Abstract
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 Θ(k) and, also, in calculating of a volume integral of the product of function Θ(k) and Green′s function in heat conduction. Using the proposed approach, it is possible to extend the obtained results not only for any canonical Cartesian domain, but also for any orthogonal one.
Highlights
The main objective of this paper is to prove a theorem Section 2.2 about deriving a Poisson’s type integral formula for a thermoelastic half-space with the homogeneous locally mixed mechanical boundary conditions and with the nonhomogeneous Dirichlet’s boundary condition for temperature
The most developed theory, which is widely used in practical calculations, is the theory of thermal stresses, that is, the theory of uncoupled thermoelasticity, where the temperature field does not depend on the field of elastic displacements
We give a theorem for determining the thermoelastic displacements for a halfspace in the form of volume and surface integrals, which is a particular case of the general integral formula in 1.3
Summary
The main objective of this paper is to prove a theorem Section 2.2 about deriving a Poisson’s type integral formula for a thermoelastic half-space with the homogeneous locally mixed mechanical boundary conditions and with the nonhomogeneous Dirichlet’s boundary condition for temperature. To prove this theorem we need some equations for Green’s integral formula in stationary thermoelasticity Section 1.1 and thermoelastic influence functions Sections 1.2 and 1.3 for a solid body, suggested and published by the first-named author earlier.
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