Shear instabilities for the buckling of non-linearly elastic plates
Author:
Pablo V. Negrón-Marrero
Journal:
Quart. Appl. Math. 49 (1991), 477-493
MSC:
Primary 73H05; Secondary 35B32, 58C27, 73K10
DOI:
https://doi.org/10.1090/qam/1121680
MathSciNet review:
MR1121680
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Abstract: In this work we apply singularity theory and global bifurcation theory to the problem of the axisymmetric buckling of circular plates. We show that a certain shear parameter can be used as an unfolding parameter and we get detailed local and global structure of the solution set. We get conditions for a singular point to be of codimension one.
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M. A. Naimark, Linear Differential Operators. I, II, Ungar, New York, 1968
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J. C. Alexander and S. S. Antman, Global behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems, Arch. Rational Mech. Anal. 76, 339–354 (1981)
J. C. Alexander and S. S. Antman, Global behavior of solutions of nonlinear equations depending on infinite-dimensional parameters, Indiana Univ. Math. J. 32, 39–62 (1983)
S. S. Antman, Buckled states of nonlinearly elastic plates, Arch. Rational Mech. Anal. 67, 111–149 (1978)
S. S. Antman and J. F. Pierce, The intricate global structure of buckled states of compressible columns, SIAM J. Appl. Math. 50, 395–419 (1990)
S. S. Antman and P. V. Negron-Marrero, The remarkable nature of radially symmetric equilibrium states of aelotropic nonlinearly elastic bodies, J. Elasticity 18, 131–164 (1987)
K. A. Cliffe and A. Spence, The calculation of high order singularities in the finite Taylor problem, Proc. Conf. on Numerical Methods for Bifurcation Problems (Dortmund, August 22–26, 1983), Birkhaüser, Basel, 1984
M. G. Crandell and P. H. Rabinowitz, Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech. 19, 1083–1102 (1970),
M. Golubitsky and D. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I, Springer-Verlag, New York, 1985
T. J. Healey, Global bifurcation and continuation in the presence of symmetry with an application to solid mechanics, SIAM J. Math. Anal. 19, 824–840 (1988)
T. J. Healey, Symmetry and equivariance in nonlinear elastostatics. I, Arch. Rational Mech. Anal. 105, 205–228 (1989)
E. Jahnke and F. Emde, Tables of Functions—With Formulas and Curves, 4th ed., Dover, New York, 1945
H. G. Kaper, M. K. Kwong, and A. Zettl, Regularizing transformations for certain singular Sturm-Liouville boundary value problems, SIAM J. Math. Anal. 15, 957–963 (1984)
J. Leray and J. Schauder, Topologie et equations fonctionelles, Ann. Sci. Ecole Norm. Sup. (3) 51, 45–78 (1934)
R. J. Magnus, A generalization of multiplicity and the problem of bifurcation, Proc. London Math. Soc. (3) 32, 251–278 (1976)
M. A. Naimark, Linear Differential Operators. I, II, Ungar, New York, 1968
P. V. Negron-Marrero, Necked states of nonlinearly elastic plates , Proc. Roy. Soc. Edinburgh, Sect. A 112, 277–291 (1989)
P. V. Negron-Marrero and S. S. Antman, Singular global bifurcation problems for the buckling of anisotropic plates, Proc. Roy. Soc. London Ser. A 427, 95–137 (1990)
P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7, 487–513 (1971)
D. H. Sattinger, Branching in the presence of symmetry, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 40, SIAM, Philadelphia, 1983
G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd Ed., Cambridge Univ. Press, 1962
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Article copyright:
© Copyright 1991
American Mathematical Society
