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Formulation of implicit finite element methods for multiplicative finite deformation plasticity. (English) Zbl 0724.73221

Summary: Some constitutive and computational aspects of finite deformation plasticity are discussed. Attention is restricted to multiplicative theories of plasticity, in which the deformation gradients are assumed to be decomposable into elastic and plastic terms. It is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator nonsymmetric even for associated plastic flow. Both explicit (i.e. no equilibrium iteration) and implicit finite element formulations are considered. An assumed strain formulation is used to accommodate the near-incompressibility associated with fully developed isochoric plastic flow. As an example of explicit integration, the rate tangent modulus method is reviewed in some detail. An implicit scheme is derived for which the consistent tangents, resulting in quadratic convergence of the equilibrium iterations, can be written out in closed form for arbitrary material models. All the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly. Examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
74C20 Large-strain, rate-dependent theories of plasticity
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[1] Anand, Int. J. Plasticity 1 pp 213– (1985)
[2] Asaro, Adv. Appl. Mech. 23 pp 1– (1983)
[3] Asaro, J. Appl. Mech. ASME 50 pp 1– (1984)
[4] Asaro, J. Mech. Phys. Solids 25 pp 309– (1977)
[5] ’An overview of semidiscretization and time integration procedures’, in and (eds.), Computational Methods for Transient Analysis, Elsevier Science Publishers, Amsterdam, 1983, pp. 1-65.
[6] Belytschko, Comp. Methods Appl. Mech. Eng. 29 pp 313– (1981)
[7] Budiansky, Problems of Hydrodynamics and Continuum Mechanics 77 pp 77– (1969)
[8] Casey, J. Appl. Mech. ASME 47 pp 672– (1980)
[9] ’A missing link in the formulation and numerical implementation of finite deformation elastoplasticity’, in (ed.), Constitutive Equations: Macro and Computational Aspects, ASME, New York, 1984, pp. 25-40.
[10] Dafalias, J. Appl. Mech. ASME 52 pp 865– (1985)
[11] Dashner, J. Appl. Mech. ASME 53 pp 55– (1986)
[12] and , ’Nonlinear elasticity’, in Advances in Applied Mechanics, Vol. 4, 1956.
[13] Goudreau, Comp. Methods Appl. Mech. Eng. 33 pp 725– (1982)
[14] and , Large Elastic Deformations, Oxford University Press, London, 1970.
[15] Green, Arch. Rational Mech. Anal. 18 pp 251– (1965)
[16] Green, Int. J. Eng. Sci. 9 pp 1219– (1971)
[17] Hibbitt, Int. J. Solids Struct. 6 pp 1069– (1970)
[18] Hill, J. Mech. Phys. Solids 14 pp 95– (1966)
[19] Hill, J. Mech. Phys. Solids 15 pp 79– (1967)
[20] Hill, J. Mech. Phys. Solids 16 pp 229– (1968)
[21] Hill, Adv. Appl. Mech. 18 pp 1– (1978)
[22] Hill, J. Mech. Phys. Solids 30 pp 5– (1982)
[23] Hill, J. Mech. Phys. Solids 20 pp 401– (1972)
[24] Hill, SIAM, J. Appl. Math. 25 pp 448– (1973)
[25] and (eds.), Finite Elements in Plasticity: Theory and Practice, Pineridge Press, Swansea, Wales, 1980. · Zbl 0482.73051
[26] Hughes, Int. j. numer. methods eng. 15 pp 1413– (1980)
[27] and , ’Unconditionally stable algorithms for quasi-static elasto/visco-plastic finite element analysis’, Comp. Struct., 169-173 (1978). · Zbl 0365.73029
[28] Hughes, Int. j. numer. methods eng. 15 pp 1862– (1980)
[29] ’Finite strain analysis of elastic-plastic solids and structures’, in (ed.), Numerical Solution of Nonlinear Structural Problems, AMD- Vol. 6, ASME, New York, 1973, pp. 17-29.
[30] Kim, Comp. Methods Appl. Mech. Eng. 53 pp 277– (1985)
[31] Lee, J. Appl. Mech. ASME 36 pp 1– (1969) · Zbl 0179.55603 · doi:10.1115/1.3564580
[32] Lubarda, J. Appl. Mech. ASME 48 pp 35– (1981)
[33] and , Mathematical Foundations of Elasticity, Prentice-Hall, Englewood Cliffs, N.J., 1983.
[34] McMeekine, J. Mech. Phys. Solids 25 pp 357– (1977)
[35] McMeeking, Int. J. Solids Struct. 11 pp 601– (1975)
[36] Nagtegaal, Int. j. numer. methods eng. 17 pp 15– (1981)
[37] Nagtegaal, Comp. Methods Appl. Mech. Eng. 4 pp 153– (1974)
[38] Needleman, J. Mech. Phys. Solids 20 pp 111– (1972)
[39] ’Finite elements for finite strain plasticity problems’, in and (eds.), Plasticity of Metals at Finite Strain: Theory, Computation and Experiment, 1982, pp. 387-436.
[40] Needleman, Comp. Struct. 20 pp 247– (1985)
[41] and , ’Finite element analysis of localization in plasticity’, in and (eds.), Finite Elements–Special Problems in Solid Mechanics, Vol. 5, 1983, pp. 94-157.
[42] Needleman, J. Mech. Phys. Solids 32 pp 461– (1984)
[43] Nemat-Nasser, Int. J. Solids Struct. 18 pp 857– (1982)
[44] and , ’A tangent modulus method for rate dependent solids’, Comp. Struct., 875-887 (1984). · Zbl 0531.73057
[45] Rice, J. Appl. Mech. ASME 37 pp 728– (1970) · doi:10.1115/1.3408603
[46] ’Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms’, in (ed.), Constitutive Equations in Plasticity, M.I.T. Press, 1975.
[47] and , ’The role of large crack tip geometry changes in plane strain fracture’, in et al. (eds.), Inelastic Behavior of Solids, McGraw-Hill, New York, 1970, pp. 641-670.
[48] Rubinstein, Comp. Methods Appl. Mech. Eng. 36 pp 277– (1983)
[49] Simo, Comp. Methods Appl. Mech. Eng. 66 pp 199– (1988)
[50] Simo, Comp. Methods Appl. Mech. Eng. 49 pp 221– (1985)
[51] Simo, Comp. Methods Appl. Mech. Eng. 46 pp 201– (1984)
[52] Simo, Comp. Methods Appl. Mech. Eng. 48 pp 101– (1985)
[53] Simo, Comp. Methods Appl. Mech. Eng. 51 pp 177– (1985)
[54] ’A dynamic theory of dislocations and its applications to the theory of the elastic-plastic continuum’, in et al. (eds.), Fundamental Aspects of Dislocation Theory, Vol. II, National Bureau of Standards (U.S.) Special Publication, 1970, pp. 837-876.
[55] Teodosiu, Int. J. Eng. Sci. 14 pp 713– (1976)
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