Axisymmetric Stress-Strain State of a Body with Thin Rigid Disk-Shaped Heat-Resistant Inclusion

Abstract

UDC 539.3 We construct a solution of the problem of thermoelasticity for a body with thin rigid heat-resistant inclusion in the class of functions specifying the stress-strain state with constant displacements normal to the plane of inclusion at infinity. The inclusion is modeled by a boundary layer corresponding, from the mathematical viewpoint, to a sheet of moment dipoles and forces and the jump of radial displacements and stresses normal to the plane of inclusion serves as its mechanical manifestation. The solution of the heat-conduction and thermoelasticity equations with satisfying the requirement of continuous dependence of the solutions on boundary conditions is reduced to integral equations of the first kind and realized by the method of Neumann generalized series. We also determine the jump of normal stresses on the surfaces of the inclusion, which guarantees the realization of perfect mechanical contact between the rigid inclusion and the elastic matrix. The plane problems of thermoelasticity for bodies with rigid striplike inclusions were investigated in numerous works by the methods of the theory of functions of complex variables. A survey of these works can be found in the monograph [11]. However, there are only few works devoted to the study of three-dimensional problems of thermoelasticity for bodies with thin rigid plane inclusions. The problems of thermoelasticity for bodies with inclusions in the case of known temperature displacements and stresses in the domain of location of the inclusions are reduced to the problems of elasticity with the corresponding conditions of contact interaction between thin rigid diskshaped inclusions and the elastic matrix under force loading [3, 9, 12, 13]. These problems, in turn, can be reduced by the methods of potential theory to the solution of integral equations of the first kind. Their solution specifies the singular distribution of the characteristics of the stress-strain state at the edge of the inclusion. The problems of thermoelasticity for a body with heat-generating thin rigid disk-shaped inclusion were solved in [4, 6]. In the presence of a heat-resistant inclusion, the temperature jump and tangential stresses is formed on its surfaces. This is explained by the unequal heating of the surfaces. Under the conditions of perfect mechanical contact between the rigid inclusion and the elastic matrix, a jump of normal stresses (which should be determined) is formed on the surfaces of he inclusions [5]. In [1], we proposed a model of heat-resistant inclusion as a sheet of heat dipoles distributed in its plane with a certain density and determined the temperature displacements and stresses induced by these dipoles. In the present work, we apply this model for the solution of the problem of interaction of a thin rigid diskshaped inclusion under the conditions of perfect mechanical contact with the elastic matrix in the class of functions specifying the component of the vector of heat flux normal to the plane of inclusion and its normal displacements (with constant value at infinity).