Refined dimensional reduction for isotropic elastic Cosserat shells with initial curvature

TL;DR

This paper develops a refined two-dimensional shell model for isotropic elastic Cosserat shells with initial curvature, incorporating curvature effects and providing explicit energy density expressions up to order h^5.

Contribution

It introduces a geometrically motivated 8-parameter ansatz for Cosserat shells, enabling analytical integration through the thickness and explicit energy density formulation.

Findings

Explicit form of elastic strain energy density up to order h^5

Closed-form expressions for thickness stretch and shift in bending

Inclusion of initial curvature effects in the shell model

Abstract

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytically and a generalized plane stress condition allows for a closed-form expression of the thickness stretch and the nonsymmetric shift of the midsurface in bending. We obtain an explicit form of the elastic strain energy density for Cosserat shells, including terms up to order $O(h^5)$ in the shell thickness $h$. This energy density is expressed as a quadratic function of the nonlinear elastic shell strain tensor and the bending-curvature tensor, with coefficients depending on the initial curvature of the shell.