Yoffe-type moving edge crack in a piezoelectric block

Summary We consider an anti-plane edge moving crack problem with the constant velocity in a piezoelectric ceramic block. The far-field anti-plane shear mechanical and in-plane electrical loads are applied to the piezoelectric block. It is expressed to a Fredholm integral equation of the second kind. Expressions for the dynamic field intensity factors and the dynamic energy release rate are obtained. The dynamic stress intensity factor and the dynamic energy release rate depend on the crack propagation speed. Numerical results for several piezoelectric materials are also presented.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Investigation of anti-plane shear behavior of a Griffith crack in a piezoelectric material by using the non-local theory

In this paper, the behavior of a Griffith crack in a piezoelectric material under anti-plane shear loading is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with two pairs of dual integral equations. These equations are solved using Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present at the crack tip.     

A moving Griffith crack at the interface of two dissimilar elastic layers

The anti-plane shear problem of a Griffith crack traveling with a constant velocity at the interface of two dissimilar isotropic elastic layers is considered. Integral transform method is used to reduce the problem to the solution of a singular integral equation which is further reduced, by using Chebyshev polynomials, to a system of algebraic equations. The results for the particular cases of a moving Griffith crack at the interface of a layer and a half-space and two half-spaces are derived. Numerical results for the stress intensity factor are displayed graphically.     

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。     

Investigation of anti-plane shear behavior of a Griffith permeable crack in functionally graded piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading for the permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of dual-integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tips, thus allows us to using the maximum stress as a fracture criterion. The finite hoop stresses at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Non-Local Theory Solution for an Anti-Plane Shear Permeable Crack in Functionally Graded Piezoelectric Materials

In this paper, the non-local theory solution of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading is obtained for the permeable electric boundary conditions, in which the material properties vary exponentially with coordinate parallel to the crack. The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variable is the jump of displacement across the crack surfaces, not the dislocation density function. To solve the dual integral equations, the jump of the displacement across the crack surfaces is directly expanded in a series of Jacobi polynomials. From the solution of the present paper, it is found that no stress and electric displacement singularities are present near the crack tips. The stress fields are finite near the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses and the electric displacements at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively. On the other hand, the angular variations of the strain energy density function are examined to associate their stationary value with locations of possible fracture initiation.     

压电复合材料中幂函数型曲线裂纹的反平面问题

通过构造新保角映射, 利用Stroh公式研究了远场受反平面剪应力和面内电载荷共同作用下无限大压电复合材料中幂函数型曲线裂纹的断裂行为。给出了电不可渗透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解。该解析解在幂函数的幂次为零时, 可退化为已有文献中无限大压电复合材料含直线裂纹的结果, 证明了其合理性。由解析解可知, 裂纹几何形状一定时, 电场分布将不受机械载荷的影响。最后, 通过数值算例讨论了幂函数的幂次、 系数及其在 x₁轴上的投影长度对机械应变能释放率的影响。结果表明, 当压电体仅受 x₂方向载荷作用时, 对于给定幂次与开口的曲线裂纹, 在 x₁轴上的投影长度存在一临界值使其最容易开裂; 而对于给定投影长度与幂次的曲线裂纹, 开口越大裂纹越容易扩展。      

Towards a fracture mechanics for brittle piezoelectric and dielectric materials

A theoretical fracture mechanics for brittle piezoelectric and dielectric materials is developed consistent with standard features of elasticity and dielectricity. The influence of electric field and mechanical loading is considered in this approach and a Griffith style energy balance is used to establish the relevant energy release rates. Results are given for a finite crack in an infinite isotropic dielectric and for steady state cracking in a piezoelectric strip. In the latter problem, the effect of charge separation in the material and discharge in the crack are considered. Observations of crack behavior in piezoelectrics under combined mechanical and electrical load are discussed to assess which features of the theory are useful.     

Moving Dugdale crack along the interface of two dissimilar magnetoelectroelastic materials

In this paper, the concept of Dugdale crack model and Yoffe model is extended to propose a moving Dugdale interfacial crack model, and the interfacial crack between dissimilar magnetoelectroelastic materials under anti-plane shear and in-plane electric and magnetic loadings is investigated considering the magneto-electro-mechanical nonlinearity. It is assumed that the constant moving crack is magneto-electrically permeable and the length of the crack keeps constant. Fourier transform is applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are solved exactly. The explicit expression of the size of the yield zone is derived, and the crack sliding displacement (CSD) is explicitly expressed. The result shows that the stress, electric and magnetic fields in the cracked magnetoelectroelastic material are no longer singular and the CSD is dependent on the loading, material properties and crack moving velocity. The current model can be reduced to the static interfacial crack case when the crack moving velocity is zero.     

一维六方压电准晶中带三条不对称裂纹的圆形孔口的反平面问题

本文利用复变函数方法和保角映射,研究一维六方压电准晶材料中带不对称三裂纹的圆形孔口的的断裂问题。根据准晶压电材料基本方程的基础上,利用点群的对称性和一维六方准晶的线性压电效应,导出了一维六方准晶压电材料反平面问题的控制方程,并结合Cauchy积分公式,得到电非渗透与电渗透边界条件下的裂纹尖端场强度因子的解析表达式。当改变裂纹长度和孔口半径时,所得结果可以模拟出一些新裂纹模型。在不考虑电载荷作用时,所得结果和原有结果是一致的。通过数值算例讨论了材料的几何参数对场强度因子的影响,得出水平裂纹长度和圆半径可以促进裂纹增长。本研究为工程中材料的制备与应用将提供可靠的理论价值。     

Crack expanding with constant velocity in an anisotropic solid under anti-plane strain

An analytic solution is given for a crack expanding with constant velocity from zero length in an anisotropic material under anti-plane strain. Not all anisotropic materials can support anti-plane strain, and the study is therefore by necessity limited to a certain class of materials, including monoclinic materials. A double Laplace transform is used and the inversion technique is based on the self-similarity of the problem. The result shows that the crack shape is elliptic, as in the corresponding isotropic case. The displacement on the crack plane outside the crack is found to be zero. Expressions are given for the stresses, the stress intensity factor and the energy flux into the crack edge. In contrast to the isotropic case a transverse normal stress may appear, singular at the crack edge. This revised version was published online in July 2006 with corrections to the Cover Date.     

Scattering of anti-plane shear waves by a partially debonded piezoelectric circular cylindrical inclusion

Summary This paper presents an analysis of the scattering of anti-plane shear waves by a single piezo-electric cylindrical inclusion partially bonded to an unbounded matrix. The anti-plane governing equations for piezoelectric materials are reduced to Helmholtz and Laplacian equations. The fields of scattered waves are obtained by means of the wave function expansion method when the bonded interface is perfect. When the interface is partially debonded, the region of the debonding is modeled as an interface crack with non-contacting faces. The electric permeable boundary conditions are adopted, i.e. the normal electric displacement and electric potential are continuous across the crack faces. The crack opening displacement is represented by Chebyshev polynomials and a system of equations is derived and solved for the unknown coefficients.     

Mechanical and electrical yielding for an electrically insulated crack in an interlayer between piezoelectric materials

A plane problem for a crack in a thin ductile layer between two piezoelectric materials under remote electromechanical loading is considered. The piezoelectric substrates are of the same material. They are modeled by half-spaces. For the interlayer thickness tending to zero, electrical and mechanical yielding zones of different unknown lengths are introduced at the crack continuations. These zones are modeled by jumps of electric potential and mechanical displacement jumps, respectively. A vector Hilbert problem is formulated and solved exactly. The lengths of the yield zones are found from the conditions of finiteness of the normal stress and of the electrical displacement. The crack opening and the electric potential jumps at the initial crack tip are found in closed form for the cases of electrical yield zone longer and shorter than the mechanical one. Simple formulae for energy release rate are presented for Griffith crack and the developed electromechanical yield models. Also, an approximate model for eliminating the mechanical and electrical singularities is suggested. Numerical results are presented for different values of yield parameters and of remote electromechanical loading. A good agreement between the energy release rate values obtained for a Griffith crack, exact and approximate yield zone models are established.     

TRANSIENT RESPONSE OF A CRACKED INFINITE PIEZOELECTRIC STRIP UNDER ANTI-PLANE IMPACT

By using the well-established integral transform methodology, the dynamic response of stress and electric displacement around a finite crack in an infinite piezoelectric strip are investigated under anti-plane impact. The dynamic intensity factors of stress and electric displacement are obtained analytically. The results show that the dynamic electric field will promote or retard the propagation of the crack at different stages of the loading process. On the other hand, the response of the electric field is coherent with the applied electric load and independent of the external mechanical load. The result obtained for the anti-plane impact of a cracked infinite piezoelectric ceramic can be regarded as a special case of the present work when the width of the strip tends to infinity.     

压电复合材料中圆孔边Ⅲ型非对称裂纹的场强度因子

研究了压电复合材料中圆孔边4个非对称裂纹在远处受面内电载荷和面外力载荷共同作用下的断裂行为。利用复变函数方法和新映射函数将问题转化为Cauchy积分方程组。通过求解Cauchy积分方程组,得到了电非渗透型和电渗透型两种边界条件下裂纹尖端电弹性场和场强度因子的解析解。所得结果不仅可退化为已有解,而且可模拟出若干新的缺陷构型,如压电复合材料中圆孔边三裂纹、半无限压电复合材料中半圆孔边单裂纹及半无限压电体中边界裂纹。将所得结果与有限元结果进行比较,吻合很好,证实了文中方法的正确性和有效性。数值算例分析了缺陷的几何参数对场强度因子的影响规律。     

Theory of dynamic crack branching in brittle materials

The problem of dynamic symmetric branching of a tensile crack propagating in a brittle material is studied within Linear Elastic Fracture Mechanics theory. The Griffith energy criterion and the principle of local symmetry provide necessary conditions for the onset of dynamic branching instability and for the subsequent paths of the branches. The theory predicts a critical velocity for branching and a well defined shape described by a branching angle and a curvature of the side branches. The model rests on a scenario of crack branching based on reasonable assumptions and on exact dynamic results for the anti-plane branching problem. Our results reproduce within a simplified 2D continuum mechanics approach the main experimental features of the branching instability of fast cracks in brittle materials.     

High-speed self-similar debonding of an interface between two orthotropic materials: The anti-plane shear case

The anti-plane problem of the transient debonding of an interface between two orthotropic materials is examined. The material principal axes are allowed to be oblique to the interface. The debonding is modeled as an interface crack propagating self-similarly from zero-length. The extending speed is assumed to be subsonic, transonic or supersonic. We first consider the dynamic debonding under the moving concentrated loading. The moving dislocation model of self-similar propagation of an interface crack is used to formulate the problem in a singular integral equation which is solved analytically. The stress singularity at the crack tips is discussed. The order of singularity is found to be one-half for subsonic debonding and to vary between zero and one-half depending on the crack speeds for transonic debonding. The dynamic stress intensity factors/coefficients for these two situations are presented in closed-form. The paper also concludes that supersonic debonding is impossible unless the loads are directly applied to the crack tips. Finally, the results for dynamic debonding under xⁿ-type loads are presented by using the superposition method. This revised version was published online in August 2006 with corrections to the Cover Date.     

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 for a Griffith crack interacting with a coated inclusion in piezoelectric solid

A Mode III Griffith crack interacting with a coated inclusion in piezoelectric media is investigated. The crack, the coated inclusion are embedded in an infinitely extended piezoelectric matrix media, with the crack being along the radial direction of the inclusion. In the study, three different piezoelectric material phases are involved: the inclusion, the coating layer, and the matrix. A far-field loading condition is considered. During the solution procedure, the crack is simulated as a continuous distribution of screw dislocations. By using the solution of a screw dislocation near a coated inclusion in piezoelectric media as the Green function, the problem is formulated into a set of singular integral equations, which are solved by numerical method. The stress and electric displacement intensity factors are derived in terms of the asymptotic values of the dislocation density functions evaluated from the integral equations. Numerical examples are given for various material constants combinations and geometric parameters.