A Cosserat Model of Elastic Solids Reinforced by a Family of Curved and Twisted Fibers

A Cosserat theory for fiber-reinforced elastic solids developed in Steigmann (2012) is generalized to accommodate initial curvature and twist of the fibers. The basic variables of the theory are a conventional deformation field and a rotation field that describes the local fiber orientation. Constraints on these fields are introduced to model the materiality of the fibers with respect to the underlying matrix deformation. A variational argument delivers the relevant equilibrium equations and boundary conditions and furnishes the interpretation of the Lagrange multipliers associated with the constraints as shear tractions acting on the fiber cross sections. Finally, the theory of material symmetry for such solids is developed and applied to the classification of some explicit constitutive functions.