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

Kateryna Buryachenko, Annegret Glitzky, Matthias Liero, Barbara Zwicknagl

TL;DR

The paper develops a rigorous dimension-reduction framework for a finite-deformation electro-elastic plate with prestrain by formulating the problem as a saddle-point energy and applying a tailored Γ-convergence approach. It introduces a two-dimensional bending model coupled to electrostatic effects, derived as the ε→0 limit of the three-dimensional theory, with precise limiting energies and an effective tensor that captures the coupling. The main contribution is a general convergence theory for saddle-point problems (epi/hypo-convergence) and its application to show that saddle points of the 3D problem converge to saddle points of the 2D limit, including detailed bounds and recovery sequences. The results provide a mathematically rigorous basis for reduced electro-elastic plate models, enabling more efficient analysis and simulation of electromechanically coupled thin structures under large deformations.

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 $Γ$-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.

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

TL;DR

The paper develops a rigorous dimension-reduction framework for a finite-deformation electro-elastic plate with prestrain by formulating the problem as a saddle-point energy and applying a tailored Γ-convergence approach. It introduces a two-dimensional bending model coupled to electrostatic effects, derived as the ε→0 limit of the three-dimensional theory, with precise limiting energies and an effective tensor that captures the coupling. The main contribution is a general convergence theory for saddle-point problems (epi/hypo-convergence) and its application to show that saddle points of the 3D problem converge to saddle points of the 2D limit, including detailed bounds and recovery sequences. The results provide a mathematically rigorous basis for reduced electro-elastic plate models, enabling more efficient analysis and simulation of electromechanically coupled thin structures under large deformations.

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 -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.

Paper Structure

This paper contains 17 sections, 16 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Setup for the coupled electro-elastic model
  3. Notation
  4. Three-dimensional energy functional
  5. Scaling assumptions and non-dimensionalization
  6. The two-dimensional limit model
  7. Assumptions for the bending model
  8. Preliminary analytical results
  9. Properties for fixed thickness
  10. Boundedness and converging subsequences
  11. Main result: dimension reduction for the electro-elastic problem
  12. Abstract convergence result for saddle point problems
  13. Dimension reduction
  14. Proof of asymptotic upper and lower bounds
  15. Some convergence results for subproblems
  16. ...and 2 more sections

Key Result

Lemma 3.1

We assume assu:W -- eq:AssuElectric2. Let $\varepsilon>0$ be fixed and consider a deformation $y\in \mathcal{Y}_1$ with finite mechanical energy, i.e., $\mathcal{M}_\varepsilon(y)< \infty$. Then, there is a unique weak solution $\varphi=\varphi(y,\varepsilon)\in \mathcal{V}_1$ to Moreover, this $\varphi$ is the unique minimizer of the electrostatic energy $\mathcal{E}_\varepsilon(y,\cdot):\mathca

Theorems & Definitions (35)

  • Definition 2.1
  • Remark 2.2: Isotropic case
  • Remark 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 25 more