Relative bending energy for weakly prestrained shells

Abstract

We derive a dimensionally reduced model for a thin film prestrained with a given incompatible Riemannian metric:

$$G^h\left(x^{\prime },x_3\right)=I_3+2h^{\gamma }S\left(x^{\prime }\right)+2h^{\gamma ∕2}{x}_{3}B\left(x^{\prime }\right)+\text{ higher order terms},\gamma >2,$$

where $0<h\ll 1$ is the thickness of the film. The problem is studied rigorously by using a variational approach and establishing the $\mathrm{\Gamma }$-convergence of the non-Euclidean version of the nonlinear elasticity functional. It is shown that the residual nonlinear elastic energy scales as $O\left(h^{\gamma +2}\right)$ as $h\to 0$.