Contact problems and dual variational inequality of 2-D elastoplastic beam theory. (English) Zbl 0889.73064
Summary: In order to study the frictional contact problems of the elastoplastic beam theory, an extended two-dimensional beam model is established, and a second order nonlinear equilibrium problem with both internal and external complementarity conditions is proposed. The external complementarity condition provides the free boundary condition, while the internal complementarity condition gives the interface of the elastic and plastic regions. We prove that this bicomplementarity problem is equivalent to a nonlinear variational inequality. The dual variational inequality is also developed. It is shown that the dual variational problem is much easier than the original variational problem. An application to limit analysis illustrates the method.
MSC:
| 74A55 | Theories of friction (tribology) |
| 74M15 | Contact in solid mechanics |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 49J40 | Variational inequalities |
Keywords:
frictional contact; second order nonlinear equilibrium problem; external complementarity condition; free boundary condition; internal complementarity condition; bicomplementarity problem; limit analysisReferences:
| [1] | J. P. Aubin, Applied Functional Analysis, Wiley-Interscience, New York (1979). · Zbl 0424.46001 |
| [2] | G. Allen, Variational inequalities, complementarity problems, and duality theorems, J. Math. Analy. Appl., 58 (1977), 1-10. · Zbl 0383.49005 · doi:10.1016/0022-247X(77)90222-0 |
| [3] | J. Chakrabarty, Theory of Plasticity, McGraw-Hill Book Company (1987), 791. · Zbl 0631.01024 |
| [4] | I. Ekeland and R. Temam, Convex Analysis and Variational Problems, North-Holland (1976). · Zbl 0322.90046 |
| [5] | G. Fichera, Existence theorems in elasticity-boundary value problems of elasticity with unilateral constraints, in Encyclopedia of Physics, Ed. by S. Flügger, Vol. VI a/2, Mechanics of Solids (II), Ed. by C. Truesdell, Springer-Verlag (1972), 347-427. |
| [6] | Y. Gao, Convex analysis and mathematical theory of plasticity, in Modern Mathematics and Mechanics, Ed. by Guo Zhongheng, Peking University Press (1987), 165-185. (in Chinese) |
| [7] | Y. Gao, On the complementary bounding theory for limit analysis, Int. J. Solids Structures, 24 (1988), 545-556. · Zbl 0629.73086 · doi:10.1016/0020-7683(88)90028-5 |
| [8] | Y. Gao, Panpenalty finite element programming for limit analysis, Computers & Structures, 28 (1988), 749-755. · Zbl 0632.73031 · doi:10.1016/0045-7949(88)90415-4 |
| [9] | David Yang Gao and David I., Russell, A finite element approach to optimal control of a ?smart? beam, in Int. Conf. Computational Methods in Structural and Geotechnical Engineering, Edited by P. K. K. Lee, L. G. Tham and Y. K. Cheung, December 12-14. 1994, Hong Kong (1994), 135-140. |
| [10] | David Yang Gao and David L. Russell, An extended beam theory for smart materials applications, Part I. Extended beam models, duality theory and finite element simulations, J. Applied Math. and Optimization, 34, 3 (1996). · Zbl 0856.73026 |
| [11] | R. Glowinski, J. L. Lions and R. Trémolières, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, New York, Oxford (1981). |
| [12] | S. Karamardian, Generalized complementarity problems, J. of Optimization Theory and Applications, 6, 19 (1971), 161-168. · doi:10.1007/BF00932464 |
| [13] | N. Kikuchi and J. T. Oden, Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM, Philadelphia (1988), 495. · Zbl 0685.73002 |
| [14] | D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York (1980). · Zbl 0457.35001 |
| [15] | A. Klarbring, A. Mikeli? and M. Shillor, Duality applied to contact problems with friction. Appl. Math. Optim., 22 (1990), 211-226. · Zbl 0717.49031 · doi:10.1007/BF01447328 |
| [16] | J. Lovísek, Duality in the obstacle and unilateral problem for the biharmonic operator, Aplikace Mathematiky, Svazek, 26 (1981), 291-303. |
| [17] | U. Mosco, Dual variational inequalities, J. Math. Analy. Appl., 40 (1972), 202-206. · Zbl 0262.49003 · doi:10.1016/0022-247X(72)90043-1 |
| [18] | S. T. Yau and Y. Gao, Obstacle problem for von Kármán equations, Adv. Appl. Math., 13 (1992), 123-141. · Zbl 0766.35087 · doi:10.1016/0196-8858(92)90005-H |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
