Derivation of systems of fundamental equations for a three-dimensional thermoelastic field with nonhomogeneous material properties and pts application to a semi-infinite body

A method of analytical development of three-dimensional thermoelastic problems for a medium with nonhomogeneous material properties is developed in this article. Assuming that the shear modulus elasticity G, the thermal conductivity lambda, and the coefficient of linear thermal expansion alpha vary with the power product form of axial coordinate variable z and introducing two kinds of displacement functions and the thermoelastic displacement function, the system of fundamental differential equations for such a three-dimensional field is established. As an illustrative example, we consider the thermoelastic problem of a semi-infinite body. The three-dimensional temperature solution in a steady state is obtained and the associated components of thermal displacement and stress are evaluated theoretically. Numerical calculations are carried out for several cases taking into account the variety of the nonhomogeneous material properties of G, lambda, and alpha, and these results are shown graphically.