Two-phase piecewise homogeneous plane deformations of a fibre-reinforced neo-Hookean material with application to fibre kinking and splitting

Abstract

Two-phase piecewise homogeneous plane deformations are examined in respect of a neo-Hookean matrix material reinforced with embedded aligned fibres characterized by a single stiffness parameter. The deformations are interpreted in terms of fibre kinking and fibre splitting. Previous work has shown that such a transversely isotropic material can lose ellipticity if the reinforcing stiffness is sufficiently large and the fibre direction is sufficiently compressed. In particular, it was shown that the associated failure modes are characterised by the emergence of weak surfaces of discontinuity that are normal to the fibre direction (the onset of fibre kinking) or parallel to the fibre direction (the onset of fibre splitting). Here, the analysis of strong surfaces of discontinuity, developing from weak ones, is studied. The considered model can give rise to piecewise smooth plane deformations separated by a plane stationary surface of discontinuity, interpreted as either kinking or splitting. Attention is restricted to (plane) deformations in which, on one side of the surface of discontinuity, the load axis is aligned with the fibre axis. Then the fibre stretch on this side of the discontinuity is a natural load parameter. The ellipticity status of the two-phase piecewise homogeneous plane deformations is shown to span all four possible ellipticity/non-ellipticity permutations. If both deformation states are elliptic, then a suitable intermediate deformation is shown to be non-elliptic. Moreover, it is shown that the mechanism is dissipative, and maximally dissipative quasi-static failure motion is examined in respect of both kinking and splitting. It follows that, firstly, surfaces of discontinuity perpendicular to the fibre direction, associated with fibre kinking, are nucleated followed by surfaces of discontinuity parallel to the fibre direction, associated with fibre splitting. With respect to kinking, such maximally dissipative kinks nucleate only in compression as weak surfaces of discontinuity, with the subsequent motion converting non-elliptic deformation to elliptic deformation.