Numerical Solution of a Three-Dimensional Coupled Dynamic Thermoelasticity Problem in Terms of Strains

Abstract

A three-dimensional coupled dynamic thermoelasticity problem is considered for a homogeneous isotropic parallelepiped. The primary unknowns are the six independent components of the strain tensor and temperature. Based on the equations of motion, the Duhamel–Neumann relations, and the Cauchy kinematic relations, a system of dynamic equations in terms of strains is derived and supplemented by the heat-conduction equation. Thermal effects are incorporated into the mechanical equations, while the reverse coupling is determined by the volumetric strain rate. This formulation allows strains to be computed directly without recovery from displacement derivatives. Explicit and implicit finite-difference schemes on a rectangular spatial grid are constructed for the numerical solution. The numerical example considers a sinusoidal initial temperature field with zero initial strains and strain rates. Temperature and normal strain are compared in different cross-sections of the parallelepiped. The comparison is performed using identical input data at a fixed computational time. Both schemes reproduce similar spatial distributions and the same location of the central maximum. According to the tabulated data, the maximum absolute temperature difference is 0.002, or 0.0108 percent of the maximum implicit solution value. For normal strain, the maximum difference is 0.001, while the maximum value of 0.046 is identical in both implementations. The results confirm the internal consistency of the algorithms in jointly determining the temperature and strain fields.