A general theory of finite deformation of viscoplastic thin shells

In this paper we develop a general theory of thin inelastic shells for finite deformations. In a first step we present a general three-dimensional theory of inelastic material behavior. Any inelastic constitutive model with internal state variables can be used which is formulated without the explicit notion of a yield surface. In a next step we investigate the simplifications of our constitutive theory resulting from the assumption of small elastic strains which is well established for metallic materials. Furthermore, we introduce the assumption of persistent isotropy. Finally, we present the weak form of the balance of equilibrium and indicate its linearization. To formulate our shell theory we first present the geometrical description of thin shells in the reference and current configuration, respectively. Next we introduce strain measures for the shell where we include transverse shear strains and also thickness changes of the shell. The shell theory is formulated by application of the projection method, i.e. we integrate all relevant equations of the three-dimensional theory over the shell thickness. This projection leads to a strictly two-dimensional shell theory which is formulated entirely on the shell midsurface. Finally, we indicate the numerical implementation of our shell theory.