Variational Formulation

  • Géry de Saxcé
Part of the International Centre for Mechanical Sciences book series (CISM, volume 432)


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.


Variational Principle Admissibility Condition Dissipation Power Function Shakedown Analysis Lagrangean Multiplier Technique 
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  1. 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.Google Scholar
  2. 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.Google Scholar
  3. 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.Google Scholar
  4. Ekeland, I. and Temam, R. (1975). Convex analysis and variational problems, NY: North Holland.Google Scholar
  5. Hill, R. (1948). A variational principle of maximum plastic work in classical plasticity. Quarterly Journal of Mechanics and Applied Mathematics 1: 18.MathSciNetCrossRefMATHGoogle Scholar
  6. Koiter, W.T. (1960). General theorems for elasto-plastic solids. In Progress in Solid Mechanics, Volume 1.Google Scholar
  7. Mandel, J. (1966). Cours de Mécanique des Milieux Continus, Tome 2: Mécanique des Solides. Paris: Gauthier-Villars.MATHGoogle Scholar
  8. Markov, A.A. (1947). On variational principles in theory of plasticity. Prek. Math. Mech. 11: 339.MATHGoogle Scholar
  9. Martin, J.B. (1975). Plasticity, fundamentals and general results. MA: MIT Press.Google Scholar
  10. 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.Google Scholar
  11. 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.Google Scholar
  12. Save, M., Massonnet, C. and de Saxcé, G. (1997). Plastic Limit Analysis of Plates, Shells and Disks, NY: North Holland.MATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 2002

Authors and Affiliations

  • Géry de Saxcé
    • 1
  1. 1.University of LilleLilleFrance

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