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Fourier based methodology for simulating 2D-random shapes in heterogeneous materials

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Abstract

Gaining insights into the effects of microstructural details on materials behavior may be achieved by incorporating their attributes into numerical modeling. This requires us to make considerable efforts to feature heterogeneity morphology distributions and their spatial arrangement. This paper focuses on modeling the scatter observed in materials heterogeneity geometry. The proposed strategy is based on the development of a 1D-shape signature function representing the 2D-section of a given shape, on Fourier basis functions. The Fourier coefficients are then considered as random variables. This methodology has been applied to flax fibers which are gradually introduced into composite materials as a potential alternative to synthetic reinforcements. In this contribution, the influence of some underlying assumptions regarding the choice of one 1D-shape signature function, its discretization scheme and truncation level, and the best way of modeling the associated random variables is also investigated. Some configurations coming from the combination of these tuning parameters are found to be sufficiently relevant to render efficiently the morphometric factors of the observed fibers statistically speaking.

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References

  1. Abou-Chakra Guéry A, Cormery F, Shao JF, Kondo D (2010) A comparative micromechanical analysis of the effective properties of a geomaterial: effect of mineralogical compositions. Comput Geotech 37:585–593

    Article  Google Scholar 

  2. Ashmawy A, Sukumaran B, Hoang A (2003) Evaluating the influence of particle shape on liquefaction behavior using discrete element method. In: Proceedings of the thirteenth international offshore and polar engineering conference (ISOPE 2003), Honolulu

  3. Das N (2007) Modeling three-dimensional shape of sand grains using Discrete Element Method. Ph.D. thesis, University of South Florida

  4. Escoda J, Willot F, Jeulin D, Sanahuja J, Toulemonde C (2011) Estimation of local stresses and elastic properties of a mortar sample by fft computation of fields on a 3d image. Cem Concr Res 41:542–556

    Article  Google Scholar 

  5. Fakhimi A (2009) A hybrid discrete-finite element model for numerical simulation of geomaterials. Comput Geotech 36:386–395

    Article  Google Scholar 

  6. Geißndörfer M, Liebscher A, Proppe C, Redenbach C, Schwarzer D (2014) Stochastic multiscale modeling of metal foams. Probab Eng Mech 37:132–137

    Article  Google Scholar 

  7. Gosh S, Nowak Z, Lee K (1997) Tessellation-based computational methods for the characterization and analysis of heterogeneous microstructures. Compos Sci Technol 57:1187–1210

    Article  Google Scholar 

  8. Heimbs S (2009) Virtual testing of sandwich core structures using dynamic finite element simulations. Comput Mater Sci 45:205–216

    Article  Google Scholar 

  9. Kang D, Yun T, Matthew Evans T (2014) Pore orientation of granular materials during biaxial compression. Comput Geotech 59:1–11

    Article  Google Scholar 

  10. Mattrand C, Béakou A, Charlet K (2014) Numerical modeling of the flax fiber morphology variability. Composites 63:10–20

    Article  Google Scholar 

  11. Mollon G, Zhao J (2012) Fourier–Voronoi-based generation of realistic samples for discrete modelling of granular materials. Granular Matter 14:621–638

    Article  Google Scholar 

  12. Okereke M, Akpoyomare A, Bingley M (2014) Virtual testing of advanced composites, cellular materials and biomaterials: a review. Composites 60:637–662

    Article  Google Scholar 

  13. Pennec F, Alzina A, Tessier-Doyen N, Naït-ali B, Mati-Baouche N, De Baynast H, S DS (2013) A combined finite-discrete element method for calculating the effective thermal conductivity of bio-aggregates based materials. Int J Heat Mass Transf 60:274–283

    Article  Google Scholar 

  14. Persson K (2000) Micromechanical modelling of wood and fibres properties. Ph.D. thesis, Lund University

  15. Romanov V, Lomov S, Swolfs Y, Orlova S, G L, V I (2013) Statistical analysis of real and simulated fibre arrangements in unidirectional composites. Compos Sci Technol 87:126–134

    Article  Google Scholar 

  16. Sallam A (2004) Studies on modeling angular soil particles using the discrete element method. Ph.D. thesis, University of South Florida, Tampa

  17. Savvas D, Stefanou G, Papadrakakis M, Deodatis G (2014) Homogenization of random heterogeneous media with inclusions of arbitrary shape modeled by xfem. Comput Mech 54:1221–1235

    Article  MathSciNet  MATH  Google Scholar 

  18. Singh H, Gokhale A, Mao Y, Spowart J (2006) Computer simulations of realistic microstructures of discontinuously reinforced aluminum alloy (DRA) composites. Acta Mater 54:2131–2143

    Article  MATH  Google Scholar 

  19. Smith S (1998) The scientist and engineer’s guide to digital signal processing. www.DSPguide.com

  20. Spahn J, Andrä H, Kabel M, Müller R (2014) A multiscale approach for modeling progressive damage of composite materials using fast fourier transforms. Comput Methods Appl Mech Eng 268:871–883

    Article  Google Scholar 

  21. Sreeranganathan A, Gokhale A, Young P (2010) Realistic micromechanical modeling of discontinuously reinforced composites. Comput Mater Sci 49:407–413

    Article  Google Scholar 

  22. Stefanou G, Nouy A, Clement A (2009) Identification of random shapes from images through polynomial chaos expansion of random level-set functions. Int J Numer Methods Eng 79(2):127–155

    Article  MathSciNet  Google Scholar 

  23. Thomason J, Carruthers J, Kelly J, Johnson G (2011) Fibre cross-section determination and variability in sisal and flax and its effects on fibre performance characterisation. Compos Sci Technol 71:1008–1015

    Article  Google Scholar 

  24. Timm F, Martinetz T (2010) Statistical Fourier descriptors for defect image classification. In: 2010 international conference on pattern recognition

  25. Xu H, Li Y, Brinson C, Chen W (2014) A descriptor-based design methodology for developing heterogeneous microstructural materials system. J Mech Design 136(5):051007

    Article  Google Scholar 

  26. Yadav R, Nishchal N, Gupta A, Rastogi V (2007) Retrieval and classification of shape-based objects using fourier, generic fourier, and wavelet-fourier descriptors technique: a comparative study. Opt Lasers Eng 45:695–708

    Article  Google Scholar 

  27. Yin X, Chen W, To A, McVeigh C, Liu WK (2008) Statistical volument element method for predicting microstructure-constitutive property relations. Comput Methods Appl Mech Eng 197:3516–3529

    Article  MathSciNet  Google Scholar 

  28. Zeman J (2003) Analysis of Composite Materials with Random Microstructure. Ph.D. thesis, Czech Technical University in Prague, Klokner Institute

  29. Zhang D, Lu G (2002) A Comparative Study of Fourier Descriptors for Shape Representation and Retrieval. In: ACCV2002: the 5th Asian conference on computer vision, Melbourne, 23–25 Jan 2002

  30. Zhang D, Lu G (2004) Review of shape representation and description techniques. Pattern Recognit 37:1–19

    Article  MATH  Google Scholar 

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Acknowledgments

The authors wish to thank Brigitte Gaillard-Martinie from the INRA of Theix for the sample preparation and the optical micrographs of flax elementary fibers and bundles.

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Authors and Affiliations

  1. Clermont Université, IFMA, Institut Pascal, UMR 6602 UBP/CNRS/IFMA, BP 10448, 63000, Clermont-Ferrand, France

    C. Mattrand, A. Béakou & K. Charlet

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  1. C. Mattrand
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  2. A. Béakou
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  3. K. Charlet
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Correspondence to C. Mattrand.

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Mattrand, C., Béakou, A. & Charlet, K. Fourier based methodology for simulating 2D-random shapes in heterogeneous materials. Comput Mech 56, 371–388 (2015). https://doi.org/10.1007/s00466-015-1176-8

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  • DOI: https://doi.org/10.1007/s00466-015-1176-8

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