The coupled Legendre-Hadamard condition for fiber-reinforced materials: three-dimensional solids and two-dimensional shells

Abstract Detailed derivations of the Legendre-Hadamard necessary conditions for energy-minimizing states of fiber-reinforced three-dimensional solids and two-dimensional shells are presented. The underlying conceptual framework is Cosserat elasticity theory in which the Cosserat rotation field controls the orientation of the embedded fibers. This is partially coupled to the continuum deformation gradient by the requirement that the fibers convect as material curves with respect to the matrix material in which they are embedded. The conditions obtained combine the effects of deformation and rotation and subsume previously obtained decoupled inequalities involving these effects separately.