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 哈尔滨工程大学学报  2019, Vol. 40 Issue (8): 1433-1439  DOI: 10.11990/jheu.201808067 0

### 引用本文

QI Hui, ZHANG Yang, CHEN Hongying. Scattering of SH waves with a partially degummed elliptical inclusion and a circular inclusion[J]. Journal of Harbin Engineering University, 2019, 40(8), 1433-1439. DOI: 10.11990/jheu.201808067.

### 文章历史

Scattering of SH waves with a partially degummed elliptical inclusion and a circular inclusion
QI Hui , ZHANG Yang , CHEN Hongying
College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: The complex function method, Green function method, and conformal mapping are combined to solve the problem of SH-wave scattering in a half-space with a partially degummed elliptical inclusion and circular inclusion. First, the elliptical inclusion is mapped into circular inclusions to derive the displacement and stress field of scattered waves in accordance with the principle of conformal mapping. The virtual point source of the degummed component is derived in accordance with the Green function method, and the basic solutions of displacement and stress are obtained under the continuous boundary conditions of peripheral stress and displacement with elliptical and circular inclusions. Furthermore, the degumming model is structured by applying forces that are equal in strength and opposite in direction on the degumming component to obtain the total displacement field of the SH-wave scattering in the half-space with partially degummed elliptical and circular inclusions. Numerical examples show that different medium parameters, including the SH-wave incident angle, incident frequency, wave-number ratio, medium parameter, embedded depth, distance between defects, and degumming angle, have different influences on the dynamic stress coefficient.
Keywords: SH-wave    degummed elliptical inclusion    circular inclusion    dynamic stress concentration factor    Green function    half-space    conformal mapping

1 理论模型

 Download: 图 1 半空间含有部分脱胶的椭圆夹杂及圆形夹杂的模型 Fig. 1 The model of elliptical inclusion with a partially debond curve and circular cavity in half space
 $\left\{ \begin{array}{l} {x_1} = x,{y_1} = y + {h_1}\\ {x_2} = x - d,{y_2} = y + {h_1} + {h_2} \end{array} \right.$ (1)

 $\frac{{{\partial ^2}G}}{{\partial {x^2}}} + \frac{{{\partial ^2}G}}{{\partial {y^2}}} + {k^2}G = 0$ (2)

 ${\tau _{rz}} = \frac{\mu }{{R\left| {\omega '\left( \eta \right)} \right|}}\left( {\eta \frac{{\partial G}}{{\partial \eta }} + \bar \eta \frac{{\partial G}}{{\partial \bar \eta }}} \right)$ (3)
 ${\tau _{\theta z}} = \frac{{{\rm{i}}\mu }}{{R\left| {\omega '\left( \eta \right)} \right|}}\left( {\eta \frac{{\partial G}}{{\partial \eta }} - \bar \eta \frac{{\partial G}}{{\partial \bar \eta }}} \right)$ (4)
2 散射波

 $W_{\rm{I}}^{\left( s \right)}\left( {z,\bar z} \right) = W_{{\rm{I}}C}^{\left( s \right)} + W_{{\rm{I}}T}^{\left( s \right)}$

 $\begin{array}{*{20}{c}} {W_{{\rm{IC}}}^{(s)} = \sum\limits_{n = - \infty }^{ + \infty } {{A_n}} \left\{ {H_n^{(1)}\left[ {{k_1}\left| {{z_1}} \right|} \right]{{\left[ {\frac{{{z_1}}}{{\left| {{z_1}} \right|}}} \right]}^n} + } \right.}\\ {H_n^{(1)}\left[ {{k_1}\left| {{z_1} - 2i{h_1}} \right|} \right]{{\left[ {\frac{{{z_1} - 2i{h_1}}}{{\left| {{z_1} - 2i{h_1}} \right|}}} \right]}^{ - n}}\} } \end{array}$ (5)

 $W_{{\rm{I}}T}^{(s)} = \sum\limits_{n = - \infty }^{ + \infty } {{B_n}} \left\{ {{s_1} + {s_2}} \right\}$ (6)

 $s_{2}=H_{n}^{(1)}\left[k_{1}\left|m_{1}-2 i h_{1}-i h_{2}\right|\right]\left[\frac{m_{1}-2 i h_{1}-i h_{2}}{\left|m_{1}-2 i h_{1}-i h_{2}\right|}\right]^{-n}$

AnBn是待定系数，椭圆夹杂产生的散射波式(5)代入式(3)、(4)可得椭圆夹杂的应力表达式：

 $\tau _{rz,{\rm{I}}C}^{\left( s \right)} = \frac{{{\mu _1}{k_1}}}{2}\sum\limits_{n = - \infty }^{ + \infty } {{A_n}\left[ {{\phi _1} - {\phi _2}} \right]}$ (7)
 $\tau _{\theta z,{\rm{I}}C}^{\left( s \right)} = \frac{{i{\mu _1}{k_1}}}{2}\sum\limits_{n = - \infty }^{ + \infty } {{A_n}\left[ {{\phi _1} + {\phi _2}} \right]}$ (8)
 $\begin{array}{*{20}{c}} {{\phi _1} = \left[ {H_{n - 1}^{\left( 1 \right)}\left[ {{k_1}\left| {{z_1}} \right|} \right]{{\left[ {\frac{{{z_1}}}{{\left| {{z_1}} \right|}}} \right]}^{n - 1}} - } \right.}\\ {\left. {H_{n + 1}^{\left( 1 \right)}\left[ {{k_1}\left| {{z_1} - 2i{h_1}} \right|} \right]{{\left[ {\frac{{{z_1} - 2i{h_1}}}{{\left| {{z_1}} \right| - 2i{h_1}}}} \right]}^{ - \left( {n + 1} \right)}}} \right]\frac{{\eta {{z_1}'}}}{{R\left| {{z_1}'} \right|}}} \end{array}$
 $\begin{array}{*{20}{c}} {{\phi _2} = \left[ {H_{n + 1}^{\left( 1 \right)}\left[ {{k_1}\left| {{z_1}} \right|} \right]{{\left[ {\frac{{{z_1}}}{{\left| {{z_1}} \right|}}} \right]}^{n + 1}} - } \right.}\\ {\left. {H_{n - 1}^{\left( 1 \right)}\left[ {{k_1}\left| {{z_1} - 2i{h_1}} \right|} \right]{{\left[ {\frac{{{z_1} - 2i{h_1}}}{{\left| {{z_1}} \right| - 2i{h_1}}}} \right]}^{ - \left( {n - 1} \right)}}} \right]\frac{{\bar \eta \overline {{z_1}^\prime }}}{{R\left| {{z_1}'} \right|}}} \end{array}$

 $\tau _{rz,{\rm{I}}T}^{\left( s \right)} = \frac{{{\mu _1}{k_1}}}{2}\sum\limits_{n = - \infty }^{ + \infty } {{B_n}\left[ {{\phi _3} - {\phi _4}} \right]\frac{{\eta {{z_1}'}}}{{R\left| {{z_1}'} \right|}}}$ (9)
 $\tau _{\theta z,{\rm{I}}T}^{\left( s \right)} = \frac{{i{\mu _1}{k_1}}}{2}\sum\limits_{n = - \infty }^{ + \infty } {{B_n}\left[ {{\phi _3} + {\phi _4}} \right]\frac{{\bar \eta \overline {{z_1}^\prime }}}{{R\left| {{z_1}'} \right|}}}$ (10)

 $\begin{array}{*{20}{c}} {{\phi _3} = \left[ {H_{n - 1}^{\left( 1 \right)}\left[ {{k_1}\left| {{m_1}} \right|} \right]{{\left[ {\frac{{{m_1} + i{h_2}}}{{\left| {{m_1} + i{h_2}} \right|}}} \right]}^{n - 1}} - } \right.}\\ {\left. {H_{n + 1}^{\left( 1 \right)}\left[ {{k_1}\left| {{m_1} - 2i{h_1} - i{h_2}} \right|} \right]{{\left[ {\frac{{{m_1} - 2i{h_1} - i{h_2}}}{{\left| {{m_1} - 2i{h_1} - i{h_2}} \right|}}} \right]}^{ - \left( {n + 1} \right)}}} \right]} \end{array}$
 $\begin{array}{*{20}{c}} {{\phi _4} = \left[ {H_{n + 1}^{\left( 1 \right)}\left[ {{k_1}\left| {{m_1}} \right|} \right]{{\left[ {\frac{{{m_1} - i{h_2}}}{{\left| {{m_1} - i{h_2}} \right|}}} \right]}^{n + 1}} - } \right.}\\ {\left. {H_{n - 1}^{\left( 1 \right)}\left[ {{k_1}\left| {{m_1} - 2i{h_1} - i{h_2}} \right|} \right]{{\left[ {\frac{{{m_1} - 2i{h_1} - i{h_2}}}{{\left| {{m_1} - 2i{h_1} - i{h_2}} \right|}}} \right]}^{ - \left( {n - 1} \right)}}} \right]} \end{array}$

 $\begin{array}{*{20}{c}} {W_{\Pi C}^{\left( s \right)} = \sum\limits_{n = - \infty }^{ + \infty } {{A_n}\left\{ {H_n^{\left( 1 \right)}\left[ {{k_1}\left| {{m_2} - i{h_2}} \right|} \right]{{\left[ {\frac{{{m_2} - i{h_2}}}{{\left| {{m_2} - i{h_2}} \right|}}} \right]}^n} + } \right.} }\\ {\left. {H_n^{\left( 1 \right)}\left[ {{k_1}\left| {{m_2} - 2i{h_2} - i{h_2}} \right|} \right]{{\left[ {\frac{{{m_2} - 2i{h_1} - i{h_2}}}{{\left| {{m_2} - 2i{h_1} - i{h_2}} \right|}}} \right]}^{ - n}}} \right\}} \end{array}$ (11)

 $\begin{array}{*{20}{c}} {W_{\Pi T}^{\left( s \right)} = \sum\limits_{n = - \infty }^{ + \infty } {{B_n}\left\{ {H_n^{\left( 1 \right)}\left[ {{k_1}\left| {{z_2}} \right|} \right]{{\left[ {\frac{{{z_2}}}{{\left| {{z_2}} \right|}}} \right]}^n} + } \right.} }\\ {\left. {H_n^{\left( 1 \right)}\left[ {{k_1}\left| {{z_2} - 2i\left( {{h_1} + {h_2}} \right)} \right|} \right]{{\left[ {\frac{{{z_2} - 2i\left( {{h_1} + {h_2}} \right)}}{{\left| {{z_2} - 2i\left( {{h_1} + {h_2}} \right)} \right|}}} \right]}^{ - n}}} \right\}} \end{array}$ (12)

 $\tau _{rz,\prod C }^{\left( s \right)} = \frac{{{k_1}{\mu _1}}}{2}\sum\limits_{n = - \infty }^\infty {{A_n}\left[ {{\varphi _1}{{\rm{e}}^{{\rm{i}}{\theta _1}}} - {\varphi _2}{{\rm{e}}^{{\rm{i}}{\theta _1}}}} \right]}$ (13)
 $\tau _{\theta z,\prod C }^{\left( s \right)} = \frac{{i{k_1}{\mu _1}}}{2}\sum\limits_{n = - \infty }^\infty {{A_n}\left[ {{\varphi _1}{{\rm{e}}^{{\rm{i}}{\theta _1}}} + {\varphi _2}{{\rm{e}}^{ - {\rm{i}}{\theta _1}}}} \right]}$ (14)

 $\begin{array}{*{20}{c}} {{\varphi _1} = H_{n - 1}^{\left( 1 \right)}\left( {{k_1}\left| {{m_2} - i{h_2}} \right|} \right){{\left[ {\frac{{{m_2} - i{h_2}}}{{\left| {{m_2} - i{h_2}} \right|}}} \right]}^{n - 1}} - }\\ {H_{n + 1}^{\left( 1 \right)}\left( {{k_1}\left| {{m_2} - 2i{h_1} - i{h_2}} \right|} \right){{\left[ {\frac{{{m_2} - 2i{h_1} - i{h_2}}}{{\left| {{m_2} - 2i{h_1} - i{h_2}} \right|}}} \right]}^{ - \left( {n + 1} \right)}}} \end{array}$
 $\begin{array}{*{20}{c}} {{\varphi _2} = H_{n + 1}^{\left( 1 \right)}\left( {{k_1}\left| {{m_2} - i{h_2}} \right|} \right){{\left[ {\frac{{{m_2} - i{h_2}}}{{\left| {{m_2} - i{h_2}} \right|}}} \right]}^{n + 1}} - }\\ {H_{n - 1}^{\left( 1 \right)}\left( {{k_1}\left| {{m_2} - 2i{h_1} - i{h_2}} \right|} \right){{\left[ {\frac{{{m_2} - 2i{h_1} - i{h_2}}}{{\left| {{m_2} - 2i{h_1} - i{h_2}} \right|}}} \right]}^{ - \left( {n - 1} \right)}}} \end{array}$

 $\tau _{rz,\prod T }^{(s)} = \frac{{{k_1}{\mu _1}}}{2}\sum\limits_{n = - \infty }^\infty {{B_n}} \left[ {{\varphi _3}{{\rm{e}}^{{\rm{i}}{\theta _1}}} - {\varphi _4}{{\rm{e}}^{ - {\rm{i}}{\theta _1}}}} \right]$ (15)
 $\tau _{\theta z,\prod T }^{(s)} = \frac{{i{k_1}{\mu _1}}}{2}\sum\limits_{n = - \infty }^\infty {{B_n}} \left[ {{\varphi _3}{{\rm{e}}^{{\rm{i}}{\theta _1}}} + {\varphi _4}{{\rm{e}}^{ - {\rm{i}}{\theta _1}}}} \right]$ (16)

 $\begin{array}{*{20}{c}} {{\varphi _3} = H_{n - 1}^{(1)}\left( {{k_1}\left| {{z_2}} \right|} \right){{\left[ {\frac{{{z_2}}}{{\left| {{z_2}} \right|}}} \right]}^{n - 1}} - }\\ {H_{n + 1}^{(1)}\left( {{k_1}\left| {{z_2} - 2i\left( {{h_1} + {h_2}} \right)} \right|} \right){{\left[ {\frac{{{z_2} - 2i\left( {{h_1} + {h_2}} \right)}}{{\left| {{z_2} - 2i\left( {{h_1} + {h_2}} \right)} \right|}}} \right]}^{ - (n + 1)}}} \end{array}$
 $\begin{array}{*{20}{c}} {{\varphi _4} = H_{n + 1}^{(1)}\left( {{k_1}\left| {{z_2}} \right|} \right){{\left[ {\frac{{{z_2}}}{{\left| {{z_2}} \right|}}} \right]}^{n + 1}} - }\\ {H_{n - 1}^{(1)}\left( {{k_1}\left| {{z_2} - 2i\left( {{h_1} + {h_2}} \right)} \right|} \right){{\left[ {\frac{{{z_2} - 2i\left( {{h_1} + {h_2}} \right)}}{{\left| {{z_2} - 2i\left( {{h_1} + {h_2}} \right)} \right|}}} \right]}^{ - (n - 1)}}} \end{array}$
3 椭圆夹杂及圆夹杂内的驻波

 $W_\Pi ^{\left( {st} \right)} = \sum\limits_{n = - \infty }^{ + \infty } {{C_n}\left\{ {{J_n}\left( {{k_2}\left| {{z_1}} \right|} \right){{\left[ {\frac{{{z_1}}}{{\left| {{z_1}} \right|}}} \right]}^n}} \right\}}$ (17)

 $\tau_{r z}^{(s t)}=\frac{\mu_{2} k_{2}}{2} \sum\limits_{n=-\infty}^{+\infty} C_{n}\left(\psi_{1}-\psi_{2}\right)$ (18)
 $\tau_{\theta z}^{(s t)}=\frac{i \mu_{2} k_{2}}{2} \sum\limits_{n=-\infty}^{+\infty} C_{n}\left(\psi_{1}+\psi_{2}\right)$ (19)

 $\psi_{1}=J_{n-1}\left[k_{2}\left|z_{1}\right|\right]\left[\frac{z_{1}}{\left|z_{1}\right|}\right]^{n-1} \frac{\eta {z_1}^{\prime}}{R\left|{z_1}^{\prime}\right|}$
 $\psi_{2}=J_{n+1}\left[k_{2}\left|z_{1}\right|\right]\left[\frac{z_{1}}{\left|z_{1}\right|}\right]^{n+1} \frac{\overline{\eta} \overline{{z_1}^{\prime}}}{R\left|{{z^\prime}_1}\right|}$

 $W_{\mathrm{O}}^{(\mathrm{sto})}=\sum\limits_{n=-\infty}^{+\infty} D_{n}\left\{J_{n}\left(k_{3}\left|z_{2}\right|\right)\left[\frac{z_{2}}{\left|z_{2}\right|}\right]^{n}\right\}$ (20)

 $\tau_{r z}^{(\mathrm{sto})}=\frac{\mu_{3} k_{3}}{2} \sum\limits_{n=-\infty}^{+\infty} D_{n}\left[\psi_{1} \mathrm{e}^{i \theta}-\psi_{2} \mathrm{e}^{-i \theta}\right]$ (21)
 $\tau_{\theta z}^{(\mathrm{sto})}=\frac{i \mu_{3} k_{3}}{2} \sum\limits_{n==-\infty}^{+\infty} D_{n}\left[\psi_{1} \mathrm{e}^{i \theta}+\psi_{2} \mathrm{e}^{-i \theta}\right]$ (22)

 $\psi_{1}=J_{n-1}\left[k_{3}\left|z_{2}\right|\right]\left[\frac{z_{2}}{\left|z_{2}\right|}\right]^{n-1}$
 $\psi_{2}=J_{n+1}\left[k_{3}\left|z_{2}\right|\right]\left[\frac{z_{2}}{\left|z_{2}\right|}\right]^{n+1}$

4 Green函数的构造

 Download: 图 2 SH波垂直入射时τθz*随着k1a的分布 Fig. 2 Distribution of τθz* with k1a by SH-wave vertically
 $\left\{\begin{array}{ll}{G_{1}=G_{\Pi}^{\mathrm{st}},} & {\tau_{r z, \mathrm{I}}=\tau_{r z, \Pi}^{\mathrm{st}}} \\ {G_{\Pi}=G_{\Pi}^{\mathrm{sto}},} & {\tau_{r z, \Pi}=\tau_{r z, \Pi}^{\mathrm{sto}}}\end{array}\right.$ (23)

 $G_{\mathrm{I}}^{(i)}=\frac{i}{4 \mu_{1}} H_{0}^{(1)}\left(k_{1}\left|z_{1}-z_{0}\right|\right)$ (24)

 $G_{\mathrm{I}}^{(r)}=\frac{i}{4 \mu_{1}} H_{0}^{(1)}\left(k_{1}\left|z_{1}-\overline{z_{0}}-2 i h_{1}\right|\right)$ (25)

 $\tau_{r z, \mathrm{I}}^{(i)}=-\left(i k_{1} / 8\right)\left(s_{1}+s_{2}\right)$ (26)
 $\tau_{\theta z, \mathrm{I}}^{(i)}=\left(k_{1} / 8\right)\left(s_{1}-s_{2}\right)$ (27)
 $\tau_{r z, \mathrm{I}}^{(r)}=-\left(i k_{1} / 8\right)\left(s_{3}+s_{4}\right)$ (28)
 $\tau_{\theta z, \mathrm{I}}^{(r)}=\left(k_{1} / 8\right)\left(s_{3}-s_{4}\right)$ (29)

 $s_{1}=H_{1}^{(1)}\left(k_{1}\left|z_{1}-z_{0}\right|\right) \frac{\overline{z_{1}}-\overline{z_{0}}}{\left|z_{1}-z_{0}\right|} \frac{\eta z_{1}^{\prime}}{R\left|z_{1}^{\prime}\right|}$
 $s_{2}=H_{1}^{(1)}\left(k_{1}\left|z_{1}-z_{0}\right|\right) \frac{z_{1}-z_{0}}{\left|z_{1}-z_{0}\right|} \frac{\overline{\eta} \overline{z_{1}^{\prime}}}{R\left|z_{1}^{\prime}\right|}$
 $s_{3}=H_{1}^{(1)}\left(k_{1}\left|z_{1}-\overline{z_{0}}-2 i h_{1}\right|\right) \frac{\overline{z_{1}}-z_{0}+2 i h_{1}}{\left|z_{1}-\overline{z_{0}}-2 i h_{1}\right|} \frac{\eta z_{1}^{\prime}}{R\left|z_{1}^{\prime}\right|}$
 $s_{4}=H_{1}^{(1)}\left(k_{1}\left|z_{1}-\overline{z_{0}}-2 i h_{1}\right|\right) \frac{z_{1}-\overline{z_{0}}-2 i h_{1}}{\left|z_{1}-\overline{z_{0}}-2 i h_{1}\right|} \frac{\overline{\eta} \overline{z_{1}^{\prime}}}{R\left|z_{1}^{\prime}\right|}$

 $\left\{ \begin{array}{l} G_{\rm{I}}^{(i)} + G_{\rm{I}}^{(r)} + G_{{\rm{I}}C}^{(s)} + G_{{\rm{I}}T}^{(s)} = W_\Pi ^{({\rm{st}})}\\ \tau _{rz,{\rm{I}}}^{(i)} + \tau _{rz,{\rm{I}}}^{(r)} + \tau _{rz,{\rm{I}}C}^{(s)} + \tau _{rz,{\rm{I}}T}^{(s)} = \tau _{rz,\Pi }^{({\rm{st}})}\\ {\bf{ \pmb{\mathsf{ τ}} }}_{rz,\Pi }^{(i)} + {\bf{ \pmb{\mathsf{ τ}} }}_{rz,\Pi }^{(r)} + {\bf{ \pmb{\mathsf{ τ}} }}_{rz,\Pi C}^{(s)} + {\bf{ \pmb{\mathsf{ τ}} }}_{rz,\Pi T}^{(s)} = {\bf{ \pmb{\mathsf{ τ}} }}_{rz,\Pi }^{({\rm{sto}})}\\ G_\Pi ^{(i)} + G_\Pi ^{(r)} + G_{\Pi C}^{(s)} + G_{\Pi T}^{(s)} = W_\Pi ^{({\rm{sto}})} \end{array} \right.$ (30)

 $\left\{ \begin{array}{l} \sum\limits_{n = - \infty }^\infty {{A_n}} \xi _n^{(11)} + {B_n}\xi _n^{(12)} + {C_n}\xi _n^{(13)} = {\xi ^{(1)}}\\ \sum\limits_{n = - \infty }^\infty {{A_n}} \xi _n^{(21)} + {B_n}\xi _n^{(22)} + {C_n}\xi _n^{(23)} = {\xi ^{(2)}}\\ \sum\limits_{n = - \infty }^\infty {{A_n}} \xi _n^{(31)} + {B_n}\xi _n^{(32)} + {D_n}\xi _n^{(34)} = {\xi ^{(3)}}\\ \sum\limits_{n = - \infty }^\infty {{A_n}} \xi _n^{(41)} + {B_n}\xi _n^{(42)} + {D_n}\xi _n^{(44)} = {\xi ^{(4)}} \end{array} \right.$ (31)

 $\begin{array}{*{20}{c}} {W_{\rm{I}}^{(i)} = {W_0}\exp \left\{ {\left( {{\rm{i}}{k_1}/2} \right)\left[ {(z - {\rm{i}}h){{\rm{e}}^{ - {\rm{i}}{\alpha _0}}} + } \right.} \right.}\\ {(\bar z + {\rm{i}}h){{\rm{e}}^{{\rm{i}}{\alpha _0}}}]\} } \end{array}$ (32)
 $\begin{array}{c}{W_{\mathrm{I}}^{(r)}=W_{0} \exp \left\{\left(\mathrm{i} k_{1} / 2\right)\left[(z-\mathrm{i} h) \mathrm{e}^{\mathrm{i} \alpha_{0}}+\right.\right.} \\ {(\overline{z}+\mathrm{i} h) \mathrm{e}^{-\mathrm{i} \alpha_{0}} ] \}}\end{array}$ (33)

 ${W_{\rm{I}}} = W_{\rm{I}}^{(i)} + W_{\rm{I}}^{(r)} + W_{{\rm{I}}C}^{(s)} + W_{{\rm{I}}T}^{(s)}$ (34)

 $\begin{array}{*{20}{c}} {W_{\rm{I}}^{(ct)} = W_{\rm{I}}^{(i)} + W_{\rm{I}}^{(r)} + W_{{\rm{I}}C}^{(s)} + W_{{\rm{I}}T}^{(s)} - }\\ {\int_{{\theta _1}}^{{\theta _2}} {{{\left. {{\tau _{\theta z,{\rm{I}}}}} \right|}_{\eta = {\eta _0}}}} {G_{\rm{I}}}\left( {\eta ,{\eta _0},\theta ,{\theta _0}} \right){\rm{d}}\theta } \end{array}$ (35)

 $\tau _{\theta z}^* = \left| {\tau _{\theta z,{\rm{I}}}^{(ct)}/{\tau _0}} \right|$ (36)

5 算例及分析

 Download: 图 3 SH波以不同的角度α入射时τθz*的分布 Fig. 3 Distribution of τθz* with different incident angles by SH-wave horizontally

 Download: 图 4 SH波水平入射随着k1*的变化τθz*的分布情况 Fig. 4 Distribution of τθz* with k1* by SH-wave horizontally

 Download: 图 5 SH波水平入射时τθz*随着k1*及μ1*的分布情况 Fig. 5 Distribution of τθz* with k1*andμ1*by SH-wave horizontally

 Download: 图 6 SH波水平入射时τθz*的分布 Fig. 6 Distribution of τθz* with h2/r by SH-wave horizontally

 Download: 图 7 SH波水平入射时随着θ1*的不同τθz*的分布情况 Fig. 7 Distribution of τθz* with θ1* by SH-wave horizontally
6 结论

1) 入射频率、入射波数比越大，动应力集中系数越大，在达到一定数值后趋于稳定，但变化更加复杂，容易出现明显的震荡。

2) 缺陷埋深越小，2个缺陷之间的距离越近，对动应力集中系数的影响越大。

3) 脱胶角度对浅埋缺陷的动应力影响较大，脱胶部分比未脱胶部分的动应力集中更大，应该引起重视。

4) 根据介质参数的不同，对动应力集中系数产生的不同影响，可以在工程实际中选择合适的材料进行施工或者结构设计。

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