A moving thermal dielectric crack in piezoelectric ceramics with a shearing force applied on its surface

This article conducts an exact analysis of a thermal dielectric crack moving in piezoelectric materials. Self-generating thermal and electric loadings by the crack interior are exerted on the crack surfaces as well as various external loadings including a shearing force. Fundamental solutions of the thermal and electro-elastic coupling fields are given by determining a temperature function and a harmonic function with eigenvalues properties due to material properties considered. Analytical expressions are obtained benefiting evaluation of key parameters. Numerical analysis is done and some interesting observations are found. There is a critical crack velocity within and beyond which the electric loading exerts different influences on the thermal flux of crack interior and the thermal stress intensity factor.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.     

Static and extending interface cracks with contact zones in anisotropic as well as in piezoelectric bimaterials

An interface crack with an electrically permeable and mechanically frictionless contact zone in a piezoelectric bimaterial under the action of a remote mixed mode mechanical loading as well as thermal and electrical fields is considered in the first part of this paper. By use of the matrix‐vector representations of thermal, mechanical and electrical fields via sectionally‐holomorphic functions the problems of linear relationships are formulated and solved exactly both for an electrically permeable and an electrically impermeable interface crack. For these cases the transcendental equations and clear analytical formulas are derived for the determination of the contact zone lengths and the associated fracture mechanical parameters. A plane strain problem for a crack with a frictionless contact zone at the leading crack tip extending stationary along an interface of two semi‐infinite anisotropic spaces with a subsonic speed under the action of various loading is considered in the second part of this paper. By introducing of a moving coordinate system connected with the crack tip and by using the formal similarity of static and propagating crack problems the combined Dirichlet‐Riemann boundary value problem is formulated and solved exactly for this case as well and a transcendental equation is obtained for the determination of the real contact zone length. It is found that the increase of the crack speed leads to an increase of the real contact zone length and the correspondent stress intensity factors which increase significantly for a quasi‐Rayleigh wave speed.     

正交异性双材料的Ⅱ型界面裂纹尖端场

通过引入含16个待定实系数和两个实应力奇异指数的应力函数,再借助边界条件,得到了两个八元非齐次线性方程组.求解该方程组,在双材料工程参数满足适当条件下,确定了两个实应力奇异指数.根据极限唯一性定理,求出了全部系数,得到了应力函数的表示式.代入相应的力学公式,推出了当特征方程组两个判别式都小于0时,每种材料的裂纹尖端应力强度因子、应力场和位移场的理论解.裂纹尖端附近的应力和位移有混合型断裂特征,但没有振荡奇异性和裂纹面相互嵌入现象作为特例,当两种正交异性材料相同时,可以推出正交异性单材料Ⅱ型断裂的应力奇异指数、应力强度因子公式、应力场、位移场表示式.     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.     

压电材料中两个非对称平行裂纹的基本解

采用Schmidt方法分析压电材料中非对称平行的双可导通裂纹的断裂性能．利用Fourier变换使问题的求解转换为求解两对以裂纹面位移之差为未知变量的对偶积分方程．为了求解对偶积分方程,直接把裂纹面位移差函数展开成Jacobi多项式形式．最终得到了裂纹的应力强度因子与电位移强度因子之间的关系．数值结果表明,应力强度因子和电位移强度因子与裂纹间的距离、裂纹的几何尺寸有关;与不可导通裂纹有关结果相比,可导通裂纹的电位移强度因子远小于相应问题不可导通裂纹的电位移强度因子．同时可以发现裂纹间的“屏蔽”效应也在压电材料中出现．     

压电压磁复合材料中界面裂纹对弹性波的散射

利用Schmidt方法分析了压电压磁复合材料中可导通界面裂纹对反平面简谐波的散射问题．经过富里叶变换得到了以裂纹面上的间断位移为未知变量的对偶积分方程A·D2在求解对偶积分方程的过程中,裂纹面上的间断位移被展开成雅可比多项式的形式．数值模拟分析了裂纹长度、波速和入射波频率对应力强度因子、电位移强度因子、磁通量强度因子的影响A·D2从结果中可以看出,压电压磁复合材料中可导通界面裂纹的反平面问题的应力奇异性形式与一般弹性材料中的反平面问题应力奇异性形式相同．     

Basic solutions of a 3D rectangular limited-permeable crack or two 3D rectangular limited-permeable cracks in the piezoelectric/piezomagnetic composite materials

The solutions of a three-dimensional rectangular limited-permeable crack or two three-dimensional rectangular limited-permeable cracks in the piezoelectric/piezomagnetic composite materials were investigated by using the generalized Almansi’s theorem and the Schmidt method. Finally, the relations among the electric field, the magnetic flux field and the stress field near the crack tips were obtained and the effects of the electric permittivity, the magnetic permeability of the air inside the crack, the shape of the rectangular crack on the stress, the electric displacement and magnetic flux intensity factors in the piezoelectric/piezomagnetic composite materials were analyzed.     

Problem of thermoelectromagnetoelasticity for a piezoelectric/piezomagnetic bimaterial with interface crack

We consider a plane strain problem for a piezoelectric/piezomagnetic bimaterial space with a crack in the region of the interface of the materials. At infinity, tensile and shear stresses and heat, electric, and magnetic flows are set. Using representations for all mechanical, thermal, and electromagnetic factors in terms of piecewise analytic functions, we formulate problems of linear conjugation that correspond to a model of an open crack and models taking into account the contact zone in the vicinity of a crack tip. Exact analytic solutions of the indicated problems are constructed. Expressions for stresses, the electric and magnetic inductions, jumps of derivatives of displacements, and electric and magnetic potentials on the interface are written. The coefficients of intensities of the indicated factors are presented. We derive a transcendental equation for the determination of the real length of the contact zone. The dependences of this length and the coefficients of intensity on the set external influences are investigated.     

A periodic system of electrically permeable cracks at the interface between two piezoelectric materials

We constructed a closed solution for a bimaterial plane consisting of two dissimilar piezoelectric half-planes with a periodic system of electrically permeable cracks at the interface between these materials. The presence of zones of smooth contact of the crack lips near their tips was taken into account. By representing the characteristics of electromechanical fields via piecewise analytic functions, we reduced the problem to a Dirichlet–Riemann periodic problem, which was solved exactly. As a result of numerical analysis of the derived solution, we studied the dependence of the relative length of the contact zones and stress intensity factors on the ratio between the crack length and period for different combinations of piezoelectric materials.     

Crack – interface crack interactions in functionally graded/homogeneous composite bimaterials subjected to a heat flux

A bimaterial containing an interface crack and consisting of a homogeneous material and a functionally graded material (FGM) with a system of small internal cracks is considered. The thermal fracture of the biomaterial under the action of a heat flux applied to it at infinity is investigated. The problem is studied in the case where the interface crack is much larger than the internal ones. It is assumed that the thermal properties of the FGM are continuous functions of the thickness coordinate. Asymptotic analytical formulas for the thermal stress intensity factors (TSIFs) at the tips of the interface crack are obtained as series in a small parameter (the ratio between sizes of the internal and interface cracks). Then, the critical heat fluxes and the angles of propagation direction of the interface crack are calculated using the criterion of maximum circumferential stress. A parametric analysis shows that the propagation direction of the interface crack depends on the location and orientation of the system of internal cracks. The parameters of inhomogeneity of the FGM affect the value of TSIFs and, hence, the deflection angle of the interface crack.     

具有双裂纹的弹性复合材料板承受剪切冲击时的动应力强度因子

本文研究了两个材半限弹性的接合面附近存在与接合面平行的双裂纹，并承受剪切冲击时的瞬态应力，运用付里叶（Ｆｏｕｒｉｅｒ）和拉普拉斯（Ｌａｐｌａｃｅ）变换，将问题归结为求解二元积分方程，求解时将裂纹所在面上，下的位移差展成级数，并让其自动满足裂纹面外的位移差为零的条件，利用裂纹面上的边界条件和施密特（Ｓｃｈｍｉｄｔ）方法求解级数中的待定系数，在拉普拉斯像空间中，求得动应力强度因子，并将其数值地逆变换至     

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.     

A new boundary element method for the solution of plane steady-state thermoelastic fracture mechanics problems

An efficient numerical technique is developed for plane, homogeneous, isotropic, steady-state thermoelasticity problems involving arbitrary internal smooth and/or kinkedcracks. The thermal stress intensity factors and relative crack surface displacements due to steady-state temperature distributions are determined and compared to available solutions obtained by other methods. In these analyses the thermal boundary conditions across the crack surface are assumed to be insulated. The present approach involves coupling the direct boundary integral equations to newly developed crack integral equations.     

A BEM for transient thermoelastic fracture analysis of FGMs

A boundary element method for the transient thermoelastic fracture analysis in isotropic, continuously non-homogeneous and linear elastic functionally graded materials subjected to a thermal shock is presented. The material parameters are assumed to be continuous functions of the Cartesian coordinates. Laplace-domain fundamental solutions of linear coupled thermoelasticity for infinite, isotropic, homogeneous and linear elastic solids are applied to derive the boundary-domain integral equation formulation. The numerical implementation is performed by using a collocation method for the spatial discretization. Numerical results for the dynamic stress intensity factors are presented and discussed. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Contact Problems for Interface Cracks under Harmonic Loading

The linear crack located at the bimaterial interface between dissimilar homogeneous linearly elastic materials under normally incident harmonic tension-compression wave is considered in the study. The problem is solved by the method of boundary integral equations using an iterative algorithm. The dynamic stress intensity factors are computed as functions of the loading frequency taking the contact interaction of the opposite crack's faces into account. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effects of the material gradation on the SIFs of a crack in a 2-D FGM plate under thermal shock

Crack analysis of linear coupled thermoelasticity in two-dimensional, isotropic, non-homogeneous and linear elastic functionally graded materials subjected to thermal shock is performed by using a boundary-domain element method. The material parameters are assumed to be continuous functions of the spatial coordinates. Fundamental solutions of linear coupled thermoelasticity for the corresponding isotropic, homogeneous and linear elastic solids in Laplace-transformed domain are applied in the boundary-domain integral formulation. A collocation method is implemented for the numerical solution. Numerical examples for an exponential gradation of the material parameters are presented and discussed. The influences of the material gradation on the stress intensity factors are investigated. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress distribution in a transversely isotropic body with a thermally insulating parabolic crack subjected to a uniform heat flux

An explicit static thermoelastic solution is constructed for an infinite transversely isotropic body containing a thermally insulating parabolic crack in the plane of isotropy. The surface of the crack is free of stress. A uniform thermal flux is incident on the crack perpendicular to its surface. Formulas are obtained for the stress intensity factors near the tip of the crack. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; Catholic University, Portugal. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 54–66, 1999.     

Closed-form solutions for a rectangular layer with central crack under anti-plane shear stress

We consider the problem of determining the stress distributionin a finite rectangular elastic layer containing a Griffithcrack which is opened by internal shear stress acting alongthe length of the crack. The mode III crack is assumed to belocated in the middle plane of the rectangular layer. The followingtwo problems are considered: (A) the central crack is perpendicularto the two fixed lateral surfaces and parallel to the othertwo stress-free surfaces; (B) all the lateral surfaces of therectangular layer are clamped and the central crack is parallelto the two lateral surfaces. By using Fourier transformations,we reduce the solution of each problem to the solution of dualintegral equations with sine kernels and a weight function whichare solved exactly. Finally, we derive closed-form expressionsfor the stress intensity factor at the tip of the crack andthe numerical values for the stress intensity factor at theedges of the cracks are presented in the form of tables.