A non-incremental approach for elastoplastic plates basing on the Brezis-Ekeland-Nayroles principle. (English) Zbl 1481.74083

Summary: This paper is devoted to the numerical simulations of elastoplastic plates at small strains by using the non-incremental Brezis-Ekeland-Nayroles (BEN) principle. This space-time variational principle is based on the energy dissipation and transforms the boundary value problem into a null minimization one under constraints. Unlike the standard incremental procedure, the BEN principle characterizes the entire trajectory and delivers the mechanical fields at all time steps simultaneously. Two kinds of plates: the thin plate Love-Kirchhoff and the thick Reissner-Mindlin circular plates undergoing uniform pressure are studied respectively. The description of the dicretization of the principle using the mixed finite element method and the implementation are detailed. The obtained results are compared against analytical solutions provided in the literature and numerical predictions derived by the classical incremental procedure.


74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74K20 Plates
49J52 Nonsmooth analysis
Full Text: DOI


[1] Zienkiewicz, O. C.; Valliapan, S. N.; King, I. P., Elasto-plastic solutions of engineering problems. initial stress, finite element approach, Znt. J. Num. Meth. Eng., 1, 75-100 (1969) · Zbl 0247.73087
[2] Zienkiewicz, O. C., The Finite Element in Engineering Science (1971), McGraw-Hill: McGraw-Hill London · Zbl 0237.73071
[3] Oden, J. T., Finite Elements of Nonlinear Continua (1972), McGraw-Hill · Zbl 0235.73038
[4] Simo, J. C.; Hughes, T. J.R., Computational inelasticity (2000), Springer-Verlag New York Inc. · Zbl 0934.74003
[5] Belytschko, T.; Velebit, M., Finite element method for elastic plastic plates, Proc. ASCE, J. of Engng. Mech. Diu., EM 1, 227-242 (1972)
[6] Crisfield, M. A., Non-linear Finite Element Analysis of Solids and Structures, vol. 2 (1996), Wiley: Wiley Advanced Topics
[7] Ibrahimbegovic, A., Nonlinear Solid Mechanics: Theoretical Formulations and Finite Element Solution Methods (2009), Springer · Zbl 1168.74002
[8] Hadri, T., A mixed finite element method for the elastoplastic plate bending and buckling, Comput. Methods Appl. Mech. Eng., 50, 1-23 (1985) · Zbl 0552.73067
[9] Hadri, T., A model for the buckling and the stability of thin elastoplastic plates, J. Math. Anal. App, 74-115 (1985) · Zbl 0584.73050
[10] Turco, E.; Caracciolo, P., Elasto-plastic analysis of kirchhoff plates by high simplicity finite elements, Comput. Methods Appl. Mech. Engrg., 190, 691-706 (2000) · Zbl 1007.74080
[11] Jia, H. X.; Liu, X. L., Large increment method for elastic and elastoplastic analysis of plates, Finite Elem. Anal. Des., 88, 16-24 (2014)
[12] Owen, D. R.J.; Figueiras, J. A., Anisotropic elasto-plastic finite element analysis of thick and thin plates and shells, Int. J. Numer. Methods Eng., 19, 4, 521-539 (1983) · Zbl 0508.73066
[13] Karam, V. J.; Telles, J. C.F., The BEM applied to plate bending elastoplastic analysis using Reissner’s theory, Eng. Anal. Bound. Elem., 9, 4, 351-375 (1992)
[14] Moshaiov, A.; Vorus, W. S., Elasto-plastic plate bending analysis by a boundary element method with initial plastic moments, Int. J. Solids Struct., 22, 11, 1213-1229 (1986) · Zbl 0602.73083
[15] Oliveira, S. R.C.; Karam, V. J., Elastoplastic analysis of Reissner’s plates by the boundary element method, Eng. Anal. Boundary Elem., 64, 247-254 (2016) · Zbl 1403.74207
[16] Providakis, C. P.; Beskos, D. E., Dynamics analysis of elasto-plastic flexural plates by the D/BEM, Eng. Anal. Bound. Elem., 14, 1, 75-80 (1994)
[17] Providakis, C. P.; Beskos, D. E., Inelastic transient dynamics analysis of Reissner-Mindlin plates by the D/BEM, Int. J. Numer. Methods. Eng., 49, 383-397 (2000) · Zbl 0976.74077
[18] Fortiu, P. A.; Irschik, H.; Ziegler, F., Modal analysis of elastic-plastic plate vibrations by integral equations, Eng. Anal. Bound. Elem., 14, 1, 81-97 (1994)
[19] Ladeveze, P., Nouveaux algorithmes: cadre mcanique et d’veloppements, No. 57. Report (1985)
[20] Ladeveze, P., Sur une famille d’algorithmes en mécanique des structures. Comptes-rendus des séances de l’académie des sciences. série 2, mécanique-physique, chimie, sciences de l’univers, sciences de la terre, 300, 2, 41-44 (1985)
[21] Cognard, J.-Y.; Ladevèze, P., A large time increment approach for cyclic viscoplasticity, Int. J. Plast., 9, 2, 114-157 (1993) · Zbl 0772.73028
[22] Giacoma, A.; Dureisseix, D.; Gravouil, A.; Rochette, M., A multiscale large time increment/FAS algorithm with time-space model reduction for frictional contact problems, Int. J. Numer. Methods Eng., 97, 3, 207-230 (2014) · Zbl 1352.74205
[23] Cremonesi, M.; Néron, D.; Guidault, P.-A.; Ladevèze, P., A PGD-based homogenization technique for the resolution of nonlinear multiscale problems, Comput. Methods Appl. Mech. Eng., 267, 275-292 (2013) · Zbl 1286.74084
[24] Mielke, A.; Ortiz, M., A class of minimum principles for characterizing the trajectories and the relaxation of dissipative systems, ESAIM Control Optim. Calc. Var., 14, 494-516 (2008) · Zbl 1357.49043
[25] Davoli, E.; Stefanelli, U., Dynamic perfect plasticity as convex minimzation, SIAM J. Math. Anal., 51, 2, 672-730 (2019) · Zbl 1409.70011
[26] Cao, X.; Oueslati, A.; Nguyen, A. D.; de Saxcé, G., Numerical simulation of elastoplastic problems by Brezis-Ekeland-Nayroles non-incremental variational principle, Comput. Mech., 1-14 (2020) · Zbl 1462.74151
[27] Brézis, H.; Ekeland, I., Un principe variationnel associé á certaines equations paraboliques. Le cas independant du temps, CR Acad. Sci. Paris Sér. A, 282, 971-974 (1976) · Zbl 0332.49032
[28] Brézis, H.; Ekeland, I., Un principe variationnel associé á certaines equations paraboliques. Le cas dependant du temps, CR Acad. Sci. Paris Sér. A, 282, 1197-1198 (1976) · Zbl 0334.35040
[29] Nayroles, B., Deux théoremes de minimum pour certains systemes dissipatifs, C R Acad Sci Paris Sér AB, 282, 17, A1035-A1038 (1976)
[30] Buliga, M.; de Saxcé, G., A symplectic Brezis-Ekeland-Nayroles principle, Math. Mech. Solids, 22, 6, 1288-1302 (2017) · Zbl 1371.74051
[31] Love, A. E.H., XVI. the small free vibrations and deformation of a thin elastic shell, Philos. Trans. R. Soc. London (A.), 179, 491-546 (1888) · JFM 20.1075.01
[32] R.D. Mindlin, Influence of rotatory inertia and shear flexural motions of isotropic elastic plates, 1951. · Zbl 0044.40101
[33] Bartels, R. H.; Beatty, J. C.; Barsky, B. A., An Introduction to Splines for Use in Computer Graphics and Geometric Modeling (1995), Morgan Kaufmann
[34] Halphen, B.; Nguyen, Q. S., Sur les matériaux standards généralisés, J. Mécanique, 14, 1-37 (1975) · Zbl 0308.73017
[35] Cao, X.; Oueslati, A.; Shirafkan, N.; Bamer, F.; Markert, B.; de Saxcé, G., A non-incremental numerical method for dynamic elastoplastic problems by the symplectic Brezis-Ekeland-Nayroles principle, Comput. Methods Appl. Mech. Eng., 384, 2 (2021)
[36] Oñate, E., Structural Analysis with the Finite Element Method Linear Statics, Beams, Plates and Shells, vol. 2 (2013), Springer · Zbl 1279.74004
[37] Leonetti, L.; Magisano, D.; Madeo, A.; Garcea, G.; Kiendl, J.; Reali, A., A simplified Kirchhoff-Love large deformation model for elastic shells and its effective isogeometric formulation, Comput. Methods Appl. Mech. Eng., 354, 369-396 (2019) · Zbl 1441.74115
[38] Donnell, L. H., Beams, Plates, and Shells(Book), 1976, 563 (1976), McGraw-Hill Book Co.: McGraw-Hill Book Co. New York
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