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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R. S. Rivlin, Plane strain of a net formed by inextensible cords, J. Rational Mech. Anal. 4, 951–974 (1955)
T. G. Rogers and A. C. Pipkin, Holes in inextensible networks, Quart. J. Mech. Appl. Math. 33, 447–462 (1980)
A. C. Pipkin, Some developments in the theory of inextensible networks, Quart. Appl. Math. 38, 343–355 (1980)
A. C. Pipkin, Plane traction problems for inextensible networks, Quart. J. Mech. Appl. Math. 34, 415–429 (1981)
A. C. Pipkin, Inextensible networks with slack, Quart. Appl. Math. 40, 63–71 (1982)
A. C. Pipkin, Energy minimization for nets with slack, Quart. Appl. Math. 44, 249–253 (1986)
A. C. Pipkin and T. G. Rogers, Infinitesimal plane wrinkling of inextensible networks, J. Elasticity 17, 35–52 (1987)
S. M. Genensky and R. S. Rivlin, Infinitesimal plane strain in a network of elastic cords, Arch. Rational Mech. Anal. 4, 30–44 (1959–60)
W. A. Green and J. Shi, Plane deformations of membranes formed with elastic cords, Quart. J. Mech. Appl. Math. 43, 317–333 (1990)
J. Shi, Elastic networks and membranes, Ph.D. Thesis, Nottingham University, U.K., 1988
R. Hill, On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solids 5, 229–241 (1957)
R. W. Ogden, Nonlinear elastic deformations, Ellis Horwood Limited, Chichester, 1984
A. J. M. Spencer, Continuum mechanics, Longman, New York, 1980
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Article copyright:
© Copyright 1992
American Mathematical Society