Finite Elasticity and Weak Diffeomorphisms

Abstract

Let Ω ⊂ ℝ³ be a bounded domain which is taken to be the reference configuration or the rest state of a perfectly elastic body. Classically a deformation of the body, which in its rest position occupies the region Ω, is described by a smooth map u : $\Omega\to \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Omega }\subset {\mathbb{R}^3}$ that is orientation preserving and globally invertible. According to our idealized situation of a perfectly elastic body no heat transfer occurs in the process of loading and unloading a perfectly elastic material, and such loading and unloading process is completely reversible. Therefore it is reasonable to assume that all mechanical properties of a perfectly elastic material are characterized by a stored energy function W(x, G),which depends on the infinitesimal deformations G = Du(x), and that in terms of it the total internal energy stored by the body which undergo the deformation u is given by

$$\varepsilon (u,\Omega ): = \int\limits_\Omega W (x,Du(x))dx$$

(1)

Materials whose mechanical properties are characterized by a stored energy functions are often called hyperelastic materials.