Nonlinear dynamic equations for elastic micromorphic solids and shells. Part I

The present paper develops a general approach to deriving nonlinear equations of motion for solids whose material points possess additional degrees of freedom. The essential characteristic of this approach is the account of incompatible deformations that may occur in the body due to distributed defects or in the result of the some kind of process like growth or remodelling. The mathematical formalism is based on least action principle and Noether symmetries. The peculiarity of such formalism is in formal description of reference shape of the body, which in the case of incompatible deformations has to be regarded either as a continual family of shapes or some shape embedded into non-Euclidean space. Although the general approach yields equations for Cosserat-type solids, micromorphic bodies and shells, the latter differ significantly in the formal description of enhanced geometric structures upon which the action integral has to be defined. Detailed discussion of this disparity is given.