# Heterogeneous elastic plates with in-plane modulation of the target curvature and applications to thin gel sheets

## TL;DR

This paper derives a rigorous 2D Kirchhoff plate theory for heterogeneous elastic sheets with in-plane modulated target curvature, and applies it to model shape changes in swelling gel sheets, advancing understanding of thin elastic structures.

## Contribution

It introduces a novel derivation of a Kirchhoff plate model from 3D elasticity for heterogeneous materials with in-plane modulated spontaneous strains, using $ ext{Gamma}$-convergence.

## Key findings

- Derived a 2D Kirchhoff plate model from 3D elasticity for heterogeneous sheets.
- Characterized the energy penalizing deviations from target curvature.
- Applied the model to swelling-induced shape changes in gel sheets.

## Abstract

We rigorously derive a Kirchhoff plate theory, via $\Gamma$-convergence, from a three-di\-men\-sio\-nal model that describes the finite elasticity of an elastically heterogeneous, thin sheet. The heterogeneity in the elastic properties of the material results in a spontaneous strain that depends on both the thickness and the plane variables $x'$. At the same time, the spontaneous strain is $h$-close to the identity, where $h$ is the small parameter quantifying the thickness. The 2D Kirchhoff limiting model is constrained to the set of isometric immersions of the mid-plane of the plate into $\mathbb{R}^3$, with a corresponding energy that penalizes deviations of the curvature tensor associated with a deformation from a $x'$-dependent target curvature tensor. A discussion on the 2D minimizers is provided in the case where the target curvature tensor is piecewise constant. Finally, we apply the derived plate theory to the modeling of swelling-induced shape changes in heterogeneous thin gel sheets.