Material homogeneity and strain compatibility in thin elastic shells

TL;DR

This paper explores the relationship between material inhomogeneity and strain incompatibility in thin elastic shells using 3D and 2D Cosserat theories, deriving explicit measures and governing equations for residual stresses.

Contribution

It introduces explicit forms of dislocation density tensors and formulates governing equations linking inhomogeneity and residual stresses in thin shells.

Findings

Derived explicit dislocation density tensors for inhomogeneous shells

Formulated governing equations for residual stress fields

Simplified equations under Kirchhoff-Love assumptions

Abstract

We discuss several issues regarding material homogeneity and strain compatibility for materially uniform thin elastic shells from the viewpoint of a 3-dimensional theory, with small thickness, as well as a 2-dimensional Cosserat theory. A relationship between inhomogeneity and incompatibility measures under the two descriptions is developed. More specifically, we obtain explicit forms of intrinsic dislocation density tensors characterising inhomogeneity of a dislocated Cosserat shell. We also formulate a system of governing equations for the residual stress field emerging out of strain incompatibilities which in turn are related to inhomogeneities. The equations are simplified for several cases under the Kirchhoff-Love assumption.