# Forward Analysis of Love-Wave Scattering due to a Cavity-Like Defect

Authors: Chen Yang; Bin Wang; Zhenghua Qian
Journal: Shock and Vibration
 (2018)

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## Abstract

This paper presents a modified boundary element method (BEM) to solve the scattering problem of Love surface wave from a two-dimensional cavity defect. Because of the truncation of BEM models at a far distance from the cavity, spurious reflected waves are generated. In order to eliminate the unwanted reflections, the guided Love-wave displacement patterns are assumed on the far-field infinite boundaries previously omitted by model truncation, and they are incorporated into the BEM equation set as modified items. The numerical results are verified by theoretical solutions of far-field Green’s functions. Additional parametric studies are performed to find out the influence of truncation distance and defects’ geometric characters on the accuracy of scattered wave solutions.


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## Body

## 1. Introduction

The ultrasonic nondestructive testing (NDT) techniques have wide applications for quantitative characterizations of mechanical properties and detection and characterization of cracks and defects. Traditional ultrasonic testing techniques using bulk waves are very time-consuming, since these techniques need an overall inspection of the structure. However, ultrasonic guided waves are attractive for inspection of long-range or wide area structures because they can travel considerable distances and therefore scan large regions for defects in shorter testing time [1, 2].The current NDT applications of guided waves include pitch-catch [3] or pulse-echo [4], flaw detection method, phased array configuration [5], and diffraction tomography [6]. Generally, these methods make use of time-of-flight (TOF) of the reflected waves from inner defects to locate their approximate positions.However, further information (e.g., defect shapes or depths) cannot be further utilized because of the complexity of guided waves. Hence, the quantitative nondestructive testing requires a thorough understanding of surface wave scattering in forward and inverse aspects. For the forward problem, we need to solve the near- and far-fields accurately and obtain the scattering coefficients for following inverse reconstruction [7–9].The Love-wave is a special kind of guided waves that travels along the surface of elastic layer covered on top of an elastic half-plane. The scattered Love-waves are relied on to investigate underground information in geotechnique engineering, earthquake engineering [10], or detecting flaws and cracks at the bounding interface in nondestructive testing applications [11]. An effective utilization of the Love-wave requires a thorough understanding of its scattering phenomenon.For the calculation of scattered wave field over a finite domain, various technologies can be implied, like finite element method (FEM) [12], BEM [13], mode-exciting method [14], matrix theory [15], and so on. However, for the forward analysis of a half-plane, the BEM is especially effective, since only the interfaces and flaw boundaries need to be meshed. There are BEM approaches using two kinds of Green’s functions: half-space and full-space. Using the former one, only the flawed portion needs to be meshed; however, Green’s function cannot be written in a closed form. Conversely, using the latter one, the whole interface should be meshed; however, Green’s function is much simpler. Thus, for the forward analysis of Love-wave, we adopt the latter one.However, in traditional BEM approaches, due to the inevitable artificial truncation of BEM model, spurious reflected waves are introduced in the final results of scattered wave field, which causes considerable error. Another big challenge to solve the scattering problem is the existence of multiple dispersive modes of Love-waves at a certain frequency along with the modal conversion, due to the interaction at the damage location.Here, a modified BEM for calculating scattered Love-waves is introduced. In this paper, the guided Love-wave displacement patterns are assumed on the far-field infinite boundaries previously omitted, and they are incorporated into BEM equation sets as the modified items. With this improvement, the spurious reflected waves are eliminated. The numerical results are verified by theoretical far-field Green’s functions [16, 17]. Furthermore, various parametric studies of the influence of defect locations and geometrical shapes and size on the calculations of Love-wave scattered fields are carried out in the later sections, which have potential values for investigating forward problem or inverse problem of flaw reconstruction based on surface waves.


## 2. Statement of the Problem

The Love surface wave propagates along the surface of elastic layer of thicknessH covered on top of a homogeneous, elastic half-plane, containing a cavity of arbitrary shape on the bonding interface of the x1-x2 plane (see Figure 1(a)). Here, we consider an incident Love-wave propagating in the x1-direction, which interacts with the cavity generating forward-scattered and back-scattered surface wave.Figure 1
Linear superposition principle: (a) the total field; (b) the incident field; (c) the scattered field.By virtue of linear superposition principle, the total field in the flawed structure defined by Figure1(a) can be considered as the superposition of the incident and the scattered waves. The incident wave can be treated in the intact (or reference) structure without cavity, as shown by Figure 1(b), and the scattered field is analyzed in the flawed configuration in Figure 1(c). The scattered field is equivalent to the field generated by the contribution of the tractions exerted on the actual surface of the cavity. Furthermore, these tractions are equal in magnitude but opposite in sign to the corresponding tractions produced by the incident Love-wave field on the surface of the fictitious cavity as shown by Figure 1(b). Thus, these tractions can be obtained by calculating the stress components and the outward normal vectors along the fictitious cavity surface using the Cauchy’s formula from the incident field. The dynamic reciprocal theorem is then applied to calculate the scattered wave field equivalent to the radiated field generated by these tractions.


## 3. Equations

### 3.1. The Elastodynamic Reciprocal Theorem

The dynamic reciprocal theorem relates two elastodynamic states 1 and 2 of the same bounded or unbounded body, which can be stated as(1)∫Vfk1x,ωuk2x,ω-fk2x,ωuk1x,ωdVx=∫Aτkl2x,ωnkul1x,ω-τkl1x,ωnkul2x,ωdAx,where fk1,2, uk1,2, and τkl1,2 represent body forces, displacements, and stresses, respectively, and nk is thekth component of unit vector outward surface normal to A.Let us consider two-dimensional elastodynamic problems in an isotropic half-plane with a different homogeneous and isotropic layer covered with boundaryL. The boundary integral equation of antiplane motion for a source point ξ taken on L, in the absence of body forces, is developed from (1) and derived as(2)12uξ,ω=∫Lu∗ξ,x,ωtx,ω-t∗ξ,x,ωux,ωdLx,where the factor 1/2 is valid only if the boundary L is smooth at point ξ and u∗ and t∗ are the full-space frequency domain elastodynamic antiplane fundamental solution displacement and traction tensors, respectively, which are derived [18] as(3)u∗ξ,x,ω=i4μIH01kTBr0≤ξ2≤Hori4μIIH01kTArξ2≤0t∗ξ,x,ω=-ikTB4H11kTBr∂r∂n0≤ξ2≤Hor-ikTA4H11kTAr∂r∂nξ2≤0,where Hn1 is the Hankel function of the nth order of the first kind; μA, μB, kTA, and kTB stand for the elastic constants and the wave-numbers of the shear wave at current frequency, for the upper and lower materials, respectively, where kTJ=ω/μJ/ρJJ=A,B, in which ρA and ρB are material densities; r represents the distance between ξ and x; u∗ and t∗ are the displacement and boundary traction, respectively, at the point x, respectively, due to a unit line force exerted at ξ. For current antiplane problem, both the line force and Green’s function -- u∗ and t∗ only have the x3 component.Let us assume that, except the flaw regionL1 and L5, both the free-traction surface and the interface are flat. Let L0 and L3 be the free upper surface and the interface between upper-layer and half-plane, respectively, and L∞a and L∞b represent the remaining infinite part of upper and lower boundary, respectively, which will be omitted by truncation in traditional BEM (see Figure 1).By substituting all boundaries divided in Figure2 into (2), the BIE of the layered media and half-plane are derived as(4)12uξ,ω+∫L0∩L3∩L1tBξ,x,ωux,ωdLx+∫L∞a∪L∞btBξ,x,ωux,ωdLx-∫L∞auBξ,x,ωtx,ωdLx=∫L3∩L1uBξ,x,ωtx,ωdLx,(5)12uξ,ω+∫L3∩L5tAξ,x,ωux,ωdLx+∫L∞atAξ,x,ωux,ω-uAξ,x,ωtx,ωdLx=∫L3∩L5uAξ,x,ωtx,ωdLx,respectively, where the superscripts A and B indicate the Green functions of half-plane and the layer, respectively.Figure 2
Schematic definition of the computational domain of the Love-wave scattered field, with divided interface boundariesL=L±∞b∪L0±∪L3±∪L1±∪L5±∪L±∞a.


### 3.2. Far-Field Assumption

Since body waves geometrically attenuate in the propagating direction, the far-field displacement solution can be approximated by a series of Love surface waves, neglecting the contribution of body waves.Therefore, we assume that if the truncated points are located far enough from the source regions, the displacement solutions of the infinite boundary at each side can be expressed as(6)ux,ω≈R1-ωu1-x,ω+R2-ωu2-x,ω+⋯+Rn-ωun-x,ωForx∈L-∞a∪L-∞b,ux,ω≈R1+ωu1+x,ω+R2+ωu2+x,ω+⋯+Rn+ωun+x,ωForx∈L+∞a∪L+∞b,where the coordinate vector x is in the form of x1x2, Ri±ω are defined as the unknown complex amplitudes of the far-field solutions of the ith order mode Love-wave. Here, n is the number of modes, and ui±x,ω represent the ith order mode displacement of unit amplitude Love-wave propagating in the positive and negative direction of axis x1. (Note that Love surface waves are dispersive.)By virtue of assumptions in (6), (4) and (5) can be rewritten as (7)12uξ,ω+∫L0∩L3∩L1tBξ,x,ωux,ωdLx-∫L3∩L1uBξ,x,ωtx,ωdLx=∑iRi-∫L-∞a∪L-∞btBξ,x,ωui-x,ωdLx-∫L-∞auBξ,x,ωti-x,ωdLx+Ri+∫L+∞a∪L+∞btBξ,x,ωui+x,ωdLx-∫L+∞auBξ,x,ωti+x,ωdLxi=1,2,…,n,12uξ,ω+∫L3∩L1tAξ,x,ωux,ωdLx-∫L3∩L1uAξ,x,ωtx,ωdLx=-∑iRi-∫L-∞atAξ,x,ωui-x,ωdLx-∫L-∞auAξ,x,ωti-x,ωdLx+Ri+∫L+∞atAξ,x,ωui+x,ωdLx-∫L+∞auAξ,x,ωti+x,ωdLxi=1,2,…,n,respectively. From (7), we define(8)AiB±ξ=∫L±∞a∪L±∞btBξ,x,ωui±x,ωdLx-∫L±∞auBξ,x,ωti±x,ωdLx,AiA±ξ=∫L±∞atAξ,x,ωui±x,ωdLx-∫L±∞auAξ,x,ωti±x,ωdLxwhich represent the corrected items accounting for the contribution of the omitted boundary. Thus, (7) are simplified as(9)12uξ,ω+∫L0∩L3∩L1tBξ,x,ωux,ωdLx-∫L3∩L1uBξ,x,ωtx,ωdLx=∑iRi-AiB-+Ri+AiB+i=1,2,…,n.12uξ,ω+∫L3∩L1tAξ,x,ωux,ωdLx-∫L3∩L1uAξ,x,ωtx,ωdLx+∑iRi-AiA-+Ri+AiA+=∫L3uAξ,x,ωtx,ωdLxi=1,2…,nNote that2n unknown parameters Ri±ω are introduced into the BIEs, which will add degrees of freedom to the final BEM system of the BIEs.


### 3.3. Correction over the Omitted Part of the Infinite Boundary

In traditional BEM approaches, the contribution of integral terms on the infinite boundary, that is, the fourth term on the right-hand side of (4) and the third term on the right-hand side of (5), is omitted, which introduces considerable error. In order to separately determine the integral terms over infinite boundaries such as L±∞a and L±∞b, a multidomain approach is applied, which involves the division of the whole interfaces and boundaries into four regions by introducing two fictitious boundaries L2 and L4, as shown in Figure 2. Here, an incident Love-wave mode with unit amplitude is introduced propagating along the upper free surface in the positive or negative direction of x1, respectively (see Figure 3).Figure 3
Schematic diagram of the multidomain approach: (a) calculateAiB-; (b) calculate AiB+; (c) calculate AiA-; (d) calculate AiA+.Let us choose the Love surface wave of unit amplitude as elastodynamic state 1 and the full-space fundamental solution as elastodynamic state 2. For instance, by virtue of reciprocal theorem seen from (2), the BIE for region 1 is given as(10)12ui±ξ,ω=∫Lu∗ξ,x,ωti±x,ω-t∗ξ,x,ωui±x,ωdLx.By simplifying (10), we arrive at(11)AiB+ξ=-12ui+ξ,ω-∫L0+∪L1+∪L3+∪L2tBξ,x,ωui+x,ωdLx+∫L1+∪L3+∪L2uBξ,x,ωti+x,ωdLx.By implying an analogous approach for other regions, we can getAiB-ξ, AiA-ξ and AiA+ξ, which are expressed as(12)AiB-ξ=-12ui-ξ,ω-∫L0-∪L1-∪L3-∪L2tBξ,x,ωui-x,ωdLx+∫L1-∪L3-∪L2uBξ,x,ωti-x,ωdLx(13)AiA-ξ=-12ui-ξ,ω-∫L3-∪L5-∪L4tAξ,x,ωui-x,ωdLx+∫L3-∪L5-∪L4uAξ,x,ωti-x,ωdLx(14)AiA+ξ=-12ui+ξ,ω-∫L3+∪L5+∪L4tAξ,x,ωui+x,ωdLx+∫L3+∪L5+∪L4uAξ,x,ωti+x,ωdLx.Note that for the calculation ofAiB-ξ and AiA-ξ, the incident Love-wave is assumed to propagate in the positive direction, while for AiA+ξ and AiB+ξ the propagating direction is opposite. The fictitious boundary L4 is, in principle, infinite. However, since the integrand over L4 attenuates rapidly in exponential form away from the interface of half-plane, without loss of accuracy, we consider the boundary L4 as a distance of about two Love wavelengths.


## 3.1. The Elastodynamic Reciprocal Theorem

The dynamic reciprocal theorem relates two elastodynamic states 1 and 2 of the same bounded or unbounded body, which can be stated as(1)∫Vfk1x,ωuk2x,ω-fk2x,ωuk1x,ωdVx=∫Aτkl2x,ωnkul1x,ω-τkl1x,ωnkul2x,ωdAx,where fk1,2, uk1,2, and τkl1,2 represent body forces, displacements, and stresses, respectively, and nk is thekth component of unit vector outward surface normal to A.Let us consider two-dimensional elastodynamic problems in an isotropic half-plane with a different homogeneous and isotropic layer covered with boundaryL. The boundary integral equation of antiplane motion for a source point ξ taken on L, in the absence of body forces, is developed from (1) and derived as(2)12uξ,ω=∫Lu∗ξ,x,ωtx,ω-t∗ξ,x,ωux,ωdLx,where the factor 1/2 is valid only if the boundary L is smooth at point ξ and u∗ and t∗ are the full-space frequency domain elastodynamic antiplane fundamental solution displacement and traction tensors, respectively, which are derived [18] as(3)u∗ξ,x,ω=i4μIH01kTBr0≤ξ2≤Hori4μIIH01kTArξ2≤0t∗ξ,x,ω=-ikTB4H11kTBr∂r∂n0≤ξ2≤Hor-ikTA4H11kTAr∂r∂nξ2≤0,where Hn1 is the Hankel function of the nth order of the first kind; μA, μB, kTA, and kTB stand for the elastic constants and the wave-numbers of the shear wave at current frequency, for the upper and lower materials, respectively, where kTJ=ω/μJ/ρJJ=A,B, in which ρA and ρB are material densities; r represents the distance between ξ and x; u∗ and t∗ are the displacement and boundary traction, respectively, at the point x, respectively, due to a unit line force exerted at ξ. For current antiplane problem, both the line force and Green’s function -- u∗ and t∗ only have the x3 component.Let us assume that, except the flaw regionL1 and L5, both the free-traction surface and the interface are flat. Let L0 and L3 be the free upper surface and the interface between upper-layer and half-plane, respectively, and L∞a and L∞b represent the remaining infinite part of upper and lower boundary, respectively, which will be omitted by truncation in traditional BEM (see Figure 1).By substituting all boundaries divided in Figure2 into (2), the BIE of the layered media and half-plane are derived as(4)12uξ,ω+∫L0∩L3∩L1tBξ,x,ωux,ωdLx+∫L∞a∪L∞btBξ,x,ωux,ωdLx-∫L∞auBξ,x,ωtx,ωdLx=∫L3∩L1uBξ,x,ωtx,ωdLx,(5)12uξ,ω+∫L3∩L5tAξ,x,ωux,ωdLx+∫L∞atAξ,x,ωux,ω-uAξ,x,ωtx,ωdLx=∫L3∩L5uAξ,x,ωtx,ωdLx,respectively, where the superscripts A and B indicate the Green functions of half-plane and the layer, respectively.Figure 2
Schematic definition of the computational domain of the Love-wave scattered field, with divided interface boundariesL=L±∞b∪L0±∪L3±∪L1±∪L5±∪L±∞a.


## 3.2. Far-Field Assumption

Since body waves geometrically attenuate in the propagating direction, the far-field displacement solution can be approximated by a series of Love surface waves, neglecting the contribution of body waves.Therefore, we assume that if the truncated points are located far enough from the source regions, the displacement solutions of the infinite boundary at each side can be expressed as(6)ux,ω≈R1-ωu1-x,ω+R2-ωu2-x,ω+⋯+Rn-ωun-x,ωForx∈L-∞a∪L-∞b,ux,ω≈R1+ωu1+x,ω+R2+ωu2+x,ω+⋯+Rn+ωun+x,ωForx∈L+∞a∪L+∞b,where the coordinate vector x is in the form of x1x2, Ri±ω are defined as the unknown complex amplitudes of the far-field solutions of the ith order mode Love-wave. Here, n is the number of modes, and ui±x,ω represent the ith order mode displacement of unit amplitude Love-wave propagating in the positive and negative direction of axis x1. (Note that Love surface waves are dispersive.)By virtue of assumptions in (6), (4) and (5) can be rewritten as (7)12uξ,ω+∫L0∩L3∩L1tBξ,x,ωux,ωdLx-∫L3∩L1uBξ,x,ωtx,ωdLx=∑iRi-∫L-∞a∪L-∞btBξ,x,ωui-x,ωdLx-∫L-∞auBξ,x,ωti-x,ωdLx+Ri+∫L+∞a∪L+∞btBξ,x,ωui+x,ωdLx-∫L+∞auBξ,x,ωti+x,ωdLxi=1,2,…,n,12uξ,ω+∫L3∩L1tAξ,x,ωux,ωdLx-∫L3∩L1uAξ,x,ωtx,ωdLx=-∑iRi-∫L-∞atAξ,x,ωui-x,ωdLx-∫L-∞auAξ,x,ωti-x,ωdLx+Ri+∫L+∞atAξ,x,ωui+x,ωdLx-∫L+∞auAξ,x,ωti+x,ωdLxi=1,2,…,n,respectively. From (7), we define(8)AiB±ξ=∫L±∞a∪L±∞btBξ,x,ωui±x,ωdLx-∫L±∞auBξ,x,ωti±x,ωdLx,AiA±ξ=∫L±∞atAξ,x,ωui±x,ωdLx-∫L±∞auAξ,x,ωti±x,ωdLxwhich represent the corrected items accounting for the contribution of the omitted boundary. Thus, (7) are simplified as(9)12uξ,ω+∫L0∩L3∩L1tBξ,x,ωux,ωdLx-∫L3∩L1uBξ,x,ωtx,ωdLx=∑iRi-AiB-+Ri+AiB+i=1,2,…,n.12uξ,ω+∫L3∩L1tAξ,x,ωux,ωdLx-∫L3∩L1uAξ,x,ωtx,ωdLx+∑iRi-AiA-+Ri+AiA+=∫L3uAξ,x,ωtx,ωdLxi=1,2…,nNote that2n unknown parameters Ri±ω are introduced into the BIEs, which will add degrees of freedom to the final BEM system of the BIEs.


## 3.3. Correction over the Omitted Part of the Infinite Boundary

In traditional BEM approaches, the contribution of integral terms on the infinite boundary, that is, the fourth term on the right-hand side of (4) and the third term on the right-hand side of (5), is omitted, which introduces considerable error. In order to separately determine the integral terms over infinite boundaries such as L±∞a and L±∞b, a multidomain approach is applied, which involves the division of the whole interfaces and boundaries into four regions by introducing two fictitious boundaries L2 and L4, as shown in Figure 2. Here, an incident Love-wave mode with unit amplitude is introduced propagating along the upper free surface in the positive or negative direction of x1, respectively (see Figure 3).Figure 3
Schematic diagram of the multidomain approach: (a) calculateAiB-; (b) calculate AiB+; (c) calculate AiA-; (d) calculate AiA+.Let us choose the Love surface wave of unit amplitude as elastodynamic state 1 and the full-space fundamental solution as elastodynamic state 2. For instance, by virtue of reciprocal theorem seen from (2), the BIE for region 1 is given as(10)12ui±ξ,ω=∫Lu∗ξ,x,ωti±x,ω-t∗ξ,x,ωui±x,ωdLx.By simplifying (10), we arrive at(11)AiB+ξ=-12ui+ξ,ω-∫L0+∪L1+∪L3+∪L2tBξ,x,ωui+x,ωdLx+∫L1+∪L3+∪L2uBξ,x,ωti+x,ωdLx.By implying an analogous approach for other regions, we can getAiB-ξ, AiA-ξ and AiA+ξ, which are expressed as(12)AiB-ξ=-12ui-ξ,ω-∫L0-∪L1-∪L3-∪L2tBξ,x,ωui-x,ωdLx+∫L1-∪L3-∪L2uBξ,x,ωti-x,ωdLx(13)AiA-ξ=-12ui-ξ,ω-∫L3-∪L5-∪L4tAξ,x,ωui-x,ωdLx+∫L3-∪L5-∪L4uAξ,x,ωti-x,ωdLx(14)AiA+ξ=-12ui+ξ,ω-∫L3+∪L5+∪L4tAξ,x,ωui+x,ωdLx+∫L3+∪L5+∪L4uAξ,x,ωti+x,ωdLx.Note that for the calculation ofAiB-ξ and AiA-ξ, the incident Love-wave is assumed to propagate in the positive direction, while for AiA+ξ and AiB+ξ the propagating direction is opposite. The fictitious boundary L4 is, in principle, infinite. However, since the integrand over L4 attenuates rapidly in exponential form away from the interface of half-plane, without loss of accuracy, we consider the boundary L4 as a distance of about two Love wavelengths.


## 4. Numerical Computation

Numerical solutions of (9) require the discretization of the boundary Li into elements. After the discretization of the boundary and interpolation of the displacements and tractions, the discretized BIEs for the layer and half-plane can be written for each of nodes ξj and υj, respectively, as(15)12uξj,ω+∑e∈L0∪L1∪L3∑k=1Ne∫LetBξj,η,ωϕkηdLηuξk,ω+∑i=1nAiB-ξjRi-ω+∑i=1nAiB+ξjRi+ω=∑e∈L1∪L3∑k=1Ne∫LeuBξj,η,ωϕkηdLηtξj,ωj=1,2,…,N1,12uυj,ω+∑e∈L3∑k=1Ne∫ΓetAυj,η,ωϕkηdLηuυk,ω+∑i=1nAiA-ξjRi-ω+∑i=1nAiA+ξjRi+ω=∑e∈L3∑k=1Ne∫LeuAυj,η,ωϕkηdLηtξk,ωj=1,2,…,N2,where N1, N2 are the total number of nodes for the layer and half-plane, respectively, Ne is the number of nodes per element, ϕk is the same shape function for each element, and η∈-1,1 represents the intrinsic coordinate of the parent element. It is noted that the calculation of corrected coefficients AiB±ξ and AiA±ξ is performed in the previous section.Equations (15) can be expressed in a more concise manner by defining(16)TjkB=∫LetBξj,η,ωϕkηdLηj≠k∫LetBξj,η,ωϕkηdLη+12j=kGjkB=∫LeuBξj,η,ωϕkηdLη,(17)TjkA=∫LetAυj,η,ωϕkηdLηj≠k∫LetAυj,η,ωϕkηdLη+12j=kGjkA=∫LeuAυj,η,ωϕkηdLη,where the subscripts represent the collocation points ξj and υj with the node k of element e. Then the above equations are rewritten as(18)∑e∈L0∪L1∪L3∑k=1NeTjkBuξj,ω+∑i=1nAiB-ξjRi-ω+∑i=1nAiB+ξjRi+ω=∑e∈L1∪L3∑k=1NeGjkBtξj,ωj=1,2,…,N1,(19)∑e∈L3∑k=1NeTjkAuυk,ω+∑i=1nAiA-υjRi-ω+∑i=1nAiA+υjRi+ω=∑e∈L3∑k=1NeGjkAtυk,ωj=1,2,…,N2.Then, let us assemble the local elementsTjkB, GjkB into global matrices HB, GB, the node displacement uξj,ω and node traction tξj,ω into global matrices UB, TB, and the correction AiB±ξj and the unknown amplitudes Ri±ω into the correction matrices AB± and the amplitude matrices R±. Equation (18) can be written as(20)HBUB+AB±R±=GBTB,where(21)UB=uξ1,ωuξ2,ω⋯uξN1,ωTTB=tξ1,ωtξ2,ω⋯tξN1,ωTAB±=A1B±ξ1A2B±ξ1⋯AnB±ξ1A1B±ξ2A2B±ξ2⋯AnB±ξ2⋮⋮⋮⋮A1B±ξN1A2B±ξN1⋯AnB±ξN1R±=R1±ωR2±ω⋯Rn±ωT.Conveniently, the corrected BEM system can be rewritten as(22)T11BT12BT13BT14BT21BT22BT23BT24BT31BT32BT33BT34BT41BT42BT43BT44BU0U3-U1U3++AB±R±=G11BG12BG13BG14BG21BG22BG23BG24BG31BG32BG33BG34BG41BG42BG43BG44BT0T3-T1T3+,where TijB, GijB are block matrices of TB, GB and Uα±, Tα± are the node displacement vectors and node traction vectors corresponding to Lα±, respectively.Analogously, (19) can be expressed in matrix form:(23)HAUA+AA±R±=GATA,where(24)UA=uυ1,ωuυ2,ω⋯uυN2,ωTTA=tυ1,ωtυ2,ω⋯tυN2,ωT(25)AA±=A1A±υ1A2A±υ1⋯AnA±υ1A1A±υ2A2A±υ2⋯AnA±υ2⋮⋮⋮⋮A1A±υN1A2A±υN1⋯AnA±υN1,(26)T11AT12AT13AT21AT22AT23AT31AT32AT33AU3-U5U3++AA±R±=G11AG12AG13AG21AG22AG23AG31AG32AG33AT3-T5T3+,where TijB, GijB are block matrices of TB, GB and Uα±, Tα± are the node displacement vectors and node traction vectors corresponding to Lα±, respectively.It should be pointed out that the unknown coefficient matricesR± which are assembled into the modified BEM system ((20) and (23)), will add 2n degrees of freedom into the final BEM system of equations. Here, we propose a modified method for Love-wave multimode by introducing finite sequence truncated points on far-field regions. Based on the far-field assumption (see (6)), far-field displacements of 2n sequence points ξm+ii=1,2,…,n and ξN+i-1i=1,2,…,n (see Figure 1) are written as(27)uξm+i,ω=∑j=1nuj-ξm+i,ωRj-ωuξN+i-1,ω=∑j=1nuj+ξN+i-1,ωRj+ωi=1,2,…,n,which can also be expressed as the form of matrix(28)IR-U3-=UR-R-,where(29)IR-=-In000(30)UR-=u1-ξm+1,ωu2-ξm+1,ω⋯un-ξm+1,ωu1-ξm+2,ωu2-ξm+2,ω⋯un-ξm+2,ω⋮⋮⋮⋮u1-ξm+n,ωu2-ξm+n,ω⋯un-ξm+n,ω(31)IR+U3+=UR+R+,where(32)IR+=000-InUR+=u1+ξN-n+1,ωu2+ξN-n+1,ω⋯un+ξN-n+1,ωu1+ξN-n+2,ωu2+ξN-n+2,ω⋯un+ξN-n+2,ω⋮⋮⋮⋮u1+ξN,ωu2+ξN,ω⋯un+ξN,ω.Then, by virtue of boundary conditions of two kinds, continuity of displacements and stresses, among the boundaryL3, (22) and (26) and (28) and (31) are finally assembled into global BEM system, to obtain the scattering coefficients and displacements directly; thus(33)T11BT12BT13BT14B-G12B0-G14BA0B-A0B+T21BT22BT23BT24B-G22B0-G24BA3-B-A3-B+T31BT32BT33BT34B-G32B0-G34BA1B-A1B+T41BT42BT43BT44B-G42B0-G44BA3+B-A3+B+0T11A0T13AG11AT12AG13AA3-A-A3-A+0T21A0T23AG21AT22AG23AA5A-A5A+0T31A0T33AG31AT32AG33AA3+A-A3+A+0IR-00000UR-0000IR+0000UR+U0U3-U1U3+T3-U5T3+R-R+=G11BG12BG13BG14B00000G21BG22BG23BG24B00000G31BG32BG33BG34B00000G41BG42BG43BG44B00000000T13A0G12A000000T23A0G22A000000T33A0G32A000000000000000000000T00T100T5000,where(34)AB±=A0B±A3-B±A1B±A3+B±AA±=A3-A±A5A±A3+A±.


## 5. Numerical Results

In this section, some numerical examples are illustrated to show the validity and effectiveness of this modified BEM for 2D Love-wave model. In the following numerical examples, the material parameters of the layer and half-plane are dimensionless, which have a shear modulus ratio ofμB/μA=1.8 and a longitudinal wave velocity ratio of cTB/cTA=0.78, and the dimensionless frequency is taken as ω-=2ωH/cTAπ. The element size is selected to have at least 32 elements per Love wavelength λL, which provides accurate results for 2D elastodynamic problem.Firstly, the numerical results obtained by the modified BEM will be compared with theoretical far-field Green’s functions. As shown in Figure4, this numerical model is a 2D semifinite space with unit harmonic line source acting in x3 direction, with the distance d between source and lower interface of the upper-layer. The far-field amplitudes are presented in Table 1 for various frequencies ω- = 1.2, 6.5, 10.8, while a fixed height d=0.5H. The far-field coefficients of Love-waves are obtained by modified BEM and compared with theoretical results [1, 2]. The results are in excellent agreement (see Table 1), which show the validity of this modified BEM for a certain range of frequencies. From additional parametric study, it is found from Figure 5 that, as the source moved closer to the top surface, for example, d=0.1-0.9H, longer surface lengths should remain in the BEM model to ensure the accuracy, which should be kept in mind as a criterion for accurate calculations of these numerical results.Table 1
Comparisons with truncated locations atx-=±60λL/H (λL being the Love wavelength for the lowest mode) for the unit line source problem.
Frequencyω- Modal BEM results (far-field amplitudes) Theoretical far-field Green’s functions 1.2 Modal 1 0.13562i 0.13557i 6.5 Modal 1 0.01634i 0.01635i Modal 2 0.01335i 0.01335i 10.8 Modal 1 0.00709i 0.00705i Modal 2 0.00983i 0.00983i Modal 3 −0.0358i −0.0358iFigure 4
Schematic diagram for the unit line source problem.Figure 5
Proper truncation distance with variousd for the unit line source problem.Next, the lowest incident Love-wave mode for a fixed frequency is selected to impinge onto a cavity defect of arc surface on the bonding interface, with radiusR1=H and height h (see Figure 6). The transmission and reflection coefficients for each modal at various frequencies: ω-=0.8, 5, 9.5, are shown in Table 2. And the normalized displacements for the same frequency range which are here defined as Uscat±/∑i=1nRi±Ui± are plotted. It is observed from Figures 7(a)–7(c) that the scattered displacements are approximated by Love surface waves at far ends, which satisfy the assumptions of (6).Table 2
The transmission and reflection coefficients with truncated locations atx-=±60λL/H, h=0.2 (λL being the Love wavelength for the lowest mode), for a circle arc defect at the bonding interface.
Frequencyω- Modal Reflection coefficients Transmission coefficients 0.8 Modal 1 0.00039 − 0.07060i −0.00582 + 0.03457i 5 Modal 1 −0.04383 + 0.10690i 0.01638 + 0.02342i Modal 2 −0.11344 + 0.06911i 0.07724 + 0.11906i 9.5 Modal 1 −0.03315 + 0.08802i 0.00531 + 0.00383i Modal 2 −0.19330 + 0.36373i 0.04204 + 0.03600i Modal 3 −0.22045 − 0.05598i 0.07071 + 0.09606iFigure 6
Schematic diagram for Love-wave scattering problem: (a) a cavity defect of arc surface on the bonding interface, with radiusR1 and heighth; (b) a circle defect in half-plane with radius R2 and depthd.Figure 7
Normalized amplitudes of upper boundary due to a defect at the bonding interface: (a)ω-=0.8; (b) ω-=5; (c) ω-=9.5.
(a) (b) (c)For basic check purposes, propagations in the opposite directions for the same frequency range are considered, and numerical solutions show very good agreement in all cases owing to the symmetry of the defect. Furthermore, a parametric study has been carried out to analyze the influence of defect heighth = 0.2, 0.4, 0.6, 0.8, on the reflected and transmitted amplitudes which are defined as Aref=uscat-/u- and Atrans=uscat++uinc/u+. It is found from Figure 8 that as the defect becomes larger, the absolute value of the transmitted amplitude gradually decreases and the absolute value of the reflected amplitude is diverse.Figure 8
Reflected and transmitted amplitudes due to a defect at the bonding interface.Finally, we consider the lowest incident Love-wave mode for a fixed frequency impinging onto the circle defect in half-plane with radiusR2=0.2H and depth d=0.5H, in the positive direction of x1 (see Figure 6(b)). The transmission and reflection coefficients of various frequencies: ω-=0.8, 5, 9.5, are performed in Table 3. As the relative normalized displacements are plotted in Figure 9(a)–9(c), we could get the same conclusion that the scattered displacements are approximated by Love surface waves at far ends. Also, numerical results for propagation in opposite direction show very good agreement due to the symmetry of the defect.Table 3
The transmission and reflection coefficients with truncated locations atx- = ±60λL/H, R2=0.2, and d=0.5 (λL being the Love wavelength for the lowest mode) for a circle defect in half-plane.
Frequencyω- Modal Reflection coefficients Transmission coefficients 0.8 Modal 1 −0.00026 + 0.00746i −0.00026 + 0.00758i 5 Modal 1 −0.00004 + 0.00052i −0.00028 + 0.00202i Modal 2 0.00046 + 0.00494i −0.00190 + 0.01682i 9.5 Modal 1 0.00001 − 0.00001i −0.00002 + 0.00010i Modal 2 0.00003 − 0.00005i −0.00017 + 0.00090i Modal 3 0.00018 − 0.00027i −0.00060 + 0.00375iFigure 9
Normalized amplitudes of upper boundary due to a circle defect in half-plane: (a)ω-=0.8; (b) ω-=5; (c) ω-=9.5.
(a) (b) (c)


## 6. Conclusion

In this paper, we proposed a modified BEM for scattering problem of Love surface wave by a defect. The guided Love-wave displacement patterns are assumed on the far-field infinite boundaries previously omitted, and they are incorporated into the BEM system as the modified items. With this improvement, the spurious reflected waves were eliminated. The validity and effectiveness of this modified BEM were numerically checked by theoretical far-field Green’s functions. Various parametric results show that this method can be applied on the Love-wave model with a defect of arbitrary shape and location, and as the geometrical size of the defect becomes larger, the transmitted wave gradually decreases and the reflected wave is diverse.In the future, the scattering data from forward analysis by this modified BEM will be used for the inverse analysis of reconstructing both the location and specific geometric information of the debonding cavities.



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