The fundamental solutions for the plane problem in piezoelectric media with an elliptic hole or a crack

1.IntroductionItiswell-knownthatthefundame,ltalsolutionsorGreen'sfunctionsplayanimportantroleilllinearelasticity.Forexample,theycanbeusedtoconstructmanyanalyticalsolutionsofpracticalproblems.Itismoreimportantthattheyareusedasthefundamentalsolutionsintheboundaryelementmethod(BEM)tosolvesomecomplicatedproblem.Withthewidely-increasingapplicationofpiezoelectricmaterialsinengineeringproblems,thestudyregardingtheGreen'sfLlnctionsinpiezoelectricsolidshasreceivedmuchinterest.The3DGreen'sfunctionsi…     

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.     

集中载荷作用下各向异性平面内的椭圆孔或裂纹问题

应用复变函数Cauchy积分的方法,对含有椭圆孔或裂纹的各向异性平面,系统地导出了当其在面内受任意集中载荷作用时的复应力函数解或裂纹应力强度因子解析解,即基本解;并通过基本解的迭加,得到了在椭圆孔周或裂纹表面作用一般外载时的解,其特例证实了上述解的正确性。     

On the stress problem of large elliptical cutouts and cracks in circular cylindrical shells

Numerical solutions are presented for stresses around an elliptical hole in a long, thin, circular cylindrical shell subjected to axial tension for both the symmetric orientations of the hole with respect to the shell. The method of analysis involves obtaining a series solution to the governing shell equations in terms of Mathieu functions by the method of separation of variables and satisfying the boundary conditions numerically term by term in a Fourier series formulation. Results are presented in the form of charts from which stress concentration factors can be directly read over a wide range of the two parameters, namely, axis ratio of the ellipse and a curvature parameter defining the hole size with respect to dimensions of the shell.An interesting feature of the investigation is the analysis of limiting cases of circumferential and axial cracks for axial tension and internal pressure loadings respectively. The method developed involves determining the solution completely in elliptic coordinates and then determining the singular stresses by carrying out a transformation to polar coordinates with crack tip as the origin through a Taylor series expansion. Membrane and bending stress intensity factors are computed and plotted over a sufficiently wide range of the curvature parameter extending from small to large sized cracks. As an outcome of the analysis, a “hybrid” technique has been developed by which singularity conditions at the crack tip can be handled effectively in dealing with boundary conditions in crack problems.     

Influence of the viscoelastic properties of a composite on the stress distribution around an elliptic hole in a plate

The time variation in the stresses around an elliptic hole in a composite plate is studied. Solutions that characterize the effect of the time dependence of the relaxation moduli of the composite components on stresses are obtained. The solutions in the time domain are obtained from the elastic–viscoelastic analogy and the corresponding elastic solutions for the effective moduli of the composite and the stress field around an elliptic hole in an anisotropic plate. The inverse Laplace transformation is carried out by an effective numerical method     

Analysis of the stress-strain state of an anisotropic plate with an elliptic hole and thin elastic inclusions

We solve the problem of determining the stress-strain state of an anisotropic plate with an elliptic hole and a system of thin rectilinear elastic inclusions. We assume that there is a perfect mechanical contact between the inclusions and the plate. We deal with a more precise junction model with the flexural rigidity of inclusions taken into account. (The tangential and normal stresses, as well as the derivatives of the displacements, experience a jump across the line of contact.) The solution of the problem is constructed in the form of complex potentials automatically satisfying the boundary conditions on the contour of the elliptic hole and at infinity. The problem is reduced to a system of singular integral equations, which is solved numerically. We study the influence of the rigidity and geometry parameters of the elastic inclusions on the stress distribution and value on the contour of the hole in the plate. We also compare the numerical results obtained here with the known data.     

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.     

Thermally conductive elliptic inhomogeneities in 2D anisotropic solids

The problem of a thermally conductive elliptic hole embedded in an anisotropic thermoelastic solid under a remote uniform heat flow is considered. For the plane problem, solutions are derived by the use of an extended thermoelastic version of the Stroh formalism. The hoop stress around the elliptic hole is obtained in an explicit real form. We analyze the influence of the interior thermal conductivity and the cavity thickness on the temperature and stress on the boundary points of the hole. By comparison with the thermally insulated cavity, we show that the consideration of the interior conductivity may lead to significantly different thermomechanical behaviors. Ana Ursescu on leave from Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie no. 13, Bucharest, Romania.     

The General Solution for a Finite Thermopiezoelectric Plate Containing a Hole and a Crack

By application of complex variables and conformal transformation, the general solution to multiply connected domains of two dimensions is constructed in terms of multiple Laurent series for thermopiezoelectric materials. Three typical boundaries, i.e., a rectangular contour, a curvilinear hole, and a line crack are considered in the paper. Though the Green’s function of an arbitrarily shaped hole still remains unknown for anisotropic materials, the approximate solutions both for thermal and electro-mechanical fields are obtained in explicit form by the least square method. The accuracy of the approximation are investigated upon each boundary contours. It is found that the local error on the crack surface diminishes below 10⁻⁷% by adopting only 20 terms of related power series. For a rectangular plate, the precision is enhanced up to the level of 99.99% on its boundary contour by adding the supplementary function, due to the rectangular corners, into the complex solution. Considering that the singular character of a crack is retained in the solution, the stress and electric displacement intensity factors influenced by the hole width and plate size are exhibited herein.     

Stress-strain state of an anisotropic plate with an elliptic hole and thin rigid inclusions

The stress-strain state of an anisotropic plate containing an elliptic hole and thin, absolutely rigid, curvilinear inclusions is studied. General integral representations of the solution of the problem are constructed that satisfy automatically the boundary conditions on the elliptic-hole contour and at infinity. The unknown density functions appearing in the potential representations of the solution are determined from the boundary conditions at the rigid inclusion contours. The problem is reduced to a system of singular integral equations which is solved by a numerical method. The effects of the material anisotropy, the degree of ellipticity of the elliptic hole, and the geometry of the rigid inclusions on the stress concentration in the plate are studied. The numerical results obtained are compared with existing analytical solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 173–180, July–August, 2007.     

Fracture Analysis for a Frictional Fastener Hole with Edge Cracks in an Anisotropic Plate∗

Abstract

Stress intensity factors are evaluated for a singly or doubly cracked fastener hole with frictional traction in an anisotropic plate, using a special kernel boundary integral equation (BIE) approach. The integration kernel (Green's function) used in this BIE approach has already taken the presence of the crack (or cracks) into account, thus.avoiding the need for element discretization to model the stress singularity at the crack tip. The Green's function employed is that of an infinite anisotropic plate containing an elliptical hole or crack, and subjected to an arbitrarily positioned point force. Several types of normal and shear traction conditions at the pinhole interface are considered. Numerical results are obtained for various geometrical and loading conditions and are compared with known solutions, where available, for their isotropic counterparts.     

椭圆孔口端点和裂纹端点处的变动态应力分析

分析了椭圆孔口端点和裂纹端点处的变动态应力．在分析中,设带椭圆孔口的无限平板受远处应力作用,变动态应力分析指的是,令动点趋近于椭圆孔端点和椭圆孔变成裂纹这二个过程在各种相对关系下进行．在不同相对关系下,求出椭圆孔口或裂纹端点应力的极限值．分析表明,随着不同的变动状态,对于端点处的某些应力会得到不同的极限值．     

Stress and electric displacement analyses in piezoelectric media with an elliptic hole and a small crack

In this paper, the interactions between an elliptic hole and an arbitrary distributed small crack in plane piezoelectric medium, which are often happened in engineering problems, are discussed. The Green’s functions in a piezoelectric plate with an elliptic hole for a generalized line dislocation and a generalized line force are presented. The small crack is represented by unknown continuous distributed dislocations. By considering traction free conditions on the surface of the small crack, the problem is then reduced to a group of singular integral equations which are solved by using a special numerical technique. Accuracy of the present method is confirmed by comparing the numerical results with those in literatures for PZT-4 when the elliptic hole is degenerated into a crack. The generalized stress intensity factors of cracks and the generalized stress on the edge of the elliptic hole are shown graphically. It is shown that the small crack may have shielding or amplifying effects on the main elliptic hole or crack, which depends on the location and orientation of the small crack. The hole near a crack can significantly reduce the stress intensity factor of the crack. The direction of the electric field is important to shielding effect.     

双轴载荷作用下源于椭圆孔的分支裂纹的一种边界元分析

利用一种边界元方法来研究双轴载荷作用下无限大板中源于椭圆孔的分支裂纹．该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者提出的裂尖位移不连续单元构成．在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界,文中算例说明本数值方法对计算平面弹性裂纹的应力强度因子是非常有效的。该文对双轴载荷作用下无限大板中源于椭圆孔的分支裂纹的数值结果进一步证实本数值方法对计算复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示双轴载荷及裂纹体几何对应力强度因子的影响。     

Stress intensity factors for an edge interface crack in a bonded semi-infinite plate for arbitrary material combination

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary crack lengths and material combinations. In this paper the stress intensity factors of an edge interface crack in a bonded strip are considered under tension with varying the crack length and material combinations systematically. Then, the limiting solutions are provided for an edge interface crack in a bonded semi-infinite plate under arbitrary material combinations. In order to calculate the stress intensity factors accurately, exact solutions in an infinite bonded plate are also considered to produce proportional singular stress fields in the analysis of FEM by superposing specific tensile and shear stresses at infinity. The details of this new numerical solution are described with clarifying the effect of the element size on the stress intensity factor. It is found that for the edge interface crack the normalized stress intensity factors are not always finite depending upon Dunders’ parameters. This behavior can be explained from the condition of the singular stress at the end of bonded strip. Convenient formulas are also given by fitting the computed results.     

The solution of a crack emanating from an arbitrary hole by boundary collocation method

In this paper a group of stress functions has been proposed for the calculation of a crack emanating from a hole with different shape (including circular, elliptical, rectangular, or rhombic hole) by boundary collocation method. The calculation results show that they coincide very well with the existing solutions by other methods for a circular or elliptical hole with a crack in an infinite plate. At the smae time, a series of results for different holes in a finite plate has also been obtained in this paper. The proposed functions and calculation procedure can be used for a plate of a crack emanating from an arbitrary hole.     

2维各向异性弹性力学中的Stroh公式和某些不变量

本文简要综述了各向异性弹性体2维变形的Stroh公式。该公式数学上优美而应用上有力。它的最近进展使我们将已有的复数形式的解转变成实数形式,进而可去解决一些迄今尚未解出的问题。结果,发现了许多有趣的而在物理上难以预料的现象。例如,对于界面包含x_3轴的各向异性双介质体,如果线力和线位错都沿x_3轴,那末包含x_3轴的任意径向平面上的面拉力将不随径向平面的选择而变化。而且,在任何固定点上的应力与界面的方位无关。要揭示的其他一些不变性质是:双介质材料中的虚拟力,界面裂纹上的应力奇异,界面裂纹表面上的振荡,等等;另外,在具有一个有限裂纹,或一个椭圆孔,或一个刚性椭圆核的无穷大均匀介质中的某些应力和位移也是不变量。这些不变量的大部分尚未有它们的物理解释。在实数形式的解中,Barnett-Lothe张量常常出现。在绕x_3轴转动时,这些张量的分量的某些组合是不变的。这些组合将在本文中推导并列出。最后我们给出了某些各向异性弹性材料,它们的Barnett-Lothe张量是不变量,即这些张量的所有分量在绕x_3轴转动时是不变的。     

十次对称二维准晶中的热应力分析

推导了当考虑热效应时十次对称二维准晶体平面应变问题的通解表示．作为应用，采用所获得的通解首先得到了十次对称二维准晶体中的一个点热源所引起的声子场和相位子场，给出了点热源所引起的声子场和相位子场应力分量的解析表达式；接着获得了在均匀热流作用下十次对称二维准晶体中-绝缘椭圆孔洞所引起的热应力问题的弹性解答，给出了沿椭圆边界环向应力分布的解析表达式；当椭圆的短轴趋于零时，则获得了裂纹问题的解答，给出了应力强度因子、裂纹表面张开位移及能量释放率的解析表达式；推导了在任意热载荷作用下裂尖附近的渐近场．     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.