A Geometric Theory of Thermal Stresses

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.