Consistent tangent operators for rate-independent elastoplasticity. (English) Zbl 0535.73025

It is shown that for problems involving rate constitutive equations, such as rate-independent elastoplasticity, the notion of consistency between the tangent (stiffness) operator and the integration algorithm employed in the solution of the incremental problem, plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton’s method. Within the framework of closest-point-projection algorithms, a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed. To wit, associative \(J_ 2\) flow rules with general nonlinear kinematic and isotropic hardening rules, as well as a simple class of non-associative flow rule are considered. The resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of Newton’s method, whereas use of the so-called elasto-plastic tangent in conjunction with a radial return integration algorithm, a procedure often employed, results in Newton type of algorithms with suboptimal rate of convergence. Application is made to a set of numerical examples which include saturation hardening laws of exponential type.


74S30 Other numerical methods in solid mechanics (MSC2010)
74S99 Numerical and other methods in solid mechanics
74C99 Plastic materials, materials of stress-rate and internal-variable type
49M15 Newton-type methods
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[1] Carey, G.F.; Oden, J.T., ()
[2] Goudreau, G.L.; Hallquist, J.O., Recent developments in large-scale finite element Lagrangian hydrocode technology, Comput. meths. appl. mech. engrg., 33, 725-757, (1982) · Zbl 0493.73072
[3] Hinton, E.; Owen, D.R.J., Finite elements in plasticity: theory and practice, (1980), Pineridge Press Swansea, Wales · Zbl 0482.73051
[4] Hughes, T.J.R., Numerical implementation of constitutive models: rate-independent deviatoric plasticity, ()
[5] Krieg, R.D.; Key, S.W., Implementation of a time dependent plasticity theory into structural computer programs, () · Zbl 0471.73077
[6] Krieg, R.D.; Krieg, D.B., Accuracies of numerical solution methods for the elastic-perfectly plastic model, J. pressure vessel technology, ASME, 99, 510-515, (1977)
[7] Lee, R.L.; Gresho, P.M.; Sani, R.L., Smoothing techniques for certain primitive variable solutions of the Navier-Stokes equations, Internat. J. numer. meths. engrg., 14, 12, 1785-1804, (1979) · Zbl 0426.76035
[8] Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0545.73031
[9] Matthies, H.; Strang, G., The solution of nonlinear finite element equations, Internat. J. numer. meths. engrg., 14, 1613-1626, (1979) · Zbl 0419.65070
[10] 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
[11] 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, 153-178, (1974) · Zbl 0284.73048
[12] Ortiz, M., Topics in constitutive theory for nonlinear solids, ()
[13] Schreyer, H.L.; Kulak, R.L.; Kramer, J.M., Accurate numerical solutions for elastic-plastic models, J. pressure vessel technology, ASME, 101, 226-234, (1979)
[14] Pinsky, P.M.; Pister, K.S.; Taylor, R.L., Formulation and numerical integration of elastoplastic and elasto-viscoplastic rate constitutive equations, () · Zbl 0548.73027
[15] Voce, E., Metalurgia, 51, 219, (1955)
[16] Wilkins, M.L., Calculation of elastic-plastic flow, ()
[17] Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill Berkshire, England · Zbl 0435.73072
[18] Zienkiewicz, O.C.; Taylor, R.L.; Baynham, J.M.W., Mixed and irreducible formulations in finite element analysis, () · Zbl 0484.73056
[19] Yaylor, R.L.; Zienkiewicz, O.C., Mixed finite element solution of fluid flow problems, ()
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