A Riemannian approach to the membrane limit in nonEuclidean elasticity
Abstract
NonEuclidean, or incompatible elasticity is an elastic theory for prestressed materials, which is based on a modeling of the elastic body as a Riemannian manifold. In this paper we derive a dimensionallyreduced model of the socalled membrane limit of a thin incompatible body. By generalizing classical dimension reduction techniques to the Riemannian setting, we are able to prove a general theorem that applies to an elastic body of arbitrary dimension, arbitrary slender dimension, and arbitrary metric. The limiting model implies the minimization of an integral functional defined over immersions of a limiting submanifold in Euclidean space. The limiting energy only depends on the first derivative of the immersion, and for frameindifferent models, only on the resulting pullback metric induced on the submanifold, i.e., there are no bending contributions.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.2671
 Bibcode:
 2014arXiv1410.2671K
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry
 EPrint:
 Communications in Contemporary Mathematics, Vol. 16, No. 5 (2014) 1350052