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Homogenization, dimension reduction and linearization of thin elastic plate

Amartya Chakrabortty, Georges Griso, Julia Orlik

TL;DR

The paper analyzes a periodic thin composite plate under nonlinear elasticity by performing simultaneous homogenization, dimension reduction, and linearization (SHDL) as $(\varepsilon,h)\to(0,0)$, with the total energy scaling as $h^2\varepsilon^{2a+3}$ and emphasis on the von-Kármán regime $a=1$. It develops a Γ-convergence framework combined with a rescaled unfolding approach to derive a limit energy ${\bf J}$ that depends on a Kirchhoff–Love type displacement ${\mathcal U}$ and a two-scale corrector ${\widehat{\frak u}}$, and proves the existence of a unique minimizer in the limit space ${\mathbb D}$. The analysis shows that linearization commutes with SHD/SHDL in the periodic setting: the limit energy obtained by linearizing first and then SHD matches the limit from SHDL or from simultaneous vanishing limits, and this limit is characterized by a homogenized quadratic form with cell problems determining the coefficients $a^{hom}, b^{hom}, c^{hom}$. The results extend to perforated plates via an extension theorem, confirming well-posed, unique minimizers for the limit linearized homogenized problem and providing a rigorous bridge between nonlinear heterogeneous plate theories and their linearized, homogenized counterparts.

Abstract

This paper investigates the homogenization, dimension reduction, and linearization of a composite plate subjected to external loading within the framework of non-linear elasticity problem. The total elastic energy of the problem is of order $\sim h^2\varepsilon^{2a+3}$, where $a\geq1$. The paper is divided into two parts: The first part presents the simultaneous homogenization, dimension reduction and linearization ($(\varepsilon,h)\to(0,0)$) of a composite plate without any coupling assumption of $\varepsilon$ and $h$. The second part consists of the rigorous derivation of linearized elasticity as a limit of non-linear elasticity with small deformation and external loading conditions. The results obtained demonstrate that the limit energy remains unchanged when the first linearization ($h\to 0$) is performed, followed by simultaneous homogenization dimension reduction ($\varepsilon\to0$) and when both limits approach zero simultaneously, i.e. $(\varepsilon,h)\to (0,0)$. The exact form of the limit energy(s) is obtained through the decomposition of plate deformations and plate displacements. By using the $Γ$-convergence technique, the existence of a unique solution for the limit linearized homogenized energy problem is demonstrated. These results are then extended to certain periodic perforated plates.

Homogenization, dimension reduction and linearization of thin elastic plate

TL;DR

The paper analyzes a periodic thin composite plate under nonlinear elasticity by performing simultaneous homogenization, dimension reduction, and linearization (SHDL) as , with the total energy scaling as and emphasis on the von-Kármán regime . It develops a Γ-convergence framework combined with a rescaled unfolding approach to derive a limit energy that depends on a Kirchhoff–Love type displacement and a two-scale corrector , and proves the existence of a unique minimizer in the limit space . The analysis shows that linearization commutes with SHD/SHDL in the periodic setting: the limit energy obtained by linearizing first and then SHD matches the limit from SHDL or from simultaneous vanishing limits, and this limit is characterized by a homogenized quadratic form with cell problems determining the coefficients . The results extend to perforated plates via an extension theorem, confirming well-posed, unique minimizers for the limit linearized homogenized problem and providing a rigorous bridge between nonlinear heterogeneous plate theories and their linearized, homogenized counterparts.

Abstract

This paper investigates the homogenization, dimension reduction, and linearization of a composite plate subjected to external loading within the framework of non-linear elasticity problem. The total elastic energy of the problem is of order , where . The paper is divided into two parts: The first part presents the simultaneous homogenization, dimension reduction and linearization () of a composite plate without any coupling assumption of and . The second part consists of the rigorous derivation of linearized elasticity as a limit of non-linear elasticity with small deformation and external loading conditions. The results obtained demonstrate that the limit energy remains unchanged when the first linearization () is performed, followed by simultaneous homogenization dimension reduction () and when both limits approach zero simultaneously, i.e. . The exact form of the limit energy(s) is obtained through the decomposition of plate deformations and plate displacements. By using the -convergence technique, the existence of a unique solution for the limit linearized homogenized energy problem is demonstrated. These results are then extended to certain periodic perforated plates.
Paper Structure (15 sections, 18 theorems, 175 equations, 1 figure)

This paper contains 15 sections, 18 theorems, 175 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations
  3. Domain description and the non-linear elasticity problem
  4. Description of the structure and natural assumption
  5. The non-linear elasticity problem
  6. Preliminary results
  7. Decomposition of plate deformations
  8. Decomposition of plate displacements
  9. The unfolding operators
  10. Assumption of the right-hand side forces
  11. Simultaneous linearization, homogenization and dimension reduction
  12. Asymptotic behavior of a sequence of re-scaled displacements as $(\varepsilon,h)\to (0,0)$
  13. The cell problems
  14. Linearization $h\to 0$ ($\varepsilon$ being fixed)
  15. Extension to periodic perforated plates

Key Result

Lemma 1

There exists a constant $C(\omega)$ which only depends on $\omega$ such that any $v\in H^1(\Omega_\varepsilon)^3$ be a plate deformation then The quantity $V_{e} \in H^1{(\Omega_\varepsilon)}^3$ is called elementary plate deformation and is defined by with ${\mathcal{V}}\in H^1(\omega)^3$ and ${\bf R}\in H^1(\omega)^{3\times 3}$. $\overline{v} \in H^1{(\Omega_\varepsilon)}^3$ is a residual displ

Figures (1)

  • Figure 1: Relation between Linearization, SHD and SHDL for $a\geq 1$.

Theorems & Definitions (31)

  • Remark 1
  • Remark 2
  • Lemma 1: Theorem 3.3, Shell1
  • Lemma 2
  • proof
  • Lemma 3: Theorem 4.2 and Theorem 4.3 Shell1
  • Lemma 4: Theorem 6.1 in GKL
  • Remark 3
  • Lemma 5: Korn type inequalities
  • Definition 1
  • ...and 21 more