Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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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  • [1] R. S. Rivlin, Plane strain of a net formed by inextensible cords, J. Rational Mech. Anal. 4, 951-974 (1955) MR 0074998
  • [2] T. G. Rogers and A. C. Pipkin, Holes in inextensible networks, Quart. J. Mech. Appl. Math. 33, 447-462 (1980) MR 607659
  • [3] A. C. Pipkin, Some developments in the theory of inextensible networks, Quart. Appl. Math. 38, 343-355 (1980) MR 592201
  • [4] A. C. Pipkin, Plane traction problems for inextensible networks, Quart. J. Mech. Appl. Math. 34, 415-429 (1981) MR 637877
  • [5] A. C. Pipkin, Inextensible networks with slack, Quart. Appl. Math. 40, 63-71 (1982) MR 652050
  • [6] A. C. Pipkin, Energy minimization for nets with slack, Quart. Appl. Math. 44, 249-253 (1986) MR 856178
  • [7] A. C. Pipkin and T. G. Rogers, Infinitesimal plane wrinkling of inextensible networks, J. Elasticity 17, 35-52 (1987) MR 885594
  • [8] 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) MR 0108925
  • [9] W. A. Green and J. Shi, Plane deformations of membranes formed with elastic cords, Quart. J. Mech. Appl. Math. 43, 317-333 (1990) MR 1070959
  • [10] J. Shi, Elastic networks and membranes, Ph.D. Thesis, Nottingham University, U.K., 1988
  • [11] R. Hill, On uniqueness and stability in the theory of finite elastic strain, J. Mech. Phys. Solids 5, 229-241 (1957) MR 0092379
  • [12] R. W. Ogden, Nonlinear elastic deformations, Ellis Horwood Limited, Chichester, 1984 MR 770388
  • [13] A. J. M. Spencer, Continuum mechanics, Longman, New York, 1980 MR 597343

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Additional Information

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

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