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.     

Problems of Thin Inclusions in a Two-Dimensional Viscoelastic Body

Under study are the equilibrium problems for a two-dimensional viscoelastic body with delaminated thin inclusions in the cases of elastic and rigid inclusions. Both variational and differential formulations of the problems with nonlinear boundary conditions are presented; their unique solvability is substantiated. For the case of a thin elastic inclusion modelled as a Bernoulli–Euler beam, we consider the passage to the limit as the rigidity parameter of the inclusion tends to infinity. In the limit it is the problem about a thin rigid inclusion. Relationship is established between the problems about thin rigid inclusions and the previously considered problems about volume rigid inclusions. The corresponding passage to the limit is justified in the case of inclusions without delamination.     

Asymptotics for a thin elastic fiber in contact with a rigid foundation

In this work a 3-D contact elasticity problem for a thin fiber and a rigid foundation is studied. We describe the contact condition by a linear Robin-boundary-condition (by meaning of the penalized and linearized non-penetration and friction conditions). The Robin parameters are scaled differently in the longitudinal and cross-sectional directions. The dimension of the problem is reduced by a standard ([3], [4]) asymptotic approach with an additional expansion suggested to fulfil the contact conditions. The 3-D contact conditions result into 1-D Robin-boundary-conditions for corresponding ODEs. The Robin-coefficients of the 1-D problem depend on the ones from the 3-D statement and on the cross-section of the fiber. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Diffraction by a rigid,mobile inclusion

The long-wavelength problem of scattering by a rigid inclusion in an elastic medium is studied. Taking into account the mobility of the inclusion leads to non-classical boundary conditions. At infinity the solution is sought in the form of a multipole ansatz. Near the scatterer a series of static problems is obtained. In the course of the solution the integral characteristic of the rigid mobile inclusion, whose scalar analog is the tensor e_ij studied by Polya and Szegö, arises in a natural manner.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 61–68, 1986.     

圆形界面刚性线夹杂的平面问题

研究了圆弧形界面刚性线夹杂的平面弹性问题．集中力作用于夹杂或基体中的任意点,并且无穷远处受均匀载荷作用．利用复变函数方法,得到了该问题的一般解答．当只含一条界面刚性线夹杂时,获得了分区复势函数和应力场的封闭形式解答,并给出刚性线端部奇异应力场的解析表达式．结果表明,在平面荷载下界面圆弧形刚性线夹杂尖端应力场和裂纹尖端相似具有奇异应力振荡性．对无穷远加载的情况,讨论了刚性线几何条件、加载条件和材料失配对端部场的影响．     

A contact problem for a smooth rigid disc inclusion in a penny-shaped crack

The present paper examines the problem of the complete indentation of the surface of a penny-shaped crack by a smooth rigid disc inclusion. The integral equation governing the problem is solved numerically to evaluate the axial stiffness of the rigid inclusion and the stress intensity factors at the tip of the penny-shaped crack.     

Application of the method of direct cutting-out to the solution of the problem of longitudinal shear of a wedge with thin heterogeneities of arbitrary orientation

The method of direct cutting-out consists of modeling of a finite body, in particular, with thin heterogeneities, using a much simpler problem for a bounded or a partially bounded body with thin heterogeneities located in the same manner and the presence of additional cracks or absolutely rigid inclusions of fairy large length, which are modeled by the boundary conditions of a bounded body. The method is tested on the problems of antiplane deformation of a symmetrically loaded crack in a wedge with free faces and an absolutely rigid inclusion placed with some tension in a wedge with restrained faces. For an elastic inclusion, we construct generalized conditions of interaction, which enable us to unify the procedure of giving different boundary conditions in the case of using the method of direct cutting-out.     

Shape optimization of rigid inclusions for elastic plates with cracks

In the paper, we consider an optimal control problem of finding the most safe rigid inclusion shapes in elastic plates with cracks from the viewpoint of the Griffith rupture criterion. We make use of a general Kirchhoff–Love plate model with both vertical and horizontal displacements, and nonpenetration conditions are fulfilled on the crack faces. The dependence of the first derivative of the energy functional with respect to the crack length on regular shape perturbations of the rigid inclusion is analyzed. It is shown that there exists a solution of the optimal control problem.     

Joint deformation of a circular inclusion and a matrix

An elastic infinite plane containing a circular inclusion with given jumps of tractions and displacements along the interface and nonzero conditions at infinity is considered. Explicit expressions are derived for the Goursat-Kolosov complex potentials of this problem. The solution constructed can be used to examine various circular interfacial defects, including interfacial cracks and rigid parts of the interface. The problem under consideration is fundamental for the superposition method, which solves many problems in which a circular region is an element of a polyphase elastic medium. In such cases, the well-posedness of the problem, which depends on the interrelation between the jumps of tractions and displacements, follows from the very superposition method. The application techniques of this method are demonstrated for singular problems on the action of a point force and an edge dislocation located inside an inclusion or in the matrix. Computational results for the tractions arising at the interface under the action of a point force concentrated in the inclusion are given.     

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.     

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.     

The diffraction of plane elastic waves by a delaminated rigid inclusion when there is smooth contact in the delamination region

A solution of the problem of the diffraction of harmonic elastic waves by a thin rigid strip-like delaminated inclusion in an unbounded elastic medium, in which the conditions for plane deformation are satisfied, is proposed. We mean by a delaminated inclusion an inclusion, one side of which is completely bonded to the elastic medium, while the second does not interact in any way with it, or this interaction is partial. It is assumed that the conditions for smooth contact are satisfied in the delamination region. The method of solution is based on the use of previously constructed discontinuous solutions of the equations describing the vibrations of an elastic medium under plane deformation conditions. The problem therefore reduces to solving a system of three singular integral equations in the unknown stress and strain jumps at the inclusion. An approximate solution of the latter enabled formulae to be obtained that are convenient for numerical realization when investigating the stressed state in the region of the inclusion and its displacements when acted upon by incident waves.     

A rigid inclusion in the contact problem for elastic plates

A family of problems under consideration describes the contact of elastic plates situated at a given angle to each other and, in the natural condition, touching along a line. The plates are subjected only to bending. The limiting process from the elastic inclusion to the rigid one is studied. It is demonstrated that the limit problems precisely describe the contact of an elastic plate with a rigid beam and the problem of the equilibrium of an elastic plate with a rigid inclusion. The solvability of the problems is established; the boundary conditions holding on the possible contact set are found as well as their precise interpretation.     

The three-dimensional problem of a thin inclusion in a composite elastic wedge

The three-dimensional problem of a thin rigid elliptic inclusion in the middle of a composite elastic wedge is investigated. The wedge consists of three connected wedge-shaped layers connected by a sliding clamp, in which the layer containing the inclusion is incompressible. The outer faces of the composite wedge are also under sliding-clamp conditions. The inclusion is completely bonded to the elastic medium in the contact region. Using Fourier and Kontorovich–Lebedev transformations, a system of integral equations of the problems are derived for the shear contact stresses. A regular asymptotic method is used to solve this system. Calculations are carried out. The results can be used for calculations on the strength of rubber-metal articles and structures having a corner line.     

The diffraction of plane elastic unsteady waves by a delaminated inclusion in the case of smooth contact in the delamination region

A solution of the problem of the diffraction of unsteady elastic waves by a thin strip-like delaminated rigid inclusion in an unbounded elastic medium under conditions of planer strain is proposed. We have in mind an inclusion, one side of which is completely bonded with the medium while, the other side is delaminated and conditions of smooth contact are satisfied on it. The method of solution is based on the use of discontinuous solutions of the Lamé equations of motion under conditions of planer strain, which have been constructed earlier in the space of Laplace transforms. As a result, the problem reduces to solving a system of three singular integral equations for the transforms of the unknown discontinuities. The inverse transforms are found by a numerical method, based on the replacement of a Mellin integral by a Fourier series.     

An asymptotic form of the energy functional for an elastic body with a crack and a rigid inclusion. The plane problem

The plane problem in the linear theory of elasticity for a body with a rigid inclusion located within it is investigated. It is assumed that there is a crack on part of the boundary joining the inclusion and the matrix and complete bonding on the remaining part of the boundary. Zero displacements are specified on the outer boundary of the body. The crack surface is free from forces and the stress state in the body is determined by the bulk forces acting on it. The variation in the energy functional in the case of a variation in the rigid inclusion and the crack is investigated. The deviation of the solution of the perturbed problem from the solution of the initial problem is estimated. An expression is obtained for the derivative of the energy functional with respect to a zone perturbation parameter that depends on the solution of the initial problem and the form of the vector function defining the perturbation. Examples of the application of the results obtained are studied.     

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.     

刚性线夹杂与弹性圆夹杂的相互作用

本文将刚性线夹杂与弹性圆夹杂的相互作用，归为解一个标准的柯西型奇异积分方程，获得了刚性夹杂端点的应力强度因子及夹杂的界面应力．     

A contact problem for an elastic plate with a thin rigid inclusion

Under study is an equilibrium problem for a plate under the influence of external forces. The plate is assumed to have a thin rigid inclusion that reaches the boundary at the zero angle and partially contacts a rigid body. On the inclusion face, there is a delamination. We consider the complete Kirchhoff–Love model, where the unknown functions are the vertical and horizontal displacements of the middle surface points of the plate. We present differential and variational formulations of the problem and prove the existence and uniqueness of a solution.