Referential and spatial evolutions in nonlinear elasticity

Abstract

In this PhD thesis, we propose a theoretical framework for studying referential and spatial evolutions in nonlinear elasticity. We use the referential evolution-- considering an evolving reference configuration-- to formulate a geometric theory of anelasticity. Indeed, an anelasticity source (such as temperature, defects, or growth) can manifest itself such that the body would fail to find a relaxed state in the Euclidean physical space. However, a reference configuration should by essence be stress-free so that one can properly quantify the strain with respect to it-- and the stress by means of a constitutive equation. Identifying the reference configuration with an abstract manifold-- material manifold-- allows for a rational construction of such a stress-free state which can further accommodate the evolution of the source of anelasticity by allowing the material manifold to have an evolving geometry. In this work, we formulate a general geometric theory of anelasticity for three-dimensional bodies that we apply to the particular case of thermoelasticity; and we also formulate a general theory of anelastic shells that we apply to the particular case of morphoelastic shells, i.e., those subject to growth and remodeling. In the context of anelasticity, as well as in nonlinear elasticity, most exact solutions are obtained by assuming some restrictive class of symmetry for the solution. We propose a theory of small-on-large anelasticity, that is analogous to the small-on-large theory of Green et al. in classical elasticity. It can be used to find exact solutions for non-symmetric distributions of anelasticity sources that are small perturbations of symmetric ones. Finally, motivated by gaining further insights on the theory of nonlinear elasticity as well as the case of a continuum deforming in an evolving ambient space, we formulate a theory of nonlinear elasticity where the geometry of the ambient space is time-dependent.