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Efficient Multiscale FE-FFT-Based Modeling and Simulation of Macroscopic Deformation Processes with Non-linear Heterogeneous Microstructures

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Multiscale Modeling of Heterogeneous Structures

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 86))

Abstract

The purpose of this work is the prediction of micromechanical fields and the overall material behavior of heterogeneous materials using an efficient and robust two-scale FE-FFT-based computational approach. The macroscopic boundary value problem is solved using the finite element (FE) method. The constitutively dependent quantities such as the stress tensor are determined by the solution of the local boundary value problem. The latter is represented by a periodic unit cell attached to each macroscopic integration point. The local algorithmic formulation is based on fast Fourier transforms (FFT), fixed-point and Newton-Krylov subspace methods (e.g. conjugate gradients). The handshake between both scales is defined through the Hill-Mandel condition. In order to ensure accurate results for the local fields as well as feasible overall computation times, an efficient solution strategy for two-scale full-field simulations is employed. As an example, the local and effective mechanical behavior of ferrit-perlit annealed elasto-viscoplastic 42CrMo4 steel is studied for three-point-bending tests. For simplicity, attention is restricted to the geometrically linear case and quasi-static processes.

The original version of this chapter was revised: Author provided figure corrections have been incorporated. The erratum to this chapter is available at https://doi.org/10.1007/978-3-319-65463-8_19

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Notes

  1. 1.

    Note that the identity \(\varvec{\varepsilon }^{(i)}=\varvec{\varepsilon }_{\mathrm {M}}+\hat{\varvec{\Gamma }}^{(0)}*[\mathbb {C}^{(0)}\varvec{\varepsilon }^{(i)}]\) was used to arrive at (24), where \(\mathbf {f}*\mathbf {g}=\int _{-\infty }^{\infty }\mathbf {f}(\varvec{\xi })\,\mathbf {g}(\mathbf {x}-\varvec{\xi })\,d\varvec{\xi }\) denotes the convolution integral of two arbitrary fields \(\mathbf {f}\) and \(\mathbf {g}\).

References

  1. Brisard, S., Dormieux, L.: FFT-based methods for the mechanics of composites: a general variational framework. Comput. Mater. Sci 49(3), 663–671 (2010)

    Article  Google Scholar 

  2. Brisard, S., Dormieux, L.: Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites. Comput. Methods Appl. Mech. Eng. 217(220), 197–212 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Diard, O., Leclerq, S., Rousselier, G., Cailletaud, G.: 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, 691–722 (2005)

    Article  MATH  Google Scholar 

  4. 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)

    Google Scholar 

  5. Eyre, D.J., Milton, G.W.: A fast numerical scheme for computing the response of composites using grid refinement. Eur. Phys. J. Appl. Phys. 6, 41–47 (1999)

    Article  Google Scholar 

  6. Geers, M.G.D., Kouznetsova, V., Brekelmans, W.A.M.: Multi-scale computational homogenization: trends and challenges. J. Comput. Appl. Math. 234, 2175–2182 (2010)

    Article  MATH  Google Scholar 

  7. Gélébart, L., Mondon-Cancel, R.: Non-linear extension of FFT-based methods accelerated by conjugated gradients to evaluate the mechanical behavior of composite materials. Comput. Mater. Sci 77, 430–439 (2013)

    Article  Google Scholar 

  8. Hashin, Z., Shtrikman, H.: On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids 10, 335–342 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11, 127–140 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hashin, Z.: Analysis of composite materials—a survey. J. Appl. Mech. 36, 481–505 (1983)

    Article  MATH  Google Scholar 

  11. Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)

    Article  MATH  Google Scholar 

  12. Kabel, M., Böhlke, T., Schneider, M.: Efficient fixed point and Newton-Krylov solvers for FFT-based homogenization of elasticity at large deformations. Comput. Mech. 54, 1497–1514 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kabel, M., Merkert, D., Schneider, M.: Use of composite voxels in FFT-based homogenization. Comput. Methods Appl. Mech. Eng. 294, 168–188 (2015)

    Article  MathSciNet  Google Scholar 

  14. Kochmann, J., Rezaei Mianroodi, J., Wulfinghoff, S., Reese, S., Svendsen, B.: Two-scale, FE-FFT- and phase-field based computational modeling of bulk microstructure evolution and macroscopic material behavior. Comput. Methods Appl. Mech. Eng. 305, 89–110 (2016)

    Article  Google Scholar 

  15. Kochmann, J., Wulfinghoff, S., Ehle, L., Mayer, J., Svendsen, B., Reese, S.: Efficient and accurate two-scale FE-FFT-based prediction of the effective material behaviour of elasto-viscoplastic polycrystals. Comput. Mech., in press (2017). https://doi.org/10.1007/s00466-017-1476-2

  16. Michel, J.C., Moulinec, H., Suquet, P.: A computational scheme for linear and non-linear composites with arbitrary phase contrast. Int. J. Numer. Methods Eng. 52, 139–160 (2001)

    Article  Google Scholar 

  17. Michel, J.C., Moulinec, H., Suquet, P.: A computational method based on augmented Lagrangians and Fast Fourier Transforms for composites with high contrast. Comput. Model. Eng. Sci. 1, 79–88 (2000)

    MathSciNet  Google Scholar 

  18. Miehe, C., Schotte, J., Schröder, J.: Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Comput. Mater. Sci 16, 372–382 (1999)

    Article  Google Scholar 

  19. Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Comput. Methods Appl. Mech. Eng. 171, 387–418 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mika, D.P., Dawson, P.R.: Effects of grain interaction on deformation in polycrystals. Mater. Sci. Eng. A 257, 62–76 (1998)

    Google Scholar 

  21. Monchiet, V., Bonnet, G.: A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast. Int. J. Numer. Methods Eng. 89, 1419–1436 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Moulinec, H., Silva, F.: Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials. Int. J. Numer. Methods Eng. 97, 960–985 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Moulinec, H., Suquet, P.: A fast numerical method for computing the linear and nonlinear mechanical properties of composites. C.R. Acad. Sci. Ser. IIb: Mech. Phys. Chim. Astron. 318, 1417–1423 (1994)

    Google Scholar 

  24. Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructures. Comput. Methods Appl. Mech. Eng. 157(1), 69–94 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. Amsterdam (1993)

    Google Scholar 

  26. Castaneda, P.P., Suquet, P.: Advances in Applied Mechanics, vol. 34, pp. 171–302. Elsevier (1997)

    Google Scholar 

  27. Prakash, A., Lebensohn, R.A.: Simulations of micromechanical behavior of polycrystals: finite element versus Fast Fourier Transforms. Model. Simul. Mater. Sci. Eng. 17, 16pp (2009)

    Google Scholar 

  28. Raabe, D., Sachtleber, M., Zhao, Z., Roters, F.: Micromechanical and macromechanical effects in grain scale polycrystal plasticity experimentation and simulation. Acta Mater. 49, 3433–3441 (2001)

    Article  Google Scholar 

  29. Reese, S., Wriggers, P.: A stabilization technique to avoid hourglassing in finite elasticity. Int. J. Numer. Methods Eng. 48, 79–109 (2000)

    Article  MATH  Google Scholar 

  30. Roters, F., Eisenlohr, P., Hantcherli, L., Tjahjanto, D.D., Bieler, T.R., Raabe, D.: Overview on constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite element modeling: theory, experiments, applications. Acta Mater. 58, 1152–1211 (2010)

    Article  Google Scholar 

  31. Schneider, M., Ospald, F., Kabel, M.: FFT-based homogenization for microstructures discretized by linear hexahedral elements. Int. J. Numer. Methods Eng. (2016). https://doi.org/10.1002/nme.5336

  32. Schröder, J.: Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Instabilitäten. Universität Stuttgart, Habilitationsschrift (2000)

    Google Scholar 

  33. Schröder, J.: A numerical two-scale homogenization scheme: the \(FE^2\)-method. CISM Int. Centre Mech. Sci. 550, 1–64 (2014)

    Article  MATH  Google Scholar 

  34. Shanthraj, P., Eisenlohr, P., Diehl, M., Roters, F.: Numerically robust spectral methods for crystal plasticity simulations of heterogeneous materials. Int. J. Plast. 66, 31–45 (2015)

    Article  Google Scholar 

  35. Smit, R.J.M., Brekelmans, W.A.M., Meijer, H.E.H.: Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput. Methods Appl. Mech. Eng. 155, 181–192 (1998)

    Article  MATH  Google Scholar 

  36. Spahn, J., Andrae, H., Kabel, M., Müller, R.: A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Comput. Methods Appl. Mech. Eng. 268, 871–883 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Talbot, D.R.S., Willis, J.R.: Variational principles for inhomogeneous nonlinear media. Int. J. Appl. Math. 35, 39–54 (1985)

    MathSciNet  MATH  Google Scholar 

  38. Vinogradov, V., Milton, G.W.: An accelerated FFT algorithm for thermoelastic and non linear composites. Int. J. Numer. Methods Eng. 76, 1678–1695 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Willot, F.: Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields. C.R. Acad. Sci. Ser. IIb: Mech. Phys. Chim. Astron. 323(3), 232–245 (2014)

    Google Scholar 

  40. Wulfinghoff, S., Böhlke, T.: Equivalent plastic strain gradient crystal plasticity—enhanced power law subroutine. GAMM-Mitteilunge 36(2), 134–148 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wulfinghoff, S., Reese, S.: Efficient computational homogenization of simple elasto-plastic microstructures using a shear band approach. Comput. Methods Appl. Mech. Eng. 298, 350–372 (2016)

    Article  Google Scholar 

  42. Zeman, J., Vodrejc, J., Novak, J., Marek, I.: Accelerating a FFT-based solver for numerical homogenization of a periodic media by conjugate gradients. J. Comput. Phys. 229(21), 8065–8071 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zeman, J., de Geuss, T.W.J., Vondejc, J., Peerlings, R.H.J., Geers, M.G.D.: A finite element perspective on non-linear FFT-based micromechanical simulations. Int. J. Numer. Methods Eng. (2016). https://doi.org/10.1002/nme.5481

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Acknowledgements

Financial support of Subprojects M03 and C02 of the Transregional Collaborative Research Center SFB/TRR 136 by the German Science Foundation (DFG) is gratefully acknowledged.

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Authors and Affiliations

  1. Institute of Applied Mechanics, RWTH Aachen University, 52074, Aachen, Germany

    Julian Kochmann, Stephan Wulfinghoff & Stefanie Reese

  2. Central Facility for Electron Microscopy, RWTH Aachen University, 52074, Aachen, Germany

    Lisa Ehle & Joachim Mayer

  3. Material Mechanics, RWTH Aachen University, 52062, Aachen, Germany

    Bob Svendsen

  4. Microstructure Physics and Alloy Design, Max-Planck-Institut für Eisenforschung GmbH, 40237, Düsseldorf, Germany

    Bob Svendsen

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  1. Julian Kochmann
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  2. Lisa Ehle
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  5. Bob Svendsen
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Correspondence to Julian Kochmann .

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Editors and Affiliations

  1. University of Zagreb, Zagreb, Croatia

    Jurica Sorić

  2. Institute of Continuum Mechanics, Leibniz Universität Hannover, Hannover, Germany

    Peter Wriggers

  3. LMT Cachan, Cachan, France

    Olivier Allix

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Kochmann, J., Ehle, L., Wulfinghoff, S., Mayer, J., Svendsen, B., Reese, S. (2018). Efficient Multiscale FE-FFT-Based Modeling and Simulation of Macroscopic Deformation Processes with Non-linear Heterogeneous Microstructures. In: Sorić, J., Wriggers, P., Allix, O. (eds) Multiscale Modeling of Heterogeneous Structures. Lecture Notes in Applied and Computational Mechanics, vol 86. Springer, Cham. https://doi.org/10.1007/978-3-319-65463-8_7

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  • DOI: https://doi.org/10.1007/978-3-319-65463-8_7

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  • Online ISBN: 978-3-319-65463-8

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