An exact and explicit treatment of an elliptic hole problem in thermopiezoelectric media

This paper presents an exact solution for the problem of an elliptic hole or a crack in a thermopiezoelectric solid. First, based on the extended version of Eshelby–Stroh's formulation, the generalized 2D problems of an elliptical hole in a thermopiezoelectric medium subject to uniform heat flow and mechanical–electrical loads at infinity are studied according to exact boundary conditions at the rim of the hole. The complex potentials in the medium and the electric field inside the hole are obtained in closed form, respectively. Then, when the hole degenerates into a crack, the explicit solutions for the field intensity factors near the crack tip and the electric field inside the crack are presented. It is shown that the singularities of all the field are dependent on the material constants, the applied heat load and mechanical loads at infinity, but not on the applied electric loads. It is also found that the electric field inside the crack is linearly variable, which is different from the result based on the impermeable crack model.     

An exact solution of crack problems in piezoelectric materials

IntroductionInrecentyearscrackproblemsinpiezoelectricmaterialhavereceivedmuchattention.Manytheoreticalanalyseshavebeengivenby[1～16].Itshouldbe,however,notedthatalltheaboveanalysesarebasedonaso-calledimpermeablecrackassumphon,i.e.thecrackfacesareassumedtobeimpermeabletoelectricfield,sotheelectricdisplacementvanishesinsidethecrack.Usingthisassumption,onewillobtainthefollowingresultS[2'3'5,6'9'16]=whentheelectricloadsaresolelyaPPliedatinLfinity,theelectricdisplacementissquare-rootsingularatthe…     

Generalized 2D problem of piezoelectric media containing collinear cracks

The generalized 2D problem in piezoelectric media with collinear cracks is addressed based on Stroh's formulation and the exact electric boundary conditions on the crack faces. Exact solutions are obtained, respectively, for two special cases: one is that a piezoelectric solid withN collinear cracks is subjected to uniform loads at infinity, and the other is that a piezoelectric solid containing a single crack is subjected to a line load at an arbitrary point. It is shown when uniform loads are applied at infinity or on the crack faces that, the stress intensity factors are the same as those of isotropic materials, while the intensity factor of electric displacement is dependent on the material constants and the applied mechanical loads, but not on the applied electric loads. Moreover, it is found that the electric field inside any crack is not equal to zero, which is related to the material properties and applied mechanical-electric loads. The project supported by the National Natural Science Foundation of China (19772004)     

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 fundamental solutions for the plane problem in piezoelectric media with an elliptic hole or a crack

1.IntroductionItiswell-knownthatthefundame,ltalsolutionsorGreen'sfunctionsplayanimportantroleilllinearelasticity.Forexample,theycanbeusedtoconstructmanyanalyticalsolutionsofpracticalproblems.Itismoreimportantthattheyareusedasthefundamentalsolutionsintheboundaryelementmethod(BEM)tosolvesomecomplicatedproblem.Withthewidely-increasingapplicationofpiezoelectricmaterialsinengineeringproblems,thestudyregardingtheGreen'sfLlnctionsinpiezoelectricsolidshasreceivedmuchinterest.The3DGreen'sfunctionsi…     

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.     

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.     

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.     

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。     

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     

Penny shaped crack in a piezoelectric cylinder surrounded by an elastic medium subjected to combined in-plane mechanical and electrical loads

The fracture problem of a penny shaped crack in a piezoelectric ceramic cylinder surrounded by an infinite elastic medium under in-plane normal mechanical and electrical loads is considered with the electric continuous boundary conditions on the crack surface. By using the potential theory and Hankel transform, a system of dual integral equations is obtained, and expressed to a Fredholm integral equation of the second kind. The mechanical and electrical field equations and all sorts of field intensity factors of mode I are obtained, and the numerical values of various field intensity factors for PZT-6B piezoelectric ceramic surrounded by several different elastic media are graphically shown for a uniform load and a ring-shaped load, respectively. And the effects of the size of the piezoelectric cylinder and the elastic material properties on various field intensity factors are obtained.     

Three-dimensional non-axisymmetric behavior of a penny shaped crack in a piezoelectric strip subjected to in-plane loads

The behavior of a penny shaped crack in a three-dimensional piezoelectric ceramic strip under non-axisymmetric in-plane normal mechanical and electrical loads is analyzed based on the continuous electric boundary conditions of the crack surface. The potential theory, Hankel transform and Fourier series are used to obtain the systems of dual integral equations, which are then expressed as Fredholm integral equations. The singular mechanical and electric fields and all mode-I field intensity factors are obtained, and the numerical values of various field intensity factors for PZT-6B piezoelectric ceramic are shown graphically for an uniform load and a pair of concentrated load, respectively.     

压电材料平面应力状态的直线裂纹问题一般解

研究了含直线裂纹系的压电材料平面应力问题单个裂纹和双裂纹问题的封闭解答表明，在裂纹尖端，应力、电场强度和电位移有１／２阶的奇异性并与前人结果比较了产生电场奇异性的物理因素     

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.     

压电材料平面应力状态的直线裂纹问题一般解

研究了含直线裂纹系的压电材料平面应力问题单个裂纹和双裂纹问题的封闭解答表明，在裂纹尖端，应力、电场强度和电位移有１／２阶的奇异性并与前人结果比较了产生电场奇异性的物理因素     

压电压磁复合材料中正六边形孔边裂纹的反平面断裂问题

基于线性磁电弹性理论,利用Schwarz-Christoffel(CS)变换技术和Stroth公式,首次系统研究了压电压磁复合材料中含带两个不对称裂纹的正六边形孔口问题在部分渗透磁电边界条件下的解析解.当忽略磁场时,磁电非渗透裂纹和磁电渗透裂纹两种极端情况下的解析解答可退化为文献已有研究结果.数值结果揭示了正六边形孔口尺寸、裂纹长度以及力电载荷和磁载荷对能量释放率的影响规律.研究结果表明：减小孔口边长和裂纹长度可以提高材料的可靠性;机械载荷总是促进裂纹扩展;在磁电非渗透和磁电部分渗透边界条件下,负电场和负磁场会延缓裂纹的扩展,而正电场可以增强或阻碍裂纹的扩展,这取决于所施加的电场和磁场的强度以及机械载荷的水平;在磁电渗透边界条件下,电场和磁场对裂纹的扩展没有影响.     

Exact solutions of two semi-infinite collinear cracks in piezoelectric strip

Using the complex variable function method and the conformal mapping technique,the fracture problem of two semi-infinite collinear cracks in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load on the partial crack surface.Analytic solutions of the field intensity factors and the mechanical strain energy release rate are derived under the assumption that the surfaces of the crack are electrically impermeable.The results can be reduced to the well-known solutio...     

Penny-shaped crack in a piezoelectric cylinder under electro-mechanical loads

Summary The fracture behavior of a penny-shaped crack in a axisymmetrical piezoelectric ceramic cylinder of finite radius under mechanical and electrical loads is analyzed under electric continuous boundary conditions on the crack surface. The potential theory and Hankel transform are used to obtain a system of dual integral equations, which is then expressed as a Fredholm integral equation. Singular mechanical and electrical fields and field intensity factors of mode I are obtained. The numerical values of various field intensity factors for PZT-6B piezoelectric ceramics are graphically shown for a uniform load and a ring-shaped load, respectively. The effects of the radius of the cylinder on the field intensity factors are investigated. This work was supported by the Korea Research Foundation Grant (KRF-2001-041-E00057).     

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     

热电材料孔边周期裂纹问题的热电弹分析

论文研究了均匀电流密度和能量流作用下,热电材料中带4k个周期径向裂纹的圆形孔口问题.考虑非渗透型电和热边界条件,运用复变函数理论和保形映射方法,得到了热电材料中电流密度、能量密度和应力场的精确解.依据断裂力学理论,运用Cauchy积分公式得到了周期裂纹的电流、能量和应力强度因子.数值结果分析了场强度因子随各个参数的变化...