Material homogeneity and strain compatibility in thin elastic shells

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.