各向异性介质中的椭圆夹杂问题

本文研究了含椭圆夹杂的弹性体在多项式荷载作用下的二维变形问题，获得了介质和夹杂中的弹性场，证明了夹杂内部的应力变场以荷载的同阶多项式形式出现，而介质中的弹性场也能用椭圆坐标ζ＾－１２的多项式形式表示出来，并在此基础上，以受剪力作用的含夹杂或空孔的悬臂梁为例，求解了梁中的应力扰动现象，并获得了夹杂或空孔周转的环向应力。     

二维直角平面内固定圆形夹杂对稳态入射反平面剪切波的散射

利用复变函数法、多极坐标及傅立叶级数展开技术求解了二维直角平面内固定圆形夹杂对稳态入射反平面剪切（shearing horizontal, SH）波的散射问题。首先构造出介质内不存在夹杂时的入射波场和反射波场,然后建立介质内存在夹杂时由夹杂边界产生的能够自动满足直角边应力自由条件的散射波解，从而利用叠加原理写出介质内的总波场。利用夹杂边界处位移条件和傅立叶级数展开方法列出求解散射波中未知系数的无穷代数方程组，在满足计算精度的前提下通过有限项截断，得到相应有限代数方程组的解，最后通过算例具体讨论了二维直角平面水平边界点的位移幅度比和相位随量纲一波数、入射波入射角及夹杂位置的不同而变化的情况，结果表明了算法的有效实用性。     

含共线刚性线夹杂压电介质的广义二维问题

基于Stroh理论，研究了含N个共线刚性线夹杂压电介质的广义二维问题，给出了介质内的复势函数、夹杂内的电场和夹杂尖端附近场奇异性系数的解析表达式。结果表明场奇异性系数取决于无限远处的应变。     

考虑晶界滑错的金属多晶体响应的自洽方法

本文首先建立含有三种介质（各向异性基体、各向异性夹杂，界面层）的平面应变夹杂模型，将基体和夹杂位移场展开为多项式级数，假设界面层很薄，运用变分原理得出这一问题的近似解。将上述夹杂问题的解和ＨＩＬＬ自洽方法相结合，给出了考虑晶界滑错效应的金属多晶体弹塑性响应。     

直角平面内弹性圆夹杂对入射平面SH波的散射

利用复变函数法、多极坐标移动技术及傅立叶级数展开求解二维直角平面内圆形弹性夹杂对稳态入射平面SH波的散射问题。首先写出直角平面内不含夹杂时的入射波场和反射波场;其次建立直角平面内含夹杂时夹杂外的散射波解和夹杂内的驻波解,并利用叠加原理写出问题的总波场,借助夹杂边界处应力和位移的连续条件建立求解散射波解和驻波解中未知系数的无穷代数方程组并求解,通过算例具体讨论了直角平面水平边界点的位移幅度比和夹杂边界处径向应力集中系数随不同无量纲波数、入射角及圆孔位置的变化情况,结果表明了算法的有效实用性。     

平面任意形状等剪切模量异质夹杂问题的解析解

本文研究任意形状夹杂域在受到远端均匀荷载和均匀本征应变作用下的弹性场问题,其中基体和夹杂的材料不同但具有相同的剪切模量。利用等效理论将远端均匀荷载引起的扰动转化为等效均匀本征应变的作用,再利用K-M势函数表达扰动场问题的界面连续条件;借助于黎曼映射定理,用洛朗多项式将平面光滑闭合曲线外部区域映射到单位圆外部区域,借助柯西积分公式和Faber多项式求解了等剪切本征应变下夹杂和基体的K-M势函数的显式解析解,其中考虑了夹杂相对于基体的刚体位移。将得到的结果与相关文献的结果进行对比,表明了本论文的方法和结果是有效的和正确的。     

直角平面区域内固定圆形刚性夹杂问题的Green函数解

利用复变函数法、多极坐标移动技术研究了直角平面区域内含有固定圆形夹杂时的反平面问题Green函数解.首先构造出不含夹杂的完整直角平面区域内满足边界应力条件的入射位移场;其次,建立直角平面区域内固定圆形夹杂对该入射场产生的满足直角边界应力自由条件的散射波解,并由叠加原理得到介质内的总波场.最后利用夹杂边界处的位移条件确定出散射波解中的未知系数,最终得到问题的Green函数解,还通过算例讨论了夹杂边界处的径向应力和环向应力随不同波数、角度和不同夹杂位置及不同点源位置的变化情况.算例结果表明了该文方法的有效实用性.     

横观各向同性介质中椭圆夹杂受非弹性剪切变形引起的弹性场的确定

本文求解了横观各向同性介质中椭圆夹杂内受非弹性剪切变形引起的弹性场。采用各向异性弹性力学平面问题的复变函数解法,结合保角变换,获得夹杂内应变能和基体内边界的应力分布和相应的应变能的表达式。进一步,根据最小应变能原理,获得表征夹杂平衡边界的两个特征剪切应变,从而得到了弹性场的解析解。通过应力转换关系,验证了应力解满足夹杂边界上法向正应力和剪应力的连续条件,表明了该解的正确性。本文解可用于复合材料断裂强度的分析中。     

域奇异积分的解析计算

本文讨论了r~1型及lnr型二维域奇异积分的解析计算。对多项式荷载给出了域奇异积分的正确公式。而对于一般荷载,利用泰勒展开化为多项式荷载进行积分,并给出了积分误差估计。计算结果表明,本文方法是有效的。     

双周期分布圆环形截面夹杂反平面问题的解析方法

研究含双周期分布圆环形截面弹性夹杂的无限大介质在远场均匀反平面应力下的弹性响应。通过在双周期圆环形区域内引入非均匀本征应变,将双周期非均匀介质问题转化为带有双周期非均匀本征应变的均匀介质问题,结合双周期函数和双准周期Riemann边值问题理论,获得了该问题弹性场的级数形式解答。作为一个应用,利用该解答预测了含双周期圆环形截面夹杂复合材料的有效纵向剪切模量。数值结果表明,在相同夹杂体积分数下,含圆环形截面夹杂的复合材料比含圆形截面夹杂的复合材料拥有更高的有效纵向剪切模量。     

Uniform Stress Inside an Anisotropic Elliptic Inclusion with Imperfect Interface Bonding

In this paper we study the two-dimensional deformation of an anisotropic elliptic inclusion embedded in an infinite dissimilar anisotropic matrix subject to a uniform loading at infinity. The interface is assumed to be imperfectly bonded. The surface traction is continuous across the interface while the displacement is discontinuous. The interface function that relates the surface traction and the displacement discontinuity across the interface is a tensor function, not a scalar function as employed by most work in the literature. We choose the interface function such that the stress inside the elliptic inclusion is uniform. Explicit solution for the inclusion and the matrix is presented. The materials in the inclusion and in the matrix are general anisotropic elastic materials so that the antiplane and inplane displacements are coupled regardless of the applied loading at infinity. T.C.T. Ting is Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

PLANE STRAIN PROBLEM OF PIEZOELECTRIC SOLID WITH ELLIPTIC INCLUSION

By using the complex variables function theory, a plane strain electro-elastic analysis was performed on a transversely isotropic piezoelectric material containing an elliptic elastic inclusion, which is subjected to a uniform stress field and a uniform electric displacement loads at infinity. Based on the present finite element results and some related theoretical solutions, an acceptable conjecture was found that the stress field is constant inside the elastic inclusion. The stress field solutions in the piezoelectric matrix and the elastic inclusion were obtained in the form of complex potentials based on the impermeable electric boundary conditions.     

Finite anti-plane shear of an infinite slab with a traction-free elliptical cavity: Bounds on the stress concentration factor

This paper provides an analytical approach for obtaining bounds on elastic stress concentration factors in the theory of finite anti-plane shear of homogeneous, isotropic, incompressible materials. The problem of an infinite slab with traction-free elliptical cavity subject to a state of finite simple shear deformation is considered. Explicit estimates are obtained for the maximum shear stress in terms of the cavity geometry, applied stress at infinity and constitutive parameters. The analysis is based on application of maximum principles for second-order quasilinear uniformly elliptic equations.     

Bounds on stress concentration factors in finite anti-plane shear

This paper provides an analytical approach for obtaining bounds on elastic stress concentration factors in the theory of finite anti-plane shear of homogeneous, isotropic, incompressible materials. The problem of an infinite slab with traction-free circular cavity subject to a state of finite simple shear deformation is considered. Explicit estimates are obtained for the maximum shearing stress in terms of the applied stress at infinity and constitutive parameters. The analysis is based on application of maximum principles for second-order quasilinear uniformly elliptic equations.     

On determining stresses in rigid inclusions. Plane problems

Plane problems of determining the stress-strain state of an isotropic elastic domain with a rigid inclusion are considered. It is shown that the stress field in the inclusion is uniquely determined. This field is uniform for a plane with an elliptic inclusion, and the stresses at infinity and in the inclusion are related by mutually single-valued formulas. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 183–186, July–August, 2009.     

Second order elastic effects in a homogeneous isotropic compressible infinite medium with an elliptic hole and acted upon by a tensile force at infinity

Analytic expressions for the second order elasticity case of the problem of an isotropic, infinite medium of compressible material having an elliptic hole and acted upon by a tensile force at infinity are obtained. These results are improvements on the classical elasticity theory in which the elastic constants do not play any role for determination of the stresses. The expressions for the second order effects in the cases of a circular hole and a slit along the major axis have been obtained as particular cases.     

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.     

Inverse problem of deformation of a physically nonlinear inhomogeneous medium

An isotropic linearelastic (viscoelastic) plane containing various physically nonlinear elliptic inclusions is considered. It is assumed that the distances between the centers of the inclusions are much greater than their dimensions. The problem is to determine the orientation of the inclusions and the loads applied at infinity which ensure a specified value of the principal shear stress in each inclusion. Necessary and sufficient conditions of existence of the solution of the problem are formulated for a plane strain of an incompressible inhomogeneous medium.     

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．     

Reinforcement of a plate with a circular hole by an eccentric circular patch

We study the reinforcement of an infinite elastic plate with a circular hole by a larger eccentric circular patch completely covering the hole and rigidly adjusted to the plate along the entire boundary of itself. We assume that the plate and the patch are in a generalized plane stress state generated by the action of some given loads applied to the plate at infinity and on the boundary of the hole. We use the power series method combined with the conformal mapping method to find the Muskhelishvili complex potentials and study the stress state on the hole boundary and on the adhesion line. We consider several examples, study how the stresses depend on the geometric and elastic parameters, and compare the problem under study with the case of a plate with a circular hole without a patch. In scientific literature, numerous methods for reinforcing plates with holes, in particular, with circular holes, have been studied. In the monographs [1, 2], the problem of reinforcing the hole edges by stiffening ribs is solved. Methods for reinforcing a circular hole by using two-dimensional patches pasted to the entire plate surface are studied in [3, 4]. The case of a plate with a circular cut reinforced by a concentric circular patch adjusted to the plate along the boundary of itself or along some other circle was studied in [5, 6]. The reinforcement of an elliptic hole by a confocal elliptic patch was considered in [7].