Enhanced kinematics in oriented and Cosserat solids with structural inhomogeneity

Abstract

In present paper the generalized kinematics for oriented media is discussed. The approach suggested is based on the geometrical formalism, developed in the theory of connection on principle bundles. This makes it possible to take into account a range of orientations, associated with an elementary volume of underlying body, while classical Cosserat type theories involve the only one. The need for such a generalization arises, for example, in the modeling of composites, reinforced with nanosized fibers. In such a case the transformation of elementary volume to the state, free from stresses, generally is not possible. The reason for that is the fact that with any deformation one can release local stresses, associated only with a part of fibers, oriented in some specific directions. To relax fibers, oriented otherwise, one has to apply another deformation. Thus, for complete description of all relaxation transformation in some elementary volume, the continuous family of generalized deformations is required. The notion of fiber, used in differential geometry of fiber bundles, provides an excellent tool for such formalization. In present paper a way for implementation discussed ideas is shown. It is also shown that classical Cosserat approach can be obtained within generalized one as some section of a bundle.