Remarks on instabilities and ellipticity for a fiber-reinforced compressible nonlinearly elastic solid under plane deformation

In this paper we examine loss of ellipticity and associated failure for fiber-reinforced compressible nonlinearly elastic solids under plane deformation. The analysis concerns a material model that consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. We examine reinforcement that introduces additional stiffness under shear deformations. It is shown that loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. More particularly, under uniaxial loading in the fiber direction loss of ellipticity may be associated with different failure mechanisms. Under compression these include fiber kinking and fiber splitting, while under extension the relevant mechanism is matrix failure.