三角形弹性夹杂对裂纹的影响

研究了三角形弹性夹杂和裂纹之间的相互影响问题。应用Chau和Wang导出的面力边值问题的边界积分方程为基本方程,用夹杂和基体交界面上的面力和位移的连续性条件为补充方程,从而得到了一组能够解决夹杂和裂纹相互影响问题的方程,最后的方程组用一种新的边界单元法求解。计算了各种不同的夹杂和基体的材料常数以及夹杂和基体之间不同距离情况下裂纹尖端的应力强度因子。文中结果对研究新型复合材料有一定的应用价值。     

Numerical solution of an equilibrium problem for an elastic body with a thin delaminated rigid inclusion

Under consideration is a 2D-problem of elasticity theory for a body with a thin rigid inclusion. It is assumed that there is a delamination crack between the rigid inclusion and the elastic matrix. At the crack faces, the boundary conditions are set in the form of inequalities providing mutual nonpenetration of the crack faces. Some numerical method is proposed for solving the problem, based on domain decomposition and the Uzawa algorithm for solving variational inequalities.We give an example of numerical calculation by the finite element method.     

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。     

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.     

Numerical simulation of equilibrium of an elastic two-layer structure with a through crack

In this paper, a problem of equilibrium of two elastic bodies pasted together along a curve is considered. It is assumed that there is a through crack on a part of the curve. Nonlinear boundary conditions providing mutual non-penetration between the crack faces are set. The main objective of the paper is to construct and test a numerical algorithm for solving the equilibrium problem. The algorithm is based on two approaches: a domain decomposition method and Uzawa method for solving variational inequalities. A numerical experiment illustrates the efficiency of the algorithm.     

Optimal Control of Inclusion and Crack Shapes in Elastic Bodies

The paper is concerned with the control of the shape of rigid and elastic inclusions and crack paths in elastic bodies. We provide the corresponding problem formulations and analyze the shape sensitivity of such inclusions and cracks with respect to different perturbations. Inequality type boundary conditions are imposed at the crack faces to provide a mutual nonpenetration between crack faces. Inclusion and crack shapes are considered as control functions and control objectives, respectively. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry. We prove an existence of optimal solutions.     

The Method of Smooth Domains in the Equilibrium Problem for a Plate with a Crack

A free boundary problem is considered of the equilibrium of an elastic plate with a crack. We suppose that some boundary mutual nonpenetration conditions are given on the crack faces in the form of simultaneous equalities and inequalities. We suggest a new approach to posing the problem in a smooth domain although it was stated in a domain with cuts originally. We treat the constraints on the components of the displacement vector and stress tensor on the crack faces as interior constraints, i.e., constraints given on subsets of the smooth domain of a solution.     

Junction problem for elastic and rigid inclusions in elastic bodies

An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality‐type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. Copyright © 2015 John Wiley & Sons, Ltd.     

On bending an elastic plate with a delaminated thin rigid inclusion

Under study is the problem of bending an elastic plate with a thin rigid inclusion which may delaminate and form a crack. We find a system of boundary conditions valid on the faces of the crack and prove the existence of a solution. The problem of bending a plate with a volume rigid inclusion is also considered. We establish the convergence of solutions of this problem to a solution to the original problem as the size of the volume rigid inclusion tends to zero.     

An Optimal Size of a Rigid Thin Stiffener Reinforcing an Elastic Two-Dimensional Body on the Outer Edge

The equilibrium problem for a two-dimensional body with a crack is studied. We suppose that the body consists of two parts: an elastic part and a rigid thin stiffener on the outer edge of the body. Inequality-type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. For a family of variational problems, dependence of their solutions on the length of the thin rigid stiffener is investigated. It is shown that there exists a solution of an optimal control problem. For this problem, the cost functional is defined by a continuous functional on a solution space, while the length parameter serves as a control parameter.     

On an optimal control problem for the shape of thin inclusions in elastic bodies

An optimal control problem is considered for a two-dimensional elastic body with a straight thin rigid inclusion and a crack adjacent to it. It is assumed that the thin rigid inclusion delaminates and has a kink. On the crack faces the boundary conditions are specified in the form of equalities and inequalities which describe the mutual nonpenetration of the crack faces. The derivative of the energy functional along the crack length is used as the objective functional, and the position of the kink point, as the control function. The existence is proved of the solution to the optimal control problem.     

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.     

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.     

On elastic bodies with thin rigid inclusions and cracks

This paper is concerned with the analysis of equilibrium problems for two‐dimensional elastic bodies with thin rigid inclusions and cracks. Inequality‐type boundary conditions are imposed at the crack faces providing a mutual non‐penetration between the crack faces. A rigid inclusion may have a delamination, thus forming a crack with non‐penetration between the opposite faces. We analyze variational and differential problem formulations. Different geometrical situations are considered, in particular, a crack may be parallel to the inclusion as well as the crack may cross the inclusion, and also a deviation of the crack from the rigid inclusion is considered. We obtain a formula for the derivative of the energy functional with respect to the crack length for considering this derivative as a cost functional. An optimal control problem is analyzed to control the crack growth. Copyright © 2010 John Wiley & Sons, Ltd.     

The formulation of linearized boundary integral equations of the anisotropic theory of elasticity and their application in geometrical inverse problems

An elastic bounded anisotropic solid with an elastic inclusion is considered. An oscillating source acts on part of the boundary of the solid and excites oscillations in it. Zero displacements are specified on the other part of the solid and zero forces on the remaining part. A variation in the shape of the surface of the solid and of the inclusion of continuous curvature is introduced and the problem of the theory of elasticity with respect to this variation is linearized. An algorithm for constructing integral representations for such linearized problems is described. The limiting properties of the linearized operators are investigated and special boundary integral equations of the anisotropic theory of elasticity are formulated, which relate the variations of the boundary strain and stress fields with the variations in the shape of the boundary surface. Examples are given of applications of these equations in geometrical inverse problems in which it is required to establish the unknown part of the body boundary or the shape of an elastic inclusion on the basis of information on the wave field on the part of the body surface accessible for observation.     

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。     

圆外平面弹性问题的边界积分公式

将边界上的应力函数及其法向导数展开为罗朗级数,与复应力函数的罗朗级数的表达式对比,可以确定罗朗级数的各系数,再利用傅利叶级数和卷积的几个公式进行计算,得到应力函数边界积分公式．通过边界的应力函数及其法向导数的积分,直接得到圆外应力函数值,并给出几个算例,表明结果用于求解单位圆外平面弹性问题十分方便．     

Optimal rigid inclusion shapes in elastic bodies with cracks

The paper concerns the control of rigid inclusion shapes in elastic bodies with cracks. Cracks are located on the boundary of rigid inclusions and in the bulk. Inequality type boundary conditions are imposed at the crack faces to guarantee mutual non-penetration. Inclusion shapes are considered as control functions. First we provide the problem formulation and analyze the shape sensitivity with respect to geometrical perturbations of the inclusion. Then, based on Griffith criterion, we introduce the cost functional, which measures the shape sensitivity of the problem with respect to the geometry of the inclusion, provided by the energy release rate. We prove existence of optimal shapes for the problem considered.     

A Novel Time‐Domain BIEM for Wave Propagation Analysis in Anisotropic Solids With Cracks

Analysis of elastic wave propagation in anisotropic solids with cracks is of particular interest to quantitative non‐destructive testing and fracture mechanics. For this purpose, a novel time‐domain boundary integral equation method (BIEM) is presented in this paper. A finite crack in an unbounded elastic solid of general anisotropy subjected to transient elastic wave loading is considered. Two‐dimensional plane strain or plane stress condition is assumed. The initial‐boundary value problem is formulated as a set of hypersingular time‐domain traction boundary integral equations (BIEs) with the crack‐opening‐displacements (CODs) as unknown quantities. A time‐stepping scheme is developed for solving the hypersingular time‐domain BIEs. The scheme uses the convolution quadrature formula of Lubich [1] for temporal convolution and a Galerkin method for spatial discretization of the BIEs. An important feature of the present time‐domain BIEM is that it uses the Laplace‐domain instead of the more complicated time‐domain Green's functions. Fourier integral representations of Laplace‐domain Green's functions are applied. No special technique is needed in the present time‐domain BIEM for evaluating hypersingular integrals.