Asymptotic analysis for a second order curved thin film

TL;DR

This paper performs an asymptotic analysis of a second order curved thin film, deriving a reduced 2D model from a 3D energy functional involving nonlinear and second order derivatives as the film thickness approaches zero.

Contribution

It introduces a rigorous asymptotic expansion approach for second order curved thin films, extending classical models to include second derivatives of deformation.

Findings

Derived a 2D limit model from 3D energy functional

Extended thin film theories to include second derivatives

Provided mathematical justification for asymptotic expansion

Abstract

We consider a second order thin curved film whose behavior is governed by an energy made up of a first order nonlinear part depending on the gradient of the deformation augmented by a quadratic second order part depending on the tensor of second derivatives of the deformation. We carry out a 3D-2D analysis through an asymptotic expansion in powers of the thickness of the film as it tends to zero.