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Necked states of non-linearly elastic plates

Published online by Cambridge University Press:  14 November 2011

Pablo V. Negrón-Marrero
Pablo V. Negrón-Marrero
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH144AS, U.K.
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Synopsis

In this paper we study the equilibrium equations for axisymmetric deformations of isotropic circular plates in tension. We give results on the global multiplicity of solutions and study the stability of the trivial homogeneous solution for large displacements.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 112 , Issue 3-4 , 1989 , pp. 277 - 291
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1Antman, S. S.. Nonuniqueness of equilibrium states for bars in tension. J. Math. Anal. Appl. 44 (1973), 333–349.CrossRefGoogle Scholar
2Antman, S. S.. Buckled states of nonlinearly elastic plates. Arch. Rational Mech. Anal. (2) 67 (1979), 111–149.Google Scholar
3Antman, S. S.. Regular and singular problems for large elastic deformations of tubes, wedges, and cylinders. Arch. Rational Mech. Anal. 83 (1983), 1–52.CrossRefGoogle Scholar
4Antman, S. S. and Carbone, E. R.. Shear and necking instabilities in nonlinear elasticity. J. Elasticity 7 (1977), 125–151.CrossRefGoogle Scholar
5Browder, F. E.. On continuity of fixed points under deformations of continuous mappings. Summa Brasil. Mat. 4 (1960), 183–191.Google Scholar
6Crandall, M. G. and Rabinowitz, P. H.. Nonlinear Sturm-Liouville eigenvalue problems and topological degree. J. Math. Mech. 19 (1970), 1083–1102.Google Scholar
7Ericksen, J. L.. Equilibrium of bars. J. Elasticity 5 (1975), 191–201.CrossRefGoogle Scholar
8Gurtin, M. E.. Topics in Finite Elasticity (Philadelphia: Society for Industrial and Applied Mathematics, 1983).Google Scholar
9Ize, J.. Bifurcation theory for Fredholm operators. Mem. Amer. Math. Soc. 7 (1976).Google Scholar
10Kielhöfer, H.. Multiple eigenvalue bifurcation for Fredholm operators. (Preprintno. 103 Institut für Angewandte Mathematik und Statistik, Am Hubland, 8700 Würzburg, F.R.G., 1984).Google Scholar
11Leray, J. and Schauder, J.. Topologie et équations functionelles. Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78.CrossRefGoogle Scholar
12Magnus, R. J.. A generalization of multiplicity and the problem of bifurcation. Proc. London Math. Soc. (3) 32 (1976), 251–278.CrossRefGoogle Scholar
13Negrón-Marrero, P. V.. Large buckling of circular plates with singularities due to anisotropy (Doctoral Dissertation, University of Maryland at College Park, 1985).Google Scholar
14Owen, N.. Existence and stability of necking deformations for nonlinearly elastic rods. Arch. Rational Mech. Anal. 98 (1987) 357–383.CrossRefGoogle Scholar
15Rabinowitz, P. H.. Nonlinear Sturm-Liouville problems for second order ordinary differential equations. Comm. Pure Appl. Math. 23 (1970) 939–961.CrossRefGoogle Scholar
16Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Fund. Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
17Rabinowitz, P. H.. A global theorem for nonlinear eigenvalue problems and applications. In Contributions to Nonlinear Functional Analysis, ed. E. Zarantonello, H. (New York: Academic Press, 1971).Google Scholar
18Rabinowitz, P. H.. Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973), 161–202.CrossRefGoogle Scholar
19Radon, J.. Über lineare Funktionaltransformationen und Funktionalgleichungen. S.-B. Akad. Wiss. Wien 128 (1919), 1083–1121.Google Scholar
20Wang, C. C. and Truesdell, C.. Introduction to Rational Elasticity (Leyden: Noordhoff, 1973).Google Scholar