Abstract
In this chapter, we review two recently proposed methodologies, based on crystal plasticity, for the prediction of microstructure–property relations in polycrystalline aggregates. The first, known as the second-order viscoplastic self-consistent (SC) method, is a mean-field theory, while the second, known as the fast Fourier transform (FFT)-based formulation, is a full-field method. The main equations and assumptions underlying both formulations are presented, using a unified notation and pointing out their similarities and differences. Concerning mean-field SC homogenization theories for the prediction of mechanical behavior of nonlinear viscoplastic polycrystals, we carry out detailed comparisons of the different linearization assumptions that can be found in the literature. Then, after validating the FFT-based full-field formulation by comparison with available analytical results, the effective behavior of model material systems predicted by means of different SC approaches are compared with ensemble averages of full-field solutions. These comparisons show that the predictions obtained by means of the second-order SC approach– which incorporates statistical information at grain level beyond first-order, through the second moments of the local field fluctuations inside the constituent grains– are in better agreement with the FFT-based full-field solutions. This is especially true in the cases of highly heterogeneous materials due to strong nonlinearity or single-crystal anisotropy. The second-order SC approach is next applied to the prediction of texture evolution of polycrystalline ice deformed in compression, a case that illustrates the flexibility of this formulation to handle problems involving materials with highly anisotropic local properties. Finally, a full three-dimensional implementation, the FFT-based formulation, is applied to study subgrain texture evolution in copper deformed in tension, with direct input and validation from orientation images. Measurements and simulations agree in that grains with initial orientation near<110>tend to develop higher misorientations. This behavior can be explained in terms of attraction toward the two stable orientations and grain interaction. Only models like the FFT-based formulation that account explicitly for interaction between individual grains are able to capture these effects.
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Acharya A, Bassani JL, Beaudoin A (2003) Geometrically necessary dislocations, hardening, and a simple gradient theory of crystal plasticity. Scr Mater 48: 167–172.
Barbe F, Decker L, Jeulin D, Cailletaud G (2001) Intergranular and intragranular behavior of polycrystalline aggregates. Part 1: F.E. model. Int J Plast 17: 513–536.
Becker R (1991) Analysis of texture evolution in channel die compression. 1. Effects of grain interaction. Acta Metall Mater 39: 1211–1230.
Becker R, Butler JF, Hu H, Lalli LA (1991). Analysis of an aluminum single-crystal with unstable initial orientation (001)[110] in channel die compression. Metall Trans A 22:45–58.
Berveiller M, Fassi-Fehri O, Hihi A (1987) The problem of 2 plastic and heterogeneous inclusions in an anisotropic medium. Int J Eng Sci 25: 691–709.
Bhattacharyya A, El-Danaf E, Kalidindi SR, Doherty RD (2001) Evolution of grain-scale microstructure during large strain simple compression of polycrystalline aluminum with quasi-columnar grains: OIM measurements and numerical simulations. Int J Plast 17: 861–883.
Bobeth M, Diener G (1987) Static elastic and thermoelastic field fluctuations in multiphase composites. J Mech Phys Solids 35:137–149.
Castelnau O, Duval P, Lebensohn RA, Canova GR (1996) Viscoplastic modelling of texture development in polycrystalline ice with a self-consistent approach: comparison with bound estimates. J Geophys Res B 101: 13851–13868.
Castelnau O, Blackman DK, Lebensohn RA, Ponte Castañeda P (2008) Micromechanical modelling of the viscoplastic behavior of olivine. J Geophys Res B 113: B09202.
Chen LQ (2004) Introduction to the phase-field method of microstructure evolution. In: Raabe D, Roters F, Barlat F, Chen LQ (eds). Continuum scale simulations of engineering materials. Wiley, Wenheim, pp.37–51.
Cheong KS, Busso EP (2004) Discrete dislocation density modelling of single phase FCC polycrystal aggregates. Acta Mater 52: 5665–5675.
Chin GY, Mammel WL, Dolan MT (1967) Taylor’s theory of texture for axisymmetric flow in body-centered cubic metals. Trans Met Soc AIME 239: 1854–1855.
deBotton G, Ponte Castañeda P (1995) Variational estimates for the creep-behavior of polycrystals. Proc R Soc Lond A 448: 121–142.
Delaire F, Raphanel JL, Rey C (2001) Plastic heterogeneities of a copper multicrystal deformed in uniaxial tension: experimental study and finite element simulations. Acta Mater 48:1075.
Delannay L, Logé RE, Chastel Y, Signorelli JW, Van Houtte P (2003) Measurement of in-grain orientation gradients by EBSD and comparison with finite element results. Adv Eng Mater 5: 597–600.
Delannay L, Jacques PJ, Kalidindi SR (2006) Finite element modeling of crystal plasticity with grains shaped as truncated octahedrons. Int J Plast 22: 1879–1898.
Diard O, Leclercq S, Rousselier G, Cailletaud G (2005) Evaluation of finite element based analysis of 3D multicrystalline aggregates plasticity Application to crystal plasticity model identification and the study of stress and strain fields near grain boundaries. Int J Plast 21:691.
Dillamore IL, Morris PL, Smith CJE, Hutchinson WB (1972) Transition bands and recrystallization in metals. Proc R Soc Lond A 329: 405–420.
Dillamore IL, Katoh H (1974) Mechanisms of recrystallization in cubic metals with particular reference to their orientation-dependence. Met Sci J 8:73–83.
Dykhne AM (1970) Conductivity of a two-dimensional two-phase system. Dokl Akad Nauk SSSR 59: 110–115.
Hershey AV (1954) The elasticity of an isotropic aggregate of anisotropic cubic crystals. J Appl Mech 21: 236–240.
Hill R (1965) Continuum micro-mechanics of elastoplastic polycrystals. J Mech Phys Solids 13:89.
Hutchinson JW (1976) Bounds and self-consistent estimates for creep of polycrystalline materials. Proc R Soc Lond A 348: 101–127.
Hu SY, Chen LQ (2004) Spinodal decomposition in a film with periodically distributed interfacial dislocations. Acta Mater 52: 3069–3074.
Idiart MI, Ponte Castañeda P (2007) Field statistics in nonlinear composites. I. Theory. Proc R Soc Lond A 463: 183–202.
Idiart MI, Moulinec H, Ponte Castaneda P, Suquet P (2006) Macroscopic behavior and field fluctuations in viscoplastic composites: second-order estimates versus full-field simulations. JMech Phys Solids 54: 1029–1063.
Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int JSolids Struct 40: 3647–3679.
Kreher W (1990) Residual-stresses and stored elastic energy of composites and polycrystals. JMech Phys Solids 38: 115–128.
Lahellec N, Michel JC, Moulinec H, Suquet P (2001) Analysis of inhomogeneous materials at large strains using fast Fourier transforms. In: Miehe C (ed). IUTAM symposium on computational mechanics of solids materials. Kluwer Academic, Dordretcht, pp.247–268.
Laws N (1973) On the thermostatics of composite materials. J Mech Phys Solids 21:9–17.
Lebensohn RA, Tomé CN (1993) A selfconsistent approach for the simulation of plastic deformation and texture development of polycrystals: application to Zirconium alloys. Acta Metall Mater 41: 2611–2624.
Lebensohn RA, Turner PA, Signorelli JW, Canova GR, Tomé CN (1998) Calculation of intergranular stresses based on a large strain viscoplastic selfconsistent polycyrstal model. Model Simul Mater Sci Eng 6: 447–465.
Lebensohn RA (2001) N-site modelling of a 3D viscoplastic polycrystal using fast Fourier transform. Acta Mater 49: 2723–2737.
Lebensohn RA, Liu Y, Ponte Castañeda P (2004a) On the accuracy of the self-consistent approximation for polycrystals: comparison with full-field numerical simulations. Acta Mater 52: 5347–5361.
Lebensohn RA, Liu Y, Ponte Castañeda P (2004b) Macroscopic properties and field fluctuations in model power-law polycrystals: full-field solutions versus self-consistent estimates. Proc R Soc Lond A 460: 1381–1405.
Lebensohn RA, Castelnau O, Brenner R, Gilormini P (2005) Study of the antiplane deformation of linear 2-D polycrystals with different microstructures. Int J Solids Struct 42: 5441–5449
Lebensohn RA, Tomé CN, Ponte Castañeda P (2007) Self-consistent modeling of the mechanical behavior of viscoplastic polycrystals incorporating intragranular field fluctuations. Phil Mag 87: 4287–4322.
Lebensohn RA, Brenner R, Castelnau O, Rollett AD (2008) Orientation image-based micromechanical modelling of subgrain texture evolution in polycrystalline copper. Acta Mater 56: 3914–3926.
Lequeu P, Gilormini P, Montheillet F, Bacroix B, Jonas JJ (1987) Yield surfaces for textured polycrystals. 1. Crystallographic approach. Acta Metall 35: 439–451.
Liu Y, Ponte Castañeda P (2004) Second-order theory for the effective behavior and field fluctuations in viscoplastic polycrystals. J Mech Phys Solids 52:467–495.
Lurie KA, Cherkaev AV (1984). G-closure of some particular sets of admissible material characteristics for the problem of bending of thin elastic plates. J Optim Theor Appl 42: 305–316.
Masson R, Bornert M, Suquet P, Zaoui A (2000) Affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals. J Mech Phys Solids 48: 1203–1227.
Michel JC, Moulinec H, Suquet P (2000) A computational method based on augmented Lagrangians and fast Fourier transforms for composites with high contrast. Comput Model Eng Sci 1:79–88.
Mika DP, Dawson PR (1998) Effects of grain interaction on deformation in polycrystals. Mater Sci Eng A 257:62–76.
Molinari A, Canova GR, Ahzi S (1987) Self consistent approach of the large deformation polycrystal viscoplasticity. Acta Metall 35: 2983–2994.
Moulinec H, Suquet P (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites. C R Acad Sci Paris II 318: 1417–1423.
Moulinec H, Suquet P (1998) Numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Eng 157:69–94.
Moulinec H, Suquet P (2003) Intraphase strain heterogeneity in nonlinear composites: a computational approach. Eur J Mech Solids 22:751–770.
Mura T (1987) Micromechanics of defects in solids. Martinus-Nijhoff Publishers, Dordrecht.
Musienko A, Tatschl A, Schmidegg K, Kolednik O, Pippan R, Cailletaud G (2007) Three-dimensional finite element simulation of a polycrystalline copper specimen. Acta Mater 55: 4121–4136.
Nebozhyn MV, Gilormini P, Ponte Castañeda P (2000) Variational self-consistent estimates for viscoplastic polycrystals with highly anisotropic grains. C R Acad Sci Paris IIb 328:11–17.
Parton VZ, Buryachenko VA (1990) Stress fluctuations in elastic composites. Sov Phys Dokl 35(2):191–193.
Ponte Castañeda P (1991) The effective mechanical properties of nonlinear isotropic composites. J Mech Phys Solids 39:45–71.
Ponte Castañeda P (1996) Exact second-order estimates for the effective mechanical properties of nonlinear composites. J Mech Phys Solids 44: 827–862.
Ponte Castañeda P (2002) Second-order homogenization estimates for nonlinear composites incorporating field fluctuations: I- theory. J Mech Phys Solids 50: 737–757.
Raabe D, Sachtleber M, Zhao Z, Roters F, Zaefferer S (2001) Micromechanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation. Acta Mater 49: 3433–3441.
Rollett AD, Manohar P (2004) The Monte Carlo method. In: Raabe D, Roters F, Barlat F, Chen LQ (eds) Continuum scale simulations of engineering materials. Wiley, Wenheim, pp.77–111
Sachs G (1928). On the derivation of a condition of flowing. Z Verein Deut Ing 72: 734–736.
Semiatin SL, Bieler TR (2001) The effect of alpha platelet thickness on plastic flow during hot working of Ti-6Al-4V with a transformed microstructure. Acta Mater 49: 3565–3573.
Sulsky D, Zhou SJ, Schreyer HL (1995) Application of a particle-in-cell method to solid mechanics. Comput Phys Commun 87: 236–252.
Taylor GI (1938) Plastic strain in metals. J Inst Met 62: 307–324.
Tomé CN, Canova GR, Kocks UF, Christodoulou N, Jonas JJ (1984) The relation between macroscopic and microscopic strain-hardening in fcc polycrystals. Acta Metall 32: 1637–1653.
Tomé CN, Lebensohn RA (2008). Manual for Code Viscoplastic Self-Consistent (version 7).
Walpole LJ (1969) On the overall elastic moduli of composite materials. J Mech Phys Solids 17: 235–251.
Wang YU, Jin YMM, Khachaturyan AG (2002) Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid. J Appl Phys 92: 1351–1360.
Wenk HR, Tomé CN (1999) Modeling dynamic recrystallization of olivine aggregates deformed in simple shear. J Geophys Res B 104: 25513–25527.
Willis JR (1981) Variational and related methods for the overall properties of composites. Adv Appl Mech 21:1–78.
Winther G, Margulies L, Schmidt S, Poulsen HF (2004) Lattice rotations of individual bulk grains Part II: correlation with initial orientation and model comparison. Acta Mater 52: 2863–2872.
Acknowledgments
We wish to thank our colleagues Carlos Tomé (LANL, Los Alamos, USA), Tony Rollett (CMU, Pittsburgh, USA), Pierre Gilormini (ENSAM, Paris, France), and Pierre Suquet (LMA, Marseille, France) for fruitful discussions.
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Lebensohn, R.A., Castañeda, P.P., Brenner, R., Castelnau, O. (2011). Full-Field vs. Homogenization Methods to Predict Microstructure–Property Relations for Polycrystalline Materials. In: Ghosh, S., Dimiduk, D. (eds) Computational Methods for Microstructure-Property Relationships. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0643-4_11
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