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: $$G^h(x',x_3)=I_3+2h^{\gamma}\,S(x')+2h^{\gamma/2}\,x_3B(x')+h.o.t, \,\,\,\gamma>2,$$ 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 $h\to 0$.