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Scattering of a plane harmonic SH-wave and dynamic stress concentration for multiple multilayered inclusions embedded in an elastic half-space
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Scattering of a plane harmonic SH-wave and dynamic stress concentration for multiple multilayered inclusions embedded in an elastic half-space
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SCATTERING OF A PLANE HARMONIC SH-WAVE AND DYNAMIC STRESS CONCENTRATION FOR MULTIPLE MULTILAYERED INCLUSIONS EMBEDDED IN AN ELASTIC HALF-SPACE by Ramtin Sheikhhassani A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Mechanical Engineering) May 2015 Copyright 2015 Ramtin Sheikhhassani Dedication I dedicate my dissertation work to my loving parents, Dr. Gholamreza Sheikhhassani and Dr. Simin Taei, for their endless support and encouragement; my grandmother, Ms. Manzarbanoo Marvi Esfahani for her kind heart; and my sister, Dr. Yasmin Sheikhhas- sani. I also dedicate this dissertation to my dearest friend, Dr. Pasha Anvari, whose charis- matic presence inspired me to be a better person by perseverance. ii Acknowledgements First and foremost, I would like to express my sincere gratitude to my advisor, Prof. Marijan Dravinski, for his constant patience, precious guidance, and insightful comments. I am immensely thankful to Prof. Satwindar S. Sadhal, Prof. Larry Redekopp, and Prof. Vincent W. Lee for serving on my qualifying and defense committee. Furthermore, I want to thank my colleagues Dr. Dean Bergman and Dr. Giacomo Castiglioni for many helpful conversations. iii Table of Contents Dedication ii Acknowledgements iii List Of Figures vi Abstract xviii Chapter 1: Introduction 1 Chapter 2: Problem Statement 5 2.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 3: Solution of the Problem 10 3.1 Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Model Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 4: Numerical Results 27 4.1 Key Model Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Verication of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 29 4.2.1 Circular Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.2 Circular Pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Layered Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3.1 Eect of Multiple Scattering . . . . . . . . . . . . . . . . . . . . . 31 4.3.2 Eect of Layering . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3.3 Eect of the Layers' Geometry . . . . . . . . . . . . . . . . . . . 36 4.4 The Role of Impedance Contrast for Layered Pipes . . . . . . . . . . . . . 37 Chapter 5: Stress Field for Multiple Multilayered Inclusions Embedded in a Half- Space and Subjected to a Plane Harmonic SH-Wave 54 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Boundary Integral Equations for the Half-Space Problem . . . . . . . . . 57 5.3 Discretization of the Half-Space Model . . . . . . . . . . . . . . . . . . . . 65 5.4 Verication of the Half-Space Results . . . . . . . . . . . . . . . . . . . . 70 5.4.1 Single Cavity in the Half-Space (N 1L1 cav ) . . . . . . . . . . . . . 71 iv 5.4.2 Two Circular Cavities in the Half-Space (N 2L1 cav ) . . . . . . . . . 71 5.4.3 Pipe in a Half-Space (N 1L2) . . . . . . . . . . . . . . . . . . . . 72 5.5 Numerical Results for the General Half-Space Model . . . . . . . . . . . . 85 5.5.1 Eect of Multiple Scattering . . . . . . . . . . . . . . . . . . . . . 85 5.5.2 Eect of Layering . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.5.3 Eect of Scatterer Stiness upon SCF . . . . . . . . . . . . . . . . 86 5.5.4 Eect of Impedance Contrast for Layered Pipes . . . . . . . . . . 87 5.5.5 Eect of the Separation Distance between the Inclusions . . . . . 88 Chapter 6: Summary and Conclusion 99 Appendix A Singular kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A.1 Basic Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A.2 Singularities in the Boundary Integral Equation Method . . . . . . . . . 102 Appendix B Stress Formulation for the Full-space . . . . . . . . . . . . . . . . . . . . . . . . 105 B.1 Problem Statement: Full-space Problem . . . . . . . . . . . . . . . . . . . 105 B.2 Solution of the Full-space Single Inclusion Problem . . . . . . . . . . . . . 107 B.2.1 DBIEs for the Full-Space . . . . . . . . . . . . . . . . . . . . . . . 107 B.2.2 GBIE for the Full-Space . . . . . . . . . . . . . . . . . . . . . . . . 108 B.2.3 Basic Properties of the Integral Equations . . . . . . . . . . . . . . 110 B.2.3.1 Property I . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.2.3.2 Property II . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.2.3.3 Property III . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.2.4 Regularization of DBIE . . . . . . . . . . . . . . . . . . . . . . . . 113 B.2.5 Regularization of the Displacement Gradient Integral Equation . . 114 B.2.5.1 Discretization of the First Integral in Eq.B.51 . . . . . . 115 B.2.5.2 Discretization of the Second Integral in Eq. B.51 . . . . . 118 B.2.5.3 Discretization of the Third Integral in Eq. B.51 . . . . . 118 B.2.5.4 Discretization of the Fourth Integral in Eq. B.51 . . . . . 119 B.2.5.5 Discretization of the Fifth Integral in Eq. B.51 . . . . . . 119 B.2.6 Boundary Integral Equations (BIEs) for an Inclusion Subjected to an SH-wave in a Full-Space . . . . . . . . . . . . . . . . . . . . . . 120 B.3 Overhauser Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 B.4 Verication of the Full-space Numerical Results . . . . . . . . . . . . . . . 123 B.4.1 Dynamic Stress Concentration for a Cavity in a Full-Space Sub- jected to an SH-wave . . . . . . . . . . . . . . . . . . . . . . . . . . 123 B.4.1.1 Analytical Solution . . . . . . . . . . . . . . . . . . . . . 123 B.4.2 Dynamic Stress Concentration for a Circular Elastic Inclusion Em- bedded in a Full-Space Subjected to a Plane Harmonic Incident SH-wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 References 133 v List Of Figures 2.1 Problem model consisting ofN layered inclusions withL embedded within an elastic half-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Geometry of the i-th inclusion with L i layers. Here, S i j denotes the layer interfaces, i = 1 :N; j = 1 :L i . The corresponding domains are denoted by D i j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1 The half-space limit model when the observation point y is located on the boundary of the half-space S 0 or on one of the outermost layers S i 1 ; i = 1 :N of the inclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 The ring limit model for domainD i j when the observation pointy is located on the boundary S i j (or S i j+1 ), where i = 1 :N; j = 1 :L i 1. . . . . . . 25 3.3 The inclusion limit model when the observation point y is located on the boundary S i j , i = 1 :N, ;j =L i . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Discretization for the two-inclusion-double-layer, where E (s) 0 and E (e) 0 de- note the rst and last element on the half-space surface S 0 , respectively, and E (s)i j and E (e)i j , i;j = 1 : 2, are the rst and last elements on the i-th inclusion and j-th layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.1 Comparison of the surface responses of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to vertical and 60 incident plane harmonic SH waves studied by Lee [1]. Solid and dashed lines represent the results of this study, whereas the squares and triangles denote the results of Lee [1]. The parameters are: = 1, N = 1, a 1 =b 1 = 1, O = (0; 1:5), 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 . . . . . . . . . . 39 vi 4.2 Comparison of the surface responses of a pipe embedded within a half-space subjected to vertical and horizontal incident plane harmonic SH waves, as studied by Manoogian [2]. Solid and dashed lines represent the results of this study, whereas the squares and circles denote the results of Manoogian [2]. The parameters are = 2, N = 1, L = 2, a 1 =b 1 = 1, a 2 =b 2 = 0:9, O = (0; 1:5); 0 = 0 = 1, 1 = 3, 1 = 1, 2 = 1 600 , 2 = 1 2 . . . . . . . . 39 4.3 Surface displacement amplitude for the soft three-inclusion-triple-layer model vs. the one-inclusion-triple-layer model for vertical (top) and grazing (bot- tom) incident waves. The layer centers are located at O 1 = (3; 1:5), O 2 = (0; 1:5), andO 3 = (3; 1:5). Similarly, the various layer principal axes are assumed to be a i 1 = b i 1 = 1;a i 2 = b i 2 = 0:75;a i 3 = b i 3 = 0:5; i = 1 : 3. The material properties of the layers are specied by i j = i(soft) j ; i j = i(soft) j ; i;j = 1 : 3: For i(soft) j and i(soft) j , see Eq. (4.4). . . . . . . . . . 40 4.4 Surface displacement amplitude for the sti three-inclusion-triple-layer model vs. the one-inclusion-triple-layer model for vertical (top) and grazing (bot- tom) incident waves. The parameters of the problem (see Fig. 4.3) are as follows : O 1 = (3; 1:5), O 2 = (0; 1:5), O 3 = (3; 1:5), a i 1 = b i 1 = 1, a i 2 =b i 2 = 0:75, a i 3 =b i 3 = 0:5, i j = i(stiff) j , i j = i(stiff) j ;i;j = 1 : 3. . . . 41 4.5 Surface displacement amplitude for the soft ve-inclusion-three-layer model vs. the one-inclusion-three-layer model for vertical (top) and grazing (bot- tom) incident waves. The parameters of the problem are O 1 = (6; 1:5), O 2 = (3; 1:5), O 3 = (0; 1:5), O 4 = (3; 1:5), O 5 = (6; 1:5), a i 1 = b i 1 = 1, a i 2 =b i 2 = 0:75, a i 3 =b i 3 = 0:5, i j = i(soft) j , i j = i(soft) j , i = 1 : 5, j = 1 : 3. 42 4.6 Surface displacement amplitude for the sti ve-inclusion-three-layer model vs. the one-inclusion-three-layer model for vertical (top) and grazing (bot- tom) incident waves. The parameters of the problem are: O 1 = (6; 1:5), O 2 = (3; 1:5), O 3 = (0; 1:5), O 4 = (3; 1:5), O 5 = (6; 1:5), a i 1 = b i 1 = 1, a i 2 =b i 2 = 0:75, a i 3 =b i 3 = 0:5, i j = i(stiff) j ; i j = i(stiff) j , i = 1 : 5; j = 1 : 3: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.7 Surface displacement amplitude for the soft three-inclusion-triple-layer model vs. the three-inclusion-single-layer model with averaged material proper- ties for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (3; 1:5);O 2 = (0; 1:5);O 3 = (3; 1:5); a i 1 =b i 1 = 1;a i 2 = b i 2 = 0:75;a i 3 = b i 3 = 0:5; i 1 = i 2 = i 3 = 1 3 P 3 j=1 i(soft) j ; i 1 = i 2 = i 3 = 1 3 P 3 j=1 i(soft) j ;i;j = 1 : 3: . . . . . . . . . . . . . . . . . . . . 44 vii 4.8 Surface displacement amplitude for the sti three-inclusion-triple-layer model vs. the three-inclusion-single-layer model with averaged material proper- ties for vertical (top) and grazing (bottom) incident waves. The param- eters of the problem are: O 1 = (3; 1:5), O 2 = (0; 1:5), O 3 = (3; 1:5), a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, a i 3 = b i 3 = 0:5, i j = 1 3 P 3 j=1 i(stiff) j ; i j = 1 3 P 3 j=1 i(stiff) j ; i;j = 1 : 3. . . . . . . . . . . . . . . . . . . . . . . . 45 4.9 Surface displacement amplitude for soft ve-inclusion-triple-layer model vs. ve-inclusion-single-layer model with averaged material properties for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (6; 1:5), O 2 = (3; 1:5), O 3 = (0; 1:5), O 4 = (3; 1:5), O 5 = (6; 1:5), a i j = b i j = 5j 4 , i j = 1 5 P 3 j=1 i(soft) j ; i j = 1 5 P 3 j=1 i(soft) j , i = 1 : 5; j = 1 : 3: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.10 Surface displacement amplitude for the sti ve-inclusion-three-layer model vs. the ve-inclusion-single-layer model with averaged material proper- ties for vertical (top) and grazing (bottom) incident waves. The parame- ters of the problem are: O 1 = (6; 1:5), O 2 = (3; 1:5), O 3 = (0; 1:5), O 4 = (3; 1:5), O 5 = (6; 1:5), a i j = b i j = 5j 4 , i j = 1 5 P 3 j=1 i(stiff) j ; i j = 1 5 P 3 j=1 i(stiff) j , i = 1 : 5, j = 1 : 3. . . . . . . . . . . . . . . . . . . . 47 4.11 Surface displacement amplitude for the softN 3L3 circular vs. elliptical models for vertical (top) and grazing (bottom) incident waves. The pa- rameters of the problem are: O 1 = (3; 1:5), O 2 = (0; 1:5), O 3 = (3; 1:5), i j = i(soft) j , i j = i(soft) j ; i;j = 1 : 3. For the elliptical model the princi- pal axes are assumed to be : a i 1 = 1;b i 1 = 0:75;a i 2 = 0:75;b i 2 = 0:56;a i 3 = 0:5;b i 3 = 0:38, andi = 1 : 3: For the circular model the corresponding radii are a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5. . . . . . . . . . . . . . 48 4.12 Surface displacement amplitude for the stiN 3L3 circular vs. elliptical models for vertical (top) and grazing (bottom) incident waves. The pa- rameters of the problem are: O 1 = (3; 1:5);O 2 = (0; 1:5);O 3 = (3; 1:5), i j = i(stiff) j ; i j = i(stiff) j ; i;j = 1 : 3, For the elliptical model, the principal axes are assumed to be a i 1 = 1;b i 1 = 0:75, a i 2 = 0:75;b i 2 = 0:56, a i 3 = 0:5;b i 3 = 0:38, andi = 1 : 3: For the circular model the corresponding radii are a i 1 =b i 1 = 1;a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5. . . . . . . . . . . 49 4.13 The seven-pipe model (N = 7) conguration, with pipe centers located at O 1 = (1:25; 5:83), O 2 = (2:5; 3:67), O 3 = (1:25; 1:5), O 4 = (0; 3:67), O 5 = (1:25; 5:83), O 6 = (2:5; 3:67), and O 7 = (1:25; 1:5). The principal axes are assumed to be a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, a i 3 = b i 3 = 0:5 for, i = 1 : 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 viii 4.14 The surface response for the seven-pipe model (see Fig. 17) for sti-soft vs. soft-sti layers for vertical (top) and grazing (bottom) incident waves. For the sti-soft model: i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600; i(stiff) 1 = 12=9, i(soft) 2 = 2=3; i 3 = 1=2;i = 1 : 7. For the soft-sti model: i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2; i = 1 : 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.15 The nine-pipe model (N = 9) conguration with the pipes centers at O 1 = (2:5; 6:5), O 2 = (2:5; 4), O 3 = (2:5; 1:5), O 4 = (0; 6:5), O 5 = (0; 4), O 6 = (0; 1:5), O 7 = (2:5; 6:5), O 8 = (2:5; 4), and O 9 = (2:5; 1:5). The principal axes are a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 = b i 3 = 0:5 for i = 1 : 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.16 Surface displacement amplitude for the nine-pipe model (see Fig. 4.15) for sti-soft and soft-sti layers for vertical (top) and grazing (bottom) incident waves. . For the sti-soft model: i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600, i(stiff) 1 = 12=9, i(soft) 2 = 2=3, i 3 = 1=2; i = 1 : 9. For the soft-sti model: i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2; i = 1 : 9. . . . . . . . . . . . . . . . . . . . . . . 53 5.1 Geometry of the half-space containingN multilayered inclusions withL i ;i = 1 :N layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 The ring limit model for domainD i j when the observation pointy is located on the boundary S i j (or S i j+1 ), where i = 1 :N, j = 1 :L i 1. . . . . . . 65 5.3 The inclusion limit model when the observation point y is located on the boundary S i j , i = 1 :N, ;j =L i . . . . . . . . . . . . . . . . . . . . . . . . 65 5.4 Comparison of the normalized stressj z = 0 j of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to a vertical incident plane harmonic SH -wave. Solid lines represent the results of Lin and Liu [3], and the circles denote the results of this study. The parameters are as follows: normalized wavenumber ka = 0:1, number of inclusionsN = 1, principal radii of the inclusiona 1 =b 1 = 1, and center location O = (0; 1:5). The material properties of the half-space and the inclusion are 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 , and 0 is the maximum stress amplitude of the stress eld along the cavity. . . . . . . . . . . . . . . . . 73 ix 5.5 Comparison of the normalized stressj z = 0 j of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to = 45 incident plane harmonic SH -wave. Solid lines represent the results of Lin and Liu [3], and the circles denote the results of this study. The parameters are as follows: normalized wavenumber ka = 0:1, number of inclusionsN = 1, principal radii of the inclusiona 1 =b 1 = 1, and center location O = (0; 1:5). The material properties of the half-space and the inclusion are 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 , and 0 is the maximum stress amplitude of the stress eld along the cavity. . . . . . . . . . . . . . . . . 74 5.6 Comparison of the normalized stressj z = 0 j of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to = 45 incident plane harmonic SH -wave. Solid lines represent the results of Lin and Liu [3], and the circles denote the results of this study. The parameters are: normalized wavenumber ka = 0:1, number of inclusions N = 1, principal radii of the inclusion a 1 = b 1 = 1, and center location O = (0; 12). The material properties of the half-space and the inclusion are 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 , and 0 is the maximum stress amplitude of the stress eld along the cavity. . . . . . . . . . . . . . . . . . . . . . . 75 5.7 Comparison of the normalized stressj z = 0 j of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to a vertical incident plane harmonic SH -wave. Solid lines represent the results of Lin and Liu [3], and the circles denote the results of this study. The parameters are: normalized wavenumber ka = 0:1, number of inclu- sions N = 1, principal radii of the inclusion a 1 = b 1 = 1, and center location O = (0; 12). The material properties of the half-space and the inclusion are 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 , and 0 is the maximum stress amplitude of the stress eld along the cavity. . . . . . . . . . . . . . . . . 76 5.8 Comparison of the normalized stressj z = 0 j for two circular cavities em- bedded within a half-space for dierent wavenumbers (N 2L1 cav model). The stresses are evaluated on the surface of the left circular cavity for a 45 incident plane harmonic SH-wave. The solid and dashed lines represent the results of Wang and Liu [4], and the circles, crosses and stars denote the results of this study. The parameters are the following: number of inclusions N = 2, principal axes a i = b i = 1, and location of the centers O 1 = (2:5; 1:5),O 2 = (2:5; 1:5). The material properties of the half-space and the inclusion are 0 = 0 = 1, i = 1 600 , i = 1 2 ,i = 1; 2, and 0 is the maximum stress amplitude of the stress eld along the cavity. . . . . . . . 77 x 5.9 Comparison of the normalized stressj z = 0 j for two circular cavities em- bedded within a half-space for dierent wavenumbers (N 2L1 cav model). The stresses are evaluated on the surface of the left circular cavity for a grazing incident plane harmonic SH-wave. Solid and dashed lines represent the results of Wang and Liu [4], and the circles, crosses and stars denote the results of this study. The parameters are the following: number of inclusions N = 2, principal axes a i = b i = 1, and location of the centers O 1 = (2:5; 1:5),O 2 = (2:5; 1:5). The material properties of the half-space and the inclusion are 0 = 0 = 1, i = 1 600 , i = 1 2 ,i = 1; 2; and 0 is the maximum stress amplitude of the stress eld along the cavity. . . . . . . . 78 5.10 Comparison of the normalized stressj z = 0 j for two circular cavities em- bedded within a half-space for dierent wavenumbers (N 2L1 cav model). The stresses are evaluated on the surface of the left circular cavity for a vertical incident plane harmonic SH-wave. Solid and dashed lines repre- sent the results of Wang and Liu [4], and the circles, crosses and stars denote the results of this study. The parameters are: number of inclu- sions N = 2, principal axes a i = b i = 1, and location of the centers O 1 = (2:5; 12),O 2 = (2:5; 12). The material properties of the half-space and the inclusion are 0 = 0 = 1, i = 1 600 , i = 1 2 ,i = 1; 2; and 0 is the maximum stress amplitude of the stress eld along the cavity. . . . . . . 79 5.11 Comparison of the normalized stressj z = 0 j for two circular cavities em- bedded within a half-space for dierent wavenumbers (N 2L1 cav model). The stresses are evaluated on the surface of the left circular cavity for a = 45 incident plane harmonic SH-wave. Solid and dashed lines rep- resent the results of Wang and Liu [4], and the circles, crosses and stars denote the results of this study. The parameters are: number of inclu- sions N = 2, principal axes a i = b i = 1, and location of the centers O 1 = (2:5; 12),O 2 = (2:5; 12). The material properties of the half-space and the inclusion are 0 = 0 = 1, i = 1 600 , i = 1 2 ,i = 1; 2; and 0 is the maximum stress amplitude of the stress eld along the cavity. . . . . . . . 80 5.12 Comparison of the normalized stressj z = 0 j at the outer surface of the circular pipe embedded within a half-space when subjected to a vertical incident plane harmonic SH-wave. The asterisks represent the results of this study, and the solid lines denote the results of Lee and Trifunac [5]. The parameters are the following: dimensionless frequency = 1, number of inclusions N = 1, number of layers L = 2, principal axes of the layers a 1 =b 1 = 1, a 2 =b 2 = 0:9, and layers' centers O = (0; 1:5). The material properties of the half-space and the layers are 0 = 0 = 1, 1 = 3, 1 = 1, 2 = 1 600 , 2 = 1 2 ; and 0 is the maximum stress amplitude of the stress eld along the pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 xi 5.13 Comparison of the normalized stressj z = 0 j at the inner surface of the circular pipe embedded within a half-space when subjected to a vertical incident plane harmonic SH-wave. The asterisks represent the results of this study, and the solid lines denote the results of Lee and Trifunac [5]. The parameters are the following: dimensionless frequency = 1, number of inclusions N = 1, number of layers L = 2, principal axes of the layers a 1 =b 1 = 1, a 2 =b 2 = 0:9, and layers' centers O = (0; 1:5). The material properties of the half-space and the layers are 0 = 0 = 1, 1 = 3, 1 = 1, 2 = 1 600 , 2 = 1 2 ; and 0 is the maximum stress amplitude of the stress eld along the pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.14 Comparison of the normalized stressj z = 0 j at the outer surface of the circular pipe embedded within a half-space when subjected to a vertical incident plane harmonic SH-wave. The asterisks represent the results of this study, and the solid lines denote the results of Lee and Trifunac [5]. The parameters are the following: dimensionless frequency = 1, number of inclusions N = 1, number of layers L = 2, principal axes of the layers a 1 =b 1 = 1, a 2 =b 2 = 0:9, and layers' centers O = (0; 1:5). The material properties of the half-space and the layers are 0 = 0 = 1, 1 = 0:35, 1 = 1, 2 = 1 600 , 2 = 1 2 ; and 0 is the maximum stress amplitude of the stress eld along the pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.15 Comparison of the normalized stressj z = 0 j at the inner surface of the circular pipe embedded within a half-space when subjected to a vertical incident plane harmonic SH-wave. The asterisks represent the results of this study, and the solid lines denote the results of Lee and Trifunac [5]. The parameters are the following: dimensionless frequency = 1, number of inclusions N = 1, number of layers L = 2, principal axes of the layers a 1 =b 1 = 1, a 2 =b 2 = 0:9, and layers' centers O = (0; 1:5). The material properties of the half-space and the layers are 0 = 0 = 1, 1 = 0:35, 1 = 1, 2 = 1 600 , 2 = 1 2 ; and 0 is the maximum stress amplitude of the stress eld along the pipe. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.16 Three-inclusion model (N = 3) conguration with the inclusion centers at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). The principal axes are a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3. . . . . . . . . 89 xii 5.17 Case of soft materials: stress concentration factorj z = 0 j on the outer surface of a middle inclusion in a soft three-inclusion-triple-layer model vs. a one-inclusion-triple-layer model for a vertical incident SH-wave. The layer centers are located atO 1 = (3; 1:5),O 2 = (0; 1:5), andO 3 = (3; 1:5). Similarly, the various layer principal axes are assumed to be a i 1 =b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 = b i 3 = 0:5, i = 1 : 3. The material properties of the layers are specied by i j = i(soft) j , i j = i(soft) j ; i;j = 1 : 3, where i(soft) j and i(soft) j are dened by Eq. (4.4), andj 0 j is the maximum stress amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.18 Case of sti materials: stress concentration factorj z = 0 j on the outer surface of a middle inclusion in a sti three-inclusion-triple-layer model vs. a one-inclusion-triple-layer model for a vertical incident SH-wave. The layer centers are located atO 1 = (3; 1:5),O 2 = (0; 1:5), andO 3 = (3; 1:5). Similarly, the various layer principal axes are assumed to be a i 1 =b i 1 = 1, a i 2 = b i 2 = 0:75, a i 3 = b i 3 = 0:5, i = 1 : 3. The material properties of the layers are specied by i j = i(stiff) j , i j = i(stiff) j ; i;j = 1 : 3, where i(stiff) j and i(stiff) j are dened by Eq. (4.5), andj 0 j is the maximum stress amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.19 Case of sti materials: stress concentration factorj z = 0 j for a sti three- inclusion-triple-layer model vs. a three-inclusion-single-layer model with average material properties for a vertical incident wave. The centers are located at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). The layer principal axes are assumed to be a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 = b i 3 = 0:5. The material properties of the layered inclusions are i j = i(stiff) j , i j = i(stiff) j , whereas for the average model, the mate- rial properties are i j = 1 3 P 3 j=1 i(stiff) j ; i j = 1 3 P 3 j=1 i(stiff) j ; i;j = 1 : 3, andj 0 j is the maximum stress amplitude. . . . . . . . . . . . . . . . . . . 91 5.20 Case of soft materials: stress concentration factorj z = 0 j for a soft three- inclusion-triple-layer model vs. a three-inclusion-single-layer model with average material properties for a vertical incident wave. The centers are located at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). The layer principal axes are assumed to be a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 = b i 3 = 0:5. The material properties of the layered inclusions are i j = i(soft) j , i j = i(soft) j , whereas for the average model, the material properties are i j = 1 3 P 3 j=1 i(soft) j ; i j = 1 3 P 3 j=1 i(soft) j ; i;j = 1 : 3, andj 0 j is the maximum stress amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.21 The three-inclusion model (N 3L1) conguration with the inclusion centers at O 1 = (2:5; 1:5), O 2 = (0; 1:5), and O 3 = (2:5; 1:5), The principal axes are a i 1 =b i 1 = 1, for i = 1 : 3: . . . . . . . . . . . . . . . . . . . . . . . . . 93 xiii 5.22 Normalized stress (j z = 0 j) at the surface of a cavity that is located in between two-sti vs. two-soft inclusions for a vertical incident wave. The center locations are O 1 = (2:5; 1:5), O 2 = (0; 1:5), and O 3 = (2:5; 1:5), The material properties are 1(stiff) = 3(stiff) = 14=9, 2 = 1=600, 1(stiff) = 3(stiff) = 12=9, 2 = 1=2 for the sti-cavity-sti model and 1(soft) = 3(soft) = 4=9, 2 = 1=600; 1(soft) = 3(soft) = 2=3, 2 = 1=2 for the soft-cavity-soft model. Here,j 0 j is the maximum stress amplitude. 93 5.23 The three-pipe layered model (N 3L3) conguration with the pipe centers at O 1 = (2:5; 1:5), O 2 = (0; 1:5), and O 3 = (2:5; 1:5), The principal axes are a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3: . . . . . . 94 5.24 Normalized stress (j z = 0 j) at the outer surface of the middle pipe in a three pipe model for sti-soft vs. soft-sti layers for a vertical incident wave. The parameters of the problem are O 1 = (2:5; 1:5), O 2 = (0; 1:5), and O 3 = (2:5; 1:5). The material properties for the sti-soft model are i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600, i(stiff) 1 = 12=9, i(soft) 2 = 2=3, i 3 = 1=2, for i = 1 : 3: For the soft-sti model, they are i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2, for i = 1 : 3. The principal axes are a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3. Here,j 0 j is the maximum stress amplitude. . 94 5.25 The nine-pipe layered model (N 9L3) conguration with the pipe centers at O 1 = (2:5; 6:5),O 2 = (2:5; 4),O 3 = (2:5; 1:5),O 4 = (0; 6:5),O 5 = (0; 4), O 6 = (0; 1:5), O 7 = (2:5; 6:5), O 8 = (2:5; 4), and O 9 = (2:5; 1:5). The principal axes are a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 9: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.26 Normalized stress (j z = 0 j) at the outer surface of the centered pipe (S 5 1 ) in aN 9L3 layered pipe model for sti-soft vs. soft-sti layers for a vertical incident wave. The centers of of the pipes are located at O 1 = (2:5; 6:5), O 2 = (2:5; 4), O 3 = (2:5; 1:5), O 4 = (0; 6:5), O 5 = (0; 4), O 6 = (0; 1:5), O 7 = (2:5; 6:5), O 8 = (2:5; 4), and O 9 = (2:5; 1:5). The principal axes are a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 = b i 3 = 0:5 i = 1 : 9. The material properties of the half-space are 0 = 0 = 1 and for the sti-soft model i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600, i(stiff) 1 = 12=9, i(soft) 2 = 2=3, i 3 = 1=2, for i = 1 : 9. For the soft-sti model, they are i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2, for i = 1 : 9. Here,j 0 j is the maximum amplitude of the incidence stress-eld along the corresponding surface. . . . . . . . . . . . . . . . . . . . . . . . 96 xiv 5.27 Normalized stress (j z = 0 j) at the outer surface of the middle pipe in a three pipe model for soft-sti layers for a vertical incident wave for dif- ferent distances between outer surfaces of the pipes. The pipes' centers are located at O 1 = (2 +d; 1:5), O 2 = (0; 1:5), and O 3 = (2 +d; 1:5), where d = [0:5; 1; 1:5] is the distance between the outer surfaces of the pipes. The material properties for the soft-sti model: are i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2, for i = 1 : 3. The principal axes are a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3. Here,j 0 j is the maximum stress amplitude. . 97 5.28 Normalized stress (j z = 0 j) at the outer surface of the middle pipe in a three-pipe model with sti-soft layers for a dierent distances between outer surfaces of the pipes and vertical incident wave. The parameters of the problem are:. Th O 1 = (2 +d; 1:5), O 2 = (0; 1:5), and O 3 = (2+d; 1:5), whered = [0:5; 1; 1:5] is the distance between the outer surfaces of the pipes. The material properties for the sti-soft model are i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600, i(stiff) 1 = 12=9, i(soft) 2 = 2=3, i 3 = 1=2, for i = 1 : 3. The principal axes are a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3. Here,j 0 j is the maximum stress amplitude. . 98 A.1 A single cavity embedded in full space and subjected to an SH wave. Here, S,, @S, n, D 0 , and u inc denote the cavity surface, the indentation, the part of the boundary removed due to the indentation, the outward unit normal, the full-space domain, and the incident wave, respectively. . . . . 103 B.1 An elastic cylindrical inclusion embedded in a full-space and subjected to a plane harmonic incident SH-wave. The domains of the full-space and inclusion are denoted by D 0 and D 1 , respectively, and n represents the outward unit normal to the interface S 1 . . . . . . . . . . . . . . . . . . . 106 B.2 Normalized stress eldj z = 0 j as a function of for a cavity of unit radius subjected to a low-frequency (ka = 0:1) plane harmonic incident SH-wave. Solid lines represent the exact solution [6], and the solid dots indicate the results obtained with the BIE method in this study using 64 Overhauser elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 B.3 Normalized stress eld j z = 0 j as a function of for a cavity of unit radius and subjected to plane harmonic incident SH-wave with ka = 1, Solid lines represent the exact solution [6], and the solid dots denote the results obtained with the BIE method in this study using 64 Overhauser elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 xv B.4 Normalized stress eldj z = 0 j as a function of for cavity of unit radius and subjected to plane harmonic incident SH-wave with ka = 2. Solid lines represent the exact solution [6], and the solid dots denote the results obtained with the BIE method in this study using 64 Overhauser elements. 126 B.5 Comparison of the numerical results and the exact solution for a single point on the cavity surface and for dierent frequencies (Rej z = 0 j and Imj z = 0 j vs. ka at = =2) for a unit cylinder embedded in a full- space subjected to a plane harmonic SH-wave. The BIE solution uses 64 Overhauser elements over the cavity surface. . . . . . . . . . . . . . . . . 127 B.6 Comparison of the stress concentration factor (j z = 0 j vs ) along the interface of a soft, elastic, circular inclusion of unit radius embedded in a full-space and subjected to a plane harmonic SH-wave with a single frequency (ka = 2). Solid lines denote the exact solution, and the asterisks represent the BIE results. The material properties of the full-space are 0 = 0 = 1. The inclusion's properties are 1 = 1=6 and 1 = 1=2. For the BIE solution, 64 Overhauser elements are used and 0 = 1 k 0 denotes the maximum stress in the absence of the inclusion. . . . . . . . . . . . . 129 B.7 Comparison of the stress eldj rz = 0 j vs along the interface of a soft, elastic, circular inclusion of unit radius embedded in a full-space and sub- jected to a plane harmonic SH-wave with a single frequency (ka = 2). Solid lines denote the exact solution, and the asterisks represent the BIE results. The material properties of the full-space are 0 = 0 = 1. The inclusion's properties are 1 = 1=6 and 1 = 1=2. For the BIE solution 64 Overhauser elements are used, and 0 = 1 k 0 denotes the maximum stress in the absence of the inclusion. . . . . . . . . . . . . . . . . . . . . . . . . 130 B.8 Comparison of the stress eldj z = 0 j vs along the interface of a sti, elastic, circular inclusion of unit radius embedded in a full-space and sub- jected to a plane harmonic SH-wave with a single frequency (ka = 2). Solid lines denote the exact solution, and the asterisks represent the BIE results. The material properties of the full-space are 0 = 0 = 1. The inclusion's properties are 1 = 8 and 1 = 2. For the BIE solution 64 Overhauser elements are used, and 0 = 1 k 0 denotes the maximum stress in the absence of the inclusion. . . . . . . . . . . . . . . . . . . . . . . . . 131 xvi B.9 Comparison of the stress eldj rz = 0 j vs along the interface of a sti, elastic, circular inclusion of unit radius embedded in a full-space and sub- jected to a plane harmonic SH-wave with a single frequency (ka = 2). Solid lines denote the exact solution, and the asterisks represent the BIE results. The material properties of the full-space are 0 = 0 = 1. The inclusion's properties are 1 = 8 and 1 = 2. For the BIE solution 64 Overhauser elements are used, and 0 = 1 k 0 denotes the maximum stress in the absence of the inclusion. . . . . . . . . . . . . . . . . . . . . . . . . 132 xvii Abstract The scattering of a plane harmonic SH-wave by an arbitrary number of layered inclu- sions in a half-space is investigated by a direct boundary integral equation method. The inclusions, which have arbitrary shape and arrangement, are embedded within an elastic half-space. The eects of multiple scattering, the geometry, and the impedance contrast of the materials for layered inclusions and pipes, on the surface motion, are considered in detail. Additionally, a non-hypersingular technique is employed to calculate the dynamic stress concentration factor along the interfaces of inclusions. Various contributing factors that can in uence the stress concentration, including multiple scattering, the impedance contrast of layers, and the separation distance between the scatterers, are investigated. The numerical results are presented for a wide range of parameters present in the problem. Keywords: Multiple scattering, dynamic stress concentration factor, wave propa- gation, layered inclusions, elastic waves, anti-plane strain, non-hypersingular boundary element method. xviii Chapter 1 Introduction Diraction of waves by multiple scatterers has been the subject of numerous investi- gations. This problem arises in the elds of elastodynamics, acoustics, electromagnetics, and optics. Many applications, such as non-destructive evaluation of materials [7, 8], medical imaging [9, 10], and seismology [11, 12] incorporate these types of models. The methods used to study these problems are either analytical or numerical [13]. The earliest classical theory was pioneered by Foldy [14] who considered a random distribution of isotropic point scatterers and derived an eective wave number for scalar waves propagating through an inhomogeneous medium. He found that this eective wave number diers in the presence of scatterers. Lax [15] followed Foldy's procedure to include anisotropic and inelastic scattering. He introduced an eective medium in which scattering uctuations were imbedded. The parameters were determined by averaging, similar to Foldy's method. Twersky [16] employed the wave series expansion method for scattering of a plane wave by an arbitrary conguration of two parallel cylinders and, later, spheres [17] for acoustics and electromagnetics. 1 Waterman and Truell [18] considered nite size scatterers and obtained a second-order correction to Foldy's formula in terms of the scatterer density. Waterman [19] devised the T-matrix method, which expands the components of the scattered and incident elds by a set of orthonormal vectors. Subsequently, Peterson and Str om [20, 21, 22, 23] used the T-matrix approach to determine the scattering of multiple and multilayered objects for the acoustic and electromagnetic theory. Thereafter, Bose and Mal [24] and Varadan et al. [25] advanced the multiple scattering theory of scalar waves in an elastic full-space. Specically, they described the elastic waves and the interactions between two particles. Berryman [26], Sabina and Willis [27], Yang and Mal [28], and Kanaun [29] approxi- mated the interaction among particles by assuming that each particle is embedded within an eective medium. Yang and Mal [28, 30] implemented a method developed by Christensen [31] to evaluate the eective elastic properties of composite materials. They applied it to the Waterman-Truell [18] model for the scattering of randomly distributed bers in a com- posite via statistical averaging. Kanaun and Levin [32] examined the eective medium method for axial elastic shear wave propagation through ber composites. Wei and Huang [33] studied the dynamic eective properties of particle-reinforced composites by modeling a thin homogeneous viscoelastic interphase between the inclusion and the matrix. Biwa et al. [34] used the eigenfunction expansion and collocation methods with the assumed periodicity of the ber arrangement in the direction normal to the propagation direction for SH and subsequently 2 for P and SV waves. Ou and Lee [35] adopted the eigenfunction expansion method for the scattering of S and P waves by a single nano-sized coated ber. The derivation of analytical solutions that satisfy both the dierential equations and boundary conditions is limited to only very simple problems [36]. For more realistic problems, one must rely on numerical procedures. In continuum mechanics, numerical techniques such as the, nite dierence method (FDM), the nite element method (FEM), and the boundary integral equation method (BEM) are often used. For unbounded media, the use of BEM is greatly preferred over the other computa- tional methods. In BEM, the dierential equations are represented as integral equations. The advantage is that only the boundaries are discretized [37], instead of the volume of the problem, which considerably reduces the size of the system of the equations, compared with the FEM. In addition, the radiation conditions at innity are satised exactly. In solving wave propagation problems in unbounded media using FDM and FEM, articial absorbing boundary conditions are introduced [38] to improve the accuracy of the solution. A number of researchers have applied boundary integral equations to the problem of wave propagation (e.g., [39{43]). Benites et al. [44, 45] implemented the boundary integral equation method (BEM) for the scattering of multiple cavities in a full-space and a half-space by SH, P, and SV waves. DeSanto [46] used boundary integral equations for an N-layered single scatterer of arbitrarily shaped surfaces embedded in a full-space. 3 Dravinski and Yu [47] investigated scattering of SH waves by an arbitrary number of homogeneous inclusions of general shape and placement within an elastic half-space by using a direct boundary integral equation approach. They examined the eect of inclusion locations for two, three, and nine inclusion models and demonstrated the signicance of multiple scattering on the peak surface amplication. Dravinski and Sheikhhassani [48] showed the importance of layering for a single layered obstacle embedded within a half-space, especially for soft materials. In addition, they showed that the impedance contrast of the layers has a signicant impact on the surface motion. The intent of this study is to extend Dravinski and Sheikhhassani's [48] approach to include multiple layered scatterers for the evaluation of both displacement and stress elds. This thesis is divided into six chapters. Chapter 2 presents the statement of the prob- lem. Chapter 3 considers the solution used to nd the displacements, which is formulated using integral equations. Chapter 4 presents the numerical results. Chapter 5 presents a framework for stress evaluation. Finally, Chapter 6 summarizes the presented work. 4 Chapter 2 Problem Statement The scattering of a plane harmonic SH wave by an arbitrary number of layered inclu- sions that are, completely embedded within an elastic half-space (Fig. 2.1) is investigated by using the direct boundary integral equations method. Each inclusion consists of a nite number of layers that are perfectly bonded together. The interfaces between the layers are assumed to be C (1) continuous without any sharp corners. The geometry of the problem is described in Cartesian coordinatesfjxj<1; 0z< 1g with thez-axis pointing downward (Fig. 2.1). The half-space surface and the domain are denoted as S 0 and D 0 , respectively. The outward unit normal on various surfaces is denoted by n. The interfaces and domains for dierent layers are dened by considering a generic i-th inclusion, 1iN, as shown by Fig. 2.2. From the outermost to the innermost layers the corresponding domains are denoted by D i 1 to D i Li , respectively, where L i represents the number of layers for the inclusion. Therefore,D i j denotes the domain of the j-th layer of the i-th inclusion. The corresponding layer interfaces are represented by S i 1 ;:::;S i Li . 5 All of the layers, except the innermost one, are bounded by two interfaces. The innermost layer D i Li is bounded by only one interface, S i Li . 2.1 Equations of Motion The steady-state SH waves are governed by the Helmholtz equation [49] r 2 + [k (i) j ] 2 u (i) j = 0 k (i) j = ! (i) j (i) j = v u u t (i) j (i) j i = 0; 1 to N j = 0; 1 to L i (2.1) whereu (i) j ;!;k (i) j ; (i) j ; (i) j , and (i) j are the displacement eld, the circular frequency, the wave number, the shear wave speed, the shear modulus, and the density of the domain D i j , respectively. Here, D i j ;i = 0; 1;:::;N; j = 0; 1;:::;L i represents the domains of the half-space and the layers, respectively. Thus, the superscript (i) refers to the inclusion number, and the subscript j denotes the layer number within that inclusion. For convenience, the following notation is adopted for the domain and displacement elds in the half-space: D 0 =D 0 0 andu 0 =u (0) 0 . The remaining displacements are labeled u (i) j ; i = 1 :N; j = 1 :L i . 6 Finally, the Laplacian operator is given by r 2 = @ 2 @x 2 + @ 2 @z 2 (2.2) The incident wave is assumed to be of the form u inc (x;!) =e i[k 0 (x sin z cos )!t] ;x2D 0 (2.3) where is the o-vertical angle of incidence (Fig. 2.1). The exponential term e iwt and the frequency dependence will be omitted throughout. 2.2 Boundary Conditions As the incident wave strikes the inclusions, it generates the scattered wave eld. Therefore, the total displacement eld in each domain can be written as u 0 =u ff +u sc 0 ;x2D 0 u (i) j =u sc(i) j ;x2D i j ;i = 1 :N;j = 1 :L i (2.4) Here, u ff denotes the free eld created by the half-space surface in the absence of the inclusions. In addition, u sc 0 and u sc(i) j represent the unknown scattered wave eld within the half-space and layers, respectively. The traction-free boundary condition of the half-space surface is given by t 0 (x) = 0 @u 0 @z = 0; x2S 0 (2.5) 7 The continuity of the displacements and tractions between the half-space and the outermost layers of the dierent inclusions are given by u 0 =u (i) 1 ;x2S (i) 1 ;i = 1 :N t 0 =t (i) 1 ;x2S (i) 1 ;i = 1 :N (2.6) Similarly, the continuity conditions along the interfaces between the dierent layers in the inclusions can be stated as u (i) j1 =u (i) j ;x2S (i) j ;i = 1 :N;j = 2 :L i f (i) j1 =f (i) j ;x2S (i) j ;i = 1 :N;j = 2 :L i (2.7) where the traction on the S (i) j surface is dened by t (i) j = (i) j @u (i) j @n (2.8) Here,@()=@n =r()n andn is the outward unit normal. Finally, the scattered waves in the half-space are required to satisfy the radiation conditions at innity. 8 n γ u inc n n n n n n n S 0 D 0 D 1 1 D 2 1 D 1 i D j i D i D 1 N Li D LN N S 1 1 S 2 1 S 1 i S j i S i Li x z Figure 2.1: Problem model consisting ofN layered inclusions withL embedded within an elastic half-space. D 2 i D 1 i S 1 i S 2 i S Li i S j i D j i D Li i Figure 2.2: Geometry of the i-th inclusion with L i layers. Here,S i j denotes the layer interfaces, i = 1 :N; j = 1 :L i . The corresponding domains are denoted by D i j . 9 Chapter 3 Solution of the Problem 3.1 Integral Equations The problem under consideration is solved by using a direct boundary integral equa- tions (BIE) technique. Following Dravinski and Sheikhhassani [48], three limit models for the problem are introduced. These limit models refer to the half-space, the generic ring, and the elastic inclusion. The half-space limit model consists of the half-space boundary S 0 and the outer- most inclusion boundaries S i 1 , i = 1 : N; together with the corresponding semicircular indentations 00 and i 10 of radius "! 0; centered at the boundary point y (Fig. 3.1). Thus, 00 refers to the indentation approaching the boundary point y2S 0 from within the domain D 0 . Similarly, i 10 denotes the indentation approaching the boundary point y2 S i 1 , i = 1 : N; from within the half-space. The introduction of the indentations 00 and i 10 will exclude the boundary points y2 S 0 or y2 S i 1 ;i = 1 : N; from the half-space domain. Therefore, for y2 S 0 , the boundary of the limit half-space model becomes S 0 + P N i=1 S i 1 +S 1 + 00 @S 0 . Here, @S 0 represents the portion of boundary 10 S 0 that is being removed by introducing the indentation 00 . In addition,S 1 denotes the semicircular surface of the innite radius (Fig. 3.1). Analogously, for the observation point on the outermost inclusion boundaries, i.e., y2 S i 1 ;i = 1 : N, the boundary of the half-space limit model becomes S 0 + P N i=1 S i 1 + S 1 @S i 1 + i 10 . Here,@S i 1 represents the portion of boundaryS i 1 that has to be removed in the presence of the indentations i 10 . Then, fory2S 0 (ory2S i 1 ), Betti's external formula [50] yields the following results for the scattered waves in the half-space 0 = lim !0;r!1 Z S 0 + P N i=1 S i 1 +S1+@S U (0) (x;y)t sc 0 (x;n) T (0) (x;y;n)u sc 0 (x) dS x y2S 0 or y2S i 1 ;i = 1 :N (3.1) Here = 8 > > > > < > > > > : 00 ;y2S 0 i 10 ;y2S i 1 (3.2) @S = 8 > > > > < > > > > : @S 0 ;y2S 0 @S i 0 ;y2S i 1 (3.3) where =jxyj and r = p x 2 +z 2 is the radius of the half-space, while U (0) and T (0) are the full-space displacement and traction Green's function [51], respectively dened for the material of domain D 0 . Furthermore, u sc 0 and t sc 0 are the unknown scattered 11 displacements and tractions in D 0 . Invoking the radiation conditions at innity [52] eliminates the integral over S 1 , and Eq. (3.1) becomes 0 = lim !0 Z S 0 + P N i=1 S i 1 @S U (0) (x;y)t sc 0 (x;n)T (0) (x;y;n)u sc 0 (x) dS x + Z U (0) (x;y)t sc 0 (x;n)T (0) (x;y;n)u sc 0 (x) dS x y2S 0 or y2S i 1 ;i = 1 :N (3.4) Assuming that the displacement, u sc 0 , and traction, t sc 0 , are H older continuous and implementing the traction-free boundary conditions on the half-space surface S 0 leads to the following result u 0 (y)c sc(0) (y)+ Z S 0 T (0) (x;y;n)u 0 (x)dS x + N X i=1 Z S i 1 T (0) (x;y;n)u 0 (x)dS x N X i=1 Z S i 1 U (0) (x;y)t i 1 (x;n)dS x =F (y) y2S 0 or y2S i 1 ;i = 1 :N (3.5) where R denotes the principal value integral and 12 F (y) =u ff (y)c sc(0) + Z S 0 T (0) (x;y;n)u ff (x)dS x + N X i=1 Z S i 1 T (0) (x;y;n)u ff (x)dS x N X i=1 Z S i 1 U (0) (x;y)t ff (x;n)dS x y2S 0 or y2S i 1 ;i = 1 :N (3.6) Here, u ff and t ff are the free-eld displacements and tractions, respectively. The free terms are dened by c sc(0) (y) = 8 > > > > > < > > > > > : lim !0 R 00 T (0) (x;y;v)dS x ; y2S 0 lim !0 R i 10 T (0) (x;y;v)dS x ; y2S i 1 (3.7) where T (0) represents the static part of the traction Green's function T (0) , while v is the unit normal vector along the indentation surface (Fig. 3.1). The limit model for a generic ring, depicted by Fig. 3.2, is obtained by introducing innitesimal indentations i j;j on the surface S i j or i j+1;j on the surface S i j+1 . Here, i j;j denotes the semicircular region incised on the S i j surface from domain D i j , and i j+1;j denotes the semi-circular region removed from S i j+1 in the domain D i j . Meanwhile, the 13 direction of normal vector v on @S i j is opposite from that on @S i j+1 . Therefore, the boundary integral equation for a generic ring becomes u (i) j (y)c sc(i) j (y)+ Z S i j T (i) j (x;y;n)u (i) j (x)dS x Z S i j+1 T (i) j (x;y;n)u (i) j+1 (x)dS x Z S i j U (i) j (x;y)t (i) j (x;n)dS x + Z S i j+1 U (i) j (x;y)t (i) j+1 (x;n)dS x = 0; y2S i j or S i j+1 1iN 1jL i 1 (3.8) Here,U (i) j (x;y) andT (i) j (x;y;n) denote the full-space displacement and traction Green's functions, respectively, with the material properties corresponding to domain D i j . More- over,u (i) j (y) andt (i) j (x;n) represent the unknown displacements and tractions for domains D i j , i = 1 :N, j = 1 :L i 1, respectively. In addition, the free terms are given by c sc(i) j (y) = 8 > > > > > > < > > > > > > : lim !0 R i j;j T (i) j (x;y;v)dS x ; y2S i j lim !0 R i j+1j T (i) j (x;y;v)dS x ; y2S i j+1 (3.9) 14 where T (i) j represents the static part of the full-space traction's Green's function T (i) j for the domain D i j . A similar procedure for the innermost layer limit model (Fig. 3.3) leads to the following integral equation u (i) Li (y)c sc(i) Li (y) Z S i Li U (i) Li (x;y)t (i) Li (x;n)dS x +PV Z S i Li T (i) Li (x;y;n)u (i) Li (x)dS x = 0; y2S i Li (3.10) c sc(i) Li (y) = lim !0 Z i Li;Li T (i) Li (x;y;v)dS x ;y2S i Li (3.11) with T (i) Li denoting the full-space static traction Green's function with the material prop- erties of domain D (i) L i . This completes the formulation of the integral equations of the problem. The model discretization is considered next. 3.2 Model Discretization The half-space domain is considered rst. The corresponding boundaries S 0 and S i 1 , i = 1 :N; are divided intoP 0 andP i 1 linear elements, respectively. It is convenient to intro- duce the starting and the ending element/node numbers along each surface. Thus, along S 0 , the starting and ending element/node numbers are called E (s) 0 /N (s) 0 and E (e) 0 /N (e) 0 , 15 respectively. Similarly, along the surfaces S i 1 , the starting and ending element/node numbers are E (s)i 1 /N (s)i 1 and E (e)i 1 /N (e)i 1 , respectively (see Fig. 3.4). Consequently, the unknown displacements of all the nodes associated with the half-space domain are [u N (s) 0 ;:::;u N (e) 0 ;u N (s)1 1 ;:::;;u N (e)1 1 ;:::;u N (s)N 1 ;:::;;u N (e)N 1 ] (3.12) A similar notation applies to the unknown tractions. Consequently, the discretized form of Eq.(3.5) becomes u l c sc(0) (l)+ E (e) 0 X k=E (s) 0 A (0) 1lk A (0) 2lk 2 6 6 4 u k u k+1 3 7 7 5 + N X i=1 E (e)i 1 X k=E (s)i 1 A (0) 1lk A (0) 2lk 2 6 6 4 u k+1 u k+2 3 7 7 5 N X i=1 E (e)i 1 X k=E (s)i 1 B (0) 1lk B (0) 2lk 2 6 6 4 t k+1 t k+2 3 7 7 5 =F l (3.13) 16 where F l =u ff l c sc(0) (l)+ E (e) 0 X k=E (s) 0 A (0) 1lk A (0) 2lk 2 6 6 4 u ff k u ff k+1 3 7 7 5 + N X i=1 E (e)i 1 X k=E (s)i 1 A (0) 1lk A (0) 2lk 2 6 6 4 u ff k+1 u ff k+2 3 7 7 5 N X i=1 E (e)i 1 X E (s)i 1 B (0) 1lk B (0) 2lk 2 6 6 4 t ff k+1 t ff k+2 3 7 7 5 l =N (s) 0 :N (e) 0 and N (s)i 1 :N (e)i 1 ;i = 1 :N (3.14) In the above equations, u k and f k are the respective displacements and tractions at node k; and l represents the collocation node. The integration constants A (0) qlk and B (0) qlk ; q = 1; 2; are dened by A (0) qlk = 1 Z 1 T (0) (;l;n k ) q ()J k d;q = 1; 2 (3.15) B (0) qlk = 1 Z 1 U (0) (;l) q ()J k d;q = 1; 2 (3.16) Here, q and J k are the shape function and the Jacobian of the global-to-local element coordinate transformation, specied by 17 1 () = 1 2 (1) (3.17) 2 () = 1 2 (1 +) (3.18) J k = L k 2 (3.19) where L k denotes the element length. It must be noted that the rst and last nodes coincide with each other along each of the closed surfaces S i 1 , i = 1 :N. The discretization of the generic ring is considered next. Using the notation introduced for the half-space domain and the linear elements, the discretized version of Eq. (3.8) becomes u l c sc(i) j (l)+ E (e)i j X k=E (s)i j A (i;j) 1lk A (i;j) 2lk 2 6 6 4 u k+1 u k+2 3 7 7 5 E (e)i j+1 X k=E (s)i j+1 A (i;j) 1lk A (i;j) 2lk 2 6 6 4 u k+1 u k+2 3 7 7 5 E (e)i j X k=E (s)i j B (i;j) 1lk B (i;j) 2lk 2 6 6 4 t k+1 t k+2 3 7 7 5 + E (e)i j+1 X k=E (s)i j+1 B (i;j) 1lk B (i;j) 2lk 2 6 6 4 t k+1 t k+2 3 7 7 5 = 0 l =N (s)i j :N (e)i j+1 ;i = 1 :N;j = 1 :L i1 (3.20) 18 For the generic innermost inclusion,D i Li ,i = 1 :N, the discretized Eq. (3.10) becomes c sc(i) j (l)u l + E (e)i Li X k=E (s)i Li A (i;j) 1lk A (i;j) 2lk 2 6 6 4 u k+1 u k+2 3 7 7 5 E (e)i Li X k=E (s)i Li B (i;j) 1lk B (i;j) 2lk 2 6 6 4 t k+1 t k+2 3 7 7 5 = 0; l =N (s)i Li :N (e)i Li ;i = 1 :N (3.21) For domains D i j , 1 : N, j = 1 : L i , the integration constants A (i;j) qlk and B (i;j) qlk are obtained by replacing T (0) and U (0) in Eq. (3.15) and (3.16) with T (i) j and U (i) j , respec- tively. Therefore, the discretized integral equations (3.13), (3.20) and (3.21) can be assembled into a matrix form as AU =F (3.22) A = [A;B] (3.23) U =fu;tg T (3.24) u =fu 0 ;u 1 1 ;u 1 2 ;:::;u 1 L 1 ;u 2 1 ;:::;u 2 L 2 ;:::;u N 1 ;:::;u N L N g T (3.25) t =ft 1 1 ;t 1 2 ;:::;t 1 L 1 ;t 2 1 ;:::;f 2 L 2 ;:::;t N 1 ;:::;t N L N g T (3.26) F =fA 0 u ff B 0 t ff ;0g (3.27) 19 Here,u andt represent the unknown displacement and traction vectors. All the remaining matrices in Eq. (3.22) are explicitly known. The sizes of various matrices in Eq. (3.22) are dened as A2C (n+2m)(n+2m) U2C (n+2m)1 F2C (n+2m)1 A 0 2C (n+p)(n+m) B 0 2C (n+p)m u ff 2C (n+m)1 t ff 2C m1 n = 1 +P 0 m = N X i=1 L i X j=1 P i j p = N X i=1 P i 1 (3.28) Here,C mn denotes anmn dimensional complex vector space, andP 0 andP i j represent the number of nodes along the surfaces S 0 and S i j . At this point, it is useful to examine the structure of Eq. (3.22) in more detail. This structure is illustrated for the case of N inclusions with L layers, where L 1 = 2. Let i = i P k=1 L k , then 20 A = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 A (0) 11 A (0) 12 0 : : : A (0) 1;2+ i1 0 : : : A (0) 1;2+ N1 : : : 0 0 A (1;1) 22 A (1;1) 23 : : : 0 0 : : : 0 : : : 0 0 0 A (1;2) 33 : : : . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 : : : A (i;1) 2+ i1 ;2+ i1 A (i;1) 2+ i1 ;3+ i1 : : : . . . : : : 0 . . . . . . . . . . . . . . . 0 0 0 : : : 0 A (i;j) 2+j+ i1 ;2+j+ i1 A (i;j) 2+j+ i1 ;3+j+ i1 0 : : : 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 : : : 0 0 A (i;Li) 1+ i ;1+ i : : : 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 : : : 0 : : : 0 : : : 0 A (N;1) 2+ N1 ;2+ N1 A (N;1) 2+ N1 ;3+ N1 . . . . . . . . . . . . : : : . . . . . . 0 0 0 : : : 0 : : : : : : 0 0 A (N;L N ) 1+ N ;1+ N 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (3.29) 21 B = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 B (0) 11 0 B (0) 1;1+ i1 0 : : : 0 0 B (0) 1; N1 0 B (1;1) 21 B (1;1) 22 0 0 : : : 0 0 0 0 0 B (1;2) 32 0 0 : : : 0 0 0 0 0 0 . . . 0 : : : 0 0 0 0 0 0 B (i;1) 2+ i1 ;1+ i1 B (i;1) 2+ i1 ;2+ i1 : : : 0 0 0 0 . . . . . . . . . . . . . . . 0 0 0 0 B (i;j) 2+j+ i1 ;1+j+ i1 B (i;j) 2+j+ i1 ;2+j+ i1 : : : 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 0 0 B (i;Li) i +1; i 0 0 0 0 0 0 0 0 0 0 B (N;1) 2+ N1 ;1+N1 B (N;1) 2+ N1 ;2+ N1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 0 0 0 0 B (N;L N ) 1+ N ; N 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 (3.30) 22 where all of the sub-matrices in Eq. (3.29 and 3.30) are explicitly known. Matrices A and B have a dened structure and they can be constructed for an arbitrary number of inclusions and layers. Once the integral equations are solved for the unknown displacements and tractions along the boundaries, the displacements at any location can be determined by the follow- ing equations Domain D 0 u 0 (y) =u ff (y)+ Z N P i=1 S i 1 U (0) (x;y) h t 0 (x;n)t ff (x;n) i dS x Z S 0 + N P i=1 S i 1 T (0) (x;y;n) h u 0 (x;n)u ff (x;n) i dS x y2D 0 &y = 2S 0 jy = 2S i 1 (3.31) Domain D i j , 1iN, 1jL i1 u sc(i) j (y) = Z S i j [U (i) j (x;y)t sc(i) j (x;n)T (i) j (x;y;n)u sc(i) j (x)]dS x + Z S i j+1 [U (i) j (x;y)t (i) j (x;n)T (i) j (x;y;n)u (i) j (x)]dS x y2D i j &(y = 2S i j jy = 2S i j+1 ) (3.32) 23 Domain D i L i u sc(i) L i (y) = Z S (i) L i [U (i) L i (x;y)t sc(i) L i (x;n)T (i) L i (x;y;n)u sc(i) L i (x)]dS x y2D (i) L i &y = 2S (i) L i (3.33) This completes the discretization of the integral equations. The numerical results are considered next. 24 D 1 S 1 1 δ 00 y ∂S 0 S 0 δ 1 10 ∂S 1 1 y S ! D 2 S 1 2 δ 2 10 ∂S 1 2 y D N S 1 N δ N 10 ∂S 1 N y v v v Figure 3.1: The half-space limit model when the observation pointy is located on the boundary of the half-spaceS 0 or on one of the outermost layersS i 1 ;i = 1 :N of the inclusions. δ i jj v v n n n δ i j+1 j ∂S j i y S j+1 i D j i S j i D j+1 i ∂S j+1 i y Figure 3.2: The ring limit model for domain D i j when the observation point y is located on the boundary S i j (or S i j+1 ), where i = 1 :N; j = 1 :L i 1. δ i LiLi v n ∂S Li i y D Li i S Li i Figure 3.3: The inclusion limit model when the observation point y is located on the boundary S i j , i = 1 :N, ;j =L i . 25 E (s) 0 E (e) 0 E (s)1 1 E (e)1 1 E (s)1 2 E (s)1 2 E (s)2 1 E (e)2 1 E (s)2 2 E (s)2 2 Figure 3.4: Discretization for the two-inclusion-double-layer, where E (s) 0 and E (e) 0 denote the rst and last element on the half-space surfaceS 0 , respectively, andE (s)i j andE (e)i j , i;j = 1 : 2, are the rst and last elements on the i-th inclusion and j-th layer. 26 Chapter 4 Numerical Results 4.1 Key Model Properties Without loss of generality, the layer interfaces are assumed to be of an elliptical shape, and a dimensionless frequency is introduced as = 2a inc a =max(a i 1 );i = 1 :N (4.1) where inc is the wavelength of the incident wave and a i 1 is the major principal axis of the inclusion number i. Furthermore, the following parameters are assumed S 0 =x L x R x L =x O 1 10 x R =x O N + 10 P 0 = 20S 0 P i j = 128; i = 1 :N;j = 1 :L i (4.2) 27 Here,S 0 is the length of the half-space surface, andP 0 andP i j are the number of elements used to discretize S 0 and S i j surfaces, respectively. In addition, x O 1 and x O N are the x- coordinates of the rst and last inclusion centers, respectively. Due to the large number of parameters present in the problem, the following assump- tions in regards to the materials, the geometry, the number of inclusions, and the layers are proposed. The material properties of the half-space are set to be 0 = 0 = 1 (4.3) where 0 and 0 denote the shear modulus and the wave speed, respectively. The material properties of the layers are denoted by i j and i j , i = 1 : N, j = 1 : L i , where N denotes the number of layered inclusions and L i represent the number of layers for the i-th inclusion. The layer materials are assumed to be either of the sti or the soft types. For the sti layers, the impedance contrast is i j i j i j+1 i j+1 > 1, ( j = 1 :L i 1) whereas for the soft layers, the impedance contrast is i j i j i j+1 i j+1 < 1. The shear modulus and the shear wave velocity for the soft layers are dened by [48] i(soft) j = 0 j (i) L i ;j = 1 :L i ; i = 1 :N i(soft) j = 0 j (i) L i ;j = 1 :L i ; i = 1 :N (i) = 5 6 ; (i) = 1 2 (4.4) 28 It is apparent from Eq. (4.4) that the layer properties for dierent inclusions are assumed to be the same. Equivalently, for the sti layers, [48] i(stiff) j = 0 +j (i) L i ;j = 1 :L i ; i = 1 :N i(stiff) j = 0 +j (i) L i ;j = 1 :L i ; i = 1 :N (i) = 5 6 ; (i) = 1 2 (4.5) For example, for the three-inclusion-triple-layered soft inclusions model, the shear modu- lus of layers 1 to 3 are given by ( i 1 ; i 2 ; i 3 ) = ( 13 = 18 ; 4 = 9 ; 1 = 6 ), i = 1 : 3. The correspond- ing velocities are ( i 1 ; i 2 ; i 3 ) = ( 5 = 6 ; 2 = 3 ; 1 = 2 ). Similarly, for the three-inclusion-triple- layered sti inclusions model the shear moduli of the layers are given by ( i 1 ; i 2 ; i 3 ) = ( 23 = 18 ; 14 = 9 ; 11 = 6 ),i = 1 : 3, and the corresponding velocities are ( i 1 ; i 2 ; i 3 ) = ( 7 = 6 ; 8 = 6 ; 3 = 2 ). To simplify the analysis of the numerical results further, the number of inclusions is assumed to be N = 3; 5; 7; or 9. The number of inclusion layers is chosen to be L i = 3 or 5. Consequently, the following notation is adopted in denoting the models: NNLL, where N and L represent the number of inclusions and layers, respectively. For example, N 3L3 denotes a three-inclusion-triple-layered model. Finally, only elliptical layers are considered in evaluating the numerical results. 4.2 Verication of Numerical Results To assess the validity of the proposed method, the BIE numerical results are compared with the available analytical results. 29 4.2.1 Circular Cavity For a circular cavity embedded in a half-space, the numerical results obtained by the present method are compared with the analytical results of Lee [1]. The circular cavity BIE problem is modeled by a very soft single inclusion, which has a high impedance con- trast with the surrounding material, i.e., 1 = 1=600, 1 = 1=2, leading to an impedance contrast between the inclusion and the half-space of 1 1 0 0 = 1 1200 . Figure 4.1 shows a comparison between the two results for two angles of incidence. Apparently, the results are in very good agreement. 4.2.2 Circular Pipe For a circular pipe embedded in the half-space, the numerical results obtained by this model are compared with the analytical results obtained by Manoogian [2]. The circular pipe BIE problem is modeled using a two-layer inclusion, in which the innermost layer has the material properties of a very soft inclusion with high impedance contrast with respect to the half-space material. Therefore, 2 = 1 = 600 ; 2 = 1 = 2 . The material properties of the outermost layer are given by 1 = 3 , 1 = 1. Figure 4.2 illustrates very good agreement between the two results for both vertical and horizontal incident waves. This concludes the testing of the numerical results. Various layered inclusion cases are considered next. 30 4.3 Layered Inclusions 4.3.1 Eect of Multiple Scattering The eect of the multiple scattering of elastic waves is investigated by comparing the surface response of a multiple multilayered inclusions model with that of a single-layered inclusion. As stated earlier, the models involving three, ve, seven, and nine elliptical inclusions with three layers are considered. The layer materials are assumed to be either of the soft type or the sti type (see Eq. 4.4 and 4.5). Three-Inclusion-Three-Layer Model vs. Single-Inclusion-Three-Layer Model For the vertical and the grazing incidences, the surface response of the soft three- inclusions-three-layer (N 3L3) and the single-inclusion-three-layer (N 1L3) circular models are illustrated in Fig. 4.3. It is apparent that for the vertical incidence, the peak surface displacement amplitude (PSDA) of theN 3L3 model is greater than that of theN 1L3 model. For the grazing incidence, one clearly distinguishes the illuminated region (x<5), the shadow region (x > 5), and the region directly atop the inclusions (jxj < 5). No signicant eect of multiple scattering can be seen in the illuminated region. Atop the inclusions, however, a strong eect of multiple scattering can be observed. In particular, the PSDA of the multi-inclusion model is greater than that of the single inclusion model. Similarly, the surface responses in the shadow region demonstrate signicant dierences for the single- and multiple-inclusion models. Therefore, the presented results clearly demonstrate that multiple scattering may become very important for the surface motion amplication atop the soft inclusion model. 31 As the material properties change from soft to sti, the surface displacement am- plication is depicted by Fig. 4.4. For the vertical incidence, there is a reduction of oscillatory motion amplitude in the far-eld region in contrast to the soft model. More- over, the PSDA for the multiple inclusion (N 3L3) model is greater than that of the single (N 1L3) model. For the grazing incidence, the PSDAs for the two models are similar. However, for theN 3L3 model, the location of the PSDA is shifted to the left compared with theN 1L3 model. It is interesting to observe that there are negligible oscillations in the illuminated regions of both models. In contrast, in the shadow region the responses for the two models attenuate similarly with distance. Therefore, the results of theN 3L3 andN 1L3 models can be summarized as follows: Multiple scattering of the elastic waves by dierent inclusions may strongly aect the surface motion. The PSDA for the three-inclusion-three-layer (N 3L3) model is larger than or equal to that of the single-inclusion-three-layer model (N 1L3). A signicant reduction of oscillatory motion is present in the illuminated regions for both models. A considerable dierence in the surface motion can be observed between the illu- minated and the shadow regions. Comparison of the surface response between the ve-inclusion and single-inclusion models is considered next. 32 Five-Inclusion-Three-LayerModelvs. Single-Inclusion-Three-LayerModel As the number of inclusions increases to ve and the number of layers remains at three, the corresponding surface responses are depicted in Fig. 4.5. It should be noted that the inclusion distribution for the ve-inclusion model diers from that of the three-inclusion model. In the former, the inclusions are placed at the corners and the center of a square. For the latter, the inclusions are distributed at a constant depth within the half-space. For the vertical incidence, the PSDA for theN 5L3 model is similar to that of the N 1L3 model. Highly oscillatory surface motion can be observed in the far-eld regions. Likewise, the peak surface displacement amplitudes for the two models are similar for the grazing incidence as well. As before, the strong attenuation of the surface response with the distance is observed in the shadow region of the model. At this point, it is of interest to compare the response of the three- and ve-inclusion models (Fig. 4.3 and 4.5). It is apparent that the PSDAs of the two models are very similar for vertical and grazing incidences. Furthermore, the oscillatory motion in the illuminated regions and the attenuation in the shadow regions can be observed for both models as well. The surface responses of the sti ve-inclusion-three-layer and single-inclusion-three- layer sti circular models are illustrated in Fig. 4.6. For the vertical incidence wave, atop the inclusions, the PSDAs are similar, while in the far-eld region, they are dierent. In addition, signicant oscillatory motion can be observed in the far-eld region. 33 For the grazing incident wave, the illuminated region in both models displays a very small oscillatory motion. Atop the inclusions, the PSDA for theN 5L3 model is greater than that of theN 1L3 model. In addition, dierent attenuation in the shadow region can be observed for the two models. Therefore, the results for the ve-inclusion model can be summarized as follows Multiple scattering of waves may have a signicant eect on the oscillatory motion in the illuminated region and the attenuating motion in the shadow regions. A signicant reduction of oscillatory motion takes place in the illuminated region for the sti inclusions. The key observations of the surface response for the three-inclusion model agree with those of the ve-inclusion model. This concludes the analysis of the eect of multiple scattering in the model. The role of inclusion layering on surface motion is considered next. 4.3.2 Eect of Layering To study the eect of inclusion layering on the surface motion, the response of multiple layered inclusions is compared with that of the corresponding homogeneous inclusions models. The averaged material properties are assumed for the latter model. The results for three and ve inclusions with one and three layers are considered next. Three-Inclusion-Three-Layer Model vs. Three-Inclusion-Single-Layer Av- eraged Model 34 When the layers are made of the soft materials, the surface responses of the half-space for theN 3L3 andN 3L1 avg models are shown in Fig. 4.7. For the vertical incidence, atop the inclusions, the PSDA for theN 3L3 model is higher than that of theN 3L1 avg model. However, in the far-eld (jxj > 5), the PSDAs of the two models are quite similar. For the grazing incidence, however, the PSDA for the averaged model is much higher than that of the multilayered model, in the illuminated region, and slightly higher atop the inclusions. Conversely, the PSDA of the multilayered model is overall higher than that of the averaged model in the shadow region. The case of layers with sti materials is depicted in Fig. 4.8. It is evident that for both the vertical and the grazing incidences no signicant dierence between the layered and homogeneous models is observed. A comparison of the surface response between the ve-inclusion-three-layer and ve- inclusion-single-layer model is considered next. Five-Inclusion-Three-Layer Model vs. Five-Inclusion-Single-Layer Aver- aged Model As shown in Fig. 4.9 for soft layers and the vertical incidence the PSDAs for the layered and homogenous models are rather similar, whereas for the grazing incidence, the PSDA of the homogeneous model is higher than that of the layered model in the illuminated region and atop the inclusions. It is interesting to observe that the PSDA of the layered model exceeds that of the homogeneous model in the shadow region. When the layers are selected from the sti materials (Fig. 4.10), the surface motions of the layered and homogeneous models are very similar. 35 Therefore, the results presented demonstrate that the surface response of the model strongly depends upon the inclusion layering for soft materials. For the sti materials, however, the eect of layering upon PSDA is rather small. 4.3.3 Eect of the Layers' Geometry The in uence of the geometry of the layers can be studied by comparing the response of circular and elliptical models. The three-inclusion-three-layer model (N 3L3) with soft materials is considered. The location of the inclusion centers is the same for both of these models. In addition, the major axes of the elliptical layers are the same as the radii of the corresponding circles. It is evident from Fig. 4.11 that the PSDA of the two models is similar in the far-eld and atop the inclusions for the vertical incidence. However, for the grazing incidence, the PSDAs of the elliptical inclusions are overall higher than those of the circular inclusions in the illuminated region. The PSDAs are similar atop the inclusions, and dierent attenuation behavior is observed in the shadow region. In the case of sti materials, the dierence between the PSDAs of the responses of circular and elliptical multilayered models is negligible for both incidences (Fig. 4.12). In summary, the presented results show that the inclusion shape aects the surface motion more for the soft materials than for the sti materials. This completes the analysis of the surface response for multilayered multiple inclu- sions. The cases of layered pipes are considered next. 36 4.4 The Role of Impedance Contrast for Layered Pipes The layered inclusion models can be extended to solve the diraction problem by layered pipes as well. For that purpose, the innermost inclusions are assumed to be com- posed of an extremely soft material. Throughout, only two-layer pipes will be considered. The outer layers involve both soft and sti materials. Thus, the impedance contrast between the layers is assumed to be 0 0 i 3 i 3 > 1000 i 1 i 1 i 2 i 2 > 1; for sti-soft layers, i=1:N, i 1 i 1 i 2 i 2 < 1; for soft-sti layers, i=1:N. (4.6) The responses of seven and nine pipe congurations are investigated in this study. The seven-pipe model geometry is depicted by Fig. 4.13. The pipes are arranged in a hexagonal conguration with an additional pipe located at the center. The corresponding surface responses for soft-sti and sti-soft layers are shown by Fig. 4.14. For the vertical incidence, the PSDAs of the soft-sti and sti-soft models are very similar. For the grazing incidence and5<x< 0, the PSDA of the sti-soft pipes is higher than that of the soft-sti pipes. In addition, high-frequency content is observed in the illuminated region, and attenuated motion is present in the shadow region. At this point, the response of a nine-pipe model in a lattice array (Fig. 4.15) is considered. 37 For sti-soft and soft-sti layers, the corresponding surface displacements are shown in Fig. 4.16. The results show that the surface responses of the two models may signicantly dier from each other for both incident waves. It is apparent from the results of Fig. 4.16 that the surface motion strongly depends upon the impedance contrast of the layers. The overall PSDA is observed for sti-soft pipe materials. As before, the surface motion changes considerably from the near to the far-eld. The presented results clearly demonstrate the importance of the impedance contrast between layers in the pipes upon surface motion. This section concludes the investigation of scattered waves for layered pipes. 38 −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 γ = 60° γ =90° LEE, γ = 60° LEE, γ =90° Figure 4.1: Comparison of the surface responses of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to vertical and 60 incident plane harmonic SH waves studied by Lee [1]. Solid and dashed lines represent the results of this study, whereas the squares and triangles denote the results of Lee [1]. The parameters are: = 1, N = 1, a 1 = b 1 = 1, O = (0; 1:5), 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 . −3 −2 −1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 γ = 0 ° γ =90° MANOOGIAN, γ = 0° MANOOGIAN, γ =90° Figure 4.2: Comparison of the surface responses of a pipe embedded within a half-space sub- jected to vertical and horizontal incident plane harmonic SH waves, as studied by Manoogian [2]. Solid and dashed lines represent the results of this study, whereas the squares and circles denote the results of Manoogian [2]. The parameters are = 2, N = 1, L = 2, a 1 =b 1 = 1, a 2 =b 2 = 0:9, O = (0; 1:5); 0 = 0 = 1, 1 = 3, 1 = 1, 2 = 1 600 , 2 = 1 2 39 −10 −5 0 5 10 0 1 2 3 4 γ=0, η=1, S 0 =26 x Displacement |u 3 | N1L3 soft N3L3 soft −10 −5 0 5 10 0 0.5 1 1.5 2 2.5 3 3.5 γ=90, η=1, S 0 =26 x Displacement |u 3 | N1L3 soft N3L3 soft Figure 4.3: Surface displacement amplitude for the soft three-inclusion-triple-layer model vs. the one-inclusion-triple-layer model for vertical (top) and grazing (bottom) incident waves. The layer centers are located at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). Similarly, the various layer principal axes are assumed to be a i 1 = b i 1 = 1;a i 2 = b i 2 = 0:75;a i 3 = b i 3 = 0:5; i = 1 : 3. The material properties of the layers are specied by i j = i(soft) j ; i j = i(soft) j ; i;j = 1 : 3: For i(soft) j and i(soft) j , see Eq. (4.4). 40 −10 −5 0 5 10 0 0.5 1 1.5 2 2.5 3 3.5 γ=0, η=1, S 0 =26 x Displacement |u 3 | N1L3 stiff N3L3 stiff −10 −5 0 5 10 1 1.5 2 2.5 3 3.5 γ=90, η=1, S 0 =26 x Displacement |u 3 | N1L3 stiff N3L3 stiff Figure 4.4: Surface displacement amplitude for the sti three-inclusion-triple-layer model vs. the one-inclusion-triple-layer model for vertical (top) and grazing (bottom) incident waves. The parameters of the problem (see Fig. 4.3) are as follows : O 1 = (3; 1:5), O 2 = (0; 1:5), O 3 = (3; 1:5), a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, a i 3 = b i 3 = 0:5, i j = i(stiff) j , i j = i(stiff) j ;i;j = 1 : 3. 41 −15 −10 −5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 γ=0, η=1, S 0 =32 x Displacement |u 3 | N1L3 soft N5L3 soft −15 −10 −5 0 5 10 15 0 1 2 3 4 5 γ=90, η=1, S 0 =32 x Displacement |u 3 | N1L3 soft N5L3 soft Figure 4.5: Surface displacement amplitude for the soft ve-inclusion-three-layer model vs. the one-inclusion-three-layer model for vertical (top) and grazing (bottom) inci- dent waves. The parameters of the problem are O 1 = (6; 1:5), O 2 = (3; 1:5), O 3 = (0; 1:5), O 4 = (3; 1:5), O 5 = (6; 1:5), a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, a i 3 =b i 3 = 0:5, i j = i(soft) j , i j = i(soft) j , i = 1 : 5, j = 1 : 3. 42 −15 −10 −5 0 5 10 15 0.5 1 1.5 2 2.5 3 γ=0, η=1, S 0 =32 x Displacement |u 3 | N1L3 stiff N5L3 stiff −15 −10 −5 0 5 10 15 1 1.5 2 2.5 3 3.5 4 4.5 γ=90, η=1, S 0 =32 x Displacement |u 3 | N1L3 stiff N5L3 stiff Figure 4.6: Surface displacement amplitude for the sti ve-inclusion-three-layer model vs. the one-inclusion-three-layer model for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (6; 1:5), O 2 = (3; 1:5), O 3 = (0; 1:5), O 4 = (3; 1:5), O 5 = (6; 1:5), a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, a i 3 =b i 3 = 0:5, i j = i(stiff) j ; i j = i(stiff) j , i = 1 : 5; j = 1 : 3: 43 −10 −5 0 5 10 0 1 2 3 4 γ=0, η=1, S 0 =26 x Displacement |u 3 | N3L1 avg N3L3 soft −10 −5 0 5 10 0 1 2 3 4 5 γ=90, η=1, S 0 =26 x Displacement |u 3 | N3L1 avg N3L3 soft Figure 4.7: Surface displacement amplitude for the soft three-inclusion-triple-layer model vs. the three-inclusion-single-layer model with averaged material properties for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (3; 1:5);O 2 = (0; 1:5);O 3 = (3; 1:5); a i 1 = b i 1 = 1;a i 2 = b i 2 = 0:75;a i 3 = b i 3 = 0:5; i 1 = i 2 = i 3 = 1 3 P 3 j=1 i(soft) j ; i 1 = i 2 = i 3 = 1 3 P 3 j=1 i(soft) j ;i;j = 1 : 3: 44 −10 −5 0 5 10 0 0.5 1 1.5 2 2.5 3 3.5 γ=0, η=1, S 0 =26 x Displacement |u 3 | N3L1 avg N3L3 stiff −10 −5 0 5 10 1 1.5 2 2.5 3 3.5 γ=90, η=1, S 0 =26 x Displacement |u 3 | N3L1 avg N3L3 stiff Figure 4.8: Surface displacement amplitude for the sti three-inclusion-triple-layer model vs. the three-inclusion-single-layer model with averaged material properties for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (3; 1:5), O 2 = (0; 1:5), O 3 = (3; 1:5), a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, a i 3 =b i 3 = 0:5, i j = 1 3 P 3 j=1 i(stiff) j ; i j = 1 3 P 3 j=1 i(stiff) j ; i;j = 1 : 3. 45 −15 −10 −5 0 5 10 15 0 0.5 1 1.5 2 2.5 3 3.5 γ=0, η=1, S 0 =32 x Displacement |u 3 | N5L1 avg N5L3 soft −15 −10 −5 0 5 10 15 0 1 2 3 4 5 γ=90, η=1, S 0 =32 x Displacement |u 3 | N5L1 avg N5L3 soft Figure 4.9: Surface displacement amplitude for soft ve-inclusion-triple-layer model vs. ve- inclusion-single-layer model with averaged material properties for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (6; 1:5), O 2 = (3; 1:5), O 3 = (0; 1:5), O 4 = (3; 1:5), O 5 = (6; 1:5), a i j =b i j = 5j 4 , i j = 1 5 P 3 j=1 i(soft) j ; i j = 1 5 P 3 j=1 i(soft) j , i = 1 : 5; j = 1 : 3: 46 −10 −5 0 5 10 0.5 1 1.5 2 2.5 3 γ=0, η=1, S 0 =26 x Displacement |u 3 | N5L1 avg N5L3 stiff −10 −5 0 5 10 1.5 2 2.5 3 3.5 4 4.5 γ=90, η=1, S 0 =26 x Displacement |u 3 | N5L1 avg N5L3 stiff Figure 4.10: Surface displacement amplitude for the sti ve-inclusion-three-layer model vs. the ve-inclusion-single-layer model with averaged material properties for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (6; 1:5), O 2 = (3; 1:5), O 3 = (0; 1:5), O 4 = (3; 1:5), O 5 = (6; 1:5), a i j =b i j = 5j 4 , i j = 1 5 P 3 j=1 i(stiff) j ; i j = 1 5 P 3 j=1 i(stiff) j , i = 1 : 5, j = 1 : 3. 47 −10 −5 0 5 10 0 1 2 3 4 γ=0, η=1, S 0 =26 x Displacement |u 3 | N3L3 soft crc N3L3 soft elp −10 −5 0 5 10 0 1 2 3 4 5 γ=90, η=1, S 0 =26 x Displacement |u 3 | N3L3 soft crc N3L3 soft elp Figure 4.11: Surface displacement amplitude for the softN 3L3 circular vs. elliptical models for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (3; 1:5), O 2 = (0; 1:5), O 3 = (3; 1:5), i j = i(soft) j , i j = i(soft) j ; i;j = 1 : 3. For the elliptical model the principal axes are assumed to be : a i 1 = 1;b i 1 = 0:75;a i 2 = 0:75;b i 2 = 0:56;a i 3 = 0:5;b i 3 = 0:38, and i = 1 : 3: For the circular model the corresponding radii are a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 =b i 3 = 0:5. 48 −10 −5 0 5 10 0 0.5 1 1.5 2 2.5 3 3.5 γ=0, η=1, S 0 =26 x Displacement |u 3 | N3L3 stiff crc N3L3 stiff elp −10 −5 0 5 10 0.5 1 1.5 2 2.5 3 3.5 γ=90, η=1, S 0 =26 x Displacement |u 3 | N3L3 stiff crc N3L3 stiff elp Figure 4.12: Surface displacement amplitude for the stiN 3L3 circular vs. elliptical models for vertical (top) and grazing (bottom) incident waves. The parameters of the problem are: O 1 = (3; 1:5);O 2 = (0; 1:5);O 3 = (3; 1:5), i j = i(stiff) j ; i j = i(stiff) j ; i;j = 1 : 3, For the elliptical model, the principal axes are assumed to be a i 1 = 1;b i 1 = 0:75, a i 2 = 0:75;b i 2 = 0:56, a i 3 = 0:5;b i 3 = 0:38, and i = 1 : 3: For the circular model the corresponding radii are a i 1 = b i 1 = 1;a i 2 = b i 2 = 0:75, and a i 3 =b i 3 = 0:5. 49 x z O Figure 4.13: The seven-pipe model (N = 7) conguration, with pipe centers located at O 1 = (1:25; 5:83), O 2 = (2:5; 3:67), O 3 = (1:25; 1:5), O 4 = (0; 3:67), O 5 = (1:25; 5:83), O 6 = (2:5; 3:67), and O 7 = (1:25; 1:5). The principal axes are assumed to be a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, a i 3 =b i 3 = 0:5 for, i = 1 : 7. 50 −10 −5 0 5 10 0 0.5 1 1.5 2 2.5 3 3.5 γ=0, η=1, S 0 =25 x Displacement |u 3 | N7−soft−stiff pipe N7−stiff−soft pipe −10 −5 0 5 10 0 1 2 3 4 5 6 7 γ=90, η=1, S 0 =25 x Displacement |u 3 | N7−soft−stiff pipe N7−stiff−soft pipe Figure 4.14: The surface response for the seven-pipe model (see Fig. 17) for sti-soft vs. soft- sti layers for vertical (top) and grazing (bottom) incident waves. For the sti-soft model: i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600; i(stiff) 1 = 12=9, i(soft) 2 = 2=3; i 3 = 1=2; i = 1 : 7. For the soft-sti model: i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2; i = 1 : 7. 51 x z O Figure 4.15: The nine-pipe model (N = 9) conguration with the pipes centers at O 1 = (2:5; 6:5), O 2 = (2:5; 4), O 3 = (2:5; 1:5), O 4 = (0; 6:5), O 5 = (0; 4), O 6 = (0; 1:5), O 7 = (2:5; 6:5), O 8 = (2:5; 4), and O 9 = (2:5; 1:5). The principal axes are a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 9. 52 -10 -5 0 5 10 0 0.5 1 1.5 2 2.5 3 3.5 γ=0, η=1, S 0 =25 x Displacement |u 3 | N9-soft-stiff pipe N9-stiff-soft pipe -10 -5 0 5 10 0 1 2 3 4 5 6 7 γ=90, η=1, S 0 =25 x Displacement |u 3 | N9-soft-stiff pipe N9-stiff-soft pipe Figure 4.16: Surface displacement amplitude for the nine-pipe model (see Fig. 4.15) for sti-soft and soft-sti layers for vertical (top) and grazing (bottom) incident waves. . For the sti-soft model: i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600, i(stiff) 1 = 12=9, i(soft) 2 = 2=3, i 3 = 1=2; i = 1 : 9. For the soft-sti model: i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2; i = 1 : 9. 53 Chapter 5 Stress Field for Multiple Multilayered Inclusions Embedded in a Half-Space and Subjected to a Plane Harmonic SH-Wave 5.1 Introduction The concentration of dynamic stresses in the scattering and diraction of elastic waves plays a prominent role in many elds of science and engineering, including fracture me- chanics, seismology, material science, non-destructive testing and the design of compos- ites. The associated dynamic stress factor can be obtained analytically or numerically. Analytical solutions describing excitations by a horizontal shear (SH) wave plane har- monic wave are only available for simple geometries, such as a cavity in a full-space [6], a cavity in a half-space [3], and two circular cavities in a half-space [4]. Of the available nu- merical methods, the most prominent technique for the problem at hand is the boundary element method (BEM). The basis of the BEM is an appropriate fundamental solution, which, for displacement elds, leads to strongly singular integral equations (Eq. A.2). 54 However, in computing the stresses, the kernels of the integral equations become hyper- singular when the gradient of the displacement boundary integral equation (DBIE) is taken (see Appendix A). To overcome this obstacle several techniques based on the integration by parts [53], Stokes' theorem [54] , integral identities [55, 56], and numerical methods using special quadratures [57] have been developed. The basic idea behind the regularization of stress boundary integral equations (BIEs) involves writing the hypersingular kernel, which occurs in the integral equation, as a sum of singular and free terms. Consequently, it can be shown that the hypersingular integrals cancel out, leaving only weakly singular integrals that can be evaluated numerically. Guiggiani [58{60] demonstrated that the singular terms cancel out by considering the analytical limit and expanding the singular terms using local coordinates. Gray [61, 62] implemented a symbolic manipulator to take the analytical limit for singular terms to show that they cancel out. Krishnasamy et al. [54] invoked Stokes' theorem to express the singular surface inte- grals in terms of regular line integrals. This method enabled numerical evaluation of the required integrals. Liu and Rizzo [55, 56] presented a weakly singular form of the stress BIE using integral identities [63] of the fundamental solutions. They represented the displacement gradients as tangential derivatives, which required using the C (1) continuous elements. Hildenbrand and Kuhn [57] employed Overhauser elements along with Kutt's quadra- ture [64] to avoid the limiting process and satisfy the C (1) continuity requirement. 55 One of the earliest analytical stress eld results was obtained by Mow and Pao [6], who solved the scattering of harmonic SH-waves by a circular cavity embedded in a full- space using the method of wave function expansion (see Appendix B). They computed the concentration of dynamic stress along the cavity for a wide range of frequencies. These results will be used in the present work for testing purposes. Recently, numerical results based on BIEM for cavities and inclusions in a nite elastic solid and subjected to a plane harmonic SH-wave were obtained by Parvanova et al. [65, 66]. These results reveal the dependence of the stress concentration on the number, location and shape of the scatterers. In this investigation, to evaluate the stress eld in the scattering of the elastic plane harmonic SH-waves by elastic obstacles in a half-space, the BIE is formulated in terms of the weak form of the equations of motion. This method was rst suggested by Okada et al. [67, 68, 69] for elastostatics and later extended by Qian et al. [70] for acoustics. The corresponding integral equation involves both weakly and strongly singular integrals. The strongly singular integrals can be expressed in terms of weakly singular integrals using the divergence theorem and properties of integral equations. This technique completely avoids evaluating hypersingular integrals [70{72] and does not impose any continuity requirements during discretization. Therefore, even linear elements can be used without relying on special numerical quadratures. The fundamental basis of this method is outlined in Appendix B, where the problem of the scattering of a plane harmonic SH-wave by a circular inclusion/cavity embedded in a full-space is considered. Available analytical solutions to this problem enabled extensive 56 testing of the proposed method. This method is now extended to consider the scattering of plane harmonic SH-waves by multiple multilayered inclusions embedded in a half-space. The problem of evaluating the stress eld in the half-space problem is organized as follows. First, the corresponding BIEs for the displacement gradients are introduced. Next, the verication of the results is presented for the half-space cases available in the literature. Finally, the general results for multiple multilayered inclusions within the half-space are provided. 5.2 Boundary Integral Equations for the Half-Space Problem The geometry of the problem at hand is depicted by Fig. 5.1. The problem consists of multiple multilayered inclusions embedded in a half-space and subjected to a plane harmonic incident SH-wave. Here, S i j and D i j , i = 0 : N, j = 0 : N, are the layer interfaces and domains, respectively. The half-space surface and domain are denoted by S 0 S 0 0 andD 0 D 0 0 , respectively. Furthermore, all of the displacements for domainD i j are represented byu (i) j . Consistent with the convention adopted earlier, the displacement eld in the half-space is denoted by u 0 u 0 0 . 57 n γ u inc n n n n n n n S 0 D 0 D 1 1 D 2 1 D 1 i j i i D 1 N Li D LN N S 1 1 S 2 1 S 1 i S j i S i Li S 1 N S 2 N n n D 1 i-1 D 2 i-1 S 1 i-1 S 2 i-1 n n D 1 i+1 D 2 i+1 S 1 i+1 S 2 i+1 y x D D Figure 5.1: Geometry of the half-space containing N multilayered inclusions with L i ;i = 1 :N layers. The displacement boundary integral equation for the scattered wave eld in the half- space is given by Sheikhhassani and Dravinski [73] 1 2 u (0) (y)+ Z S 0 T (0) (x;y;n)u (0) (x)dS x + N X i=1 Z S i 1 T (0) (x;y;n)u (0) (x)dS x N X i=1 Z S i 1 U (0) (x;y)t (i) 1 (x;n)dS x =F (y) y2S 0 or y2S i 1 ;i = 1 :N (5.1) where the forcing term is dened by 58 F (y) = 1 2 u ff (y)+ Z S 0 T (0) (x;y;n)u ff (x)dS x + N X i=1 Z S i 1 T (0) (x;y;n)u ff (x)dS x N X i=1 Z S i 1 U (0) (x;y)t ff (x;n)dS x y2S 0 or y2S i 1 ;i = 1 :N (5.2) Here, U (0) and T (0) are the full-space displacement and traction Green's functions [51], respectively, dened for the material in domain D 0 . Furthermore, u 0 is the unknown scattered wave eld in the half-space, t (i) 1 is the traction on the surface S i 1 , and u ff and t ff are the free-eld displacements and tractions, respectively. 59 For a generic ring (Fig. 5.2) for domain D i j bounded by the interfaces S i j and S i j+1 , the BIE for the unknown scattered wave elds is given by Sheikhhassani and Dravinski [73] as 1 2 u (i) j (y)+ Z S i j T (i) j (x;y;n)u (i) j (x)dS x Z S i j+1 T (i) j (x;y;n)u (i) j+1 (x)dS x Z S i j U (i) j (x;y)t (i) j (x;n)dS x + Z S i j+1 U (i) j (x;y)t (i) j+1 (x;n)dS x = 0; y2S i j or S i j+1 1iN 1jL i 1 (5.3) Here,u (i) j (y) andt (i) j (x;n) represent the unknown scattered displacements and tractions, respectively, for the domain D i j , i = 1 : N, j = 1 : L i 1. As before, R denotes the principal value integrals. 60 Similarly, in the innermost layer in each of the inclusions (Fig 5.3), the corresponding displacement integral equation is given by Sheikhhassani and Dravinski [73] as 1 2 u (i) Li (y) Z S i Li U (i) Li (x;y)t (i) Li (x;n)dS x + Z S i Li T (i) Li (x;y;n)u (i) Li (x)dS x = 0; y2S i Li i = 1 :N (5.4) Here, u (i) Li and t (i) Li denote the displacement and traction along the surface S i Li (Fig. 5.3), respectively, while U (i) Li and T (i) Li are the displacement and traction Green's functions for the full-space domain with the material property of D i Li . The displacement gradient boundary integral equation (GBIE) for a generic ring can be obtained by generalizing the results for the full-space single-inclusion problem discussed in Appendix B. Therefore, by utilizing integral equation properties IIII (Eqs. B.29, 61 B.34, B.44), the displacement gradient BIEs for the generic domain D i j ,i = 1 :N;j = 1 : L i 1 (Fig. 5.2) can be written in the following form: 0 = Z S i j (i) j [k (i) j ] 2 n q (x)U (i) j (x;y)u (i) j (x)dS x Z S i j+1 (i) j [k (i) j ] 2 n q (x)U (i) j (x;y)u (i) j+1 (x)dS x + Z S i j U (i) j;q (x;y)[t (i) j (x)t (i) j (y)]dS x Z S i j+1 U (i) j;q (x;y)[t (i) j+1 (x)t (i) j (y)]dS x Z S i j T (i) j (x;y)[u (i) j;q (x)u (i) j;q (y)]dS x + Z S i j+1 T (i) j (x;y)[u (i) j+1;q (x)u (i) j;q (y)]dS x + Z S i j (i) j n q (x)U (i) j;l (x;y)[u (i) j;l (x)u (i) j;l (y)]dS x Z S i j+1 (i) j n q (x)U (i) j;l (x;y)[u (i) j+1;l (x)u (i) j;l (y)]dS x Z S i j T (i) j (x;y)u (i) j;q (y)dS x + Z S i j+1 T (i) j (x;y)u (i) j;q (y)dS x + 1 2 u (i) j;q (y) i = 1 :N;j = 1 :L i 1;q = 1; 2;l = 1; 2 y2D i j &(y2S i j or S i j+1 ) (5.5) where u (i) j , t (i) j , and u (i) j;q denote the displacement, traction, and displacement gradient along the surface S i j (Fig. 5.2), respectively. In addition, U (i) j and T (i) j are the full-space displacement and traction Green's functions for domain D i j . Additionally, n q is the q th component of the unit normal vector for various interfaces. 62 Note that the above equation contains only weakly singular integrals that can be eval- uated numerically. Thus, the problem of evaluating hypersingular integrals is completely eliminated. Similarly, for the innermost layer of the domain D i L i for each inclusion (Fig. 5.3), the displacement gradient is given by 0 = Z S i L i (i) L i [k (i) L i ] 2 n q (x)U (i) L i (x;y)u (i) L i (x)dS x + Z S i L i U (i) L i ;q (x;y)[t (i) L i (x)t (i) L i (y)]dS x Z S i L i T (i) L i (x;y)[u (i) L i ;q (x)u (i) L i ;q (y)]dS x + Z S i L i (i) L i n q (x)U (i) L i ;l (x;y)[u (i) L i ;l (x)u (i) L i ;l (y)]dS x Z S i L i T (i) L i (x;y)u (i) L i ;q (y)dS x + 1 2 u (i) L i ;q (y) i = 1 :N;q = 1; 2;l = 1; 2 y2D i L i &y2S i L i (5.6) Here, u (i) L i , t (i) L i , and u (i) L i ;q are the displacement, traction, and displacement gradient along the surface S i L i , respectively. In addition, U (i) L i and T (i) L i are the corresponding displacement and traction Green's functions for domain D i L i . 63 The structure of Eqs. (5.5) and (5.6) enables the numerical evaluation of the displace- ment gradient throughout the elastic domain. The discretization of these equations for dierent congurations of layered inclusions is considered next. 64 D j i S j i D j+1 i S j+1 i n n Figure 5.2: The ring limit model for domain D i j when the observation point y is located on the boundary S i j (or S i j+1 ), where i = 1 :N, j = 1 :L i 1. n S Li i D Li i Figure 5.3: The inclusion limit model when the observation point y is located on the boundary S i j , i = 1 :N, ;j =L i . 5.3 Discretization of the Half-Space Model The discretization procedure of the model for displacements was discussed in detail in Section 3.2. Consequently, for the half-space displacement, the discretized version of Eqs. (5.1), (5.3), and (5.4) are given by Eqs. (3.13), (3.20), and (3.21). In addition, the gradient integral equations for the generic domains in the half-space, (Eqs. 5.6 and 5.5) are discretized based on Eqs. (B.60), (B.61), (B.64), (B.67), and (B.69), as discussed in Appendix B. 65 Therefore, for a general conguration of multiple multilayered inclusions embedded in a half-space, the discretized system of equations can be assembled into the following form: MU =F M2C (n+4m)(n+4m) U2C (n+4m)1 F2C (n+4m)1 n =P 0 + 1 m = N X i=1 Li X j=1 P i j (5.7) where n is the number of nodes on the half-space surface S 0 , and m is the total number nodes on all the inclusion surfaces and layers. Moreover, P i j represents the number of elements along the surfaceS i j , andP 0 denotes the number of elements along the half-space surface. The matrix equation (Eq. 5.7) can be partitioned in the following manner: 2 6 6 4 A B 0 C E D 3 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 u t u ;1 u ;2 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 4 F 0 3 7 7 5 (5.8) 66 where the sub-matrices A;B;C;D;E, and F are all explicitly known. The sub-matrices A,B, and F are associated with the displacement BIE; the sub-matrices C, D, and E correspond to the gradient BIE. To illustrate the structure of Eq. (5.8), the case of two inclusions embedded within the half-space is considered. The rst inclusion has one layer, and the second inclusion has two layers. Therefore, in the notation introduced when studying the displacements (Chapter 3), this model is denoted byN 2L(1; 2). The unknowns consist of the displacements along the half-space surface and the displacements, tractions and stresses along the layered surfaces of the inclusions. All of the unknowns in the partitioned form of the matrix equation can be written as follows: u =fu (0) ;u (1) ;u (2) 1 ;u (2) 2 g T t =ft (1) ;t (2) 1 ;t (2) 2 g T u ;1 =fu 1 ;1 ; (u ;1 ) 2 1 ; (u ;1 ) 2 2 g T u ;2 =fu 1 ;2 ; (u ;2 ) 2 1 ; (u ;2 ) 2 2 g T (5.9) By determining the displacement gradients, the stresses can be evaluated from the relation 3q =u ;q q = 1; 2 (5.10) where is the shear modulus. 67 Sub-matrix A is the assembled matrix of all of the coecients of the displacements in the discretized displacement BIE; The size of this submartix is (n + 2m) (n +m). Specically, matrix A can be written as A = 2 6 6 6 6 6 6 6 6 6 6 4 A 0 A 0 1 A 0 2 0 0 A 1;1 1 0 0 0 0 A 2;1 1 A 2;1 2 0 0 0 A 2;2 2 3 7 7 7 7 7 7 7 7 7 7 5 where the matrices A 0 and A 0 j ;j = 1; 2; are associated with the surfaces S 0 , S i 1 ;i = 1; 2; and domain D 0 . Similarly, the matrices A i;j k correspond to the surfaces S i k ;i = 1; 2;k = 1; 2; and domain D i j ;j = 1; 2. Sub-matrix B in Eq. (5.8) is the assembled form of all of the coecients of tractions in the DBIE and has a size of (n + 2m)m. The matrix can be written explicitly as B = 2 6 6 6 6 6 6 6 6 6 6 4 B 0 1 B 0 2 0 B 1;1 1 0 0 0 B 2;1 1 B 2;1 2 0 0 B 2;2 2 3 7 7 7 7 7 7 7 7 7 7 5 where the matrices B 0 j ;j = 1; 2 are associated with the surfaces S i 1 ;i = 1; 2 and domain D 0 . Similarly, the matrices B i;j k correspond to the surfaces S i k ;i = 1; 2;k = 1; 2; and domain D i j ;j = 1; 2. 68 Sub-matrix C is the assembled form of all of the coecients of the displacements in the GBIE and has a size of 2m (p +m). Explicitly, the matrix assumes the form C = 2 6 6 6 6 6 6 6 6 6 6 4 0 C 1 1 0 0 0 C 1 2 0 0 0 C 2;1 11 C 2;1 12 0 0 C 2;1 21 C 2;1 22 0 3 7 7 7 7 7 7 7 7 7 7 5 where all of the sub-matrices are known. Sub-matrixE in Eq. (5.8) is the assembled form of all of the coecients of the tractions in the discretized gradient BIE and has a size of 2mm. It assumes the form E = 2 6 6 6 6 6 6 6 6 6 6 4 E 1 1 0 0 E 1 2 0 0 0 E 2;1 11 E 2:1 12 0 E 2;1 21 E 2:1 22 3 7 7 7 7 7 7 7 7 7 7 5 where all the sub-matrices are explicitly known. Sub-matrix D in Eq. (5.8) is the assembled form of all of the coecients of the displacement gradients in the GBIE, and has a size of 2m 2m. The sub-matrix can be written as 69 D = 2 6 6 6 6 6 6 6 6 6 6 4 G 1 11 0 0 D 1 12 0 0 D 1 21 0 0 G 1 22 0 0 0 G 2;1 11;1 G 2;1 11;2 0 D 2;1 12;1 D 2;1 12;2 0 D 2;1 21;1 D 2;1 21;2 0 G 2;1 22;1 G 2;1 22;2 3 7 7 7 7 7 7 7 7 7 7 5 where all of the element matrices are known. The above procedure reveals that these discretized matrices have a certain structure. Based on that structure, one can write the discretized form of the integral equations for dierent numbers of inclusions and inclusion layers. In general, the discretized form of the BIE for evaluating the stresses will take the form 2 6 6 4 A B 0 C E D 3 7 7 5 2 6 6 6 6 6 6 6 6 6 6 4 u t u ;1 u ;2 3 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 4 F 0 3 7 7 5 (5.11) where various block matrices depend on the number of inclusions and the number of layers in each inclusion. This step completes the formulation of the problem. The verication of the proposed numerical results is considered next. 5.4 Verication of the Half-Space Results To verify the accuracy of the proposed method, the numerical results obtained here are compared with those available in the literature. The following models are considered: a circular cavity, two circular cavities, and a pipe embedded in a half-space. Key material 70 properties are discussed in Section 4.1. Similarly, 128 Overhauser elements per surface are used throughout for all of the inclusions. 5.4.1 Single Cavity in the Half-Space (N 1L1 cav ) The numerical results produced by the present method are compared with the ana- lytical results of Lin and Liu [3] for a single circular cavity embedded at a certain depth within a half-space. The cavity has a unit radius and is subjected to an oblique plane harmonic incident SH-wave. The cavity in this study can be modeled as a very soft inclusion (see Section 4.2.1). Figures 5.4-5.7 show a comparison between the two results for dierent angles of incidence and cavity depths. The results are clearly in excellent agreement for a wide range of parameters present in the problem. The case of two circular cavities in the half-space is considered next. 5.4.2 Two Circular Cavities in the Half-Space (N 2L1 cav ) The numerical results obtained by the present method are compared with the analyti- cal results of Wang and Liu [4] for two circular cavities with unit radii embedded within a half-space and subjected to a plane harmonic incident SH-wave. Figures 5.8-5.11 compare the two results for dierent angles of incidence and cavity depths. It is clear from these gures that the results obtained by the two dierent studies are in complete agreement. 71 5.4.3 Pipe in a Half-Space (N 1L2) The numerical calculations for a pipe embedded in a half-space subjected to a plane harmonic SH-wave, produced by the present method are compared with the analytical results of Lee and Trifunac [5]. Figures 5.12-5.15 compare the the two results. Clearly, the two results are in excellent agreement for dierent material properties of the pipe. Testing of the results revealed that the proposed method produces accurate results for stresses along dierent interfaces present in the problem. Thorough the testing of the results, it was established that for the range of frequencies considered here, the method requires 128 Overhauser elements (see Appendix B.3) for each inclusion surface. For number of the elements on the surface of the half-space the reader is referred to Eq. (4.2). This section concludes the verication of the method. The stress elds for various general models are considered next. 72 0.2 0.4 0.6 0.8 30 210 60 240 90 270 120 300 150 330 180 0 Distribution of | / | around a cavity edge σ σ θz 0 ka=0.1, =0, h=1.5 γ Lin & Liu BEM Figure 5.4: Comparison of the normalized stressj z = 0 j of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to a vertical incident plane harmonic SH -wave. Solid lines represent the results of Lin and Liu [3], and the circles denote the results of this study. The parameters are as follows: normalized wavenumber ka = 0:1, number of inclusions N = 1, principal radii of the inclusion a 1 =b 1 = 1, and center location O = (0; 1:5). The material properties of the half- space and the inclusion are 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 , and 0 is the maximum stress amplitude of the stress eld along the cavity. 73 1 2 3 4 30 210 60 240 90 270 120 300 150 330 180 0 . Lin & Liu BEM ka=0.1, =45˚, h=1.5 γ Distribution of | / | around a cavity edge σ σ θz 0 Figure 5.5: Comparison of the normalized stressj z = 0 j of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to = 45 incident plane harmonic SH -wave. Solid lines represent the results of Lin and Liu [3], and the circles denote the results of this study. The parameters are as follows: normalized wavenumber ka = 0:1, number of inclusions N = 1, principal radii of the inclusion a 1 =b 1 = 1, and center location O = (0; 1:5). The material properties of the half- space and the inclusion are 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 , and 0 is the maximum stress amplitude of the stress eld along the cavity. 74 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 Lin & Liu BEM Distribution of | / | around a cavity edge σ σ θz 0 ka=0.1, =45, h=12 γ Figure 5.6: Comparison of the normalized stressj z = 0 j of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to = 45 incident plane harmonic SH -wave. Solid lines represent the results of Lin and Liu [3], and the circles denote the results of this study. The parameters are: normalized wavenumber ka = 0:1, number of inclusions N = 1, principal radii of the inclusion a 1 =b 1 = 1, and center location O = (0; 12). The material properties of the half-space and the inclusion are 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 , and 0 is the maximum stress amplitude of the stress eld along the cavity. 75 1 2 3 4 30 210 60 240 90 270 120 300 150 330 180 0 Lin & Liu BEM ka=0.1, =0, h=12 γ Distribution of | / | around a cavity edge σ σ θz 0 Figure 5.7: Comparison of the normalized stressj z = 0 j of an extremely soft circular inclusion and the circular cavity embedded within a half-space subjected to a vertical incident plane harmonic SH -wave. Solid lines represent the results of Lin and Liu [3], and the circles denote the results of this study. The parameters are: normalized wavenumber ka = 0:1, number of inclusions N = 1, principal radii of the inclusion a 1 =b 1 = 1, and center location O = (0; 12). The material properties of the half-space and the inclusion are 0 = 0 = 1, 1 = 1 600 , 1 = 1 2 , and 0 is the maximum stress amplitude of the stress eld along the cavity. 76 1 2 3 4 30 210 60 240 90 270 120 300 150 330 180 0 d=2.5a, h=1.5a, =45 γ ° Wang & Liu, ka=.1 Wang & Liu, ka=1 Wang & Liu, ka=2 BEM, ka=.1 BEM, ka=1 BEM, ka=2 Distribution of | / | vs σ σ θ θz 0 Figure 5.8: Comparison of the normalized stressj z = 0 j for two circular cavities embedded within a half-space for dierent wavenumbers (N 2L1 cav model). The stresses are evaluated on the surface of the left circular cavity for a 45 incident plane harmonic SH-wave. The solid and dashed lines represent the results of Wang and Liu [4], and the circles, crosses and stars denote the results of this study. The parameters are the following: number of inclusions N = 2, principal axes a i = b i = 1, and location of the centers O 1 = (2:5; 1:5),O 2 = (2:5; 1:5). The material properties of the half-space and the inclusion are 0 = 0 = 1, i = 1 600 , i = 1 2 , i = 1; 2, and 0 is the maximum stress amplitude of the stress eld along the cavity. 77 2 4 6 30 210 60 240 90 270 120 300 150 330 180 0 d=2.5a, h=1.5a, =90 γ ° Wang & Liu, ka=.1 Wang & Liu, ka=1 Wang & Liu, ka=2 BEM, ka=.1 BEM, ka=1 BEM, ka=2 Distribution of | / | vs σ σ θ θz 0 Figure 5.9: Comparison of the normalized stressj z = 0 j for two circular cavities embedded within a half-space for dierent wavenumbers (N 2L1 cav model). The stresses are evaluated on the surface of the left circular cavity for a grazing incident plane har- monic SH-wave. Solid and dashed lines represent the results of Wang and Liu [4], and the circles, crosses and stars denote the results of this study. The parameters are the following: number of inclusions N = 2, principal axes a i = b i = 1, and location of the centers O 1 = (2:5; 1:5),O 2 = (2:5; 1:5). The material properties of the half-space and the inclusion are 0 = 0 = 1, i = 1 600 , i = 1 2 , i = 1; 2; and 0 is the maximum stress amplitude of the stress eld along the cavity. 78 2 4 6 8 30 210 60 240 90 270 120 300 150 330 180 0 d=2.5a, h=12a, =0 γ ° Wang & Liu, ka=.1 Wang & Liu, ka=1 Wang & Liu, ka=2 BEM, ka=.1 BEM, ka=1 BEM, ka=2 Distribution of | / | vs σ σ θ θz 0 Figure 5.10: Comparison of the normalized stressj z = 0 j for two circular cavities embedded within a half-space for dierent wavenumbers (N 2L1 cav model). The stresses are evaluated on the surface of the left circular cavity for a vertical incident plane harmonic SH-wave. Solid and dashed lines represent the results of Wang and Liu [4], and the circles, crosses and stars denote the results of this study. The parameters are: number of inclusions N = 2, principal axes a i = b i = 1, and location of the centers O 1 = (2:5; 12),O 2 = (2:5; 12). The material properties of the half-space and the inclusion are 0 = 0 = 1, i = 1 600 , i = 1 2 , i = 1; 2; and 0 is the maximum stress amplitude of the stress eld along the cavity. 79 1 2 3 4 30 210 60 240 90 270 120 300 150 330 180 0 d=2.5a, h=12a, =45 γ ° Wang & Liu, ka=.1 Wang & Liu, ka=1 Wang & Liu, ka=2 BEM, ka=.1 BEM, ka=1 BEM, ka=2 Distribution of | / | vs σ σ θ θz 0 Figure 5.11: Comparison of the normalized stressj z = 0 j for two circular cavities embedded within a half-space for dierent wavenumbers (N 2L1 cav model). The stresses are evaluated on the surface of the left circular cavity for a = 45 incident plane harmonic SH-wave. Solid and dashed lines represent the results of Wang and Liu [4], and the circles, crosses and stars denote the results of this study. The parameters are: number of inclusions N = 2, principal axes a i = b i = 1, and location of the centers O 1 = (2:5; 12),O 2 = (2:5; 12). The material properties of the half-space and the inclusion are 0 = 0 = 1, i = 1 600 , i = 1 2 , i = 1; 2; and 0 is the maximum stress amplitude of the stress eld along the cavity. 80 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 Lee BEM Stress | / | at outer surface of pipe (r=1) σ σ θz 0 =0, h=1.5a, =3, =1 γ μ η 1 Figure 5.12: Comparison of the normalized stressj z = 0 j at the outer surface of the circular pipe embedded within a half-space when subjected to a vertical incident plane harmonic SH-wave. The asterisks represent the results of this study, and the solid lines denote the results of Lee and Trifunac [5]. The parameters are the following: dimensionless frequency = 1, number of inclusions N = 1, number of layers L = 2, principal axes of the layers a 1 = b 1 = 1, a 2 = b 2 = 0:9, and layers' centers O = (0; 1:5). The material properties of the half-space and the layers are 0 = 0 = 1, 1 = 3, 1 = 1, 2 = 1 600 , 2 = 1 2 ; and 0 is the maximum stress amplitude of the stress eld along the pipe. 81 1 2 3 30 210 60 240 90 270 120 300 150 330 180 0 Lee BEM Stress | / | at inner pipe (r=0.9) surface σ σ θz 0 =0, h=1.5a, =3, =1 γ μ η 1 Figure 5.13: Comparison of the normalized stressj z = 0 j at the inner surface of the circular pipe embedded within a half-space when subjected to a vertical incident plane harmonic SH-wave. The asterisks represent the results of this study, and the solid lines denote the results of Lee and Trifunac [5]. The parameters are the following: dimensionless frequency = 1, number of inclusions N = 1, number of layers L = 2, principal axes of the layers a 1 = b 1 = 1, a 2 = b 2 = 0:9, and layers' centers O = (0; 1:5). The material properties of the half-space and the layers are 0 = 0 = 1, 1 = 3, 1 = 1, 2 = 1 600 , 2 = 1 2 ; and 0 is the maximum stress amplitude of the stress eld along the pipe. 82 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 Lee BEM Stress | / | at outer pipe (r=1) surface σ σ θz 0 =0, h=1.5a, =0.35, =1 γ μ η 1 Figure 5.14: Comparison of the normalized stressj z = 0 j at the outer surface of the circular pipe embedded within a half-space when subjected to a vertical incident plane harmonic SH-wave. The asterisks represent the results of this study, and the solid lines denote the results of Lee and Trifunac [5]. The parameters are the following: dimensionless frequency = 1, number of inclusions N = 1, number of layers L = 2, principal axes of the layers a 1 = b 1 = 1, a 2 = b 2 = 0:9, and layers' centers O = (0; 1:5). The material properties of the half-space and the layers are 0 = 0 = 1, 1 = 0:35, 1 = 1, 2 = 1 600 , 2 = 1 2 ; and 0 is the maximum stress amplitude of the stress eld along the pipe. 83 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 Lee BEM Stress | / | at inner pipe (r=0.9) surface σ σ θz 0 =0, h=1.5a, =0.35, =1 γ μ η 1 Figure 5.15: Comparison of the normalized stressj z = 0 j at the inner surface of the circular pipe embedded within a half-space when subjected to a vertical incident plane harmonic SH-wave. The asterisks represent the results of this study, and the solid lines denote the results of Lee and Trifunac [5]. The parameters are the following: dimensionless frequency = 1, number of inclusions N = 1, number of layers L = 2, principal axes of the layers a 1 = b 1 = 1, a 2 = b 2 = 0:9, and layers' centers O = (0; 1:5). The material properties of the half-space and the layers are 0 = 0 = 1, 1 = 0:35, 1 = 1, 2 = 1 600 , 2 = 1 2 ; and 0 is the maximum stress amplitude of the stress eld along the pipe. 84 5.5 Numerical Results for the General Half-Space Model To investigate the eects of multiple scattering, layering, impedance contrast and the distance between scatterers, on the stress concentration factor (SCF), the following cases are presented. 5.5.1 Eect of Multiple Scattering To study the eect of multiple scattering, the SCF of a three-inclusion-triple-layered model (N 3L3) is compared with that of a single-inclusion-triple-layered model (N 1L3). For a vertical incidence and a soft material, the introduction of two adjacent in- clusions results in an increase in the SCF on the outer surface of the middle inclusion (S 2 1 ) compared with the same surface of the single-inclusion-triple-layer model (Fig. 5.17). However, adding two sti inclusions to a single sti-layered inclusion resulted in a minimal increase in the SCF on the outer surface of the middle inclusion of the three-inclusion model compared with the single-inclusion model (Fig. 5.18). These results show that scattering from the multiple inclusions may have a pronounced in uence on the SCF, particularly for soft materials. Based on Figs. 5.17 and 5.18, the amplitude of the SCF can vary signicantly along the surface of the scatterer. This fact clearly demonstrates the importance of the scattering of waves in determining SCF. 5.5.2 Eect of Layering The eect of inclusion layering can be studied by comparing the SCF of a multiple multilayered inclusion model with that of the corresponding multiple inclusion model 85 with the average material properties. The average material properties have been dened in Section 4.1. For this purpose, the SCF of a three-inclusion-triple-layer model is compared with that of a three-inclusion-single-layer model. The results for sti and soft materials are shown in Figs. 5.19 and 5.20, respectively. Figure 5.19 shows that for sti materials and a vertical incidence the SCF of the layered inclusion is very similar to that of the inclusions with the average material prop- erties. However, for soft materials (Fig. 5.20), both the amplitude and the pattern of the SCF for the layered inclusions are signicantly dierent from those of the corresponding three-inclusion model with the average material properties. Therefore, the results of Figs. 5.19 and 5.20 clearly demonstrate the importance of layering upon the SCF. 5.5.3 Eect of Scatterer Stiness upon SCF In this case, the model of a single cavity with two adjacent inclusions (N 3L1) is considered (Fig. 5.21). The materials of the inclusions are either soft or sti. The eect of the scatterer stiness on the SCF for the cavity is depicted by Fig. 5.22. These results show that the SCF along the cavity can be strongly aected by the stiness of the adjacent inclusions. The inclusions' stiness may in uence both the amplitude and the pattern of the SCF along the surface of the cavity. The eect of impedance contrast on the SCF is considered next. 86 5.5.4 Eect of Impedance Contrast for Layered Pipes Here, three circular pipes with two layers are considered (N 3L3) rst (Fig. 5.23). The geometry and material properties of the layered pipes are assumed to be the same. The impedance contrast between the pipe layers corresponds to either soft-sti or sti- soft materials. Subsequently, nine-layered circular pipes are considered (N 9L3) as well (Fig. 5.25). For the three-pipe model, the eect of the impedance contrast between the layers is investigated by comparing the stress concentration on the outer surface of the middle pipe (S 2 1 ) for sti-soft and soft-sti layers. The SCF on the outer surface of the middle pipe of theN 3L3 model is depicted for the two materials in Fig. 5.24. Apparently, the SCF for the soft-sti pipes is larger than that of the sti-soft pipes. Therefore, the impedance contrast of the layers may produce signicant changes in the SCF. The nine-pipe layered model (N 9L3) is shown in Fig 5.25. The SCF for this model with dierent impedance contrast values is depicted in Fig. 5.26, where the model with soft-sti layers is compared with that of the sti-soft model. For the sake of deniteness, the stresses are evaluated over the outer surface of S 5 1 . It is apparent that the SCF is considerably higher on the surface of the central inclusion (S 5 1 ) for pipes with soft-sti layers than for the pipes with soft-sti layers. In addition, the change in the impedance contrast of the materials resulted in a signicant change in the SCF distribution along the surface. It is of interest to note that the SCF values are distributed symmetrically over the surface S 5 1 further conrming the validity of the results. 87 5.5.5 Eect of the Separation Distance between the Inclusions To investigate the eect of separation distance between the inclusions on the SCF, double-layered pipes are considered (N 3L3). The pipes are either of a sti-soft or sti- soft conguration. The SCF is evaluated for dierent separation distances along the outer surface of the middle inclusion. These results are shown for three separation distances in Figs. 5.27 and 5.28. The SCF is aected signicantly more by a soft-sti combination of materials than by sti-soft materials. These results demonstrate that the separation dis- tance between inclusions may strongly in uence the SCF, dependent upon the impedance contrast of the material. Both the geometry and material properties of the model aect the constructive/destructive interference of the scattered wave eld, resulting in changes in the SCF along the surface of the scatterer. 88 o S 2 1 d Figure 5.16: Three-inclusion model (N = 3) conguration with the inclusion centers at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). The principal axes are a i 1 = b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3. 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 N3L3 soft N1L3 soft Stress | / | vs σ σ θ θz 0 =0, =1 γ η Figure 5.17: Case of soft materials: stress concentration factorj z = 0 j on the outer surface of a middle inclusion in a soft three-inclusion-triple-layer model vs. a one-inclusion- triple-layer model for a vertical incident SH-wave. The layer centers are located at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). Similarly, the various layer principal axes are assumed to be a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5, i = 1 : 3. The material properties of the layers are specied by i j = i(soft) j , i j = i(soft) j ; i;j = 1 : 3, where i(soft) j and i(soft) j are dened by Eq. (4.4), and j 0 j is the maximum stress amplitude. 89 0.5 1 1.5 2 30 210 60 240 90 270 120 300 150 330 180 0 N3L3 stiff N1L3 stiff Stress | / | vs σ σ θ θz 0 =0, =1 γ η Figure 5.18: Case of sti materials: stress concentration factorj z = 0 j on the outer surface of a middle inclusion in a sti three-inclusion-triple-layer model vs. a one-inclusion- triple-layer model for a vertical incident SH-wave. The layer centers are located at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). Similarly, the various layer principal axes are assumed to be a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, a i 3 = b i 3 = 0:5, i = 1 : 3. The material properties of the layers are specied by i j = i(stiff) j , i j = i(stiff) j ; i;j = 1 : 3, where i(stiff) j and i(stiff) j are dened by Eq. (4.5), andj 0 j is the maximum stress amplitude. 90 0.5 1 1.5 2 30 210 60 240 90 270 120 300 150 330 180 0 N3L3 sitff N3L1 avg Stress | / | vs σ σ θ θz 0 =0, =1 γ η Figure 5.19: Case of sti materials: stress concentration factor j z = 0 j for a sti three- inclusion-triple-layer model vs. a three-inclusion-single-layer model with aver- age material properties for a vertical incident wave. The centers are located at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). The layer principal axes are assumed to be a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 = b i 3 = 0:5. The material properties of the layered inclusions are i j = i(stiff) j , i j = i(stiff) j , whereas for the average model, the material properties are i j = 1 3 P 3 j=1 i(stiff) j ; i j = 1 3 P 3 j=1 i(stiff) j ; i;j = 1 : 3, andj 0 j is the maximum stress amplitude. 91 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 , N3L3 soft N31L1 avg Stress | / | vs σ σ θ θz 0 =0, =1 γ η Figure 5.20: Case of soft materials: stress concentration factorj z = 0 j for a soft three-inclusion- triple-layer model vs. a three-inclusion-single-layer model with average material properties for a vertical incident wave. The centers are located at O 1 = (3; 1:5), O 2 = (0; 1:5), and O 3 = (3; 1:5). The layer principal axes are assumed to be a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 = b i 3 = 0:5. The material properties of the layered inclusions are i j = i(soft) j , i j = i(soft) j , whereas for the average model, the material properties are i j = 1 3 P 3 j=1 i(soft) j ; i j = 1 3 P 3 j=1 i(soft) j ; i;j = 1 : 3, andj 0 j is the maximum stress amplitude. 92 o O 1 O 3 O 2 S 2 Figure 5.21: The three-inclusion model (N 3L1) conguration with the inclusion centers atO 1 = (2:5; 1:5), O 2 = (0; 1:5), and O 3 = (2:5; 1:5), The principal axes are a i 1 =b i 1 = 1, for i = 1 : 3: 1 2 30 210 60 240 90 270 120 300 150 330 180 0 N3L1, siff-cav-stiff N3L1, soft-cav-soft Stress | | vs σ σ θ θz/ 0 =0, =1 γ η Figure 5.22: Normalized stress (j z = 0 j) at the surface of a cavity that is located in between two-sti vs. two-soft inclusions for a vertical incident wave. The center locations are O 1 = (2:5; 1:5), O 2 = (0; 1:5), and O 3 = (2:5; 1:5), The material properties are 1(stiff) = 3(stiff) = 14=9, 2 = 1=600, 1(stiff) = 3(stiff) = 12=9, 2 = 1=2 for the sti-cavity-sti model and 1(soft) = 3(soft) = 4=9, 2 = 1=600; 1(soft) = 3(soft) = 2=3, 2 = 1=2 for the soft-cavity-soft model. Here,j 0 j is the maximum stress amplitude. 93 o S 2 1 d Figure 5.23: The three-pipe layered model (N 3L3) conguration with the pipe centers atO 1 = (2:5; 1:5), O 2 = (0; 1:5), and O 3 = (2:5; 1:5), The principal axes are a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3: 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 N3 soft-stiff pipe N3 stiff-soft pipe Figure 5.24: Normalized stress (j z = 0 j) at the outer surface of the middle pipe in a three pipe model for sti-soft vs. soft-sti layers for a vertical incident wave. The parameters of the problem are O 1 = (2:5; 1:5), O 2 = (0; 1:5), and O 3 = (2:5; 1:5). The material properties for the sti-soft model are i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600, i(stiff) 1 = 12=9, i(soft) 2 = 2=3, i 3 = 1=2, for i = 1 : 3: For the soft- sti model, they are i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2, for i = 1 : 3. The principal axes are a i 1 = b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3. Here,j 0 j is the maximum stress amplitude. 94 o S 5 1 Figure 5.25: The nine-pipe layered model (N 9L3) conguration with the pipe centers at O 1 = (2:5; 6:5), O 2 = (2:5; 4), O 3 = (2:5; 1:5), O 4 = (0; 6:5), O 5 = (0; 4), O 6 = (0; 1:5), O 7 = (2:5; 6:5), O 8 = (2:5; 4), and O 9 = (2:5; 1:5). The principal axes are a i 1 =b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 9: 95 0.5 1 1.5 2 30 210 60 240 90 270 120 300 150 330 180 0 N9-soft-stiff pipe N9-stiff-soft pipe Stress | / | vs σ σ θ θz 0 =0, =1 γ η Figure 5.26: Normalized stress (j z = 0 j) at the outer surface of the centered pipe (S 5 1 ) in a N 9L3 layered pipe model for sti-soft vs. soft-sti layers for a vertical incident wave. The centers of of the pipes are located at O 1 = (2:5; 6:5), O 2 = (2:5; 4), O 3 = (2:5; 1:5), O 4 = (0; 6:5), O 5 = (0; 4), O 6 = (0; 1:5), O 7 = (2:5; 6:5), O 8 = (2:5; 4), and O 9 = (2:5; 1:5). The principal axes are a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 = b i 3 = 0:5 i = 1 : 9. The material properties of the half-space are 0 = 0 = 1 and for the sti-soft model i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600, i(stiff) 1 = 12=9, i(soft) 2 = 2=3, i 3 = 1=2, for i = 1 : 9. For the soft- sti model, they are i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2, for i = 1 : 9. Here,j 0 j is the maximum amplitude of the incidence stress-eld along the corresponding surface. 96 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 d=.5 d=1 d=1.5 Figure 5.27: Normalized stress (j z = 0 j) at the outer surface of the middle pipe in a three pipe model for soft-sti layers for a vertical incident wave for dierent distances between outer surfaces of the pipes. The pipes' centers are located at O 1 = (2 +d; 1:5), O 2 = (0; 1:5), and O 3 = (2 +d; 1:5), where d = [0:5; 1; 1:5] is the distance between the outer surfaces of the pipes. The material properties for the soft-sti model: are i(soft) 1 = 4=9, i(stiff) 2 = 14=9, i 3 = 1=600; i(soft) 1 = 2=3, i(stiff) 2 = 12=9, i 3 = 1=2, for i = 1 : 3. The principal axes are a i 1 = b i 1 = 1, a i 2 = b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3. Here,j 0 j is the maximum stress amplitude. 97 0.5 1 1.5 2 30 210 60 240 90 270 120 300 150 330 180 0 d=.5 d=1 d=1.5 Figure 5.28: Normalized stress (j z = 0 j) at the outer surface of the middle pipe in a three- pipe model with sti-soft layers for a dierent distances between outer surfaces of the pipes and vertical incident wave. The parameters of the problem are:. Th O 1 = (2+d; 1:5),O 2 = (0; 1:5), andO 3 = (2+d; 1:5), whered = [0:5; 1; 1:5] is the distance between the outer surfaces of the pipes. The material properties for the sti-soft model are i(stiff) 1 = 14=9, i(soft) 2 = 4=9, i 3 = 1=600, i(stiff) 1 = 12=9, i(soft) 2 = 2=3, i 3 = 1=2, for i = 1 : 3. The principal axes are a i 1 = b i 1 = 1, a i 2 =b i 2 = 0:75, and a i 3 =b i 3 = 0:5 for i = 1 : 3. Here,j 0 j is the maximum stress amplitude. 98 Chapter 6 Summary and Conclusion This study presents a direct boundary integral equation approach to examine the steady-state scattering of SH-waves by multiple multilayered inclusions embedded in a half-space. The analysis assumes smooth interfaces with perfect bonding between the layers. Several tests were performed to verify the accuracy of the proposed method with analytical solutions available in the literature. Specically, the results of this study, in- volving a cavity and a pipe, are found to be in excellent agreement with the corresponding analytical results. The numerical results, presented here for surface motion of a half-space with layered inclusions and pipes, can be summarized as follows: The surface motion may be strongly aected by the multiple scattering of elastic waves by the inclusions. In addition, the surface response strongly depends on layering for the soft materials. For the sti inclusion materials, this dependence is not as strong. 99 The eect of inclusion shape on the surface motion is more pronounced for the soft inclusion layers than for the sti inclusion layers. For the grazing incidence, the highly oscillatory motion in the illuminated portion of the far-eld is greatly reduced when the soft inclusion layers are replaced by the sti layers. For the layered pipes, the surface motion is found to be strongly dependent on the impedance contrast between the layers. In addition, the surface response changes considerably from the near to the far-eld. A method for the evaluation of the stress concentration factor (SCF) for multiple mul- tilayered inclusions has been developed. The method avoids evaluation of hypersingular integrals that arise in the problem. Extensive testing was performed to verify the numerical results. These results were compared with analytical solutions for both full-space and half-space problems available in literature. All tests show excellent agreement between the two results. The numerical results presented here show the SCF, over the surfaces of the layered inclusions and pipes embedded within a half-space. These results can be summarized as follows: The scattering from the multiple inclusions may have a pronounced eect on the SCF, especially for soft materials. The results clearly demonstrate the importance of the layering of the inclusions on SCF. Specically, the SCF for the layered inclusion may be very dierent from that for the homogeneous inclusion. 100 It was shown that the separation distance between the inclusions may greatly in- uence the SCF, depending upon the impedance contrast of the materials. Therefore, the results show that accurate evaluation of the surface response and the SCF in complex models, may require incorporation of multiple layers and multiple scatterers. 101 Appendix A Singular kernel The boundary element method for displacements involves the evaluation of Cauchy principle value integrals. If the derivative of the displacement BIE is taken, the inte- gral formulations for the stresses lead to the computation of the so-called hypersingular integrals. A.1 Basic Denitions In singular integral equations, the kernel becomes unbounded in the range of integra- tion [74, 75]. Dierent solutions exist depending on the order of singularity. For one-dimensional case, when the kernel is of the form K(x;y) = q(x;y) p(x;y) , as x!y; based on the p(x,y), the integral equations are classied as weakly singular, strongly singular, and hypersingular. Weakly singular: If p(x;y) = O(jxyj n ), 0 < n < 1 or p(x;y) = O(lnjxyj), as x! y, then K(x;y) is unbounded for all q(x;y)6= 0. The integral R b a K(x;y)(x)dx exists for any continuous and dierentiable (x). Strongly singular: Ifp(x;y) =O(xy), asx!y,K(x;y) is unbounded, whenq(x;y) is dierentiable and q(x;y)6= 0. Then, the integral R b a K(x;y)(x)dx will be dened in the sense of the Cauchy principle value integral PV Z b a K(x;y)(x)dx = lim !0 Z y a K(x;y)(x)dx + Z b y+ K(x;y)(x)dx where (x) is a continuous and dierentiable function. Hypersingular: If p(x;y) = O(jxyj n );n > 1, q(x;y) is continuous and q(x;y)6= 0: Then, the integral R b a K(x;y)(x)dx is dened as hypersingular. A similar denition can be extended to 2D and 3D models. A.2 Singularities in the Boundary Integral Equation Method To illustrate the problem, the case of a simple cavity of surface S, embedded within a full space (D 0 ), is considered. The cavity is subjected to an incident plane harmonic SH wave (Fig. A.1). 102 u inc S D 0 n SH-wave δ ∂S Figure A.1: A single cavity embedded in full space and subjected to an SH wave. Here,S,,@S, n,D 0 , andu inc denote the cavity surface, the indentation, the part of the boundary removed due to the indentation, the outward unit normal, the full-space domain, and the incident wave, respectively. The scattered wave displacement eld is governed by the following equation [76] u(y) = Z S [U (0) (x;y)t(x)T (0) (x;y;n)u(x)]dS x ; y2D 0 ;y = 2S (A.1) whereU (0) andT (0) denote the full space displacement and traction Green's function, re- spectively (see Eq. 3.1 and A.5). Here,u andt represent the scattered wave displacement and traction elds on the surface S, respectively. As the source point (y) approaches the eld point (x) on the surface S, the distance between these two points goes to zero (i.e., r =jyxj! 0). The kernel of the rst integral in (Eq. A.1) shows weak singularity, whereas the kernel of the second integral is of the strong singularity type [77], which results in the integral equation [50] c(y)u(y) = Z S U (0) (x;y)t(x)dS x PV Z S T (0) (x;y;n)u(x)dS x c = lim !0 Z T (0) (x;y;v)dS x (A.2) Here,c 3 represents the free term (see Eq. 3.7) and PV denotes the principal value integral dened by Z S T (0) (x;y;n)u(x)dS x = lim !0 Z S@S T (0) (x;y;n)u(x)dS x (A.3) where, @S denotes the removed portion of the surface S after incising the indentation and is the indentation along the boundary (see Fig. A.1). 103 The dierentiation of Eq. (A.1) with respect to y results in the following integral equation for the stresses t(y) = 3q (y)n q = Z S U (0) ;q (x;y)n q t(x)dS x Z S T (0) ;q (x;y;n)n q u(x)dS x q = 1; 2; y2D 0 ;y = 2S (A.4) where [51] U (0) (x;y) = i 4 H 1 0 (k s r) T (0) (x;y;n) = ik s 4 H 1 1 (k s r) @r @n r =jyxj (A.5) Here, H 1 0 and H 1 1 are the Hankel functions of the rst kind and of order zero and one, respectively, and n denotes the outward unit normal to S. The integrals in Eq. (A.4) become singular as r! 0. In particular, the rst integral on the RHS of Eq. (A.4) is a CPV integral with the kernel behaving as O(r 1 ). However, the second integral is hypersingular. Namely, asr! 0, the asymptotic form of the Hankel function of the rst kind and order m is given by [78] H 1 m (r) 2i (m 1)! r m Therefore, as r! 0, U (0) ;q ik s r q 1 4r 2i k s r = 1 2 r q r 2 T (0) ;q = ik s 4 n q H 1 1 (k s r) r [k s r q (r:n)] H 1 2 (k s r) r 2 1 2 (n q )[ 1 r 2 ] + 1 2 r q (r:n)[ 1 r 4 ] (A.6) Thus, the integrand in R S T (0) ;q (x;y;n)n q u 3 (x)dS x has a singularity ofO(r 4 ) asr! 0, which leads to a hypersingular integral. 104 Appendix B Stress Formulation for the Full-space B.1 Problem Statement: Full-space Problem First, the case of a single elastic cylinder embedded within a full space is considered. A cylinder with surfaceS 1 is subjected to an incident plane harmonic SH-wave (Fig. B.1). For simplicity, the cylinder surface is assumed to be C (1) continuous without sharp cor- ners. The problem is dened in Cartesian coordinatesfjxj <1;jyj <1g; where the origin is set at the center of the cylinder (Fig. B.1). As the incident wave strikes the cylinder, it creates a scattered wave eld within the full-space and the cylinder. This results in stress elds along the cylinder interface. The method employed in this investigation is demonstrated for the case of a single inclusion. The method originates in the works of Okada et al. [69] and Qian et al. [70]. The steady-state wave equations are given by r 2 u (j) +k 2 j u (j) = 0 k j = ! j ;j = 0; 1 (B.1) wherer 2 is the Laplacian operator, andk j ;!; j , andu (j) are the wave number, circular frequency, the shear wave velocity and displacement eld of the domain D j , respectively. The incident plane wave propagating in the positive x-direction (Fig. B.1) is of the form u inc (x;!) =e i(k 0 x!t) ; x2D 0 (B.2) Throughout, the time-dependent exponential term e i!t will be omitted. When the incident wave u inc impinges on the inclusion's surface, a scattered wave eldu sc(0) is created. The total displacement wave eld in the full-space domain is given by u (0) (x) =u inc (x) +u sc(0) (x); x2D 0 (B.3) Additionally, the scattered wave eldu sc(0) must satisfy Sommerfeld's radiation condition [6] lim R!1 R 1=2 j @u sc(0) @R ik 0 u sc(0) j = 0 (B.4) 105 u inc S 1 D 0 SH-wave D 0 D 0 D 1 x y n Figure B.1: An elastic cylindrical inclusion embedded in a full-space and subjected to a plane harmonic incident SH-wave. The domains of the full-space and inclusion are de- noted byD 0 andD 1 , respectively, andn represents the outward unit normal to the interface S 1 . where R is the radius of the circle in the xyplane. The displacement wave eld inside the inclusion D 1 is specied by u (1) (x) =u sc(1) (x); x2D 1 (B.5) where u sc(1) denotes the scattered wave eld. The boundary conditions constitute continuity of the displacement and traction elds along the cylindrical inclusion surface: u (0) (x) =u (1) (x) t (0) (x;n) =t (1) (x;n) x2S 1 (B.6) where t (j) (for j = 0; 1) is the traction eld, which is dened as t (j) = j ru (j) :n; j = 0; 1 (B.7) Here, n is the outward unit normal, is the shear modulus,r() = () ;i e i , and e i denotes the base vectors. This section completes the statement of the problem. Next, the solution of the problem is considered. 106 B.2 Solution of the Full-space Single Inclusion Problem The fundamental displacement solution for the full-space problemU(x;y) is the solu- tion to the steady-state wave equation, where the forcing term is the Dirac delta function (x;y), with x = (x 1 ;x 2 ) and y = (y 1 ;y 2 ) denoting the source and load points, respec- tively [51]. U ;ii (x;y) +k 2 U(x;y) =(x;y) U(x;y) = i 4 H (1) 0 (kr) r =jxyj (B.8) Here, () ;i implies dierentiation with respect to x i , where summation is understood over the repeated indices; i = 1; 2; i is the imaginary unit; H (1) 0 is the Hankel function of the rst kind and order zero; and r is the distance between x and y. To derive the displacement boundary integral equations (DBIEs), the method em- ployed by Qian et al. [70] and Han and Atluri [79] will be used. B.2.1 DBIEs for the Full-Space Let the function U =U, where U is the Green's function dened by Eq. (B.8), be used as a test function in the weak form of Eq. (B.1) Z V (u ;ii +k 2 u)U dV x = 0 (B.9) where the superscripts denoting the domains have been omitted. For the product function uv, the divergence theorem can be stated as follows Z V (uv) ;i dV = Z S n i uvdS (B.10) where V is the volume, S is the corresponding surface, and n i are the components of an outward unit normal of the surface S. The last equation can be written as Z V (uv ;i +u ;i v)dV = Z S n i uvdS Z V uv ;i dV = Z S n i uvdS Z V u ;i vdV (B.11) Now, Eq. (B.9) can be expanded as Z V u ;ii U dV x + Z V k 2 uU dV x = 0 (B.12) 107 Applying the divergence theorem to the rst integral of the last equation results in Z S n i u ;i U dS x Z V u ;i U ;i dV x + Z V k 2 uU dV x = 0 (B.13) By applying the divergence theorem to the second integral of the above equation, it follows that Z S n i u ;i U dS x Z S n i uU ;i dS x + Z V u(U ;ii +k 2 U )dV x = 0 (B.14) Using Eq. (B.8) the last equation yields u(y) = Z S U (x;y)n i u ;i (x)dS x Z S n i U ;i (x;y)u(x)dS x ; y2D (B.15) All dierentiations in integrands in the last equation are with respect to the variable x i , and summation over repeated indices is understood. The following relationships are used t(x) =n i u ;i (x) T =n i U ;i d dz H (1) m (z) = mH (1) m (z) z H (1) m+1 (z) (B.16) where T is the fundamental traction solution of the problem [51] and H 1 m is the Hankel function of the rst kind and order m. Consequently, Eq. (B.15) becomes u(y) = Z S U(x;y)t(x)dS x Z S T (x;y)u(x)dS x ;y2D 0 ;y = 2S (B.17) Equation (B.17) represents Betti's internal integral formula for displacements in elas- todynamics [80]. To calculate the stress eld, the gradient of the displacements has to be evaluated. A gradient boundary integral equation (GBIE) can be derived in a similar manner as the displacement integral equation. B.2.2 GBIE for the Full-Space To obtain the GBIE equation, the derivative of the fundamental solution is employed as the test function according to [70] Z V (u ;ii +k 2 u)U ;q dV x = 0 (B.18) that is, Z V (u ;ii U ;q +k 2 uU ;q )dV = 0 (B.19) By applying the divergence theorem to the rst term in the last equation, it follows that 108 Z S n i u ;i U ;q dS Z V u ;i U ;qi dV + Z V k 2 uU ;q dV = 0 (B.20) Invoking the divergence theorem again results in the equation Z S n i u ;i U ;q dS Z S n q u ;i U ;i dS + Z V u ;iq U ;i dV + Z V k 2 uU ;q dV = 0 (B.21) Additional application of the divergence theorem in the above equation leads to Z S n i u ;i U ;q dS Z S n q u ;i U ;i dS + Z S n i u ;q U ;i dS Z V u ;q U ;ii dV + Z V k 2 uU ;q dV = 0 (B.22) Finally, applying the divergence theorem to Eq. (B.22) produces the following Z S n i u ;i U ;q dS x Z S n q u ;i U ;i dS x + Z S n i u ;q U ;i dS x + Z S n q k 2 uU dS x Z V u ;q (U ;ii +k 2 U )dV x = 0 (B.23) Using Eq. (B.8), the last equation becomes u ;q (y) = Z S n i u ;i U ;q dS x Z S n q u ;i U ;i dS x + Z S n i u ;q U ;i dS x + Z S n q k 2 uU dS x (B.24) Furthermore, in terms of tractions, Eq. (B.24) can be written as u ;q (y) = Z S U ;q (x;y)t(x)dS x + Z S U ;i (x;y)[n i u ;q (x)n q u ;i (x)]dS x + Z S k 2 n q U(x;y)u(x)dS x y2D 0 ; y = 2S (B.25) Finally, Eq. (B.25) assumes the form 109 u ;q (y) = Z S U ;q (x;y)t(x)dS x Z S n q U ;i (x;y)u ;i (x)dS x + Z S T (x;y)u ;q (x)dS x + Z S k 2 n q U(x;y)u(x)dS x y2D 0 ;y = 2S (B.26) The last equation represents the gradient integral equation of the problem for the full-space. Therefore, Eqs. (B.17) and (B.26) represent the integral equations for the displace- ment and displacement gradient elds, respectively. In both Eqs. (B.17) and (B.26), the observation point is located away from the surface of the inclusion. Bringing the observa- tion point to the surface the inclusion is considered in the following (regularization). For this. purpose one must investigate the following relationships in the integral equations. B.2.3 Basic Properties of the Integral Equations B.2.3.1 Property I Considering the constant c as the test function, the weak form of the fundamental displacement solution equation becomes Z V [U ;ii (x;y) +k 2 U (x;y) +(x;y)]cdV = 0; y2D (B.27) Applying the divergence theorem to the rst term in the above equation leads to Z S n i cU ;i (x;y)dS Z V c ;i U ;i (x;y)dV + Z V ck 2 U (x;y)dV +c = 0 y2D (B.28) Because c is a constant, the last equation can be written as Z S cn i U ;i (x;y)dS + Z V ck 2 U (x;y)d +c = 0; y2D (B.29) Equation (B.29) represents the rst property of the integral equation. B.2.3.2 Property II Using a rigid body motion (u(y) =const:) as the test function the weak form of the fundamental solution can be written as 110 Z V [U ;ii (x;y) +k 2 U (x;y) +(x;y)]u(y)dV x = 0; y2D (B.30) Applying the divergence theorem to the rst integral leads to Z S n i U ;i (x;y)u(y)dS Z V U ;i (x;y)u ;i (y)dV + Z V k 2 U (x;y)u(y)dV +u(y) = 0 Because u ;i (y) = 0, as u(y) =constant, one obtains Z S T (x;y)u(y)dS + Z V k 2 U (x;y)u(y)dV +u(y) = 0; y2D;y = 2S (B.31) As the observation point approaches the surface of the scatterer, it follows that lim y!S Z S T (x;y)u(y)dS = Z T (x;y)u(y)dSc(y)u(y) (B.32) where c = 8 > < > : 0 for y2D 0 ;y = 2S 1 for y2D 1 ;y = 2S 1=2 for y2S 1 and S 1 is smooth (B.33) and R denotes the principal value integral. By assuming that the surface S 1 is smooth, one obtains the following Z S T (x;y)u(y)dS + Z V k 2 U (x;y)u(y)dV + 1 2 u(y) = 0 y2S 1 (B.34) The principal value integral ( R ) is dened by Par s and Ca~ nas [80] as Z T (x;y)u(y)dS = lim "!0 Z SS" T (x;y)u(y)dS (B.35) where S " is a small semicircular indentation of radius " around the point y. Equation (B.34) represents the second property of the integral equations. 111 B.2.3.3 Property III Withu ;q (x) =constant chosen as the test function, the weak form of the fundamental solution can be written as Z V U ;ii (x;y) +k 2 U (x;y) +(x;y) u ;q (x)dV x = 0 (B.36) or Z V U ;ii (x;y) +k 2 U (x;y) u ;q (x)dV +u ;q (y) = 0 (B.37) where, as stated previously, U (x;y) is the displacement Green's function. Utilizing the divergence theorem, the last equation yields Z S n i U ;i (x;y)u ;q (y)dS Z U ;i (x;y)u ;iq (y)dV + Z V k 2 U (x;y)u ;q (y)dV +u ;q (y) = 0 (B.38) Because u ;iq (y) = 0, it follows from Eq. (B.38) that Z S T (x;y)u ;q (y)dS + Z V k 2 U (x;y)u ;q (y)dV +u ;q (y) = 0; y = 2S (B.39) where T (x;y) is the traction Green's function. Equation (B.39) represents the third property of the integral equation. Next, the structure of that property is considered, as the observation point approaches the collocation points (y!x). From the divergence theorem, it can be shown that Z S n i (x)u ;i (y)U ;q dS x = Z V u ;ii (y)U ;q dV x + Z V u ;i (y)U ;qi dV x = Z V u ;i (y)U ;qi dV x (B.40) and Z S n q (x)u ;i (y)U ;i dS x = Z V u ;i (y)U ;qi V x (B.41) Consequently, Z S n i (x)u ;i (y)U ;q dS x Z S n q (x)u ;i (y)U ;i dS x = 0 (B.42) 112 By combining Eqs. (B.39) to (B.42), one obtains u ;q (y) = Z S n i (x)u ;i (y)U ;q dS x + Z S D iq u(y)U ;i (x;y)dS x + Z V k 2 u ;q (y)U (x;y)dV x ; y = 2S (B.43) where D iq u(y) = [n i (x)u ;q (y)n q (x)u ;i (y)] (B.44) Considering the limit as the observation point approaches the surface S, one obtains from Eq. (B.39) Z S T (x;y)u ;q (y)dS x +c(y)u ;q (y) = Z V k 2 U (x;y)u ;q (y)dV x ; y2S (B.45) Next, the regularization of the integral equations is considered. B.2.4 Regularization of DBIE Regularization requires taking the observation point to the surface of the scatterer S. Combining Betty's formula (Eq. B.17) to Eq. (B.31) leads to Z S U(x;y)t(x)dS Z S T (x;y)[u(x)u(y)]dS + Z V k 2 U(x;y)u(y)dV = 0; y2D (B.46) Using Eq. (B.34) in the last equation, one obtains the following for the points y2S Z S U(x;y)t(x)dS Z S T (x;y)[u(x)u(y)]dS = 1 2 u(y) + Z S T (x;y)u(y)dS; y2S (B.47) The last equation is the regularized integral equation for the displacements. Next, the regularization of the integral equation for the displacement gradient is considered . 113 B.2.5 Regularization of the Displacement Gradient Integral Equation By subtracting Eq. (B.43) from Eq. (B.25), one obtains 0 = Z S U ;q (x;y)[t(x)n i u ;i (y)]dS + Z S U ;i (x;y)[D iq u(x)D iq u(y)]dS + Z S k 2 U (x;y)[n q u(x)u ;q (y)]dS; y2D;y = 2S (B.48) Note that for observation and source points that are close together one can use the Taylor series expansion u(x) u(y) + (x q y q )u ;q (y), where higher-order terms have been neglected. Consequently, for the points on the boundary, Eq. (B.48) becomes 0 = Z S U ;q (x;y)[t(x)n i u ;i (y)]dS x + Z S U ;i (x;y)[D iq u(x)D iq u(y)]dS x + Z S k 2 U (x;y)n q u(x)dS x + Z S T (x;y)u ;q (y)dS + 1 2 u ;q (y) (B.49) Note that linear (C 0 ) elements can be used for the discretization of the last equation [70]. By substituting the values ofD iq u from Eq. (B.44) into the last equation, one obtains the following result: 0 = Z S U ;q (x;y)[t(x)n i u ;i (y)]dS x + Z S U ;i (x;y)[n i u ;q (x)n i u ;q (y)]dS x Z S U ;i (x;y)[n q u ;i (x)n q u ;i (y)]dS x + Z S k 2 U (x;y)n q u(x)dS x + Z S T (x;y)u ;q (y)dS x + 1 2 u ;q (y) (B.50) 114 By introducing the traction fundamental solution in the second integral of the last equation, one obtains 0 = Z S T (x;y)[u ;q (x)u ;q (y)]dS x + Z S U ;q (x;y)[t(x)t(y)]dS x Z S n q (x)U ;i (x;y)[u ;i (x)u ;i (y)]dS x + Z S k 2 n q (x)U(x;y)u(x)dS x + Z S T (x;y)u ;q (y)dS x + 1 2 u ;q (y) (B.51) The last equation represents the regularized integral equation for the displacement gra- dient. The integrals in Eq. (B.51) are at most weakly singular. To investigate the dis- cretization of the specic integrals in Eq. (B.51) in further detail and to evaluate these integrals, a procedure proposed in elastostatics by Matsumoto et al. [81] is extended to the elastodynamics case. B.2.5.1 Discretization of the First Integral in Eq.B.51 As the collocation point y approaches the surface S, one can proceed as follows. Assuming that surface S is divided into N segments, S = P N p=1 S p . For the sake of deniteness, one can assume that the collocation point falls on the rst node ( l = 1,y = [ N l=1 y l , y l 2 S) with the singular elements being the rst and N th elements. Therefore, from Z S T (x;y)[u ;q (x)u ;q (y)]dS x = Z S 1 T (x;y)[u ;q (x)u ;q (y)]dS x + N1 X p=2 Z Sp T (x;y)[u ;q (x)u ;q (y)]dS x + + Z S N T (x;y)[u ;q (x)u ;q (y)]dS x (B.52) it follows that 115 Z S T (x;y)[u ;q (x)u ;q (y)]dS x = Z S 1 T (x; 1)[u ;q (x)u ;q (1)]dS x + N1 X p=2 Z Sp T (x; 1)[u ;q (x)u ;q (1)]dS x + Z S N T (x; 1))[u ;q (x)u ;q (1)]dS x (B.53) Consequently, the integrals over S 1 and S N are singular, whereas all the others are non- singular. In the rst integral on the right-hand side of Eq. (B.53), one can use linear shape functions to write u ;q (x) = 1 ()u ;q (1) + 2 ()u ;q (2) 1 1 (B.54) where 1 and 2 are the linear shape functions. The integral over S 1 then becomes Z S 1 T (x; 1)[u ;q (x)u ;q (1)]dS x = Z S 1 T (x; 1)[ 1 u ;q (1) + 2 u ;q (2)u ;q (1)]dS x = Z S 1 T (x; 1)[( 1 1)u ;q (1) + 2 u ;q (2)]dS x = AS 111 A 211 u ;q (1) u ;q (2) (B.55) where AS 111 = Z 1 1 T (; 1)[ 1 () 1]J 1 d A 211 = Z 1 1 T (; 1) 2 ()J 1 d (B.56) Here, J 1 = L 1 2 represents the Jacobian of the rst element with L 1 denoting the length of this element. Similarly, for the last integral over S N , it follows that 116 Z S N T (x; 1)[u ;q (x)u ;q (1)]dS x = Z S N T (x; 1)[ 1 u ;q (N) + 2 u ;q (1)u ;q (1)]dS x = Z S N T (x; 1)[ 1 u ;q (N) + ( 2 1)u ;q (1)]dS x = A 11N AS 21N u ;q (N) u ;q (1) (B.57) A 11N = Z 1 1 T (; 1) 1 J N d AS 21N = Z 1 1 T (; 1)[ 2 1]J N d (B.58) In general the integration constants are dened as AS mlp 8 > < > : m : 1st/2nd node of the element p l : the collocation point p : the element number (B.59) and index q in the derivatives u ;q , is related to the stress component 3q . Equations (B.55) and (B.57) are discretized versions of the singular integrals, which appear in the rst integral of the integral equation when the collocation point falls on the rst node of the rst element or the second node of the N th element. This process can be repeated for all other nodes of the elements (p = 2 :N). Therefore, in general, when the collocation point is at node l (the rst node of the element l or the last node of the element l 1, l = 1 :N), one obtains the following for the rst singular integral Z S T(x;y)[u ;q (x)u ;q (l)]dS x = AS 1ll A 2ll u ;q (l) u ;q (l + 1) + A 1l(l1) AS 2l(l1) u ;q (l 1) u ;q (l) + N X k=1;p6=l;l1 A 1lp A 2lp u ;q (p) u ;q (p + 1) (B.60) where no summation over repeated superscripts is understood and continuity of the in- clusion surface implies that u ;q (N + 1) = u ;q (1) and u ;q (0) = u ;q (N) . This completes the discretization process for the rst singular integral. The second singular integral in Eq. (B.51) is considered next. 117 B.2.5.2 Discretization of the Second Integral in Eq. B.51 The second singular integral in Eq. (B.51) can be discretized as Z S U ;q (x;y)[t(x)t(l)]dS x = ES 1ll q E 2ll q t(l) t(l + 1) + h E 1l(l1) q ES 2l(l1) q i t(l 1) t(l) + N X k=1;p6=l&l1 h E 1lp q E q 2lp i t(p) t(p + 1) (B.61) where t(N + 1) = t(1) and t(0) = t(N). Furthermore, the integration constants are dened by ES 1ll q = Z 1 1 U ;q (;l)[ 1 () 1]J l d; p =l E 2ll q = Z 1 1 U ;q (;l) 2 ()J l d; p =l E 1l(l1) q = Z 1 1 U ;q (;l) 1 ()J l1 d; p =l 1 ES 2l(l1) q = Z 1 1 U ;q (;l)[ 2 () 1]J l1 d; p =l 1 (B.62) and E jlp q = Z 1 1 U ;q (;l) 1 J p d j = 1; 2;l = 1 :N; q = 1; 2 p6=l &l 1 (B.63) B.2.5.3 Discretization of the Third Integral in Eq. B.51 Using the above procedure, the third integral in Eq. (B.51) leads to the following expression Z S n q U ;i (x;y)[u ;i (x)u ;i (l)]dS x = DS 1ll qi D 2ll qi u ;i (l) u ;i (l + 1) + h D 1l(l1) qi DS 2l(l1) qi i u ;i (l 1) u ;i (l) + N X k=1;p6=l&l1 h D 1lp qi D qi 2lp i u ;i (p) u ;i (p + 1) (B.64) 118 where u ;i (N + 1) =u ;i (1) and u ;i (0) = u ;i (N): Furthermore, when the collocation point belongs to the element, the integration constants are dened by the following integrals: DS 1ll qi = Z 1 1 n q U ;i (;l)[ 1 () 1]J l d; p =l D 2ll qi = Z 1 1 n q U ;i (;l) 2 ()J l d; p =l D 1l(l1) qi = Z 1 1 n q U ;i (;l) 1 ()J l1 d; p =l 1 DS 2l(l1) qi = Z 1 1 n q U ;i (;l)[ 2 () 1]J l1 d; p =l 1 (B.65) Similarly, when the collocation is not on the element, the integration constants are dened by D jlp qi = Z 1 1 n q U ;i (;l) j J p d p6=l &l 1;j = 1; 2 (B.66) This completes the discretization of the third integral in Eq. (B.51). The discretization of the fourth integral in Eq. (B.51) is investigated next. B.2.5.4 Discretization of the Fourth Integral in Eq. B.51 For this weakly singular integral, the discretized form of the integral can be written as Z S k 2 n q U(x;y)u(x)dS x = N X p=1 h B 1lp q B 2lp q i u(p) u(p + 1) ; (B.67) where u(N + 1) =u(1), and the integration constants are dened by Bn jlp q = Z 1 1 k 2 n q U (;l) j J p d j = 1; 2;l = 1 :N;p = 1 :N;q = 1; 2 (B.68) B.2.5.5 Discretization of the Fifth Integral in Eq. B.51 The discretized form of the fth integral in Eq. (B.51) has the form Z S T (x;y)u ;q (y)dS x = N X p=1 A lp u ;q (l) (B.69) 119 where A lp = Z 1 1 T (;l)J p d l = 1 :N;p = 1 :N;q = 1; 2 (B.70) This completes the discretization of all the integrals that appear in the gradient integral equation (Eq. B.51). The procedure for evaluating stresses is summarized in the following for the case of a circular inclusion embedded in a full-space and subjected to a plane harmonic SH-wave. B.2.6 Boundary Integral Equations (BIEs) for an Inclusion Subjected to an SH-wave in a Full-Space The displacement integral equations (Eq. B.47) in domain D 0 can be extended to include the incident wave u inc (y) 1 2 u(y) + Z S T (0) (x;y)u(x)dS x Z S U (0) (x;y)t(x)dS x =u inc (y) y2S;y2D 0 (B.71) where u (inc) =e ik (0) x . For domain D 1 , the displacement boundary integral is given by 1 2 u(y) + Z S T (1) (x;y)u(x)dS x Z S U (1) (x;y)t(x)dS x = 0;y2S;y2D 1 (B.72) With the adjustment of the outward normal vector andr =jxyj for the interior domain, the displacement gradient boundary equation in domain D 1 is written in the form 0 = Z S 1 k 2 1 n q (x)U (1) (x;y)u(x)dS x + Z S U (1) ;q (x;y)[t(x)t(y)]dS x Z S T (1) (x;y)[u ;q (x)u ;q (y)]dS x + Z S 1 n q (x)U (1) ;i (x;y)[u ;i (x)u ;i (y)]dS x Z S T (1) (x;y)u ;q (y)dS x + 1 2 u ;q (y) (B.73) 120 Equations (B.71), (B.72), and (B.73) can be discretized and then assembled in a matrix MU =F U =fu;t;u ;1 ;u ;2 g T ; F =fu inc ;0;0;0g T u;t;u ;1 and u ;2 2C N1 M2C 4N4N U2C 4N1 F2C 4N1 (B.74) where M = 0 B B B @ A (0) +:5I B (0) 0 0 A (1) :5I B (1) 0 0 Bn (1) 1 E (1) 1 As (1) +:5I +D (1) 11 D (1) 12 Bn (1) 2 E (1) 2 D (1) 21 As (1) +:5I +D (1) 12 1 C C C A (B.75) with all the block sub-matrices known explicitly andC N1 denoting the complex vector space of size N 1. To evaluate matrix elements in Eq. (B.75), one can use either linear or other elements. To increase the accuracy using the same number of elements, it is particularly useful to consider the Overhauser element of type C 1 . B.3 Overhauser Element An Overhauser element, also known as a Catmull-Rom element [82, 83] is a cubic spline. It can be implemented in the interpolation of the displacement, traction, and displacement gradients. Higher-order elements increase accuracy and reduce the error and the total number of required elements relative to linear elements [82, 83]. The displacement eld can be written as u() = 4 X j=1 j ()u j 0 1 (B.76) where the four shape functions are dened as 1 () =( 2 2 + 3 )=2 2 () = (2 5 2 + 3 3 )=2 3 () = ( + 4 2 3 3 )=2 4 () =( 2 3 )=2 (B.77) 121 Let Q i = (Q 1 i ;Q 2 i ), i = 1 : 4, be four subsequent points on the Overhauser curve, where Q 1 i and Q 2 i are the horizontal and vertical coordinates of the points, respectively. The following relationships are applicable [82] Q j () = 4 X i=1 Q j i i ();j = 1; 2 dQ j () d = 4 X i=1 Q j i d i () d jJj = dQ 1 () d 2 + dQ 2 () d 2 1 2 dS =jJjd jJjn 1 = dQ 2 () d jJjn 2 = dQ 1 () d (B.78) where J is the Jacobian term and n 1 and n 2 are the components of the normal vector. For the sake of illustration, the discretization of the integral equations using Over- hauser elements for a closed surface S divided into N elements is described only for the fourth integral in Eq. (B.51). Z S k 2 n q U(x;y)u(x)dS x = N X p=1 h B 1lp q B 2lp q B 3lp q B 4lp q i 8 > > < > > : u 1 p u 2 p u 3 p u 4 p 9 > > = > > ; = (B 1l2 q +B 2l1 q +B 3lN q +B 4l(N1) q )u(1) + (B 3l1 q +B 2l2 q +B 1l3 q +B 4lN q )u(2) +::: + (B 4l(k2) q +B 3l(k1) q +B 2lk q +B 1l(k+1) q )u(k) +::: + (B 1lN q +B 4l(N1) q +B 3l(N2) q +B 2lN3 q )u(N 1) + (B 1l1 q +B 4l(N2) q +B 3l(N1) q +B 2lN q )u(N) (B.79) with B jlp q = Z 1 0 n q U(;l) j ()jJ p jd l = 1 :N;p = 1 :N;j = 1 : 4 (B.80) Similar results can be obtained for other integrals in Eq. (B.51). Therefore, the discretized integral equation can be obtained using either linear or Overhauser elements. Specically, 122 for the full-space problem the discretization is performed using either linear or Overhauser elements. Subsequently, for the half-space problem discretization utilizes both linear and Overhauser elements. To verify the proposed numerical solution, the full-space stress eld is evaluated and compared with the existing results in the literature; this is considered next. B.4 Verication of the Full-space Numerical Results B.4.1 Dynamic Stress Concentration for a Cavity in a Full-Space Subjected to an SH-wave B.4.1.1 Analytical Solution An incident SH-wave with unit amplitude, in a cylindrical coordinate system is dened by u x = 0 u y = 0 u z =e ikrcos (B.81) Here,k is the wave number, r = p x 2 +y 2 and =tan 1y x . Note that the radius of the circular cavity is assumed to be equal to a. Throughout, the material properties of the full-space are assumed to be 0 = 0 = 1. In terms of Bessel functions, the incident wave can be rewritten as [84] w (i) = 1 X n=0 a n J n (kr) cosn (B.82) where J n is the Bessel function of the rst kind and order n, a n =" n i n ( " 0 = 1 " n = 2 ;n> 0 (B.83) and i = p 1. The scattered wave eld is given by [6] w (s) = 1 X n=0 A n H (1) n (kr)cosn (B.84) where H (1) n are the Hankel functions of the rst kind and order zero. The unknowns A n are found using the boundary conditions. 123 The surface of the cavity is stress free; therefore rz r=a = @ @r (w i +w s ) r=a = 0 (B.85) or a n @J n (kr) @r r=a +A n @H (1) n (kr) @r r=a cosn = 0 (B.86) From the recursion relations for Bessel functions [78], the unknown constants A n are given by [6] A n =" n i n nJ n (ka)kaJ n+1 (ka) nH n (ka)kaH n+1 (ka) (B.87) In the absence of the cavity, the stress is induced only by the incident wave eld (i) rz = @w (i) @r = 0 cose i(kx!t+=2) (i) z = 1 r @w (i) @ = 0 sine i(kx!t+=2) (B.88) Here, 0 =k is the maximum value of the stress, and ! is the circular frequency. In the presence of the cavity, the boundary conditions are rz r=a = 0 z r=a = a 1 X n=0 n[a n J n (ka) +A n H n (ka)] sinn (B.89) where rz and z are total stress elds. Using Eq. (B.87), the stress eld z on the boundary is given by z r=a = 0 1 X n=0 1 ka " n i n J n (ka) [nJ n (ka)kaJ n+1 (ka)] [nH n (ka)kaH n+1 (ka)] H n (ka) n sinn (B.90) This exact solution for the stress eld is compared with the stress obtained using the BIEs developed in the previous section. This is shown as a function of the angle for three dierent frequencies in Figs. B.2 through B.4. Similarly, the stress eld is compared in Fig. B.5 for a wide range of frequencies and at a single point (r =a; ==2) It is apparent from the results of Figs. B.2-B.5 that the numerical solution obtained using the BIE method shows excellent agreement with the exact solution for a wide range of frequencies and locations. 124 This concludes testing of the results for a circular cavity embedded in a full-space when subjected to a plane harmonic incident SH-wave. Testing the results for the corresponding case of an elastic inclusion is considered next. 0.5 1 1.5 2 2.5 30 210 60 240 90 270 120 300 150 330 180 0 err=8.0578e−05,k=0.1 non−singular sBIE OH exact Figure B.2: Normalized stress eldj z = 0 j as a function of for a cavity of unit radius sub- jected to a low-frequency (ka = 0:1) plane harmonic incident SH-wave. Solid lines represent the exact solution [6], and the solid dots indicate the results obtained with the BIE method in this study using 64 Overhauser elements. 125 0.6 1.2 1.8 30 210 60 240 90 270 120 300 150 330 180 0 err=7.9005e−05,k=1 non−singular sBIE OH exact Figure B.3: Normalized stress eldj z = 0 j as a function of for a cavity of unit radius and subjected to plane harmonic incident SH-wave with ka = 1, Solid lines represent the exact solution [6], and the solid dots denote the results obtained with the BIE method in this study using 64 Overhauser elements. 0.4 0.8 1.2 1.6 30 210 60 240 90 270 120 300 150 330 180 0 err=0.00010588,k=2 non−singular sBIE OH exact Figure B.4: Normalized stress eldj z = 0 j as a function of for cavity of unit radius and subjected to plane harmonic incident SH-wave with ka = 2. Solid lines represent the exact solution [6], and the solid dots denote the results obtained with the BIE method in this study using 64 Overhauser elements. 126 0 1 2 3 4 5 6 7 8 1.4 1.6 1.8 2 2.2 ka Real |σ θ z /σ 0 | for θ=π /2 Exact BIE N64 OH non−HS 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 Imag |σ θ z /σ 0 | Exact BIE N64 OH non−HS Figure B.5: Comparison of the numerical results and the exact solution for a single point on the cavity surface and for dierent frequencies (Rej z = 0 j and Imj z = 0 j vs. ka at ==2) for a unit cylinder embedded in a full-space subjected to a plane harmonic SH-wave. The BIE solution uses 64 Overhauser elements over the cavity surface. B.4.2 Dynamic Stress Concentration for a Circular Elastic Inclusion Embedded in a Full-Space Subjected to a Plane Harmonic Incident SH-wave As for the cavity problem in the full-space, the incident and the scattered wave elds for this problem are given by Eqs. (B.82) and (B.84). The elastic cylinder of unit radius consists of an isotropic material with shear modulus and velocity, 1 and 1 , respectively. The standing wave inside the cylinder can be written as w (1) = 1 X n=0 B n J n (k 1 r) cosn (B.91) whereB n are the unknown coecients that can be determined by applying the continuity conditions and J n are the Bessel functions of the rst kind and order n. To ensure the continuity of the displacement and stress elds at the surface of the cylinder, namely r =a, the following conditions must be satised: w (i) +w (s) = w (1) ;r =a (B.92) @ @r (w (i) +w (s) ) = 1 @w (1) @r ;r =a (B.93) 127 Substituting the scattered wave eld into the continuity of the displacement eld results in the following equation X a n J n (ka) +A n H (1) n (ka)B n J n (k 1 a) cosn = 0 (B.94) which holds for arbitrary . Here, the coecients a n are known, and the scattered wave coecients A n and B n are to be determined. Therefore, it follows that a n J n (ka) +A n H (1) n (ka) =B n J n (k 1 a) (B.95) Similarly, the continuity of the stress eld (B.93) gives a n @J n (kr) @r +A n @H (1) n (kr) @r r=a = 1 B n @J n (k 1 r) @r r=a (B.96) The last two equations can be combined into the following system of equations for the unknowns A n and B n . 2 4 H (1) n (ka) J n (k 1 a) @H (1) n (kr) @r ) r=a 1 @Jn(k 1 r) @r r=a 3 5 A n B n = 8 < : a n J n (ka) a n @Jn(kr) @r r=a 9 = ; (B.97) The above system of equations can be solved for the unknown coecients for the scat- tered wave elds. Using these equations, one can evaluate the displacements and stresses throughout the elastic media. Next, the exact stress eld along the circular interface is compared with the results obtained using the BIE method. The stress components, z and rz are used to calculate stress concentration factorsj z = 0 j andj rz = 0 j, where 0 = 1 k 0 is the maximum stress in the absence of the inclusion. Figures B.6 through B.9 depict the stress concentration factors for a soft inclusion and a sti inclusion. The results show excellent agreement between the exact solution and the BIE results for a wide range of parameters present in the problem. This completes the formulation and verication of the results for the full-space cavity and inclusion problem. 128 0.6 1.2 1.8 30 210 60 240 90 270 120 300 150 330 180 0 N1L1 σ θz BIE inc exact inc Figure B.6: Comparison of the stress concentration factor (j z = 0 j vs ) along the interface of a soft, elastic, circular inclusion of unit radius embedded in a full-space and subjected to a plane harmonic SH-wave with a single frequency (ka = 2). Solid lines denote the exact solution, and the asterisks represent the BIE results. The material properties of the full-space are 0 = 0 = 1. The inclusion's properties are 1 = 1=6 and 1 = 1=2. For the BIE solution, 64 Overhauser elements are used and 0 = 1 k 0 denotes the maximum stress in the absence of the inclusion. 129 2 4 6 8 30 210 60 240 90 270 120 300 150 330 180 0 N1L1 σ rz BIE inc exact inc Figure B.7: Comparison of the stress eldj rz = 0 j vs along the interface of a soft, elastic, circular inclusion of unit radius embedded in a full-space and subjected to a plane harmonic SH-wave with a single frequency (ka = 2). Solid lines denote the exact solution, and the asterisks represent the BIE results. The material properties of the full-space are 0 = 0 = 1. The inclusion's properties are 1 = 1=6 and 1 = 1=2. For the BIE solution 64 Overhauser elements are used, and 0 = 1 k 0 denotes the maximum stress in the absence of the inclusion. 130 0.05 0.1 0.15 0.2 0.25 30 210 60 240 90 270 120 300 150 330 180 0 N1L1 σ θz BIE inc exact inc Figure B.8: Comparison of the stress eldj z = 0 j vs along the interface of a sti, elastic, circular inclusion of unit radius embedded in a full-space and subjected to a plane harmonic SH-wave with a single frequency (ka = 2). Solid lines denote the exact solution, and the asterisks represent the BIE results. The material properties of the full-space are 0 = 0 = 1. The inclusion's properties are 1 = 8 and 1 = 2. For the BIE solution 64 Overhauser elements are used, and 0 = 1 k 0 denotes the maximum stress in the absence of the inclusion. 131 0.05 0.1 0.15 0.2 0.25 30 210 60 240 90 270 120 300 150 330 180 0 N1L1 σ rz BIE inc exact inc Figure B.9: Comparison of the stress eldj rz = 0 j vs along the interface of a sti, elastic, circular inclusion of unit radius embedded in a full-space and subjected to a plane harmonic SH-wave with a single frequency (ka = 2). Solid lines denote the exact solution, and the asterisks represent the BIE results. The material properties of the full-space are 0 = 0 = 1. The inclusion's properties are 1 = 8 and 1 = 2. For the BIE solution 64 Overhauser elements are used, and 0 = 1 k 0 denotes the maximum stress in the absence of the inclusion. 132 References [1] VW Lee. On deformations near circular underground cavity subjected to incident plane sh waves. In Proceedings of the Application of Computer Methods in Engineer- ing Conference, volume 2, pages 951{962. 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Abstract (if available)
Abstract
The scattering of a plane harmonic SH-wave by an arbitrary number of layered inclusions in a half-space is investigated bya direct boundary integral equation method. The inclusions, which have arbitrary shape and arrangement, are embedded within an elastic half-space. The effects of multiple scattering, the geometry, and the impedance contrast of the materials for layered inclusions and pipes, on the surface motion, are considered in detail. ❧ Additionally, a non-hypersingular technique is employed to calculate the dynamic stress concentration factor along the interfaces of inclusions. Various contributing factors that can influence the stress concentration, including multiple scattering, the impedance contrast of layers, and the separation distance between the scatterers, are investigated. The numerical results are presented for a wide range of parameters present in the problem.
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Scattering of a plane harmonic SH-wave and dynamic stress concentration for multiple multilayered inclusions embedded in an elastic half-space
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anti-plane strain
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elastic waves
layered inclusions
multiple scattering
non-hypersingular boundary element method
wave propagation