An Eulerian formulation of a constrained variable thickness growing Cosserat shell

Abstract

Constitutive equations for Cosserat shell theory are usually developed using a Lagrangian formulation based on a zero-stress reference configuration. However, for growing shells a zero-stress state, if it exists, can change. The main objective of this paper is to propose an Eulerian formulation of constitutive equations for a growing shell. The director normal to the shell’s mid-surface is constrained to remain normal to that surface, but is allowed to have a variable length to model non-uniform thickness changes of the shell. Elastic measures of dilatation, distortional deformation, mean and Gaussian curvatures are determined by evolution equations that are independent of a reference or intermediate configuration. These evolution equations model homeostasis, which is the process of growth causing a tendency for these variables to approach their homeostatic values. Examples of a growing spherical shell are used to examine aspects of the proposed theory.