×

Eigenstrain based reduced order homogenization for polycrystalline materials. (English) Zbl 1423.74796

Summary: In this manuscript, an eigenstrain based reduced order homogenization method is developed for polycrystalline materials. A two-scale asymptotic analysis is used to decompose the original equations of polycrystal plasticity into micro- and macroscale problems. Eigenstrain based representation of the inelastic response field is employed to approximate the microscale boundary value problem using an approximation basis of much smaller order. The reduced order model takes into account the grain-to-grain interactions through influence functions that are numerically computed over the polycrystalline microstructure. The proposed approach is also endowed with a hierarchical model improvement capability that allows accurate representation of stress and deformation state within subgrains. The proposed approach was implemented and its performance was assessed against crystal plasticity finite element simulations. Numerical studies point to the capability to efficiently compute the mechanical response of the polycrystal RVEs with good accuracy and the ability to capture stress risers near grain boundaries.

MSC:

74Q05 Homogenization in equilibrium problems of solid mechanics
74E15 Crystalline structure

Software:

DREAM.3D; Neper
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Staroselsky, A.; Anand, L., Inelastic deformation of polycrystalline face centered cubic materials by slip and twinning, J. Mech. Phys. Solids, 46, 671-696, (1998) · Zbl 0971.74024
[2] Thamburaja, P.; Anand, L., Polycrystalline shape-memory materials: effect of crystallographic texture, J. Mech. Phys. Solids, 49, 709-737, (2001) · Zbl 1011.74049
[3] Quey, R.; Dawson, P. R.; Barbe, F., Large-scale 3d random polycrystals for the finite element method: generation, meshing and remeshing, Comput. Methods Appl. Mech. Engrg., 200, 1729-1745, (2011) · Zbl 1228.74093
[4] Groeber, M.; Jackson, M., Dream.3d: A digital representation environment for the analysis of microstructure in 3d, Integr. Mater. Manuf. Innov., 3, 5-19, (2014)
[5] Rice, J. R., Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity, J. Mech. Phys. Solids, 19, 433-455, (1971) · Zbl 0235.73002
[6] Meier, F.; Schwarz, C.; Werner, E., Crystal-plasticity based thermo-mechanical modeling of al-components in integrated circuits, Comput. Mater. Sci., 94, 0, 122-131, (2014)
[7] Anand, L.; Su, C., A theory for amorphous viscoplastic materials undergoing finite deformations, with application to metallic glasses, J. Mech. Phys. Solids, 53, 1362-1396, (2005) · Zbl 1120.74361
[8] Li, H.; Wu, C.; Yang, H., Crystal plasticity modeling of the dynamic recrystallization of two-phase titanium alloys during isothermal processing, Int. J. Plast., 51, 0, 271-291, (2013)
[9] Dahlberg, C. F.O.; Faleskog, J.; Niordson, C. F.; Legarth, B. N., A deformation mechanism map for polycrystals modeled using strain gradient plasticity and interfaces that slide and separate, Int. J. Plast., 43, 0, 177-195, (2013)
[10] Dunne, F. P.E.; Wilkinson, A. J.; Allen, R., Experimental and computational studies of low cycle fatigue crack nucleation in a polycrystal, Int. J. Plast., 23, 2, 273-295, (2007) · Zbl 1127.74315
[11] Peirce, D.; Asaro, R. J.; Needleman, A., An analysis of nonuniform and localized deformation in ductile single crystals, Acta Metall., 30, 1087-1119, (1982)
[12] Peirce, D.; Asaro, R. J.; Needleman, A., Material rate dependence and localized deformation in crystalline solids, Acta Metall., 31, 1951-1976, (1983)
[13] Tian, S.; Xie, J.; Zhou, X.; Qian, B.; Jun, J.; Yu, L.; Wang, W., Microstructure and creep behavior of FGH95 nickel-base superalloy, Mater. Sci. Eng. A, 528, 2076-2084, (2011)
[14] Knezevic, M.; Drach, B.; Ardeljan, M.; Beyerlein, I. J., Three dimensional predictions of grain scale plasticity and grain boundaries using crystal plasticity finite element models, Comput. Methods Appl. Mech. Engrg., 277, 239-259, (2014) · Zbl 06927988
[15] Bieler, T. R.; Eisenlohr, P.; Roters, F.; Kumar, D.; Mason, D. E.; Crimp, M. A.; Raabe, D., The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals, Int. J. Plast., 25, 1655-1683, (2009) · Zbl 1168.74047
[16] Thomas, J.; Groeber, M.; Ghosh, S., Image-based crystal plasticity FE framework for microstructure dependent properties of ti-6al-4V alloys, Mater. Sci. Eng. A, 553, 164-175, (2012)
[17] Guedes, J.; Kikuchi, N., Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Comput. Methods Appl. Mech. Engrg., 83, 2, 143-198, (1990) · Zbl 0737.73008
[18] Oskay, C., Variational multiscale enrichment for modeling coupled mechano-diffusion problems, Internat. J. Numer. Methods Engrg., 89, 6, 686-705, (2012) · Zbl 1242.74020
[19] Oskay, C., Variational multiscale enrichment method with mixed boundary conditions for modeling diffusion and deformation problems, Comput. Methods Appl. Mech. Engrg., 264, 0, 178-190, (2013) · Zbl 1286.74072
[20] Zhang, S.; Oskay, C., Variational multiscale enrichment method with mixed boundary conditions for elasto-viscoplastic problems, Comp. Mech., 55, 4, 771-787, (2015) · Zbl 1334.74088
[21] E, W.; Engquist, B., The heterognous multiscale methods, Commun. Math. Sci., 1, 1, 87-132, (2003) · Zbl 1093.35012
[22] Hou, T. Y.; Wu, X., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134, 169-189, (1997) · Zbl 0880.73065
[23] Aboudi, J., A continuum theory for fiber-reinforced elastic-viscoplastic composites, Int. Eng. Sci., 20, 5, 605-621, (1982) · Zbl 0493.73067
[24] Moulinec, H.; Suquet, P., A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Methods Appl. Mech. Engrg., 157, 69-94, (1998) · Zbl 0954.74079
[25] Michel, J. C.; Moulinec, H.; Suquet, P., A computational scheme for linear and non-linear composites with arbitrary phase contrast, Internat. J. Numer. Methods Engrg., 52, 1-2, 139-160, (2001)
[26] Michel, J. C.; Moulinec, H.; Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach, Comput. Methods Appl. Mech. Engrg., 172, 109-143, (1999) · Zbl 0964.74054
[27] Dvorak, G. J.; Benveniste, Y., On transformation strains and uniform fields in multiphase elastic media, Proc. R. Soc. Lond. Ser. A, 437, (1992), 291-10 · Zbl 0748.73003
[28] Michel, J. C.; Suquet, P., Nonuniform transformation field analysis, Int. J. Solids Struct., 40, 25, 6937-6955, (2003), Special issue in Honor of George J. Dvorak · Zbl 1057.74031
[29] Michel, J. C.; Suquet, P., Computational analysis of nonlinear composite structures using the nonuniform transformation field analysis, Comput. Methods Appl. Mech. Engrg., 193, 48-51, 5477-5502, (2004), Advances in Computational Plasticity · Zbl 1112.74471
[30] Fritzen, F.; Leuschner, M., Reduced basis hybrid computational homogenization based on a mixed incremental formulation, Comput. Methods Appl. Mech. Engrg., 260, 143-154, (2013) · Zbl 1286.74081
[31] Krysl, P.; Lall, S.; Marsden, J. E., Dimensional model reduction in non-linear finite element dynamics of solids and structures, Internat. J. Numer. Methods Engrg., 51, 479-504, (2001) · Zbl 1013.74071
[32] Yvonnet, J.; He, Q.-C., The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains, J. Comput. Phys., 223, 1, 341-368, (2007) · Zbl 1163.74048
[33] Hernandez, J. A.; Oliver, J.; Huespe, A. E.; Caicedo, M. A.; Cante, J. C., High-performance model reduction techniques in computational multiscale homogenization, Comput. Methods Appl. Mech. Engrg., 276, 149-189, (2014) · Zbl 1423.74785
[34] Yvonnet, J.; Gonzalez, D.; He, Q.-C., Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials, Comput. Methods Appl. Mech. Engrg., 198, 2723-2737, (2009) · Zbl 1228.74067
[35] Oskay, C.; Fish, J., Eigendeformation-based reduced order homogenization for failure analysis of heterogeneous materials, Comput. Methods Appl. Mech. Engrg., 196, 1216-1243, (2007) · Zbl 1173.74380
[36] Yuan, Z.; Fish, J., Multiple scale eigendeformation-based reduced order homogenization, Comput. Methods Appl. Mech. Engrg., 198, 2016-2038, (2009) · Zbl 1227.74051
[37] Crouch, R.; Oskay, C., Symmetric mesomechanical model for failure analysis of heterogeneous materials, Int. J. Multiscale. Com., 8, 5, 447-461, (2010)
[38] Fish, J.; Filonova, V.; Yuan, Z., Reduced order computational continua, Comput. Methods Appl. Mech. Engrg., 221-222, 104-116, (2012) · Zbl 1253.74085
[39] Marin, E. B.; Dawson, P. R., On modelling the elasto-viscoplastic response of metals using polycrystal plasticity, Comput. Methods Appl. Mech. Engrg., 165, 1-21, (1998) · Zbl 0952.74012
[40] Houtte, P. V., On the equivalence of the relaxed Taylor theory and the Bishop-Hill theory for partially constrained plastic deformation of crystals, Mater. Sci. Eng. A, 55, 1, 69-77, (1982)
[41] Houtte, P. V.; Li, S.; Seefeldt, M.; Delannay, L., Deformation texture prediction: from the Taylor model to the advanced lamel model, Int. J. Plast., 21, 3, 589-624, (2005) · Zbl 1154.74320
[42] S. Turteltaub, S. Yadegari, A. Suiker, Grain cluster mthod for multiscale simulations of multiphase steels, in: 11th World Congress on Computational Mechanicy, 2014. · Zbl 1329.74057
[43] Kroner, E., Berechnung der elastischen konstanten des vielkristalls aus den konstanten des einkristalls, Z. Phys, 151, 504-518, (1958), (in German)
[44] Lebensohn, R. A.; Tome, C. A., A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys, Acta Metall. Mater., 41, 2611-2624, (1993)
[45] Segurado, J.; Lebensohn, R. A.; LLorca, J.; Tome, C. N., Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements, Int. J. Plast., 28, 1, 124-140, (2012)
[46] Knezevic, M.; McCabe, R. J.; Lebensohn, R. A.; Tome, C. N.; Liu, C.; Lovato, M. L.; Mihaila, B., Integration of self-consistent polycrystal plasticity with dislocation density based hardening laws within an implicit finite element framework: application to low-symmetry metals, J. Mech. Phys. Solids, 61, 10, 2034-2046, (2013)
[47] Eshelby, J. D., The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. R. Soc. Lond. Ser. A, 241, 1226, 376-396, (1957) · Zbl 0079.39606
[48] Lebensohn, R. A., N-site modeling of a 3d viscoplastic polycrystal using fast Fourier transform, Acta Mater., 49, 2723-2737, (2001)
[49] Lebensohn, R. A.; Kanjarla, A. K.; Eisenlohr, P., An elasto-viscoplastic formulation based on fast Fourier transforms for the prediction of micromechanical fields in polycrystalline materials, Int. J. Plast., 32-33, 59-69, (2012)
[50] Eisenlohr, P.; Diehl, M.; Lebensohn, R. A.; Roters, F., A spectral method solution to crystal elasto-viscoplasticity at finite strains, Int. J. Plast., 46, 37-53, (2013)
[51] Knezevic, M.; Al-Harbi, H. F.; Kalidindi, S. R., Crystal plasticity simulations using discrete Fourier transforms, Acta Mater., 57, 6, 1777-1784, (2009)
[52] Zecevic, M.; McCabe, R. J.; Knezevic, M., Spectral database solutions to elasto-viscoplasticity within finite elements: application to a cobalt-based fcc superalloy, Int. J. Plast., 70, 151-165, (2015)
[53] Zecevic, M.; McCabe, R. J.; Knezevic, M., A new implementation of the spectral crystal plasticity framework in implicit finite elements, Mech. Mater., 84, 114-126, (2015)
[54] Knezevic, M.; Savage, D. J., A high-performance computational framework for fast crystal plasticity simulations, Comput. Mater. Sci., 83, 101-106, (2014)
[55] Roters, F.; Eisenlohr, P.; Hantcherli, L.; Tjahjanto, D. D.; Bieler, T. R.; Raabe, D., Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications, Acta Mater., 58, 4, 1152-1211, (2010)
[56] Knezevic, M.; Zecevic, M.; Beyerlein, I. J.; Bingert, J. F.; McCabe, R. J., Strain rate and temperature effects on the selection of primary and secondary slip and twinning systems in HCP zr, Acta Mater., 88, 55-73, (2015)
[57] Anand, L., Constitutive equations for the rate-dependent deformation of metals at elevated temperatures, J. Eng. Mater. Technol., 104, 1, 12-17, (1982)
[58] Bensoussan, A.; Lions, J.-L.; Papanicolaou, G., Asymptotic analysis for periodic structures, (1978), North-Holland Amsterdam · Zbl 0411.60078
[59] Sanchez-Palencia, E., Lecture Notes in Physics, vol. 127, (1980), Springer-Verlag Berlin
[60] Hui, T.; Oskay, C., A nonlocal homogenization model for wave dispersion in dissipative composite materials, Int. J. Solids Struct., 50, 1, 38-48, (2013)
[61] Hui, T.; Oskay, C., A high order homogenization model for transient dynamics of heterogeneous media including micro-inertia effects, Comput. Methods Appl. Mech. Engrg., 273, 181-203, (2014) · Zbl 1296.74091
[62] Hui, T.; Oskay, C., Laplace-domain, high-order homogenization for transient dynamic response of viscoelastic composites, Internat. J. Numer. Methods Engrg., (2015) · Zbl 1352.74271
[63] Feyel, F.; Chaboche, J. L., FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre sic/ti composite materials, Comput. Methods Appl. Mech. Engrg., 183, 309-330, (2000) · Zbl 0993.74062
[64] Feyel, F., A multilevel finite element method (FE2) to describe the response of highly non-linear structures using generalized continua, Comput. Methods Appl. Mech. Engrg., 192, 3233-3244, (2003) · Zbl 1054.74727
[65] Courant, R.; Hilbert, D., Methods of Mathematical Physics, vol. 1, (1991), Wily-VCH · Zbl 0729.00007
[66] Cerrone, A.; Tucker, J.; Stein, C.; Rollett, A.; Ingraffea, A., Micromechanical modeling of rené88DT: from characterization to simulation, (2012 Joint Conference of the Engineering Mechanics Institute and the 11th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, (2012))
[67] Nguyen, V. D.; Bechet, E.; Geuzaine, C.; Noels, L., Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation, Comput. Mater. Sci., 55, 390-406, (2012)
[68] Meissonnier, F. T.; Busso, E. P.; O’Dowd, N. P., Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains, Int. J. Plast., 17, 4, 601-640, (2001) · Zbl 1052.74054
[69] Mathur, K.; Dawson, P., On modeling the development of crystallographic texture in bulk forming processes, Int. J. Plast., 5, 1, 67-94, (1989)
[70] Groh, S.; Marin, E. B.; Horstemeyer, M. F.; Zbib, H. M., Multiscale modeling of the plasticity in an aluminum single crystal, Int. J. Plast., 25, 1456-1473, (2009) · Zbl 1165.74013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.