On the F\"oppl-von K\'arm\'an theory for elastic prestrained films with varying thickness

TL;DR

This paper extends the F"oppl-von K"arm"an theory to prestrained thin films with variable thickness, deriving the limiting variational model and analyzing equilibrium convergence.

Contribution

It introduces a $ ext{Gamma}$-convergence approach for variable thickness films, deriving new limiting equations and linking deformations with growth tensors.

Findings

Derived the variational limit for prestrained films with non-constant thickness.

Calculated Euler-Lagrange equations for the limiting energy.

Analyzed convergence of equilibrium states.

Abstract

We derive the variational limiting theory of thin films, parallel to the F\"oppl-von K\'arm\'an theory in the nonlinear elasticity, for films that have been prestrained and whose thickness is a general non-constant function. Using $\Gamma$-convergence, we extend the existing results to the variable thickness setting, calculate the associated Euler-Lagrange equations of the limiting energy, and analyze the convergence of equilibria. The resulting formulas display the interrelation between deformations of the geometric mid-surface and components of the growth tensor.