Exact solutions for anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material

Based on the complex variable method and the technique of conformal mapping, the anti-plane problem of two asymmetrical edge cracks emanating from an elliptical hole in a piezoelectric material is studied. The exact solutions of field intensity factors and energy release rate are presented in closed-form with the assumption that the surfaces of the cracks and the elliptical hole are electrically impermeable. With the variation of the hole-size and the crack length, the present results can be reduced to the cases of two symmetrical edge cracks and a single edge crack emanating from a circular hole given by Wang and Gao [Wang, Y.J., Gao, C.F., 2008. The mode III cracks originating from the edge of a circular hole in a piezoelectric solid. International Journal of Solids and Structures 45, 4590–4599]. Moreover, new models used for simulating more practical defects in a piezoelectric solid are obtained, such as two symmetrical edge cracks and a single edge crack emanating from an elliptical hole, two asymmetrical edge cracks emanating from a circular hole, T-shaped crack, cross-shaped crack and semi-infinite plane with an edge crack. Numerical examples are then conducted to reveal the effects of the hole-size and the crack length on the field intensity factors and the energy release rate.     

The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material

In this paper, a general and simple way was found to solve the problem of an arbitrary hole with edge cracks in transversely isotropic piezoelectric materials based on the complex variable method and the method of numerical conformal mapping. Firstly, the approximate mapping function which maps the outside of the arbitrary hole and the cracks into the outside of a circular hole is derived after a series of conformal mapping process. Secondly, based on the assumption that the surface of the cracks and hole is electrically impermeable and traction-free, the approximate expressions for the complex potential, fields intensity factors and energy release rates are presented, respectively. Thirdly, under the in-plane electric loading together with the out-plane mechanical loading, the influences of the hole size, crack length and mechanical/electric loading on the fields intensity factors and energy release rates are analyzed. Finally, some particular holes with edge cracks are studied in numerical analysis. The result shows that, the mechanical loading always promotes crack growth, while the electric loading may retard crack growth.     

压电体中考虑表面效应时开裂孔洞的断裂特征

研究了反平面机械载荷和面内电载荷作用下压电体中考虑表面效应时孔边双裂纹问题的断裂特征。基于Gurtin-Murdoch表面理论模型，通过构造映射函数，利用复势电弹理论获得了应力场和电位移场的闭合解答。给出了裂纹尖端应力强度因子、电位移场强因子和能量释放率的解析解。讨论了开裂孔洞几何参数和施加力电载荷对电弹场强因子和能量释放率的影响。     

The mode III cracks emanating from an elliptical hole in the piezo-electro-magneto-elastic materials

The mode III crack problem in a medium possessing coupled electro-magneto-elastic is considered. Two asymmetrical edge cracks emanate from an elliptical hole. Combined stress, electric and magnetic loads are considered. The elliptical hole and cracks are assumed to be either magneto-electrically impermeable or permeable. The closed-form solution for stress, electric and magnetic intensity factors are derived explicitly. Also the solution for energy release rate is given in closed form. The solution is based on the complex variable method combined with the method of conformal mapping. Numerical computations are given to illustrate the effect of variable geometrical and material parameters on stress, electric and magnetic intensity factors and energy release rate.     

The mode III cracks originating from the edge of a circular hole in a piezoelectric solid

A mode III fracture problem of edge cracks originating from a circular hole in an infinite piezoelectric solid is studied based on complex variable method combined with the method of conformal mapping. Explicit and exact expressions for the complex potentials, field intensity factors and energy release rates are presented under the assumption that the surface of the cracks and hole is electrically impermeable. Numerical analysis is then conducted to discuss the influences of crack length and applied mechanical/electric loads on the field intensity factors and energy release rate for one and two edge cracks, respectively. It is found that for the case of a single edge crack, the field intensity factors are greater than those of double edge cracks, and moreover the electric loads can either promote or retard crack growth, depending on the magnitude and direction of the applied electric loads.     

双压电材料内含椭圆孔孔边界面裂纹的反平面问题

利用复变函数方法,研究了横观各向同性压电双材料中椭圆孔孔边界面裂纹的反平面问题．首先,利用保角变换函数将椭圆孔保角变换到一直线裂纹;其次,基于孔边及裂纹表面均电不可穿透并且自由的假设,利用Stroh公式分别得到了本问题的复势函数、裂尖场集中系数的解析表达式;最后,在面内电载荷及面外机械载荷的作用下,分析了椭圆孔尺寸、裂纹长度和外载对裂尖场集中系数的影响,并得到了一个有意义的结论：椭圆孔一边裂纹长度的改变对另一边裂纹裂尖场的影响有限,然而一旦椭圆孔退化为竖直裂纹,该影响将变得非常显著．     

电可通纳米尺度孔边均布多裂纹的电弹断裂分析

解析研究了面内电载荷和反平面机械载荷作用下压电体中纳米尺度圆孔边均布电可通多裂纹问题的断裂性能。基于Gurtin-Murdoch表面弹性理论，利用保角映射方法和复变弹性理论给出了裂纹尖端电弹场分布、电弹场强度因子及能量释放率的解析结果。阐述了无量纲电弹场强度因子、无量纲能量释放率的尺寸依赖效应，讨论了裂纹数量和缺陷几何参数对无量纲场强度因子和无量纲能量释放率的影响。结果表明：无量纲电弹场强度因子和无量纲能量释放率具有显著的尺寸依赖效应；考虑表面效应，孔径和裂纹长度相当时，电弹场强度因子达到最大；裂纹/孔径比对电弹场强度因子随裂纹数量变化的制约会随着裂纹数量的增加而逐渐消失；过大或过小的裂纹孔径比会削弱裂纹长度对能量释放率的影响。     

电可通纳米尺度孔边均布多裂纹的电弹断裂分析

解析研究了面内电载荷和反平面机械载荷作用下压电体中纳米尺度圆孔边均布电可通多裂纹问题的断裂性能。基于Gurtin-Murdoch表面弹性理论,利用保角映射方法和复变弹性理论给出了裂纹尖端电弹场分布、电弹场强度因子及能量释放率的解析结果。阐述了无量纲电弹场强度因子、无量纲能量释放率的尺寸依赖效应,讨论了裂纹数量和缺陷几何参数对无量纲场强度因子和无量纲能量释放率的影响。结果表明：无量纲电弹场强度因子和无量纲能量释放率具有显著的尺寸依赖效应;考虑表面效应,孔径和裂纹长度相当时,电弹场强度因子达到最大;裂纹/孔径比对电弹场强度因子随裂纹数量变化的制约会随着裂纹数量的增加而逐渐消失;过大或过小的裂纹孔径比会削弱裂纹长度对能量释放率的影响。     

压电复合材料中正n边形孔边裂纹的反平面问题研究

本文基于Cauchy积分理论和Schwarz-Christoffel(SC)变换技术,针对压电复合材料中带一条裂纹的正n边形孔口缺陷的反平面断裂力学进行了探究．假设满足电不可通边界条件,利用Cauchy积分公式和留数定理,获得了任意正n边形裂尖处应力和电位移两个场强度因子以及全能量释放率的封闭形式的显式解．当正n边形边数取定时,所得解可退化为已有结果,以此验证方法的有效性．并通过数值算例,对比分析了n=3, n=4, n=5三种特殊情形对应的等效场强度因子和无量纲能量释放率与孔口边长、裂纹长度和受到的力、电载荷之间的曲线图．数值结果显示：正n边形孔洞的尺寸和裂纹长度均会促进裂纹扩展,且前者的影响更显著一些;正n边形边的数量增加会阻止裂纹的扩展;在电不可通边界条件下,机械载荷对裂纹的扩展始终贡献显著,电场对断裂行为的影响取决于机械载荷．本研究结果具有一般性,适用于任意正n边形孔边裂纹问题的求解,为压电复合材料元器件的优化设计和断裂特性分析提供了新思路．     

Anti-plane analysis of semi-infinite crack in piezoelectric strip

Using the complex variable function method and the technique of the conformal mapping, the fracture problem of a semi-infinite crack in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load. The analytic solutions of the field intensity factors and the mechanical strain energy release rate are presented under the assumption that the surface of the crack is electrically impermeable. When the height of the strip tends to infinity, the analytic solutions of an infinitely large piezoelectric solid with a semi-infinite crack are obtained. Moreover, the present results can be reduced to the well-known solutions for a purely elastic material in the absence of the electric loading. In addition, numerical examples are given to show the influences of the loaded crack length, the height of the strip, and the applied mechanical/electric loads on the mechanical strain energy release rate.     

Exact solutions for piezoelectric screw dislocations near two asymmetrical edge cracks emanating from an elliptical hole

The interaction between piezoelectric screw dislocations and two asymmetrical interfacial cracks emanating from an elliptic hole under combined mechanical and electric load at infinity is dealt with. The closed-form solutions are derived for complex potentials and generalized stress fields. In the limiting cases, some well-known results can be obtained from the present solutions. Moreover, some new exact solutions are shown. The stress intensity factor and the energy release rate at the right tip due to a screw dislocation near the right interfacial crack are also calculated. The results show that the shielding effect of dislocation on crack expanding decreases with the increase in dislocation azimuth angle and the distance between the dislocation and the crack tip, and the repulsion acting on the dislocation from the other half plane demotes crack propagation. The increasing of the length of the other crack promotes crack growth, but the increasing of the minor semi-axis demotes it.     

含圆孤裂纹系的压电材料反平面应变问题

应用复变函数解析延展原理，并通过求解Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ问题，得到了含圆弧裂纹压电材料反平面应变问题的一般解，对单个圆弧裂纹的情形，给出了封闭形式的复函数解和场强度因子，结果表明，当无限远处或裂纹表面同时受机械载荷（应力τ＾∞或Ｔｚ）和电载荷（电位移Ｄ＾∞或电荷ｑ）联合作用时，应力强度因子仅与机械载荷有关，而电位移动强度因子仅与电载荷有关。     

压电压磁材料中正n边形孔边裂纹分析

基于电磁复合材料力学,运用Stroh型公式和复变函数方法,针对压电压磁材料中含正n边形孔边裂纹反平面问题进行了研究。利用Schwarz-Christoffel变换技术,结合Cauchy积分公式和留数定理,导出了磁电全非渗透型边界条件下任意正n边形裂纹尖端场强度因子和能量释放率的解析解。当缺失磁场时,所得解退化为已有结果,以此验证方法的有效性。通过数值算例,对比分析了n=3,n=4和n=5三种特殊情形对应的孔口边长、裂纹长度和受到的力、电和磁载荷对等效场强度因子和无量纲能量释放率的影响规律。研究结果发现,正n边形孔洞的尺寸和裂纹长度均会促进裂纹扩展,且前者的影响更显著一些;正n边形边的数量增加会阻止裂纹的扩展;在磁电全非渗透型边界条件下,机械载荷始终促进裂纹的扩展,电位移载荷可以促进或抑制裂纹的扩展,磁载荷对裂纹的扩展贡献较少。本研究结果适用于任意正n边形孔边裂纹求解问题,为压电压磁材料元器件的优化设计和断裂特性分析提供了新思路。     

Closed form solution for two unequal collinear semi-permeable straight cracks in a piezoelectric media

The problem of two unequal collinear straight cracks weakening a poled transversely isotropic piezoelectric ceramic is addressed under semi-permeable electric boundary conditions on the crack faces. The plate has been subjected to combined in-plane normal(to the faces of the cracks) mechanical and electric loads. Problem is formulated employing Stroh formalism and solved using complex variable technique. The elastic field, electric field and energy release rate are obtained in closed analytic form. A case study is presented for poled PZT-5H cracked plate to study the effect of prescribed mechanical load, electric load, inter-crack distance and crack lengths on crack arrest parameters stress intensity factor (SIF), electric displacement intensity factor (EDIF) and mechanical and total energy release rates (ERR). Moreover a comparative study is done of impermeable and semi-permeable crack face boundary conditions on SIF, EDIF and ERR, and results obtained is presented graphically. It is observed that the effect of dielectric medium in the crack gap cannot be ignored.     

Anti-plane analysis of multiple cracks originating from a circular hole in a magnetoelectroelastic solid

The anti-plane problem of multiple cracks originating from a circular hole in a magnetoelectroelastic solid is investigated under remotely uniform anti-plane mechanical loading and in-plane electromagnetic loadings. The boundary value problem is reduced to a Cauchy integral equation by a new mapping function and the complex variable method, which is further solved exactly. The analytic expressions of the complex potentials, the field intensity factors and the energy release rate are derived in closed-form under the assumption that the surfaces of the cracks and hole are both electrically and magnetically impermeable. The effects of crack configurations and combined loadings on the energy release rate are shown graphically. Several useful results which may have potential applications to the design and fracture analysis of magnetoelectroelastic structures are given.     

Closed-form solution for two collinear cracks in a piezoelectric strip

Under the permeable electric boundary condition, the problem of two collinear anti-plane shear cracks lying at the mid-plane of a piezoelectric strip is investigated. By using the Fourier transform, the associated problem is reduced to a singular integral equation. Solving the resulting equation analytically, the electro-elastic field intensity factors and energy release rates at the inner and outer crack tips can be determined in explicit form. Numerical results for PZT-5H piezoelectric ceramic are also presented graphically. The results reveal that the effect of electric field on crack growth in piezoelectric materials is dependent on applied elastic displacement.     

Multiple parallel cracks interaction problem inpiezoelectric ceramics

This paper has two goals. First, we propose the pseudo-traction–electric displacementmethod for solving the interaction problem of multiple parallel cracks in transversely isotropicpiezoelectric ceramics. Second, we present a fundamental understanding for the role that theelectric displacement loading plays in the interaction problem. Detailed comparisons between theresults under the compound mechanical–electric loading conditions and those derived underpurely mechanical loading conditions are performed. It is shown that the mechanical fractureparameters such as the stress intensity factors are no longer independent of the electric loading asthey would be in single crack problems. Quite contrary, the electric displacement loading has asignificant influence on the stress intensity factors, the total potential energy release rate and themechanical strain energy release rate. This important conclusion is mainly due to the interactioneffect, i.e., one of the multiple cracks releases the stresses and disturbs the electric fields near theother crack. It is also found that there are some special relative locations for the multiple parallelcracks at which the electric displacement loading has no effect on the Mode I stress intensityfactor. However, the mechanical strain energy release rate has no such a property.     

压电效应下一维六方准晶双材料孔边裂纹的反平面问题研究

利用复变函数方法和保角变换技术研究了压电效应下一维六方准晶双材料中圆孔边单裂纹的反平面问题．考虑电不可渗透型边界条件,运用保角变换和Stroh公式得到了弹性体受远场剪切力和面内电载荷作用下裂纹尖端应力强度因子和能量释放率的解析解. 数值算例分析了几何参数、远场受力、电位移载荷对能量释放率的影响．结果表明：裂纹长度、耦合系数和远场剪切力的减小可以抑制裂纹的扩展．不考虑电场时,声子场应力对能量释放率的影响较小．本文的研究结果可作为研究一维六方压电准晶双材料孔边裂纹问题的理论基础,同时为压电准晶及其复合材料的设计、制备、优化和性能评估提供理论依据．     

一维六方准晶中纳米尺度开裂孔洞的III型断裂力学

研究了一维六方准晶中纳米尺度开裂孔洞的Ⅲ型断裂力学问题。基于复变弹性理论和表面弹性理论获得了考虑表面效应时椭圆孔边裂纹的应力场、应力强度因子和能量释放率的解析表达；讨论了缺陷尺寸、裂纹/孔洞比、耦合系数和施加载荷对应力强度因子和能量释放率的影响。研究表明：考虑表面效应且缺陷的尺寸在纳米尺度时，声子场和相位子场的无量纲应力强度因子以及无量纲能量释放率具有明显的尺寸依赖；裂纹相对尺寸较小时，表面效应对声子场和相位子场的无量纲应力强度因子影响较小；纳米尺度时无量纲能量释放率随耦合系数的增加而增大；耦合系数一定时，无量纲能量释放率受到椭圆孔尺寸影响；随着声子场载荷的增大，无量纲能量释放率先减小后增加，最后趋于稳定；无量纲能量释放率随相位子场载荷的增大单调减小，非常小和非常大的声子场载荷(或相位子场载荷)屏蔽了相位子场载荷(或声子场载荷)的影响。     

一维六方准晶中纳米尺度开裂孔洞的III型断裂力学

研究了一维六方准晶中纳米尺度开裂孔洞的Ⅲ型断裂力学问题。基于复变弹性理论和表面弹性理论获得了考虑表面效应时椭圆孔边裂纹的应力场、应力强度因子和能量释放率的解析表达;讨论了缺陷尺寸、裂纹/孔洞比、耦合系数和施加载荷对应力强度因子和能量释放率的影响。研究表明：考虑表面效应且缺陷的尺寸在纳米尺度时,声子场和相位子场的无量纲应力强度因子以及无量纲能量释放率具有明显的尺寸依赖;裂纹相对尺寸较小时,表面效应对声子场和相位子场的无量纲应力强度因子影响较小;纳米尺度时无量纲能量释放率随耦合系数的增加而增大;耦合系数一定时,无量纲能量释放率受到椭圆孔尺寸影响;随着声子场载荷的增大,无量纲能量释放率先减小后增加,最后趋于稳定;无量纲能量释放率随相位子场载荷的增大单调减小,非常小和非常大的声子场载荷(或相位子场载荷)屏蔽了相位子场载荷(或声子场载荷)的影响。