半空间双相压电介质垂直边界附近圆孔对SH波的散射

张希萌; 齐辉; 项梦

doi:10.11883/1001-1455(2017)04-0591-09

摘要: 利用“Green函数法”和“镜像法”对垂直边界附近含圆孔的半空间双相压电介质对SH波的散射问题进行分析，得到其稳态解。利用镜像法得到满足水平边界应力自由与电位移自由的波函数解析表达式。根据垂直边界连续性条件，利用“契合法”建立第一类Fredholm型积分方程组，得到圆孔周边的动应力集中系数与电场强度集中系数解析表达式。数值算例分析了入射波频率、入射角度、介质参数等对动应力集中系数与电场强度集中系数的影响，并与已有文献进行比较。计算表明，高频SH波垂直入射危害较大。

关键词:

Abstract: The scattering of the SH-wave by a circular cavity near the vertical boundary in the piezoelectric bi-material half-space was analyzed using the Green function method and the mirror method to obtain the steady state response. The analytical expression of the wave function was obtained on the horizontal boundaries using the mirror method. This function was the stress-free and electric-displacement-free. According to the continuity condition on the vertical boundary, the first kind of Fredholm integral equations were established, thereby obtaining the analytical expression of the dynamic stress concentration factor and the electric field intensity concentration factor around the edge of the circular cavity by the conjunction method. The influence of the frequencies of the incident wave, the incident angle and the media parameter, etc., on the dynamic stress concentration factor and the electric field intensity concentration factor was examined and compared with existing literatures using calculating examples. The numerical results show that serious damage occurs when the high-frequency incident SH wave comes in vertically.

Key words:

half space /
piezoelectric bi-material /
circular cavity /
SH wave /
dynamic stress concentration factor /
electric field intensity concentration factor

压电介质具有机-电耦合效应，广泛应用于智能结构和传感器元件中，实现结构的自我诊断、自我修复等功能，因此其在未来航空航天飞行器设计中占有重要地位。由于加工工艺、环境变化等因素，复合材料会产生圆孔等缺陷，这些缺陷存在于界面附近(材料性质变化最剧烈)时，会引起材料失效、破坏等问题，例如压电元件在生产加工过程中形成的圆孔，其动应力集中问题比一般材料更复杂。许多学者对缺陷问题进行了研究并取得了丰富成果^[1-11]。近年来，舒小平等^[1]利用等效单层理论求解了正交压电复合材料层板在各类边界条件下的解析解；王永健等^[2]利用理论对各项同性压电双材料中椭圆圆孔孔边裂纹的反平面问题进行分析；C.F.Gao等^[3]利用复变函数法研究了压电介质中椭圆形孔洞进行了断裂力学分析；K.L.Lee等^[4]对压电介质中斜椭圆孔的断裂问题进行了分析；J.K.Du等^[5]利用波函数展开法对部分脱胶夹杂对反平面剪切波的散射问题进行了研究；W.J.Feng等^[6]利用奇异积分方程技术研究了压电材料中脱胶夹杂对SH波的散射问题；宋天舒等^[7-9]研究了全空间双相压电介质中水平边界附近圆孔的动力学问题。本文中，利用“Green函数法”和“镜像法”构造出满足水平边界应力与电位移自由、垂直边界连续性条件的波函数。根据直角域垂直边界上连续性条件，利用“契合法”建立第一类Fredholm型积分方程组并进行求解。通过具体算例和数值结果，讨论入射角度、入射频率、介质参数等对压电材料力学和电学性质的影响。

1. 问题的描述

如图 1所示，介质Ⅰ为含圆孔的直角域，其质量密度、弹性常数、压电系数和介电常数分别为ρ₁、c₄₄^Ⅰ、e₁₅^Ⅰ和κ₁₁^Ⅰ，其水平边界、垂直边界分别为Γ_H、Γ_V，圆孔中心位置与垂直边界Γ_V距离为d，与水平边界Γ_H距离为h，其边界为Γ_C；介质Ⅱ为无缺陷的直角域，其质量密度、弹性常数、压电系数和介电常数分别为ρ₂、c₄₄^Ⅱ、e₁₅^Ⅱ和κ₁₁^Ⅱ；圆孔内空气的压电常数和介电常数分别为e₁₅^c和κ₁₁^c。本文采用坐标变换法，建立坐标系xOy与x′O′y′，对应的复坐标系分别为η=x+iy=re^iθ与η′=x′+iy′=r′e^iθ′，两坐标系之间关系为：

图 1 含圆孔的半空间双相压电介质模型

Figure 1. Model of a piezoelectric bi-material in half space with a circular cavity

下载: 全尺寸图片幻灯片

${x^\prime } = x - d,\quad \;\;\;{y^\prime } = y - h$

(1)

2. Green函数

设z轴为压电材料的电极化方向，则反平面动力学问题的稳态控制方程(忽略时间因子e^-iωt)为：

$\left\{ \begin{array}{*{35}{l}} {{c}_{44}}\ {{\nabla }^{2}}w+{{e}_{15}}\ {{\nabla }^{2}}\varphi +\rho {{\omega }^{2}}w=0 \\ {{e}_{15}}\ {{\nabla }^{2}}w-{{\kappa }_{11}}\ {{\nabla }^{2}}\varphi =0 \\ \end{array} \right.$

(2)

式中：w和φ分别为压电材料的平面位移和电势，ω为SH波的圆频率。令φ =e₁₅(w+f)/κ₁₁，则式(1)可以简化为：

$\left\{\begin{array}{l}{\nabla^{2} w+k^{2} w=0} \\ {\nabla^{2} f=0}\end{array}\right.$

(3)

式中：波数 $k=\rho \omega^{2} / c^{*}, c^{*}=c_{44}+e_{15}^{2} / \kappa_{11}$ 。

利用复变函数法，令 $\eta =x+\text{i}y,\bar{\eta }=x-\text{i}y$ ，在复平面(η, η)中，控制方程转化为：

$\left\{ {\begin{array}{*{20}{l}} {\frac{{{\partial ^2}w}}{{\partial \eta \partial \bar \eta }} + \frac{1}{4}{k^2}w = 0}\\ {\frac{{{\partial ^2}f}}{{\partial {\eta ^\partial }\bar \eta }} = 0} \end{array}} \right.$

(4)

在复平面(η, )内采用极坐标系，令 $\eta =r{{\text{e}}^{\text{i}\theta }},\bar{\eta }=r{{\text{e}}^{-\text{i}\theta }}$ ，则本构方程为：

$\left\{ \begin{matrix} {{\tau }_{rz}}=\left( {{c}_{44}}+\frac{e_{15}^{2}}{{{\kappa }_{11}}} \right)\left( \frac{\partial w}{\partial \eta }{{\text{e}}^{\text{i}\theta }}+\frac{\partial w}{\partial \bar{\eta }}{{\text{e}}^{-\text{i}\theta }} \right)+\frac{e_{15}^{2}}{{{\kappa }_{11}}}\left( \frac{\partial f}{\partial \eta }{{\text{e}}^{\text{i}\theta }}+\frac{\partial f}{\partial \bar{\eta }}{{\text{e}}^{-\text{i}\theta }} \right)\ \ \ \ \\ {{\tau }_{\theta z}}=\text{i}\left( {{c}_{44}}+\frac{e_{15}^{2}}{{{\kappa }_{11}}} \right)\left( \frac{\partial \tau }{\partial \eta }{{\text{e}}^{\text{i}\theta }}-\frac{\partial \tau }{\partial \bar{\eta }}{{\text{e}}^{-\text{i}\theta }} \right)+\text{i}\frac{e_{15}^{2}}{{{\kappa }_{11}}}\left( \frac{\partial f}{\partial \eta }{{\text{e}}^{\text{i}\theta }}-\frac{\partial f}{\partial \bar{\eta }}{{\text{e}}^{-\text{i}\theta }} \right) \\ {{D}_{r}}=-{{e}_{15}}\left( \frac{\partial f}{\partial \eta }{{\text{e}}^{\text{i}\theta }}+\frac{\partial f}{\partial \bar{\eta }}{{\text{e}}^{-\text{i}\theta }} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ {{D}_{\theta }}=-\text{i}{{e}_{15}}\left( \frac{\partial f}{\partial \eta }{{\text{e}}^{\text{i}\theta }}-\frac{\partial f}{\partial \bar{\eta }}{{\text{e}}^{-\text{i}\theta }} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\ \right.$

(5)

式中：τ_rz和τ_θz分别的压电介质的径向和切向应力，D_r和D_θ分别为圆孔中电场的径向和切向电位移。直角域介质Ⅰ在线源荷载δ(η-η₀)作用下的模型如所示，其中 ${{\eta }_{0}}=d+\text{i}y(y\le h)$ ，表示某个位于介质Ⅰ垂直边界Γ_V上的点。

图 2 受线源荷载作用的直角域模型

Figure 2. Right-angle plane model impacted by a line source force

下载: 全尺寸图片幻灯片

直角域介质Ⅰ的边界条件可以表示为：

$\left\{ {\begin{array}{*{20}{l}} {{\mathit{\Gamma }_{\rm{H}}}:\;\;{{\left. {\tau _{yz}^{\rm{Ⅰ}}} \right|}_{y = h}} = 0, \quad {{\left. {D_y^{\rm{Ⅰ}}} \right|}_{y = h}} = 0}\\ {{\mathit{\Gamma }_{\rm{V}}}:\;\;{{\left. {\tau _{xz}^{\rm{Ⅰ}}} \right|}_{x = d}} = \delta \left( {\eta - {\eta _0}} \right)}\\ {{\mathit{\Gamma }_{\rm{C}}}:\;\;{{\left. {\tau _{rz}^{\rm{Ⅰ}}} \right|}_{r = a, - \pi \le \theta \le \pi }} = 0}\\ {{\mathit{\Gamma }_{\rm{C}}}:\;\;{{\left. {G_\varphi ^{\rm{Ⅰ}}} \right|}_{r = a, - \pi \le \theta \le \pi }} = {{\left. {G_\varphi ^{\rm{c}}} \right|}_{r = a, - \pi \le \theta \le \pi }}}\\ {{\mathit{\Gamma }_{\rm{C}}}:\;\;{{\left. {D_r^{\rm{Ⅰ}}} \right|}_{r = a, - \pi \le \theta \le \pi }} = {{\left. {D_r^{\rm{c}}} \right|}_{r = a, - \pi \le \theta \le \pi }}} \end{array}} \right.$

(6)

式中：G_w^Ⅰ和G_φ^Ⅰ分别表示介质Ⅰ中平面位移和电势的Green函数，上标“Ⅰ”表示与介质Ⅰ相关的物理量； $G_{\varphi}^{\mathit{\boldsymbol{c}}}$ 与 $D_{r}^{\mathit{\boldsymbol{c}}}$ 分别表示圆孔内的电势与电位移的Green函数，上标“c”表示圆孔内物理量。

求解线源荷载δ(η-η₀)产生的扰动可得入射波的位移Green函数G_wⁱⁿ。本文中利用“镜像法”构造满足水平垂直边界应力与电位移自由的入射波与散射波，其中与入射波相关的位移Green函数(G_wⁱⁿ)和电势Green函数(G_φⁱⁿ)表达式为：

$\left\{ {\begin{array}{*{20}{c}} {G_w^{{\rm{in}}} = \frac{{\rm{i}}}{{2c_{44}^{\rm{Ⅰ}}\left( {1 + {\lambda ^{\rm{Ⅰ}}}} \right)}}\left[ {H_0^{(1)}\left( {{k_1}\left| {\eta - {\eta _0}} \right|} \right) + H_0^{(1)}\left( {{k_1}\left| {\eta - {{\bar \eta }_0} - 2{\rm{i}}h} \right|} \right)} \right]}\\ {G_\varphi ^{{\rm{in}}} = \frac{{e_{15}^{\rm{Ⅰ}}}}{{\kappa _{11}^{\rm{Ⅰ}}}}G_w^{{\rm{in}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}} \right.$

(7)

式中： $\lambda^{\mathit{\boldsymbol{Ⅰ}}}=\left(e_{15}^{\mathit{\boldsymbol{Ⅰ}}}\right)^{2} /\left(c_{44}^{\mathit{\boldsymbol{Ⅰ}}} \kappa_{11}^{\mathit{\boldsymbol{Ⅰ}}}\right)$ 为量纲一压电参数，上标“in”表示与入射波相关；k₁为SH波在介质Ⅰ中的波数，H⁽¹⁾为第一类Hankel函数，其下标表示阶数。令上标“s”表示与散射波相关，则与散射波相关的位移Green函数(G_w^s)和电势Green函数(G_φ^s)表达式为：

$\left\{ \begin{matrix} G_{w}^{\text{s}}=\frac{\text{i}}{2c_{44}^{\text{Ⅰ}}\left( 1+{{\lambda }^{\text{Ⅰ}}} \right)}\sum\limits_{n=-\infty }^{+\infty }{{{A}_{n}}}\sum\limits_{j=1}^{4}{S_{n}^{(j)}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ G_{\varphi }^{\text{s}}=\frac{e_{15}^{\text{Ⅰ}}}{\kappa _{11}^{\text{Ⅰ}}}\left( G_{w}^{\text{s}}+{{f}^{\text{s}}} \right),\ \ \ \ \ \ {{f}^{\text{s}}}=\sum\limits_{n=1}^{+\infty }{\left[ {{B}_{n}}\sum\limits_{j=1}^{4}{\varphi _{1n}^{(j)}}+{{C}_{n}}\sum\limits_{j=1}^{4}{\varphi _{2n}^{(j)}} \right]}\ \ \\ \end{matrix} \right.$

(8)

式中：

$\begin{array}{l} S_n^{(1)} = H_n^{(1)}\left( {{k_1}|\eta |} \right){[\eta /|\eta |]^n}, \quad S_n^{(2)} = H_n^{(1)}\left( {{k_1}\left| {{\eta _1}} \right|} \right){\left[ {{\eta _1}/\left| {{\eta _1}} \right|} \right]^{ - n}}\\ S_n^{(3)} = {( - 1)^n}H_n^{(1)}\left( {{k_1}\left| {{\eta _z}} \right|} \right){\left[ {{\eta _z}/\left| {{\eta _z}} \right|} \right]^n}, \quad S_n^{(4)} = {( - 1)^n}H_n^{(1)}\left( {{k_1}\left| {{\eta _3}} \right|} \right){\left[ {{\eta _3}/\left| {{\eta _3}} \right|} \right]^{ - n}}\\ \varphi _{1n}^{(1)} = {\eta ^{ - n}}, \quad \;\;\;\;\;\;\varphi _{1n}^{(2)} = {(\bar \eta + 2{\rm{i}}h)^{ - n}}\\ \varphi _{1n}^{(3)} = {( - 1)^n}{(\bar \eta - 2d)^{ - n}}, \quad \varphi _{1n}^{(4)} = {( - 1)^n}{(\eta - 2d - 2{\rm{i}}h)^{ - n}}\\ \varphi _{2n}^{(1)} = {{\bar \eta }^{ - n}}, \quad \varphi _{2n}^{(n)} = {(\eta - 2{\rm{i}}h)^{ - n}}\\ \varphi _{2n}^{(3)} = {( - 1)^n}{(\eta - 2d)^{ - n}}, \quad \varphi _{2n}^{(4)} = {( - 1)^n}{(\bar \eta - 2d + 2{\rm{i}}h)^{ - n}}\\ {\eta _1} = \eta - 2{\rm{i}}h, \quad {\eta _2} = {\eta _1} - 2d, \quad {\eta _3} = \eta - 2d \end{array}$

(8)

根据以上结果，可以得到介质Ⅰ中位移Green函数G_w^Ⅰ与电势Green函数G_φ^Ⅰ，即

$G_w^{\rm{Ⅰ}} = G_w^{{\rm{in}}} + G_w^{\rm{s}}, \quad \;\;\;\;\;G_\varphi ^{\rm{Ⅰ}} = G_\varphi ^{{\rm{in}}} + G_\varphi ^{\rm{s}}$

(9)

对于圆孔内部可以形成电场，其电势Green函数(G_φ^c)的表达式为：

$G_\varphi ^{\rm{c}} = \frac{{e_{15}^{\rm{c}}}}{{\kappa _{11}^{\rm{c}}}}{f^{\rm{c}}}, \quad {f^{\rm{c}}} = {D_0} + \sum\limits_{n = 1}^{ + \infty } {\left( {{D_n}{\eta ^n} + {E_n}{{\bar \eta }^n}} \right)}$

(10)

利用边界条件式(6)建立关于A_n、B_n、C_n、D_n、E_n的方程组如下：

$\left\{ {\begin{array}{*{20}{c}} {{\xi ^{(1)}} = \sum\limits_{n = - \infty }^{ + \infty } {{A_n}} \xi _n^{(1)} + \sum\limits_{n = 1}^{ + \infty } {{B_n}} \xi _n^{(12)} + \sum\limits_{n = 1}^{ + \infty } {{C_n}} \xi _n^{(13)} + \sum\limits_{n = 0}^{ + \infty } {{D_n}} \xi _n^{(14)} + \sum\limits_{n = 1}^{ + \infty } {{E_n}} \xi _n^{(15)}}\\ {{\xi ^{(2)}} = \sum\limits_{n = - \infty }^{ + \infty } {{A_n}} \xi _n^{(2)} + \sum\limits_{n = 1}^{ + \infty } {{B_n}} \xi _n^{(21)} + \sum\limits_{n = 1}^{ + \infty } {{C_n}} \xi _n^{(22)} + \sum\limits_{n = 0}^{ + \infty } {{D_n}} \xi _n^{(24)} + \sum\limits_{n = 1}^{ + \infty } {{E_n}} \xi _n^{(25)}}\\ {{\xi ^{(3)}} = \sum\limits_{n = 1}^{ + \infty } {{B_n}} \xi _n^{(32)} + \sum\limits_{n = 1}^{ + \infty } {{C_n}} \xi _n^{(33)} + \sum\limits_{n = 0}^{ + \infty } {{D_n}} \xi _n^{(34)} + \sum\limits_{n = 1}^{ + \infty } {{E_n}} \xi _n^{(35)}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}} \right.$

(11)

式中：

$\hat{\xi }_{n}^{(11)}=\frac{\text{i}{{k}_{1}}}{4}\left[ \sum\limits_{j=1}^{4}{\chi _{n}^{(j)}}\exp (\text{i}\theta )+\sum\limits_{j=1}^{4}{\mathsf{\gamma }_{n}^{(j)}}\exp (-\text{i}\theta ) \right]$

$\xi _{n}^{(12)}=\frac{{{\left( e_{15}^{\text{Ⅰ}} \right)}^{2}}}{\kappa _{11}^{\text{Ⅰ}}}\left[ \sum\limits_{j=1}^{2}{\zeta _{n}^{(j)}}\exp (\text{i}\theta )+\sum\limits_{j=1}^{2}{\vartheta _{n}^{(j)}}\exp (-\text{i}\theta ) \right]$

$\xi _{n}^{(13)}=\frac{{{\left( e_{15}^{\text{Ⅰ}} \right)}^{2}}}{\kappa _{11}^{\text{Ⅰ}}}\left[ \sum\limits_{j=1}^{2}{v_{n}^{(j)}}\exp (\text{i}\theta )+\sum\limits_{j=1}^{2}{\psi _{n}^{(j)}}\exp (-\text{i}\theta ) \right]$

$\xi _n^{(14)} = - \frac{{{{\left( {e_{15}^{\rm{c}}} \right)}^2}}}{{\kappa _{11}^{\rm{c}}}}n{\eta ^n}\exp ({\rm{i}}\theta ), \quad \quad \;\;\;\;\hat \xi _n^{(15)} = - \frac{{{{\left( {e_{15}^{\rm{c}}} \right)}^2}}}{{\kappa _{11}^{\rm{c}}}}n{\bar \eta ^n}\exp ( - {\rm{i}}\theta )$

$\xi_{n}^{(21)}=\frac{e_{15}^{\mathit{\boldsymbol{Ⅰ}}} \mathit{{\rm{i}}}}{2 c_{44}^{\mathit{\boldsymbol{Ⅰ}}}\left(1+\lambda^{\mathit{\boldsymbol{Ⅰ}}}\right) \kappa_{11}^{\mathit{\boldsymbol{Ⅰ}}}} \sum\limits_{j=1}^{4} S_{n}^{(j)}, \quad \xi_{n}^{(22)}=\frac{e_{15}^{\mathit{\boldsymbol{Ⅰ}}}}{\kappa_{11}^{\mathit{\boldsymbol{Ⅰ}}}} \sum\limits_{j=1}^{4} \varphi_{1 n}^{(j)}$

$\xi_{n}^{(23)}=\frac{e_{15}^{Ⅰ}}{\kappa_{11}^{Ⅰ}} \sum\limits_{j=1}^{4} \varphi_{2 n}^{(j)}, \quad \xi_{n}^{(24)}=-\frac{e_{15}^{\rm{c}}}{\kappa_{11}^{\rm{c}}} \eta^{n}, \quad \xi_{n}^{(25)}=-\frac{e_{15}^{\rm{c}}}{\kappa_{11}^{\rm{c}}} \overline{\eta}^{n}$

$\xi_{n}^{(32)}=-e_{15}^{\mathit{\boldsymbol{Ⅰ}}}\left[\sum\limits_{j=1}^{2} \zeta_{n}^{(j)} \exp (\mathit{\rm{i}} \theta)+\sum\limits_{j=1}^{2} \vartheta_{n}^{(j)} \exp (-\mathit{\rm{i}} \theta)\right]$

$\hat{\xi}_{n}^{(33)}=-e_{15}^{\mathit{\boldsymbol{Ⅰ}}}\left[\sum\limits_{j=1}^{2} v_{n}^{(j)} \exp (\mathit{\rm{i}} \theta)+\sum\limits_{j=1}^{2} \psi_{n}^{(j)} \exp (-\mathit{\rm{i}} \theta)\right]$

$\xi_{n}^{(34)}=e_{15}^{\rm{c}} n \eta^{n-1} \exp (\mathit{\rm{i}} \theta), \quad \quad \xi_{n}^{(35)}=e_{15}^{\mathit{\rm{c}}} n \overline{\eta}^{n-1} \exp (-\mathit{\rm{i}} \theta)$

$\begin{array}{l} {\xi ^{(1)}} = - \frac{{{\rm{i}}{k_1}}}{4}\left\{ {\left[ {H_{ - 1}^{(1)}\left( {{k_1}\left| {\eta - {\eta _0}} \right|} \right)} \right.} \right.\frac{{\bar \eta - {{\bar \eta }_0}}}{{\left| {\eta - {\eta _0}} \right|}} + H_{ - 1}^{(1)}\left( {{k_1}\left| {\eta - {{\bar \eta }_0} - 2{\rm{i}}h} \right|} \right)\frac{{\bar \eta - {\eta _0} + 2{\rm{i}}h}}{{\left| {\eta - {{\bar \eta }_0} - 2{\rm{i}}h} \right|}}]{{\rm{e}}^{{\rm{i}}\theta }} + \\ \left[ {H_{ - 1}^{(1)}\left( {{k_1}\left| {\eta - {\eta _0}} \right|} \right)\frac{{\eta - {\eta _0}}}{{\left| {\eta - {\eta _0}} \right|}}} \right. + H_{ - 1}^{(1)}\left( {{k_1}\left| {\eta - {{\bar \eta }_0} - 2{\rm{i}}h} \right|} \right)\frac{{\eta - {{\bar \eta }_0} - 2{\rm{i}}h}}{{\left| {\eta - {{\bar \eta }_0} - 2{\rm{i}}h} \right|}}]{{\rm{e}}^{ - {\rm{i}}\theta }}\} \end{array}$

${\xi ^{(2)}} = - \frac{{{\rm{i}}e_{15}^{\rm{Ⅰ}}}}{{2{c_{44}}\left( {1 + {\lambda ^{\rm{Ⅰ}}}} \right)\kappa _{11}^{\rm{Ⅰ}}}}\left[ {H_0^{(1)}\left( {{k_1}\left| {\eta - {\eta _0}} \right|} \right)} \right. + H_0^{(1)}\left( {{k_1}\left| {\eta - {{\bar \eta }_0} - 2{\rm{i}}h} \right|} \right)], \quad \;\;\;\;{\xi ^{(3)}} = 0$

$\chi _n^{(1)} = H_{n - 1}^{(1)}\left( {{k_1}|\eta |} \right){[\eta /|\eta |]^{n - 1}}, \quad \chi _n^{(2)} = - H_{n + 1}^{(1)}\left( {{k_1}\left| {{\eta _1}} \right|} \right){\left[ {{\eta _1}/\left| {{\eta _1}} \right|} \right]^{ - n - 1}}$

$\chi _n^{(3)} = {( - 1)^n}H_{n - 1}^{(1)}\left( {{k_1}\left| {{\eta _2}} \right|} \right){\left[ {{\eta _2}/\left| {{\eta _2}} \right|} \right]^{n - 1}}, \quad \chi _n^{(4)} = - H_{n + 1}^{(1)}\left( {{k_1}\left| {{\eta _1}} \right|} \right){\left[ {{\eta _1}/\left| {{\eta _1}} \right|} \right]^{ - n - 1}}$

$\gamma_{n}^{(1)}=-H_{n+1}^{(1)}\left(k_{1}|\eta|\right)[\eta /|\eta|]^{n+1}, \quad \gamma_{n}^{(2)}=H_{n-1}^{(1)}\left(k_{1}\left|\eta_{1}\right|\right) \quad\left[\eta_{1} /\left|\eta_{1}\right|\right]^{-n+1}$

$\gamma_{n}^{(3)}=-(-1)^{n} H_{n+1}^{(1)}\left(k_{1}\left|\eta_{2}\right|\right)\left[\eta_{2} /\left|\eta_{2}\right|\right]^{n+1}, \quad \gamma_{n}^{(4)}=(-1)^{n} H_{n-1}^{(1)}\left(k_{1}\left|\eta_{3}\right|\right)\left[\eta_{3} /\left|\eta_{3}\right|\right]^{-n+1}$

$\zeta_{n}^{(1)}=-n \eta^{-n-1}, \quad \zeta_{n}^{(2)}=-(-1)^{n} n(\eta-2 d-2 \mathit{\rm{i}} h)^{-n-1}$

$\vartheta_{n}^{(1)}=-n(\overline{\eta}+2 \mathit{\rm{i}} h)^{-n-1}, \quad \vartheta_{n}^{(2)}=-(-1)^{n} n(\overline{\eta}-2 d)^{-n-1}$

$v_{n}^{(1)}=-n(\eta-2 \mathit{\rm{i}} h)^{-n-1}, \quad v_{n}^{(2)}=-n(-1)^{n}(\eta-2 d)^{-n-1}$

$\psi_{n}^{(1)}=-n \overline{\eta}^{-n-1}, \quad \psi_{n}^{(2)}=-(-1)^{n} n(\overline{\eta}-2 d+2 \mathit{\rm{i}} h)^{-n-1}$

将式(11)中各等式两端同时乘以e^-imθ(m=0, ±1, ±2, ±3…)，在边界Γ_C对(-π, π)区间积分，截取有限项，从而得到关于待定系数的线性方程组，求解即可得出A_n、B_n、C_n、D_n、E_n。

与介质Ⅰ类比，设k₂为SH波在介质Ⅱ中的波数，λ^Ⅱ为介质Ⅱ的量纲一压电参数，则其Green函数的表达式为：

$\left\{ {\begin{array}{*{20}{c}} {G_w^Ⅱ = \frac{\rm{i}}{{2c_{44}^Ⅱ\left( {1 + {\lambda ^Ⅱ}} \right)}}\left[ {H_0^{(1)}\left( {{k_2}\left| {\eta - {\eta _0}} \right|} \right) + H_0^{(1)}\left( {{k_2}\left| {\eta - {{\bar \eta }_0} - 2{\rm{i}}h} \right|} \right)} \right]}\\ {G_\varphi ^Ⅱ = \frac{{e_{15}^Ⅱ}}{{\kappa _{11}^Ⅱ}}G_w^Ⅱ\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} \end{array}} \right.$

(12)

3. SH波的散射

根据文献[9-11]中方法，由入射波、反射波、折射波、和散射波引起的压电材料位移函数wⁱⁿ、w^r、w^f、w^s及其激发的电势函数wⁱⁿ、w^r、w^f和w^s表达式分别为：

$\begin{array}{l} {w^{{\rm{in}}}} = {w_0}\exp \left\{ {\frac{{{\rm{i}}{k_1}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( { - {\rm{i}}{\alpha _0}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\} + {w_0}\exp \left\{ {\frac{{{\rm{i}}{k_1}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( {{\rm{i}}{\alpha _0}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\}\\ {\varphi ^{{\rm{in}}}} = {\varphi _0}\exp \left\{ {\frac{{{\rm{i}}{k_1}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( { - {\rm{i}}{\alpha _0}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\} + {\varphi _0}\exp \left\{ {\frac{{{\rm{i}}{k_1}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( {{\rm{i}}{\alpha _0}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\}\\ {w^{\rm{r}}} = {w_1}\exp \left\{ {\frac{{{\rm{i}}{k_1}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( { - {\rm{i}}{\beta _0}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\} + {w_1}\exp \left\{ {\frac{{{\rm{i}}{k_1}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( {{\rm{i}}{\beta _0}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\}\\ {\varphi ^{\rm{r}}} = {\varphi _1}\exp \left\{ {\frac{{{\rm{i}}{k_1}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( { - {\rm{i}}{\beta _0}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\} + {\varphi _1}\exp \left\{ {\frac{{{\rm{i}}{k_1}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( {{\rm{i}}{\beta _0}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\}\\ {w^{\rm{f}}} = {w_2}\exp \left\{ {\frac{{{\rm{i}}{k_2}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( { - {\rm{i}}{\alpha _2}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\} + {w_2}\exp \left\{ {\frac{{{\rm{i}}{k_2}}}{2}\left[ {(\eta - d - {\rm{i}}h)\exp \left( {{\rm{i}}{\alpha _2}} \right) + {\rm{c}}.{\rm{c}}} \right]} \right\}\\ {w^{\rm{s}}} = \frac{{\rm{i}}}{{2c_{44}^{\rm{Ⅰ}}\left( {1 + {\lambda ^Ⅰ}} \right)}}\sum\limits_{n = - \infty }^{ + \infty } {{K_n}} \sum\limits_{j = 1}^4 {S_n^{(j)}} , \quad {\varphi ^{\rm{s}}} = \frac{{e_{15}^{\rm{Ⅰ}}}}{{\kappa _{11}^{\rm{Ⅰ}}}}\left( {{w^{\rm{s}}} + f_w^{\rm{s}}} \right), \quad f_w^{\rm{s}} = \sum\limits_{n = 1}^\infty {\left[ {{P_n}\sum\limits_{j = 1}^4 {\varphi _{1n}^{(j)}} + {Q_n}\sum\limits_{j = 1}^4 {\varphi _{2n}^{(j)}} } \right]} \\ {\varphi ^{\rm{c}}} = \frac{{e_{15}^{\rm{c}}}}{{\kappa _{11}^{\rm{c}}}}f_w^{\rm{c}}, \quad f_w^{\rm{c}} = {S_0} + \sum\limits_{n = 1}^{ + \infty } {\left( {{S_n}{\eta ^n} + {T_n}{{\bar \eta }^n}} \right)} \end{array}$

(13)

式中：c.c表示取前一项的复共轭；β₀为反射角度，且β₀=π-α₀；α₀为入射角度；α₂为折射角度；w₀、w₁、w₂、φ₁、φ₂为常数，满足连续性条件：

${w_0} + {w_1} = {w_2}, \quad \;\;{\varphi _0} + {\varphi _1} = {\varphi _2}, \quad \;\;{k_1}\sin {\alpha _0} = {k_2}\sin {\alpha _2}$

(14)

待定系数K_n、P_n、Q_n、S_n、T_n可以根据边界条件(应力自由，电势和电位移连续)进行求解，与上节确定Green函数中系数的方法相同。

4. 契合法

如图 3所示，利用“契合法”将两直角域模型介质Ⅰ和介质Ⅱ在垂直边界上“契合”起来，形成半空间模型，其中坐标系x′O′y′与xOy的关系为η=η′+d+ih。为满足垂直边界上的连续性，根据文献[]中方法，在垂直边界Γ_V上施加一对反平面外力系 $f_{1}\left(r_{0}^{\prime}, \theta_{0}^{\prime}\right)$ 和 $f_{2}\left(r_{0}^{\prime}, \theta_{0}^{\prime}\right)$ 及一对平面内电场 $f_{3}\left(r_{0}^{\prime}, \theta_{0}^{\prime}\right)$ 和 $f_{4}\left(r_{0}^{\prime}, \theta_{0}^{\prime}\right)$ 。

图 3 含圆孔的半空间双相压电介质垂直界面的契合

Figure 3. Conjunction of piezoelectric bi-material vertical interface in half space with a circular cavity

下载: 全尺寸图片幻灯片

在介质Ⅰ中：

$\left\{ {\begin{array}{*{20}{c}} {{w^{\rm{Ⅰ}}} = {w^{{\rm{in}}}} + {w^{\rm{r}}} + {w^{\rm{s}}}}\\ {\tau _{\theta z}^{\rm{Ⅰ}} = \tau _{\theta z}^{{\rm{in}}} + \tau _{\theta z}^{\rm{r}} + \tau _{\theta z}^{\rm{s}}}\\ {{\varphi ^{\rm{Ⅰ}}} = {\varphi ^{{\rm{in}}}} + {\varphi ^{\rm{r}}} + {\varphi ^{\rm{s}}}}\\ {D_\theta ^{\rm{Ⅰ}} = D_\theta ^{{\rm{in}}} + D_\theta ^{\rm{r}} + D_\theta ^{\rm{s}}} \end{array}} \right.$

(15)

在介质Ⅱ中：

${w^{{\rm{Ⅱ}}}} = {w^{\rm{f}}}, \quad \tau _{\theta z}^{{\rm{Ⅱ}}} = \tau _{\theta z}^{\rm{f}}, \quad {\varphi ^Ⅱ} = {\varphi ^{\rm{f}}}, \quad D_\theta ^{{\rm{Ⅱ}}} = D_\theta ^{\rm{f}}$

(16)

在垂直边界 $\theta_{0}^{\prime}=-\pi / 2$ 上，由连续性条件可知：

$\left\{ {\begin{array}{*{20}{c}} {\tau _{\theta z}^{\rm{Ⅰ}}\sin \theta _0^\prime + {f_1}\left( {r_0^\prime , \theta _0^\prime } \right) = \tau _{\theta z}^Ⅱ\sin \theta _0^\prime + {f_2}\left( {r_0^\prime , \theta _0^\prime } \right), \quad {w^{\rm{Ⅰ}}} + {w^{{f_1}}} = {w^Ⅱ} + {w^{{f_2}}}}\\ {D_\theta ^{\rm{Ⅰ}}\sin \theta _0^\prime + {f_3}\left( {r_0^\prime , \theta _0^\prime } \right) = D_\theta ^Ⅱ\sin \theta _0^\prime + {f_4}\left( {r_0^\prime , \theta _0^\prime } \right), \quad {\varphi ^{\rm{Ⅰ}}} + {\varphi ^{{f_3}}} = {\varphi ^Ⅱ} + {\varphi ^{{f_4}}}} \end{array}} \right.$

(17)

式中：w^f₁和w^f₂分别为外力系 $f_{1}\left(r_{0}^{\prime}, \theta_{0}^{\prime}\right)$ 和 $f_{2}\left(r_{0}^{\prime}, \theta_{0}^{\prime}\right)$ 引起的位移，φ^f₃和φ^f₄分别为外电场 $f_{3}\left(r_{0}^{\prime}, \theta_{0}^{\prime}\right)$ 和 $f_{4}\left(r_{0}^{\prime}, \theta_{0}^{\prime}\right)$ 引起的电势。

垂直边界Γ_V上连续性条件为：

$\left\{ {\begin{array}{*{20}{c}} {{w^{{\rm{in}}}} + {w^{\rm{r}}} = {w^{\rm{f}}}, \quad \tau _{\theta z}^{{\rm{in}}} + \tau _{\theta z}^{\rm{r}} = \tau _{\theta z}^{\rm{f}}}\\ {{\varphi ^{{\rm{in}}}} + {\varphi ^{\rm{r}}} = {\varphi ^{\rm{f}}}, \quad D_\theta ^{{\rm{in}}} + D_\theta ^{\rm{r}} = D_\theta ^{\rm{f}}} \end{array}} \right.$

(18)

利用式(17)对式(16)进行简化，得到关于外力系的积分方程如下：

${f_1}\left( {r_0^\prime , \theta _0^\prime } \right) = {f_2}\left( {r_0^\prime , \theta _0^\prime } \right), \quad \int_0^\infty {{f_1}} \left( {r_0^\prime , \theta _0^\prime } \right)\left[ {G_w^{(1)}\left( {r_0^\prime , \theta _0^\prime ;r, \theta } \right) + G_w^{(2)}\left( {r_0^\prime , \theta _0^\prime ;r, \theta } \right)} \right]{\rm{d}}r_0^\prime = - {w^{\rm{s}}}$

(19)

${f_3}\left( {r_0^\prime , \theta _0^\prime } \right) = {f_4}\left( {r_0^\prime , \theta _0^\prime } \right), \quad \int_0^\infty {{f_3}} \left( {r_0^\prime , \theta _0^\prime } \right)\left[ {G_\varphi ^{(1)}\left( {r_0^\prime , \theta _0^\prime ;r, \theta } \right) + G_\varphi ^{(2)}\left( {r_0^\prime , \theta _0^\prime ;r, \theta } \right)} \right]{\rm{d}}r_0^\prime = - {\varphi ^{\rm{s}}}$

(20)

积分方程式(19)~(20)为含弱奇异性的第一类Fredholm型积分方程，可以采用直接离散法进行求解。

5. 动应力集中系数

根据文献[10]，在SH波作用下圆孔周边的环向剪切应力可以表示为：

${\tau _{\theta z}} = \tau _{\theta z}^{\rm{Ⅰ}} + {\rm{i}}c_{44}^{\rm{Ⅰ}}\int_0^\infty {{f_1}} \left( {\eta _0^\prime } \right)\left( {\frac{{\partial G_w^{\rm{Ⅰ}}}}{{\partial \eta }}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial G_w^{\rm{Ⅰ}}}}{{\partial \bar \eta }}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right){\rm{d}}\left| {\eta _0^\prime } \right| + {\rm{i}}e_{15}^{\rm{Ⅰ}}\int_0^\infty {{f_3}} \left( {\eta _0^\prime } \right)\left( {\frac{{\partial G_\varphi ^{\rm{Ⅰ}}}}{{\partial \eta }}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial G_\varphi ^{\rm{Ⅰ}}}}{{\partial \bar \eta }}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right){\rm{d}}\left| {\eta _0^\prime } \right|$

(21)

动应力集中系数τ_θz^*(dynamic stress concentration factor, DSCF)可表示为：

$\tau _{\theta z}^* = \left| {{\tau _{\theta z}}/{\tau _0}} \right|, \quad {\tau _0} = {\rm{i}}{k_1}\left[ {c_{44}^{\rm{Ⅰ}} + {{\left( {e_{15}^{\rm{Ⅰ}}} \right)}^2}/\kappa _{11}^{\rm{Ⅰ}}} \right]{w_0}$

(22)

6. 电场强度集中系数

根据文献[10]，在SH波作用下圆孔周边电场强度可以表示为：

${E_\theta } = E_\theta ^{\rm{Ⅰ}} - {\rm{i}}\int_0^\infty {{f_3}} \left( {\eta _0^\prime } \right)\left( {\frac{{\partial G_\varphi ^{\rm{Ⅰ}}}}{{\partial \eta }}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial G_\varphi ^{\rm{Ⅰ}}}}{{\partial \bar \eta }}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right){\rm{d}}\left| {\eta _0^\prime } \right|, \quad E_\theta ^{\rm{Ⅰ}}= - {\rm{i}}\left( {\frac{{\partial {\varphi ^{\rm{Ⅰ}}}}}{{\partial \eta }}{{\rm{e}}^{{\rm{i}}\theta }} - \frac{{\partial {\varphi ^{\rm{Ⅰ}}}}}{{\partial \bar \eta }}{{\rm{e}}^{ - {\rm{i}}\theta }}} \right)$

(23)

由此可得，电场强度集中系数E_θ^*(electric field intensity concentration factor, EFICF)的表达式为：

$E_\theta ^* = \frac{{{E_\theta }}}{{{E_0}}},\quad {E_0} = \frac{{{k_1}e_{15}^{\rm{Ⅰ}}{w_0}}}{{k_{11}^{\rm{Ⅰ}}}}$

(24)

7. 具体算例

当 $\lambda^{\mathit{\boldsymbol{Ⅰ}}}=\lambda^{\mathbb{Ⅱ}}=0, \quad c_{44}^{\mathit{\boldsymbol{Ⅰ}}}=c_{44}^{\mathbb{Ⅱ}}, k_{1}=k_{2}, \rho_{1}=\rho_{2}$ 时，本文模型退化为含圆孔的半空间弹性介质。当参数取值与文献[12]相同时，该模型中动应力集中系数τ_θz^*的分布情况如图 4(a)所示。对比可知，计算结果与文献[12]中结果吻合较好。当λ^Ⅰ=λ^Ⅱ=0，c₄₄^Ⅱ=0，k₂=0，ρ₂=0时，本文模型退化为含圆孔的直角域弹性介质。采用与文献[13]中相同的参数求解得到τ_θz^*的分布情况，如图 4(b)所示。对比可知，计算结果与文献[13]中结果吻合较好。因此本文所采用的计算方法是可行的。以下令k₁=k，构造量纲一参数k₁^*=k₂/k₁，h^*=h/a，d^*=d/a对计算模型进行分析，并设k₁^*=1，c₄₄^Ⅱ/c₄₄^Ⅰ=2，κ₁₁^Ⅰ/κ₁₁^c=1000，其中κ₁₁^c为圆孔内部空气的介电常数，a为圆孔半径。

图 4 方法验证(与文献[12-13]比较)

Figure 4. Vertification of the present method (Compared to reference [12-13])

下载: 全尺寸图片幻灯片

图 5给出了低频SH波以不同角度入射时圆孔周边动应力集中系数的分布情况。图 5显示：SH波水平入射时，τ_θz^*最大值分布在圆孔上、下两侧；垂直入射时，τ_θz^*最大值分布在圆孔左、右两侧。当SH波垂直入射时，τ_θz^*最大值为2(θ=0°)，比水平入射时τ_θz^*的最大值1.67(θ=73°)提高约19.7%，可见入射角度对τ_θz^*存在影响。

图 5 SH波水平入射时圆孔周边动应力集中系数随λ^Ⅰ的分布

Figure 5. DSCF around circular cavity edge vs. λ^Ⅰ by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 6给出了SH波水平入射时圆孔周边动应力集中系数随参数ka的变化情况。由图 6可知：当ka=0.1时，τ_θz^*的最大值为1.67(θ=73°)；当ka=2时，τ_θz^*最大值为2(θ=90°)，提高了约19.7%。因此ka对τ_θz^*影响显著。综合图 5和图 6结果可知，高频SH波垂直入射对τ_θz^*的影响较大。

图 6 SH波水平入射时圆孔周边动应力集中系数随参数ka的分布

Figure 6. DSCF around circular cavity edge vs. ka by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 7给出了SH波水平入射时圆孔周边动应力集中系数随λ^Ⅰ分布情况。由图 7可知：当λ^Ⅰ=0.3时，τ_θz^*最大值为1.2(θ=90°)；当λ^Ⅰ=1时，τ_θz^*最大值为2(θ=90°)，约为前者的1.6倍。

图 7 SH波水平入射时圆孔周边动应力集中系数随λ^Ⅰ的分布

Figure 7. DSCF around circular cavity edge vs. λ^Ⅰ by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 8给出了SH波水平入射时圆孔周边动应力集中系数随λ^Ⅱ的变化情况。由图 8可知：当λ^Ⅱ=1时，τ_θz^*最大值为5.2(θ=17°)；λ^Ⅱ=0.3或0.5时，τ_θz^*的分布基本一致，最大值为2(θ=90°)，约为前者的38%。由此可见，λ^Ⅱ对τ_θz^*的影响比λ^Ⅰ更显著。

图 8 SH波水平入射时圆孔周边动应力集中系数随λ^Ⅱ的分布

Figure 8. DSCF around circular cavity edge vs. λ^Ⅱ by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 9给出了SH波水平入射时圆孔θ=-π处动应力集中系数随ka的变化情况。由图 9可知：τ_θz^*随ka增大振荡变化；当0≤ka < 1.1时，λ^Ⅰ=0.3对应的τ_θz^*比较大；当1.1≤ka < 2时，λ^Ⅰ=1对应的τ_θz^*较大；当λ^Ⅰ=1、ka=1.9时，τ_θz^*达到最大值，约为2.46。由此可见，当参数ka相同时，λ^Ⅰ对τ_θz^*的分布存在影响。

图 9 SH波水平入射时圆孔周边应力集中系数随参数ka的变化

Figure 9. DSCF around circular cavity edge vs. ka by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 10给出了SH波以不同角度入射时圆孔周边电场强度系数的分布情况。由图 10可知：SH波水平入射时，E_θ^*最大值分布在圆孔上下两侧；垂直入射时，E_θ^*最大值分布在圆孔左右两侧，与图 5中τ_θz^*的分布趋势一致。SH波水平入射时，E_θ^*的最大值为0.81(θ=-108°)；垂直入射时，E_θ^*的最大值为2(θ=0°)，约为前者的2.4倍。由此可见，入射角度对E_θ^*存在影响。

图 10 SH波以不同角度入射时圆孔周边电场强度集中系数的分布

Figure 10. EFICF around circular cavity edge by SH-wave with different incident angles

下载: 全尺寸图片幻灯片

图 11给出了SH波水平入射时圆孔周边电场强度系数随参数ka的变化情况。由图 11可知，ka对E_θ^*影响显著。当ka=0.1时，E_θ^*最大值为0.81(θ=－108°)；当ka=2时，E_θ^*最大值为2(θ=90°)，约为前者的2.4倍。综合图 10和图 11结果可知，高频SH波垂直入射对E_θ^*影响较大。

图 11 SH波水平入射时圆孔周边电场强度集中系数随参数ka的分布

Figure 11. EFICF around circular cavity edge vs.ka by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 12给出了θ=-π处SH波水平入射时电场强度系数随ka的变化情况。由图 12可知：E_θ^*随着ka的增大振荡变化；当0≤ka < 1.7时，λ^Ⅰ=0.3对应的E_θ^*较大；当1.7≤ka < 2时，λ^Ⅰ=1对应的E_θ^*较大；当ka=1.9、λ^Ⅰ=1时，E_θ^*达到最大值，约为2.95。由此可见，在参数ka相同的情况下，λ^Ⅰ对E_θ^*的分布存在一定的影响。

图 12 SH波水平入射时电场强度集中系数随参数ka的分布

Figure 12. Variation of EFICF around circular cavity edge vs.ka by horizontal SH-wave

下载: 全尺寸图片幻灯片

8. 结论

利用Green函数法、“镜像法”和“契合法”对半空间双压电介质垂直边界附近圆孔对SH波的散射进行分析研究。计算结果表明：入射角度、入射波频率、量纲一压电参数对圆孔周边的动应力强度系数与电场强度集中系数存在影响，且高频SH波垂直入射对压电材料的危害较大；随着入射波频率的增加，圆孔周边θ=-π处的动应力集中系数与电场强度集中系数均随着ka的增大而振荡变化。该结果为压电元件的设计制造及工程应用提供有益的参考

图 1 含圆孔的半空间双相压电介质模型

Figure 1. Model of a piezoelectric bi-material in half space with a circular cavity

下载: 全尺寸图片幻灯片

图 2 受线源荷载作用的直角域模型

Figure 2. Right-angle plane model impacted by a line source force

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图 3 含圆孔的半空间双相压电介质垂直界面的契合

Figure 3. Conjunction of piezoelectric bi-material vertical interface in half space with a circular cavity

下载: 全尺寸图片幻灯片

图 4 方法验证(与文献[12-13]比较)

Figure 4. Vertification of the present method (Compared to reference [12-13])

下载: 全尺寸图片幻灯片

图 5 SH波水平入射时圆孔周边动应力集中系数随λ^Ⅰ的分布

Figure 5. DSCF around circular cavity edge vs. λ^Ⅰ by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 6 SH波水平入射时圆孔周边动应力集中系数随参数ka的分布

Figure 6. DSCF around circular cavity edge vs. ka by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 7 SH波水平入射时圆孔周边动应力集中系数随λ^Ⅰ的分布

Figure 7. DSCF around circular cavity edge vs. λ^Ⅰ by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 8 SH波水平入射时圆孔周边动应力集中系数随λ^Ⅱ的分布

Figure 8. DSCF around circular cavity edge vs. λ^Ⅱ by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 9 SH波水平入射时圆孔周边应力集中系数随参数ka的变化

Figure 9. DSCF around circular cavity edge vs. ka by horizontal SH-wave

下载: 全尺寸图片幻灯片

图 10 SH波以不同角度入射时圆孔周边电场强度集中系数的分布

Figure 10. EFICF around circular cavity edge by SH-wave with different incident angles

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图 11 SH波水平入射时圆孔周边电场强度集中系数随参数ka的分布

Figure 11. EFICF around circular cavity edge vs.ka by horizontal SH-wave

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图 12 SH波水平入射时电场强度集中系数随参数ka的分布

Figure 12. Variation of EFICF around circular cavity edge vs.ka by horizontal SH-wave

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