Cosserat-Type Shells

In this chapter we discuss the Cosserat-type theories of plates and shells. We call Cosserat-type shell theories various theories of shells based on the consideration of a shell base surface as a deformable directed surface, that is the surface with attached deformable or non-deformable (rigid) vectors (directors), or based on the derivation of two-dimensional (2D) shell equations from the three-dimensional (3D) micropolar (Cosserat) continuum equations. Originally the first approach of such a kind belongs to Cosserat brothers who considered a shell as a deformable surface with attached three unit orthogonal directors. In the literature are known theories of shells which kinematics is described by introduction of the translation vector and additionally p deformable directors or one deformable director or three unit orthogonal to each other directors. Additional vector fields of directors describe the rotational (in some special cases additional) degrees of freedom of the shell. The most popular theories use the one deformable director or three unit directors. In both cases the so-called direct approach is applied. Another approach is based on the 3D-to-2D reduction procedure applied to the 3D motion or equilibrium equations of the micropolar shell-like body. In the literature the various reduction methods are known using for example asymptotic methods, the-through-the thickness integration procedure, expansion in series, etc. The aim of the chapter is to present the various Cosserat-type theories of plates of shells and discuss the peculiarities and differences between these approaches.