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

## TL;DR

This paper analyzes the asymptotic behavior of prestrained thin shells by studying the energy deficits of their immersions, deriving a hierarchy of models based on Riemannian curvature conditions and energy scalings.

## Contribution

It completes the scaling analysis of non-Euclidean energies for thin shells, identifying all possible energy regimes and their geometric conditions, extending previous results.

## Key findings

- Energy quantization occurs at even powers of thickness h.
- Scaling regimes are characterized by vanishing Riemann curvatures.
- Asymptotic behavior of minimizing immersions is established.

## 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 $\Omega^1$, 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 $G_{\mid \Omega^h}$ on $\Omega^h,$ over all weakly regular immersions. This deficit is measured by the non-Euclidean energies $\mathcal{E}^h$, which can be seen as modifications of the classical nonlinear three-dimensional elasticity. Building on our previous results, we complete the scaling analysis of $\mathcal{E}^h$ and the derivation of $\Gamma$-limits of the scaled energies $h^{-2n}\mathcal{E}^h$, 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 $h\to 0$.