General treatment of mode III interfacial crack problems in piezoelectric materials

Summary  The anti-plane problem of N collinear interfacial cracks between dissimilar transversely isotropic piezoelectric media, which are subjected to piecewise uniform out-of-plane mechanical loading combined with in-plane electric loading at infinity, and also a line loading at an arbitrary point, is addressed by using the complex function method. In comparison with other relevant works, the present study has two features: one is that the analysis is based on the permeable crack model, i.e. the cracks are considered as permeable thin slits, and, thus, both the normal component of electric displacement and the tangential component of electric field are assumed to be continuous across these slits. The other feature is that explicit closed-form solutions are given not only in piezoelectric media, but also inside cracks when the media are subjected to the most general loading. It is shown that the singularities of electric displacement and electric field in the media are always dependent on that of stress for the general case of loading, and all the singularities of field variables are independent of the applied uniform electric loads at infinity. For the interfacial cracks the electric field is square-root singular at the crack tips and shows jumps across the interface, while the normal component of the electric field is linearly variable inside the crack, but the tangential component is square-root singular. However, for a homogeneous medium with collinear cracks, the electric field is always nonsingular in the medium while the electric displacement exhibits square-root singularity. Moreover, in this case, the electric field inside any crack is equal to a constant when uniform loads are applied at infinity. Received 22 November 1999; accepted for publication 20 July 2000     

Limited permeable crack moving along the interface of a piezoelectric bi-material

A plane problem for a crack moving with a subsonic speed along the interface of two piezoelectric semi-infinite spaces is considered. The crack is assumed to be free from mechanical loading. The limited permeable electric condition with an account of electric traction is adopted at its faces. A uniformly distributed mixed mode mechanical loading and an electric flux are prescribed at infinity. The problem is reduced to the Riemann–Hilbert problem by means of introducing a moving coordinate system and assuming that the electric flux is uniformly distributed along the crack region. An exact solution of this problem is proposed. It permits to find in closed form all necessary electromechanical characteristics at the interface and to formulate the equation for the determination of the electric flux. Analysis of this equation confirms the correctness of the assumption concerning the uniform distribution of the electric flux in the crack region. The values of the electric flux are determined by solving the obtained equation. Thereafter, the stress and electric intensity factors as well as their asymptotic fields at the crack tip are also found. The particular case of a crack moving in a homogeneous piezoelectric material is considered. The values of the electric flux and the fracture parameters are found exactly in a simple form for this case. Also, a numerical analysis is performed for a crack propagating with a subsonic speed between PZT4 and PZT5 materials and for a crack moving in PZT4 material. The electric flux in the crack region, stress and electric intensity factors, crack opening and the energy release rate (ERR) are found as functions of the crack speed, loading and electric permeability of the crack medium. The influence of the electric traction on the crack faces upon the mentioned parameters is demonstrated.     

Electrical nonlinear anti-plane shear crack in a functionally graded piezoelectric strip

The electrical nonlinear behavior of an anti-plane shear crack in a functionally graded piezoelectric strip is studied by using the strip saturation model within the framework of linear electroelasticity. The analysis is conducted on the electrically unified crack boundary condition with the introduction of the electric crack condition parameter that can describe all the electric crack boundary condition in accordance with the aspect ratio of an ellipsoidal crack and the permittivity inside the crack, in particular, including traditional permeable and impermeable crack boundary conditions. The resulting mixed boundary value problem is analysed and near tip field is obtained by using the integral transform techniques. Numerical results for the normalized five kinds of energy release rates under the small scale electric saturation condition are presented and compared to show the influences of the electric crack condition parameter with the variation of the ellipsoidal crack parameters, electric loads, functionally graded piezoelectric material gradation, crack length, electromechanical coupling coefficient, and crack location. It reveals that there are considerable differences between the results obtained from the traditional electric crack models and those obtained from the current unified crack model.     

Green's function for thermopiezoelectric plates with holes of various shapes

Summary Thermoelectroelastic problems for holes of various shapes embedded in an infinite matrix are considered in this paper. Based on Lekhnitskii's formalism, the technique of conformal mapping and the exact electric boundary conditions on the hole boundary, the thermoelectroelastic Green's function has been obtained analytically in terms of a complex potential. As an application of the proposed function, the problem of an infinite plate containing a crack and a hole is analysed. A system of singular integral equations for the unknown temperature discontinuity and the discontinuity of elastic displacement and electric potential (EDEP) defined on crack faces is developed and solved numerically. Numerical results for stress and electric displacement (SED) intensity factors of the crack-hole system are presented to illustrate the application of the proposed formulation. Received 7 October 1998; accepted for publication 26 January 1999     

Isolated crack in three-dimensional piezoelectric solid. Part II: Stress intensity factors for circular crack

This is part II of the work concerned with finding the stress intensity factors for a circular crack in a solid with piezoelectric behavior. The method of solution involves reducing the problem to a system of hypersingular integral equations by application of the unit concentrated displacement discontinuity and the unit concentrated electric potential discontinuity derived in part I [1]. The near crack border elastic displacement, electric potential, stress and electric displacement are obtained. Stress and electric displacement intensity factors can be expressed in terms of the displacement and the potential discontinuity on the crack surface. Analogy is established between the boundary integral equations for arbitrary shaped cracks in a piezoelectric and elastic medium such that once the stress intensity factors in the piezoelectric medium can be determined directly from that of the elastic medium. Results for the penny-shaped crack are obtained as an example.     

利用焦耳效应提高含裂纹金属构件抗裂性能问题的研究

设一无限大金属薄板中含有一个线裂纹,对金属板施加恒定的电流场,在两个裂尖处产生的热量远远大于其余地方产生的热量,可简化成两个点热源．经求解得到了问题的解析解,包括裂纹尖端附近区域温度、应力、应变、应变能密度因子的解析表达式．计算结果表明,裂纹尖端处的材料发生熔化而形成一个焊点,裂纹尖端明显纯化,可抑制裂纹的进一步扩展,提高含裂纹金属构件的抗裂性能．     

压电材料裂纹顶端条状电饱和区模型的力学分析

在线性压电本构方程框架下,对裂纹顶端条状电饱和区模型进行了严格的数学分析．完整地考虑了各向异性力电耦合效应．建立了电饱和区尺寸与外加电场的依赖关系．证实了当裂纹垂直极化轴时,压电材料的断裂应力随着外加正电场的增加而减小,随着外加负电场的增加而增加．当裂纹平行于极化轴时,与极化轴平行的外加电场对断裂应力无影响     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

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.     

压电材料中心裂纹问题

以电位移法向分量及电势连通过裂纹面为边界条件,对均匀电材料的裂纹问题及两种不同压材料界面裂纹问题进行了系统分析,得到了含中心裂纹无限大体封闭形的全场解。证实了裂纹引起的非均匀扰动场只信赖于外加场而外加电场无关。     

On the influence of the electric permeability on an interface crack in a piezoelectric bimaterial compound

An interface crack between two semi-infinite piezoelectric spaces under the action of remote mixed mode loading and electric flux is considered. The properties of the materials, loading and crack geometry admit to consider a two-dimensional problem in the plane perpendicular to the crack front. The crack is assumed to be free from mechanical loading and the limited permeable electric condition holds true. Assuming the electric flux is constant along the crack area, using the known presentations of all electromechanical fields via a piecewise holomorphic vector function, the problem is reduced to a vector Hilbert problem and solved in an analytical way. Clear analytical expressions for stresses and electric displacement as well as for stress and electric intensity factors are derived. As a particular case, a crack in a homogeneous piezoelectric material is considered and exact analytical formulae are presented for this case. The numerical analysis of the obtained formulae showed that for small values of the electric flux the model of a completely permeable crack can be used for any real crack permeability’s. The validity of such an approximation decreases with increase in the mechanical loading and especially of the electric flux.     

Corrigendum to “Solution of a crack in an electrostrictive solid” [International Journal of Solids and Structures 47 (2010) 444–453]

In our previous work [Gao, C.F., Mai, Y.W., Zhang, N., 2010. Solution of a crack in an electrostrictive solid. International Journal of Solids and Structures 47, 444–453.] the intensity factor of the total stress for an impermeable crack is directly written by using the corresponding result of a permeable crack. This is based on the fact that an impermeable crack can be considered as a special case of a permeable crack where the electric field is not zero. However, the singularity of total stresses for the impermeable crack can also be analyzed directly from the complex potentials. In this Corrigendum, the singularity of the total stresses is further studied for the impermeable crack, and the intensity factors are re-derived by using the obtained complex potentials. It is shown that for an impermeable crack, the total stresses still have an inverse square-root singularity but their intensity factor is different from that obtained by the solution of a permeable crack. Therefore, it is concluded that solutions for impermeable cracks cannot be obtained directly from those of permeable cracks, since the assumption of the electric boundary condition has not only influenced the electric fields on the crack-faces but also on the electric body force inside the material.     

三相压电复合本构模型中的弧形界面裂纹

深入研究了三相同心圆柱压电复合本构模型中的弧形绝缘界面裂纹问题。采用复势方法获得了该问题的级数形式的解答,并给出了应力、应变、电位移和电场强度等物理量在全场及界面上的分布,同时推导了裂尖处广义强度因子及裂面张开位移和裂面上电势差的表达式。具体计算表明该级数解答收敛迅速,同时显示出第三相混杂区的影响是不能忽略的。由于裂尖处应力奇异性为－1／2,则这种解答不会出现平面应变状态下界面裂纹裂尖处的振荡奇异性,从而不会产生违反物理实际的裂面相互嵌入现象,则该弹性解答也是建立了坚实的物理基础之上。     

Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric half-planes

Summary  In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous piezoelectric materials bonded through a functionally graded interfacial region is considered. Integral transforms are employed to reduce the problem to Cauchy singular integral equations. Numerical results illustrate the effect of the loading combination parameter λ, material property distribution and crack configuration on the dynamic stress and electric displacement intensity factors. It is found that the presence of the dynamic electric field could impede of enhance the crack propagation depending on the time elapsed and the direction of applied electric impact. Received 4 December 2001; accepted for publication 9 July 2002 This work is supported by the National Natural Science Foundation of China through Grant No. 10132010.     

Pre-kinking of a moving crack in a magnetoelectroelastic material under in-plane loading

A constant moving crack in a magnetoelectroelastic material under in-plane mechanical, electric and magnetic loading is studied for impermeable crack surface boundary conditions. Fourier transform is employed to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. Steady-state asymptotic fields near the crack tip are obtained in closed form and the corresponding field intensity factors are expressed explicitly. The crack speed influences the singular field distribution around the crack tip and the effects of electric and magnetic loading on the crack tip fields are discussed. The crack kinking phenomena is investigated using the maximum hoop stress intensity factor criterion. The magnitude of the maximum hoop stress intensity factor tends to increase as the crack speed increases.     

Permeable cracks between two dissimilar piezoelectric materials

An interfacial crack with electrically permeable surfaces between two dissimilar piezoelectric ceramics under electromechanical loading is investigated. An exact expression for singular stress and electric fields near the tip of a permeable crack between two dissimilar anisotropic piezoelectric media are obtained. The interfacial crack-tip fields are shown to consist of both an inverse square root singularity and a pair of oscillatory singularities. It is found that the singular fields near the permeable interfacial crack tip are uniquely characterized by the real valued stress intensity factors proposed in this paper. The energy release rate is obtained in terms of the stress intensity factors. The exact solution of stress and electric fields for a finite interfacial crack problem is also derived.     

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.     

Mode-III interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials

Summary  The problem of an interface edge crack between two bonded quarter-planes of dissimilar piezoelectric materials is considered under the conditions of anti-plane shear and in-plane electric loading. The crack surfaces are assumed to be impermeable to the electric field. An integral transform technique is employed to reduce the problem under consideration to dual integral equations. By solving the resulting dual integral equations, the intensity factors of the stress and the electric displacement and the energy release rate as well as the crack sliding displacement and the electric voltage across the crack surfaces are obtained in explicit form for the case of concentrated forces and free charges at the crack surfaces and at the boundary. The derived results can be taken as fundamental solutions which can be superposed to model more realistic problems. Received 10 November 2000; accepted for publication 28 March 2001     

压电材料中的微裂纹屏蔽问题分析

分析当主裂纹与一个微裂纹在远场I型力（KI）和远场电位移（Ke)作用下的相互干涉问题,得出了在微裂纹的位置角和方向角周时独立变化时,微裂纹对主裂纹的屏蔽作用的全局使命主裂纹扩展,通过电算还发现Ortiz在各向同性材料和各向异性材料中得出的“微裂纹群对主裂纹最大屏蔽效应产生在微裂纹方向与最大主应力垂直的方向”在压电材料中不再成立,进而提出除Hutchinson指出微裂纹屏蔽效应两个来源（即：材料有效刚度的降低和残余应力的释放）外的另一个来源,微裂纹对主裂砂电场的扰动,在对主微裂纹J积分分析时发现J2积分与J1积分具有同等重要的地位。     

Heat conduction and thermal stress induced by an electric current in an infinite thin plate containing an elliptical hole with an edge crack

A uniform electric current at infinity was applied to a thin infinite conductor containing an elliptical hole with an edge crack. The electric current gives rise to two states, i.e., uniform and uneven Joule heat. These two states must be considered to analyze the heat conduction problem. The uneven Joule heat gives rise to uneven temperature and thus to heat flux, and to thermal stress.Using a rational mapping function, problems of the electric current, the Joule heat, the temperature, the heat flux, the thermal stress are analyzed, and each of their solutions is obtained as a closed form. The distributions of the electric current, the Joule heat, the temperature, the heat flux and the stress are shown in figures.The heat conduction problem is solved as a temperature boundary value problem. Solving the thermal stress problem, dislocation and rotation terms appear, which complicates this problem. The solutions of the Joule heat, the temperature, the heat flux and the thermal stress are nonlinear in the direction of the electric current. The crack problems are also analyzed, and the singular intensities at the crack tip of each problem are obtained. Mode II (sliding mode) stress intensity factor (SIF) is produced as well as Mode I (opening mode) SIF, for any direction of the electric current. The relations between the electric current density and the melting temperature and between the electric current density and SIF are investigated for some crack lengths in an aluminum plate.