Thermally nonlinear generalized thermoelasticity investigation of a functionally graded thick hollow cylinder based on the finite difference method

Abstract

In the present paper, the propagation and reflection of the transient displacement, temperature, and stress waves are numerically explored in a functionally graded thick-walled hollow cylinder, based on the Lord–Shulman generalized thermoelectricity theory and using the finite difference method. The motion and thermoelastic energy balance equations are first established based on the dimensionless radial displacement and temperature fields. It is assumed that the elasticity modulus, Poisson’s ratio, mass density, thermal expansion coefficient, thermal conductivity coefficient, and specific heat follow power functions of the dimensionless radius while the relaxation time is assumed to be constant within the FGM cylinder. Unlike many previous studies in which the energy balance equation was linearized by assuming small temperature rises, the present research accounts for the significant temperature variations and employs a nonlinear energy balance equation. Nondimensional forms of energy balance and motion equations are derived by introducing dimensionless quantities and discretized via the finite difference method. The Newmark method with constant average acceleration was employed to solve the highly nonlinear equations found from the finite difference method. Two different boundary conditions were used to investigate the propagation and reflection of the displacement, temperature, and stress transient waves in the radial direction of the cylinder. The results revealed the variable speed of the displacement wave and the significant role of the changes in the properties along the radius of the cylinder and the boundary conditions in altering the results.