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 | References | Similar Articles | Additional Information

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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Additional Information

DOI:
https://doi.org/10.1090/qam/1178430

Article copyright:
© Copyright 1992
American Mathematical Society