The incompressible von K\'arm\'an theory for thin prestrained plates

TL;DR

This paper develops a new incompressible von Kármán theory for thin prestrained plates, deriving energy and equations using $ ext{Gamma}$-convergence, advancing the mathematical understanding of such materials.

Contribution

It introduces a novel incompressible von Kármán model for prestrained plates, extending previous theories with new derivations and mathematical techniques.

Findings

Derived a new von Kármán energy for prestrained plates

Established Euler-Lagrange equations in the incompressible setting

Utilized $ ext{Gamma}$-convergence to rigorously justify the model

Abstract

We derive a new version of the von K\'arm\'an energy and the corresponding Euler-Langrange equations, in the context of thin prestrained plates, under the condition of incompressibility relative to the given prestrain. Our derivation uses the theory of $\Gamma$-convergence in the calculus of variations, building on prior techniques in [Conti, Dolzmann (2009)] and [Lewicka, Mahadevan, Pakzad (2011)].