Return Mapping Algorithms (RMAs) for Two-Yield Surface Thermoviscoplastic Models Using the Consistent Tangent Operator
-
Grégory Antoni
,
Frédéric Lebon
Abstract
The return mapping algorithms (RMAs) presented here are designed for use with pressure-dependent thermoviscoplastic constitutive models involving irreversible effects associated with solid–solid phase transformations. During the volume solid–solid phase transformations occurring under mechanical loads, an “anomalous” plasticity, the so-called “TRansformation Induced Plasticity” (TRIP), is generated at much lower stress levels than those related to the yield stress of the material in the context of the classical plasticity. TRIP mechanisms are superimposed on the classical plasticity which is liable to occur in the case of metallic materials. Based on a non-standard generalized material framework, two different models are presented in which an “associative” plastic flow is introduced in the context of classical plasticity and a “non-associative” flow rule in the context of TRIP-like plasticity. In this paper, a complete algorithmic treatment of these two rate-dependent constitutive models is therefore proposed with the associated consistent tangent operator for dealing the quasi-surface irreversible solid–solid transformations which can appear in metal alloys during specific thermomechanical solicitations. The predictive abilities of the presented numerical procedure for modelling this kind of the irreversible solid–solid transformations involving two plasticity processes are tested and assessed by performing a two-dimensional finite-element analysis on some numerical examples.
References
[1] Q. S. Nguyen, On the elastic-plastic initial-boundary value problem and its numerical integration, Int. J. Numer. Methods Eng. 11 (1977), 817–832.10.1002/nme.1620110505Search in Google Scholar
[2] J. C. Simo, R. L. Taylor, Consistent tangent operators for rate-independent elastoplasticity, Comput. Meth. Appl. Mech. Eng. 48 (1985), 101–118.10.1016/0045-7825(85)90070-2Search in Google Scholar
[3] J. C. Simo, R. L. Taylor, A return mapping algorithm for plane stress elastoplasticity, Int. J. Numer. Methods Eng. 22 (1986), 649–670.10.1002/nme.1620220310Search in Google Scholar
[4] J. C. Simo, T. R. J. Hughes, Computational Inelasticity, Springer Verlag, New York, 1997.Search in Google Scholar
[5] M. Ortiz, E. P. Popov, Accuracy and stability of integration algorithms for elastoplastic constitutive relations, Int. J. Numer. Meth. Eng. 21 (1985), 1561–1576.10.1002/nme.1620210902Search in Google Scholar
[6] M. Ortiz, J. C. Simo, An analysis of a new class of integration algorithms for elastoplastic constitutive relations, Int. J. Numer. Meth. Eng. 23 (1986), 353–366.10.1002/nme.1620230303Search in Google Scholar
[7] M. Bonnet, S. Mukherjee, Implicit BEM formulations for usual and sensitivity problems in elasto-plasticity using the consistent tangent operator concept, Int. J. Solids Struct. 33 (1996), 1395–1416.10.1016/0020-7683(95)00279-0Search in Google Scholar
[8] H. Poon, S. Mukherjee, M. Bonnet, Numerical implementation of a CTO-base implicit approach for the BEM solution of usual and sensitivity problems in elasto-plasticity, Eng. Anal. Boundary Elem. 22 (1998), 257–269.10.1016/S0955-7997(98)00030-7Search in Google Scholar
[9] C. A. Vidal, R. B. Haber, Design sensitivity analysis for rate-independent elastoplasticity, Comput. Meth. Appl. Mech. Eng. 107 (1993), 393–431.10.1016/0045-7825(93)90074-8Search in Google Scholar
[10] N. Aravas, On the numerical integration of a class of pressure-dependent plasticity models, Int. J. Numer. Meth. Eng. 24 (1987), 1395–1416.10.1002/nme.1620240713Search in Google Scholar
[11] Z. L. Zhang, On the accuracies of numerical implementation algorithms for Gurson-based pressure-dependent elastoplastic constitutive models, Comput. Meth. Appl. Mech. Eng. 121 (1995), 15–28.10.1016/0045-7825(94)00706-SSearch in Google Scholar
[12] Z. L. Zhang, Explicit consistent tangent moduli with a return mapping algorithm for pressure-dependent elastoplasticity models, Comput. Meth. Appl. Mech. Eng. 121 (1995), 29–44.10.1016/0045-7825(94)00707-TSearch in Google Scholar
[13] J. Clausen, L. Damkilde, L. Andersen, An efficient return algorithm for non-associated plasticity with linear yield criteria in principal stress space, Comput. Struct. 85 (2007), 1795–1807.10.1016/j.compstruc.2007.04.002Search in Google Scholar
[14] M. Kleiber, Computational coupled non-associative thermoplasticity, Comput. Meth. Appl. Mech. Eng. 90 (1991), 943–967.10.1016/0045-7825(91)90192-9Search in Google Scholar
[15] J. C. Ju, Consistent tangent moduli for a class viscoplasticity, J. Eng. Mech.} 116 (1990), 1764–1779.10.1061/(ASCE)0733-9399(1990)116:8(1764)Search in Google Scholar
[16] A. Eleöd, F. Oucherif, J. Devecz, Y. Berthier, Conception of numerical and experimental tools for study of the Tribological Transformation of Surface (TTS), Proceedings of Lubrication at the Frontier: The Role of the Interface and Surface Layers in the Thin Film and Boundary Regime 673–682, 1999.10.1016/S0167-8922(99)80087-XSearch in Google Scholar
[17] G. Antoni, T. Désoyer, F. Lebon, A combined thermo-mechanical model for Tribological Surface Transformations, Mech. Mater. 49 (2012), 92–99.10.1016/j.mechmat.2011.12.005Search in Google Scholar
[18] G. Baumann, H. J. Fecht, S. Liebelt, Formation of white-etching layers on rail treads, Wear 191 (1996), 133–140.10.1016/0043-1648(95)06733-7Search in Google Scholar
[19] W. Österle, H. Rooch, A. Pyzalla, L. Wang, Investigation of white etching layers on rails by optical microscopy, electron microscopy, X-ray and synchrotron X-ray diffraction, Mater. Sci. Eng.: A 303 (2001), 150–157.10.1016/S0921-5093(00)01842-6Search in Google Scholar
[20] G. Antoni, T. Désoyer, F. Lebon, A thermo-mechanical modelling of the Tribological Transformations of Surface, C.R. Mec. 337 (2009), 653–658.10.1016/j.crme.2009.09.003Search in Google Scholar
[21] A. Boudiaf, L. Taleb, M. E. A. Belouchrani, Effects of the austenite grain size on transformation plasticity in a 35 NCD 16 steel, Int. J. Microstruct. Mater. Prop. 5 (2010), 338–353.10.1504/IJMMP.2010.037611Search in Google Scholar
[22] M. Dalgic, A. Irretier, H-W Zoch, G. Lowisch, Transformation plasticity at different phase transformation of a through hardening bearing steel, Int. J. Microstruct. Mater. Prop. 3 (2008), 49–64.10.1504/IJMMP.2008.016943Search in Google Scholar
[23] G. W. Greenwood, R. H. Johnson, The deformation of metals under small stresses during phase transformations, Proc. R. Soc. London, Ser. A: Math. Phys. Sci. 293 (1965), 403–422.10.1098/rspa.1965.0029Search in Google Scholar
[24] T. Inoue, Mechanism of transformation plasticity and the unified constitutive equation for transformation-thermo-mechanical plasticity with some applications’, Int. J. Microstruct. Mater. Prop. 5 (2010), 319–327.10.1504/IJMMP.2010.037607Search in Google Scholar
[25] R. Quey, F. Barbe, H. Hoang, L. Taleb, Effect of the random spatial distribution of nuclei on the transformation plasticity in diffusively transforming steel, Int. J. Microstruct. Mater. Prop. 5 (2010), 354–364.10.1504/IJMMP.2010.037612Search in Google Scholar
[26] J. B. Leblond, J. Devaux, J. C. Devaux, Mathematical modelling of transformation plasticity in steels – Case of ideal-plastic phases. Int. J. Plast. 5 (1989), 551–571.10.1016/0749-6419(89)90001-6Search in Google Scholar
[27] J. B. Leblond, J. Devaux, J. C. Devaux, Mathematical modelling of transformation plasticity in steels – Coupling with strain hardening phenomena, Int. J. Plast. 5 (1989), 573–591.10.1016/0749-6419(89)90002-8Search in Google Scholar
[28] L. Taleb, F. Sidoroff, A micromechanical modeling of the Greenwood-Johnson mechanism in transformation induced plasticity, Int. J. Plast. 19 (2003), 1821–1842.10.1016/S0749-6419(03)00020-2Search in Google Scholar
[29] J. C. Videau, G. Cailletaud, A. Pineau, Modélisation des effets mécaniques des transformations de phases pour le calcul de structures, J. de Physique IV, Colloque C3, supplément au Journal de Physique III 4 (1994), 227–232.10.1051/jp4:1994331Search in Google Scholar
[30] J. C. Videau, G. Cailletaud, A. Pineau, Experimental study of the transformation induced plasticity in a Cr-Ni-Mo-Al-Ti steel, Journal de Physique IV, Colloque C1, supplément au Journal de Physique III, 6 (1996), 465–474.10.1051/jp4:1996145Search in Google Scholar
[31] G. Antoni, T. Désoyer, F. Lebon, Tribological Transformations of Surface: a thermo-mechanical modelling, Proceedings The Tenth International Conference on Computational Structures Technology, CD-ROM, 14p, 2010.Search in Google Scholar
[32] G. Antoni, Transformation Tribologique de Surface: une approche thermo-mécanique, PhD thesis}, Université de Provence, France, 2010.Search in Google Scholar
[33] J. Besson, G. Cailletaud, J. L. Chaboche, S. Forest ’M&’ecanique non lin&’eaire des mat&’eriaux’, Hermès Science Publications, France, 2001.Search in Google Scholar
[34] J. Lemaitre, J. L. Chaboche, Mechanics of Solid Materials, Cambridge University Press, United Kingdom, 1990.10.1017/CBO9781139167970Search in Google Scholar
[35] J. Garrigues, Fondements de la m&’ecanique des milieux continus Hermes (2007).Search in Google Scholar
[36] J. C. Simo, S. Govindjee, Exact closed-form solution of the return mapping algorithm in plane stress elasto-viscoplasticity, Eng. Comput. 5 (1988), 254–25810.1108/eb023744Search in Google Scholar
[37] J. C. Simo, S. Govindjee, B-Stability and symmetry preserving return mapping algorithms in plasticity and viscoplasticity, Int. J. Numer. Methods Eng. 31 (1991), 151–176.10.1002/nme.1620310109Search in Google Scholar
[38] J. C. Simo, C. Miehe, Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation, Comput. Meth. Appl. Mech. Eng. 98 (1992), 41–104.10.1016/0045-7825(92)90170-OSearch in Google Scholar
[39] M. Bonnet, A. Frangi, Analyse des solides d&’eformables par la m&’ethode des &’el&’ements finis, Editions de l’&’ecole Polytechnique Palaiseau, France, 2007.10.1080/17797179.2007.9737308Search in Google Scholar
[40] M. Abbas, ’Algorithme non lin&’eaire quasi-statique’, Code\_Aster Technical Note, R50301, France, 2013.Search in Google Scholar
[41] R. Halama, Z. Poruba, Tangent modulus in numerical integration of constitutive relations and its influence on convergence of N-R method, Appl. Comput. Mech. 3 (2009), 27–38.Search in Google Scholar
[42] R. Blaheta, O. Jakl, J. Stary, Displacement decomposition and parallelisation of the PCG method for elasticity problems, Int. J. Comput. Sci. Eng. 1 (2/3/4) (2005), 183–191.10.1504/IJCSE.2005.009702Search in Google Scholar
[43] I. Ficza, M. Vaverka;, M. Hartl, Numerical solution of contact pressure in lubricated non-smooth point contact using convolution algorithms, Int. J. Comput. Sci. Eng. 9 (5/6) (2014), 457–464.10.1504/IJCSE.2014.064530Search in Google Scholar
[44] H. Busing, J. Willkomm, C. H. Bischof, C. Clauser, Using exact Jacobians in an implicit Newton method for solving multiphase flow in porous media, Int. J. Comput. Sci. Eng. 9 (5/6) (2014), 499–508.10.1504/IJCSE.2014.064535Search in Google Scholar
[45] M. R. Gosz, Finite Element Method: Applications in Solids, Structures, and Heat Transfer, CRC Press, 2005.Search in Google Scholar
[46] G. H. Paulino, Y. Liu, Implicit consistent and continuum tangent operators in elastoplastic boundary element formulations, Comput. Meth. Appl. Mech. Eng. 190 (2001), 2157–2179.10.1016/S0045-7825(00)00228-0Search in Google Scholar
[47] G. Widlak, Radial Return method applied in thick-walled cylinder analysis, J. Theor. Appl Mech. 48 (2010), 381–395.Search in Google Scholar
[48] Nukala, P. K. V. V., A return mapping algorithm for cyclic viscoplastic constitutive models. Comput Meth. Appl. Mech. Eng. 195 (2006), 148–178.10.1016/j.cma.2005.01.009Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Numerical Treatment of the Modified Burgers’ Equation via Backward Differentiation Formulas of Orders Two and Three
- Return Mapping Algorithms (RMAs) for Two-Yield Surface Thermoviscoplastic Models Using the Consistent Tangent Operator
- Synchronization of Multiple Mechanical Oscillators Under Noisy Measurements Signals and Mismatch Parameters
- Nonlinear Wave Modulation in Nanorods Using Nonlocal Elasticity Theory
- Asymptotic Behavior of the Fractional Order three Species Prey–Predator Model
- Continuous Dependence on Data for Solutions of Fractional Extended Fisher–Kolmogorov Equation
- L-stable Explicit Nonlinear Method with Constant and Variable Step-size Formulation for Solving Initial Value Problems
- Weakness and Mittag–Leffler Stability of Solutions for Time-Fractional Keller–Segel Models
- Stability Analysis of Multi-point Boundary Value Problem for Sequential Fractional Differential Equations with Non-instantaneous Impulses
- Existence and Uniqueness of Classical and Mild Solutions of Generalized Impulsive Evolution Equation
- A Collocation Method Based on Jacobi and Fractional Order Jacobi Basis Functions for Multi-Dimensional Distributed-Order Diffusion Equations
- Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients
Articles in the same Issue
- Frontmatter
- Numerical Treatment of the Modified Burgers’ Equation via Backward Differentiation Formulas of Orders Two and Three
- Return Mapping Algorithms (RMAs) for Two-Yield Surface Thermoviscoplastic Models Using the Consistent Tangent Operator
- Synchronization of Multiple Mechanical Oscillators Under Noisy Measurements Signals and Mismatch Parameters
- Nonlinear Wave Modulation in Nanorods Using Nonlocal Elasticity Theory
- Asymptotic Behavior of the Fractional Order three Species Prey–Predator Model
- Continuous Dependence on Data for Solutions of Fractional Extended Fisher–Kolmogorov Equation
- L-stable Explicit Nonlinear Method with Constant and Variable Step-size Formulation for Solving Initial Value Problems
- Weakness and Mittag–Leffler Stability of Solutions for Time-Fractional Keller–Segel Models
- Stability Analysis of Multi-point Boundary Value Problem for Sequential Fractional Differential Equations with Non-instantaneous Impulses
- Existence and Uniqueness of Classical and Mild Solutions of Generalized Impulsive Evolution Equation
- A Collocation Method Based on Jacobi and Fractional Order Jacobi Basis Functions for Multi-Dimensional Distributed-Order Diffusion Equations
- Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients
