Linear Strain Tensors on Hyperbolic Surfaces and Asymptotic Theories for Thin Shells

Abstract

We perform a detailed analysis of the solvability of linear strain equations on hyperbolic surfaces under a technical assumption (noncharacteristic). For regular enough hyperbolic surfaces, we prove that smooth infinitesimal isometries are dense in the $W^{2,2}$ infinitesimal isometries. Then the matching property for noncharacteristic regions is established; that is, smooth enough infinitesimal isometries can be matched with higher-order infinitesimal isometries. Those results are applied to the elasticity of thin shells for the $\Gamma$-limits, where the recovery sequences for noncharacteristic regions are obtained when the elastic energy density scales like $h^{\beta}$, $\beta\in(2,4)$, where $h$ is the thickness of a shell.