Convergence of Equilibria for Incompressible Elastic Plates in the von K\'arm\'an Regime

TL;DR

This paper proves that critical points of nonlinear elastic energies for thin incompressible plates converge to solutions of the limiting incompressible von Kármán equations as the plate thickness approaches zero.

Contribution

It establishes the convergence of critical points of 3D elastic energies to the 2D incompressible von Kármán model under the specified scaling.

Findings

Critical points of elastic energies converge to the limiting functional

Validation of the incompressible von Kármán model for thin plates

Mathematical proof of convergence under the von Kármán scaling

Abstract

We prove convergence of critical points $u^h$ of the nonlinear elastic energies $E^h$ of thin incompressible plates $\Omega^h=\Omega \times (-h/2, h/2)$, which satisfy the von K\'arm\'an scaling: $E^h(u^h)\leq Ch^4$, to critical points of the appropriate limiting (incompressible von K\'arm\'an) functional.