# Exact constructions in the (non-linear) planar theory of elasticity: From elastic crystals to nematic elastomers

## TL;DR

This paper characterizes conditions for stress-free configurations in non-linear planar elasticity, unifying the treatment of crystals and nematic elastomers, and extends known deformation patterns with new theoretical insights.

## Contribution

It generalizes previous results to the n-well problem, linking non-linear and linear theories, and introduces new constructions for stress-free deformations in complex materials.

## Key findings

- Derived necessary and sufficient conditions for stress-free configurations.
- Unified treatment of crystal and nematic elastomer models.
- Extended known deformation patterns to broader classes.

## Abstract

In this article we deduce necessary and sufficient conditions for the presence of `Conti-type', highly symmetric, exactly-stress free constructions in the geometrically non-linear, planar $n$-well problem, generalising results of [CKZ17]. Passing to the limit $n\rightarrow \infty$, this allows us to treat solid crystals and nematic elastomer differential inclusions simultaneously. In particular, we recover and generalise (non-linear) planar tripole star type deformations which were experimentally observed in [MA80,MA80a,KK91]. Further we discuss the corresponding geometrically linearised problem.