Antiplane electro-mechanical field of a piezoelectric finite wedge under shear loading and at fixed-grounded boundary conditions

This paper presents the general solutions of antiplane electro-mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as fixed and grounded. Employing the finite Mellin transform method, the stress and electrical displacement at all fields of the piezoelectric finite wedge are derived analytically. In addition, the singularity orders and intensity factors of stress and electrical displacement can also be obtained. These parameters can be applied to examine the fracture behavior of the wedge structure. After being reduced to the problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.     

Some problems in the antiplane shear deformation of bi-material wedges

The antiplane shear deformation of a bi-material wedge with finite radius is studied in this paper. Depending upon the boundary condition prescribed on the circular segment of the wedge, traction or displacement, two problems are analyzed. In each problem two different cases of boundary conditions on the radial edges of the composite wedge are considered. The radial boundary data are: traction–displacement and traction–traction. The solution of governing differential equations is accomplished by means of finite Mellin transforms. The closed form solutions are obtained for displacement and stress fields in the entire domain. The geometric singularities of stress fields are observed to be dependent on material property, in general. However, in the special case of equal apex angles in the traction–traction problem, this dependency ceases to exist and the geometric singularity shows dependency only upon the apex angle. A result which is in agreement with that cited in the literature for bi-material wedges with infinite radii. In part II of the paper, Antiplane shear deformation of bi-material circular media containing an interfacial edge crack is considered. As a special case of bi-material wedges studied in part I of the paper, explicit expressions are derived for the stress intensity factor at the tip of an edge crack lying at the interface of the bi-material media. It is seen that in general, the stress intensity factor is a function of material property. However, in special cases of traction–traction problem, i.e., similar materials and also equal apex angles, the stress intensity factor becomes independent of material property and the result coincides with the results in the literature.     

The effect of a fiber loss in periodic composites

Three types of analyses are combined to investigate the effect of missing fibers in periodic continuous fiber composites that are subjected to thermomechanical loadings. The representative cell method is employed in the first analysis for the construction of Green’s functions elastic fields for the fiber–matrix interfacial jumps problem. As a result, the infinite domain problem is reduced, in conjunction with the discrete Fourier transform, to a finite domain problem on which Born–von Karman type boundary conditions are applied. In the second analysis, the transformed elastic field is determined by a second-order expansion of the displacement vector in terms of local coordinates, and by imposing the equilibrium equations, the interfacial traction and displacement conditions, and the Born–von Karman type boundary conditions. The actual non-periodic elastic field at any point is obtained from the Fourier-transformed fields by a numerical inversion. In the third one, a micromechanical analysis for periodic continuous fiber composites in which all fibers are perfectly bonded to the surrounding matrix provides the elastic field within the phases. A superposition of the thermoelastic fields obtained from the first and third analysis provides the traction-free boundary conditions at the interface of the missing fibers. The accuracy of the offered approach is verified by comparison with analytical solutions that exist in some special cases. Results show the effect of a missing fiber in boron/epoxy and glass/epoxy composites that are subjected to various types of thermomechanical loadings.     

Extended Kantorovich method for local stresses in composite laminates upon polynomial stress functions

The extended Kantorovich method is employed to study the local stress concentrations at the vicinity of free edges in symmetrically layered composite laminates subjected to uniaxial tensile load upon polynomial stress functions. The stress fields are initially assumed by means of the Lekhnitskii stress functions under the plane strain state. Applying the principle of complementary virtual work, the coupled ordinary differential equations are obtained in which the solutions can be obtained by solving a generalized eigenvalue problem. Then an iterative procedure is estab-lished to achieve convergent stress distributions. It should be noted that the stress function based extended Kantorovich method can satisfy both the traction-free and free edge stress boundary conditions during the iterative processes. The stress components near the free edges and in the interior regions are calculated and compared with those obtained results by finite element method (FEM). The convergent stresses have good agreements with those results obtained by three dimensional (3D) FEM. For generality, various layup configurations are considered for the numerical analysis. The results show that the proposed polynomial stress function based extended Kan-torovich method is accurate and efficient in predicting the local stresses in composite laminates and computationally much more efficient than the 3D FEM.     

ANTI-PLANE FRACTURE ANALYSIS OF FUNCTIONALLY GRADIENT MATERIAL INFINITE STRIP WITH FINITE WIDTH

The special case of a crack under mode III conditions was treated, lying parallel to the edges of an infinite strip with finite width and with the shear modulus varying exponentially perpendicular to the edges. By using Fourier transforms the problem was formulated in terms of a singular integral equation. It was numerically solved by representing the unknown dislocation density by a truncated series of Chebyshev polynomials leading to a linear system of equations. The stress intensity factor (SIF) results were discussed with respect to the influences of different geometric parameters and the strength of the non-homogeneity. It was indicated that the SIF increases with the increase of the crack length and decreases with the increase of the rigidity of the material in the vicinity of crack. The SIF of narrow strip is very sensitive to the change of the non-homogeneity parameter and its variation is complicated. With the increase of the non-homogeneity parameter, the stress intensity factor may increase, decrease or keep constant, which is mainly determined by the strip width and the relative crack location. If the crack is located at the midline of the strip or if the strip is wide, the stress intensity factor is not sensitive to the material non-homogeneity parameter.     

Axially symmetric thermal stress of an external circular crack under general thermal conditions

Summary The paper presents a solution for the linear thermoelastic problem of determining axisymmetric stress and displacement fields in an isotropic elastic solid of infinite extent weakened by an external circular crack under general mechanical loadings and general thermal conditions. The mechanical loadings and thermal conditions applied on the crack faces are axisymmetric, being non-symmetric about the crack plane. In similar lines of [7], equations of equilibrium of an elastic solid conducting heat have been solved using Hankel transforms and Abel operators of the first kind. Expressions for stress, displacement, temperature and heat flux functions are obtained in terms of Abel transforms of the first kind of the jumps of stress, displacement, temperature and heat flux at the crack plane. Two types of thermal conditions, that is, general surface temperatures and general heat flux on faces of the crack are considered. In both the cases, closed form solutions have been obtained for the unknown functions solving Abel type of integral equations. Explicit expressions for stresses, displacements, temperature fields, stress intensity factors have been obtained. Two special cases of thermal conditions in which: (i) crack faces are subjected to constant non-symmetric temperatures over a circular ring area, (ii) crack faces are subjected to constant non-symmetric heat flux over a circular ring area, have been considered. In some special cases, results have been compared with those from the literature.     

Mixed method and convex optimization for limit analysis of homogeneous Gurson materials: a kinematical approach

A fully kinematical, mixed finite element approach based on a recent interior point method for convex optimization is proposed to solve the limit analysis problem involving homogeneous Gurson materials. It uses continuous or discontinuous quadratic velocity fields as virtual variables, with no hypothesis on a stress field. Its modus operandi is deduced from the Karush–Kuhn–Tucker optimality conditions of the mathematical problem, providing an example of cross-fertilization between mechanics and mathematical programming. This method is used to solve two classical problems for the von Mises plasticity criterion as a test case, and for the Gurson criterion for which analytical solutions do not exist. Using only the original plasticity criterion as material data, the method proposed appears robust and efficient, providing very tight bounds on the limit loadings investigated.     

Analysis of anisotropic sector with a radial crack under anti-plane shear loading

In this paper, the anti-plane shear deformation of an anisotropic sector with a radial crack is investigated. The traction–traction boundary conditions are imposed on the radial edges and the traction-free condition is considered on the circular segment of the sector. A novel mathematical technique is employed for the solution of the problem. This technique consists of the use of some recently proposed finite complex transforms (Shahani, 1999), which have complex analogies to the standard finite Mellin transforms of the first and second kinds. However, it is essential to state the traction-free condition of the crack faces in the form of a singular integral equation which is done in this paper by describing an exact analytical method. The resultant dual integral equations are solved numerically to determine the stress intensity factors at the crack tips. In the special cases, the obtained results coincide with those cited in the literature.     

GREEN''''S FUNCTIONS FOR A 2D ANISOTROPIC MEDIUM WITH A HYPERBOLIC BOUNDARY

By using Stroh's formalism and the conformal mapping technique, this paper derives simple explicit Green's functions of a piezoelectric anisotropic body with a free or fixed hyperbolic boundary. The corresponding elastic fields in the medium are obtained, too. In particular, degenerated solutions of an external crack from those of a hyperbolic problem are analysed in detail. Then the singular stress fields and the fracture mechanics parameters are found. The solutions obtained are valid not only for plane and antiplane problems but also for the coupled ones between inplane and outplane deformations.     

STRESS INTENSITY FACTOR OF AN ANTI-PLANE CRACK PARALLEL TO THE WEAK/MICRO-DISCONTINUOUS INTERFACE IN A BI-FGM COMPOSITE

The problem considered is a mode Ⅲ crack lying parallel to the interface of an exponential-type functional graded material （FGM） strip bonded to a linear-type FGM substrate with infinite thickness. By applying the Fourier integral transform, the problem was reduced as a Cauchy singular integral equation with an unknown dislocation density function. The collocation method based on Chebyshev polynomials proposed by Erdogan and Gupta was used to solve the singular integral equation numerically. With the numerical solution, the effects of the geometrical and physical parameters on the stress intensity factor （SIF） were analyzed and the following conclusions were drawn： （a） The region affected by the interface or free surface varies with the material rigidity, and higher material rigidity will lead to bigger affected region. （b） The SIF of the crack in the affected region and parallel to the micro-discontinuous interface is lower than those of the weak discontinuous cases. Reducing the weak-discontinuity of the interface will be beneficial to decrease the SIF of the interface-parallel crack in the region affected by the interface. （c） The effect of the free surface on SIF is more remarkable than that of the interface, and the latter is still more notable than that of the material rigidity. When the effects of the interface and free surface are fixed, increase of the material rigidity will enhance the value of SIF.     

Anisotropic Elasticity and Multi-Material Singularities

Multi-material wedges associated with convergence of geometrical and material discontinuity lines generally show singular stress fields around the vertex of the wedge. In this paper, the eigenvalue problem for a multi-material wedge composed of several anisotropic elastic sectors is formulated in a completely generally manner, including the cases of degenerate and extra-degenerate material sectors, and various types of edge conditions for both open and closed wedges. General representation of the elasticity solution in a degenerate or extra-degenerate anisotropic sector requires higher-order eigenmodes (generalized eigenfunctions) in addition to zeroth-order eigenmodes. Such higher-order eigenmodes are obtained from appropriate analytical expressions of the zeroth-order eigenmode by using the derivative rule. The analysis is applied to one bisector wedge and one trisector wedge in a three-layer cracked composite model to obtain accurate elasticity solutions of the singular stress fields. These solutions were determined using the traction data generated on a circular collocation path by a conventional finite element analysis.     

Influence of boundary conditions on the solution of a hyperbolic thermoelasticity problem

We consider a series of problems with a short laser impact on a thin metal layer accounting various boundary conditions of the first and second kind. The behavior of the material is modeled by the hyperbolic thermoelasticity of Lord–Shulman type. We obtain analytical solutions of the problems in the semi-coupled formulation and numerical solutions in the coupled formulation. Numerical solutions are compared with the analytical ones. The analytical solutions of the semi-coupled problems and numerical solutions of the coupled problems show qualitative match. The solutions of hyperbolic thermoelasticity problems are compared with those obtained in the frame of the classical thermoelasticity. It was determined that the most prominent difference between the classical and hyperbolic solutions arises in the problem with fixed boundaries and constant temperature on them. The smallest differences were observed in the problem with unconstrained, thermally insulated edges. It was shown that a cooling zone is observed if the boundary conditions of the first kind are given for the temperature. Analytical expressions for the velocities of the quasiacoustic and quasithermal fronts as well as the critical value for the attenuation coefficient of the excitation impulse are verified numerically.     

In-plane perturbation of the tunnel-crack under shear loading II: Determination of the fundamental kernel

For any plane crack in an infinite isotropic elastic body subjected to some constant loading, Bueckner–Rice's weight function theory gives the variation of the stress intensity factors due to a small coplanar perturbation of the crack front. This variation involves the initial SIF, some geometry independent quantities and an integral extended over the front, the “fundamental kernel” of which is linked to the weight functions and thus depends on the geometry considered. The aim of this paper is to determine this fundamental kernel for the tunnel-crack. The component of this kernel linked to purely tensile loadings has been obtained by Leblond et al. [Int. J. Solids Struct. 33 (1996) 1995]; hence only shear loadings are considered here. The method consists in applying Bueckner–Rice's formula to some point-force loadings and special perturbations of the crack front which preserve the crack shape while modifying its size and orientation. This procedure yields integrodifferential equations on the components of the fundamental kernel. A Fourier transform in the direction of the crack front then yields ordinary differential equations, that are solved numerically prior to final Fourier inversion.     

A coupled FEM/BEM approach and its accuracy for solving crack problems in fracture mechanics

The finite element (FEM) and the boundary element methods (BEM) are well known powerful numerical techniques for solving a wide range of problems in applied science and engineering. Each method has its own advantages and disadvantages, so that it is desirable to develop a combined finite element/boundary element method approach, which makes use of their advantages and reduces their disadvantages. Several coupling techniques are proposed in the literature, but until now the incompatibility of the basic variables remains a problem to be solved. To overcome this problem, a special super-element using boundary elements based on the usual finite element technique of total potential energy minimization has been developed in this paper. The application of the most commonly used approaches in finite element method namely quarter-point elements and J-integrals techniques were examined using the proposed coupling FEM–BEM. The accuracy and efficiency of the proposed approach have been assessed for the evaluation of stress intensity factors (SIF). It was found that the FEM–BEM coupling technique gives more accurate values of the stress intensity factors with fewer degrees of freedom.     

Closed-form solution for a mode-III interfacial edge crack between two bonded dissimilar elastic strips

A closed-form solution is obtained for the problem of a mode-III interfacial edge crack between two bonded semi-infinite dissimilar elastic strips. A general out-of-plane displacement potential for the crack interacting with a screw dislocation or a line force is constructed using conformal mapping technique and existing dislocation solutions. Based on this displacement potential, the stress intensity factor (SIF, K_III) and the energy release rate (ERR, G_III) for the interfacial edge crack are obtained explicitly. It is shown that, in the limiting special cases, the obtained results coincide with the results available in the literature. The present solution can be used as the Green’s function to analyze interfacial edge cracks subjected to arbitrary anti-plane loadings. As an example, a formula is derived correcting the beam theory used in evaluation of SIF (K_III) and ERR (G_III) of bimaterials in the double cantilever beam (DCB) test configuration.     

四角点支承四边自由矩形薄板屈曲问题的新解析解

角点支承矩形薄板的屈曲问题是板壳力学的一类重要课题，控制方程和边界条件的复杂性导致寻求该类问题的解析解十分困难。虽然各类近似/数值方法可用于解决此类难题，但作为基准的精确解析解在公开文献中鲜有报道。本文基于近年来提出的辛叠加方法，解析求解了四角点支承四边自由矩形薄板的屈曲问题。首先将问题拆分为两个子问题，接着利用分离变量与辛本征展开推导出子问题的解析解，最后通过叠加获得原问题的解。由于求解过程从基本控制方程出发，逐步严格推导，无需假定解的形式，因此本文解法是一种理性的解析方法。数值算例给出了不同长宽比和不同面内载荷比情况下，四角点支承四边自由矩形薄板的屈曲载荷和典型屈曲模态，并经有限元方法验证，确认了解析解的正确性。     

Generalized Timoshenko–Reissner model for a multilayer plate

A multilayer plate with isotropic (or transversally isotropic) layers strongly differing in rigidity is considered. This plate is reduced to an equivalent homogeneous transversally isotropic Timoshenko–Reissner plate whose deflections and free transverse vibration frequencies are close to those of the multilayer plate. By comparison with the exact solution of test three-dimensional problems of elasticity, the error of the proposed method is estimated both for the static problem and for free vibrations. This comparison can readily be carried out for the hinged edges of the plate, and explicit approximate formulas are obtained for the vibration frequencies. The scope of the proposed model turned out to be rather wide (the Young moduli of soft and rigid layers can differ by a factor of 1000). In the case of boundary conditions other than hinged support, a closed-form solution cannot be constructed in general. For rigidly fixed edges, the asymptotic method proposed by V. V. Bolotin is generalized to the case of a Timoshenko–Reissner plate.     

Generalized solutions of transient thermal shock problem with bounded boundaries

In this work, the generalized thermoelastic solutions with bounded boundaries for the transient shock problem are proposed by an asymptotic method. The governing equations are taken in the context of the generalized thermoelasticity with one relaxation time (L–S theory). The general solutions for any set of boundary conditions are obtained in the physical domain by the Laplace transform techniques. The corresponding asymptotic solutions for a thin plate with finite thickness, subjected to different sudden temperature rises in its two boundaries, are obtained by means of the limit theorem of Laplace transform. In the context of these asymptotic solutions, two specific problems with different boundary conditions have been conducted. The distributions of displacement, temperature and stresses, as well as the propagations, intersections and reflections of two elastic waves, named as thermoelastic wave and thermal wave separately, are obtained and plotted. These results are agreed with the results obtained in the existing literatures.     

有限长滚子黏滑接触分析

有限长线接触在齿轮传动与滚子轴承中广泛存在,滚子端部的应力集中严重影响机械零件的疲劳寿命. 本文中基于矩阵法和重叠半空间模型方法建立了有限长滚子黏滑接触的数值计算模型,利用共轭梯度法和快速傅里叶变换方法对模型进行了求解;并且分析了滚子自由端面和不同切向力对滚子黏滑接触的影响. 结果表明:当两有限长滚子的自由端面完全重合时,切应力在端面不会出现应力集中现象,端面的黏着区域相比于中间截面的黏着区会缩小. 此外,随着切向力的增大,端面黏滑区域的切应力增大.      

混合Trefftz有限元法反平面断裂问题的探讨

基于修正变分原理,采用满足控制微分方程的应力和位移混合Trefftz函数,满足裂纹单元断裂性质的特殊Trefftz函数以及满足裂纹尖端条件的附加试函数,推导出混合Trefftz有限元法反平面断裂问题公式,给出应力集中因子解析表达式.同时,给出单个边裂纹、单个曲折裂纹和三个边裂纹反平面裂纹问题三个算例,探讨特殊Trefftz函数个数、破裂单元个数、高斯点数以及不同附加试函数对结果的影响.最后,将计算结果与一般有限元算法或其它方法结果进行对比,分析了混合Trefftz有限元法的精确性和高效性.