  We present a detailed study of rectilinear shear deformation in the framework
of orthotropic nonlinear elasticity, under Dirichlet and mixed-boundary
conditions. We take a slab made of a soft matrix, reinforced with two families
of extensible fibers. We consider the case where the shear occurs along the
bissectrix of the angle between the two privileged directions aligned with the
fibers. We show that if the two families of parallel fibers are mechanically
equivalent, then only smooth solutions are possible, whereas if the mechanical
differences among the two families of fibers is pronounced, then strain
singularities may develop. We determine the precise conditions for the
existence of such singular solutions for the standard reinforcing orthotropic
model. We then extend our findings to some orthotropic models of interest in
biomechanical applications, and we discuss the possible relevance of the
singular solutions to biomechanics.
