Homogenization, dimension reduction and linearization of thin elastic plate

TL;DR

This paper rigorously derives the effective behavior of thin composite plates under external loads through homogenization, dimension reduction, and linearization, revealing that the order of elastic energy remains consistent across limits.

Contribution

It provides a comprehensive mathematical framework for simultaneous homogenization, dimension reduction, and linearization of nonlinear elasticity in thin plates without coupling assumptions.

Findings

Limit energy remains unchanged under different limit procedures.

Existence of a unique solution for the limit linearized homogenized problem.

Extension of results to periodic perforated plates.

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 $\Gamma$-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.