Kinematics of Classical and Cosserat Continua

Abstract

The aim of this chapter is the presentation of the kinematics of classical (Boltzmann ) and polar (Cosserat ) continuous mixtures. The motions of material points of constituent \(\alpha \) are first mathematically introduced for a classical mixture as mappings from separate constituent points onto a single point in the present configuration, Fig. 22.3 . This guarantees that material points in physical space are a merger of all constituents. This motion function then yields through spatial and temporal differentiations the well-known definitions of the classical deformation measures: deformation gradient, right and left Cauchy–Green deformation tensors, Euler –Lagrange strains, and associated strain rates. Of importance is the polar decomposition, which splits the deformation gradient into a sequence of pure strain and rotation or vice versa. Whereas the classical stretch and stretching measures are obtained by inner products of the constituent vectorial line element with itself, deformation measures of Cosserat kinematics are generated by inner products between vectorial material line increments and the directors. The mappings of the latter between the reference and present configurations are postulated to be pure rotations (Fig. 22.5 ). This then yields the various Cosserat strain measures, which are analogous to, but not the same as those of the classical theory. The kinematically independent rotation of the directors gives rise to the introduction of skew-symmetric rank-3 and full rank-2 curvature tensors, quasi as measures of the spatial variation of the micro rotation. Analogous to the additive decomposition of the velocity gradient into stretching and vorticity tensors in the classical formulation, two additional decompositions of the velocity gradient are introduced using the polar decomposition and leads to non-symmetric strain rate and the so-called gyration tensors, and objective time derivatives of the Cosserat version of the Almansi tensor and the curvature tensors. All these quantities are also written relative to the natural basis system. The chapter ends with the presentation of the balance law of micro-inertia. It is based on the assumption that material points of micro-polar continua move like rigid bodies.