Abstract
The restrictive framework of generalised standard materials can be suitably extended to model non-associative constitutive equations, by exploiting the concept of implicit standard materials, based on the use of a bipotential of dissipation. As presented in this study, it allows one to incorporate, in an easy and elegant way, non-linear kinematic and isotropic hardening in the constitutive equations and to recover useful flow rule normality for these non-associative behaviours.
Sommario. Il ristretto ambito dei materiali standard generalizzati è adeguatamente esteso nel lavoro utilizzando il concetto di materiale standard implicito e di bipotenziale di dissipazione. Ciò consente di incorporare nelle equazioni costitutive, in modo semplice ed elegante, leggi non lineari di incrudimento cinematico ed isotropo e di riottenere formalmente, per questi comportamenti di tipo non associato, la condizione di normalità nella legge di flusso.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lemaitre, J. and Chaboche, J.L., Mécanique des matériaux solides, 2ème éd., Bordas, Paris, 1988, Dunod, Paris, 1996, pp. 544.
Chaboche, J.L. and Nouailhas D., ‘A unified constitutive model for cyclic viscoplasticity and its application to various stainless steels’ J. Engng Mater. Technol. 111 (1989) 424–430.
Ohno, N., ‘Recent topics in constitutive modeling of cyclic plasticity and viscoplasticity’ Appl. Mech. Rev. 43(11) (1990) 283–295.
Ohno, N. and Wang, J.D., ‘Kinematic hardening rules for simulation of ratchetting behavior’ Eur. J. Mech. A/Solids 13(4) (1994) 519–531.
Jiang, Y. and Kurath, P., ‘Characteristics of the Armstrong–Frederick type plasticity models’ Int. J. Plast. 12(3) (1996) 387–415.
Ohno, N., ‘Recent progress in constitutive modeling for ratchetting’ Mater. Sci. Res. Int. 3(1) (1997) 1–9.
Ohno, N., ‘Constitutive modeling of cyclic plasticity with emphasis on ratchetting’ Int. J. Mech. Sci. 40(2–3) (1998) 251–261.
Germain, P., Nguyen, Q.S. and Suquet, P., ‘Continuum thermodynamics’ J. Appl. Mech. 50 (1983) 1010–1020.
Freed, A.D., Chaboche, J.L. and Walker, K.P., ‘A viscoplastic theory with thermodynamic considerations’ Acta Mech. 90 (1991) 155–174.
Chaboche, J.L., ‘Cyclic viscoplastic constitutive equations, Part I: A thermodynamically consistent formulation, Part II: Stored energy – comparison between models and experiments’ J. Appl. Mech. 60 (1993) 813–821, 822–828.
Chaboche, J.L., ‘Thermodynamic formulation of constitutive equations and application to the viscoplasticity and viscoelasticity of metals and polymers’ Int. J. Solids Struct. 34(18) (1997) 2239–2254.
Moreau, J.J., ‘On unilateral constraints, friction and plasticity’ In: Capriz, G. and Stampacchia, G. (Coords), New Variational Techniques in Mathematical Physics, CIME, Cremonese, Roma, 1974, pp. 171–322.
Rockafellar, R.T., Convex Analysis, Princeton University Press, Princeton, 1970, 1997 (paperback), 451 pp.
Eve, R.A., Reddy, B.D. and Rockafellar, R.T., ‘An internal variable theory of elastoplasticity based on the maximum plastic work inequality’ Quart. App. Math. 48(1) (1990) 59–83.
Romano, G., Rosati, L. and Marotti de Sciarra, F., ‘A variational theory for finite-step elasto-plastic problems’ Int. J. Solids Struct. 30(17) (1993) 2317–2334.
Romano, G., Rosati, L. and Marotti de Sciarra, F., ‘An internal variable theory of inelastic behaviour from the uniaxial rigid-perfectly plastic law’ Int. J. Engng Sci. 31(8) (1993) 1105–1120.
Halphen, B. and Nguyen, Q.S., ‘Sur les matériaux standards généralisés’ J. Méc. 14(1) (1975) 39–63.
Hopperstad, O.S. and Remseth, S., ‘A return mapping algorithm for a class of cyclic plasticity models’ Int. J. Numer. Meth Eng. 38 (1995) 549–564.
Chaboche, J.L. and Cailletaud, G., ‘Integration methods for complex plastic constitutive equations’ Comput. Met. Appl. Mech. Engng 133 (1996) 125–155.
Mahnken, R., ‘Improved implementation of an algorithm for non-linear isotropic/kinematic hardening in elastoplasticity’ Commun. Numer. Meth. Engng 15 (1999) 745–754.
de Saxcé, G., ‘Une généralisation de l'inégalité de Fenchel et ses applications aux lois constitutives’ C. R. Acad. Sci. Paris 314 Série II (1992) 125–129.
Ladevèze, P., ‘Sur une famille d'algorithmes en mécanique des structures’ C. R. Acad. Sc. Paris 300(2) Série II (1985) 41–44.
Ladevèze, P. and Rougée, P., ‘Plasticité et viscoplasticité sous chargement cyclique: propriétés et calcul du cycle limite’ C. R. Acad. Sc. Paris 301(13) Série II (1985) 891–894.
Ladevèze, P., Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation, Springer-Verlag, New York, Berlin, Heidelberg, 1999, 220 pp.
Cailletaud, G. and Pilvin, P., ‘Identification and inverse problems related to material behaviour’ In: Bui, H.D., Tanaka, M. et al. (eds), Inverse Problems in Engineering Mechanics, Balkema, Rotterdam, 1994, pp. 79–86.
Hjiaj, M., Bodovillé, G. and de Saxcé, G., ‘Matériaux viscoplastiques et lois de normalité implicites’ C. R. Acad. Sci. Paris 328 Série IIb (2000) 519–524.
Bodovillé, G. and de Saxcé, G., ‘Plasticity with non-linear kinematic hardening: modelling and shakedown analysis by the bipotential approach’ Eur. J. Mech. A/Solids 20 (2001) 99–112.
Bodovillé, G., ‘Sur l'endommagement et les matériaux standard implicites’ C. R. Acad. Sci. Paris 327 Série IIb (1999) 715–720.
Berga, A. and de Saxcé, G., ‘Elastoplastic finite element analysis of soil problems with implicit standard material constitutive laws’ Revue Européenne des Éléments Finis 3(3) (1994) 411–456.
de Saxcé, G. and Feng, Z.Q., ‘The bipotential method: a constitutive approach to design the complete contact law with friction and improved numerical algorithms’ Math. Comput. Model. 28(4–8) (1998) 225–245.
de Saxcé, G. and Bousshine, L., ‘Limit analysis theorems for implicit standard materials: application to the unilateral contact with dry friction and the non-associated flow rules in soils and rocks’ Int. J. Mech. Sci. 40(4) (1998) 387–398.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bodovillé, G. On Generalised and Implicit Normality Hypotheses. Meccanica 36, 273–290 (2001). https://doi.org/10.1023/A:1013902908951
Issue Date:
DOI: https://doi.org/10.1023/A:1013902908951