The Dynamic Problems About the Definition of Stress State Near Thin Elastic Inclusions Under the Conditions of Perfect Coupling

Abstract

The isotropic unbounded elastic body (matrix), which is in the condition of plane strain and which contains a thin elastic inclusion in the form of a strip is considered. It occupies the area: $\left| x \right| \leqslant a, \left| y \right| \leqslant \tfrac{h} {2}$ in the plane Oxy. It is necessary to determine the stress state in the matrix caused by the non-stationary or harmonic plane waves interacting with the inclusion. The problem is reduced to the construction of the solution of the Lame equations for plane strain, which satisfies the given boundary conditions on the inclusion. It is considered that under these conditions the inclusion is so thin that the displacements of any point of it coincide with the displacements of the appropriate point of a middle plane. The method of the solution is based on the presentation of the displacements and stresses caused by scattered waves in the form of the discontinuous solution of the Lame equations (in the non-stationary case in the space of the Laplace images). Stress intensity factors (SIF) are taken as amounts characterizing the stress state near the inclusion, as in a number of publications, where analogous problems were considered in the static formulation. The transition from the Laplace images to the originals is implemented numerically for non-stationary problems calculations. The numerical research of the dependence of SIF on time or frequency and the ratio of elastic constants of the matrix and the inclusion has been done. The possibility of the consideration of inclusions of large rigidity as absolutely rigid ones is analyzed.