A direct approach to the formulation of constitutive equations for rods and shells

Today many variants of the rod and shell theory exist. Most of them can be characterized by two facts: (a) they are based on one- or two-dimensional equations and (b) they operate with forces and moments (higher-order moments are mostly ignored). Restricting ourself, the objective of this presentation is to recall the governing equations of the direct theory of shells. Based on the energy balance law we demonstrate that arguments of the strain energy density (the strain measures) can be introduced as integrals of a system of first order partial differential equations. Furthermore, we discuss the symmetry groups of the elasticity tensors and the corresponding symmetry transformations. For several examples, including laminated anisotropic shells, we illustrate the general structure of the elasticity tensors.