Relative Bending Energy for Weakly Prestrained Shells

TL;DR

This paper develops a reduced mathematical model for thin, prestrained shells with incompatible metrics, rigorously analyzing the elastic energy scaling as the shell thickness approaches zero.

Contribution

It introduces a novel dimensionally reduced model for weakly prestrained shells using $ ext{Gamma}$-convergence, extending nonlinear elasticity theory.

Findings

Residual elastic energy scales as $O(h^{ ext{ extgamma}+2})$

Model captures effects of incompatible prestrain on shell behavior

Provides rigorous mathematical foundation for thin shell elasticity with prestrain

Abstract

In this paper, we derive a dimensionally reduced model for a thin film prestrained with a given incompatible Riemannian metric: $$G^h(x',x_3)=I_3+2h^{\gamma}\,S(x')+2h^{\gamma/2}\,x_3B(x')+h.o.t, \,\,\,\gamma>2,$$ where $0<h\ll 1$ is the thickness of the film. The problem is studied rigorously by using a variational approach and establishing the $\Gamma$-convergence of the non-Euclidean version of the nonlinear elasticity functional. It is shown that the residual nonlinear elastic energy scales as $O(h^{\gamma+2})$ as $h\to 0$.