Abstract

This work is devoted to the study of the existence of “regular” solutions for a one-dimensional problem with unilateral constrained gradient in Perfect-Plasticity. The particularity of this problem consists in the fact, that the tools usually employed to prove the Inf-Sup equality between the displacement problem and the stress problem do not work. In a first part, we establish this equality by the mean of a penalty method which employs the theory of the convex functions of measures. In a second part, we find the regular limit loads, between which the displacement problem possesses at least a solution which is in W1,1W^{1,1}, verifiyng the boundary conditions and the constraint on the gradient. We give an example where these loads are infinite. Ce travail est consacré à l’étude de l’existence de solutions régulières pour un problème unidimensionnel, avec un gradient contraint unilatéral en Plasticité Parfaite. La particularité de ce problème réside en ceci que, les outils habituellement employés pour montrer l'égalité entre les problèmes en déplacement et en contrainte ne fonctionnent plus. Dans une première partie, on établit cette égalité au moyen d’une méthode de pénalité, qui utilise la théorie des fonctions convexes de mesure. Dans un deuxième temps, on trouve les charges limites régulières, en deça desquelles le problème en déplacement admet des solutions qui sont dans W1,1W^{1,1}, et qui vérifient les conditions aux bords et la contrainte sur le gradient. On donne aussi un exemple où ces charges limites sont infinies.

Author information

Contact details are reproduced from the original publication and may be historical.

Thierry Astruc

Laboratoire d’Analyse Non-Linéaire Appliquée, UFR Sciences et Techniques Université de Toulon et du Var, BP 132, 83130 La Garde cedex, France

References

  1. T. Astruc: Thèse, Université de Paris-Sud, 1994.Google Scholar
  2. T. Astruc: Existence of Regular Solution for a one-dimensional Simplified Perfect-Plastic Problem, Mathematical Modeling and Numerical Analysis, Vol. 29 (1995) 63-95.https://doi.org/10.1051/m2an/1995290100631Google Scholar
  3. G. Buttazzo, L. Faina: Proc. Roy. Soc. Edinburgh 123-A, 1993.https://doi.org/10.1017/S0308210500030912Google Scholar
  4. G. Buttazzo, L. Faina: Portugaliae Math. 51, 1994.Google Scholar
  5. F. Demengel: Compactness theorems for spaces of functions with bounded derivatives and applications to limit analysis problems in plasticity, Archive for Rational Mechanics and Analysis, 1989.https://doi.org/10.1007/BF00250834Google Scholar
  6. F. Demengel: Problèmes variationnels de la mécanique, Cours de DEA de Paris-Sud, 1990.Google Scholar
  7. F. Demengel, R. Temam: Convex functions of a measure, Indiana University Math. J., Vol. 33, n' 5, 1984.https://doi.org/10.1512/iumj.1984.33.33036Google Scholar
  8. F. Demengel, R. Temam: Duality and limit analysis in plasticity, Archive for Rational Mechanics and Analysis, 1989.Google Scholar
  9. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam, New-York, 1976.Google Scholar
  10. Giaquinta, Modica: Non-linear systems of the type of the stationary Navier-Stokes system, J.-Reine-Angew.-Math, 1982.https://doi.org/10.1515/crll.1982.330.173Google Scholar
  11. R.V. Kohn, R. Temam: Dual spaces of stresses and strains with applications to hencky plasticity, Appl.Math.Optimization 10 (1983) 1-35.https://doi.org/10.1007/BF01448377Google Scholar
  12. M. A. Krasnosel'skii: Topological Method in the Theory of Nonlinear Integral Equations, Pergamon Student Editions, Oxford, London, New-York, Paris, 1963.Google Scholar
  13. P. Sternberg, G. William and P. Ziemer: Existence, uniqueness, and regularity for functions of least gradient, J.-reine-Angew.-Math., 1992.https://doi.org/10.1515/crll.1992.430.35Google Scholar
  14. P. Suquet: Existence and regularity of solutions for plasticity problems, Variational Methods in Solid mechanics, 1980.https://doi.org/10.1016/B978-0-08-024728-1.50053-7Google Scholar
  15. R. Temam: Problèmes variationnels en plasticité, Gauthier-Villars, 1985. english version.Google Scholar
  16. R. Temam, G. Strang: Functions of bounded deformations, ARMA 75 (1980) 7-21.https://doi.org/10.1007/BF00284617Google Scholar

Suggested citation

T. Astruc. “Existence of Regular Solutions for a One-Dimensional Simplified Perfect-Plastic Problem with a Unilateral Gradient Constraint.” Journal of Convex Analysis 4 (1997), No. 2, 257–280. https://doi.org/10.68381/jca04013

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 1997. Rights now held by Banach Press.