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Comparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization scheme

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Abstract

In numerical strategies developed for determining the effective macroscopic properties of heterogeneous media, the efficient and robust computation of macroscopic tangent moduli represents an essential step to achieve. Indeed, these tangent moduli are usually required in several numerical applications, such as the FE2 method and the prediction of the onset of material and structural instabilities in heterogeneous media by loss of ellipticity approaches. This paper presents a comparative study of three numerical techniques for the computation of such tangent moduli in the context of periodic homogenization: the perturbation technique, the condensation technique and the fluctuation technique. The practical implementations of these techniques within ABAQUS/Standard finite element (FE) code are especially underlined. These implementations are based on the development of a set of Python scripts, which are connected to the finite element computations to handle the computation of the tangent moduli. The extension of these techniques to mechanical problems exhibiting symmetry properties is also detailed in this contribution. The reliability, accuracy and ease of implementation of these techniques are evaluated through some typical numerical examples. It is shown from this numerical and technical study that the condensation method reveals to be the most reliable and efficient. Also, this paper provides valuable reference guidelines to ABAQUS/Standard users for the determination of the homogenized tangent moduli of linear or nonlinear heterogeneous materials, such as composites, polycrystalline aggregates and porous solids.

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Notes

  1. It is referred to ABAQUS/Standard formulation without user subroutines for nonlinear incremental analysis in this paper. ABAQUS/Explicit adopts different fundamental solving technique, as well as the theoretical formulation.

References

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  3. Ponte Castaneda P, Willis JR (1988) On the overall properties of nonlinearly viscous composites. Proc R Soc A Math Phys Eng Sci 416:217–244

    MathSciNet  MATH  Google Scholar 

  4. Suquet PM (1993) Overall potentials and extremal surfaces of power law or ideally plastic composites. J Mech Phys Solids 41:981–1002

    MathSciNet  MATH  Google Scholar 

  5. Teply JL, Dvorak GJ (1988) Bounds on overall instantaneous properties of elastic-plastic composites. J Mech Phys Solids 36:29–58

    MATH  Google Scholar 

  6. Willis JR (1981) Variational and related methods for the overall properties of composites. Adv Appl Mech 21:1–78

    MathSciNet  MATH  Google Scholar 

  7. Mura T, Shodja HM, Hirose Y (1988) Inclusion problems. Appl Mech Rev 41:118–127

    Google Scholar 

  8. Nemat-Nasser S, Hori M (1996) Micromechanics: overall properties of heterogeneous materials. J Appl Mech 63:561

    MATH  Google Scholar 

  9. Ghossein E, Lévesque M (2014) A comprehensive validation of analytical homogenization models: the case of ellipsoidal particles reinforced composites. Mech Mater 75:135–150

    Google Scholar 

  10. Mercier S, Molinari A, Berbenni S, Berveiller M (2012) Comparison of different homogenization approaches for elasticviscoplastic materials. Model Simul Mater Sci Eng 20:1–22

    Google Scholar 

  11. Moulinec H, Suquet P (1994) A fast numerical method for computing the linear and nonlinear mechanical properties of composites. Comptes Rendus Académie Sci Série II Mécanique, Phys Chim Astron 318:1417–1423

    MATH  Google Scholar 

  12. Moulinec H, Suquet P (1995) A FFT-based numerical method for computing the mechanical properties of composites from images of their microstructures. Solid Mech Appl 37:235–246

    Google Scholar 

  13. Moulinec H, Suquet P (2003) Comparison of FFT-based methods for computing the response of composites with highly contrasted mechanical properties. Phys B Condens Matter 338:58–60

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Michel JC, Moulinec H, Suquet P (1999) Effective properties of composite materials with periodic microstructure: a computational approach. Comput Methods Appl Mech Eng 172:109–143

    MathSciNet  MATH  Google Scholar 

  16. Miehe C (2002) Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation. Int J Numer Methods Eng 55:1285–1322

    MathSciNet  MATH  Google Scholar 

  17. Miehe C (2003) Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy. Comput Methods Appl Mech Eng 192:559–591

    MathSciNet  MATH  Google Scholar 

  18. 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 J Solids Struct 40:3647–3679

    MATH  Google Scholar 

  19. Ben Bettaieb M, Lemoine X, Duchêne L, Habraken AM (2011) On the numerical integration of an advanced Gurson model. Int J Numer Meth Eng 85:1049–1072

    MATH  Google Scholar 

  20. Feyel F (1999) Multiscale FE2 elastoviscoplastic analysis of composite structures. Comp Mater Sci 16:344–354

    Google Scholar 

  21. Lejeunes S, Bourgeois S (2011) Une Toolbox Abaqus pour le calcul de propriétés effectives de milieux hétérogènes. In: 10ème Colloq. Natl. en Calc. des Struct., pp 1–9

  22. Omairey SL, Dunning PD, Sriramula S (2018) Development of an ABAQUS plugin tool for periodic RVE homogenisation. Eng Comput 35:567–577

    Google Scholar 

  23. Tchalla A, Belouettar S, Makradi A, Zahrouni H (2013) An ABAQUS toolbox for multiscale finite element computation. Compos Part B Eng 52:323–333

    Google Scholar 

  24. Özdemir I, Brekelmans WAM, Geers MGD (2008) FE2 computational homogenization for the thermo-mechanical analysis of heterogeneous solids. Comput Methods Appl Mech Eng 198:602–613

    MATH  Google Scholar 

  25. Asada T, Ohno N (2007) Fully implicit formulation of elastoplastic homogenization problem for two-scale analysis. Int J Solids Struct 44:7261–7275

    MATH  Google Scholar 

  26. Terada K, Kikuchi N (2001) A class of general algorithms for multi-scale analyses of heterogeneous media. Comput Methods Appl Mech Eng 190:5427–5464

    MathSciNet  MATH  Google Scholar 

  27. Matsui K, Terada K, Yuge K (2004) Two-scale finite element analysis of heterogeneous solids with periodic microstructures. Comput Struct 82:593–606

    Google Scholar 

  28. Ladevèze P, Loiseau O, Dureisseix D (2001) A micro–macro and parallel computational strategy for highly heterogeneous structures. Int J Numer Methods Eng 52:121–138

    Google Scholar 

  29. Miehe C, Bayreuther CG (2007) On multiscale FE analyses of heterogeneous structures: from homogenization to multigrid solvers. Int J Numer Methods Eng 71:1135–1180

    MathSciNet  MATH  Google Scholar 

  30. Tadano Y, Yoshida K, Kuroda M (2013) Plastic flow localization analysis of heterogeneous materials using homogenization-based finite element method. Int J Mech Sci 72:63–74

    Google Scholar 

  31. Michel JC, Lopez-Pamies O, Ponte Castañeda P, Triantafyllidis N (2007) Microscopic and macroscopic instabilities in finitely strained porous elastomers. J Mech Phys Solids 55:900–938

    MathSciNet  MATH  Google Scholar 

  32. Miehe C, Schroder J, Becker M (2002) Computational homogenization analysis in finite elasticity. Material and structural instabilities on the micro- and macroscales of periodic composites and their interaction. Comput Methods Appl Mech Eng 191:4971–5005

    MATH  Google Scholar 

  33. Bruno D, Greco F, Lonetti P, Nevone Blasi P, Sgambitterra G (2010) An investigation on microscopic and macroscopic stability phenomena of composite solids with periodic microstructure. Int J Solids Struct 47:2806–2824

    MATH  Google Scholar 

  34. Temizer I, Wriggers P (2008) On the computation of the macroscopic tangent for multiscale volumetric homogenization problems. Comput Methods Appl Mech Eng 198:495–510

    MathSciNet  MATH  Google Scholar 

  35. Kiran R, Khandelwal K (2014) Complex step derivative approximation for numerical evaluation of tangent moduli. Comput Struct 140:1–13

    Google Scholar 

  36. Miehe C, Koch A (2002) Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Arch Appl Mech 72:300–317

    MATH  Google Scholar 

  37. Miehe C, Schröder J, Bayreuther C (2002) On the homogenization analysis of composite materials based on discretized fluctuations on the micro-structure. Acta Mech 155:1–16

    MATH  Google Scholar 

  38. Miehe C, Schotte J, Lambrecht M (2002) Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals. J Mech Phys Solids 50:2123–2167

    MathSciNet  MATH  Google Scholar 

  39. Ji W, Waas AM, Bazant ZP (2013) On the importance of work-conjugacy and objective stress rates in finite deformation incremental finite element analysis. J Appl Mech 80:041024

    Google Scholar 

  40. Ben Bettaieb M, Debordes O, Dogui A, Duchene L (2012) Averaging properties for periodic homogenization and large deformation. Int J Multiscale Comput Eng 10:281–293

    Google Scholar 

  41. Debordes O (1986) Homogenization computations in the elastic or plastic range; applications to unidirectional composite and perforated sheets. Computational Mechanics Publications, Springer, Atlanta, pp 453–458

    Google Scholar 

  42. Miehe C, Bayreuther CG (2006) Multilevel FEM for heterogeneous structures: from homogenization to multigrid solvers. In: Lecture notes in applied and computational mechanics, pp 361–397

  43. Geers MGD, Kouznetsova V, Matouš K, Yvonnet J (2017) Homogenization methods and multiscale modeling: nonlinear problems. In: Stein E, de Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, 2nd edn, vol 198, pp 1–34

  44. Salahouelhadj A, Abed-Meraim F, Chalal H, Balan T (2012) Application of the continuum shell finite element SHB8PS to sheet forming simulation using an extended large strain anisotropic elastic-plastic formulation. Arch Appl Mech 82:1269–1290

    MATH  Google Scholar 

  45. Léné F (1984) Contribution à l’étude des materiaux composites et de leurs endommagements. PhD thesis, Universite Paris VI

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Acknowledgements

The first author is grateful to the China Scholarship Council for providing him a PhD grant during the preparation of this work.

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Authors and Affiliations

  1. CNRS, Arts et Metiers Institute of Technology, LEM3, Université de Lorraine, 57070, Metz, France

    J. C. Zhu, M. Ben Bettaieb & F. Abed-Meraim

  2. DAMAS, Laboratory of Excellence on Design of Alloy Metals for Low-mAss Structures, Université de Lorraine, Metz, France

    M. Ben Bettaieb & F. Abed-Meraim

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  1. J. C. Zhu
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  2. M. Ben Bettaieb
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Correspondence to M. Ben Bettaieb.

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Appendices

Appendix A: Typical input file to output elementary stiffness matrices

In this example, the statements in red are required to be added to the input file. In this additional part, ‘Nel’ and ‘Frequency = n’ denote the total number of elements and the frequency of record of the elementary stiffness matrices (each ‘n’ increments).

figure d

Appendix B: Some sections of Python scripts

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Zhu, J.C., Bettaieb, M.B. & Abed-Meraim, F. Comparative study of three techniques for the computation of the macroscopic tangent moduli by periodic homogenization scheme. Engineering with Computers 38, 1365–1394 (2022). https://doi.org/10.1007/s00366-020-01091-y

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