Quantitative immersability of Riemann metrics and the infinite hierarchy of prestrained shell models

Abstract: This paper concerns the variational description of prestrained materials, in the context of dimension reduction for thin films $$\Omega^{h=\omega\times} (-\frac{h}{2}, \frac{h}{2})$$. Given a Riemann metric $G$ on $\Omega<sup>1$, we study the question of what is the infimum of the averaged pointwise deficit of a given immersion from being an orientation-preserving isometric immersion of $G_{\mid \Omega<sup>h}$ on $\Omega<sup>h,$ over all weakly regular immersions. This deficit is measured by the non-Euclidean energies $\mathcal{E}<sup>h$, which can be seen as modifications of the classical nonlinear three-dimensional elasticity. Building on our previous results, we complete the scaling analysis of $\mathcal{E}<sup>h$ and the derivation of $\Gamma$-limits of the scaled energies $h<sup>{-2n}\mathcal{E}<sup>h$, for all $n\geq 1$. We show the energy quantisation in the sense that the even powers $2n$ of $h$ are indeed the only possible ones (all of them are also attained). For each $n$, we identify the equivalent conditions for the validity of the corresponding scaling, in terms of the vanishing of appropriate Riemann curvatures of $G$ to certain orders, and in terms of the matched isometry expansions. We also establish the asymptotic behaviour of the minimizing immersions as $h\to 0$.