Annotation

A spatial problem of the theory of elasticity about stress distribution in a transversal-isotropic medium containing the same inclusion at the boundary of the phase separation is considered. The studies show that, at the interface, voltages are local in nature and rapidly attenuate with distance from the inclusion surface.

In the mechanics of a deformed solid, spatial problems of the theory of elasticity and thermal elasticity, which relate to stress distribution in the neighborhood of inhomogeneities with structural composites, occupy an important place. With respect to the strength of such materials, their production requires information on the achievement of stress components of extreme values in certain zones (zones of destruction). Such extreme values are generally reached at the interface.

The problem of obtaining reliable information about stress distribution in materials or structural elements is very important with consideration of the real picture of interfacial interaction, which is related to the use of the effective methods of solving spatial problems of elasticity theory.

The study of spatial problems of thermoelasticity for homogeneous isotropic and anisotropic bodies in general formulation is fraught with great mathematical difficulties because of the complexity of constructing the solution of a partial differential equation that satisfies certain boundary conditions.

One of the effective methods for solving the problems of elasticity theory is the Fourier method, which is based on the representation of the general solutions of equilibrium equations through potential functions. A feature of the Fourier method is the use of different representations of the Lame equation solution through harmonic functions, which allows one to search for a series solution.

The paper deals with the problem of stress distribution of an unlimited transversely isotropic medium, which contains anisotropic, relatively mechanical and thermal properties, inclusion in the form of a compressed spheroid with linear uniaxial heating. At the boundary of the phase section, conditions of non-perfect mechanical and thermal contacts are proposed.

The solution of the spatial problem under given boundary conditions on the surface of inclusion of the linear force and temperature fields is reduced to the development of the sought potential functions in trigonometric series by the connected Legendre functions of the first and second genera.

Satisfying the boundary conditions, an infinite system of linear algebraic equations for determining the expansion coefficients, which has a convergent solution, is obtained.

The obtained results show that the presence of inclusion in the form of a compressed spheroid in an elastic transverse isotropic medium at imperfect contact in the case of longitudinal stretching along the OZ axis depending on the ratio of spheroid axes does not significantly affect the stress concentration towards nominal values; the concentration of normal and circular stresses has a compressive nature, going to the nominal values with the increasing ratio of the spheroid axes.