Numerical analysis of doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic medium using the boundary integral equation method

In this paper, the boundary integral equation approaches are used to study the doubly periodic array of cracks/rigid-line inclusions in an infinite isotropic plane medium. For the doubly periodic rigid-line inclusion problems, the special integral equation containing the axial and shear forces within the rigid-line inclusion is used. The doubly periodic crack problems are dealt with using the displacement discontinuous integral equation approach. Stress intensity factors, effective elastic properties for doubly periodic array of cracks/rigid-line inclusions are calculated and compared with the available numerical solutions.     

Stroh formalism based boundary integral equations for 2D magnetoelectroelasticity

This paper presents a novel approach for obtaining boundary integral equations of 2D anisotropic magnetoelectroelasticity. This approach is based on the holomorphy theorems and the Stroh formalism and allows developing of the integral equations for the aperiodic, singly and doubly periodic problems of magnetoelectroelasticity. Obtained equations contain the unknown discontinuities of displacement, electric and magnetic potentials and also traction, electric displacement and magnetic induction that allow adopting the existing boundary element procedures for their solution. Analytical solutions for systems of collinear permeable or impermeable cracks are obtained. Numerical boundary element solutions are obtained for the singly and doubly periodic sets of permeable and impermeable cracks in the magnetoelectroelastic medium and a half-plane. Comparison with analytical solutions and other available results validate the present formulations and numerical computation.     

Interaction between periodic cracks and periodic rigid-line inclusions in piezoelectric materials

Piezoelectric materials with a doubly periodic array of cracks and rigid-line inclusions under far-field antiplane mechanical load and inplane electric load are investigated. By employing the conformal mapping technique and the elliptical function theory, an exact solution of the whole-field stress and electrical displacement is obtained. The closed-form formulae for the stress and electrical displacement intensity factors at the tips of cracks and rigid-line inclusions and the effective electroelastic moduli of the composites are presented. Some existing solutions can be regarded as degenerated cases of the present results. Numerical examples are provided to show the interesting electroelastic interaction between multiple cracks and multiple rigid-line inclusions.     

双周期电磁弹性纤维增强复合材料的纵向剪切问题

利用等效夹杂法并结合双 (准) 周期解析函数理论,为双周期电磁弹性圆截面纤维复合材料的纵向剪切问题发展了实用有效的解析方法,获得了全场的级数解,给出了应力、电位移和磁感应强度的显式结果。利用数值算例讨论了该类非均匀材料微结构对其电磁弹性性能的影响。本文方法为电磁弹性复合材料的性能分析和微结构设计提供了一个实用有效的计算工具。      

Coupled 2D electric and mechanical fields in piezoelectric solids containing cracks and thin inhomogeneities

This paper develops integral equations and boundary element method for determination of 2D electro-elastic state of solids containing cracks, thin voids and inclusions. It proves that stress and electric displacement field near the tip of thin inhomogeneity possesses square root singularity. Thus, for determination of electromechanical fields near thin defects new special base functions are introduced. The interpolation quadratures along with the polynomial transformations are adopted for efficient numerical evaluation of singular and hypersingular integrals. Presented numerical examples show high efficiency and accuracy of the proposed approach.     

A comprehensive study on the 2D boundary element method for anisotropic thermoelectroelastic solids with cracks and thin inhomogeneities

This paper develops Somigliana type boundary integral equations for 2D thermoelectroelasticity of anisotropic solids with cracks and thin inclusions. Two approaches for obtaining of these equations are proposed, which validate each other. Derived boundary integral equations contain domain integrals only if the body forces or distributed heat sources are present, which is advantageous comparing to the existing ones. Closed-form expressions are obtained for all kernels. A model of a thin pyroelectric inclusion is obtained, which can be also used for the analysis of solids with impermeable, permeable and semi-permeable cracks, and cracks with an imperfect thermal contact of their faces. The paper considers both finite and infinite solids. In the latter case it is proved, that in contrast with the anisotropic thermoelasticity, the uniform heat flux can produce nonzero stress and electric displacement in the unnotched pyroelectric medium due to the tertiary pyroelectric effect. Obtained boundary integral equations and inclusion models are introduced into the computational algorithm of the boundary element method. The numerical analysis of sample and new problems proved the validity of the developed approach, and allowed to obtain some new results.     

A general method of solution of two-dimensional problems in the theory of cracks

Original systems of singular integral equations of plane problems in the theories of elasticity, heat conduction and thermoelasticity for a solid containing a set of straight thermally isolated arbitrarily oriented cracks are developed.The problems such as those of a set of cracks in infinite and semi-infinite plates, a set of cracks in a circular disk and an infinite plane with a circular hole, a periodic and doubly periodic set of arbitrarily oriented cracks in an infinite plate, a periodic row of cracks in a half-plane are considered.Analytical solution of the equations obtained is performed by using a perturbation technique when cracks are wide apart or remote at a large distance from the boundary of a region; otherwise the equations are solved numerically.     

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。     

The contact problem of a rigid stamp with friction on a functionally graded magneto-electro-elastic half-plane

We consider in this study the frictional sliding contact problem between a functionally graded magneto-electro-elastic material and a perfectly conducting rigid punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the unknowns are the normal contact stress, the electric displacement and the magnetic induction. These integral equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the non-homogeneity parameter, the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat and circular punch profiles.     

An Interaction Integral Method for Computing Fracture Parameters in Functionally Graded Magnetoelectroelastic Composites

A contour integral method is developed for the computation of stress intensity, electric and magnetic intensity factors for cracks in continuously nonhomogeneous magnetoelectroelastic solids under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium are the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for the computation of the physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations (LIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LIEs. The accuracy of the present method for computing the stress intensity factors (SIF), electrical displacement intensity factors (EDIF) and magnetic induction intensity factors (MIIF) are discussed by comparison with numerical solutions for homogeneous materials.     

An integral equation approach to the inclusion-crack interactions in three-dimensional infinite elastic domain

 In this paper, an integral equation method to the inclusion-crack interaction problem in three-dimensional elastic medium is presented. The method is implemented following the idea that displacement integral equation is used at the source points situated in the inclusions, whereas stress integral equation is applied to source points along crack surfaces. The displacement and stress integral equations only contain unknowns in displacement (in inclusions) and displacement discontinuity (along cracks). The hypersingular integrals appearing in stress integral equation are analytically transferred to line integrals (for plane cracks) which are at most weakly singular. Finite elements are adopted to discretize the inclusions into isoparametric quadratic 10-node tetrahedral or 20-node hexahedral elements and the crack surfaces are decomposed into discontinuous quadratic quadrilateral elements. Special crack tip elements are used to simulate the variation of displacements near the crack front. The stress intensity factors along the crack front are calculated. Numerical results are compared with other available methods. Received: 28 January 2002 / Accepted: 4 June 2002 The work described in this paper was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region, China (Project No.: HKU 7101/99 E).     

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.     

Thermoelectroelastic response of a functionally graded piezoelectric strip with two parallel axisymmetric cracks

A mixed-mode thermoelectroelastic fracture problem of a functionally graded piezoelectric material strip containing two parallel axisymmetric cracks, such as penny-shaped or annular cracks, is considered in this study. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. Using integral transform techniques, the problem is reduced to that of solving two systems of singular integral equations. Systematic numerical calculations are carried out, and the variations of the stress and electric displacement intensity factors are plotted for various values of dimensionless parameters representing the crack size, the crack location and the material non-homogeneity.     

An eigenfunction expansion-variational method based on a unit cell in analysis of a generally doubly periodic array of cracks

A new variational functional for a unit cell of a heterogeneous solid with periodic microstructures is constructed by incorporating the quasi-periodicity of the displacement field and the periodicity of the stress and strain fields into the strain energy functional. The functional can accommodate a broad class of periodic structures including the case where symmetry or antisymmetry properties of the unit cell may not exist. Then the functional is applied to deal with a doubly periodic array of cracks under plane and anti-plane loading. By combining with the eigenfunction expansions of the complex potentials satisfying the traction-free condition on the crack surfaces, an eigenfunction expansion-variational method based on a unit cell is developed. Numerical examples are presented and compared with existing results to demonstrate the high accuracy and efficiency, and wide application scope of the present method. Some interesting phenomena of multi-crack interaction, which do not occur in the case of symmetrical arrays of cracks, are revealed and discussed.     

Boundary integral equations for 2D thermoelectroelasticity of a half-space with cracks and thin inclusions

The paper presents a rigorous and straightforward approach for obtaining the 2D boundary integral equations for a thermoelectroelastic half-space containing holes, cracks and thin foreign inclusions. It starts from the Cauchy integral formula and Stroh orthogonality relations to obtain the integral formulae for the Stroh complex functions, which are piecewise-analytic in the complex half-plane with holes and opened mathematical cuts. Further application of the Stroh formalism allows derivation of the Somigliana type integral formulae and boundary integral equations for a thermoelectroelastic half-space. The kernels of these equations correspond to the fundamental solutions of heat transfer, electroelasticity and thermoelectroelasticity for a half-space. It is shown that the difference between the obtained fundamental solution of thermoelectroelasticity and those presented in literature is due to the fact, that present solution additionally accounts for extended displacement and stress continuity conditions, thus, it is physically correct. Obtained integral equations are introduced into the boundary element approach. Numerical examples validate derived boundary integral equations, show their efficiency and accuracy.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

Electroelastic fields and effective moduli of a medium containing cavities or rigid inclusions of arbitrary shape under anti-plane mechanical and in-plane electric fields

Summary We examine the two-dimensional problem of an infinite piezoelectric medium containing a solitary cavity or rigid inclusion of arbitrary shape, subjected to a coupled anti-plane mechanical and in-plane electric load at the remote boundary of the matrix. Conformal mapping techniques are employed to analyze the boundary value problems. Specific results are given for elliptical, polygonal and star-shape inclusions. Local fields of this type are used to estimate the overall moduli of a medium containing voids or rigid inclusions. This is accomplished with the help of an extension of Eshelby's formula which evaluates the total electric enthalpy by a particular line integral. Explicit estimates of the effective moduli are derived for dilute as well as for moderate area fractions of inclusions. The formulae depend solely on the cross area of the inclusion, area fraction and one particular coefficient of the mapping function. In addition, the stress and electric displacement singularities around the sharp corners of the inclusion are examined. The existence of uniform fields inside the inclusion is also envisaged. The present results, with appropriate modifications, apply equally well to those of thermoelectric and magnetoelectric effects.     

Complex hypersingular integral equation for the piece-wise homogeneous half-plane with cracks

New complex hypersingular integral equation (CHSIE) is derived for the half-plane containing the inclusions (which can have the different elastic properties), holes, notches and cracks of the arbitrary shape. This equation is obtained by superposition of the equations for each homogeneous region in a half-plane. The last equations follow from the use of complex analogs of Somigliana's displacement and stress identities (SDI and SSI) and Melan's fundamental solution (FS) written in a complex form. The universal numerical algorithm suggested before for the analogous problem for a piece-wise homogeneous plane is extended on case of a half plane. The unknown functions are approximated by complex Lagrange polynomials of the arbitrary degree. The asymptotics for the displacement discontinuities (DD) at the crack tips are taken into account. Only two types of the boundary elements (straight segments and circular arcs) are used to approximate the boundaries. All the integrals involved in CHSIE are evaluated in a closed form. A wide range of elasticity problems for a half-plane with cracks, openings and inclusions are solved numerically.     

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.