Monotonicity of nonlinear boundary value problems related  to deformation theory of plasticity

Hasanov, Alemdar

doi:https://doi.org/10.1155/MPE/2006/58143

Mathematical Problems in Engineering

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Volume 2006 | Article ID 058143 | https://doi.org/10.1155/MPE/2006/58143

Monotonicity of nonlinear boundary value problems related to deformation theory of plasticity

Alemdar Hasanov¹

Received01 Mar 2005

Accepted20 Jun 2005

Published23 Feb 2006

Abstract

We study nonlinear boundary value problems arising in the deformation theory of plasticity. These problems include 3D mixed problems related to nonlinear Lame system, elastoplastic bending of an incompressible hardening plate, and elastoplastic torsion of a bar. For all these different problems, we present a general variational approach based on monotone potential operator theory and prove solvability and monotonicity of potentials. The obtained results are illustrated on numerical examples.

References

R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Mathematics, Academic Press, New York, 1975.
View at: Zentralblatt MATH | MathSciNet
J. H. Argyris and S. Kelsey, Energy Theorems and Structural Analysis, Butterworths, London, 1960.
View at: MathSciNet
P. G. Ciarlet, The Finite Element Method for Elliptic Problems, vol. 4 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, 1978.
View at: Zentralblatt MATH | MathSciNet
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, vol. 219 of Grundlehren der mathematischen Wissenschaften, Springer, Berlin, 1976.
View at: Zentralblatt MATH | MathSciNet
H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, vol. 38 of Mathematische Lehrbücher und Monographien, II. Abteilung, Mathematische Monographien, Akademie, Berlin, 1974.
View at: Zentralblatt MATH | MathSciNet
A. Hasanov, “An inverse problem for an elastoplastic medium,” SIAM Journal on Applied Mathematics, vol. 55, no. 6, pp. 1736–1752, 1995.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Hasanov, “Convexity argument for monotone potential operators and its application,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 41, no. 7-8, pp. 907–919, 2000.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Hasanov and Z. Seyidmamedov, “Determination of unknown elastoplastic properties in Signorini problem,” International Journal of Non-Linear Mechanics, vol. 33, no. 6, pp. 979–991, 1998.
View at: Publisher Site | Google Scholar | MathSciNet
A. A. Il'yušin, Plasticity. Part 1: Elasticity-Plastic Deformations, OGIZ, Moscow, 1948.
View at: MathSciNet
L. M. Kachanov, Fundamentals of the Theory of Plasticity, Mir, Moscow, 1974.
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, vol. 49 of Applied Mathematical Sciences, Springer, New York, 1985.
View at: Zentralblatt MATH | MathSciNet
A. Langenbach, “Elastisch-plastische deformationen von platten,” Zeitschrift für Angewandte Mathematik und Mechanik. Ingenieurwissenschaftliche Forschungsarbeiten, vol. 41, pp. 126–134, 1961 (German).
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Langenbach, “Verallgemeinerte und exakte Lösungen des Problems der elastisch-plastischen Torsion von Stäben,” Mathematische Nachrichten, vol. 28, pp. 219–234, 1964/1965 (German).
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S. G. Mikhlin, Variational Methods in Mathematical Physics, Macmillan, New York, 1964.
View at: Zentralblatt MATH | MathSciNet
J. Nečas and I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, Elsevier, Amsterdam, 1980.
View at: Zentralblatt MATH | MathSciNet
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, vol. 49 of Mathematical Surveys and Monographs, American Mathematical Society, Rhode Island, 1997.
View at: Zentralblatt MATH | MathSciNet
K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon, New York, 2nd edition, 1981.

Copyright

Copyright © 2006 Alemdar Hasanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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