148Runesson, K., see Munz, T., 74, 78Russel, H. G., 4

Saouma, V. E., 13Schellekens, J. C. J., 30, 31, 105, 110Schellekens, J. C. J., see Rots, J. G., 110Schlangen, E., 60, 61, 118, 120, 121, 123Sejnoha, M., see Audi, M., 29Shah, S. P., 120Shah, S. P., see Jenq, Y. S., 12Simo, J. C., 11–13, 16, 72, 73, 76, 117Simone, A., 13, 41, 71, 105, 125, 135Simone, A., see Alfaiate, J., 16, 19Simons, D., 120Sluys, L. J., 1, 71, 72, 87, 99, 117, 118, 146, 149Sluys, L. J., see Alfaiate, J., see Askes, H., see

Borst, R. de, see Georgin, J. F., seeIacono, C., see Simone, A., seeTijssens, M. G. A., see Wang, W. M.,see Wells, G. N., see De Proft, K., 1,3, 11–17, 19, 21–24, 28, 31, 32, 34,39, 41, 51, 52, 71, 74, 87, 117–119,121, 125, 135, 146

Souza, J. L., see Bittencourt, T. N., 13Speziale, C. G., see Eringen, A. C., 141

170 Author/editor index

Stakgold, I., 115Stang, H., 92, 120Stang, H., see Li, V. C., 92Steinmann, P., see Dietsche, A., 118Suarez, B., see Owen, D. R. J., see Onate, E., 1,

71, 119Swartz, S. E., see Shah, S. P., 120

Tabbara, M., see Pijaudier-Cabot, G., 117Tailhan, J. L., see Bode, L., 1Taylor, R. L., see Goodman, R. E., see Simo, J.

C., see Zienkiewicz, O. C., 11, 13,132

Tijssens, M. G. A., 12, 39Trangenstein, J., see Garaizar, F. X., 146Tvergaard, V., 146Tvergaard, V., see Needleman, A., 146

Valli, A., see Quarteroni, A., 131, 132Vardoulakis, I., see Muhlhaus, H.-B., see

Zervos, A., 118, 146Vermeer, P. A., 50Vliet, M. R. A. van, 120Vree, J. H. P. de, 7Vree, J. H. P. de, see Peerlings, R. H. J., 8, 9,

41–43, 47, 50, 72, 96, 118, 121,125–127, 130, 133, 135, 136, 150

Wang, M. L., see Shah, S. P., 120Wang, W. M., 82, 117Wanhill, R. J. H., see Ewalds, H. L., 138Ward, R. J., 32Waszczyszyn, Z., 29, 120Wawrzynek, P. A., see Bittencourt, T. N., 13Wells, G. N., 1, 3, 14, 15, 17, 21–24, 28, 31, 32,

34, 45, 51, 52, 71, 86, 87, 121, 146Wells, G. N., see Remmers, J. J. C., see Simone,

A., see De Proft, K., 13, 29, 41, 71,105, 119

Willam, K., see Carol, I., see Dietsche, A., seeRizzi, E., see Munz, T., 74, 78, 118,145, 146, 148

Williams, J. G., 22Wills, J., see Crisfield, M. A., 2Wittmann, F. H., 119, 120Wittmann, F. H., see Roelfstra, P. E., 119Wu, H. C., see Li, V. C., 5, 34, 38Wu, K., see Zhang, D., 120Wunderlich, W., 3, 4, 41, 53, 67

Xu, X.-P., 12, 39

Yamaguchi, E., see Kabele, P., 1

Zavattieri, P. D., see Espinosa, H., 1

Zeman, J., see Audi, M., 29Zeng, L. F., 74Zervos, A., 146Zhang, D., 120Zhao, C., 24Zienkiewicz, O. C., 132Zienkiewicz, O. C., see Oden, J. T., 16Zimmermann, T., see Jirasek, M., 1, 23, 69

Subject index

applicationscomposite compact-tension specimen,

56–60, 87–89concrete beam in four-point bending,

53–56damage characterisation in mode I

problems, 137damage characterisation in shear band

problems, 145discontinuities

‘dummy stiffness’, 28–31cohesive, 32–38, 92–95traction-free, 53–69, 87–98

linear-elastic analysis of a notched beam,28–31

single-edge notched beam, 60–64,121–123

steel fibre-reinforced concrete, 92–95strip footing near a slope, 89–92

Babuska-Brezzi condition, 132bumps, see discontinuity and bumps

comparisonsDuvaut-Lions vs Perzyna

rate-dependentdamage-elastoplasticity, 78, 82

interface vs PU-based elements, 105–111Mazars vs modified von Mises

equivalent strain, 121–123rate-dependent model vs

gradient-enhanced model, 89–91,96–98, 121–123, 145–150

continuum modelsconsiderations on numerical modelling

of concrete, 119–121damage, 7–9for softening materials, 8–9, 12, 117gradient-enhanced damage, 42limitations, 1–4, 53–56, 103, 119–121,

135–151plasticity, 9–12rate-dependent damage-elastoplasticity

Duvaut-Lions, 73, 102general formulation, 72Perzyna, 76

rate-dependent elastoplasticity, 86reassessment of model parameters, 56,

61, 102coupled problem, 130–133crack propagation

shock-wise, 65–69, 98smooth, 98

damage evolution lawsexponential softening, 8, 44, 53, 64linear softening, 8, 127modified power softening, 8, 57, 64, 67

discontinuityand boundary conditions, 16, 17, 41, 43and bumps, 59, 67, 68, 98and drops, 55, 61, 64, 87, 88and non-locality, 65–69, 96–98, 102, 137and rate-dependence, 96–98cohesive, 94direction, 22, 51, 85, 90in problem fields, 14, 42insertion, 21, 50, 85motivations, 1–4quality of the discontinuous

enhancement, 103, seediscontinuity and bumps, seediscontinuity and drops

requirements on the underlyingcontinuum model, 6

distributed failure in discontinuous setting,102

drops, see discontinuity and drops

element size vs length scale, 67energy

complementary energy, 131total potential energy, 131

equivalent strain definitionvon Mises, 139, 146

modified, 7, 53, 61, 121positive principal strains, 7, 57, 121, 142

fracture energy, 120

Helmholtz equation, 9, 43, 45, 125, 137homogenisation, 119

172 Subject index

inf-sup condition, see Babuska-Brezzicondition

interpolationlinear-linear, 127quadratic-linear, 130requirements for gradient-enhanced

media, 125–133inverse methods, 119, 120

knowledge, imperfect, 119

Lagrangian functional, 132Lagrangian multiplier, 132length scale

constant, 53variable, 68, 69

localisation limiters, see continuum models,forsoftening materials

mixed method, 130–133modelling

macromechanical approach, 119micromechanical approach, 119

numerical integration, 23, 28–31, 105–113

oscillationsin stress, 125, 127in non-local equivalent strain, 66–69,

96–98in traction profile, 28

panacea for discrete cracking, 102

realistic physical situation?, 1–4, 64–69, 71,135–151

regularised models, see continuum models,forsoftening materials

softening, 117strategies for failure analysis, 4–6

traction-separation law, 12transfer of history data, 24, 52

yield functionvon Mises, 10, 78, 82, 89, 149Rankine, 11, 87, 94, 98, 121

Summary

Continuous-Discontinuous Modelling of Failureby A. Simone

The foundations of a safe structural design lie on the understanding of fail-ure processes of engineering materials and in their correct representation.In a numerical context, failure representation in engineering materials canbe pursued either in a continuous or in a discontinuous setting. Both ap-proaches can model certain failure modes, but in some cases do not reflectthe physical processes behind failure properly. To some extent, continuousfailure representation can be improved by enriching the standard kinemat-ics with displacement discontinuities, which can be thought of as a naturalconsequence of material failure processes.

In this work, a continuous-discontinuous approach is applied to elas-tic, strain-hardening and regularised strain-softening media. It is shownthat the success of a continuous-discontinuous analysis depends largelyon the underlying continuous model. Analyses performed with elastic andstrain-hardening media gave satisfactory results; conversely, when strain-softening media were considered, the performance of the approach was re-lated to the nature of the regularisation employed. It is shown that, in acontinuous-discontinuous setting, softening models in which the underly-ing continuum description is enriched through a temporal regularisationformalism (rate-dependence) perform better than models obeying a spatialregularisation concept (non-locality). This issue is examined with respect toa differential version of a non-local model (implicit gradient-enhanced dam-age continuum model) and a rate-dependent elastoplastic-damage model.More specifically, it is shown that a class of regularised models based ona non-local dissipation driving variable is not adequate for failure descrip-tion in arbitrary loading scenarios. Several applications illustrate the im-proved flexibility of a continuous-discontinuous approach to failure whencompared to a continuous approach alone.

Samenvatting

Continue-Discontinue Modellering van Bezwijkgedragdoor A. Simone

Een veilig constructief ontwerp is gebaseerd op een goed begrip vanbezwijkprocessen van technische materialen en op hun correcte represen-tatie. Het bezwijkgedrag van technische materialen kan in een numeriekecontext worden beschreven in een continu of in een discontinu kader. Beideaanpakken zijn ideaal voor de modellering van bepaalde bezwijkmechanis-men, maar weerspiegelen in sommige gevallen de fysische processen nietcorrect. Continue representaties van bezwijken kunnen tot op zekere hoogteworden verbeterd door de standaard kinematica te verrijken met verplaat-singsdiscontinuıteiten, hetgeen kan worden beschouwd als een natuurlijkeconsequentie van bezwijkprocessen in materialen.

In deze studie is een continue-discontinue aanpak toegepast van elasti-sche, strain-hardening en strain-softening media. Aangetoond is dat het suc-ces van een continue-discontinue analyse grotendeels afhangt van het on-derliggende continue model. Analyses met elastische en strain-hardeningmedia vertoonden bevredigende resultaten; wanneer echter strain-softeningmedia worden beschouwd hangt de prestatie van de aanpak af van hettype regularisatie dat is gebruikt. Voor een continu-discontinu raamwerkis aangetoond dat softening modellen, waarvan het onderliggende con-tinuum is verrijkt via een tijdsregularisatietechniek (snelheidsafhankelijk-heid), beter presteren dan modellen met een ruimtelijk regularisatieconcept(niet-localiteit). Dit aspect is onderzocht voor een differentiaalversie van eenniet-locaal model (impliciet gradientverrijkte schade continuummodel) eneen snelheidsafhankelijk elastoplastisch schademodel. In het bijzonder isgetoond dat een bepaalde klasse van geregulariseerde modellen, gebaseerdop een niet-locale dissipatiesturende variabele, niet geschikt zijn voor hetbeschrijven van bezwijkgedrag in willekeurige belastingsscenario’s. Di-verse toepassingen illustreren de verbeterde flexibiliteit van een continue-discontinue benadering van bezwijkgedrag vergeleken met een continue be-nadering als zodanig.

Curriculum vitae

Dec. 19th, 1973 born in Taranto, Italy, as Angelo Simone

June 1992 maturita scientifica

Oct. 1998 laurea in ingegneria civile, Politecnico di Milano

June 1999 – Sep. 2003 research assistant, Faculty of Civil Engineering andGeosciences, Delft University of Technology

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