矩形弹性夹杂与裂纹相互干扰的边界元分析

使用边界元法研究了无限弹性体中矩形弹性夹杂对曲折裂纹的影响,导出了新的复边界积分方程．通过引入与界面位移密度和面力有关的未知复函数H(t),并使用分部积分技巧,使得夹杂和基体界面处的面力连续性条件自动满足,而边界积分方程减少为2个,且只具有1/r阶奇异性．为了检验该边界元法的正确性和有效性,对典型问题进行了数值计算．所得结果表明:裂纹的应力强度因子随着夹杂弹性模量的增大而减小,软夹杂有利于裂纹的扩展,而刚性较大的夹杂对裂纹有抑制作用．     

裂纹与弹性夹杂的相互影响*

本文利用无限域上单根弹性夹杂和单根裂纹产生的位移和应力,将裂纹与弹性夹杂的相互影响问题归为解一组柯西型奇异积分方程,然后用此对夹杂分枝裂纹解答的奇性性态作了理论分析,并求得了振荡奇性界面应力场,对于不相交的夹杂裂纹问题,具体计算了端点的应力强度因子及夹杂上的界面应力,结果令人满意。     

多裂纹问题计算分析的本征COD边界积分方程方法

针对多裂纹问题，若采用常规的数值求解技术，计算效率较低.为实现多裂纹问题的大规模数值模拟，建立了本征裂纹张开位移（crack opening displacement, COD）边界积分方程及其迭代算法，并引入Eshelby矩阵的定义，将多裂纹分为近场裂纹和远场裂纹来处理裂纹间的相互影响.以采用常单元作为离散单元的快速多极边界元法为参照，对提出的计算模型和迭代算法进行了数值验证.结果表明，本征COD边界积分方程方法在处理多裂纹问题时取得较大的改进，其计算效率显著高于传统的边界元法和快速多极边界元法.     

压电螺型位错和含界面裂纹圆形夹杂的电弹干涉效应

研究了在无穷远反平面剪切和面内电场共同作用下压电材料基体中一个压电螺型位错与含界面裂纹圆形弹性夹杂的电弹耦合干涉作用．运用复变函数方法,获得了该问题的一般解答．作为典型算例,求出了界面含一条裂纹时,基体和夹杂区域复势函数的封闭形式解以及裂纹尖端应力和电位移场强度因子．应用扰动技术和广义Peach-Koehler公式,导出了位错力的解析表达式．数值结果表明,界面裂纹对压电螺型位错与夹杂的干涉具有强烈扰动效应,当裂纹长度达到临界值时,可以改变其干涉机理．同时,分析说明压电材料中软夹杂可以排斥基体中的位错．     

裂纹与线夹杂垂直接触问题的奇性分析

以短纤维复合等作为工程背景,采用现有文献中单裂纹和单夹杂的基本解,对于限平项（基体）上,裂纹和线夹杂的垂直接触问题从断理解力学的角度作了研究,得到了问题的积分方程,推出了接触点的性线指数,奇性应力及以此表示的接触点附近三个区域内的应力强度因子表达式,并给出一些数值结果,可供工程实际参考。     

应用边界元法模拟纤维增强复合材料平面弹性问题

将含有随机分布多种夹杂相复合材料的二维弹性力学问题归结为复连通区域的边界积分方程,进而转化成矩阵方程进行求解和分析．根据同类夹杂相外在边界上的面力与位移之间关系矩阵完全相同的特点,使得最后的矩阵方程阶数得到大规模减少,这正是此处提出改进的边界元方法的主要思路．数值算例表明,对于此类问题,与常规的边界元分域解法相比更加有效．以该方法为基础,可以详细给出纤维增强复合材料二维条件下的宏观等效力学性质．     

非对称载荷作用的外部圆形裂纹问题

使用边界积分方程方法,研究了三维无限弹性体中受非对称载荷作用的外部圆形裂纹问题。通过使用Fourier级数和超几何函数,将问题的二维边界奇异积分方程简化为Abel型方程,获得了一般非对称载荷作用的外部圆形裂纹问题的应力强度因子精确解,比用Hankel变换法得到的结果更为一般。结果表明:边界积分方程法在解析分析方面还有很大的潜力。     

含曲线裂纹圆柱扭转问题的新边界元法

研究含曲线裂纹圆柱的Saint-Venant扭转,将问题化归为裂纹上边界积分方程的求解．利用裂纹尖端的奇异元和线性元插值模型,给出了扭转刚度和应力强度因子的边界元计算公式．对圆弧裂纹、曲折裂纹以及直线裂纹的典型问题进行了数值计算,并与用Gauss-Chebyshev求积法计算的直裂纹情形结果进行了比较,证明了方法的有效性和正确性．     

压电复合材料圆形夹杂中螺型位错和界面裂纹的干涉分析

研究了无穷远纵向剪切和面内电场共同作用下,压电复合材料圆形夹杂中螺型位错与界面裂纹的电弹耦合干涉作用．运用Riemann-Schwarz 对称原理,并结合复变函数奇性主部分析方法,获得了该问题的一般解答．作为典型算例,求出了界面含一条裂纹时基体和夹杂区域复势函数和电弹性场的封闭形式解．应用广义Peach-Koehler公式,导出了位错力的解析表达式．分析了裂纹几何参数和材料的电弹性常数对位错力的影响规律．结果表明,界面裂纹对位错力和位错平衡位置有很强的扰动效应,当界面裂纹长度达到临界值时,可以改变位错力的方向．该结果可以作为格林函数研究圆形夹杂内裂纹和界面裂纹的干涉效应．其公式的退化结果与已有文献完全一致．     

压电材料椭圆夹杂界面局部脱粘问题的分析

利用复变函数方法,研究在反平面剪切和面内电场共同作用下压电材料椭圆夹杂的界面脱粘问题．假定夹杂界面脱粘导致了界面电绝缘型裂纹的产生．通过保角变换和解析延拓,将原问题化为两个黎曼-希尔伯特问题,获得了夹杂和基体复势的级数解,进而求得应力变形场以及夹杂-基体界面脱粘的能量释放率的一般表达式．通过理想粘结的椭圆夹杂、完全脱粘的椭圆夹杂、局部脱粘的刚性导体椭圆夹杂、局部脱粘的圆形夹杂等特例的分析说明了该解的有效性和通用性．     

Mathematical and Numerical Simulation of Equilibrium of an Elastic Body Reinforced by a Thin Elastic Inclusion

A boundary value problem describing the equilibrium of a two-dimensional linear elastic body with a thin rectilinear elastic inclusion and possible delamination is considered. The stress and strain state of the inclusion is described using the equations of the Euler–Bernoulli beam theory. Delamination means the existence of a crack between the inclusion and the elastic matrix. Nonlinear boundary conditions preventing crack face interpenetration are imposed on the crack faces. As a result, problem with an unknown contact domain is obtained. The problem is solved numerically by applying an iterative algorithm based on the domain decomposition method and an Uzawa-type algorithm for solving variational inequalities. Numerical results illustrating the efficiency of the proposed algorithm are presented.     

夹杂和裂纹的相互作用及端点相交的奇性性态分析

利用单根裂纹和单根夹杂的基本解,通过弹性力学的线性叠加原理,将平面裂纹和夹杂相互作用的问题归结为解一组带有柯西型奇异积分的积分方程组,计算了裂纹和夹杂端点的应力强度因子,给出了一些数值例子,并对夹杂和裂纹水平接触时的情形作了奇性分析,结果可作为研究夹杂尖端引起的裂纹及其扩展的工程分析的计算模型。     

The interaction between a penny-shaped crack and a spherical inclusion under torsion

The axisymmetric interaction problem of an elastic spherical inclusion with a penny-shaped crack in an elastic space under torsion is considered. The superposition and reflection methods [3]-[4] are used to solve the mixed boundary value problem in question. With the help of the dual integral equations technique and appropriate re-expansion of the eigenfunction, the problem is reduced to an infinite system of linear algebraic equations of the second kind. The matrix elements of that system decrease exponentially along the rows and the columns. Its unique solution is proved to exist in a proper class of sequences and is shown to be represented by a convergent, in the vicinity of the origin, power series in a geometric parameter, equal to the ratio of the radius of the inclusion to its distance from the crack. This procedure provides an efficient formula for the stress intensity factor.     

In-plane loading of a cracked elastic solid by a disc inclusion with a Mindlin-type constraint

The paper examines the problem related to the axisymmetric interaction between an external circular crack and a centrally placed penny-shaped rigid inclusion located in the plane of the crack. The interface between the inclusion and the elastic medium exhibits a Mindlin-type imperfect bi-lateral contact. Analytical results presented in the paper illustrate the manner in which the lateral translational stiffness of the inclusion and the stress intensity factor at the boundary of the external circular crack are influenced by the inclusion/crack radii ratio.     

On inverse crack problems in elastostatics

In solid mechanics, nondestructive testing has been an important technique in gathering information about unknown cracks, or defects in material. From a mathematical point of view, this is described as an inverse problem of partial differential equations, that is, the problem is to extract information about the location and shape of an unknown crack from the surface displacement field and traction on the boundary of the elastic material. By using the enclosure method introduced by Prof. Ikehata we can derive the extraction formula of an unknown linear crack from a single set of measured boundary data. Then, we need to have precise properties of a solution of the corresponding boundary value problem; for instance, an expansion formula around the crack tip. In this paper we consider the inverse problem concentrating on this point. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastic stress concentration near boundary defects in the case of a longitudinal shear deformation

We study the concentration of stresses due to two boundary defects located in an elastic half-space with stress-free boundary loaded at infinity with a constant shear load. The problem is reduced to solving singular integral equations for the cases in which the half-space contains defects of different types: two cracks, two inclusions, and a crack and an inclusion, whose solutions are sought by the method of mechanical quadratures. The interaction of the defects as they approach each other and the influence of their relative sizes are studied numerically. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

The inverse scattering problem by an elastic inclusion

In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.     

Stress analysis in a sheet with multiple cracks

The stress field due to the presence of a Volterra dislocation in an isotropic elastic sheet is obtained. The stress components exhibit the familiar Cauchy type singularity at dislocation location. The solution is utilized to construct integral equations for elastic sheets weakened by multiple embedded or edge cracks. The cracks are perpendicular to the sheet boundary and applied traction is such that crack closing may not occur. The integral equations are solved numerically and stress intensity factors (SIFs) are determined on a crack edges.