A Riemannian Approach to Reduced Plate, Shell, and Rod Theories
Abstract
We derive a dimensionallyreduced limit theory for an $n$dimensional nonlinear elastic body that is slender along $k$ dimensions. The starting point is to view an elastic body as an $n$dimensional Riemannian manifold together with a not necessarily isometric $W^{1,2}$immersion in $n$dimensional Euclidean space. The equilibrium configuration is the immersion that minimizes the average discrepancy between the induced and intrinsic metrics. The dimensionally reduced limit theory views the elastic body as a $k$dimensional Riemannian manifold along with an isometric $W^{2,2}$immersion in $n$dimensional Euclidean space and linear data in the normal directions. The equilibrium configuration minimizes a functional depending on the average covariant derivatives of the linear data. The dimensionallyreduced limit is obtained using a $\Gamma$convergence approach. The limit includes as particular cases plate, shell, and rod theories. It applies equally to "standard" elasticity and to "incompatible" elasticity, thus including as particular cases socalled nonEuclidean plate, shell, and rod theories.
 Publication:

arXiv eprints
 Pub Date:
 January 2012
 arXiv:
 arXiv:1201.3565
 Bibcode:
 2012arXiv1201.3565K
 Keywords:

 Mathematics  Differential Geometry;
 Condensed Matter  Soft Condensed Matter;
 Mathematics  Functional Analysis;
 74B20;
 53C42 (Primary) 74K10;
 74K20;
 74K25 (Secondary)
 EPrint:
 61 pages, added references, fixed typos