Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. (English) Zbl 0554.73036

This paper focuses on the treatment of volume constraints which in the context of elasto-plasticity typically arise as a result of assuming volume-preserving plastic flow. Projection methods based on the modification of the discrete gradient operator B, often proposed on an ad-hoc basis, are systematically obtained in the variational context furnished by a three-field Hu-Washizu principle. The fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume-preserving and dilatational parts, without relying on rate forms of the weak form of momentum balance. This approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient, and enables one to exactly enforce the condition of volume-preserving plastic flow. Within the framework proposed in this paper, rate forms and incrementally objective algorithms are entirely bypassed.


74B99 Elastic materials
74C99 Plastic materials, materials of stress-rate and internal-variable type
74D99 Materials of strain-rate type and history type, other materials with memory (including elastic materials with viscous damping, various viscoelastic materials)
74S30 Other numerical methods in solid mechanics (MSC2010)
74B20 Nonlinear elasticity
49S05 Variational principles of physics


Full Text: DOI


[1] Argyris, J., An excursion through large rotations, Comput. meths. appl. mech. engrg., 32, 85-155, (1982) · Zbl 0505.73064
[2] Argyris, J.H.; Doltsinis, J.St., On the large strain inelastic analysis in natural formulation — part I. quasistatic problems, Comput. meths. appl. mech. engrg., 20, 213-252, (1979) · Zbl 0437.73065
[3] Argyris, J.H.; Doltsinis, J.St., On the large strain inelastic analysis on natural formulation—part II. dynamic problems, Comput. meths. appl. mech. engrg., 21, 91-126, (1980) · Zbl 0437.73067
[4] Argyris, J.H.; Doltsinis, J.St.; Pimenta, P.M.; Wüstenberg, H., Thermomechanical response of solids at high strains-natural approach, Comput. meths. appl. mech. engrg., 32, 3-57, (1982) · Zbl 0505.73062
[5] Arnold, V.I., Mathematical methods of classical mechanics, (1980), Springer New York
[6] Belytschko, T.; Ong, J.S.; Liu, W.K.; Kennedy, J.M., Hourglass control in linear and nonliner problems, Comput. meths. appl. mech. engrg., 43, 251-276, (1984) · Zbl 0522.73063
[7] Carey, J.; Oden, J.T., Texas series in computational mechanics, () · Zbl 0297.00017
[8] Carroll, M.M., Radial expansion of hollow spheres of elastic-plastic hardening material, (1985), preprint · Zbl 0576.73026
[9] Flory, P.J., Thermodynamic relations for high elastic materials, Trans. Faraday soc., 57, 829-838, (1961)
[10] M. Fortin, Old and new elements for incompressible flows, Rept. No. MPA21, Collection Mathematique, Université Laval, Quebec, Canada. · Zbl 0467.76030
[11] Fortin, M.; Glowinsky, R., Resolution numerique de problemes aux limites par des methodes de lagrangient augmente, ()
[12] J.O. Hallquist, NIKE2D: An implicit, finite deformation, finite element code for analyzing the static and dynamic response of two dimensional solids, Rept. UCRL-52678, Lawrence Livermore National Laboratory, University of California, Livermore.
[13] Herrmann, L.R., Elasticity equations for incompressible and nearly incompressible materials by a variational theorem, Aiaa j., 3, 10, (1965)
[14] Hughes, T.J.R., Equivalence of finite elements for nearly incompressible elasticity, J. appl. mech., 181-183, (1977)
[15] Hughes, T.J.R., Generalization of selective integration procedures to anisotropic and nonlinear media, Internat. J. numer. meths. engrg., 15, 9, 1413-1418, (1980) · Zbl 0437.73053
[16] Hughes, T.J.R.; Liu, W.K.; Brooks, A., Review of finite element analysis of incompressible viscous flows by the penalty function formulation, J. comput. phys., 30, 1-60, (1979) · Zbl 0412.76023
[17] Hughes, T.J.R.; Malkus, D.S., A general penalty/mixed equivalence theorem for anisotropic incompressible elements, ()
[18] Hughes, T.J.R.; Winget, J., Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis, Internat. J. numer. meths. engrg., 15, 9, 1413-1418, (1980)
[19] Iwan, D.W.; Yoder, P.J., Computational aspects of strain-space plasticity, J. engrg. mech., ASCE, 109, 231-243, (1983)
[20] Key, S., A variational principle for incompressible and nearly incompressible anisotropic elasticity, Internat. J. solids structures, 5, 951-964, (1969) · Zbl 0175.22101
[21] Kikuchi, N.; Song, Y.J., Remarks on relations between penalty and mixed finite element methods for a class of variational inequalities, Internat. J. numer. meths. engrg., 15, 1557-1579, (1980) · Zbl 0458.65061
[22] Krieg, R.D.; Krieg, D.B., Accuracies of numerical solution methods for the elastic-perfectly plastic model, J. pressure vessel tech., ASME, 99, (1977)
[23] Lee, E.H., Elastic-plastic deformation at finite strains, J. appl. mech., 36, (1969) · Zbl 0179.55603
[24] Malkus, D.S., Finite elements with penalties in nonlinear elasticity, Internat J. numer. meths. engrg., 16, 121-136, (1980) · Zbl 0441.73103
[25] Malkus, D.S., Eigenproblems associated with the discrete condition for incompressible finite elements, Internat. J. engrg. sci., 19, 1299-1310, (1981) · Zbl 0457.73051
[26] Malkus, D.S.; Hughes, T.J.R., Mixed finite element methods-reduced and selective integration techniques: A unification of concepts, Comput. meths. appl. mech. engrg., 15, 63-81, (1978) · Zbl 0381.73075
[27] Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0545.73031
[28] Moss, W.C., On the computational significance of the strain space formulation of plasticity theory, Internat. J. numer. meths. engrg., 20, 1703-1709, (1984) · Zbl 0544.73047
[29] Nagtegaal, J.C., On the implementation of inelastic constitutive equations with special reference to large deformation problems, Comput. meths. appl. mech. engrg., 33, 469-484, (1982) · Zbl 0492.73077
[30] Nagtegaal, J.C.; Parks, D.M.; Rice, J.R., On numerically accurate finite element solutions in the fully plastic range, Comput. meths. appl. mech. engrg, 4, 153-177, (1974) · Zbl 0284.73048
[31] J.T. Oden, RIP-methods for Stokesian flows, TICOM Rept. 80-11, The University of Texas at Austin.
[32] Oden, J.T.; Kikuchi, N., Finite element methods for constrained problems in elasticity, Internat. J. numer. meths. engrg., 701-725, (1982) · Zbl 0486.73068
[33] Oden, J.T.; Kikuchi, N.; Song, Y.J., Penalty-finite element methods for the analysis of Stokesian flows, Comput. meths. appl. mech. engrg., 31, 297-329, (1982) · Zbl 0478.76041
[34] Ogden, R.W., Elastic deformations of rubber-like solids, (), 499-537
[35] Pinsky, P.M.; Ortiz, M.; Pister, K.S., Numerical integration of rate constitutive equations in finite deformation analysis, Comput. meths. appl. mech. engrg., 40, 137-158, (1983) · Zbl 0504.73057
[36] J. Pitkaranta and R. Stenberg, Error bounds for the approximation of the Stokes problem using bilinear/constant elements on irregular quadrilateral meshes. Rept. MAT-A222, Helsinki University of Technology. · Zbl 0598.76036
[37] Reddy, J.N., On penalty function methods in the finite element analysis of flow problems, Internat. J. numer. meths. fluids, 2, 151-171, (1982) · Zbl 0483.76008
[38] Reissner, E., On a variational principle for elastic displacements and pressure, J. appl. mech., 51, 444-445, (1983) · Zbl 0564.73019
[39] Rubinstein, R.; Atluri, S.N., Objectivity of incremental constitutive relations over finite time steps in computational finite deformation analyses, Comput. meths. appl. mech. engrg., 36, 277-290, (1983) · Zbl 0486.73081
[40] Schreyer, H.L.; Kulak, R.L.; Kramer, J.M., Accurate numerical solutions for elastic-plastic models, J. pressure vessel tech, ASME, 101, (1979)
[41] Schutz, B.F., Geometrical methods of mathematical physics, (1980), Cambridge University Press Cambridge · Zbl 0462.58001
[42] Simo, J.C.; Marsden, J.E., Stress tensors, Riemannian metrics and alternative representations of elasticity,, () · Zbl 0557.73003
[43] Simo, J.C.; Marsen, J.E., On the rotated stress tensor and the material version of the doyle-ericksen formula, Arch. rat. mech. anal., 86, 3, 213-231, (1984) · Zbl 0567.73003
[44] Simo, J.C.; Ortiz, M., A unified approach to finite deformation elastoplasticity based on the use of hyperelastic constitutive equations, Comput. meths. appl. mech. engrg., 49, 221-245, (1985) · Zbl 0566.73035
[45] Simo, J.C.; Pister, K.S., Remarks on rate constitutive equations for finite deformation problems: computational implications, Comput. meths. appl. mech. engrg., 46, 201-215, (1984) · Zbl 0525.73042
[46] Simo, J.C.; Taylor, R.L., Consistent tangent operators for rate independent elasto-plasticity, Comput. meths. appl. mech. engrg., 48, 101-118, (1985) · Zbl 0535.73025
[47] Simo, J.C.; Wriggers, P.; Taylor, R.L., A perturbed Lagrangian formulation for the finite element solution of contact problems, Comput. meths. appl. mech. engrg., 50, 163-180, (1985) · Zbl 0552.73097
[48] Y.J. Song, J.T. Oden and N. Kikuchi, Discrete LBB-conditions for RIP-finite element methods, TICOM Rept. 80-7, The University of Texas at Austin.
[49] Taylor, L.M.; Becker, E.B., Some computational aspects of large deformation, rate-dependent plasticity problems, Comput. meths. appl. mech. engrg., 41, 251-278, (1983) · Zbl 0509.73046
[50] Taylor, R.L.; Pister, K.S.; Hermann, L.R., A variational principle for incompressible and nearly-incompressible orthotropic elasticity, Internat. J. solids structures, 4, 875-883, (1968) · Zbl 0167.52804
[51] Truesdell, C.; Noll, W., The non-linear field theories of mechanics, () · Zbl 0779.73004
[52] Zienkiewicz, O.C.; Nakazawa, S., The penalty function method and its applications to the numerical solution of boundary value problems, () · Zbl 0533.65074
[53] Zienkiewicz, O.C.; Taylor, R.L.; Baynham, J.M.W., Mixed and irreducible formulations in finite element analysis, () · Zbl 0484.73056
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.