Superposing plane strain on anti-plane shear deformations in a special class of fiber-reinforced incompressible hyperelastic materials

The purpose of the present research is to investigate how the nonlinearity and the boundary conditions have the power to influence the coupling of the various modes of deformation when a plane strain deformation is superimposed on anti-plane shear deformation for a class of fiber-reinforced incompressible hyperelastic materials described by a strain energy density W. Attention is confined when W depends only on the first invariant of the strain tensor and on the square of the stretch in the direction of the fibers. We are able to write down the governing equations of equilibrium as a coupled system of three nonlinear partial differential equations for three displacement fields. Two displacements are the in-plane components and one displacement is the anti-plane state. The system that we are able to deduce in a compact form is always compatible at variance with the case in which the anti-plane shear problem is analyzed. As explicit example of our findings we study the problem of the helical shear and we investigate into details the coupling of its axial and azimuthal components.