Abstract
The aim of this article is to study the quasistatic evolution of a three-dimensional elastic-perfectly plastic solid which satisfies the Prandtl-Reuss law. The evolution of the field of stresses σ—which solves a time dependent variational inequality—and that of the field of displacements u, have been described in previous works [15], [26], [35], [36], [37] but it was not shown there that σ and u satisfy indeed the Prandtl-Reuss constitutive law. In this article we find σ and u in a class of functions which are sufficiently regular for the Prandtl-Reuss law to make sense and we prove that σ and u satisfy the constitutive law. This result is attained by considering the elastic-perfectly plastic model as the limit of a family of elastic-visco-plastic models like those of Norton and Hoff. The Norton-Hoff type models which we introduce depend on a viscosity parameter α > 0; we study the perturbed models (i.e. α > 0 fixed) and then we pass to the limit α → 0.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
R. S. ADAMS, Sobolev Spaces, Academic Press, New-York, 1975.
G. ANZELOTTI, On the existence of the rates of stress and displacement for PrandtlReuss plasticity, Quart. Appl. Math., (1983), p. 181–208.
G. ANZELOTTI & M. GIAQUINTA, On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium, J. Math. Pures Appl., 61 (1982), p. 219–244.
F. BROWDER, Nonlinear maximal monotone operators in Banach spaces, Math. Ann., 175 (1968), 89–113.
L. CATTABRIGA, Suun problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Math. Univ. Padova, 31 (1961), 308–340.
F. DEMENGEL, Fonctions à hessien borné, Ann. Institut Fourier Grenoble, XXXIV (2) (1984), 155–190.
G. DUVAUT et J. L. LIONS, Les inéquations en mécanique et en physique, Dunod, Paris, 1972.
I. EKELAND & R. TEMAM, Convex analysis and variational problems, North-Holland, Amsterdam, New-York, 1976.
A. FRIAâ, La loi de Norton-Hoff généralisée en pasticité et viscoplasticité, Thèse, Université de Paris V I, 1979.
P. GERMAIN, Cours de mécanique des milieux continus, Tome I, Masson, Paris, 1973.
P. GERMAIN, Cours de mécanique, Ecole Polytechnique, 1979.
P. GERMAIN, Commentaires sur l’article “Duality and relaxation in the variational problems of plasticity de R. Temam et G. Strang”, J. Mécanique, 19 (1980), 529–538.
H. HENCKY, Z. Angew. Math. Phys., 4 (1924), 323.
H. J. HOFF, Approximate analysis of structures in presence of moderately large creep deformations, Quart. Appl. Math., 12 (1) (1954), p. 49.
C. JOHNSON, Existence theorems for plasticity problems, J. Math. Pures Appl., 55 (1976), 431–444.
C. JOHNSON, On plasticity with hardening, J. Math. Anal. Appl., 62 (1978), 325–336.
R. KOHN & R. TEMAM, Dual spaces of stresses and strains with applications to Hencky plasticity, App. Math. Optim., 10 (1983), p. 1–35.
J. L. LIONS, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier-Villars, Paris, 1969.
J. L. LIONS & E. MAGENES, Non homogeneous boundary value problems and applications, Spinger-Verlag, Berlin, Heidelberg, New-York, 1972.
G. J. MINTY, On the generalization of the direct method of the calculus of variations, Bull. Amer. Math. Soc., 73 (1967), 315–321.
J. J. MOREAU, La notion de sur-potentiel et les liaisons unilatérales en élastostatique, C. R. Acad. Sci. Paris, série A, 267 (1968), 954–957.
J. J. MOREAU, Application of convex analysis to the treatment of elastoplastic systems, dans Applications of Methods of Functional Analysis to Problems of Mechanics, P. GERMAIN, B. NAYROLES Ed., Lecture Notes in Math., vol. 503, Springer-Verlag, 1976, 56–89.
J. J. MOREAU, Fonctionnelles convexes, Séminaire Equations aux Dérivées Partielles, Collège de France, 1966.
W. NOLL, On the continuity of the solid and fluid states, J. Rational Mech. Anal., 4 (1955), p. 3–81.
W. NOLL, The Foundations of Mechanics and Thermodynamics, Springer-Verlag, 1974.
F. H. NORTON, The creep of steel at high temperature, McGraw Hill, New-York, 1929.
W. PRAGER, Introduction to mechanics of continua, Ginn and Company, 1961.
W. PRAGER, Problèmes de plasticité théoriques, Dunod, Paris, 1961.
W. PRAGER & P. HODGE, Theory of perfectly plastic solids, J. Wiley, New-York, 1951.
A. REUSS, Zeitschrift für Angewandte Mathematik and Physik, 10 (1939), p. 266.
R. T. ROCKAFELLAR, Convex analysis, Princeton University Press, 1970.
J. SERRIN, The problem of Dirichlet for quasilinear elliptic equations with many independent variables, Phil. Trans. Royal Soc. London, A 264 (1969), 413–496.
G. STRANG et R. TEMAM, Existence de solutions relaxées pour les équations de la placticité: étude d’un espace fonctionnel, C. R. Acad. Sci. Paris, 287, série A (1978), 515–518
G. STRANG et R. TEMAM, Functions of bounded deformations, Arch. Rational Mech. Anal., 75 (1980), 7–21.
G. STRANG & R. TEMAM, Duality and relaxation in the variational problems of plasticity, J. Mécanique, 19 (1980), 493–527.
P. SUQUET, Existence and regularity of solutions for plasticity problems, dans Variational Methods in Solid Mechanics, S. Nemat Nasser Ed., Pergamon Press, 1980.
P. SUQUET, Sur les équations de la plasticité: existence et régularité des solutions, J. Mécanique, 20 (1981), 3–39.
P. SUQUET, Plasticité et homogénéisation, Thèse, Université de Paris V I, 1982.
R. TEMAM, Mathematical Problems in Plasticity, Dunod, Paris, 1983 (in French), and Gauthier-Villars, Paris-New York, 1984 (in English).
R. TEMAM, Navier-Stokes equations, Theory and numerical analysis, 3rd ed., North-Holland, Amsterdam, New York, 1984.
R. TEMAM, Principe de Dissipation maximale pour la loi de Prandtl Reuss en plasticité, C. R. Acad. Sci. Paris, 302, Série I, (1986) p. 79–82.
C. TRUESDELL, A first course in rational continuum mechanics, vol. I., Academic Press, New York, 1977.
C. TRUESDELL & W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, vol. 111.3, Springer-Verlag, 1965.
E. H. ZARANTONELLO, Projections on convex sets in Hilbert spaces and spectral theory, in Contributions to Nonlinear Functional Analysis, E. H. ZARANTONELLO Ed., Academic Press, New York, 1971.
Author information
Authors and Affiliations
Additional information
Dedicated to James Serrin on the occasion of his 60th Birthday
Rights and permissions
Copyright information
© 1989 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Temam, R. (1989). A generalized Norton-Hoff Model and the Prandtl-Reuss Law of Plasticity. In: Analysis and Continuum Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83743-2_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-83743-2_17
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-50917-2
Online ISBN: 978-3-642-83743-2
eBook Packages: Springer Book Archive