On the inflation, bulging/necking bifurcation and post-bifurcation of a cylindrical membrane under limited extensibility of its constituents

Abstract

We consider the bifurcation and post-bifurcation of an extended and inflated circular cylindrical membrane under limited extensibility of its constituents. First, for illustration of the limited extensibility effect, a membrane made of the (isotropic) Gent model is briefly analyzed. Second, the membrane is considered to be made of an isotropic ground substance reinforced with fibers symmetrically arranged in two helically distributed families which are mechanically equivalent. The mechanical behavior of the fibers in a similar way to the (isotropic) Gent model is taken to reflect fiber limited extensibility. In particular, the materials under consideration are NH models augmented with two functions called reinforcing models, each one accounting for unidirectional reinforcement. For a specific material, since both families are mechanically equivalent, the reinforcing models are equal. The nature of the anisotropy considered, i.e., the reinforcing model, can depend only on the stretch in the fiber direction or can depend on the fiber stretch and also can have an influence on the shear response of the material (i.e., it also captures shearing in the fiber direction). The limitation associated with the material anisotropic deformability as well as the arrangement of the material constituents is discussed with respect to the initiation of bulging and necking for the membrane. The subsequent consequences for configurations in equilibrium during post-bifurcation are also studied in detail and a variety of results are given in terms of the limited extensibility (deformability) of the fibers as well as their mechanical response and arrangement (fiber winding angle).