狭长磁电弹性体中Ⅲ型半无限裂纹的场强度因子

通过引入合适的保角映射, 利用复变函数方法研究了裂纹面上受反平面剪应力和面内磁电载荷共同作用下狭长磁电弹性体中半无限裂纹的断裂行为, 给出了磁电全非渗透型边界条件下裂纹尖端场强度因子和能量释放率的解析解。当狭长体高度趋于无限大时, 可得到无限大磁电弹性体中半无限裂纹的解析解。若不考虑磁场或电场作用, 所得解可退化为已知解。通过数值算例, 分析了裂纹面上受载长度、狭长体高度以及磁、电和机械载荷对能量释放率的影响。     

Collinear unequal crack series in magnetoelectroelastic materials: Mode I case solved via new real fundamental solutions

For the first time, real fundamental solutions are proposed for in-plane magnetoelectroelastic governing equations. Dislocation simulation method is used to derive singular integral equations for collinear unequal crack series under mode I loadings. Special cases of a single crack and two collinear equal cracks are used to validate the proposed solution. The effects of crack space, crack length and magnetic/electric loading are discussed. Real fundamental solutions have an advantage over complex fundamental solutions, because real analyses are more straightforward than complex analyses. Moreover, the model of collinear unequal cracks is more practical than that of collinear equal cracks.     

An Annular Crack in a Magnetoelectroelastic Layer

A flat annular crack in a magnetoelectroelastic layer subjected to mechanical, electric and magnetic loadings is investigated under magnetoelectrically impermeable boundary condition on the crack surface. Using Hankel transform technique, the mixed boundary value problem is reduced to a system of singular integral equations. With the aid of Gauss-Chebyshev integration technique, the integral equations are further reduced to a system of algebraic equations. The field intensity factor and energy release rate are determined. Numerical results reveal the effects of electric and magnetic loadings and crack configuration on crack propagation and growth.     

有限高磁电弹性体中双半动态与静态裂纹分析

狭长体中的裂纹是断裂力学中经常采用的研究模型。含有共线无限长裂纹的条形磁电弹性体,当面内的力电磁和反平面的剪应力作用在左边裂纹尖端附近的一段裂纹面上时,往往会产生动态断裂。利用复变函数法中的拱形变换公式,导出了磁电全非渗透型边界条件下左裂纹尖端动态的应力强度因子以及机械应变能释放率的解析解。当运动速度趋于零时退化为静止状态下的解。通过数值算例分析了断裂机理,讨论了静止状态下狭长体和裂纹的几何尺寸、外力、电场和磁场分别对能量释放率的影响,为相关器件的设计与制造提供了帮助。     

Fracture analysis of a magnetoelectroelastic solid with a penny-shaped crack by considering the effects of the opening crack interior

A magnetoelectroelastic analysis for a penny-shaped crack embedded in an infinite piezoelectromagnetic material is made. Taking into account the fact that electric and magnetic fields can permeate through the opening crack, the electric and magnetic boundary conditions at the crack surfaces are assumed to be semi-permeable, or depend nonlinearly on the crack opening displacement. For the case of a circular crack normal to the poling direction, the associated mixed boundary value problem is reduced to solving dual integral equations by applying the Hankel transform technique. An entire magnetoelectroelastic field is obtained in simple and explicit form. Numerical results for a cracked BaTiO₃-CoFe₂O₄ material reveal the dependence of the electric displacement and magnetic induction at the crack surfaces with applied mechanical loading. The influences of applied electric and magnetic loadings on normalized fracture parameters are illustrated graphically for a vacuum circular crack. The impermeable and permeable cracks can be treated as two limiting cases of the present.     

Two collinear unequal cracks in a poled piezoelectric plane: Mode I case solved by a new approach of real fundamental solutions

The purpose of the present work is to study the problem of two collinear unequal cracks in a piezoelectric plane under mode I electromechanical loadings via a new approach. For the first time, real fundamental solutions are derived for in-plane piezoelectric governing equations. The cracks are simulated by continuously distributed generalized dislocations and Cauchy singular integral equations are established from the solution of a generalized point dislocation. Both the theorectical derivation and numerical computations are validated by the exact solution in a special case. Parametric studies are conducted to reveal the effects of crack space, crack length, electric loading and remanent electric displacement on energy release rate. It is found that negative electric displacement loading can decrease both the total energy release rate (TERR) and the mechanical strain energy release rate (MSERR), implying that it has a shielding effect on cracks definitely. Positive electric displacement loading can enhance MSERR, but meanwhile it can enhance or reduce TERR depending on the magnitude of the electric loading factor. The effect of a remanent electric displacement along the poling direction is equivalent to that of a positive electric field loading and should be considered in engineering design.     

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。     

Surface crack problem for functionally graded magnetoelectroelastic coating-homogeneous elastic substrate system under anti-plane mechanical and in-plane electric and magnetic loading

A functionally graded magnetoelectroelastic material layer bonded to a homogeneous elastic substrate is investigated. The functionally graded magnetoelectroelastic layer contains a surface crack that is perpendicular to the surface of the medium. The structure is subjected to anti-plane mechanical and in-plane electric and magnetic loads, the crack problem involves the anti-plane elastic field coupled with the in-plane electric and magnetic field. The elastic layer can be an ideal insulator or an ideal conductor. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. Numerical results show the influences of the material gradient parameter and crack configuration on field intensity factors and energy release rates of the functionally graded magnetoelectroelastic coating-homogeneous elastic substrate structure.     

Multiple parallel cracks in a magnetoelectroelastic strip under in-plane magnetic/electric field and clamping-induced antiplane pre-deformation

Many magnetoelectroelastic devices are mechanically clamped during service, and the clamping may lead to pre-deformation, which will in return affect the magnetoelectroelastic behavior. In the present paper, a theoretical model is constructed to formulate the fracture problem of a clamped magnetoelectroelastic strip loaded magnetically or electrically. Two groups of multiple unequal-length cracks, parallel to the lateral boundaries, are assumed to exist in the strip. In the case of antiplane deformation, Green’s functions of the shear stress components are derived by the method of dislocation simulation, and then they are employed to construct singular integral equations (SIEs) for the cracks. The SIEs are numerically solved to get the stress intensity factors (SIFs). It is found that the SIFs are linearly dependent on the applied magnetic/electric field and the clamping-induced antiplane pre-deformation. In addition, parametric studies are performed to reveal the effects of geometrical parameters on the SIFs, and the crack shielding and interference are regarded as the underlying mechanisms for the fracture behavior.     

磁电弹性功能梯度板共线Griffith裂纹断裂行为

该文考察了磁电弹性功能梯度板的反平面问题。该板具有多个垂直于边界的共线裂纹。裂纹表面采用磁电不穿透或可穿透假设。应用积分变换和位错密度函数将问题化为柯西奇异积分方程求解。导出和分析了场强度因子和能量释放率。数值结果表明了载荷组合参数、材料梯度指数及裂纹构形对裂尖断裂行为的影响。     

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.     

Fracture analysis of magnetoelectroelastic solids by using path independent integrals

A solution scheme based on the fundamental solution for a generalized edge dislocation in an infinite magnetoelectroelastic solid is presented to analyze problems involving single, multiple and slowly growing impermeable cracks. The fundamental solution for a generalized dislocation is obtained by extending the complex potential function formulation used for anisotropic elasticity. The solution for a continuously distributed dislocation is derived by integrating the solution for an edge dislocation. The problem of a system of cracks subjected to remote mechanical, electric and magnetic loading is formulated in terms of set of singular integral equations by applying the principle of superposition and the solution for a continuously distributed dislocation. The singular integral equation system is solved by using a numerical integration technique based on Chebyshev polynomials. The J_i and M-integrals for single crack and multi-cracks problems are derived and their dependence on the coordinate system is investigated. Selected numerical results for the M-integral, total energy release rate and mechanical energy release rate are presented for single, double and multiple crack problems. The case of a slowly growing crack interacting with a stationary crack is also considered. It is found that M-integral presents a reliable and physically acceptable measure for assessment of fracture behaviour and damage of magnetoelectroelastic materials.     

压电-压磁板条中反平面裂纹的电-磁-弹性分析

基于线性电磁弹性理论,获得了压电-压磁板条中反平面裂纹尖端附近的奇异应力、电场和磁场。假设裂纹位于和板条边界平行的中心位置,并且裂纹是电磁渗透型的。利用Fourier变换,将裂纹面的混合边值问题化为对偶积分方程,即而归结为第二类Fredholm积分方程。通过渐近分析,得到了裂纹尖端附近应力、应变、电位移、电场、磁场和磁感的封闭表达式。结果表明,对于电磁渗透裂纹,电场强度因子和磁场强度因子总为0;板条的宽度对应力强度因子有显著的影响;能量释放率总为正值。     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semiinfinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semi-infinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

The strip dielectric breakdown model

This paper reports on the analysis of the strip dielectric breakdown (DB) model for an electrically impermeable crack in a piezoelectric medium based on the general linear constitutive equations. The DB model assumes that the electric field in a strip ahead of the crack tip is equal to the dielectric breakdown strength, which is in analogy with the classical Dugdale model for plastic yielding. Using the Stroh formalism and the dislocation modeling of a crack, we derived the relationship between the DB strip size and applied mechanical and electrical loads, the intensity factors of stresses and electric displacement, and the local energy release rate. Based on the results, we discussed the effect of electric fields on fracture of a transversely isotropic piezoelectric ceramic by applying the local energy release rate as a failure criterion. It is shown that for an impermeable crack perpendicular to the poling direction, a positive electric field will assist an applied mechanical stress to propagate the crack, while a negative electric field will retard crack propagation. However, for an impermeable crack parallel to the poling direction, it is found that the applied electric field does not change the mode I stress intensity factor and the local energy release rate, i.e., the applied electric field has no effect on the crack growth.     

Three Collinear Antiplane Interfacial Cracks in Dissimilar Piezoelectric Materials

Three collinear impermeable interfacial cracks in bonded dissimilar transversely isotropic piezoelectric materials under electromechanical loadings is analyzed. A single antiplane mechanical and inplane electrical loads are applied at a point on crack surface. The problem is formulated by the complex function method, and reduced to the vector Hilbert problem. Solving this problem, a closed form solution for the stress intensity and electric displacement intensity factor is obtained. This solution can be used as a Green??s function for different loading conditions.     

The transient response of bonded piezoelectric and elastic half space with multiple interfacial collinear cracks

Summary This paper investigates the dynamic behavior of a bonded piezoelectric and elastic half space containing multiple interfacial collinear cracks subjected to transient electro-mechanical loads. Both the permeable and impermeable boundary conditions are examined and discussed. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which can be solved using Chebyshev polynomial expansions. Numerical results are provided to show the effect of the geometry of interacting collinear cracks, the applied electric fields, and the electric boundary conditions along the crack faces on the resulting dynamic stress intensity and electric displacement intensity factors.     

Antiplane Permeable Edge Cracks in a Piezoelectric Strip of Finite Width

A piezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

Antiplane permeable edge cracks in a piezoelectric strip of finite width

A priezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.