Tensile instabilities and ellipticity in fiber-reinforced compressible non-linearly elastic solids

In this paper we examine loss of ellipticity and associated failure for fiber-reinforced compressible non-linearly elastic solids under uniaxial plane deformations. We consider first fiber reinforcement that endows the material with additional stiffness only in the fiber direction. It is shown, in particular, that loss of ellipticity under tensile loading in the fiber direction corresponds to a turning point of the nominal stress and may require concavity of the Cauchy stress-stretch curve. Secondly we examine fiber reinforcement that introduces additional stiffness under shear deformations. In this case we find that loss of ellipticity again occurs at a turning point of the nominal stress, in contrast to the situation for incompressible materials.