# A hierarchy of multilayered plate models

## TL;DR

This paper derives a hierarchy of multilayered plate models from 3D nonlinear elasticity using $ extGamma$-convergence, capturing effects of heterogeneity, pre-stress, and different elastic regimes in thin films.

## Contribution

It introduces a systematic derivation of layered plate theories from 3D elasticity, including effects of heterogeneity, pre-stress, and multiple elastic regimes, using $ extGamma$-convergence.

## Key findings

- Derivation of linearised Kirchhoff, von Kármán, and linear plate theories with spontaneous curvature.
- Effective elastic constants expressed via moments of material properties.
- Inclusion of pre-stress effects in the derived plate models.

## Abstract

We derive a hierarchy of plate theories for heterogeneous multilayers from three dimensional nonlinear elasticity by means of $\Gamma$-convergence. We allow for layers composed of different materials whose constitutive assumptions may vary significantly in the small film direction and which also may have a (small) pre-stress. By computing the $\Gamma$-limits in the energy regimes in which the scaling of the pre-stress is non-trivial, we arrive at linearised Kirchhoff, von K{\'a}rm{\'a}n, and fully linear plate theories, respectively, which contain an additional spontaneous curvature tensor. The effective (homogenised) elastic constants of the plates will turn out to be given in terms of the moments of the pointwise elastic constants of the materials.