Analysis of a crack in a half-plane piezoelectric solid with traction-induction free boundary

In this paper, the problem of a crack embedded in a half-plane piezoelectric solid with traction-induction free boundary is analyzed. A system of singular integral equations is formulated for the materials with general anisotropic piezoelectric properties and for the crack with arbitrary orientation. The kernel functions developed are in complex form for general anisotropic piezoelectric materials and are then specialized to the case of transversely isotropic piezoelectric materials which are in real form. The obtained coupled mechanical and electric real kernel functions may be reduced to those kernel functions for purely elastic problems when the electric effects disappear. The system of singular integral equations is solved numerically and the coupling effects of the mechanical and electric phenomena are presented by the generalized stress intensity factors for transversely isotropic piezoelectric materials.     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Analysis of crack moving and curving in anisotropic solids

The general equations for a dynamically curved crack in an anisotropic solid are derived, and the asymptotic fields of a moving crack under arbitrary distributed loading on the crack surface are calculated from them. For a moving crack under mixed-mode loading conditions a general Muskhelishvili type approach is proposed to calculate intensity factors due to crack surface loading in anisotropic materials. The kinking and curving caused by dynamic loading in anisotropic materials are calculated using the maximum normal stress ratio criterion. The results show that cracks in anisotropic solids may deviate from the straight path and approach a direction parallel to the stiff axis even under symmetric loading and that a crack will tend to deviate more from the crack path to the direction of the stiff axis as the crack speed becomes higher.     

垂直磁压电材料界面三维裂纹的超奇异积分法

应用有限部积分概念和广义位移基本解，垂直于磁压电双材料界面三维复合型裂纹问题被转 化为求解一组以裂纹表面广义位移间断为未知函数的超奇异积分方程问题. 进而，通过主部 分析法精确地求得裂纹尖端光滑点附近的奇性应力场解析表达式. 然后，通过将裂纹表面 位移间断未知函数表达为位移间断基本密度函数与多项式之积，使用有限部积分法对超奇异 积分方程组建立了数值方法. 最后，通过典型算例计算，讨论了广义应力强度因子的变化规 律.     

The stress intensity factors near the tips of two collinear cracks in composite under in-plane shear

In this paper, the stress-intensity factors for two collinear cracks in a composite bonded by an isotropic and an anisotropic half-plane were calculated. The cracks are paralell to the interface, and the crack surfaces are loaded by uniform shear stresses. By using Fourier transform, the mixed boundary value problem is reduced to a set of singular integral equations. For solving the integral equations, the crack surface displacements are expanded in triangular series and the unknown coefficients in the series are determined by the Schmidt method. The stress intensity factors for the cracks in the boron-fibre plastics and aluminium joined composite and in carbon-fibre reinforced plastics were calculated numerically.     

On interface crack in laminated anisotropic medium

In this paper, the interface crack problems of a multilayered anisotropic medium under a state of generalized plane deformation are considered within the framework of anisotropic theory. A general solution procedure is introduced such that it can be uniformly applied to media with transversely isotropic, orthotropic, monoclinic, etc. layers. The problem is reduced to the solution of a system of singular integral equations by means of Fourier transform method and the stiffness matrix formulation. A Jacobi polynomial technique is then used to solve the integral equations numerically. The stress intensity factors are provided. The stress intensity factors have been calculated numerically and displayed graphically.     

Hypersingular integral equation method for a three-dimensional crack in anisotropic electro-magneto-elastic bimaterials

Using Green’s functions, the extended general displacement solutions of a three-dimensional crack problem in anisotropic electro-magneto-elastic (EME) bimaterials under extended loads are analyzed by the boundary element method. Then, the crack problem is reduced to solving a set of hypersingular integral equations (HIE) coupled with boundary integral equations. The singularity of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of HIE, and the exact analytical solutions of the extended singular stresses and extended stress intensity factors (SIFs) near the crack front in anisotropic EME bimaterials are given. Also, the numerical method of the HIE for a rectangular crack subjected to extended loads is put forward with the extended crack opening dislocation approximated by the product of basic density functions and polynomials. At last, numerical solutions of the extended SIFs of some examples are obtained.     

Wave scattering from an interface crack in multilayered piezoelectric plate

This paper focuses on the theoretical basis for the study of wave scattering from an interface crack in multilayered piezoelectric media. The materials are taken to be anisotropic with arbitrary symmetry. Based on the Fourier transform technique together with the aid of the stiffness matrix approach, the boundary value problem of wave scattering is reduced to solving a system of Cauchy-type singular equations. The intensity factors and crack opening displacements are defined in terms of the solutions of the corresponding integral equations for any incident frequencies and incident angles. Numerical results are presented. The effects of incident frequencies and crack location on both the major and coupling intensity factors are illustrated. The influence of the piezoelectricity is also shown.     

复合材料承受剪切时双裂纹周围的应力强度因子

本文研究了由各向同性和各向异性半无限接合而成的复合材料中的应力强度因子问题，在复合材料的接合面附近处具有与接合面平行且共线的两个Ｇｒｉｆｆｉｔｈ裂纹，裂纹面上作用有剪应力，本文利用付利叶变换将混合边值问题归毕为求解奇异积分方程问题，为求解这些方程，将裂纹面上，下的位移差展成级数，并满足理解纹面外侧边界条件，级数中的待定系数利用裂纹面内的边界条件和施密特方法求得，本文对硼纤维塑料和铝板接合的复合材料     

On some crack problems in anisotropic thermoelasticity

Some basic equations recently derived by Clements are used to consider crack problems in anisotropic thermoelasticity. The problems concern a single crack in an anisotropic material in which the displacement and stress are independent of one Cartesian coordinate. No symmetry elements of the material are assumed and the temperature, displacement and stress fields are determined for an arbitrary distribution of temperature or heat flux over the crack faces.     

An embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper focuses on the study of the influence of a mixed-mode crack on the coupled response of a functionally graded magnetoelectroelastic material (FGMEEM). The crack is embedded at the center of a 2D infinite medium subjected to magnetoelectromechanical loads. The material is graded in the direction orthogonal to the crack plane and is modeled as a nonhomogeneous medium with anisotropic constitutive laws. Using Fourier transform, the resulting plane magnetoelectroelasticity equations are converted analytically into singular integral equations which are solved numerically to yield the crack-tip mode I and II stress intensity factors, the electric displacement intensity factors and the magnetic induction intensity factors. The main objective of this paper is to study the influence of material nonhomogeneity on the fields’ intensity factors for the purpose of gaining better understanding on the behavior of graded magnetoelectroelastic materials.     

Determining Prefracture Zones at a Crack Tip Between Two Elastic Orthotropic Bodies

The model of prefracture zones is generalized to a crack between two anisotropic materials. An exact analytical solution is found on the assumptions that the prefracture zone is localized on the crack continuation and that only the normal displacements undergo discontinuity in the zone. From this solution, equations for determining the prefracture length and crack tip opening are derived     

Buckling of cracked composite columns

A crack in a structural element introduces a significant local flexibility which enhances the instability. Buckling of an edge-notched beam is studied for isotropic and anisotropic composites. The local compliance due to the presence of cracks in an anisotropic medium is formulated as a function of the crack-tip stress intensity factors and the elastic constants of the material. The general integration of the non-linear differential equations expressing the buckling model of an eccentrically loaded composite beam is derived for two different types of hinged supports ; namely freely approaching and fixed span. The effect of reducing rigidity on the load-carrying capacity and the post-buckling behavior of the beam is discussed and exemplary numerical solutions are provided. The study indicates that the instability increases with the beam slenderness and the crack length. In addition, the material anisotropy conspicuously reduces the load-carrying capacity of an externally cracked member.     

Interface crack problem in layered orthotropic materials under thermo-mechanical loading

In the present paper, the behavior of an interface crack for a homogeneous orthotropic strip sandwiched between two different functionally graded orthotropic materials subjected to thermal and mechanical loading is considered. It is assumed that interface crack is partly insulated, and the temperature drop across the crack surfaces is the result of the thermal resistance due to the heat conduction through the crack region. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The complicated mixed boundary problems of equations of heat conduction and elasticity are converted analytically into singular integral equations, which are solved numerically. The main objective of the paper is to study the effects of material nonhomogeneity parameters and the dimensionless thermal resistance on the thermal stress intensity factors for the purpose of gaining better understanding of the thermal behavior of graded layer.     

Oblique edge crack in an anisotropic material under antiplane shear

An oblique edge crack in an anisotropic material under antiplane shear loadings is investigated. The antiplane problems are formulated based on a linear transformation method. An anisotropic solid containing an edge crack subjected to concentrated forces is first considered. The stress intensity factor for the edge crack with concentrated forces is obtained from the solution of the transformed edge crack in an isotropic material which is solved by using conformal mapping technique and complex function theory. The solution of the edge crack under concentrated loads is used to construct the stress intensity factor for the oblique edge crack in the anisotropic material subjected to antiplane distributed loads. Some numerical computations are carried out to calculate the stress intensity factors for the edge crack in inclined orthotropic materials subjected to point forces as well as distributed tractions.     

Thermoelastic problem of steady-state heat flows disturbed by a crack at an arbitrary angle to the graded interfacial zone in bonded materials

Plane thermoelasticity solutions are presented for the problem of a crack in bonded materials with a graded interfacial zone. The interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermoelastic moduli between dissimilar, homogeneous half-planes. The crack is assumed to exist in one of the half-planes at an arbitrary angle to the graded interfacial zone, disturbing uniform steady-state heat flows. The Fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic thermoelasticity equations. Formulation of the current nonisothermal crack problem lends itself to the derivation of two sets of Cauchy-type singular integral equations for heat conduction and thermal stress analyses. The heat-flux intensity factors and the thermal-stress intensity factors are defined and evaluated in order to quantify the singular characters of temperature gradients and thermal stresses, respectively, in the near-tip region. Numerical results include the variations of such crack-tip field intensity factors versus the crack orientation angle for various combinations of material and geometric parameters of the dissimilar media bonded through the thermoelastically graded interfacial zone. The dependence of the near-tip thermoelastic singular field on the degree of crack-surface partial insulation is also addressed.     

Thermopiezoelectric interaction of macro- and micro-cracks in piezoelectric medium

Considered is the interaction of macro-and micro-cracks in an anisotropic piezoelectric solid. The Green’s function and principle of superposition are used to formulate a system of singular integral equations for solving the unknown temperature discontinuity and elastic displacement-electric potential. The residual heat flux, stress and electric displacement on the microcrack are evaluated directly from the near-tip field of main crack. Numerical results for stress and electric displacement intensity factors in a three-crack system are obtained to illustrate the application of the method.     

M-integral analysis for microcrack-damaged anisotropic composite materials

Summary  This paper presents an M-integral analysis for the microcracked anisotropic composite materials. By using an elementary solution derived for a single finite crack subjected to a concentrated force on crack faces, the problem of strong interacting, arbitrarily oriented and located microcracks in an anisotropic composite materials is reduced to a system of Fredholm integral equations. The crack-tip fracture parameters, such as the stress intensity factors, are evaluated from a numerical solution of the system of integral equations. Its dependence on the coordinate system, calculation, and physical interpretation of the M-integral are discussed in the interaction problem. Finally, a numerical example of the damage evaluation by the M-integral analysis is given. Received 24 September 1999; accepted for publication 8 February 2000     

The mixed mode crack problem in an FGM layer bonded to a homogeneous half-plane

In this paper, the plane elasticity problem of an arbitrarily oriented crack in a FGM layer bonded to a homogeneous half-plane is considered. The problem is modeled by assuming that the elastic properties of the FGM layer are exponential functions of the thickness coordinate and are continuous at the interface of the FGM layer and the half-plane.The Fourier transform technique is used to reduce the problem to the solution of a system of Cauchy-type singular integral equations, which are solved numerically. The stress intensity factors are computed for various crack orientations, crack locations and material parameters. The results show that crack length, crack orientation and the non-homogeneity parameter of the strip material have significant effect on the fracture of the FGM layer.     

Mode III Stress-intensity factors in cracked orthotropic plates—An analogy with propagating cracks in isotropic media

In this paper, the method of reflected caustics—which was used to evaluate Mode III SIF's in stationary cracks in isotropic plates—was extended to deal with stationary cracks in orthotropic plates. Furthermore, a correspondence between the anisotropic stationary case and the case of a Mode III dynamic crack, traversing an isotropic plate, is developed by analyzing appropriately the governing equations of the two problems. For this purpose the singulardisplacement field for rectilinearly orthotropic cracked bodies was combined with either Yoffé's model for steady-state, or Broberg's model for transient-crack propagation. Graphs are given where the equivalence between these cases can be established. In this way, the dynamic problem of the propagating crack in an isotropic medium can be readily simulated by considering the experimentally easier anisotropic stationary case.