석사 학위논문

Master's Thesis

유한요소방법을 이용한 얇은 판 구조물에서의

레이저 유도 초음파의 열기계학적 해석

Thermomechanical Analysis of Laser-generated Ultrasonics in

Thin Structures by Finite Element Method

임 형 욱 (林 亨 旭 Lim, Hyeong Uk)

건설 및 환경공학과

Department of Civil and Environmental Engineering

KAIST

2014

유한요소방법을 이용한 얇은 판 구조물에서의

레이저 유도 초음파의 열기계학적 해석

Thermomechanical Analysis of Laser-generated Ultrasonics in

Thin Structures by Finite Element Method

Thermomechanical Analysis of Laser-generated Ultrasonics in Thin

Structures by Finite Element Method

Advisor: Professor Jung-Wuk Hong

by

Hyeong Uk Lim

Department of Civil and Environmental Engineering

KAIST

A thesis submitted to the faculty of KAIST in partial fulfillment of the re-

quirements for the degree of Master of Science and Engineering in the Depart-

ment of Civil and Environmental Engineering. The study was conducted in ac-

cordance with Code of Research Ethics1

2013. 12. 18

Approved by

Professor Jung-Wuk Hong

Seal or sign ature

1 Declaration of Ethical Conduct in Research: I, as a graduate student of KAIST, hereby declare that I have not

committed any acts that may damage the credibility of my research. These include, but are not limited to: falsi-fication, thesis written by someone else, distortion of research findings or plagiarism. I affirm that my thesis contains honest conclusions based on my own careful research under the guidance of my thesis advisor.

유한요소방법을 이용한 얇은 판 구조물에서의 레이저

유도 초음파의 열기계학적 해석

임 형 욱

위 논문은 한국과학기술원 석사학위논문으로

학위논문심사위원회에서 심사 통과하였음.

2013 년 12 월 18 일

심사위원장

심사위원

심사위원

홍 정 욱 (인)

손 훈 (인)

김 준 희 (인)

항상 존경하는 아버지, 사랑하는 어머니, 자상한 누나에게.

i

MCE

20123586

임 형 욱. Lim, Hyeong Uk. Thermomechanical Analysis of Laser-generated Ultra-sonics in Thin Structures by Finite Element Method. 유한요소방법을 이용한 얇은

판 구조물에서의 레이저 유도 초음파의 열기계학적 해석. Department of Civil

and Environmental Engineering. 2014. 82 p. Advisor Prof. Hong, Jung-Wuk. Text in

English

ABSTRACT

Ultrasonic guided waves have been widely utilized for the structural health monitoring (SHM) of

structural components such as plates and pipes; Lamb wave as one of means for nondestructive test-

ing and evaluation has emerged to provide a new way for the inspection of thin structures. In partic-

ular, the noncontact excitation of the pipe surfaces using laser pulses has shown several advantages

in experiments by eliminating the bonding process of the dielectric patches on the curved surfaces

and the complicated interpretation of the temperature effect on the bonding layers. However, the

numerical simulation of the methodology requires thermo-mechanical coupling and large-scale

computation. Therefore, the numerical efficiency of the spatial partitioning by deploying thermo-

mechanical elements and mechanical elements is investigated. The formation and propagation of the

guided waves are also represented numerically.

A multi-scale simulation technique for three-dimensional wave propagation through pipe-like struc-

tures is developed considering the noncontact excitation by a short duration laser pulse. Using this

technique, the displacement fields in pipes, both with and without a notch, are calculated and ana-

lyzed. In particular, the heat flux is applied to a micro-scale model, and then the temperature profile

is obtained. The obtained temperature profile is imposed as the thermal boundary of the macro-scale

global model. By applying the temperature profile to an intact pipe, the displacement fields are

compared with experimental results in a literature. On the other hand, it is found that the existence

of a notch makes effects on the propagation of waves. Therefore, the interaction causes some devia-

tion of the displacement fields. The dissimilarity in the displacement fields with different notch

depths is investigated using a statistical signal processing technique. Based on the investigation, a

damage detection technique that evaluates the existence of a notch in a pipe is proposed.

ii

The generation of the waves in a plate is induced by thermal expansion utilizing non-contact laser

pulses, and the waves propagate along the media. Three-dimensional finite element analysis is per-

formed to predict Lamb wave propagation precisely, which is enabled by computational power and

efficiency. Here, we focus on the study of three-dimensional Lamb wave propagation in an isotropic

plate, and thermo-elastic finite element simulation is conducted. The study consists of three steps.

Firstly, a brief description of the simulation methodology which is employed in order to describe

thermo-elastic response is presented. Secondly, through variables (e.g. the depth of the surface and

the neighborhood area of the laser pulse), the distribution of temperature field is investigated. Last-

ly, thermo-elastic responses during and after thermal expansion are studied to help better under-

standing of the Lamb waves and wave propagations. The feasibility of the proposed methodology is

verified, through the comparison of the analytical and the experimental results.

Keywords: Ultrasonic wave; Finite element analysis; Multi-scale simulation; Structural health mon-

itoring; Damage detection; Lamb waves; Thermo-mechanics; Laser ultrasound; Thin structure;

iii

CONTENTS

ABSTRACT ····················································································· i CONTENTS ··················································································· iii

LIST OF TABLES ········································································· v

LIST OF FIGURES ······································································ vi

CHAPTER 1

INTRODUCTION ···································································· 1

1.1 Research Background ···························································· 1

1.2 Literature Review ································································· 3

1.3 Research Objective ······························································· 6

1.4 Thesis Structure ··································································· 6

CHAPTER 2

THEORIES ··············································································· 8

2.1 Governing Equation of Motion for a Homogeneous Isotropic Material···· 8

2.2 Thermo-mechanically Coupled Finite Element Formulation ··············· 10

2.3 Bottom-up Analysis for a Multi-scale Modeling ····························· 14

2.3.1 Micro-scale Analysis ···················································· 15

2.3.2 Macro-scale Analysis ···················································· 16

2.4 Integral Transformation Method ··············································· 17

2.4.1 Analytical Solution of Transient Response Due to Laser Source ·· 18

2.4.2 Problem Statement ······················································· 19

2.4.3 Fourier and Hankel Transformation ···································· 20

2.4.4 Residue Theorem ························································· 20

2.5 Wavelet Transform ······························································ 21

CHAPTER 3

NUMERICAL MODEL ······················································· 23

3.1 Numerical Implementations ···················································· 23

iv

3.2 Partitioning Scheme ····························································· 24

3.3 Multi-scale Modeling ···························································· 25

3.4 Modeling of a Pipe with a Notch ··············································· 28

3.5 Validation and Verification of the Simulation Technique ·················· 29

3.6 Plate Modeling ··································································· 31

3.6.1 Numerical Implementations ············································· 31

3.6.2 Verification of the Simulation ·········································· 32

3.6.3 Case Studies······························································· 35

CHAPTER 4

ANALYTICAL MODEL ····················································· 42

4.1 Descriptions of Analytical Model ·············································· 42

4.2 Solutions of Circular-crested Waves ·········································· 42

4.3 Comparison Between the Numerical Results and the Analytic Results ··· 50

4.4 Excitation Function ······························································ 51

CHAPTER 5

APPLICATIONS ··································································· 53

5.1 Damage Detection Method ····················································· 53

5.2 Multi-resolution Analysis Using Wavelet Transform ······················· 62

CHAPTER 6

CONCLUSIONS ······························································ 686868

6.1 Concluding Remarks ···························································· 68

6.2 Contributions of the Thesis ····················································· 70

6.3 Recommendations for Further Work ·········································· 70

REFERENCES ············································································· 71

SUMMARY (IN KOREAN) ·················································· 7575

ACKNOWLEDGEMENTS (감사의글) ·································· 77

CURRICULUM VITAE ···························································· 779

v

LIST OF TABLES

Table 3.1 Material properties of the aluminum utilized in the simulations. ························ 24

Table 3.2 Material properties of the aluminum employed for numerical simulations. ············ 33

Table 3.3 Comparison of the velocities between the theoretical solution and the numerical solu-

tion. ······························································································ 36

Table 4.1 Theoretical conditions of the simulation of analytic solution of laser-generated Lamb

waves. ··························································································· 44

Table 5.1 RMSD results of signals with a notch. ······················································ 55

Table 5.2 Results of in the range of 0.42 MHz to 0.64 MHz. ·································· 57

Table 5.3 RMSD results of a signal with the notch when it lies transverse to a straight line be-

tween the laser source point and the observation point in the pipe. ······················ 59

Table 5.4 Results of in the range of 0.42 MHz to 0.64 MHz with a notch in a pipe. ········ 60

Table 5.5 Wave speeds in the pipe. ······································································ 64

vi

LIST OF FIGURES

Figure 1.1 Terrible catastrophes that caused casualties and property damages: (a) I-35W Missis-

sippi Riverbridge collapse in Minnesota, United States (2007) and (b) Seongsu

Bridge collapse in Seoul, South Korea (1994). ············································· 1

Figure 2.1 Hierarchical description of the laser pulse excitation at different scales. ·············· 15

Figure 2.2 Procedure of integral transformation method for solving the Navier’s equation. ····· 18

Figure 2.3 An isotropic plate with thickness h subjected to a normal surface load f(r,t). ········· 18

Figure 2.4 Stress boundary conditions for a pulse laser: (a) shear traction and

(b) normal traction. ·········································································· 19

Figure 3.1 Geometry of the pipe without a notch. ····················································· 23

Figure 3.2 Partition of the pipe into a thermo-mechanical region and a mechanical region. ····· 25

Figure 3.3 (a) Dimensions of the micro-scale model, (b) heat flux applied to the model, and (c)

temporal distribution of laser pulse. ······················································· 26

Figure 3.4 (a) Temperature distribution on the surface of micro-scale model and (b) applying the

temperature distribution obtained from the micro-scale model into the large scale

global model. ················································································· 27

Figure 3.5 Radial displacement around the thermally excited point. ································ 28

Figure 3.6 Pipe with a longitudinal notch. ······························································ 28

Figure 3.7 Pipe with a circumferential notch. ·························································· 29

Figure 3.8 (a) The simulation result without the band-pass filtering, (b) the simulation result with

the band-pass filtering, (c) the experimental result with the band-pass filtering, and

(d) comparison between the band-pass filtered simulation result and filtered experi-

mental result. ················································································· 30

Figure 3.9 Simulation conditions of the laser excitation and detection of Lamb waves in an alu-

minum plate: (a) geometric properties and (b) the partitioning scheme for numerical

efficiency. ····················································································· 32

vii

Figure 3.10 The experimental and the numerical result: (a) the experimental laser ultrasonic Lamb

wave signal by Shi et al. [31], (b) the simulated laser ultrasonic Lamb wave signal in

an aluminum plate, and (c) the comparison between (a) and (b). ······················ 34

Figure 3.11 Lamb wave simulations with different material properties: different elastic moduli (50

GPa, 70 GPa, 90 GPa). ······································································ 37

Figure 3.12 Lamb wave simulations with different material properties: different densities (2769

kg/m3, 3000 kg/m

3, 4000 kg/m

3). ·························································· 38

Figure 3.13 Lamb wave simulations with different material properties: different Poisson’s ratios

(0.250, 0.300, 0.345). ········································································ 39

Figure 3.14 Multi-scale modeling of hierarchical structure for predicting the thermal wave propa-

gation. ························································································· 41

Figure 3.15 Lamb waves propagation in the plate. ······················································ 41

Figure 4.1 Analytical individual Lamb wave modes in a plate: (a) A0 mode, (b) S0 mode, (c) A1

mode, and (d) S1 mode. ······································································ 45

Figure 4.2 Analytical individual Lamb wave modes in a plate: (a) A2 mode, (b) S2 mode, (c) A3

mode, and (d) S3 mode. ······································································ 46

Figure 4.3 Combined Lamb wave modes in a plate: (a) A mode, (b) S mode, and (c) theoretical

waveform. ···················································································· 47

Figure 4.4 (a) Predicted waveform of Lamb wave and (b) finite element prediction of the propa-

gation of Lamb wave (multi-scale modeling). ············································ 48

Figure 4.5 (a) Comparison between the numerical result which is performed at the normal stress

boundary condition and (b) comparison between the experimental result and the nu-

merical result where the thermal loading is applied. ····································· 49

Figure 4.6 Incidence of a pulse laser: (a) generation of quasi-static thermo-elastic surface defor-

mation and (b) formation of dynamic elastic waves. ···································· 52

Figure 4.7 Different wave modes are generated due to a pulse laser. ······························· 52

Figure 5.1 Time domain signal without a notch in a pipe. ············································ 53

Figure 5.2 Time domain signals with longitudinal notches of different depths in a pipe. ········ 54

viii

Figure 5.3 Comparison of time domain signals with different notches. ···························· 55

Figure 5.4 Magnitudes of frequency components without a notch. ································· 56

Figure 5.5 Magnitudes of frequency components with a longitudinal notch in a pipe. ··········· 56

Figure 5.6 Magnitudes of frequency components from pipes with different depths of longitudinal

notches. ························································································ 57

Figure 5.7 Time domain signals with circumferential notches of different depths in a pipe. ···· 58

Figure 5.8 Time domain signal with a notch that lies transverse to the straight line between the

laser source point and the observation point in a pipe. ·································· 59

Figure 5.9 Magnitudes of frequency components with a circumferential notch in a pipe. ········ 60

Figure 5.10 Magnitudes of frequency components from pipes with different depths of circumfer-

ential notches. ················································································ 61

Figure 5.11 Procedure of the frequency range selection for the damage detection algorithm. ···· 62

Figure 5.12 Wavelet transformation of the intact case. ················································· 63

Figure 5.13 Wavelet coefficients with the damage of the depth from (a) 1 mm, (b) 2 mm, (c) 3

mm, and (d) 4 mm. ··········································································· 65

Figure 5.14 Group velocities for each mode of (a) 0 MHz to 1.5 MHz and (b) 1.6 MHz to 1.9

MHz in a pipe. ················································································ 65

Figure 5.15 Wavelet coefficients at 1.1 MHz for (a) case 1, (b) case 2, (c) case 3 and

(d) case 4. ····················································································· 66

Figure 5.16 Wavelet coefficients for different depths at 1.1 MHz. ··································· 67

- 1 -

CHAPTER 1

INTRODUCTION

1.1 Research Background

In the wake of severe disasters such as I-35W Bridge and Seongsu Bridge collapse (Figure 1.1),

many civil engineers have intensified their efforts to develop effective techniques for detecting

damages in structures [1]. In the inspection of damages in a broad range of structures and materials

from civil infrastructure and aerospace industry, nondestructive testing has been emerged as an ef-

fective tool. Among the techniques of nondestructive testing, ultrasound based method is especially

suitable for a testing of thin structures (e.g. plates and pipes). In ultrasonic testing, elastic waves are

generated on the surface of structures, and then propagated along the media. The robustness of the

structures is investigated thoroughly by analyzing the response of the waves.

(a) (b)

Figure 1.1 Terrible catastrophes that caused casualties and property damages: (a) I-35W Mississippi

Riverbridge collapse in Minnesota, United States (2007) and (b) Seongsu Bridge collapse in Seoul,

South Korea (1994).

Ultrasonic testing can be classified into two major categories according to the way of generating

- 2 -

waves. One is the method using piezoelectric transducers for producing elastic waves. In the testing

by using piezoelectric transducers, tone-burst signals are employed; investigators does not confused

in processing the signals obtained because the narrow bandwidth of the signals can help one to ana-

lyze the response without regarding the influence of dispersion. Another method is laser-generated

ultrasound. Unlike the piezoelectric transducers, laser can produce broadband signals. It is often

desirable to utilize the broadband signals in nondestructive testing because information over the

wide range of frequency can be useful especially in the characterization of materials.

Laser can generate bulk waves such as longitudinal waves (P-waves) and transverse waves (S-

waves), and also guided waves like Rayleigh waves (surface waves) and Lamb waves [2]. The gen-

eration of different types of waves is mainly depends on the geometry of the specimen. Bulk waves

including longitudinal and transverse waves are induced by thermal diffusivity or optical penetra-

tion by a pulse laser. Rayleigh waves and Lamb waves are product as the result of the interaction of

longitudinal and transverse waves in the boundary of the specimen. Among many wave modes, of

particular interests are the generation of Lamb waves due to its availability and flexibility.

Many efforts have been made towards the researches of laser-generated Lamb waves both experi-

mentally and analytically [3]. In the experiment using laser, Nd:YAG (neodymium-doped yttrium

aluminum garnet) laser and laser Doppler vibrometer are frequently used [4]. Also, optical fiber

sensors connected to a fiber interferometer is used for a particular purpose which means the route of

laser is not straight. By contrast, small numbers of studies are found that are closely related to laser-

generated Lamb waves. In most of the works about solving laser-generated Lamb waves analytical-

ly, governing equation is solved by taking integral transformation or Green’s function formalism.

This thesis is devoted to the development of both numerical and analytical model of laser-generated

Lamb waves. In the numerical model, the finite element method is used to simulate the transient

behavior of Lamb waves. In the analytical model, double transformation method is employed to

solving the Navier’s displacement equations.

- 3 -

1.2 Literature Review

For several decades, diverse types of civil infrastructure projects, such as bridges, dams, and power

plants, have been planned and executed with structural health monitoring techniques that involve

the use of accelerometers, displacement transducers, strain gauges, lead-zirconate-titanate (PZT)

transducers and laser ultrasound devices [5]. Among these, especially aimed to detect small defects

using built-in devices [6], ultrasonic guided waves such as Lamb waves, Rayleigh waves and hol-

low-cylinder guided waves have been widely utilized for the structural health monitoring (SHM).

Especially, Lamb wave has been widely used for the inspection of thin-walled structures, while hol-

low-cylinder guided waves have been used for the inspection of tubular structures [7].

Structural health monitoring (SHM) techniques have been employed to assess and maintain the

structures for decades [5]. Especially, the ultrasonic based inspection using laser pulses and the sen-

sors such as MFC (Micro Fiber Composite) has been widely utilized. Among these techniques, the

detection method by Lamb waves, has gained significant attention because of the fact that Lamb

wave can propagate further than other bulk waves in the structures, and the decrease in the ampli-

tude of the wave modes is smaller. Furthermore, Lamb wave observed in the sensing point presents

useful information on the structure by passing through the thickness of the structural components.

Therefore, Lamb wave is of great potential for structural health monitoring [3].

Through various contact and non-contact ultrasonic transducers [8], ultrasonically guided waves can

be generated. In particular, due to its non-contact features, laser ultrasonic generators such as the

Nd:YAG laser have been used for this purpose. These devices enable researchers to generate waves

to larger ranges compared to the conventional contact approaches, and researchers can even detect

smaller defects. Given these advantages, much effort has been exerted to understand the physics of

guided wave applications [9]. Ravi et al. investigated thermo-elastic wave propagation in isotropic,

anisotropic and composite cylinders with a theory that considers the single thermal relaxation time

[10]. Sharma et al. focused on plate-like structures in the form of an aluminum-epoxy composite

- 4 -

material to study the propagation of thermo-elastic waves in the isothermal conditions [11].

For the predictions of the existence of damage in structures, there has been substantial research on

damage detection methods. Tua et al. proposed a method that uses the times of flight of propagating

Lamb waves [12]; Yu and Giurgiutiu developed an in-situ method using embedded two-dimensional

ultrasonic phased arrays [2]; Guo and Cawley discussed mode interaction in composite laminates

with delamination for long-range damage inspections [13]; and Pierce et el. investigated reflections

and interactions with defects using surface-bonded single-mode optical fiber sensors [14]. Among

the broad and deep schemes currently available for structural health monitoring, laser ultrasound in

particular has shown great promise due to its many advantages; however, few efforts have been

made to realize a precise methodology to simulate three-dimensional wave propagation. Often, the

various parameters involved in such a simulation are chosen without suitable consideration of all the

related implications.

Raghavan and Cesnik [15] derived an analytical solution of the guided ultrasonic wave field in-

duced by the piezoelectric wafer transducers, using Lamb wave for the analysis of the composite

materials. Guo and Cawley [16] investigated the usefulness of the symmetric 0S mode for the de-

tection of the delamination. Pierce et al. [17] studied the interaction of Lamb waves in the situation

where various damages such as holes or notches exist. Moulin et al. [18] developed a modeling

methodology which can make a decision of the amplitude of Lamb wave modes excited in compo-

site plates. Lamb wave can also be used to measure the thickness of the specimen and material

properties [19]. Dewhurst et al. [20] conducted a research to estimate the waveform to measure the

thickness of the metal sheet considering the sheet velocities and the phase velocities of anti-

symmetric modes. They pointed out that Lamb wave can be excited from the different sources.

Giurgiutiu [21] demonstrated the controllability of the wave propagation using piezo-electric wafer

active sensors.

Non-contact technique can be applied to the occurrence and the sensing of Lamb waves. Air-

- 5 -

coupled ultrasonic detection is suitable to the materials subjected to damage by the physical contact

[22]. As containing broadband signals, laser pulse is appropriate to generate all modes [4]. Hutchins

and Lundgren [19] presented the method of detection and generation of Lamb wave in the thin ma-

terial using the laser technique. Staszewki et al. [23] employed the laser Doppler velocimeter to per-

ceive Lamb wave in the low frequency bandwidth. After the absorption of the laser energy, thermo-

elastic deformation occurs in the specimen. Thermo-mechanical problems have been studied for a

long time. To solve the coupled thermo-mechanical problem, the finite element method has been

widely used.

Armero and Simo [24] proposed the way to solve the non-linear coupled thermo-mechanical prob-

lem. Tian et al. [25] studied generalized the finite element formulation considering two relaxation

times. Serra and Bonaldi [26] used the finite element method to analyze the thermo-elastic damping.

In this paper, the characteristics of three-dimensional wave propagation are observed. For the nu-

merical efficiency, the entire pipe area is divided into the thermo-mechanical region and the me-

chanical region. Temperature changes in the area excited by the laser source are considered, temper-

ature changes observed in the rest of the area is neglected. Four degree of freedom and only three

degree of freedom except the temperature are considered.

In this thesis, we propose a pioneering multi-scale simulation methodology which enables us to

simulate the thermo-mechanically excited waves by a very short duration laser pulse in the frame-

work of the finite element method. Firstly, an appropriate multi-scale simulation technique is devel-

oped by dividing the simulation procedures into the two stages. The temperature profile is obtained

using a micro-scale model, and the calculated temperature profile is applied to the macro-scale

global model. Secondly, to alleviate the computational cost, a partitioning scheme is introduced by

dividing the finite element model into two sub-regions (a thermo-mechanical region and a mechani-

cal region). Lastly, a damage detection method using Fourier transformation is proposed to identify

the presence of damage.

- 6 -

1.3 Research Objective

This thesis is motivated by the need to develop appropriate numerical model for accurately predict-

ing laser-generated Lamb waves in a plate and a pipe. The objectives are summarized as follows:

to develop schemes and methods for numerical simulations of Lamb waves propagation in

thin structures;

to show the analytic solution of transient response of laser-generated Lamb waves;

to develop criteria for the evaluation of the damage in structures.

1.4 Thesis Structure

This thesis consists of six chapters. Thesis outline and overview is as follows:

Chapter 1 outlines the objectives and motivation of the thesis. Necessity of the techniques

utilizing Lamb waves is pointed out. Then diverse ultrasonic NDE techniques are re-

viewed, followed by the research objectives. Finally, the outline of the thesis is introduced.

Chapter 2 deals with the theories which are basis for understating the concepts in subse-

quent chapters. The governing equation of motion in the thermo-mechanical viewpoint is

reviewed. Thermo-mechanically coupled finite element formulation is introduced. The

modeling approach for the propagation of Lamb waves is explained, from which the im-

portance of the multi-scale modeling is highlighted. Analytical method for solving the re-

sponse due to laser source is reviewed. Also, wavelet transformation which is kind of the

multi-resolution signal processing technique is introduced.

Chapter 3 provides the numerical model for predicting Lamb waves. The condition of the

numerical simulation is introduced. Partitioning scheme to reduce computational efforts is

- 7 -

explained and detail of the scheme is summarized. The notch modeling in a pipe is intro-

duced including the longitudinal and circumferential notches. Verification of the simula-

tion technique is given. Excellent agreement is observed between experimental and numer-

ical results. Lastly, the numerical model for a plate is investigated.

Chapter 4 studies an analytical model of transient Lamb waves in a plate. The analytical

model using a double integral transformation method is used. Using the analytical model,

individual modes including anti-symmetric modes and symmetric modes are obtained and

then, overlapped wave modes are analyzed. The loading condition for the circular laser

source is explained.

Chapter 5 investigates quantitatively the effect of the presence of the notch in a pipe. It is

shown that the cases of the longitudinal and circumferential notches show a different re-

sponses when a pulse laser is shot. Root-mean-square-deviation is used for the quantitative

investigation. Also shown is that multi resolution analysis using wavelet transformation

method is applied to the damage detection method.

Chapter 6 summarized the thesis and draws conclusions. Suggestions are given regarding

future works of finite element analysis for prediction of Lamb waves.

- 8 -

CHAPTER 2

THEORIES

2.1 Governing Equation of Motion for a Homogeneous

Isotropic Material

Proposed by Ravi et al. [10], the governing equation of motion for a homogeneous isotropic materi-

al is expressed as

2( ) ( ) T u u β u , (2.1)

20 0 0( ) ( )Ek T c T T T β u u+ , (2.2)

where λ and μ are Lame constants, k is the thermal conductivity, u is the displacement vector, T0 is

the reference temperature, ρ is the mass density, is the thermo-mechanical coupling tensor, Ec

is the specific heat capacity, τ0 is the thermal relaxation time and ∇ is the three-dimensional partial

derivative operator. With Helmholtz decomposition, the displacement field u is decomposed into a

scalar potential ϕ and a vector potential H and the uniqueness condition is given simultaneously as

, 0 u H H . (2.3)

Substituting Eq. (2.3) into Eqs. (2.1) and (2.2), we have following expressions.

2( 2 ) 0T , (2.4)

2 2 20 0 0( ) ( )Ek T c T T T , (2.5)

2 0 H H . (2.6)

- 9 -

Gazis’s potential wave fields in exponential terms are assumed as

( ) ( )( ) , ( )i z n t i z n tf r e T h r e , (2.7)

( )(r) [ ]i z n t r ze H H H

H g , (2.8)

where ξ is the axial wave number, n is the circumferential wave number, and ω is the circular fre-

quency. The scalar Laplacian operator ∇2 is written in the cylindrical coordinates (r, θ, z) as

2 2

2

2 2 2

1 1r

r r r r z

. (2.9)

The Laplacian operator ∇2 of H in vector form is stated as

2 2 2 22 2 2 22 2

[ ] r rr z r zH HH H

H H H H H Hr r r r

H r θ z . (2.10)

By substituting Eqs. (2.7) and (2.8) into Eqs. (2.5) and (2.6), we have

2 2 2( ) ( ) ( )ph r c f r f r

, (2.11)

2 22

4 20

2 2 2( ) ( ) ( ) 0E E

p p p

Tc cf r f r f r

c k k c kc

, (2.12)

where 2 ( 2 ) /pc , 2

0i .

Solutions of Eqs. (2.12) and (2.8) are given by

1 1 2 1 3 2 4 1( ) n n n nf r A J r A Y r A J r A Y r , (2.13)

1 2 1 2, , ( )r n n r z n ng B J r B Y r g ig g C J r C Y r , (2.14)

- 10 -

2 2

1 1

2 2

2 2

211 2 1 2

22

01 2 2

2

2 2

22 2

2

2

,

,

1, 4 ,

2 2

,

,

,

.

E

p p

E

p

s

s

r

r

br r b b

Tcb

c k k c

cb

kc

c

c

(2.15)

where Jn and Yn are the Bessel functions of the first and the second kind, respectively, and

A1,A2,A3,A4,B1,B2,C1,C2 are constants.

2.2 Thermo-mechanically Coupled Finite Element For-

mulation

The finite element method, a proven technique for dealing with a diverse range of engineering prob-

lems, is an immensely powerful numerical technique [27]. General formulation procedures of the

finite element method for analyzing problems fall into one of three categories: those in which equi-

librium equations are formulated, those in which the relationships between the displacement and

strain rate (i.e. compatibility equations) are formulated, and those in which the relationships be-

tween strain rates and stress levels (i.e. constitutive equations) are implemented. In the same vein,

the finite element formulation used in present thesis for the modeling of wave propagation induced

by laser pulses follows the same procedure. To simulate the irreversible flow of entropy due to

thermal flux [26], coupled thermo-elasticity is implemented in the finite element formulation.

Hence, the procedures for the finite element formulation of thermo-elasticity in the literature [3] are

summarized. The equation of motion in a local form [27] and the local balance equation of entropy

[26] are written, respectively as:

- 11 -

i ij ij

u fx

, (2.16)

0i

i

qT s

x

. (2.17)

Here, ij are the stresses, fi denotes the body forces, ρ is the density, iu denotes the accelera-

tions, T0 is the equilibrium temperature, s is the entropy rate per unit volume, and /i iq x are the

divergences of the thermal fluxes. As described in earlier work [24], for a well-posed boundary val-

ue problem [3], boundary conditions including both mechanical and thermal types need to be de-

scribed. The Dirichlet boundary condition (displacement boundary condition: D ) and the Neu-

mann boundary condition (traction boundary condition: D ) are written as

, onij j i i in q n Q D , (2.18)

, .i iu u T T on D (2.19)

In these equations, i denote the prescribed surface tractions, iu are the prescribed displace-

ments, Q is the prescribed heat flux, and T represents the temperature specified at the thermal

boundaries. The thermo-mechanically coupled constitutive equations are expressed as

ij ijkl kl ijC T , (2.20)

0

ij ij E

Ts c

T , (2.21)

where ijklC are the elasticity coefficients, ij are the strains, Ec is the specific heat capacity, and

ij are the thermo-mechanical coupling coefficients. The compatibility equations including the

strain-displacement relationship and Fourier’s heat conduction law are expressed as

1

( )2

jiij

j i

uu

x x

, (2.22)

- 12 -

i ijj

Tq k

x

, (2.23)

where ijk are the thermal conductivities.

To complete the finite element formulation, by the virtual work principle, the integration is trans-

formed by integration by parts and the divergence as described in Ref. [1]:

ij ijkl kl ij ij i i ij j i i iV V V VC dV TdV u u dV n u dA f u dV

, (2.24)

0 , ,ij ij E i ij j i iV V VT TdV c TdV k T dV q n TdA

(2.25)

In the finite element formulation, the displacement vector u and the temperature of an element

are functions of the nodal values U and Θ , respectively, as

= uu H U , (2.26)

= H Θ , (2.27)

where uH and H are the displacement and temperature interpolation matrices. The strain vector

ε and the temperature gradient vector ,i are expressed in terms of the nodal displacements, tem-

perature, the strain-displacement matrix uB and the temperature-gradient interpolation matrix B

as

u B U , (2.28)

,i B Θ . (2.29)

The weak forms can be rewritten in matrix form as

- 13 -

,

T T T Tu u uV V

T T T T T Tu u u uV V

dV dV

dV dA fdV

U B CB U U B H Θ

U H H U U H σ U H (2.30)

0T T T T T T T T

u EV V VT dV c dV k dV QdA

Θ H B U Θ H H Θ Θ B B Θ Θ H . (2.31)

By eliminating the virtual terms, we have

u uu MU K Θ K U F , (2.32)

u C U C Θ K Θ Q , (2.33)

where

, ,T T T

uu u u u uV V VdV dV k dV K B CB K B H K B B , (2.34)

, ,T T T

uu u u u uV V VdV dV k dV K B CB K B H K B B , (2.35)

,T T Tu uV

dA fdV QdA F H σ H Q H . (2.36)

In the next chapter, a numerical scheme for a multi-scale modeling is proposed. For a better under-

standing of the multi-scale modeling scheme, brief explanation on the finite element implementation

of Eqs. (2.32) and (2.33) is addressed. First, in the micro-scale modeling temperature distribution is

obtained by utilizing Eq. (2.33). Four degrees of freedom are used in the micro-scale model. After

that, in the macro-scale modeling displacement field is calculated using the obtained temperature in

Eq. (2.32). In the macro-scale model, degrees of freedom are selectively allocated with regard to the

solution step since the spatial partitioning is applied to entire domain. Detailed explanation is given

in Section 3.2.

- 14 -

2.3 Bottom-up Analysis for a Multi-scale Modeling

In order to obtain the comprehensive understanding of the ultrasonic waves that are thermo-

mechanically excited, one should have an insight on how ultrasonic waves are formed and devel-

oped in the thermo-mechanical point of view. Ultrasonic waves are generated by the thermal expan-

sion of the material under the excitation surface of the specimen due to heat flux provided by the

laser pulses. As the surface expands, it simultaneously interacts with its surrounding material. The

expansion and the interaction generate waves traveling from the excitation point. The concept of the

wave generation aforementioned is also applicable for the applications of the waves for the structur-

al health monitoring (SHM). However, thermal wave propagation induced by nanosecond laser

pulses in pipe-like structures is very difficult to calculate since the duration of the pulse is extremely

short, and the response (temperature) of the material is underestimated as a very small value. To

predict the response accurately, it is necessary to predict the temperature increase by the provided

heat flux for the short time, and the obtained result needs to be imposed as the temperature bounda-

ry on the surface of the macroscopic model. In fact, to model the heat flux for few nanoseconds, the

numerical model should be refined with extremely small elements, and the entire simulation re-

quires enormous amount of time.

To alleviate this numerical cost, a bottom-up analysis for a multi-scale modeling is proposed to pre-

dict thermo-mechanical wave propagation in pipe-like structures considering two different scale

models (the micro-scale and the macro-scale models), as shown in Figure 2.1. In this analysis, the

result (i.e. the temperature distribution) obtained computationally from the micro-scale model is

served as the input in the macro-scale model. The main modeling technique used in this simulation

is the thermo-mechanically coupled finite element analysis.

- 15 -

Figure 2.1 Hierarchical description of the laser pulse excitation at different scales.

2.3.1 Micro-scale Analysis

For the analysis of the micro-scale structure, the required step is to determine the form of the tem-

perature distribution at the surface of the material. In this research, the laser pulse delivers 10 mJ

energy for the duration of 8 ns. The diameter of the laser beam is set constant at 3 mm, and the cor-

responding circular area is approximately 7 mm2. The power density is obtained by the equation, P=

W/A, where W is the power and A is the heat source area. In the calculation of the power density, the

absorptivity is determined as 5.2 %. The corresponding flux value is 9 29.20 10 W/m . Now, we

can calculate the temperature distribution by the differential equation for heat flow in a semi-infinite

slab surface given [8] as

2 TAC TT

K t K

(2.37)

where T(z, t) is the temperature distribution, AT(z, t) is the heat produced per unit volume per unit

time, K is the thermal conductivity and ρ is the density. When the nanosecond-duration laser is shot,

the temperature distribution is given from Carslaw and Jaegar [28] as

- 16 -

2 2

1

0

1( , ) 2 ,

2

1 2( ) ,

z KtT z t I ierfc

CK c

ierfc e e d

(2.38)

where 0I is the corresponding flux value.

2.3.2 Macro-scale Analysis

The modeling in the macro-scale starts from the governing partial different equations for displace-

ment in an isotropic elastic medium, which is suggested by Gazis [29]:

2 2 2( ) ( / )t u u u (2.39)

where u is the displacement vector, is the mass density, and are Lame constant. The

vector u is decomposed into the scalar potential function and the vector potential function as

, ( , ).F t u H H r (2.40)

After substituting Eq. (4) into Eq. (3), we obtain the wave equations as

2 2 2 2 2 2/ , /p sc t c t H H (2.41)

where 2 ( 2 ) /pc and 2 /sc are the pressure wave and shear wave speeds, respectively.

Eq. (5) is solved by assuming the potentials [29] as

- 17 -

( )cos cos( ) ,

( )sin sin( ) ,

( )cos sin( ) ,

( )sin cos( ) .

r r

z z

f r n t z

H g r n t z

H g r n t z

H g r n t z

(2.42)

Substituting Eq. (6) into Eq. (5), one obtains

2 2 2

2 2 2

2 2 2 2 2

2 2 2 2 2

( / ) 0 ,

( / ) 0 ,

( 1 / / ) (2 / )( / ) 0 ,

( 1 / / ) (2 / )( / ) 0 .

p

s z

s r

s r

c

c H

r c H r H

r c H r H

(2.43)

The general solution of Eq. (7) is expressed as

1 1

3 3 1 3 1

1 1 1 1 1 1 1

2 2 1 1 2 1 1

( ) ( ) ,

( ) ( ) ,

2 ( ) 2 ( ) 2 ( ) ,

2 ( ) 2 ( ) 2 ( ) .

n n

n n

r n n

r n n

f AZ r BW r

g A Z r B W r

g g g A Z r B W r

g g g A Z r B W r

(2.44)

where 2 2 2 2/ pc and 2 2 2 2/ sc . Z denotes the nth-order Bessel functions nJ and

nY , and W denotes the modified Bessel functions nI and nK of arguments 1r r and

1r r .

2.4 Integral Transformation Method

To solve the Navier’s equation given in Eqs. (2.1) and (2.40), the integral transformation method is

most widely used for obtaining the response [30]. General procedure of the integral transformation

method is shown in Figure 2.2. In this section, detailed procedures about solving the Navier’s equa-

tion are stated.

- 18 -

Figure 2.2 Procedure of integral transformation method for solving the Navier’s equation.

2.4.1 Analytical Solution of Transient Response Due to Laser

Source

Among the analytical methods for solving the governing equation, the method using Fourier and

Hankel transformation is used in this thesis. Procedures to deal with the problem fall into three

steps: problem statement, Fourier and Hankel transformation, and residue theorem. Products about

the steps are summarized in this chapter.

Figure 2.3 An isotropic plate with thickness h subjected to a normal surface load f(r,t).

Navier＇s equation Ordinary differential equation

Transformed responseResponse

Analytical solution

Inverse integral transformation

Finite Element Analysis

Integral transformation

- 19 -

2.4.2 Problem Statement

A cylindrical coordinate system is considered in the problem as shown in Figure 2.3, where an iso-

tropic plate of thickness h is subjected to a normal load f(r,t) distributed in a circular area. The stress

boundary conditions are given as

( , )( , )

0

( , ) 0 .

zz

rz

f r t at z hr t

at z h

r t at z h

(2.45)

Actually, the effect of a pulse laser is considered as the sum of shear traction and normal traction as

shown in Figure 2.4. But, in this thesis a normal load condition is used for the simplicity.

(a) (b)

Figure 2.4 Stress boundary conditions for a pulse laser: (a) shear traction and (b) normal traction.

The equations of motion with the condition / 0u are stated as

2

2

2

2

( 2 ) ,

( )( 2 ) .

r

z

u

r z t

ur

z r r t

(2.46)

- 20 -

2.4.3 Fourier and Hankel Transformation

Our interests are to get the transformed displacement fields caused by the distributed normal force.

To obtain the displacement fields, double integral transformation scheme is employed [31]. Hankel

transformation is applied to the spatial domain while in the time domain Fourier transformation is

applied. After applying the transformations, the transformed displacements is defined as [32]

0

0

1

0

ˆ ( , , ) ( , , ) ( ) ,

ˆ ( , , ) ( , , ) ( ) ,

j tz z

j tr r

u z k u z r t rJ kr e drdt

u z k u z r t rJ kr e drdt

(2.47)

where ˆzu is the out-of-plane displacement field, ˆru is the in- plane displacement field, 0 ( )J kr and

1( )J kr are the Bessel functions of the zeroth and first-order, respectively.

2.4.4 Residue Theorem

Since the transformed displacement fields can contain the functions that have an infinitely large

number of poles, it is mandatory to use the residue theorem to calculate integral over the wave-

number k. By using the residue theorem, one can get the surface displacement fields expressed as

[32]

0

0

ˆ( , , ) ( , ) ( , ) ( )

ˆ( , ) ( , ) ( )

s

a

s j tz z

k

a j tz

k

u h x t H h f k kJ kr e d

H h f k kJ kr e d

(2.48)

where ( , )szH h and ( , )azH h are the material responses expressed as

- 21 -

2 2

2 2

( )sinh( )sinh( )( , ) ,

4

( )cosh( )cosh( )( , ) .

4

s sz

s

a az

a

j k h hH h

j k h hH h

(2.49)

where

2 2 2 2

2 2 2 2

( ) cosh( )sinh( ) 4 sinh( )cosh( ) ,

( ) sinh( )cosh( ) 4 cosh( )sinh( ) .

s

a

k h h k h h

k h h k h h

(2.50)

Here 2 2 2 2Lk c , 2 2 2 2

Tk c , Lc is the longitudinal wave speed and Tc is the trans-

verse wave speed.

2.5 Wavelet Transform

Wavelet transform has been applied to various engineering areas including signal processing [33],

[34], [35]. There are many kinds of wavelet bases which are utilized to convert a time history signal

to a time-frequency interpretation. For example, Haar, Morlet, Mexican hat, Daubechies are

representative wavelets widely used. Each wavelet transformation has unique advantages and

disadvantages in applying to each case. In principle, the wavelet transformation allows a raw data

signal in time domain to be transformed to the time-frequency domain providing more information

regarding the signal. It is a kind of multiresolution analysis which enables us to investigate the

signal at different frequencies and at different times. The continuous-time wavelet transform of a

signal ( )f t is defined as

,1

, ( ) ( ) ( )| |

a bf

t bWT a b f t t dt f t dt

aa

(2.51)

where ,a b is the complex conjugate of the wavelet , ( )a b t . The complex conjugate of the wavelet

is defined as

- 22 -

1

| |

t b

aa

(2.52)

where a is the scaling parameter, also called dilation variable, and b is the translation parameter

or time shift factor. The coefficient 1/ | |a must satisfy the following mathematical condition such

that

2 2,| ( ) | | ( ) |a bE t t

(2.53)

Eq. (2.53) means that the energy of a wavelet ( )t is finite, which is in contrast to a sinusoidal

function which has the infinite value.

- 23 -

CHAPTER 3

NUMERICAL MODEL

3.1 Numerical Implementations

Simulations of thermo-elastic wave propagation in a pipe are performed using FEAP (a finite ele-

ment analysis program) in a Linux-based workstation with 16 CPU cores. Due to the large number

of elements, i.e. nearly 600,000 elements, three-dimensional simulations of thermo-elastic wave

propagation are computationally expensive; therefore, it is necessary to parallelize the code for bet-

ter computational efficiency. Vector parallelization is achieved using the OPENMP library [36]. The

geometry of the pipe implemented in the FEAP simulation is shown in Figure 3.1. The geometrical

parameters of the pipe are a length of 1000 mm, a thickness of 6 mm and a radius of 57.15 mm. The

material properties of the pipe are listed in table 1. The distance from the left-end tip of the pipe to

the point excited by the laser pulse is 350 mm, the distance between the excited point and the obser-

vation point is 300 mm, and the distance between the observation point and the right-end tip of the

pipe is 350 mm. The material employed in the simulation is aluminum, which is a typical material

for pipelines. These simulation conditions correspond precisely to those of the experiment per-

formed by Lee et al. [37] for the validation of the numerical technique and the verification of calcu-

lation result.

Figure 3.1 Geometry of the pipe without a notch.

- 24 -

Table 3.1 Material properties of the aluminum utilized in the simulations.

Material properties Aluminum

Density ( -3Kgm ) 8027

Young’s modulus ( GPa ) 193

Poisson’s ratio 0.237

Thermal expansion coefficient ( -1K ) 52.31 10

Absorptivity 25.2 10

Conductivity ( -1 -1Wm K ) 223.95

Specific heat ( -1 -1JKg K ) 926.67

3.2 Partitioning Scheme

To reduce the calculation wall clock time during the thermo-elastic wave propagation simulation, a

partitioning scheme is applied. This scheme, as depicted by Figure 3.2, divides all regions of the

pipe concerned into two sub-regions, one of which is the mechanical sub-region, and the other is the

thermo-mechanical sub-region [3]. In the thermo-mechanical region, the thermal parts are solved

first and the mechanical parts follow in the next step, while in the mechanical region only the me-

chanical parts are calculated. This partitioning can reduce the calculation time significantly. Also, it

should be noted that different time integration schemes are utilized for each domain. For example, if

an implicit scheme is applied to the discretized equation without partitioning, then it is so called

monolithic. On the other hand, for the partitioned calculation, the implicit time integration scheme is

employed in the thermo-mechanical region while the explicit time integration scheme, which does

not require a stiffness matrix during the procedures, is employed in the mechanical region. Compar-

ing the wall clock times of the simulation with the partitioning scheme and with the monolithic

scheme, the partitioning scheme expedite the computational efficiency remarkably, and the total cal-

- 25 -

culation time is ten times smaller as the case with the monolithic scheme.

Figure 3.2 Partition of the pipe into a thermomechanical region and a mechanical region.

3.3 Multi-scale Modeling

The general procedures of the simulation of an ultrasonic laser have not been fully explored in sev-

eral different fields due to difficulties associated with numerous details. In this section, a micro-

scale model is developed to obtain the temperature distribution profile that arises when the laser

pulse of 10 mJ energy is shot onto a pipe for 8 nanoseconds. The micro-scale model is 2 μm in

length, 2 μm in width and 100 μm in thick, as shown in Figure 3.3(a). Firstly, to determine the shape

of the temperature distribution on the top surface of the model, heat flux is applied as shown in Fig-

ure 3.3(b). The magnitude of the heat flux, uniformly distributed on the top surface, is estimated as

3 29.196 10 MW/m . The diameter of the laser beam is 3 mm ending up with the exposed area of

27.07 mm . The power density of the laser source is obtained by dividing the laser pulse energy by

the laser source area; the corresponding heat flux is calculated by multiplying the power density

with the absorptivity, which is 0.052 for the aluminum surface. The heat flux has a distribution that

mimics the shape of the trapezoid in the graph shown in Figure 3.3(c). The power of the laser pulse

is reduced not to make a speckle on the pipe surface. The obtained temperature distribution is repre-

sented in Figure 3.4(a). The temperature promptly increases with the laser pulse duration, while it

decays smoothly after the peak. Secondly, the temperature distribution curve, which is obtained

from the above procedure, is applied to the large scale global model as shown in Figure 3.4(b). In-

stead of applying a heat flux on the surface of the large scale model, the temperature distribution of

the Dirichlet boundary is applied, and the radial displacement around the excitation point is plotted

as shown in Figure 3.5.

- 26 -

(a)

(b)

(c)

Figure 3.3 (a) Dimensions of the microscale model, (b) heat flux applied to the model, and (c)

temporal distribution of laser pulse.

- 27 -

(a)

(b)

Figure 3.4 (a) Temperature distribution on the surface of microscale model and (b) applying the

temperature distribution obtained from the microscale model into the large scale global model.

- 28 -

Figure 3.5 Radial displacement around the thermally excited point.

3.4 Modeling of a Pipe with a Notch

Four simulations are conducted to examine the effect of the existence of notches (a longitudinal

notch and a circumferential notch) with different depths. The modeling of the notch is divided into

two cases: one is the longitudinal notch described in Figure 3.6 and the other is the circumferential

notch described in Figure 3.7. A longitudinal notch is of 20 mm in length and of 2.5 mm in width; a

circumferential notch is of 20 mm in length and of 3 mm in width. Thicknesses increase from 1 mm

to 6 mm for the cases by 1 millimeter. The notch is located on a line between the laser source and

point P. The notch configuration is detailed in Figures 3.6 and 3.7. The pipe model for FEM is di-

vided into exactly 576,000 elements, which are expressed as cylindrical coordinates ( , , )r z ; the

mesh of each element has the radial size of 1 mm, the circumferential size of 1.25 mm and the lon-

gitudinal size of 1 mm.

Figure 3.6 Pipe with a longitudinal notch.

- 29 -

Figure 3.7 Pipe with a circumferential notch.

3.5 Validation and Verification of the Simulation Tech-

nique

To validate and to verify the simulation results of the pipe, a thorough comparison between the

simulated result and the experimental result is carried out; the comparison is conducted for the cases

without a notch. Figure 3.8 shows a comparison between the signal obtained by Lee et al. [37] and

that obtained by the simulation here. In the experiment by Lee et al. [37], a band-pass filter of the

range of 10-300 kHz is applied to eliminate the background noises during the experiment. In Figure

3.8(a), the shape of the response signal by the simulation shows good agreement with the

experimental result shown in Figure 3.8(c). However, high-frequency components still remain

eminent in the simulation result. Therefore, in order to maintain consistency in the response signals,

a band-pass filter of the range of 10-300 kHz is applied to the simulation result, too. The purpose of

the band-pass filter applied in the simulation is not to eliminate the background noises but to follow

the conditions of the experiment performed by Lee et al. [37]. The band-pass filter of the range of

10-300 kHz is applied in the experiment. The result in Figure 3.8(b) shows better fidelity than the

result shown in Figure 3.8(a). Based on the good agreement between the experiment and the

simulation, as shown in Figure 3.8(d), the proposed numerical methodology is validated, and the

numerical result is verified successfully.

- 30 -

(a) (b)

(c) (d)

Figure 3.8 (a) The simulation result without the band-pass filtering, (b) the simulation result with

the band-pass filtering, (c) the experimental result with the band-pass filtering, and (d) comparison

between the band-pass filtered simulation result and filtered experimental result.

- 31 -

3.6 Plate Modeling

3.6.1 Numerical Implementations

For the finite element analysis of guided waves, a Finite Element Analysis Program (FEAP), aimed

at for research and educational use [38], is utilized under the Linux-based workstation. The work-

station used has 16 quad core CPUs at 2.53 GHz, total 32 GB memories and a 2 TB hard disk drive,

operated in Ubuntu 11.10 version. Geometric parameters of the plate for simulations are presented

in Figure 3.9(a): width 300 mmw , length 300 mml and thickness 2.0 mmt . The distance

between the laser source point and the sensing point is 133 mm.x The mesh size of the aluminum

plate is set to 1.25 mm 1.00 mm 0.80 mm , which ends up with 288,000 elements. In Figure 3.9(b),

the schematic model of the methodology, called the partitioning method as in [3], is shown; the en-

tire region is divided into two sub-area, namely, the thermo-mechanical and the mechanical sub-

regions. Concerns in determining the area of the thermo-mechanical sub-region inextricably rely on

how the temperature caused by laser pulses propagates. Therefore, in the simulations, it is assumed

that the area of the thermo-mechanical sub-region corresponds to the area of

100 mm 100 mm 3.2 mm as shown in Figure 3.9(b). In the mechanical sub-region, three degrees of

freedom is considered, while in thermo-mechanical sub-region four degrees of freedom are consid-

ered.

- 32 -

(a)

(b)

Figure 3.9 Simulation conditions of the laser excitation and detection of Lamb waves in

an aluminum plate: (a) geometric properties and (b) the partitioning scheme for numeri-

cal efficiency.

3.6.2 Verification of the Simulation

From the experimental results obtained in an aluminum plate by Shi et al. [32], the comparative

study is performed. Material properties of the aluminum employed in present simulation are pre-

sented in Table 3.2. The time steps in computations are set as 1 ns for first 20 steps and 50 ns for

- 33 -

remnants. The duration of laser pulse is very short (8 ns) and the response for short duration is simi-

lar to a Dirac delta function. Source diameter of laser pulses is 0.5 mma . Finally, three dimen-

sional Lamb wave propagation induced by laser pulses in thin aluminum plate is observed in Figure

3.10(b). In verification study, the signals obtained from simulations show a good fidelity to the ex-

perimental results.

Table 3.2 Material properties of the aluminum employed for numerical simulations.

Elastic modulus (GPa) 70.2

Poisson’s ratio 0.345

Density ( 3kg/m ) 2769

Coefficient of thermal expansion (1/K ) 2.31Ⅹ10-5

Specific heat ( J/kg K ) 223.95

Thermal conductivity ( W/m K ) 926.67

The arrival times of bulk waves are 56.5 sect (the longitudinal wave), 96.2 sect (the trans-

verse wave), and 103.9 sect (Rayleigh wave). Velocity of each wave is calculated directly from

the numerical result shown in Figure 3.10(b) by using the equation /v d t where v is velocity, d is

distance between the laser source and the sensing point, and t is elapsed time. Measured velocity

of each wave are 6315.3 m/s (the longitudinal wave), 3243.9 m/s (the transverse wave), and

3036.5 m/s (Rayleigh wave), respectively. Meanwhile theoretical wave velocity for each wave in

present aluminum plate is 6310.8 m/s (the longitudinal wave), 3070.1 m/s (the transverse wave),

and 2890.5 m/s (Rayleigh wave), which are in congruence with numerical values.

- 34 -

(a)

(b)

(c)

Figure 3.10 The experimental and the numerical result: (a) the experimental laser ultrasonic Lamb

wave signal by Shi et al. [32], (b) the simulated laser ultrasonic Lamb wave signal in an aluminum

plate, and (c) the comparison between (a) and (b).

- 35 -

3.6.3 Case Studies

Experiment on ultrasonic wave propagation caused by laser pulses in a specimen such as an alumi-

num plate has been significantly executed due to versatile use of aluminum material. Hence, we re-

focus the attention on studies of characteristics of Lamb wave propagation under alteration of mate-

rial properties such as elastic modulus, Poisson’s ratio and density. In these studies, observed prop-

erties are changes of each wave velocities and shape of signals, which help us to understand a rudi-

mentary feature of waves in the plate. After the observation, the comparison between the numerical

results and the theoretical results is performed to aid in the selection of material properties of the

plate.

Different Elastic Moduli

To observe the effect of the different elastic moduli in three dimensional Lamb waves propagation,

the elastic moduli are set to 50 GPa, 70 GPa, and 90 GPa with other parameters fixed. Accordingly,

theoretical velocities of longitudinal, transverse, and Rayleigh wave are respectively (a) 5326 m/s ,

6302 m/s , and 7146 m/s ; (b) 2591 m/s , 3067 m/s , and 3476 m/s ; (c) 2439 m/s , 2886 m/s , and

3273 m/s . Formulations of theoretical velocities of longitudinal ( Lc ), transverse ( Tc ), and Rayleigh

wave ( Rc ) are given as

2 0.87 1.12

, ,1

L T R S

vc c c c

v

(3.1)

where λ is Lame’s first parameter, μ is the shear modulus, υ is Poisson’s ratio. With the large elastic

modulus, the wave velocities calculated directly from the graph in Figure 3.11 are increased; (a)

5341.9 m/s , 6315.3 m/s , and 7157.5 m/s ; (b) 2742.3 m/s , 3243.9 m/s , and 3674.0 m/s ; (c)

2572.3 m/s , 2890.5 m/s , and 3445.6 m/s . These results would provide us that linear relationship

between elastic modulus and bulk wave velocity existed.

- 36 -

Different Densities

Figure 3.12 shows the delay in the arrival of waves in the material under change of density. As den-

sity of material increase, arrival time of wave modes is slightly delayed. Shape of wave in each case

remains same through the modification in densities.

Different Poisson’s Ratios

With the different Poisson’s ratios, the signals obtained show changes of its amplitude as shown in

Figure 3.13.

Table 3.3 Comparison of the velocities between the theoretical solution and the numerical solution.

Poisson’s ratio Wave type Theoretical velocity Measured velocity

0.250 Longitudinal wave 5507.8 m/s 5519.0 m/s

Transverse wave 3179.9 m/s 3174.2 m/s

0.300 Longitudinal wave 5833.6 m/s 5846.0 m/s

Transverse wave 3118.2 m/s 3197.1 m/s

Theoretical velocities are obtained from a theoretical approach with Eq. 3.1. Table 3.3 shows theo-

retical velocities are similar to the measured velocities. This result shows the different pattern when

it compares to cases of other parameters. With changed modulus of elasticity and density, graph pat-

tern does not change, but only velocities are changed. However, graph pattern is changed through

the modification of the Poisson’s ratio. It is related to lateral property of material; actually Poisson’s

ratio is deeply related to lateral property of materials. It is necessary to solve theoretically wave

equation and understand the relationship between wave equation and physical character.

- 37 -

Figure 3.11 Lamb wave simulations with different material properties: different elastic moduli (50

GPa, 70 GPa, 90 GPa).

- 38 -

Figure 3.12 Lamb wave simulations with different material properties: different densities (2769

kg/m3, 3000 kg/m

3, 4000 kg/m

3).

- 39 -

Figure 3.13 Lamb wave simulations with different material properties: different Poisson’s ratios

(0.250, 0.300, 0.345).

- 40 -

Multiscale Modeling

Thermal wave propagation induced by ultra-short laser pulses occurs in the plate at its several struc-

ture levels due to the presence of the certain geometric and the physical entities in regard to the ul-

trasound generation. In order to obtain the comprehensive understanding of the ultrasound genera-

tion, one should get an insight on how ultrasound generation mechanisms interact with each other.

In this section, details about mechanisms of thermo-mechanical elastic wave propagation are stated

for the process of multi-scale modeling.

A bottom-up analysis for a multi-scale modeling is proposed to predict thermo-mechanical wave

propagation in plate in the aspects of the two different scales, the micro-scale and the macro-scale,

as shown in Figure 3.14. In this analysis, the result (i.e. the temperature distribution) obtained com-

putationally at the micro-scale model is served as the input in the macro-scale model. The main

modeling technique used in this simulation is the thermo-mechanically coupled finite element anal-

ysis. In the lower level (micro-scale), the temperature increase caused by the heat flux applied is

captured while in the higher level (macro-scale) the temperature distribution data is applied to the

macro-scale model.

In Table 3.2, material properties used in the simulations are listed. Simulations for Lamb wave

propagation in the plate are performed using a finite element analysis program (FEAP) in a Linux-

based workstation with 16 CPU cores.

- 41 -

Figure 3.14 Multiscale modeling of hierarchical structure for predicting the thermal wave

propagation.

Lager-generated Lamb Wave Propagation

In Figure 3.15, the out-of-plane displacement in the plate is captured using FEAP in sequential or-

der.

t = 0.02 μs t = 12.5 μs

t = 31.9 μs t = 77.3 μs

Figure 3.15 Lamb waves propagation in the plate.

- 42 -

CHAPTER 4

ANALYTICAL MODEL

4.1 Descriptions of Analytical Model

This chapter is mainly devoted to providing the analytic solution of transient Lamb wave in a plate.

Theoretical background of the analytical model is briefly explained in Section 4 in Chapter 2. We

will start with the development of the analytic solution which is provided in Ref. [32].

To get the solution, double integral transformation technique, i.e. Fourier-Hankel transformation is

used in the Navier’s displacement equations. After the transformation, prescribed stress boundary

conditions are applied to determine the arbitrary constants in the general solutions. Finally, the ana-

lytic solution of the response due to a shot on the surface in a plate is obtained.

4.2 Solutions of Circular-crested Waves

Our interest is to obtain the surface displacement field due to a shot of laser which has order of na-

noseconds. Surface displacement in a plate is considered as summation of two different wave

modes: Anti-symmetric mode and Symmetric mode. Provided by Shi et al [32], surface displace-

ment field is given as

( , , ) ( , , ) ( , , )a sz z zu h r t u h r t u h r t , (4.1)

where azu is anti-symmetric displacement field and s

zu is symmetric displacement field. These

two displacement are expressed as

- 43 -

0

ˆ( , , ) ( , ) ( , ) ( )a

a a j t

z z

k

u h r t H h f k kJ kr e d

, (4.2)

0

ˆ( , , ) ( , ) ( , ) ( )s

s s j t

z z

k

u h r t H h f k kJ kr e d

, (4.3)

where azH and s

zH are called the material responses which are dependent on material properties.

The material responses are obtained as

2 2( )cosh( )cosh( )( , , )

4

a az a

a

j k h hH k h

, (4.4)

2 2( )sinh( )sinh( )( , , )

4

s sz s

s

j k h hH k h

. (4.5)

a and s are

2 2

3 3

2 2 2 2 2 2

3 3

8 ( )sinh( )cosh( ) 8 cosh( )sinh( )

4 sinh( )sinh( ) 4 cosh( )cosh( )

( ) cosh( )cosh( ) ( ) sinh( )sinh( )

4 cosh( )sinh( ) 4 cosh( )sinh( )

a k k h h k h h

hk h h hk h h

hk k h h hk k h h

k h h k h h

(4.6)

2 2

3 3

2 2 2 2 2 2

3 3

8 ( )cosh( )sinh( ) 8 sinh( )cosh( )

4 cosh( )cosh( ) 4 sinh( )sinh( )

( ) sinh( )sinh( ) ( ) cosh( )cosh( )

4 sinh( )cosh( ) 4 sinh( )cosh( ).

s k k h h k h h

hk h h hk h h

hk k h h hk k h h

k h h k h h

(4.7)

The variables α and β are defined as

22 2

2

L

kc

, (4.8)

- 44 -

22 2

2

T

kc

, (4.9)

where cL is the longitudinal wave speed and cT is the transverse wave speed. k is wave number J0 is

the zeroth order Bessel function.

For the calculation of Eqs. (4.2) and (4.3), summation of the integrand is carried over the wave-

number k which is the solution of the Rayleigh-Lamb frequency equation. After the summation, in-

tegration is performed in a range from the cutoff frequencies and the frequencies which correspond

to the phase velocities converging to the Rayleigh wave speed and the transverse wave speed at each

wave modes. In Figures 4.1, 4.2 and 4.3, analytic Lamb wave modes are shown. Theoretical condi-

tions for the simulation of analytic solution are given in Table 4.1.

Table 4.1 Theoretical conditions of the simulation of analytic solution of laser-generated Lamb

waves.

Material Aluminum

Propagation distance (m) 0.133

Longitudinal speed (m/s) 6320

Transverse speed (m/s) 3130

Laser source diameter (m) 0.0005

Thickness (m) 0.0032

2nd Lamé parameter (GPa) 26.1

- 45 -

(a) (b)

(c) (d)

Figure 4.1 Analytical individual Lamb wave modes in a plate: (a) A0 mode, (b) S0 mode, (c)

A1 mode, and (d) S1 mode.

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)N

orm

al D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

- 46 -

(a) (b)

(c) (d)

Figure 4.2 Analytical individual Lamb wave modes in a plate: (a) A2 mode, (b) S2 mode, (c)

A3 mode, and (d) S3 mode.

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)N

orm

al D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

- 47 -

(a) (b)

(c)

Figure 4.3 Combined Lamb wave modes in a plate: (a) A mode, (b) S mode, and (c) theoretical

waveform.

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70 80 90 100-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

- 48 -

(a)

(b)

Figure 4.4 (a) Predicted waveform of Lamb wave and (b) finite element prediction of the propaga-

tion of Lamb wave (multi-scale modeling).

0 10 20 30 40 50 60 70-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

0 10 20 30 40 50 60 70-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

- 49 -

(a)

(b)

Figure 4.5 (a) Comparison between the numerical result which is performed at the normal stress

boundary condition and (b) comparison between the experimental result and the numerical result

where the thermal loading is applied.

0 10 20 30 40 50 60 70-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

Numerical Result

Analytical Result

0 10 20 30 40 50 60 70-1.00

-0.50

0.00

0.50

1.00

Time (s)

No

rma

l D

isp

lace

me

nt

(no

rma

lize

d)

Experimental Result

Numerical Result

- 50 -

4.3 Comparison Between the Numerical Results and the

Analytic Results

We are interested in investigating the obtained waveforms as shown in Figures 4.4 and 4.5. Of par-

ticular interest is the similitude between the results. Figure 4.4(a) shows the overall displacement for

the circular source, which is obtained by the summation of the individual modes shown in Figures

4.1 and 4.2. Figure 4.4(b) is the overall displacement field, obtained from the multi-scale modeling,

which has same simulation condition with Figure 4.4(a). However, these two waveforms are not

similar.

Due to the dissimilarity in the boundary conditions for solving the analytic solution, compared with

the multi-scale modeling, a little different response is shown in Figure 4.5. In fact, normal force

boundary condition is used for solving the analytic solution of the transient Lamb waves while i