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, \({\mathcal {M}}_n\rightarrow {\mathcal {M}}\), 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.
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We are grateful to the anonymous referee for various comments.
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Communicated by J. Ball.
Raz Kupferman is partially supported by the Israel-US Binational Foundation (Grant No. 2010129), by the Israel Science Foundation (Grant No. 661/13) and by a Grant from the Ministry of Science, Technology and Space, Israel and the Russian Foundation for Basic Research, the Russian Federation.
