Complementary mixed finite element formulations for elastoplasticity. (English) Zbl 0687.73064

Summary: A global formulation of the principle of maximum plastic dissipation is systematically exploited to construct complementary mixed finite element formulations for elastoplasticity. Completely general return mapping integration algorithms are obtained as Euler-Lagrange equations of a temporally discretized Lagrangian functional. The flow rule is no longer enforced point-wise, but rather at the element level in a weak fashion that couples all Gauss points within an element. The resulting algorithms can be linearized exactly in closed-form for an arbitrary functional form of the yield condition and hardening law. The present approach enables one to extend successful superconvergent quadrilaterals to elastoplastic analysis. Numerical simulations involving plane-stress elastoplasticity are presented.


74S05 Finite element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
65K10 Numerical optimization and variational techniques
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
Full Text: DOI


[1] Wilkins, M.L., Calculation of elastic-plastic flow, ()
[2] Krieg, R.D.; Krieg, D.B., Accuracies of numerical solution methods for the elastic-perfectly plastic model, ASME J. pressure vessel tech., 99, (1977)
[3] Schreyer, H.L.; Kulak, R.L.; Kramer, J.M., Accurate numerical solutions for elastic-plastic models, ASME J. pressure vessel tech., 101, 226-334, (1979)
[4] Sandler, I.S.; Rubin, D., An algorithm and a modular subroutine for the cap model, Internat. J. numer. and analyt. methods geomech., 3, 173-186, (1979) · Zbl 0393.73002
[5] Ortiz, M.; Popov, E.P., Accuracy and stability of integration algorithms for elastoplastic constitutive equations, Internat. J. numer. methods engrg., 21, 1561-1576, (1985) · Zbl 0585.73057
[6] Simo, J.C.; Taylor, R.L., Consistent tangent operators for rate independent elasto-plasticity, Comput. methods appl. mech. engrg., 48, 101-118, (1985) · Zbl 0535.73025
[7] Simo, J.C.; Taylor, R.L., A return mapping algorithm for plane stress elastoplasticity, Internat. J. numer. engrg., 22, 3, 649-670, (1986) · Zbl 0585.73059
[8] Simo, J.C.; Ortiz, M., A unified approach to finite deformation elastoplasticity based on the use of hyperelastic constitutive equations, Comput. methods appl. mech. engrg., 49, 221-245, (1985) · Zbl 0566.73035
[9] J.C. Simo, J.W. Ju and R.L. Taylor, Softening response, completeness condition, and numerical algorithms for the cap model, Internat, J. Numer. Methods Engrg., to appear.
[10] H. Matthies, A decomposition method for the integration of the elastic-plastic rate problem, Internat. J. Numer. Methods Engrg., to appear. · Zbl 0669.73022
[11] Simo, J.C.; Hughes, T.J.R., General return mapping algorithms for rate independent plasticity, (), 221-231
[12] Simo, J.C.; Kennedy, J.G.; Govindjee, S., Non-smooth multisurface plasticity and viscoplasticity. loading/unloading conditions and numerical algorithms, Internat. J. numer. methods engrg., 26, 2161-2195, (1988) · Zbl 0661.73058
[13] Moreau, J.J., Evolution problem associated with a moving convex set in a Hilbert space, J. differential equations, 26, 347, (1977) · Zbl 0356.34067
[14] Nguyen, Q.S., On the elastic-plastic initial-boundary value problem and its numerical integration, Internat. J. numer. methods engrg., 11, 817-832, (1977) · Zbl 0366.73034
[15] Matthies, H., Problems in plasticity and their finite element approximation, ()
[16] Suquet, P., Existence et regularité de solutions des equations de al plasticité parfaite, ()
[17] Duvaut, G.; Lions, J.L., LES inequations en mechanique et en physique, (1972), Dunod Paris · Zbl 0298.73001
[18] Johnson, C., Existency theorems for plasticity problems, J. math. pures appl., 55, 431-444, (1976) · Zbl 0351.73049
[19] Johnson, C., On finite element methods for plasticity problems, Numer. math., 26, 79-84, (1976) · Zbl 0355.73035
[20] Johnson, C., A mixed finite element for plasticity, SIAM J. numer. anal., 14, 575-583, (1977) · Zbl 0374.73039
[21] Johnson, C., On plasticity with hardening, J. math. anal. appl., 62, 325-336, (1978) · Zbl 0373.73049
[22] Matthies, H.; Strang, G., The solution of nonlinear finite element equations, Internat. J. numer. methods engrg., 14, 11, 1613-1626, (1979) · Zbl 0419.65070
[23] Temam, R., Mathematical problems in plasticity, (1985), Gauthier-Villars Paris, (Translation of 1983 French original edition) · Zbl 0457.73017
[24] Hill, R., The mathematical theory of plasticity, (1950), Oxford University Press Oxford · Zbl 0041.10802
[25] Maier, G., Quadratic programming and theory of elastic-perfectly plastic structures, Meccanica, 3, 265-273, (1968) · Zbl 0181.53704
[26] Maier, G., A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes, Meccanica, 5, 54-56, (1970) · Zbl 0197.23303
[27] Moreau, J.J., Application of convex analysis to the treatment of elastoplasticity systems, () · Zbl 0337.73004
[28] Pian, T.; Sumihara, Rational approach for assumed stress finite elements, Internat. J. numer. methods engrg., 20, 9, 1685-1695, (1984) · Zbl 0544.73095
[29] Mandel, J., Contribution theorique al’étude de l’ecrouissage et des lois de l’écoulement plastique, (), 502-509
[30] Lubliner, J., A maximum-dissipation principle in generalized plasticity, Acta mech., 52, 225-237, (1984) · Zbl 0572.73043
[31] Lubliner, J., Normality rules in large-deformation plasticity, Mech. of mater., 5, 29-34, (1986)
[32] Simo, J.C., A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition: part I. continuum formulation, Comput. methods appl. mech. engrg., 66, 2, 199-219, (1986) · Zbl 0611.73057
[33] Simo, J.C., A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. part II. computational aspects, Comput. methods appl. mech. engrg., 68, 1, 1-31, (1988) · Zbl 0644.73043
[34] J.C. Simo and T. Honein, Variational Formulation, Discrete Convervation Laws and Path-Domain Independent Integrals for Elasto-Viscoplasticity, to appear. · Zbl 0739.73046
[35] Luenberger, D.G., Linear and nonlinear programming, (1984), Addison-Wesley Reading · Zbl 0241.90052
[36] Strang, G., Introduction to applied mathematics, (1986), Wellesley-Cambridge Press Wellesley · Zbl 0618.00015
[37] Pinsky, P., A finite element formulation for elastoplasticity based on a three field variational principle, Comput. methods appl. mech. engrg., 61, 1, 41-60, (1987) · Zbl 0591.73082
[38] Simo, J.C.; Hughes, T.J.R., On the variational foundations of assumed strain methods, J. appl. mech., 53, 1, 51-54, (1986) · Zbl 0592.73019
[39] Simo, J.C.; Kennedy, J.G.; Taylor, R.L., Complementary mixed finite element formulations for elastoplasticity, () · Zbl 0687.73064
[40] Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill London · Zbl 0435.73072
[41] Belytschko, T.; Bachrach, W.E., Efficient implementation of quadrilaterals with high coarse-mesh accuracy, Comput. methods appl. mech. engrg., 54, 279-301, (1985) · Zbl 0623.73074
[42] Zienkiewicz, O.C.; Taylor, R.L.; Nakazawa, S., The patch test for mixed methods, Internat. J. numer. methods engrg., 23, 1873-1883, (1986) · Zbl 0614.65115
[43] Simo, J.C.; Fox, D.D.; Rifai, M.S., On a geometrically exact shell model. part II. the linear theory; computational aspects, Comput. methods appl. mech. engrg., 73, 1, 53-92, (1989) · Zbl 0724.73138
[44] Mandel, J., Thermodynamics and plasticity, ()
[45] Strang, G.; Matthies, H.; Temam, R., Mathematical and computational methods in plasticity, ()
[46] Temam, R.; Strang, G., Functions of bounded deformation, Arch. rational mech. anal., 75, 7-21, (1980) · Zbl 0472.73031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.