張希萌，齊 輝，孫學良

(哈爾濱工程大學航天與建筑工程學院，黑龍江 哈爾濱 150001)

徑向非均勻壓電介質(zhì)中圓孔對SH波的散射*

張希萌，齊 輝，孫學良

(哈爾濱工程大學航天與建筑工程學院，黑龍江 哈爾濱 150001)

利用復變函數(shù)理論對SH波作用下含圓孔徑向非均勻壓電介質(zhì)的反平面動力特性進行了研究。壓電介質(zhì)的密度沿徑向按冪函數(shù)形式變化，但壓電參數(shù)、彈性參數(shù)、介電參數(shù)均為常數(shù)。利用變量替換法將非均勻壓電介質(zhì)的變系數(shù)波動方程組轉(zhuǎn)化為標準的Helmholtz方程組，得到了滿足邊界條件的波函數(shù)解析表達式。通過數(shù)值算例分析了入射角度、入射波頻率、冪次等對應(yīng)力集中系數(shù)和電場強度集中系數(shù)的影響，并與已有文獻進行比較。結(jié)果表明，某些參數(shù)組合下，動應(yīng)力集中系數(shù)與電場強度集中系數(shù)均隨冪次增大而增大。

徑向非均勻壓電介質(zhì)；反平面動力特征；SH波；動應(yīng)力集中系數(shù)；電場強度集中系數(shù)

壓電材料可以制造成執(zhí)行器或傳感器等智能元件，廣泛應(yīng)用于國防工業(yè)與實際生活中。由于壓電材料中力學與電學性質(zhì)相互耦合，在SH波作用下壓電材料中夾雜或圓孔等缺陷處的動應(yīng)力集中及電場強度集中問題也比一般材料更復雜。近年來，許多學者對缺陷問題進行了研究，并取得了豐富的成果[1-12]。X.F.Li等[1]基于電磁材料彈性理論研究了徑向非均勻性的壓電壓磁球殼的靜態(tài)響應(yīng)問題；時朋朋等[2]利用分離變量法和Hilbert核奇異積分方程理論研究了功能梯度壓電壓磁雙材料的周期界面裂紋問題；靳靜等[3]利用積分變換法和奇異積分方程技術(shù)研究了壓電壓磁雙材料界面裂紋的二維斷裂問；舒小平[4-5]基于等效單層理論的位移場和電勢場求解了正交壓電復合材料層板在各類邊界條件下的解析解；宋天舒等[6-7]研究了雙相壓電介質(zhì)中圓孔與界面裂紋相互作用的動力學問題。但是，以上工作中大部分是關(guān)于徑向非均勻介質(zhì)的靜態(tài)響應(yīng)問題的求解，對含圓孔的壓電介質(zhì)在SH波作用下的動態(tài)響應(yīng)問題，目前仍未見報道。

1 控制方程

圖1 含圓孔徑向非均勻壓電介質(zhì)模型Fig.1 Model of the radial inhomogeneouspiezoelectric medium with a circular cavity

(1)

式中：w和φ分別為壓電材料的位移和電勢，ω為SH波的圓頻率。令φ=e15(w+f)/κ11，對式(1)化簡得：

(2)

波數(shù)滿足：

(3)

(4)

(5)

本構(gòu)方程為：

(6)

式中：τrz和τθz分別為非均勻壓電介質(zhì)的徑向應(yīng)力和切向應(yīng)力，Dr和Dθ分別為圓孔中電場的徑向電位移和切向電位移。

2 介質(zhì)中的位移場

SH波散射過程中，入射波引起的壓電材料位移win表達式為：

(7)

散射波引起的壓電材料位移ws表達式為：

(8)

(9)

散射波引起的電場附加函數(shù)fs表達式為：

(10)

式中：Bn和Cn為系數(shù)。由此得到：

(11)

式中：上標“in”、“s”分別表示物理量與入射波、反射波相關(guān)。圓孔內(nèi)部存在電場，滿足方程：

(12)

式中：fc為圓孔內(nèi)部的電場附加函數(shù)。求解式(12)可得：

(13)

式中：Dn和En為系數(shù)。由此可得：

(14)

式中：上標“c”表示物理量與圓孔中空氣形成的電場相關(guān)。

3 邊界條件與定解方程

圓孔處的邊界條件為：

(15)

利用以上邊界條件式(15)建立關(guān)于An、Bn、Cn、Dn、En的方程組：

(16)

式中：

(17)

將式(16)取有限截斷項，等式兩邊同時乘以e-imθ(m=0,±1,±2,±3,…)，從(-π,π)進行積分得到多元一次方程組，從而求解出未知系數(shù)An、Bn、Cn、Dn、En。

4 動應(yīng)力集中系數(shù)與電場強度系數(shù)

(18)

式中：

5 算例分析

圖2 方法驗證(與文獻[7]比較)Fig.2 Verification of the present method(compared with reference [7])

圖3 SH波入射角度不同時動應(yīng)力集中系數(shù)的變化Fig.3 Varition of DSCF around the circular cavity edge by SH-wave with different incident angles

圖4 SH波水平入射時圓孔周邊動應(yīng)力集中系數(shù)隨波數(shù)ka的變化情況Fig.4 DSCF around circular cavity edge vs.ka by horizontal SH-wave

圖5 SH波垂直入射時圓孔周邊動應(yīng)力集中系數(shù)隨波數(shù)ka的變化情況Fig.5 DSCF around circular cavity edge vs.ka by vertical SH-wave

圖6 SH波水平入射時圓孔周邊動應(yīng)力集中系數(shù)隨λ變化情況Fig.6 DSCF around circular cavity edge vs. λ by horizontal SH-wave

圖7 SH波水平入射時圓孔周邊動應(yīng)力集中系數(shù)隨冪次β的變化情況Fig.7 DSCF around circular cavity edge vs. β by horizontal SH-wave

圖8 SH波垂直入射時圓孔θ=π/2處動應(yīng)力集中系數(shù)隨ka的變化Fig.8 DSCF around circular cavity edge vs. ka by vertical SH-wave

圖9 SH波以不同角度入射時圓孔周邊電場強度系數(shù)的變化情況Fig.9 Variation of EFICF around circular cavity edge by SH-wave with different incident angles

圖10 SH波水平入射時圓孔周邊電場強度系數(shù)隨λ的變化情況Fig.10 EFICF around circular cavity edge vs. λ by horizontal SH-wave

圖11 SH波水平入射時圓孔周邊電場強度系數(shù)隨β變化情況Fig.11 EFICF around circular cavity edge vs. β by horizontal SH-wave

圖12 SH波水平入射時圓孔θ=π/2處電場強度系數(shù)隨波數(shù)ka變化情況Fig.12 EFICF at the circular cavity edge vs. ka by horizontal SH-wave

6 結(jié) 論

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[5] 舒小平.正交壓電復合材料層板各類邊界的解析解[J].工程力學,2013,30(10):288-295. Shu Xiaoping. Analytical solutions of cross-ply piezoelectric composite laminates with various boundary conditions[J]. Engineering Mechanics, 2013,30(10):288-295.

[6] Hassan A, Song T S. Dynamic anti-plane analysis for two symmetrically interfacial cracks near circular cavity in piezoelectric bi-materials[J]. Applied Mathematics and Mechanics, 2014,35(10):1261-1270.

[7] 宋天舒,劉殿魁,于新華.SH波在壓電材料中的散射和動應(yīng)力集中[J].哈爾濱工程大學學報,2002,23(1):120-123. Song Tianshu, Liu Diankui, Yu Xinhua. Scattering of SH-wave and dynamic stress concentration in a piezoelectric medium with a circular hole[J]. Journal of Harbin Engineering University, 2002,23(1):120-123.

[8] 楊在林,黑寶平,楊欽友.徑向非均勻介質(zhì)中圓形夾雜的動應(yīng)力分析[J].力學學報,2015,47(3):539-543. Yang Zailin, Hei Baoping, Yang Qinyou. Dynamic analysis on a circular inclusion in a radially inhomogeneous medium[J]. Chinese Journal of Theoretical and Applied Mechanics, 2015,47(3):539-543.

[9] 齊輝,楊杰.SH波入射雙相介質(zhì)半空間淺埋任意位置圓形夾雜的動力分析[J].工程力學,2012,29(7):320-327. Qi Hui, Yang Jie. Dynamic analysis for shallowly buried circular inclusions of arbitrary positions impacted by SH-wave in bi-material half space[J]. Engineering Mechanics, 2012,29(7):320-327.

[10] 林宏,劉殿魁.半無限空間中圓形孔洞周圍SH波的散射[J].地震工程與工程振動,2002,22(2):9-16. Lin Hong, Liu Diankui. Scattering of SH-wave around a circular cavity in half space[J]. Earthquake Engineering and Engineering Vibration, 2002,22(2):9-16.

[11] 丁曉浩,齊輝,趙元博.含有直線裂紋的直角域中橢圓形夾雜對SH波的散射[J].天津大學學報,2016,49(4):415-421. Ding Xiaohao, Qi Hui, Zhao Yuanbo. Scattering of SH-wave by elliptic inclusion in right-angle plane with beeline crack[J]. Journal of Tianjin University, 2016,49(4):415-421.

[12] 李冬,宋天舒.含圓孔直角域壓電介質(zhì)的動力反平面特性[J].哈爾濱工程大學學報,2010,31(12):1606-1612. Li Dong, Song Tianshu. Dynamic anti-plane behavior for a quarter-infinite piezoelectric medium with a subsurface circular cavity[J]. Journal of Harbin Engineering University, 2010,31(12):1606-1612.

(責任編輯 王玉鋒)

Scattering of SH-wave by a circular cavity in radial inhomogeneous piezoelectric medium

Zhang Ximeng, Qi Hui, Sun Xueliang

(CollegeofAerospaceandCivilEngineering,HarbinEngineeringUniversity,Harbin150001,Heilongjiang,China)

The dynamic anti-plane behavior of the radial inhomogeneous piezoelectric medium with a circular cavity under the SH-wave was investigated using the complex function theory. It was assumed that the density of the piezoelectric medium varied as a power-law function on the radial distance but the elastic parameters, the piezoelectric parameters, and the dielectric parameters all remained as constants. The wave equations of the inhomogeneous piezoelectric medium were converted to the standard Helmholtz equations by variable substitution and the analytical expression of the wave function satisfying the boundary condition was obtained. The influence of the incident angle, the frequency of incident wave and the power of the power-law function, etc. on the dynamic stress concentration factor and electric field intensity concentration factor was analyzed and compared with the existing references in the calculated example. The numerical results show that the values of the dynamic stress concentration factor and the electric field intensity concentration factor increase as the power increases with combination of certain parameters.

radial inhomogeneous piezoelectric medium; dynamic anti-plane behavior; SH wave; dynamic stress concentration factor; electric field intensity concentration factor

10.11883/1001-1455(2017)03-0464-07

2015-11-23；

2016-06-24

黑龍江自然科學基金項目(A201404)

張希萌(1989- )，男，博士研究生； 通信作者： 齊 輝，qihui205@sina.com。

O343.4 國標學科代碼： 13015

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