Dimension reduction for a coupled electro-elastic saddle-point problem at finite strains

TL;DR

This paper rigorously derives a two-dimensional reduced model for a thin electro-elastic sheet undergoing finite deformations, capturing the coupling between mechanics and electrostatics through advanced mathematical techniques.

Contribution

It introduces a novel dimension reduction approach for a coupled electro-elastic saddle-point problem at finite strains using $ ext{Gamma}$-convergence methods.

Findings

Convergence of 3D electro-elastic problems to a 2D bending model.

Validation that saddle points converge to saddle points in the limit.

Effective coupling between bending and electric effects in the reduced model.

Abstract

We study the finite deformation of a thin, elastically heterogeneous sheet subject to electrostatic coupling. The interaction between mechanics and electrostatics is formulated as a saddle-point problem involving the deformation and the electrostatic potential. Starting from a three-dimensional electro-elastic model with prestrain in the elastic energy, we rigorously derive a reduced plate model in the bending regime. To perform the dimension reduction, that is, to derive the energy of a thin object by taking a suitable limit as its thickness tends to zero, we apply $\Gamma$-convergence-type methods to the underlying saddle-point problem. In the case of bivariate functionals, this convergence is understood in an adapted epi/hypo-convergence sense. In this concept, we demonstrate the convergence of the rescaled electro-elastic problems to an effective two-dimensional bending model coupled to electric effects. We verify that cluster points of saddle points are saddle points for the limit.