A geometric theory of thermal stresses

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.

Nonlinear elastic inclusions in isotropic solids

“…The body being residually stressed means that this 3-manifold, which we call the material manifold, cannot be isometrically embedded in R 3 . This geometric framework is identical to that used for calculating residual stresses in the presence of non-uniform temperature distributions [15], bodies with bulk growth [21] and bodies with distributed defects [22][23][24]. It should also be noted that this approach is general, i.e.…”

“…a Riemannian metric. For example, a change in temperature changes the natural (relaxed) distances of material points in a solid [15]. In the case of a ball with an inclusion, the natural distances in the inclusion and the matrix are different and this induces a stress field.…”

“…In their spherical inclusion problem F * = (1 + α)I, where I is the identity tensor. Note that this implies that the material metric [15] is G = (1 + α) 2 I. In their numerical calculations for α = 0.1, Diani & Parks [14] observed that in the inclusion the Cauchy stress is uniform and hydrostatic.…”

“…These residual stresses can be described geometrically by positing that the stress-free configuration is a Riemannian manifold with a geometry explicitly depending on the anelasticity source(s) (figure 1b). The ambient space is a Riemannian manifold (S, g), and hence the computation of stresses requires a Riemannian material manifold (B, G) and a mapping ϕ : B → S. For example, in the case of non-uniform temperature changes and bulk growth [15,21], one starts with a material metric G that specifies the relaxed distances of the material points. In the case of distributed defects, the material metric is calculated indirectly [22][23][24].…”

Weyl geometry and the nonlinear mechanics of distributed point defects

“…In other words, in our geometric approach, one by-passes the notion of local intermediate configuration by using an appropriate geometry in the material manifold that automatically makes the body with distributed defects stress-free. The ambient space being a Riemannian manifold (S, g), the computation of stresses requires a Riemannian material manifold (B, G) (the underlying Riemannian material manifold) and a map 4 : B → S. For example, in the case of non-uniform temperature changes and bulk growth (Ozakin & Yavari 2010;Yavari 2010), one starts with a material metric G that specifies the relaxed distances of the material points. However, the material metric cannot always be obtained directly.…”