The extended finite element method with new crack‐tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X‐FEM) with new crack‐tip enrichment functions. In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and crack‐tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ? class and κ class, two classes of crack‐tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J‐integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.     

Stress intensification due to an edge crack in an anisotropic elastic solid

Integral transform techniques are used to determine the stress intensity factors of a crack at the edge of an anisotropic elastic half space under generalized plane strain conditions. Numerical results are given for a carbon fibre reinforced epoxy in uniaxial tension.     

Interface energy on effective elastic constant of piezoelectric composites with spherically anisotropy nano-particles under external uniform strain

The surfaces/interfaces effect is vital in nanocomposites. The electro-elastic surface/interface theory is introduced to predict the size-dependent effective elastic constants of piezoelectric composites with spherically anisotropy nano-particles under external uniform strain, and the analytical solution is obtained. New boundary conditions governing the surface piezoelectricity are used to analyze the surface piezoelectricity effect. The average electro-elastic coupling field of the randomly distributed nano-particles with surface/interface effect is derived by using the effective field method. In the numerical examples, the interface effect under different material constituents is analyzed. It is found that the interface effect on the effective elastic constants is significantly related to the material properties of nano-particles. The elastic constants and piezoelectric constants show different effects on the effective elastic constants.     

The interface crack problem for bonded piezoelectric and orthotropic layers under antiplane shear loading

The primary objective of this paper is to study the influence of the electroelastic interactions on the stress intensity factor in bonded layers of piezoelectric and orthotropic materials containing a crack along the interface under antiplane shear. Attention is given to a two-layer hybrid laminate formed by adding a layer of piezoelectric ceramic to a unidirectional graphite/epoxy composite or an aluminum layer. Electric displacement or electric field is prescribed on the surfaces of the piezoelectric layer. The problem is formulated in terms of a singular integral equation which is solved by using a relatively simple and efficient technique. A number of examples are given for various material combinations. The results show that the effect of the electroelastic interactions on the stress intensity factor and the energy release rate can be highly significant. This revised version was published online in July 2006 with corrections to the Cover Date.     

Griffith crack moving along the interface of two dissimilar piezoelectric materials

The problem of an anti-plane Griffith crack moving along the interface of dissimilar piezoelectric materials is solved by using the integral transform technique. It is shown from the result that the intensity factors of anti-plane stress and electric displacement are dependent on the speed of the Griffith crack as well as the material coefficients. When the two piezoelectric materials are identical, the present result will reduce to the result for the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

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。     

压电材料切口奇性指数计算

利用一种数值方法分析压电材料切口尖端包括奇异应力场和奇异电位移场在内的双重奇异性。基于切口尖端的位移场按幂级数渐近展开假设, 从应力平衡方程和Maxwell方程出发, 推导出关于压电材料切口奇性指数的特征方程组, 同时将切口的力学和电学边界条件转化为奇性指数和特征函数的组合表达, 从而将压电材料双重奇性分析问题转化为在相应边界条件下微分方程组的特征值求解问题, 采用插值矩阵法, 可以一次性地计算出压电材料切口的各阶奇性指数。裂纹作为切口的特例, 其尖端的电弹性奇性指数亦可以根据本法求出。     

Surface-breaking crack in an elastic half-space

A semi-infinite crack terminating at the boundary of an elastic half-space is considered. It is assumed that the crack is subjected to a mode-I load applied in a finite region remote from the crack plane, and the boundary of the half-space and the crack surfaces are free of tractions. The problem is formulated in terms of a hypersingular integral equation with respect to the relative crack-face separation defined over the region occupied by the crack. The behaviour of the solution near the corner point where the crack edge intersects the boundary is analysed and results which show the dependence of the stress singularity exponent on the angle of inclination of the crack edge are presented.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

Interface crack in periodically layered bimaterial composite

A directional crack growth prediction in a compressed homogenous elastic isotropic material under plane strain conditions is considered. The conditions at the parent crack tip are evaluated for a straight stationary crack. Remote load is a combined biaxial compressive normal stress and pure shear. Crack surfaces are assumed to be frictionless and to remain closed during the kink formation wherefore the mode I stress intensity factor K _I is vanishing. Hence the mode II stress intensity factor K _II remains as the single stress intensity variable for the kinked crack. An expression for the local mode II stress intensity factor k ₂ at the tip of a straight kink has been calculated numerically with an integral equation using the solution scheme proposed by Lo (1978) and refined by He and Hutchinson (1989). The confidence of the solution is strengthened by verifications with a boundary element method and by particular analytical solutions. The expression has been found as a function of the mode II stress intensity factor K _II of the parent crack, the direction and length of the kink, and the difference between the remote compressive normal stresses perpendicular to, and parallel with, the plane of the parent crack. Based on the expression, initial crack growth directions have been suggested. At a sufficiently high non-isotropic compressive normal stress, so that the crack remains closed, the crack is predicted to extend along a curved path that maximizes the mode II stress intensity factor k ₂. Only at an isotropic remote compressive normal stress the crack will continue straight ahead without change of the direction. Further, an analysis of the shape of the crack path has revealed that the propagation path is, according the model, required to be described by a function y=cx , where the exponent is equal to 3/2. In that case, when =3/2, predicts the analytical model a propagation path that is self-similar (i.e. the curvature c is independent of any length of a crack extension), and which can be described by a function of only the mode II stress intensity factor K _II at the parent crack tip and the difference between the remote compressive normal stress perpendicular to, and parallel with, the parent crack plane. Comparisons with curved shear cracks in brittle materials reported in literature provide limited support for the model discussed.     

Stress singularities near the tip of a two-dimensional notch formed from several elastic anisotropic materials

In this paper, we studied the stress singularities near the tip of a two-dimension notch, which could be a crack tip, formed from several elastic materials, each of them may be generally anisotropic. By introducing the dual variables in the state space, the basic equations governing the posed problem equations were established. We also proposed a numerical method to solve the governing equations. It was shown that the mathematical formulations advanced are quite simple and the numerical method proposed is easy and highly accurate. Finally, a few examples of problems in this topic were solved and presented in order to demonstrate the accuracy and the potential possibilities of applications of the present method.     

On contact zone models for an electrically impermeable interface crack in a piezoelectric bimaterial

An electrically impermeable interface crack between two semi-infinite piezoelectric planes under remote mechanical tension-shear and electrical loading is studied. Assuming the stresses, strains and displacements are independent on the coordinate x ₂ the expressions for the elastic displacement and potential jumps as well as for the stresses and electrical displacement along the interface via a sectionally holomorphic vector function are found. Introducing an artificial contact zone at the right crack tip and assuming the materials possess the symmetry class 6 mm the problem is reduced for a wide range of bimaterial compounds to a combination of combined Dirichlet–Riemann and Hilbert boundary value problems which are solved analytically. From these solutions clear analytical expressions for characteristic mechanical and electrical parameters are derived. As particular cases of the above mentioned solution the classical (oscillating) and contact zone solutions are obtained. Further, a comparison with an associated solution for an electrically permeable crack has been performed. The fracture mechanical parameters for all models via the remote loads are found analytically and important relationships between these parameters are obtained. Due to these relationships an important algorithm of a numerical method applicable for the investigation of an interface crack in a finite sized piezoelectric bimaterial is suggested.     

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.     

Electro-elastic behavior induced by an external circular crack in a piezoelectric material

The electro-elastic problem of a transversely isotropic piezoelectric material with a flat crack occupying the outside of a circle perpendicular to the poling axis is considered in this paper. By using the Hankel transform technique, a mixed boundary value problem associated with the considered problem is solved analytically. The results are presented in closed form both for impermeable crack and for permeable crack. A full field solution is given, i.e., explicit expressions for electro-elastic field at any point in the entire piezoelectric space, as well as field intensity factors near the crack front, are determined. A numerical example for a cracked PZT-5H ceramic is given, and the effects of applied electric fields on elastic and electric behaviors are presented graphically.     

Continuous near-tip fields for a dynamic crack propagating in a power-law elastic-plastic material

By use of the J ₂ flow theory and the rectangular components of field quantities, the near-tip asymptotic fields are studied for a dynamic mode-I crack propagating in an incompressible power-law elastic-plastic material under the plan strain conditions. Through assuming that the stress and strain components near a dynamic crack tip are of the same singularity, the present paper constructs with success the fully continuous dominant stress and strain fields. The angular variations of these fields are identical with those corresponding to the dynamic crack propagation in an elastic-perfectly plastic material (Leighton et al., 1987). The dynamic asymptotic field does not reduce to the quasi-static asymptotic field in the limit as the crack speed goes to zero. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mode III crack growth in linear hardening materials with strain gradient effects

The flow-theory version of couple stress strain gradient plasticity is adopted for investigating the asymptotic fields near a steadily propagating crack-tip, under Mode III loading conditions. By incorporating a material characteristic length, typically of the order of few microns for ductile metals, the adopted constitutive model accounts for the microstructure of the material and can capture the strong size effects arising at small scales. The effects of microstructure result in a substantial increase in the singularities of the skew-symmetric stress and couple stress fields, which occurs also for a small hardening coefficient. The symmetric stress field turns out to be non-singular according to the asymptotic solution for the stationary crack problem in linear elastic couple stress materials. The performed asymptotic analysis can provide useful predictions about the increase of the traction level ahead of the crack-tip due to the sole contribution of the rotation gradient, which has been found relevant and non-negligible at the micron scale.     

3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium

This paper considers the non-axisymmetric three-dimensional problem of a penny-shaped crack with permeable electric conditions imposed on the crack surfaces, subjected to a pair of point normal forces applied symmetrically with respect to the crack plane. The crack is embedded in an infinite transversely isotropic piezoelectric body with the crack face perpendicular to the axis of material symmetry. Applying the symmetry of the problem under consideration then leads to a mixed–mixed boundary value problem of a half-space, for which potential theory method is employed for the purpose of analysis. The cases of equal eigenvalues are also discussed. Although the treatment differs from that for an impermeable crack reported in literature, the resulting governing equation still has a familiar structure. For the case of a point force, exact expressions for the full-space electro-elastic field are derived in terms of elementary functions with explicit stress and electric displacement intensity factors presented. The exact solution for a uniform loading is also given.     

含不完美界面的磁电复合材料倾斜裂纹问题

研究了含非完美界面的双层压电/压磁复合材料中压电相存在一个倾斜于界面的Ⅲ型裂纹问题。采用弹簧型耦合界面模型模拟非完美界面,运用Fourier积分变换法将裂纹面条件转化为奇异积分方程,并使用Lobatto-Chebyshev方法数值求解了裂纹尖端应力强度因子（SIF）。详细地研究了裂纹尖端SIF与界面参数、压电/压磁材料参数和材料的层厚、裂纹的倾斜角、裂纹与界面的距离等几何参数的关系。结果表明：力学不完美性可以独立地增大SIF,而磁学、电学不完美性只有与力学不完美性耦合时才会减小SIF;力学-电学、力学-磁学不完美性的耦合会减小SIF,而磁学-电学不完美性的耦合不会影响SIF;磁场作用下,增大压磁层弹性模量会减小SIF,而增大压电层压电系数,减小压电层弹性模量和介电常数,均会减小SIF;界面不完美性会影响SIF随裂纹倾斜角度或裂纹与界面之间距离的变化规律;在一定范围内增加压电层或压磁层厚度可以减小SIF。     

Stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes

The stresses around a crack in an interfacial layer between two dissimilar elastic half-planes are obtained. The crack is parallel to the interfaces. The material constants of the layer vary continuously within a range from those of the upper half-plane to those of the lower half-plane. An internal gas pressure is applied to the surfaces of the crack. To derive the solution, the nonhomogeneous interfacial layer is divided into several homogeneous layers with different material properties. The boundary conditions are reduced to dual integral equations, which are solved by expanding the differences of the crack face displacements into a series. The unknown coefficients in the series are determined using the Schmidt method, and a stress intensity factor is calculated numerically for epoxy-aluminum composites.     

Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials

The near-tip field of a dynamically propagating crack in an incompressible power-law hardening material is studied using the asymptotic method as an extension of Leighton et al. (Journal of the Mechanics and Physics of Solids, 1987, Vol.35, pp.541–563). The crack is subjected to tensile loads and propagates steadily under plane strain conditions. The material deformation is described by the J ₂ flow theory of plasticity with infinitesimal displacement gradients. Results show that, in the present crack-tip field, (a) the angular variations of stresses, strains and particle velocities around the crack tip are fully continuous; (b) the stresses and strains at the crack tip are bounded; (c) there is a free parameter _eq0 which cannot be determined in the asymptotic analysis. Comparisons indicate that the present asymptotic solution matches well with full-field numerical results, and the parameter _eq0 can characterize the effects of the far field on the crack-tip field. Furthermore, the present solution approaches that of Leighton et al. (1987) in the limit as the material hardening exponent goes to infinity, but does not reduce to the accepted solution for quasi-statically growing cracks in the limit as the crack speed goes to zero.