Numerical simulation of the effect of damaged induced softening on the inflation of a circular rubber membrane

Rubbery materials can undergo deformation induced microstructural changes leading to softening of response and permanent set. In this work, these materials are modeled by a constitutive equation that incorporates the following micromechanism : when the deformation becomes large enough, network junctions break during further increments of deformation and then heal. New networks are formed by this conversion process, each unstressed in the local configuration at which it is formed. The constitutive equation is used to describe the response of the particles of an initially flat circular membrane which is inflated by lateral pressure. The original material and each newly formed network are assumed to respond as neo-Hookean elastic materials. Results of a numerical simulation show the influence of this conversion process on the distribution of stretch ratios, inflated shapes and the relation between the pressure and the crown stretch ratio. The general form of the constitutive equation for non-linear elastic solids is based on assumptions which imply that stress arises from a single unchanging material micro- mechanism at all stages of deformation. However, rubber-like materials exhibit changes in micromechanism evidenced by the occurrence of permanent set under large deformations (Hart-Smith, 1966). Rajagopal and Wineman (1992) have presented a constitutive theory which can be used to model such mechanical response. In their model, the stress is deter- mined by one micromechanism within some regime of deformation ; as deformation increases, a new micromechanism arises which affects the mechanical response. They con- sidered the particular example in which material acts as a rubbery solid if the deformations are not too large. When deformations become sufficiently large, network junctions in the original material break and then reform to produce a new network with a new unstressed local configuration. Their work allowed for continuous conversion of the original material to new networks as deformation proceeds. It was shown that the material can undergo substantial softening, and that there is permanent set when the stress is removed. There have been several applications of this constitutive theory to problems involving non-homogeneous deformations. The examples all assume that the material is incom- pressible and that new networks are generated at sufficiently large deformations. Wineman and Rajagopal (1990) studied the finite extension and torsion of a circular cylinder. They showed that as the angle of twist increases, there evolves an inner core of material which consists of the original network and an outer shell of the modified material. These regions are separated by an interface whose radius is determined by the angle of twist. They also studied the influence of the material response on the torque-twist relation. Huntley (1992) considered the problem in which a hollow concentric cylinder is fixed at its inner surface and its outer surface is rotated about the centerline. Each particle of the cylinder is subjected to a simple shear deformation, the magnitude of which decreases with