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A new Green's function and a new Poisson's type integral formula for a boundary value problem (BVP) in thermoelasticity for a half-space with mixed boundary conditions are derived. The thermoelastic displacements are generated by a heat source, applied in the inner points of the half-space and by temperature, and prescribed on its boundary. All results are obtained in closed forms that are formulated in a special theorem. A closed form solution for a particular BVP of thermoelasticity for a half-space also is included. The main difficulties to obtain these results are in deriving of functions of influence of a unit concentrated force onto elastic volume dilatation

The main objective of this paper is to prove a theorem (Section

The Green’s function plays the leading role in finding the solutions in integrals for boundary value problems (BVPs) in different fields of mathematical physics. The theory of thermoelasticity, which is a synthesis of the theory of heat conduction and elasticity theory, is one of such fields. By now, a number of theories of thermoelasticity have been developed and described in classical scientific literature [

Thus, the introduced functions of influence of a unit heat source on thermoelastic displacements

Finally, the influence functions

Take note that all influence functions in (

the formula for influence functions of a unit point heat flux

the formula for influence functions of a unit point temperature

the formula for influence functions of unit point heat exchange of the body with exterior medium

The formula in (

The advantage of the proposed integral formula in (

In this section we give a theorem for determining the thermoelastic displacements for a half-space in the form of volume and surface integrals, which is a particular case of the general integral formula in (

To obtain the functions of influence of an inner unit point heat source

Let the field of displacements

Let also the temperature field

The matrices

First, well-known Green’s function

To obtain the matrix

To get the influence functions

So, as for the BVP for

Finally, if we introduce the expressions

Now we have both functions:

At the next step of the proof of the theorem we have to check the correctness of the functions

To check (

The next step is to calculate the other influence functions in (

Introducing the influence functions in (

If in (

The obtained displacements, described by Poisson’s type integral formula (

Let us solve (

Finally, calculating the derivatives of expressions in (

The Poisson’s type integral formula, obtained in this paper, is new, useful, and completely ready to be efficiently applied for computing of the thermoelastic displacements

The most difficult problems in the proposed here method are the problems of deriving the Green’s functions

The approach presented in this paper in thermoelasticity for Cartesian canonical domains can be extended onto spherical [

The approach presented in this paper is valid also for other physical phenomena as electroelasticity, magnetoelasticity, and poroelasticity, described by the same BVP as in thermoelasticity.

The improper integral in (

the following equalities on the boundary plane of the half-space:

the relations

the following property of Dirac’s function:

the Green’s formula inside the half-space

One of the authors, Dr V. Seremet, is grateful to the University Paris-Est, Marne-la-Vallee, France, for the support of his two research visits to the university. Also he expresses many thanks to the reviewers of this paper, whose comments have contributed substantially to its improvements.