Numerical method of crack analysis in 2D finite magnetoelectroelastic media

The present paper extends the hybrid extended displacement discontinuity fundamental solution method (HEDD-FSM) (Eng Anal Bound Elem 33:592–600, 2009) to analysis of cracks in 2D finite magnetoelectroelastic media. The solution of the crack 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. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy the prescribed boundary conditions on the boundary of the domain and on the crack face. The Crouch fundamental solution for a parabolic element at the crack tip is derived to model the square root variations of near tip fields. The extended stress intensity factors are calculated under different electric and magnetic boundary conditions.     

Three-dimensional BEM for piezoelectric fracture analysis

A boundary element approach with quadratic isoparametric elements, quarter-point elements and singular quarter-point elements for three-dimensional crack problems in piezoelectric solids under mechanical and electrical loading conditions, is presented in this paper for the first time. The procedure is based on Deeg's fundamental solution for anisotropic piezoelectric materials, and the classical extended displacement boundary integral equation. Stress and electric displacement intensity factors are directly evaluated as system unknowns, and also as functions of the computed nodal displacements and electric potentials at crack faces. Special attention is paid to the fundamental solution evaluation. Several three-dimensional crack problems in transversely isotropic bodies under mechanical and electrical loading conditions are analysed. Numerical solutions computed for prismatic cracked 3D plate problems with a plane strain behaviour are in very good agreement with their corresponding 2D BE solutions. Results for a penny shape crack in a piezoelectric cylinder are presented for the first time. The proposed approach is shown to be a simple, robust and useful tool for stress and electric displacement intensity factors evaluation in piezoelectric media.     

Analysis of an arbitrarily oriented crack in a finite piezoelectric plane via the hybrid extended displacement discontinuity-fundamental solution method

In this paper, we analyze an arbitrarily oriented crack in a finite two-dimensional piezoelectric medium with the polarization saturation model near the crack tip. We first derive the extended Green’s functions corresponding to the extended point-displacement discontinuities of an arbitrarily oriented crack based on the Green’s functions of the extended point forces and the Somigliana identity. Then, the extended field intensity factors and the local J-integral near the crack tip are expressed in terms of the extended displacement discontinuity on crack faces. Finally, the nonlinear hybrid extended displacement discontinuity-fundamental solution method is proposed to analyze an electrically nonlinear crack in a finite piezoelectric medium. Numerical examples are carried out for both linear and nonlinear fracture models of the crack under electrically impermeable boundary conditions. The influence of the crack orientation and geometric size on the fracture behaviors of the crack is investigated.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Electroelastic analysis of an electrically dielectric Griffith crack in a piezoelectric layer

The fracture analysis of an electrically dielectric Griffith crack embedded in a piezoelectric layer is made under in-plane electro-mechanical loadings. To simulate an opening crack full of a dielectric interior, the energetically consistent crack-face boundary conditions are utilized. Applying the Fourier transform technique, the boundary-value problem is reduced to solving two coupling singular integral equations. The intensity factors of stress, electric displacement, crack opening displacement (COD) and electric potential are further determined by the Lobatto-Chebyshev collocation method. The variations of the electric displacement at the crack surfaces are investigated by using the energetically consistent and semi-permeable boundary conditions respectively. The observations show that the electric displacement inside the crack is decreasing with an increase of the ratio between the crack length and piezoelectric layer width. Numerical computations are further carried out to compare the intensity factors of stress and electric potential, and the energy release rate using the energetically consistent boundary conditions with those using the semi-permeable boundary conditions. The obtained results reveal that the stress induced by a dielectric inside a crack has great effects on the stress intensity factor and energy release rate, but little influence on the electric potential difference across the crack.     

On the electrical boundary conditions on the crack surfaces in piezoelectric ceramics

This paper investigates a cracked piezoelectric ceramic under remote electro-mechanical loads. The ideal crack boundary conditions for electrically impermeable and permeable crack assumptions, and the deformed crack with a yet-to-be-determined crack shape are considered. The last is referred to as the “natural boundary condition (NBC)”. Closed-form solutions to the crack-tip field intensity factors are obtained. The analysis shows that traditional approaches to the electric boundary conditions on the crack faces, that is, either the impermeable crack assumption or the permeable crack assumption, produce significantly different results for the crack-tip quantities such as electric displacement intensity factor, energy release rate and crack opening displacement. There are also considerable differences between the results obtained from traditional impermeable and permeable crack analyses and those obtained from the proposed NBC. The difference increases with applied electric loads.     

压电复合材料的共线周期性裂纹平面问题的电弹性场分析

利用复变函数知识、半逆解法及待定系数法, 研究了压电复合材料的共线周期性裂纹问题, 给出了在电不可渗透边界条件下的应力、电位移、应力强度因子、电位移强度因子和机械应变能释放率的解析解。当裂纹间距趋于无穷时, 共线周期性裂纹退化为一条单裂纹, 得到了压电复合材料一条单裂纹的结果。通过数值算例讨论了共线周期性裂纹的裂纹长度、裂纹间距和机电载荷对机械应变能释放率的影响规律。结果表明, 机械应变能释放率随着共线周期性裂纹的裂纹长度、共线周期性裂纹的裂纹间距、机械载荷和正电场的增大而增大, 随着负电场的增大而减小。     

Extended displacement discontinuity Green's functions for three-dimensional transversely isotropic magneto-electro-elastic media and applications

A versatile method is presented to derive the extended displacement discontinuity Green's functions or fundamental solutions by using the integral equation method and the Green's functions of the extended point forces. In particular, the three-dimensional (3D) transversely isotropic magneto-electro-elastic problem is used to demonstrate the method. On this condition, the extended displacement discontinuities include the elastic displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, while the extended forces include the point forces, the point electric charge and the point electric current. Based on the obtained Green's functions, the extended Crouch fundamental solutions are derived and an extended displacement discontinuity method is developed for analysis of cracks in 3D magneto-electro-elastic media. The extended intensity factors of two coplanar and parallel rectangular cracks are calculated under impermeable boundary condition to illustrate the application, accuracy and efficiency of the proposed method.     

The effective electroelastic property of cracked piezoelectric media

Summary This paper presents a study on the effective electroelastic property of piezoelectric media with parallel or randomly distributed cracks. The theoretical formulation is derived using the dilute model of distributed cracks and the solution of a single dielectric crack problem, in which the electric boundary condition along the crack surfaces is governed by the crack opening displacement. It is observed that the effective electroelastic property of such cracked piezoelectric media is nonlinear and sensitive to loading conditions. Numerical simulations are conducted to show the effects of crack distribution and electric boundary condition upon the effective electroelastic property. The transition between the commonly used electrically permeable and impermeable crack models is studied.     

The influence of Maxwell stresses on the fracture mechanics of piezoelectric materials

The influence of Maxwell stresses on the generalized 2D fracture mechanics problem of piezoelectric materials under combined mechanical and electric loads at infinity is studied. The electrically semi-permeable crack boundary condition is adopted in this paper. Based on the Stroh’s formalism, explicit and closed-form solutions of electric displacement inside the crack, stress and electric intensity factors are obtained. Numerical results are also given to discuss the effects of Maxwell stresses on the stress and electric displacement intensity factors when the interior of the crack and the surrounding space at infinity are filled with different dielectric medium. It is found that the stress intensity factor increases rapidly with increasing value of the applied electric displacement load for the case of the dielectric constant of the surrounding at infinity is smaller than that inside the crack. The electric displacement intensity factor always increases as the applied electric loads or the applied mechanical loads increase.     

Transient analysis of dynamic crack propagation in piezoelectric materials

Abstract

In this paper, the transient analysis of semi‐infinite propagating cracks in piezoelectric materials subjected to dynamic anti‐plane concentrated body force is investigated. The crack surface is assumed to be covered with an infinitesimally thin, perfectly conducting electrode that is grounded. In analyzing this problem, it has characteristic lengths and a direct attempt towards solving this problem by transform and Wiener‐Hopf techniques (Noble, 1958) is not applicable. In order to solve this problem, a new fundamental solution for propagating cracks in piezoelectric materials is first established and the transient response of the propagating crack is obtained by superposition of the fundamental solution in the Laplace transform domain. The fundamental solution to be used is the responses of applying exponentially distributed traction in the Laplace transform domain on the propagating crack surface. Taking into account the quasi‐static approximation, exact analytical transient solutions for the dynamic stress intensity factor and the dynamic electric displacement intensity factor are obtained by using the Cagniard‐de Hoop method (Cagnard, 1939; de Hoop, 1960) of Laplace inversion and are expressed in explicit forms. Numerical calculations of dynamic intensity factors are evaluated and the results are discussed in detail. The transient solutions for stationary cracks have been shown to approach the corresponding static values after the shear wave of the piezoelectric material has passed the crack tip.     

Numerical analysis for piezoelectric crack under varied boundary conditions by optimized hybrid element method

In this article, a piezoelectric hybrid element is presented and optimized by penalty equilibrium approach, and special crack surface element is suggested for exactly implementing the boundary conditions on crack surface. An iteration technique is used to treat one of the electric boundary conditions. Then, a piezoelectric material with crack is numerically studied by the optimized hybrid element method, and the results are compared with the analytical solutions. The stress and the electrical displacement fields with different crack surface conditions are studied, and the influence to those fields arisen by the far field mechanical and electric loading is also studied.     

Three-dimensional extended displacement discontinuity method for vertical cracks in transversely isotropic piezoelectric media

The conventional displacement discontinuity method is extended to study a vertical crack under electrically impermeable condition, running parallel to the poling direction and normal to the plane of isotropy in three-dimensional transversely isotropic piezoelectric media. The extended Green's functions specifically for extended point displacement discontinuities are derived based on the Green's functions of extended point forces and the Somigliana identity. The hyper-singular displacement discontinuity boundary integral equations are also derived. The asymptotical behavior near the crack tips along the crack front is studied and the ordinary 1/2 singularity is obtained at the tips. The extended field intensity factors are expressed in terms of the extended displacement discontinuity on crack faces. Numerical results on the extended field intensity factors for a vertical square crack are presented using the proposed extended displacement discontinuity method.     

压电复合材料中III型唇形裂纹问题的解析解

采用复变函数方法和保角映射技术,研究了压电复合材料中含唇形裂纹的无限大体远场受反平面机械载荷和面内电载荷作用下的反平面问题,利用复变函数中的留数定理和Cauchy积分公式,分别获得了电不可通和电可通两种边界条件下裂纹尖端场强度因子和机械应变能释放率的解析表达式。当唇形裂纹的高度趋于零时,可得到无限大压电复合材料中Griffith裂纹的解析解。若不考虑电场作用,所得解退化为经典材料的已知结果。数值算例显示了裂纹的几何尺寸和机电载荷对机械应变能释放率的影响规律。结果表明: 唇形裂纹高度的增加会阻碍裂纹的扩展;机械载荷总是促进裂纹的扩展;电载荷对裂纹扩展的影响与裂纹面电边界条件有关。     

Interaction of parallel dielectric cracks in functionally graded piezoelectric materials

In this paper, the problem of two interacting parallel cracks in functionally graded piezoelectric materials under in-plane electromechanical loads is studied. The formulation is based on using Fourier transforms and modeling the cracks as distributed dislocations, and the resulting singular integral equations are solved with Chebyshev polynomials. A dielectric crack model considering the crack filling effect is adopted to describe the electric boundary conditions along crack surfaces. Numerical simulations are made to show the effect of material gradient, the geometry of interacting cracks, and crack position upon fracture parameters such as stress intensity factors, electric displacement intensity factor, and COD intensity factor. By considering the effect of a dielectric medium inside the crack and crack deformation, the results obtained from the dielectric crack model are always between those from the traditional crack models with physical limitation.     

Dynamic response for non-homogeneous piezoelectric medium with multiple cracks

A general procedure to analyze the dynamic response of non-homogeneous piezoelectric medium containing some non-collinear cracks is developed. It is assumed that all the material properties only depend on the coordinates y (along the thickness direction). The assumption is made that the non-homogeneous medium is composed of numerous laminae with their surfaces perpendicular to the thick direction. The solution method is based upon the Fourier and Laplace transforms to reduce the boundary value problem to a system of generalized singularity equations in the Laplace transform domain. The singular integral equations for the problem are derived and numerically solved by weight residual value method. The time-dependent full field solutions are obtained in the time domain. As numerical illustration, the stress and electric displacement intensity factors for a three-layer plate specimen with two cracks are presented. It is found that the stress and electric fields are coupled in the crack plane ahead of the crack tip for non-homogenous piezoelectric materials.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

Dynamic crack interactions in magnetoelectroelastic composite materials

In this paper, the dynamic interactions among cracks embedded in a two-dimensional (2-D) piezoelectric-piezomagnetic composite material are analyzed by means of a hypersingular formulation of the boundary element method. In the numerical solution procedure, extended crack opening displacements and extended traction jumps across the crack are considered as basic unknowns, so that only the traction boundary integral equations are needed on the crack surfaces. Quadratic discontinuous boundary elements are implemented together with discontinuous quarter-point elements placed next to the crack tips to ensure a proper representation of the square root asymptotic behavior. Several impermeable cracks configurations subjected to time-harmonic incident L-waves are analyzed in order to characterize the effects of the magnetoelectromechanical coupling on the dynamic crack interactions and to illustrate the dependence on such coupling of the fracture parameters: stress intensity factors, electric displacement intensity factor and magnetic induction intensity factor.     

Basic solutions of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials

In this paper, the stress and the electric intensity factors of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials were examined by means of the Schmidt method for the permeable electric boundary conditions. The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variables are the jumps of displacements across the crack surfaces, not the dislocation density functions. To solve the dual integral equations, the displacement jumps are directly expanded in a series of Jacobi polynomials. Finally, the effects of the distance between two parallel cracks and the distance between two collinear cracks on the stress and the electric intensity factors in the piezoelectric materials are analyzed. These results can be used for the strength evaluation of the piezoelectric materials with multi-cracks.     

Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media

A complete form of stress and electric displacement fields in the vicinity of the tip of an interfacial crack, between two dissimilar anisotropic piezoelectric media, is derived by using the complex function theory. New definitions of real-valued stress and electric displacement intensity factors for the interfacial crack are proposed. These definitions are extensions of those for cracks in homogeneous piezoelectric media. Closed form solutions of the stress and electric displacement intensity factors for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media are also obtained by using the mutual integral.