A geometrically nonlinear Cosserat (micropolar) curvy shell model via Gamma convergence

TL;DR

This paper derives a nonlinear Cosserat shell model using Gamma convergence, capturing transverse shear and Cosserat-curvature effects, and compares it with existing models.

Contribution

It introduces a new membrane-like Cosserat shell model derived via Gamma convergence, highlighting differences from classical models and including Cosserat effects.

Findings

The model captures transverse shear deformation.

It accounts for Cosserat-curvature effects.

Comparison with other shell models is provided.

Abstract

Using $\Gamma$-convergence arguments, we construct a nonlinear membrane-like Cosserat shell model on a curvy reference configuration starting from a geometrically nonlinear, physically linear three-dimensional isotropic Cosserat model. Even if the theory is of order $O(h)$ in the shell thickness $h$, by comparison to the membrane shell models proposed in classical nonlinear elasticity, beside the change of metric, the membrane-like Cosserat shell model is still capable to capture the transverse shear deformation and the {Cosserat}-curvature due to remaining Cosserat effects. We formulate the limit problem by scaling both unknowns, the deformation and the microrotation tensor, and by expressing the parental three-dimensional Cosserat energy with respect to a fictitious flat configuration. The model obtained via $\Gamma$-convergence is similar to the membrane {(no $O(h^3)$ flexural terms, but still depending on the Cosserat-curvature)} Cosserat shell model derived via a derivation approach but these two models do not coincide. Comparisons to other shell models are also included.