A unified procedure for construction of theories of deformable media. II. Generalized continua

In contrast to classical theories discussed in the accompanying paper (part I), the present paper deals exclusively with generalized continua, a term that refers to a body embedded in a euclidean three-dimensional space with each of its material points endowed with additional kinematic structure. Such generalized continua are conveniently identified here by Cosserat (or directed) continua in which the additional structure is represented by independent director fields and the material domain of the body manifold is classified according to the four categories: (A) a three-dimensional volume, (B) a two-dimensional surface, (C) a one-dimensional space curve and (D) a point. In the derivation of a thermomechanical theory for each of these categories, we use the same unified procedure as in part I but now the various energies that enter the balance of energy must be modified to include additional kinetic ingredients. New theories of this kind have increasingly provided, over the last three decades, effective means of formulating and studying new behaviour of materials (in both fluid and solid mechanics) that were not previously possible by the classical theories. Our attention is first focused in the development of the theory for the first category and this is discussed in two stages: first for a Cosserat continuum with a single director and then for N directors. Our derived results include for the first time a derivation from the balance of energy of balances of director momenta and balances of entropies (in the presence of more than one independent temperature field). The treatments of categories (B) to (D) are somewhat shorter since their developments are formally similar to that of category (A).