Material instabilities in fiber-reinforced nonlinearly elastic solids under plane deformation

Abstract

Material instabilities in fiber-reinforced nonlinearly elastic solids are examined under plane deformation. In particular, the materials under consideration are isotropic nonlinearly elastic models augmented by a function that accounts for the existence of a unidirectional reinforcing. This function describes the anisotropic (transversely isotropic) character of the material and is referred to as a reinforcing model. The onset of failure is signalled by the loss of ellipticity of the governing differential equations. Previous work has dealt with the analysis of specific reinforcing models and has established that the loss of ellipticity for such augmented isotropic materials requires contraction in the reinforcing direction. The loss of ellipticity was related to fiber kinking. Here we generalize these results and establish sufficient conditions for the ellipticity of the governing equations of equilibrium for more general reinforcing models to be guaranteed. We also establish necessary conditions for failure of ellipticity. The incipient loss of ellipticity is interpreted in terms of fiber kinking, fiber de-bonding, fiber splitting and matrix failure in fiber-reinforced composite materials. Attention is restricted to incompressible materials in this paper.