Inhomogeneous shear of orthotropic incompressible non-linearly elastic solids: Singular solutions and biomechanical interpretation

The paper presents a detailed study of rectilinear shear deformation in the framework of orthotropic nonlinear elasticity, under Dirichlet and mixed boundary conditions. Here the shear takes place in a slab made of a soft matrix reinforced with two families of extensible fibers, along the bisectrix of the angle between the two privileged directions aligned with the fibers. An analytic approach coupled to a careful computational treatment reveal that if the two families of parallel fibers are mechanically equivalent, then only smooth solutions are possible, whereas if the mechanical differences between the two families of fibers are pronounced, then strain singularities may develop. For the standard reinforcing orthotropic model, it is possible to determine the precise conditions for the existence of singular solutions. For an orthotropic constitutive law used for artery modeling, the existence of singular solutions can have repercussions in biomechanical applications.