Relative Lagrangian Formulation of Finite Thermoelasticity

The motion of a body can be expressed relative to the present configuration of the body, known as the relative motion description, besides the classical Lagrangian and the Eulerian descriptions. When the time increment from the present state is small enough, the nonlinear constitutive equations can be linearized relative to the present state so that the resulting system of boundary value problems becomes linear. This formulation is based on the well-known ``small-on-large'' idea, and can be implemented for solving problems with large deformation in successive incremental manner. In fact, the proposed method is a process of repeated applications of the well-known “small deformation superposed on finite deformation” in the literature. This article presents these ideas applied to thermoelastic materials with a brief comment on the exploitation of entropy principle in general. Some applications of such a formulation in numerical simulations are briefly reviewed and a numerical result is shown.

Relative Lagrangian Formulation of Finite Thermoelasticity

Besides the Lagrangian and the Eulerian descriptions, the motion of a body can also be expressed relative to the present configuration of the body, known as the relative motion description. It is interesting to consider such a relative motion description in general to formulate the basic system of field equations for solid bodies. In doing so, when the time increment from the present state is small enough, the nonlinear constitutive equations can be linearized relative to the present state so that the resulting system becomes linear. This will be done for thermoelastic materials with a brief comment on the exploitation of entropy principle in general. Relative Lagrangian formulation is based on the well-known ``small-on-large'' idea, and can be implemented for solving problems with large deformation in successive incremental manner. Some applications of such a formulation in numerical simulations are briefly reviewed.

Small-on-large geometric anelasticity

In this paper, we are concerned with finding exact solutions for the stress fields of nonlinear solids with non-symmetric distributions of defects (or more generally finite eigenstrains) that are small perturbations of symmetric distributions of defects with known exact solutions. In the language of geometric mechanics, this corresponds to finding a deformation that is a result of a perturbation of the metric of the Riemannian material manifold. We present a general framework that can be used for a systematic analysis of this class of anelasticity problems. This geometric formulation can be thought of as a material analogue of the classical small-on-large theory in nonlinear elasticity. We use the present small-on-large anelasticity theory to find exact solutions for the stress fields of some non-symmetric distributions of screw dislocations in incompressible isotropic solids.