Classical and sequential limit analysis revisited

Jean-Baptiste Leblond; Djimédo Kondo; Léo Morin; Almahdi Remmal

doi:10.1016/j.crme.2017.12.015

Classical and sequential limit analysis revisited

Jean-Baptiste Leblond ¹ ; Djimédo Kondo ¹ ; Léo Morin ² ; Almahdi Remmal ^1,³

¹ Sorbonne Universités, Université Pierre-et-Marie-Curie (UPMC), CNRS, UMR 7190, Institut Jean-Le Rond-d'Alembert, Tour 65-55, 4, place Jussieu, 75252 Paris cedex 05, France
² Arts et Métiers-ParisTech, CNAM, CNRS, UMR 8006, Laboratoire “Procédés et ingénierie en mécanique et matériaux”, 151, boulevard de l'Hôpital, 75013 Paris, France
³ AREVA, Tour Areva, 1, place Jean-Millier, 92084 Paris La Défense cedex, France

Comptes Rendus. Mécanique, Volume 346 (2018) no. 4, pp. 336-349.

Résumé

Classical limit analysis applies to ideal plastic materials, and within a linearized geometrical framework implying small displacements and strains. Sequential limit analysis was proposed as a heuristic extension to materials exhibiting strain hardening, and within a fully general geometrical framework involving large displacements and strains. The purpose of this paper is to study and clearly state the precise conditions permitting such an extension. This is done by comparing the evolution equations of the full elastic–plastic problem, the equations of classical limit analysis, and those of sequential limit analysis. The main conclusion is that, whereas classical limit analysis applies to materials exhibiting elasticity – in the absence of hardening and within a linearized geometrical framework –, sequential limit analysis, to be applicable, strictly prohibits the presence of elasticity – although it tolerates strain hardening and large displacements and strains. For a given mechanical situation, the relevance of sequential limit analysis therefore essentially depends upon the importance of the elastic–plastic coupling in the specific case considered.

Reçu le : 2017-12-13
Accepté le : 2017-12-14
Publié le : 2018-01-05

DOI : 10.1016/j.crme.2017.12.015

Mots-clés : Classical limit analysis, Sequential limit analysis, Hill's approach, Drucker's approach, Elasticity, Strain hardening, Large displacements and strains

Affiliations des auteurs :

Jean-Baptiste Leblond ¹ ; Djimédo Kondo ¹ ; Léo Morin ² ; Almahdi Remmal ^{1, 3}

¹ Sorbonne Universités, Université Pierre-et-Marie-Curie (UPMC), CNRS, UMR 7190, Institut Jean-Le Rond-d'Alembert, Tour 65-55, 4, place Jussieu, 75252 Paris cedex 05, France
² Arts et Métiers-ParisTech, CNAM, CNRS, UMR 8006, Laboratoire “Procédés et ingénierie en mécanique et matériaux”, 151, boulevard de l'Hôpital, 75013 Paris, France
³ AREVA, Tour Areva, 1, place Jean-Millier, 92084 Paris La Défense cedex, France

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     author = {Jean-Baptiste Leblond and Djim\'edo Kondo and L\'eo Morin and Almahdi Remmal},
     title = {Classical and sequential limit analysis revisited},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {336--349},
     publisher = {Elsevier},
     volume = {346},
     number = {4},
     year = {2018},
     doi = {10.1016/j.crme.2017.12.015},
     language = {en},
}

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Jean-Baptiste Leblond; Djimédo Kondo; Léo Morin; Almahdi Remmal. Classical and sequential limit analysis revisited. Comptes Rendus. Mécanique, Volume 346 (2018) no. 4, pp. 336-349. doi : 10.1016/j.crme.2017.12.015. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.12.015/

Bibliographie
Cité par

[1] W.H. Yang Large deformation of structures by sequential limit analysis, Int. J. Solids Struct., Volume 30 (1993), pp. 1001-1013

[2] L. Corradi; N. Panzeri A triangular finite element for sequential limit analysis of shells, Adv. Eng. Softw., Volume 35 (2004), pp. 633-643

[3] S.Y. Leu Analytical and numerical investigation of strain-hardening viscoplastic thick walled-cylinders under internal pressure by using sequential limit analysis, Comput. Methods Appl. Mech. Eng., Volume 196 (2007), pp. 2713-2722

[4] S.Y. Leu; R.S. Li Exact solutions of sequential limit analysis of pressurized cylinders with combined hardening based on a generalized Holder inequality: formulation and validation, Int. J. Mech. Sci., Volume 64 (2012), pp. 47-53

[5] D. Kong; C.M. Martin; B.W. Byrne Modelling large plastic deformations of cohesive soils using sequential limit analysis, Int. J. Numer. Anal. Methods Geomech., Volume 41 (2017), pp. 1781-1806 | DOI

[6] X.P. Yuan; B. Maillot; Y.M. Leroy Deformation pattern during normal faulting: a sequential limit analysis, J. Geophys. Res., Solid Earth, Volume 122 (2017), pp. 1496-1516 | DOI

[7] D.C. Drucker; W. Prager; M.J. Greenberg Extended limit analysis theorems for continuous media, Q. Appl. Math., Volume 9 (1952), pp. 381-389

[8] R. Hill On the state of stress in a plastic-rigid body at the yield point, Philos. Mag., Volume 42 (1951), pp. 868-875

[9] J. Mandel Cours de mécanique des milieux continus, Gauthier-Villars, Paris, 1966

[10] J. Salençon Calcul à la rupture et analyse limite, Presses de l'École nationale des Ponts et Chaussées, Paris, 1983

[11] J.J. Moreau Fonctionnelles Convexes, 2003 (publication of the Consiglio Nazionale delle Richerche, Roma and the Facolta di Ingegneria di Roma “Tor Vergata”)

[12] J.-C. Michel; M. Suquet A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations, J. Mech. Phys. Solids, Volume 90 (2016), pp. 254-285

[13] V. Tvergaard; Y. Huang; J.W. Hutchinson Cavitation instabilities in a power hardening elastic–plastic solid, Eur. J. Mech. A, Solids, Volume 11 (1992), pp. 215-231

[14] A.L. Gurson Continuum theory of ductile rupture by void nucleation and growth: Part I – yield criteria and flow rules for porous ductile media, ASME J. Eng. Mater. Technol., Volume 99 (1977), pp. 2-15

[15] R. Hill The essential structure of constitutive laws for metal composites and polycrystals, J. Mech. Phys. Solids, Volume 15 (1967), pp. 79-95

[16] J. Mandel Contribution théorique à l'étude de l'écrouissage et des lois d'écoulement plastique, Proceedings of the 11th International Congress on Applied Mechanics, Springer, Munich, FRG, 1964, pp. 502-509

[17] G. Perrin; J.-B. Leblond Analytical study of a hollow sphere made of porous plastic material and subjected to hydrostatic tension – application to some problems in ductile fracture of metals, Int. J. Plast., Volume 6 (1990), pp. 677-699

[18] J.-B. Leblond; G. Perrin A self-consistent approach to coalescence of cavities in inhomogeneously voided ductile solids, J. Mech. Phys. Solids, Volume 47 (1999), pp. 1823-1841

[19] G. Perrin; J.-B. Leblond Accelerated void growth in porous ductile solids containing two populations of cavities, Int. J. Plast., Volume 16 (2000), pp. 91-120

[20] J.-B. Leblond; G. Perrin; J. Devaux An improved Gurson-type model for hardenable ductile metals, Eur. J. Mech. A, Solids, Volume 14 (1995), pp. 499-527

[21] R. Lacroix; J.-B. Leblond; G. Perrin Numerical study and theoretical modelling of void growth in porous ductile materials subjected to cyclic loadings, Eur. J. Mech. A, Solids, Volume 55 (2016), pp. 100-109

[22] L. Morin; J.-C. Michel; J.-B. Leblond A Gurson-type layer model for ductile porous solids with isotropic and kinematic hardening, Int. J. Solids Struct., Volume 118–119 (2017), pp. 167-178

[23] P. Armstrong; C. Frederick A mathematical representation of the multiaxial Bauschinger effect, Mater. High Temp., Volume 24 (1966), pp. 11-26

[24] J.-L. Chaboche Constitutive equations for cyclic plasticity and cyclic viscoplasticity, Int. J. Plast., Volume 5 (1989), pp. 247-302

[25] J. Paux; R. Brenner; D. Kondo Plastic yield criterion and hardening of porous single crystals, Int. J. Solids Struct. (2017) | DOI

[26] J. Devaux; M. Gologanu; J.-B. Leblond; G. Perrin On continued void growth in ductile metals subjected to cyclic loadings (J. Willis, ed.), Proceedings of the IUTAM Symposium on Nonlinear Analysis of Fracture, Kluwer, Cambridge, GB, 1997, pp. 299-310

[27] A. Mbiakop; A. Constantinescu; K. Danas On void shape effects of periodic elasto-plastic materials subjected to cyclic loading, Eur. J. Mech. A, Solids, Volume 49 (2014), pp. 481-499

[28] L. Cheng; K. Danas; A. Constantinescu; D. Kondo A homogenization model for porous ductile solids under cyclic loads comprising a matrix with isotropic and linear kinematic hardening, Int. J. Solids Struct., Volume 121 (2017), pp. 174-190

[29] N. Lahellec; P. Suquet Effective response and field statistics in elasto-plastic and elasto-viscoplastic composites under radial and non-radial loadings, Int. J. Plast., Volume 42 (2013), pp. 1-30

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