Plane problem for a piecewise-homogeneous anisotropic body with elastic inclusion in the form of a strip

Abstract

A plane problem of the theory of elasticity lbr a piecewise-homogeneous anisotropic plane with elastic inclusions in the form of strips is solved by the method of jump functions. Inclusions are simulated by the jumps of the vector of stresses and the derivative of the vector of displacements on the median surfaces. By using complex potentials, we obtain the dependences of the components of the stress tensor and the vector of displacements on the load and unknown jump functions. In view of the conditions of interaction of a thin inclusion with an anisotropic medium, the problem is reduced to a system of singular integral equations for the jump functions. In the general case, this system is solved by the collocation method. For the cases of a slot and a perfectly rigid inclusion, we deduce the dependences of the generalized stress intensity factors on the concentrated forces and edge dislocations.On the basis of the principle of conjugation of continua of different dimensionality [1 ], the plane problem of the mathematical theory of thin elastic inclusions was first studied for the model of a pliable orthotropic inclusion interacting with an ambient isotropic medium in [2]. An approach to the solution of problems of thin elastic interface inclusions in isotropic materials proposed in [3] made it possible to obtain solutions for a crack and a perfectly rigid film as special cases. The general method for the solution of problems of this type is presented in [4].A detailed description of the method for the solution of problems of the theory of cracks in anisotropic media and a survey of the results obtained in this field can be found in [5]. Interface cracks in anisotropic materials were considered in [6,7]. More general results concerning sharp holes in anisotropic media are presented in [8], where the shape of a hole varies from a notch to a triangular hypocycloid.There is no principal difference between the solution of the problems of the theory of cracks and the solution of the corresponding problems for perfectly rigid films. Moreover, there exists an analogy between these problems [9] that enables one to pass from the solution obtained for a perfectly rigid film to the solution of a similar problem for a crack (but not in the opposite direction). This means that solutions obtained for perfectly rigid films are, in a certain sense, more general. Nevertheless, very little attention is given to their construction in the literature. As far as anisotropic materials are concerned, we can mention the works [10][11][12][13][14][15][16][17][18].Some specific features of deformation of elastic inclusions are taken into account by the approach proposed in [19] and applied in the same work to the analysis of an orthotropic plate containing a hole with an isotropic disk inserted in it with tension through an intermediate layer (thin bar). The most general results can be found in [20] where the problem of stress concentration near thin interface elastic inclusions in stress fields uniform at infinity is studied by the method ...

Interaction of thin inclusion with circular hole in the elastic isotropic medium

“…It is proved in [10] and in Sulym and Shevchuk [15] that the stress field distribution near the tip of the thin elastic inclusion yields ⎛ ⎜ ⎝…”