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A Geometric Theory of Thermal Stresses

Arash_Yavari's picture

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.

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