Convergence of equilibria of thin elastic plates under physical growth conditions for the energy density

TL;DR

This paper investigates the limiting behavior of equilibrium configurations of thin elastic plates as their thickness approaches zero, demonstrating convergence to critical points of a derived limit functional under physically realistic energy density conditions.

Contribution

It establishes the convergence of critical points of the elastic energy to those of the Gamma-limit, considering physically relevant energy density blow-up conditions.

Findings

Critical points of the elastic functional converge as thickness tends to zero.

The convergence is shown under the assumption of energy density blow-up as det F approaches zero.

The results connect physical growth conditions to the mathematical limit behavior of elastic plates.

Abstract

The asymptotic behaviour of the equilibrium configurations of a thin elastic plate is studied, as the thickness $h$ of the plate goes to zero. More precisely, it is shown that critical points of the nonlinear elastic functional $\mathcal E^h$, whose energies (per unit thickness) are bounded by $Ch^4$, converge to critical points of the $\Gamma$-limit of $h^{-4}\mathcal E^h$. This is proved under the physical assumption that the energy density $W(F)$ blows up as $\det F\to0$.