Limits of elastic models of converging Riemannian manifolds

Abstract

In non-linear incompatible elasticity, the configurations are maps from a non-Euclidean body manifold into the ambient Euclidean space, ${\mathbb{R}}^k$. We prove the $\Gamma$-convergence of elastic energies for configurations of a converging sequence, ${\mathcal{M}}_n\to \mathcal{M}$, of body manifolds. This convergence result has several implications: (i) It can be viewed as a general structural stability property of the elastic model. (ii) It applies to certain classes of bodies with defects, and in particular, to the limit of bodies with increasingly dense edge-dislocations. (iii) It applies to approximation of elastic bodies by piecewise-affine manifolds. In the context of continuously-distributed dislocations, it reveals that the torsion field, which has been used traditionally to quantify the density of dislocations, is immaterial in the limiting elastic model.

References in corpus (4)

• Elastic theory of unconstrained non-Euclidean plates
• On the theory of relaxation in nonlinear elasticity with constraints on the determinant
• The emergence of torsion in the continuum limit of distributed edge-dislocations
• A Riemannian approach to the membrane limit in non-Euclidean elasticity

Cited by in corpus (4)

• Reshetnyak rigidity for Riemannian manifolds
• Limits of distributed dislocations in geometric and constitutive paradigms
• On material-uniform elastic bodies with disclinations and their homogenization
• From Volterra dislocations to strain-gradient plasticity