A Geometric Theory of Thermal Stresses

TL;DR

This paper develops a geometric framework for understanding thermal stresses by associating temperature distributions with Riemannian material manifolds, linking geometric changes to stress and deformation in nonlinear elasticity.

Contribution

It introduces a geometric approach to thermal stresses using Riemannian manifolds, connecting temperature changes to material metric evolution and equilibrium configurations.

Findings

Zero-stress temperature distributions identified via curvature tensor.

Change in temperature alters equilibrium configurations.

Linearized equations match classical thermal stress theory.

Abstract

In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change of temperature corresponds to a change of the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change of the material manifold, i.e. a change of the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change of temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses.