Abstract
We present the dual variational principles of the rigid perfectly plastic material due to Markov and Hill, from which ones the bound theorems of limit analysis can be deduced. Following the same kind of method, the bound theorems can be extended to variable repeated loads. They provide two dual ways to calculate the shakedown load α a by solving constrained optimization problems. Finally, the duality between the variational principles of shakedown is discussed from the viewpoint of the non smooth mechanics.
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References
Chen, H. and Ponter, A.R.S. (2000). A method for the evaluation of a ratchet limit and the amplitude of plastic strain for bodies subjected to cyclic loading. European Journal of Mechanics A/Solids, accepted for publication.
de Saxcé, G. (1986). Sur Quelques Problèmes de Mécanique des Solides Considérés Comme Matériaux à Potentiels Convexes, Doctor thesis, Université de Liège, Collection des Publications de la Faculté des Sciences Appliquées, 118.
de Saxcé, G. (1995). A variational deduction of upper and lower bound shakedown theorems by Markov’s and Hill’s principles over a cycle. In Mröz, Z., et al., eds., Inelastic Behavior of Structures Under Variable Loads, Dordrecht: Kluwer Academic Publishers.
Ekeland, I. and Temam, R. (1975). Convex analysis and variational problems, NY: North Holland.
Hill, R. (1948). A variational principle of maximum plastic work in classical plasticity. Quarterly Journal of Mechanics and Applied Mathematics 1: 18.
Koiter, W.T. (1960). General theorems for elasto-plastic solids. In Progress in Solid Mechanics, Volume 1.
Mandel, J. (1966). Cours de Mécanique des Milieux Continus, Tome 2: Mécanique des Solides. Paris: Gauthier-Villars.
Markov, A.A. (1947). On variational principles in theory of plasticity. Prek. Math. Mech. 11: 339.
Martin, J.B. (1975). Plasticity, fundamentals and general results. MA: MIT Press.
Ponter, A.R.S. and Chen, H. (2000). A minimum theorem for cyclic load in excess of shakedown with application to the evaluation of a ratchet limit. European Journal of Mechanics A/Solids, accepted for publication.
Save, M., de Saxcé, G. and Borkowski, A. (1991). Computation of shake-down loads, feasibility study, Commission of the European Communities, Nuclear Science and Technology, report EUR 13618.
Save, M., Massonnet, C. and de Saxcé, G. (1997). Plastic Limit Analysis of Plates, Shells and Disks, NY: North Holland.
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© 2002 Springer-Verlag Wien
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de Saxcé, G. (2002). Variational Formulation. In: Weichert, D., Maier, G. (eds) Inelastic Behaviour of Structures under Variable Repeated Loads. International Centre for Mechanical Sciences, vol 432. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2558-8_2
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DOI: https://doi.org/10.1007/978-3-7091-2558-8_2
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-83687-3
Online ISBN: 978-3-7091-2558-8
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