Finite Element Method for Cosserat Plates

TL;DR

This paper develops a finite element method tailored for Cosserat elastic plates, analyzing their mathematical properties and validating the approach through numerical experiments including stress concentration around holes.

Contribution

It introduces a FEM for Cosserat plates based on an optimal splitting parameter, with proofs of existence, uniqueness, and convergence, plus numerical validation and analysis of stress concentrations.

Findings

FEM shows optimal convergence order compared to analytical solutions.

Stress concentration factors are lower around holes than classical predictions.

Smaller holes result in less stress concentration, as expected.

Abstract

This paper presents the Finite Element Method for Cosserat plates. The mathematical model for Cosserat elastic plates is based on the calculation of the optimal value of the splitting parameter. We discuss the existence and uniqueness of the weak solution and the convergence of the proposed FEM. The Finite Element analysis of the clamped Cosserat plates of different shapes under different loads is provided. We present the numerical validation of the proposed FEM by estimating the order of convergence, when comparing the main kinematic variables with the analytical solution. We also consider the numerical analysis of plates with circular holes. We show that as expected the stress concentration factor around the hole is smaller than the classical value and smaller holes exhibit less stress concentration compared to larger ones.