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Homogenization of a thin linear elastic plate reinforced with a periodic mosaic of small rigid plates

Amartya Chakrabortty, Georges Griso, Julia Orlik

TL;DR

This work derives a rigorous two-dimensional limit model for a thin Kirchhoff-Love plate reinforced by a periodic mosaic of rigid inclusions, under simultaneous homogenization and dimension reduction with vanishing thickness and periodicity. By decomposing displacements to separate KL bending from rigid constraints and employing ${\cal Q}_1$-interpolations and rescaled unfolding operators, the authors obtain uniform Korn-type estimates and unfold the 3D problem to a pair of decoupled 2D plate problems. The homogenized limit is governed by a two-field variational inequality with an effective tensor $\mathcal{A}^{(\alpha)}$ computed from cell problems, yielding a strong, well-posed macroscopic model with explicit correctors and recovery sequences. The results provide a detailed, multiscale framework for designing and analyzing thin composite plates with rigid periodic inclusions, including NPC considerations and asymptotic energy convergence.

Abstract

In the framework of linearized elasticity, we study thin elastic composite plates with thickness $δ$. The plates contain small, rigid rectangular plates distributed periodically along $\varepsilon$. Between two neighboring rigid plates is an elastic beam with thickness $δ< \varepsilon/3 < 1$. Through a simultaneous process of homogenization and dimension reduction, we obtain the limit model. Our analysis yields Korn-type inequalities adapted to the rigid-elastic geometry of the structure and provides a precise characterization of the limit deformation and displacement fields. In the $2$D limit problem, the bending is the sum of two functions, each depending on only one variable. This is due to the fact that the mixed derivatives of the outer-plane displacement vanish. Finally, the limiting 2D problem is two decoupled plates or strips, each one with just three degrees of freedom: shear along the strip axis, the cross-contraction (-extension), and the cross-bending. The corresponding correctors are defined in the same way in the periodicity cell. In the linearized setting, all the correctors are decomposed.

Homogenization of a thin linear elastic plate reinforced with a periodic mosaic of small rigid plates

TL;DR

This work derives a rigorous two-dimensional limit model for a thin Kirchhoff-Love plate reinforced by a periodic mosaic of rigid inclusions, under simultaneous homogenization and dimension reduction with vanishing thickness and periodicity. By decomposing displacements to separate KL bending from rigid constraints and employing -interpolations and rescaled unfolding operators, the authors obtain uniform Korn-type estimates and unfold the 3D problem to a pair of decoupled 2D plate problems. The homogenized limit is governed by a two-field variational inequality with an effective tensor computed from cell problems, yielding a strong, well-posed macroscopic model with explicit correctors and recovery sequences. The results provide a detailed, multiscale framework for designing and analyzing thin composite plates with rigid periodic inclusions, including NPC considerations and asymptotic energy convergence.

Abstract

In the framework of linearized elasticity, we study thin elastic composite plates with thickness . The plates contain small, rigid rectangular plates distributed periodically along . Between two neighboring rigid plates is an elastic beam with thickness . Through a simultaneous process of homogenization and dimension reduction, we obtain the limit model. Our analysis yields Korn-type inequalities adapted to the rigid-elastic geometry of the structure and provides a precise characterization of the limit deformation and displacement fields. In the D limit problem, the bending is the sum of two functions, each depending on only one variable. This is due to the fact that the mixed derivatives of the outer-plane displacement vanish. Finally, the limiting 2D problem is two decoupled plates or strips, each one with just three degrees of freedom: shear along the strip axis, the cross-contraction (-extension), and the cross-bending. The corresponding correctors are defined in the same way in the periodicity cell. In the linearized setting, all the correctors are decomposed.

Paper Structure

This paper contains 24 sections, 21 theorems, 246 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Domain description
  4. Set of admissible displacements
  5. Linearized elasticity problem
  6. Decomposition of the displacements of the structure
  7. Global estimates
  8. The tools: Global fields using ${\cal Q}_1$-interpolations
  9. The fields ${\cal U}^{\diamond}$ and ${\cal R}^{\diamond}$
  10. The fields ${\cal U}$ and ${\cal R}$
  11. The functions ${\cal U}^\blacktriangle_{\alpha}$ and ${\cal U}^{\blacktriangle\blacktriangle}_{\alpha}$
  12. The applied body forces
  13. Unfolding operators and their properties
  14. Asymptotic limit of the strain tensor
  15. Limit behavior of the macroscopic fields
  16. ...and 9 more sections

Key Result

Theorem 3.1

Any displacement $u$ in ${\cal H}^1({\Omega}_{{\delta}})^3$ is decomposed as follows: where $\frak{U}_{\alpha} \in H^1({\omega}_{{\delta}})$, $\frak{U}_3 \in H^2({\omega}_{{\delta}})$, $\frak{u}\in H^1({\omega}_{\delta})$ and $\widetilde{u}\in H^1({\Omega}_{\delta})^3$ satisfies We set $\frak{U}_m=\frak{U}_1{\bf e}_1+\frak{U}_2{\bf e}_2$. The following estimates hold: The constants depend only

Figures (4)

  • Figure 1: Blue (hard) tiles placed periodically, and red (soft) rod grid.
  • Figure 2: Covering a part of $\omega_\delta$.
  • Figure 3: The different open sets: ${\cal O}^H_{\varepsilon\delta}$ (blue), ${\cal O}_\varepsilon$, ${\cal O}_{\varepsilon\delta}$ (red+blue) and ${\cal O}^E_{\varepsilon\delta}$ (empty+red+blue).
  • Figure 4: $\mathcal{O}_{{\varepsilon\delta}}$ is covered with squares; below are four strategies for defining $\mathcal{V}_3$.

Theorems & Definitions (42)

  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • proof : Proof of the estimate \ref{['EQ46']}$_4$
  • Lemma 4.2
  • proof
  • ...and 32 more