Visnyk LNAU: Architecture and Farm Building 2020 №21: 5-8


Bubniak Т., Physics and Mathematics Ph.D.
Lviv National Agrarian University


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. Studies show that, at the interface, voltages are local in nature and rapidly attenuate with the distance from the inclusion surface.

In the mechanics of a deformed solid body, spatial problems of the theory of elasticity and thermal elasticity, which are related to the stress distribution in the neighborhood of inhomogeneities with structural composites, occupy an important place. Considering 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 of great importance, taking into account the real picture of interfacial interaction, which is related to the use of 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.

The Fourier method is one of the effective methods for solving the problems of elasticity theory. It is based on the representation of 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 searching 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.

By satisfying the boundary conditions, an infinite system of linear algebraic equations to determine the coefficients of a convergent solution is obtained.

The analysis of the results of the study shows that on the surface of the transversely isotropic medium, the concentration of meridional and circular stresses rapidly decays from compression to tensile with movement from the pole to the equator.

Key words

potential functions, transversal-isotropic medium, non-ideal contact, spheroid, stress fields and thermal stresses

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