QUARTERLYOF APPLIED MATHEMATICS VOLUME L, NUMBER 3 SEPTEMBER 1992, PAGES 501-516 DEGENERATE DEFORMATIONS AND UNIQUENESS IN HIGHLY ELASTIC NETWORKS By W. A. GREEN (Nottingham University, U.K.) AND JINGYU SHI (University of Queensland, Australia) Abstract. This work deals with the continuum theory for plane deformations of a network formed of two families of highly elastic cords, under the assumption of no resistance to shearing. Following Pipkin [3] it is shown that there exists a collapse mode of deformation in which a finite region of the network collapses onto a single curve and examples are exhibited which correspond to a universal deformation and to a universal state of tension. It is further shown that the assumption that the cords can withstand no compression leads to the existence of half-slack and fully-slack regions, as defined by Pipkin [5]. The most general deformation associated with a half-slack region is determined. A variational principle is established for the general boundary value problem and it is shown that, for strain-energy functions which are quadratic in the stretches of the cords, this leads to a minimum principle and a generalized uniqueness theorem. A stability and uniqueness theorem is derived for the materials with a more general strain energy function. 1. Introduction. The continuum theory for plane deformations of networks formed by two families of continuously distributed inextensible cords was first formulated by Rivlin [1], Rivlin's theory, which assumes no shearing resistance between the cords, was subsequently applied by Rogers and Pipkin [2] to treat problems of inextensible networks with holes. An extension of Rivlin's model to include shear effects was proposed by Pipkin [3, 4], who also discussed some of the singularities that may occur in the solutions. A later paper by Pipkin [5] deals with a modification of Rivlin's theory so that the cords may shorten but not lengthen and may transmit tension but not compression. Pipkin [5, 6] also treats problems of existence and uniqueness of solutions in this modified theory, whilst Pipkin and Rogers [7] present a further extension to account for wrinkling of the network. Amongst the degenerate or singular solutions that can arise in these theories, Pipkin [3] has identified a collapse mode of deformation in which a finite region of the Received November 6, 1990. 1991 Mathematics Subject Classification. Primary 73G05. ©1992 Brown University 501