Geometric Nonlinear Thermoelasticity and the Time Evolution of Thermal Stresses

In this paper we formulate a geometric theory of nonlinear thermoelasticity that can be used to calculate the time evolution of temperature and thermal stress fields in a nonlinear elastic body. In particular, this formulation can be used to calculate residual thermal stresses. In this theory the material manifold (natural stress-free configuration of the body) is a Riemannian manifold with a temperature-dependent metric. Evolution of the geometry of the material manifold is governed by a generalized heat equation. As examples, we consider an infinitely long circular cylindrical bar with a cylindrically symmetric temperature distribution and a spherical ball with a spherically-symmetric temperature distribution. In both cases we assume that the body is made of an arbitrary incompressible isotropic solid. We numerically solve for the evolution of thermal stress fields induced by thermal inclusions in both a cylindrical bar and a spherical ball, and compare the linear and nonlinear solutions for a generalized neo-Hookean material.