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Derivation of Kirchhoff-type plate theories for elastic materials with voids

Manuel Friedrich, Leonard Kreutz, Konstantinos Zemas

TL;DR

The work rigorously derives a Blake–Zisserman–Kirchhoff-type two-dimensional theory for thin elastic plates with unprescribed voids by Γ-convergence from a three-dimensional elastic model with Willmore-type curvature penalization in the bending regime. It develops a curvature-regularized variational framework that permits general void geometries and produces a limiting energy consisting of a bending term (scaled by 1/24 via 𝒬_2(II_y)), a surface-energy contribution from void boundaries, and a doubled-count for discontinuities or folds of the limiting mid-surface. The analysis hinges on a novel piecewise rigidity approach, Sobolev replacements on good cubes, and a blow-up/approximation scheme to connect to SBV-isometric immersions, yielding compactness and matching Γ-liminf/Γ-limsup results. This provides a rigorous bridge from 3D SDRI-type elasticity with voids to an effective 2D theory that accommodates fractures and folds, with precise energy contributions from curvature, void perimeters, and geometric discontinuities.

Abstract

We rigorously derive a Blake-Zisserman-Kirchhoff theory for thin plates with material voids, starting from a three-dimensional model with elastic bulk and interfacial energy featuring a Willmore-type curvature penalization. The effective two-dimensional model comprises a classical elastic bending energy and surface terms which reflect the possibility that voids can persist in the limit, that the limiting plate can be broken apart into several pieces, or that the plate can be folded. Building upon and extending the techniques used in the authors' recent work on the derivation of one-dimensional theories for thin brittle rods with voids, the present contribution generalizes the results of Santili and Schmidt (2022), by considering general geometries on the admissible set of voids and constructing recovery sequences for all admissible limiting configurations.

Derivation of Kirchhoff-type plate theories for elastic materials with voids

TL;DR

The work rigorously derives a Blake–Zisserman–Kirchhoff-type two-dimensional theory for thin elastic plates with unprescribed voids by Γ-convergence from a three-dimensional elastic model with Willmore-type curvature penalization in the bending regime. It develops a curvature-regularized variational framework that permits general void geometries and produces a limiting energy consisting of a bending term (scaled by 1/24 via 𝒬_2(II_y)), a surface-energy contribution from void boundaries, and a doubled-count for discontinuities or folds of the limiting mid-surface. The analysis hinges on a novel piecewise rigidity approach, Sobolev replacements on good cubes, and a blow-up/approximation scheme to connect to SBV-isometric immersions, yielding compactness and matching Γ-liminf/Γ-limsup results. This provides a rigorous bridge from 3D SDRI-type elasticity with voids to an effective 2D theory that accommodates fractures and folds, with precise energy contributions from curvature, void perimeters, and geometric discontinuities.

Abstract

We rigorously derive a Blake-Zisserman-Kirchhoff theory for thin plates with material voids, starting from a three-dimensional model with elastic bulk and interfacial energy featuring a Willmore-type curvature penalization. The effective two-dimensional model comprises a classical elastic bending energy and surface terms which reflect the possibility that voids can persist in the limit, that the limiting plate can be broken apart into several pieces, or that the plate can be folded. Building upon and extending the techniques used in the authors' recent work on the derivation of one-dimensional theories for thin brittle rods with voids, the present contribution generalizes the results of Santili and Schmidt (2022), by considering general geometries on the admissible set of voids and constructing recovery sequences for all admissible limiting configurations.

Paper Structure

This paper contains 15 sections, 14 theorems, 320 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notation
  4. The models and the main results
  5. The three-dimensional model
  6. Limiting model and main result
  7. Piecewise rigidity and Sobolev modification of deformations
  8. Rigidity results
  9. Local rigidity estimates and Sobolev replacement on good cubes
  10. Construction of almost Sobolev replacements and proof of Proposition \ref{['Lipschitz_replacement']}
  11. Proof of Theorem \ref{['thm:main_kirchhoff_cptness']}
  12. Proof of Theorem \ref{['thm:main_kirchhoff_gamma_conv']}(i)
  13. Proof of Theorem \ref{['thm:main_kirchhoff_gamma_conv']}(ii)
  14. Proofs of Proposition \ref{['summary_estimates_proposition']} and Corollary \ref{['difference_rigid-motions']}
  15. A linearization argument for the elastic energy liminf.

Key Result

Theorem 2.1

$\text{(\underline {Compactness})}$ Let $(h_j)_{j\in \mathbb{N}}\subset (0,\infty)$ with $h_j\searrow 0$ and $(y_{h_j}, V_{h_j}) \in \hat{\mathcal{A}}_{h_j}$ (cf. admissible_configurations_h_level) be such that Then, there exists $(y,V)\in \mathcal{A}$ (cf. limiting_admissible_pairs) such that, up to a non-relabeled subsequence, where $(\widetilde{y}, \widetilde{V})$ is associated to $(y,V)$ via

Figures (3)

  • Figure 1: Densely packed thin spikes and microscopically small components leading to loss of rigidity. For simplicity, the figure illustrates a two-dimensional example.
  • Figure 2: Bending a plate with voids and cracks: Collapsing voids lead to discontinuity curves for $J_y$. Folds corresponding to the presence of $J_{\nabla'y}$ are not depicted for simplicity.
  • Figure 3: An illustration of the dyadic construction giving the covering $\mathcal{W}_h$. The jump set $\Gamma_h$ is depicted by the black segments.

Theorems & Definitions (27)

  • Theorem 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4: Possible extensions and variants
  • Proposition 3.1: Sobolev modification of deformations and their gradients
  • Proposition 3.2: Global thickening of sets
  • Theorem 3.3: Geometric rigidity in variable domains
  • Theorem 3.4: Korn-Poincaré inequality for functions with small jump set
  • Remark 3.5
  • Corollary 3.6
  • ...and 17 more