On the steady-state thermoelastic problem for the half-space

Abstract

Summary.This paper deals with the determination of the steady-state thermal stresses and displacements in a semi-infinite elastic medium which is bounded by a plane. The problem is treated within the classical theory of elasticity and is approached by the method of Green. It is shown that the stress field induced by an arbitrary distribution of surface temperatures is plane and parallel to the boundary. If the surface temperature is prescribed arbitrarily over a bounded "region of exposure" and is otherwise constant, the problem reduces to the determination of Boussinesq's three-dimensional logarithmic potential for a disk in the shape of the region of exposure, whose mass density is equal to the given temperature. Moreover, it is found that there exists a useful connection between the solutions to Boussinesq's and to the present problem for the half-space. An exact closed solution, in terms of complete and incomplete elliptic integrals of the first and second kind, is given for a circular region of exposure at uniform temperature. Exact solutions in terms of elementary functions are presented for a hemispherical distribution of temperature over a circular region, as well as for a rectangle at constant temperature.Introduction. Recent years have seen a revival of interest in the thermoelastic problem which has received repeated previous attention in the theory of elasticity.1 Nevertheless, the existing literature on this subject is largely confined to two-dimensional problems and to problems characterized by polar or rotational symmetry.A significant advance in connection with the general three-dimensional thermoelastic problem was made by Goodier [4], who considered a medium occupying the entire space; he reduced the computation of the thermal stresses due to an arbitrarily prescribed temperature distribution to the determination of the Newtonian potential for a mass distribution whose density coincides with the given temperature field. For domains other than the entire space, Goodier's approach merely supplies a particular solution of the thermoelastic equations and still necessitates the solution of an ordinary boundary-value problem in the theory of elasticity. A particular integral of the same form was employed earlier by Borchardt [5] in dealing with the special problem of the sphere.Goodier's method was extended by Mindlin and Cheng [6] to the problem of the half-space with a traction-free boundary, subjected to an arbitrary temperature distribution in its interior. As an application of the extended scheme, the problem of the half-space with a uniformly heated (or cooled) spherical core is solved in [6].In the present paper we return to the problem of the half-space but limit our attention

Problem of FGM TBC coated cylinder

“…The method of solution employed was the double Fourier transforms. The results confirmed solution of analogous problems, being inspiration to their work, received earlier by Sternberg and McDowell [2], based on Green's function, and by Muki [3], who used method combining the theory of Fourier series and the Hankel transforms of integral order. Recent achievements concerning application of FGM layer, treated directly as thermal barrier coating or indirectly as interface between coating and substrate, are mainly focused on plates or shells of thin or moderate thickness in which assumptions of plane stress or simplified 3D stress states are natural consequences of Kirchhoff-Love's or Reissner-Mindlin's hypotheses used.…”

On The Stress Free Deformation Of Linear FGM Interface Under Constant Temperature

“…The method of solution employed was the double Fourier transform. The results confirmed solution of analogous problems, being inspiration to their work, received earlier by Sternberg and McDowell (1957), based on Green's function, and by Muki (1957), who used method combining the theory of Fourier series and the Hankel transforms of integral order. Also there exists other more elegant way, based on application of Iljushin's potential (Iljushin et al, 1979), which is demonstrated in the present work.…”

Plane stress state of FGM thick plate under thermal loading

“…Authors introduce original concept of functionally graded Green's functions for better capturing of nature of displacement, stress and electric potential fields corresponding to functionally a graded PZT-4 half-space and a coated functionally graded PZT-4 layer over a homogeneous BaTiO 3 half-space. This work can be treated as excellent extension of paper by Sternberg and McDowell [3] since under certain assumptions there exists fully analogy between fields of electric potential and temperature.…”