Cosserat Continuum

Abstract

This chapter develops the general Cosserat continuum theories, proposed by Cosserat brothers in 1909, and addresses their implementations in motion formalism. After derived the general three-dimensional Cosserat continuum theory, the general shell-like theory and beam-like theory, the typical two-dimensional and one-dimensional Cosserat continua, are readily obtained through the reduction of geometric dimension. When all the deformations of Cosserat continuum are ignored, the general Cosserat continuum theory is simplified to the theory of rigid body. Furthermore, the general Cauchy continuum theory can also be obtained from Cosserat theory by neglecting the motion of material particles directors or neglecting the curvature strains, equivalently. The membrane theory and cable theory are obtained in the same manner by ignoring the curvature strains or the fiber directors in the theories of shell and beam, respectively. In this sense, the general Cosserat continuum theory proposed in this chapter affords the unified descriptions for three-dimensional Cosserat continuum, shell, beam, rigid body, three-dimensional Cauchy continuum, membrane and cable.