Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Degenerate deformations and uniqueness in highly elastic networks


Authors: W. A. Green and Jingyu Shi
Journal: Quart. Appl. Math. 50 (1992), 501-516
MSC: Primary 73G05; Secondary 73H05, 73K15, 73K99, 73V25
DOI: https://doi.org/10.1090/qam/1178430
MathSciNet review: MR1178430
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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.


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DOI: https://doi.org/10.1090/qam/1178430
Article copyright: © Copyright 1992 American Mathematical Society


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