The fundamental solution of the problem of thermoelasticity for a piecewise homogeneous transversely isotropic elastic space

The problem of constructing fundamental solutions to the thermoelasticity problem for a piecewise-homogeneous transversely isotropic space is reduced to the matrix Riemann problem in the space of generalized slow growth functions. As a result of the solution of which, were obtained expressions in explicit form for the components of the vector of the fundamental solution of the heat conduction problem, as well as simple representations for the components of the stress tensor and the displacement vector in plane of connection of transversely isotropic elastic half-spaces containing concentrated stationary heat sources. The temperature distribution is investigated depending on the thermophysical characteristics of the half-space materials. MSC: 74B05, 74H05, 74J20.


Introduction
The study of stress concentration in the vicinity of interfacial and internal defects such as cracks or inclusions in thermoelastic fields is of great practical importance. Many works have been devoted to this problem for various environments. In particular, in [1][2], the problems of stationary thermoelasticity for bodies with a heat-penetrating disc inclusion, between whose surfaces there is an imperfect thermal contact, as well as problems with a thin heat-active disc inclusion are considered.
The problem is reduced to hypersingular integral equations of the first and second kind, for which exact solutions are obtained.In [3][4][5][6][7][8][9] non-axisymmetric problems of elasticity and thermoelasticity for piecewise-homogeneous transversely isotropic spaces containing interfacial stress concentrators, such as cracks or rigid inclusions, using the method of singular integral relations (SIR) [10] reduced to systems of two-dimensional singular integral equations (SIR) and proposed a method for their solution. A similar approach was applied in [11][12][13][14][15] to solving problems of interfacial and internal defects in piecewise homogeneous anisotropic media.
In the mathematical formulation and solution of such problems about defects, it is necessary to set the boundary conditions on the defect itself, such as stress on the crack edges or displacement at the inclusion.Since in thу physical formulation of the problems from determining the stress and displacement fields in the vicinity of the stress concentrators, known the stresses or displacements at the boundary of the region,at some interior points or at infinity (for unbounded bodies), then the determination of the boundary conditions on the defect is a separate problem.
Within the framework of the linear theory of thermoelasticity, to solve this problem, it is necessary to know the distribution of the temperature, stress and displacement fields in the corresponding piecewise homogeneous bodies without defects in the presence of volumetric forces and concentrated heat sources.
In particular, for piecewise homogeneous isotropic and transversally isotropic spaces, such solutions are given, respectively, in [16] and [17].Green's functions for piecewise homogeneous transversally isotropic spaces in the presence of a concentrated heat source and in the absence of thermal diffusion were constructed in [18], and in the presence of thermal diffusion in [19]. In [20,21], Green's functions for a layered thermal environment were constructed.
An effective method for solving this problem is the method of fundamental solutions in the space ℑ ′ (R 3 ) of generalized functions of slow growth. In particular, in [14], the problem of constructing fundamental solutions for piecewise homogeneous two-dimensional anisotropic media is reduced to the matrix Riemann problem with respect to some variables in the space ℑ ′ (R 3 ) and suggested an approach to its solution. In this work, this approach is generalized to construct in an explicit analytical form fundamental solutions to the problem of heat conduction and thermoelasticity for a piecewise-homogeneous transversely isotropic space, which made it possible to study the temperature distribution and obtain how the temperature affects the distribution of stresses and displacements in the plane of joining materials.

Main Results
1. Statement of the problem. Let stationary heat sources, concentrated in some regions of dimension n (n = 0,1,2,3), act in an inhomogeneous space composed of two different transversely isotopic half-spaces, completely linked in the plane = 0.
Based on the equilibrium equations and the generalized Hooke's law, and also taking into account the Duhamel-Neumann relation with respect to the components of the vector v, in the space of generalized functions of slow growth ℑ ′ (R 3 ) we write the following boundary value problem Here we use the notation  We introduce the following notation w = { } =1,4 , where functions ( , , ) ∈ ℑ ′ (R 3 ) there are components of the system of fundamental solutions, that is, solutions that satisfy the following system of boundary value problems , ± −thermal conductivity coefficients for the upper > 0 and lower < 0 half-spaces, respectively.
. After applying the inverse Fourier transform, the components of will be presented as: )+ )+ Sought temperature we get like this = 4 * . 3. Construction of a fundamental solution to the problem of thermoelasticity. We apply to the matrix equation (2) the operator of the three-dimensional Fourier transform F 3 iз ℑ ′ (R 3 ), given the following representation for vector components v: Then, considering the conditions (3) and the results of [10][11][12][13][14]26,27], with respect to and also that the functions V ± ∈ ℑ ′ (R 3 ± ) admit an analytical representation [22,26,27] in the variable 3 we obtain the following matrix equation where ( 1 , 2 )− unknown functions from ℑ ′ (R 2 ) for determine which, we need to use conditions (3) after the Fourier-transformed.
We represent the sought functions as where Ψ ± , Υ ± ( = 1, 2) new unknown functions, then matrix equation (9) can be separated into two independent equations where we use the notation }︁ ,

Directly from equations (12) we obtain
. Further, using representations (11), (12) after applying the inverse Fourier transform, we obtain: 4. Stress and displacement fields in the plane of connection of half-spaces. Putting = 0 in the fundamental solutions, we obtain the distribution of normal and tangential stresses and displacements in the plane of connection of half-spaces in the presence of heat sources.

Conclusion
The problem of constructing fundamental solutions to the thermoelasticity problem for a piecewise-homogeneous transversely isotropic space is reduced to the matrix Riemann problem in the space of generalized slow growth functions.As a result of the solution of which, were obtained expressions in explicit form for the components of the vector of the fundamental solution of the heat conduction problem, as well as simple representations for the components of the stress tensor and the displacement vector in plane of connection of transversely isotropic elastic half-spaces containing concentrated stationary heat sources. The temperature distribution is investigated depending on the thermophysical characteristics of the half-space materials.