SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack

Based on the complex potential approach, the two-dimensional problems in a piezoelectric material containing an elliptic hole subjected to uniform remote loads are studied. The explicit, closed-form solutions satisfying the exact electric boundary condition on the hole surface are given both inside and outside the hole. When the elliptic hole degenerates into a crack, the field intensity factors are obtained. It is shown that the stress intensity factors are the same as that of isotropic material, while the electric displacement intensity factor depends on both the material properties and the mechanical loads, but not on the electric loads. In other words, the uniform electric loads have no influence on the field singularities. It is also shown that the impermeable crack assumption used previously to simply the electric condition is not valid to crack problems in piezoelectric materials.     

Electro-elastic fundamental solutions of anisotropic piezoelectric materials with an elliptical hole

By using Stroh' complex formalism and Cauchy's integral method, the electro-elastic fundamental solutions of an infinite anisotropic piezoelectric solid containing an elliptic hole or a crack subjected to a line force and a line charge are presented in closed form. Particular attention is paid to analyzing the characteristics of the stress and electric displacement intensity factors. When a line force-charge acts on the crack surface, the real form expression of intensity factors is obtained. It is shown through a special example that the present work is correct. The project supported by the Fund of the State Education Commission of China for Excellent Young Teachers     

Unified electrical boundary conditions for a crack interacting with a dislocation in piezoelectric media

This paper investigates the interaction problem between a dislocation and a finite crack in piezoelectric media. Analytical solutions for the generalized two-dimensional problem of a dislocation that is interacting with a finite crack in piezoelectric media are formulated via Stroh formalism. The analysis is conducted on the unified electrical crack boundary condition with the introduction of the electric crack condition parameter that can describe all the electric crack boundary conditions. The two ideal crack boundary conditions, namely, the electrically impermeable and permeable crack assumptions are obtained as two special cases for the current solutions. Based on the complex variable method and the perturbation technique, closed form solutions are obtained. The field intensity factors at the crack tip and the image forces on the dislocation due to the crack are computed and discussed.     

Stress and electric displacement analyses in piezoelectric media with an elliptic hole and a small crack

In this paper, the interactions between an elliptic hole and an arbitrary distributed small crack in plane piezoelectric medium, which are often happened in engineering problems, are discussed. The Green’s functions in a piezoelectric plate with an elliptic hole for a generalized line dislocation and a generalized line force are presented. The small crack is represented by unknown continuous distributed dislocations. By considering traction free conditions on the surface of the small crack, the problem is then reduced to a group of singular integral equations which are solved by using a special numerical technique. Accuracy of the present method is confirmed by comparing the numerical results with those in literatures for PZT-4 when the elliptic hole is degenerated into a crack. The generalized stress intensity factors of cracks and the generalized stress on the edge of the elliptic hole are shown graphically. It is shown that the small crack may have shielding or amplifying effects on the main elliptic hole or crack, which depends on the location and orientation of the small crack. The hole near a crack can significantly reduce the stress intensity factor of the crack. The direction of the electric field is important to shielding effect.     

The behavior of two non-symmetrical permeable cracks emanating from an elliptical hole in a piezoelectric solid

Based on the Stroh-type formalism, we present a concise analytic method to solve the problem of complicated defects in piezoelectric materials. Using this method and the technique of conformal mapping, the problem of two non-symmetrical collinear cracks emanating from an elliptical hole in a piezoelectric solid is investigated under remotely uniform in-plane electric loading and anti-plane mechanical loading. The exact solutions of the field intensity factors and the energy release rate are presented in closed-form under the permeable electric boundary condition. With the variation of the geometrical parameters, the present results can be reduced to the well-known results of a mode-III crack in piezoelectric materials. Moreover, new special models used for simulating more practical defects in a piezoelectric solid are obtained, such as two symmetrical edge cracks and single edge crack emanating from an elliptical hole or circular hole, T-shaped crack, cross-shaped crack, and semi-infinite plane with an edge crack. Numerical results are then presented to reveal the effects of geometrical parameters and the applied mechanical loading on the field intensity factors and the energy release rate.     

Mode I fracture analysis of a piezoelectric strip based on real fundamental solutions

In existing papers, mode I crack problems of piezoelectric ceramics are generally solved in complex domain because of the complex fundamental solutions of in-plane piezoelectric governing equations. In fact, these problems can alternatively be analyzed in real number field by recasting the solutions in real form instead. The main purpose of the present work is to develop such real fundamental solutions by detailed eigenvalue and eigenvector analyses. As an example of application, the widely studied fracture problem of a piezoelectric strip with a center-situated crack under mode I loading condition is then revisited based on the real fundamental solutions. Mixed boundary value conditions of the crack are transformed into Cauchy singular integral equations, which are then solved numerically to get fracture parameters including the energy release rate and intensity factors. Convergence behaviors of the kernel functions are surveyed. Theoretical derivation and computation are validated by the exact solution in a special case. The effect of a combined geometrical parameter on the crack is also investigated.     

Numerical solution of polarization saturation/dielectric breakdown model in 2D finite piezoelectric media

The polarization saturation (PS) model [Gao, H., Barnett, D.M., 1996. An invariance property of local energy release rates in a strip saturation model of piezoelectric fracture. Int. J. Fract. 79, R25–R29; Gao, H., Zhang, T.Y., Tong, P., 1997. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491–510], and the dielectric breakdown (DB) model [Zhang, T.Y., Zhao, M.H., Cao, C.F., 2005. The strip dielectric breakdown model. Int. J. Fract. 132, 311–327] explain very well some experimental observations of fracture of piezoelectric ceramics. In this paper, the nonlinear hybrid extended displacement discontinuity-fundamental solution method (NLHEDD-FSM) is presented for numerical analysis of both the PS and DB models of two-dimensional (2D) finite piezoelectric media under impermeable and semi-permeable electric boundary conditions. In this NLHEDD-FSM, the solution is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack and the electric yielding zone. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy certain conditions on the boundary of the domain, on the crack face and the electric yielding zone. The zero electric displacement intensity factor in the PS model or the zero electric field strength intensity factor in the DB model at the outer tips of the electric yielding zone is used as a supplementary condition to determine the size of the electric yielding zone. Iteration approaches are adopted in the NLHEDD-FSM. The electric yielding zone is determined, and the extended intensity factors and the local J-integral are calculated for center cracks in piezoelectric strips. The effects of finite domain size, saturation property and different electric boundary conditions, as well as different models on the electric yielding zone and the local J-integral, are studied.     

Anisotropic thermopiezoelectric solids with an elliptic inclusion or a hole under uniform heat flow

The two-dimensional problem of a thermopiezoelectric material containing an elliptic inclusion or a hole subjected to a remote uniform heat flow is studied. Based on the extended Lekhnitskii formulation for thermopiezoelectricity, conformal mapping and Laurent series expansion, the explicit and closed-form solutions are obtained both inside and outside the inclusion (or hole). For a hole problem, the exact electric boundary conditions on the hole surface are used. The results show that the electroelastic fields inside the inclusion or the electric field inside the hole are linear functions of the coordinates. When the elliptic hole degenerates into a slit crack, the electroelastic fields and the intensity factors are obtained. The effect of the heat flow direction and the dielectric constant of air inside the crack on the thermal electroelastic fields are discussed. Comparison is made with two special cases of which the closed solutions exist and it is shown that our results are valid.     

Piezothermoelastic analysis of a piezoelectric material with an elliptic cavity under uniform heat flow

Summary The problem of a two-dimensional piezoelectric material with an elliptic cavity under a uniform heat flow is discussed, based on the modified Stroh formalism for the piezothermoelastic problem. The exact electric boundary conditions at the rim of the hole are introduced in the analysis. Expressions for the elastic and electric variables induced within and outside the cavity are derived in closed form. Hoop stress around the hole and electric fields in the hole are obtained. The limit situation when the hole is reduced to a slit crack is discussed, and the intensity factors for the problem are obtained. Received 14 April 1998; accepted for publication 25 June 1998     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

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.     

Anti-plane dynamic hole–crack interaction in a functionally graded piezoelectric media

The anti-plane dynamic problem of a functionally graded piezoelectric plane containing a hole–crack system is treated by a non-hypersingular traction-based boundary integral equation method. The material parameters vary exponentially in the same manner in an arbitrary direction. The system is loaded by an incident SH-type wave, and impermeable boundary conditions are assumed. Using a frequency-dependent fundamental solution of the wave equation, the boundary value problem is transformed into a system of integro-differential equations along the boundary of the hole and on the crack line. Its numerical solution yields the dynamic stress intensity factors and stress concentration factors. A parametric study reveals their dependence on the hole–crack scenario and its geometry, characteristics of the dynamic load and magnitude and direction of material inhomogeneity.     

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．     

Isolated crack in three-dimensional piezoelectric solid: Part I - Solution by Hankel transform

The fundamental solutions are obtained for a unit concentrated electric potential discontinuity and unit concentrated displacement discontinuity in a three-dimensional piezoelectric medium. Displacements and stresses are derived by application of the boundary integral equation method. These expressions are used to obtain the stress intensity factors for a circular crack in Part II of the study.     

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．     

压电材料中裂纹面电边界条件的适用性

先分析理想裂纹面电边界条件(包括导通型裂纹和不导通型裂纹),然后分析缺陷的厚度及其对尖端场的影响,用封闭形式的解表示出了缺陷厚度的影响．结果指出,裂纹长厚比对计算结果是有影响的,除非裂纹极扁,不可通裂纹边界条件可以给出合理的结果,而可通边界条件只有在一些限制条件下才适用。     

压电体椭圆孔边的力学分析

基于复变函数的方法,以PZT-4材料为例,分别采用精确电边界条件和非导通电边界条件进行了远场均匀载荷作用下的横观各向同性压电体椭圆孔的力学分析并与相关结果进行对比。结果表明当椭圆孔退化为圆孔时,无论在远场作用力载荷或电载荷,两种电边界条件下的结果均能完全吻合。随着椭圆孔的愈加尖锐化,非导通电边界条件逐渐不能适用。     

Analysis method of planar cracks of arbitrary shape in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium

The hyper-singular boundary integral equation method of crack analysis in three-dimensional transversely isotropic magnetoelectroelastic media is proposed. Based on the fundamental solutions or Green’s functions of three-dimensional transversely isotropic magnetoelectroelastic media and the corresponding Somigliana identity, the boundary integral equations for a planar crack of arbitrary shape in the plane of isotropy are obtained in terms of the extended displacement discontinuities across crack faces. The extended displacement discontinuities include the displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, and correspondingly the extended tractions on crack face represent the conventional tractions, the electric displacement and the magnetic induction boundary values. The near crack tip fields and the intensity factors in terms of the extended displacement discontinuities are derived by boundary integral equation approach. A solution method is proposed by use of the analogy between the boundary integral equations of the magnetoelectroelastic media and the purely elastic materials. The influence of different electric and magnetic boundary conditions, i.e., electrically and magnetically impermeable and permeable conditions, electrically impermeable and magnetically permeable condition, and electrically permeable and magnetically impermeable condition, on the solutions is studied. The crack opening model is proposed to consider the real crack opening and the electric and magnetic fields in the crack cavity under combined mechanical-electric-magnetic loadings. An iteration approach is presented for the solution of the non-linear model. The exact solution is obtained for the case of uniformly applied loadings on the crack faces. Numerical results for a square crack under different electric and magnetic boundary conditions are displayed to demonstrate the proposed method.     

集中载荷作用下各向异性平面内的椭圆孔或裂纹问题

应用复变函数Cauchy积分的方法,对含有椭圆孔或裂纹的各向异性平面,系统地导出了当其在面内受任意集中载荷作用时的复应力函数解或裂纹应力强度因子解析解,即基本解;并通过基本解的迭加,得到了在椭圆孔周或裂纹表面作用一般外载时的解,其特例证实了上述解的正确性。