Comptes Rendus
no. 8
Computational modeling of material forming processes / Simulation numérique des procédés de mise en forme
Strain localization analysis for planar polycrystals based on bifurcation theory
Mohamed Ben Bettaieb1, 2  ; Farid Abed-Meraim1, 2
1 Arts et Métiers ParisTech, Université de Lorraine, CNRS, LEM3, 57000 Metz, France
2 DAMAS, Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures, Université de Lorraine, France
Comptes Rendus. Mécanique, Volume 346 (2018) no. 8, pp. 647-664.

In the present paper, an efficient numerical tool is developed to investigate the ductility limit of polycrystalline aggregates under in-plane biaxial loading. These aggregates are assumed to be representative of very thin sheet metals (with typically few grains through the thickness). Therefore, the plane-stress assumption is naturally adopted to numerically predict the occurrence of strain localization. Furthermore, the initial crystallographic texture is assumed to be planar. Considering the latter assumptions, a two-dimensional single-crystal model is advantageously chosen to describe the mechanical behavior at the microscopic scale. The mechanical behavior of the planar polycrystalline aggregate is derived from that of single crystals by using the full-constraint Taylor scale-transition scheme. To predict the occurrence of localized necking, the developed multiscale model is coupled with bifurcation theory. As will be demonstrated through various numerical results, in the case of biaxial loading under plane-stress conditions, the planar single-crystal model provides the same predictions as those given by the more commonly used three-dimensional single-crystal model. Moreover, the use of the two-dimensional model instead of the three-dimensional one allows dividing the number of active slip systems by two and, hence, significantly reducing the CPU time required for the integration of the constitutive equations at the single-crystal scale. Furthermore, the planar polycrystal model seems to be more suitable to study the ductility of very thin sheet metals, as its use allows us to rigorously ensure the plane-stress state, which is not always the case when the fully three-dimensional polycrystalline model is employed. Consequently, the adoption of this planar formulation, instead of the three-dimensional one, allows us to simplify the computational aspects and, accordingly, to considerably reduce the CPU time required for the numerical predictions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crme.2018.06.006
Mots clés : Forming processes, Computational modeling, Localized necking, Crystal plasticity, Planar polycrystals, Rate-independent behavior
Mohamed Ben Bettaieb 1, 2 ; Farid Abed-Meraim 1, 2

1 Arts et Métiers ParisTech, Université de Lorraine, CNRS, LEM3, 57000 Metz, France
2 DAMAS, Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures, Université de Lorraine, France 
@article{CRMECA_2018__346_8_647_0,
     author = {Mohamed Ben Bettaieb and Farid Abed-Meraim},
     title = {Strain localization analysis for planar polycrystals based on bifurcation theory},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {647--664},
     publisher = {Elsevier},
     volume = {346},
     number = {8},
     year = {2018},
     doi = {10.1016/j.crme.2018.06.006},
     language = {en},
}
TY  - JOUR
AU  - Mohamed Ben Bettaieb
AU  - Farid Abed-Meraim
TI  - Strain localization analysis for planar polycrystals based on bifurcation theory
JO  - Comptes Rendus. Mécanique
PY  - 2018
SP  - 647
EP  - 664
VL  - 346
IS  - 8
PB  - Elsevier
DO  - 10.1016/j.crme.2018.06.006
LA  - en
ID  - CRMECA_2018__346_8_647_0
ER  - 
%0 Journal Article
%A Mohamed Ben Bettaieb
%A Farid Abed-Meraim
%T Strain localization analysis for planar polycrystals based on bifurcation theory
%J Comptes Rendus. Mécanique
%D 2018
%P 647-664
%V 346
%N 8
%I Elsevier
%R 10.1016/j.crme.2018.06.006
%G en
%F CRMECA_2018__346_8_647_0
Mohamed Ben Bettaieb; Farid Abed-Meraim. Strain localization analysis for planar polycrystals based on bifurcation theory. Comptes Rendus. Mécanique, Volume 346 (2018) no. 8, pp. 647-664. doi : 10.1016/j.crme.2018.06.006. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2018.06.006/

[1] H. Badreddine; C. Labergère; K. Saanouni Ductile damage prediction in sheet and bulk metal forming, C. R. Mecanique, Volume 344 (2016), pp. 296-318

[2] H. Badreddine; K. Saanouni; A. Dogui On non-associative anisotropic finite plasticity fully coupled with isotropic ductile damage for metal forming, Int. J. Plast., Volume 26 (2010), pp. 1541-1575

[3] K. Saanouni Damage Mechanics in Metal Forming, ISTE/Wiley, London, 2012 (ISBN: 978-1-84821-3487)

[4] A. Abdul-Latif; K. Saanouni Damaged anelastic behavior of FCC polycrystalline metals with micromechanical approach, Int. J. Damage Mech., Volume 3 (1994), pp. 237-259

[5] M. Boudifa; K. Saanouni; J.-L. Chaboche A micromechanical model for inelastic ductile damage prediction in polycrystalline metals for metal forming, Int. J. Mech. Sci., Volume 51 (2009), pp. 453-464

[6] S.P. Keeler; W.A. Backofen Plastic instability and fracture in sheets stretched over rigid punches, ASM Trans. Q., Volume 56 (1963), pp. 25-48

[7] K.W. Neale; E. Chater Limit strain predictions for strain-rate sensitive anisotropic sheets, Int. J. Mech. Sci., Volume 22 (1980), pp. 563-574

[8] M. Kuroda; V. Tvergaard Forming limit diagrams for anisotropic metal sheets with different yield criteria, Int. J. Solids Struct., Volume 37 (2000), pp. 5037-5059

[9] J.W. Hutchinson; K.W. Neale Sheet necking-III. Strain-rate effects (D.P. Koistinen; N.M. Wang, eds.), Mechanics of Sheet Metal Forming, Plenum Press, New York, London, 1978, pp. 269-285

[10] M. Ben Bettaieb; F. Abed-Meraim Strain rate effects on the enhancement of localized necking in substrate-supported metal layers, Int. J. Adv. Manuf. Technol., Volume 92 (2017), pp. 3461-3480

[11] L.Z. Mansouri; H. Chalal; F. Abed-Meraim Ductility limit prediction using a GTN damage model coupled with localization bifurcation analysis, Mech. Mater., Volume 76 (2014), pp. 64-92

[12] H. Chalal; F. Abed-Meraim Hardening effects on strain localization predictions in porous ductile materials using the bifurcation approach, Mech. Mater., Volume 91 (2015), pp. 152-166

[13] K. Inal; K.W. Neale; A. Aboutajeddine Forming limit comparisons for FCC and BCC sheets, Int. J. Plast., Volume 21 (2005), pp. 1255-1266

[14] Z. Marciniak; K. Kuczynski Limit strains in processes of stretch-forming sheet metal, Int. J. Mech. Sci., Volume 9 (1967), pp. 609-620

[15] J.W. Signorelli; M.A. Bertinetti; P.A. Turner Predictions of forming limit diagrams using a rate-dependent polycrystal self-consistent plasticity model, Int. J. Plast., Volume 25 (2009), pp. 1-25

[16] C. Schwindt; F. Schlosser; M.A. Bertinetti; J.W. Signorelli Experimental and visco-plastic self-consistent evaluation of forming limit diagrams for anisotropic sheet metals: an efficient and robust implementation of the M–K model, Int. J. Plast., Volume 73 (2015), pp. 62-99

[17] R. Knockaert; Y. Chastel; E. Massoni Forming limits predictions using rate-independent polycrystalline plasticity, Int. J. Plast., Volume 18 (2002), pp. 231-247

[18] H.K. Akpama; M. Ben Bettaieb; F. Abed-Meraim Localized necking predictions based on rate-independent self-consistent polycrystal plasticity: bifurcation analysis versus imperfection approach, Int. J. Plast., Volume 91 (2017), pp. 205-237

[19] G. Franz; F. Abed-Meraim; M. Berveiller Strain localization analysis for single crystals and polycrystals: towards microstructure-ductility linkage, Int. J. Plast., Volume 48 (2013), pp. 1-33

[20] S. Stören; J.R. Rice Localized necking in thin sheets, J. Mech. Phys. Solids, Volume 23 (1975), pp. 421-441

[21] K. Yoshida; M. Kuroda Comparison of bifurcation and imperfection analyses of localized necking in rate-independent polycrystalline sheets, Int. J. Solids Struct., Volume 49 (2012), pp. 2073-2084

[22] Y. Tadano; K. Yoshida; M. Kuroda Plastic flow localization analysis of heterogeneous materials using homogenization-based finite element method, Int. J. Mech. Sci., Volume 72 (2013), pp. 63-74

[23] E. Schmid; W. Boas Plasticity of Crystals, Chapman and Hall, London, 1935

[24] A. Chenaoui; F. Sidoroff; A. Hihi The texture evolution of a planar polycrystal, J. Mech. Phys. Solids, Volume 48 (2000), pp. 2559-2584

[25] J.W. Hutchinson; K.W. Neale; A. Needleman Sheet necking-I. Validity of plane stress assumptions of the long-wavelength approximation (D.P. Koistinen; N.M. Wang, eds.), Mechanics of Sheet Metal Forming, Plenum, 1978, pp. 111-126

[26] R.J. Asaro; J.R. Rice Strain localization in ductile single crystals, J. Mech. Phys. Solids, Volume 25 (1977), pp. 309-388

[27] R.J. Asaro Geometrical effects in the homogeneous deformation of ductile single crystals, Acta Metall., Volume 27 (1979), pp. 445-453

[28] T. Iwakuma; S. Nemat-Nasser Finite elastic–plastic deformation of polycrystalline metals, Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci., Volume 394 (1984), pp. 87-119

[29] J.R. Rice The localization of plastic deformation, 14th International Congress of Theoretical and Applied Mechanics, 1976, pp. 207-220

[30] Y.F. Dafalias Planar double-slip micromechanical model for polycrystal plasticity, J. Eng. Mech., Volume 119 (1993), pp. 1260-1284

[31] A. Kumar; P.R. Dawson The simulation of texture evolution with finite elements over orientation space II. Application to planar crystals, Comput. Methods Appl. Mech. Eng., Volume 130 (1996), pp. 247-261

[32] M.M. Rashid Texture evolution and plastic response of two-dimensional polycrystals, J. Mech. Phys. Solids, Volume 40 (1992), pp. 1009-1029

[33] V.C. Prantil; J.T. Jenkins; P.R. Dawson An analysis of texture and plastic spin for planar polycrystals, J. Mech. Phys. Solids, Volume 41 (1993), pp. 1357-1382

[34] A.H. Shalaby; K.S. Havner A general kinematical analysis of double slip, J. Mech. Phys. Solids, Volume 26 (1978), pp. 79-92

[35] J. Boukadia; F. Sidoroff Simple shear and torsion of a perfectly plastic single crystal in finite transformation, Arch. Mech., Volume 40 (1988), pp. 497-513

[36] J. Boukadia; A. Chenaoui; F. Sidoroff Simple shear in FCC single crystals at large deformations, Fontainebleau, France (C. Teodosiu; J.-L. Raphanel; F. Sidoroff, eds.) (1991), pp. 109-116

[37] J. Mandel Généralisation de la théorie de la plasticité de W.T. Koiter, Int. J. Solids Struct., Volume 1 (1965), pp. 273-295

[38] R. Hill; J.R. Rice Constitutive analysis of elastic–plastic crystals at arbitrary strain, J. Mech. Phys. Solids, Volume 20 (1972), pp. 401-413

[39] R.J. Asaro Crystal plasticity, J. Appl. Mech., Volume 50 (1983), pp. 921-934

[40] R. Hill Generalized constitutive relations for incremental deformation of metal crystals by multislip, J. Mech. Phys. Solids, Volume 14 (1966), pp. 95-102

[41] R.I. Borja; J.R. Wren Discrete micromechanics of elastoplastic crystals, Int. J. Numer. Methods Eng., Volume 36 (1993), pp. 3815-3840

[42] M. Ben Bettaieb; O. Débordes; A. Dogui; L. Duchêne; C. Keller On the numerical integration of rate independent single crystal behavior at large strain, Int. J. Plast., Volume 32–33 (2012), pp. 184-217

[43] H.K. Akpama; M. Ben Bettaieb; F. Abed-Meraim Numerical integration of rate-independent BCC single crystal plasticity models: comparative study of two classes of numerical algorithms, Int. J. Numer. Methods Eng., Volume 108 (2016), pp. 363-422

[44] L. Anand; M. Kothari A computational procedure for rate-independent crystal plasticity, J. Mech. Phys. Solids, Volume 44 (1996), pp. 525-558

[45] A. Fischer A special Newton-type optimization method, Optimization, Volume 24 (1992), pp. 269-284

[46] A. Fischer Solution of monotone complementarity problems with locally Lipschitzian functions, Math. Program., Volume 76 (1997), pp. 513-532

[47] S. Elbououni; S. Bourgeois; O. Débordes; A. Dogui Simulation by periodic homogenization of the behavior of a polycrystalline material in large elastoplastic transformations, J. Phys. IV, Volume 105 (2003), pp. 123-130

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Ductile damage prediction in sheet and bulk metal forming

Houssem Badreddine; Carl Labergère; Khemais Saanouni

C. R. Méca (2016)

Interface controlled plastic flow modelled by strain gradient plasticity theory

Thomas Pardoen; Thierry J. Massart

C. R. Méca (2012)

A phenomenological model for microstructural evolution during plastic flow

John L. Bassani; Haizhen Pan

C. R. Méca (2012)