Elastic Stability of Rubber Products

Abstract

Abstract Due to severe nonlinearities, inherent in the finite-element elasticity, uniquely defined boundary-value problems of rubber elasticity may have multiple stable and unstable solutions. An early example was given by Rivlin, who considered the problem of a Neo-Hookean cube, in a state of pure homogeneous deformations, and subjected to three pairs of equal and opposite forces acting normally on the faces of the cube and distributed uniformly over them. He found that, for forces below a certain value, the only possible solution is the symmetric solution, as might be expected. Beyond that certain value, however, there are seven possible equilibrium solutions. One of these seven solutions is the symmetric solution. It is interesting to notice that the symmetric solution, which is initially stable, becomes unstable when loads have reached a certain threshold. The stability problems of homogeneous deformations of Mooney-Rivlin type of materials, under symmetric loading, for triaxial loading and for the plane stress and plane strain cases, are dealt with in Reference 3. It was shown that a finite-element method can be applied for such analyses. The stability of a sheet of Mooney-Rivlin type of material has been studied for a symmetrical loading condition. Such instability phenomenon was first observed by Treloar. In this work, the problem of a sheet of Mooney-Rivlin type of material, subject to general biaxial loading, is studied both analytically and by finite element. An energy approach to the problem is first presented. This problem represents the biaxial loading of rubber sheets or combined extension and inflation of rubber tubes, which are often used in experimental work for characterization of rubber materials. It is shown that the problem has multiple solutions for a certain domain of loading. The equilibrium state, actually attained, is dependent on the manner of quasistatic loading. Various stable solutions are obtained by finite element.