Inextensible networks with slack

A. C. Pipkin
1982 Quarterly of Applied Mathematics  
1. Introduction. Although it is easy to distort a piece of cloth, such distortions usually involve only relatively small stretching of the fibers or threads in the material. Rivlin [1] , with rather open networks in mind, formulated a continuum theory in which the cords or fibers in the network are treated as absolutely inextensible, but with no resistance to changes in the angle between two intersecting fibers. Pipkin [2] recast this theory in vector form and discussed various kinds of
more » ... us kinds of singularities that solutions can exhibit. Rogers and Pipkin [3] and Rogers [4] have applied the theory to problems involving holes or tears in sheets. Because Rivlin's theory uses the constraint that no fiber segment can change its length at all, even under compressive loading, solutions in this theory can involve compressive stresses [2], It is necessary to allow compressive stresses in order to assure existence of solutions, but when compressive stresses are admitted, solutions become highly nonunique. Often there is exactly one solution with non-negative fiber tensions, and this solution is chosen as the correct one. However, as we show in the present paper, there are problems in which no solution has non-negative fiber tensions everywhere. The purpose of the present paper is to formulate a theory in which fibers can grow shorter but not longer, and can carry tensile but not compressive loads. We also prove a lemma concerning uniqueness of solutions in the new theory. Since the extended theory presented here is stated in terms of inequalities as well as equations (Sec. 2), it is clear from the outset that the deformation is highly arbitrary in any
doi:10.1090/qam/652050 fatcat:paost7vnfja7jdkvpr35mhi5fu