The Monge-Ampere constrained elastic theories of shallow shells

Abstract

Motivated by the degree of smoothness of constrained embeddings of surfaces in ${\mathbb{R}}^3$, and by the recent applications to the elasticity of shallow shells, we rigorously derive the $\Gamma$-limit of 3-dimensional nonlinear elastic energy of a shallow shell of thickness $h$, where the depth of the shell scales like $h^{\alpha }$ and the applied forces scale like $h^{\alpha +2}$, in the limit when $h\to 0$. The main analytical ingredients are two independent results: a theorem on approximation of $W^{2,2}$ solutions of the Monge-Ampère equation by smooth solutions, and a theorem on the matching (in other words, continuation) of second order isometries to exact isometries.

References in corpus (2)

• The von Karman equations for plates with residual strain
• Models for elastic shells with incompatible strains

Cited by in corpus (5)

• Variational models for prestrained plates with Monge-Ampère constraint
• Plates with incompatible prestrain of higher order
• Plates with incompatible prestrain
• The linearized Kirchhoff theory for plates with incompatible prestrain
• Dimension reduction for thin films with transversally varying prestrain: the oscillatory and the non-oscillatory case