Relative Bending Energy for Weakly Prestrained Shells

Abstract

In this paper, we derive a dimensionally reduced model for a thin film prestrained with a given incompatible Riemannian metric: $$G^h(x^{\prime },x_3)=I_3+2h^{\gamma }S(x^{\prime })+2h^{\gamma /2}{x}_{3}B(x^{\prime })+h.o.t,\gamma >2,$$ 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 $h\to 0$.

arXiv admin note: text overlap with arXiv:1404.3298, arXiv:1002.2252 by other authors