Convexity and Quasiconvexity in a Cosserat Model for Fiber-Reinforced Elastic Solids

An inequality of the quasiconvexity type is derived in the context of a Cosserat model for fiber-reinforced elastic solids in which the Cosserat rotation field accounts for the kinematics of the embedded fibers, modeled as extensible, unshearable spatial rods. This is shown to be equivalent to the requirement that the fiber bend-twist strain vector associated with an energy minimizer be a point of convexity of the operative strain-energy function. The analysis requires careful consideration of constraints associated with the fact that fibers are convected by the continuum deformation field as material curves.