Shape sensitivity analysis of the eigenvalues of the Reissner-Mindlin system

TL;DR

This paper analyzes how the eigenvalues of the Reissner-Mindlin system change with shape variations of an elastic plate, providing estimates, formulas, and optimization insights for vibration modes.

Contribution

It offers the first quantitative estimates, analyticity results, and Hadamard-type formulas for shape sensitivity of Reissner-Mindlin eigenvalues, including shape optimization insights.

Findings

Balls are critical points for eigenvalues under volume constraints.

Quantitative estimates for eigenvalue variation with shape changes.

Analyticity and Hadamard formulas derived for eigenvalues.

Abstract

We consider the eigenvalue problem for the Reissner-Mindlin system arising in the study of the free vibration modes of an elastic clamped plate. We provide quantitative estimates for the variation of the eigenvalues upon variation of the shape of the plate. We also prove analyticity results and establish Hadamard-type formulas. Finally, we address the problem of minimization of the eigenvalues in the case of isovolumetric domain perturbations. In the spirit of the Rayleigh conjecture for the biharmonic operator, we prove that balls are critical points with volume constraint for all simple eigenvalues and the elementary symmetric functions of multiple eigenvalues.