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An improved existence theorem for rigid nonlinearly elastic plates

Trung Hieu Giang, Cristinel Mardare

TL;DR

The paper proves that the Kirchhoff-Love nonlinear plate model has a minimizer whenever the plate is rigid, decoupling existence from boundary data and extending Rabier's clamped-case result. It introduces two rigidity frameworks: one for convex domains with boundary transverse displacement vanishing, and a second for tangential boundary data on parts of the boundary, including rectangle-like domains and partially clamped configurations. By providing explicit sufficient conditions for rigidity, it broadens Rabier's result to partially clamped plates and general domains, yielding existence of minimizers for the plate energy $J$.

Abstract

A plate is rigid if its admissible displacement fields inducing vanishing two-dimensional strain tensors must vanish. We prove that the nonlinear model of Kirchhoff-Love for such a plate has a solution for any applied forces and boundary conditions. Then we give sufficient conditions on the data ensuring the rigidity of the plate. Together, these results substantially generalize an existence theorem by Rabier whereby the plate is assumed to be clamped on its entire boundary.

An improved existence theorem for rigid nonlinearly elastic plates

TL;DR

The paper proves that the Kirchhoff-Love nonlinear plate model has a minimizer whenever the plate is rigid, decoupling existence from boundary data and extending Rabier's clamped-case result. It introduces two rigidity frameworks: one for convex domains with boundary transverse displacement vanishing, and a second for tangential boundary data on parts of the boundary, including rectangle-like domains and partially clamped configurations. By providing explicit sufficient conditions for rigidity, it broadens Rabier's result to partially clamped plates and general domains, yielding existence of minimizers for the plate energy .

Abstract

A plate is rigid if its admissible displacement fields inducing vanishing two-dimensional strain tensors must vanish. We prove that the nonlinear model of Kirchhoff-Love for such a plate has a solution for any applied forces and boundary conditions. Then we give sufficient conditions on the data ensuring the rigidity of the plate. Together, these results substantially generalize an existence theorem by Rabier whereby the plate is assumed to be clamped on its entire boundary.

Paper Structure

This paper contains 5 sections, 16 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Existence theorem for rigid plates
  4. A first type of rigidity theorem
  5. A second type of rigidity theorems

Key Result

Theorem 1

Let $\omega\subset{\mathbb{R}}^2$ be a bounded and connected open set whose boundary is Lipschitz-continuous. Then the functional $J:H^1(\omega)\times H^1(\omega)\times H^2(\omega)\to{\mathbb{R}}$ defined by e1-e6 is sequentially weakly lower semicontinuous.

Theorems & Definitions (18)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Proposition 2
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Proposition 3
  • Theorem 3
  • Corollary 1
  • ...and 8 more