On thermoelastic stresses in an asymmetrically heated half-space

Abstract

A quasistatic problem of thermoelasticity is considered for a half-space in the case of convective heat exchange (boundary condition of the third kind). In the case of boundary conditions of the first and second kind all results are obtained in exactly the same manner. The exact solution of the problem is found in the form allowing the construction of an approximate solution, simple and suitable for numerical computations and based on the asymptotic expansion of the temperature and the stresses as t → 0. The problem is reduced to determining single integrals of simple functions, and in many cases the integrals can be expressed in terms of elementary functions. The error of the approximate solution is estimated. Unlike the results obtained earlier in /1–3/, the temperature distribution in the medium adjacent to the half-space is not assumed to be axisymmetric, i.e. a general asymmetric distribution is studied under certain constraints that are not significant from the physical point of view. Such asymmetric distributions are very common in practice /4/. The results of this paper can be used to study the fracture of brittle materials which can occur under the action of thermoelastic stresses /5/. It should be noted that application of the numerical methods which were successfully used in solving the symmetric problem of thermoelasticity /6/ encounters, in the case of asymmetric, obvious difficulties caused by the increased dimensionality of the problem.