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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
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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.
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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)
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항상 존경하는 아버지, 사랑하는 어머니, 자상한 누나에게.
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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.
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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;
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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-
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- 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