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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.

MSC:
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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