Numerical analysis of neck and bulge propagation in anisotropic tubes subject to axial loading and internal pressure

Bulging/necking motion in doubly fiber-reinforced incompressible nonlinearly elastic tubes subject to axial loading and internal pressure is examined using a numerical procedure based on the modified Riks method. In particular, the materials under consideration are neo-Hookean models augmented with two functions, each one of them accounting for the existence of a unidirectional reinforcement. The functions endow the material with its anisotropic character and each one is referred to as a reinforcing model. We consider two cases for the nature of the anisotropy: (i) reinforcing models that have a particular influence on the shear response of the material and (ii) reinforcing models that depend only on the stretch in the fiber direction. Bifurcation and postbifurcation results show a very different qualitative behavior for the reinforcing models. Necking and bulging motions are captured. In general, localized bulging occurs first. Nevertheless, we find necking solutions during inflation in thin-walled cylinders under axial loading and internal pressure, although both highly elongated and inflated. We show that necking, under the conditions at hand, propagates both axially and radially, i.e. it is a combination of both axial and radial stretches. Furthermore, axial stretch at the necking zone is smaller than outside the necking region. In addition, necking motion is related to a decrease of pressure beyond the onset of necking. These features are easily identified with the necking solutions found during deflation from a highly inflated tube.