Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T14:16:09.195Z Has data issue: false hasContentIssue false
  • English
  • Français

Bifurcation in the traction problem for a transversely isotropic material

Published online by Cambridge University Press:  24 October 2008

A. Danescu
A. Danescu
Affiliation:
Institute of Mathematics, Str. Academiei No. 14, 70109-Bucharest, Romania
Get access Rights & Permissions [Opens in a new window]

Summary

The traction problem for a transversely isotropic incompressible elastic material is considered, and it is shown that when only pure homogeneous deformations are considered, the problem can be formulated as a two-dimensional ℤ2-equivariant bifurcation problem in which the bifurcation parameter is the dead-load. Using imperfect bifurcation theory, conditions for bifurcation phenomena are given and, considering a general non-linear form for the stored energy function, the recognition problem is solved in the simplest cases. The last section treats transversely isotropic non-linear perturbations for a Mooney–Rivlin material and a neo-Hookean material and the corresponding bifurcations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 2 , September 1991 , pp. 385 - 394
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Rivlin, R. S.. Stability of pure homogeneous deformations of an elastic cube under dead loading. Quart. Appl. Math. 32 (1974), 265–271.CrossRefGoogle Scholar
[2]Ball, J. M. and Schaeffer, D. G.. Bifurcation and stability of homogeneous equilibrium configurations of an elastic body under dead-load tractions. Math. Proc. Cambridge Philos. Soc. 94 (1983), 315–339.CrossRefGoogle Scholar
[3]Golubitsky, M. and Schaeffer, D. G.. Imperfect bifurcation in the presence of symmetry. Comm. Math. Phys. 67 (1979), 205–232.Google Scholar
[4]Golubitsky, M. and Schaeffer, D. G.. A theory for imperfect bifurcation via singularity theory. Comm. Pure Appl. Math. 32 (1979), 21–98.CrossRefGoogle Scholar
[5]Wan, Y. H.. The traction problem for incompressible materials. (Preprint, 1983.)Google Scholar
[6]Wan, Y. H. and Marsden, J. E.. Symmetry and bifurcation in three-dimensional elasticity. Arch. Rational Mech. Anal. 84 (1983), 203–233.Google Scholar
[7]Sawyers, K. N.. Stability of an elastic cube under dead-loading: two equal forces. Internal. J. Non-Linear Mech. 11 (1976), 11–23.CrossRefGoogle Scholar
[8]Golubitsky, M. and Schaeffer, D. G.. Singularities and Groups in Bifurcation Theory, vol. 1 (Springer-Verlag, 1985).CrossRefGoogle Scholar
[9]Dangelmeyer, G. and Armbruster, D.. Classification of ℤ2-equivariant imperfect bifurcations with corank 2. Proc. London Math. Soc. (3) 46 (1983), 517–546.CrossRefGoogle Scholar
[10]Truesdell, C. and Noll, W.. The Non-Linear Field Theories of Mechanics (Springer-Verlag, 1965).Google Scholar
[11]Schwarz, G.. Smooth functions invariant under the action of a compact Lie group. Topology 14 (1975), 63–68.Google Scholar
[12]Poenaru, V.. Singularities Cx en Presence de Symmetrie (Springer-Verlag, 1976).Google Scholar
[13]Golubitsky, M., Stewart, I. and Schaeffer, D. G.. Singularities and Groups in Bifurcation Theory, vol. 2 (Springer-Verlag, 1988).CrossRefGoogle Scholar
[14]Golubitsky, M. and Langford, W. F.. Classification and unfoldings of degenerate Hopf bifurcations. J. Differential Equations 41 (1981), 375–415.CrossRefGoogle Scholar
[15]Golubitsky, M., Keyfitz, B. L. and Schaeffer, D. G.. A singularity theory analysis of a thermal-chainbranching model for the explosion peninsula. Comm. Pure Appl. Math. 32 (1981), 433–463.CrossRefGoogle Scholar
[16]Langford, W. L.. Periodic and steady-state mode interactions lead to tori. SIAM J. Appl. Math. 37 (1979), 22–48.Google Scholar
[17]Keener, J.. Secondary bifurcation in non-linear diffusion reaction equation. Stud. Appl. Math. 55 (1976), 187–211.CrossRefGoogle Scholar
[18]Boivin, J. F.. Catastrophe theory and bifurcation. M.S. thesis, McGill University (1981).Google Scholar