Deriving the System of Fundamental Equations for Three-Dimensional Thermoelastic Field with Nonhomogeneous Material Properties and Its Application to a Thick Plate

Abstract

In this study, an analytical method for deriving a system of equations for thermoelastic problems for a medium with nonhomogeneous material properties is developed. An analytical method of development for isothermal problems of such a nonhomogeneous body has already been given by Kassir under the assumption that the shear modulus of elasticity G changes with the variable z of the axial coordinate according to the relationship G(z)=G₀z^m. However, no analytical procedure has been established for the thermoelastic field up to data. In this study, an analytical method of developing the three-dimensional thermoelastic field is proposed by introducing the thermoelastic displacement potential function and two kinds of displacement functions. Assuming that the shear modulus of elasticity G, the thermal conductivity λ, and the coefficient of linear thermal expansion α vary with the variable ζ connected to the dimensionless axial coordinate according to the relationship G(ζ)=G₀ζ^m, λ(ζ)=λ₀ζ^ι, α(ζ)=α₀ζ^k, the three-dimensional temperature solution in the steady state for a thick plate is obtained and the associated thermal stress components are evaluated theoretically. Numerical calculations are carried out for several cases, taking into account the variation of nonhomogeneous material properties and the numerical results are graphically demonstrated.