Skip to main content
SpringerLink
Log in

A data-driven computational homogenization method based on neural networks for the nonlinear anisotropic electrical response of graphene/polymer nanocomposites

  • Original Paper
  • Published:
Computational Mechanics Aims and scope Submit manuscript

Abstract

In this paper, a data-driven-based computational homogenization method based on neural networks is proposed to describe the nonlinear electric conduction in random graphene-polymer nanocomposites. In the proposed technique, the nonlinear effective electric constitutive law is provided by a neural network surrogate model constructed through a learning phase on a set of RVE nonlinear computations. In contrast to multilevel (FE\(^2\)) methods where each integration point is associated with a full nonlinear RVE calculation, the nonlinear macroscopic electric field-electric flux relationship is efficiently evaluated by the surrogate neural network model, reducing drastically (by several order of magnitudes) the computational times in multilevel calculations. Several examples are presented, where the RVE contains aligned graphene sheets embedded in a polymer matrix. The nonlinear behavior is due to the modeling of the tunelling effect at the scale of graphene sheets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

Notes

  1. Synaptic weights may range in positive as well as negative values.

  2. Training of the ANN can be considered as an optimization problem, where the design variables are the synaptic weights.

References

  1. Feyel F (1999) Multiscale FE\(^2\) elastoviscoplastic analysis of composite structure. Comput Mater Sci 16(1–4):433–454

    Google Scholar 

  2. Feyel F, Chaboche JL (2000) FE\(^2\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Comput Methods Appl Mech Eng 183(3–4):309–330

    Article  MATH  Google Scholar 

  3. Smit R, Brekelmans W, Meijer H (1998) Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput Methods Appl Mech Eng 155(1–2):181–192

    Article  MATH  Google Scholar 

  4. Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171(3–4):387–418

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  6. Kouznetsova V, Brekelmans W, Baaijens F (2001) An approach to micro–macro modeling of heterogeneous materials. Comput Mech 27(1):37–48

    Article  MATH  Google Scholar 

  7. Ghosh S, Lee K, Raghavan P (2001) A multi-level computational model for multi-scale damage analysis in composite and porous materials. Int J Solids Struct 38(14):2335–2385

    Article  MATH  Google Scholar 

  8. Ibrahimbegović A, Markovič D (2003) Strong coupling methods in multi-phase and multi-scale modeling of inelastic behavior of heterogeneous structures. Comput Methods Appl Mech Eng 192(28–30):3089–3107

    Article  MATH  Google Scholar 

  9. Yvonnet J, He QC (2007) The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys 223(1):341–368

    Article  MathSciNet  MATH  Google Scholar 

  10. Ibrahimbegovic A, Papadrakakis M (2010) Multi-scale models and mathematical aspects in solid and fluid mechanics. Comput Methods Appl Mech Eng 199:1241–1241

    Article  MathSciNet  Google Scholar 

  11. Fullwood DT, Niezgoda SR, Adams BL, Kalidindi SR (2010) Microstructure sensitive design for performance optimization. Progr Mater Sci 55(6):477–562

    Article  Google Scholar 

  12. Papadopoulos V, Impraimakis M (2017) Multiscale modeling of carbon nanotube reinforced concrete. Compos Struct 182:251–260

    Article  Google Scholar 

  13. Geers MG, Kouznetsova VG, Brekelmans W (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175–2182

    Article  MATH  Google Scholar 

  14. Geers M, Yvonnet J (2016) Multiscale modeling of microstructure-property relations. MRS Bull 41(08):610–616

    Article  Google Scholar 

  15. Hernández J, Oliver J, Huespe AE, Caicedo M, Cante J (2014) High-performance model reduction techniques in computational multiscale homogenization. Comput Methods Appl Mech Eng 276:149–189

    Article  MathSciNet  MATH  Google Scholar 

  16. Zahr MJ, Avery P, Farhat C (2017) A multilevel projection-based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics. Int J Numer Methods Eng 112(8):855–881

    Article  MathSciNet  Google Scholar 

  17. Liu Z, Bessa M, Liu WK (2016) Self-consistent clustering analysis: an efficient multi-scale scheme for inelastic heterogeneous materials. Comput Methods Appl Mech Eng 306:319–341

    Article  MathSciNet  Google Scholar 

  18. Yvonnet J, Gonzalez D, He QC (2009) Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials. Comput Methods Appl Mech Eng 198(33–36):2723–2737

    Article  MATH  Google Scholar 

  19. Le BA, Yvonnet J, He QC (2015) Computational homogenization of nonlinear elastic materials using neural networks. Int J Numer Methods Eng 104(12):1061–1084

    Article  MathSciNet  MATH  Google Scholar 

  20. Bessa M, Bostanabad R, Liu Z, Hu A, Apley DW, Brinson C, Chen W, Liu WK (2017) A framework for data-driven analysis of materials under uncertainty: countering the curse of dimensionality. Comput Methods Appl Mech Eng 320:633–667

    Article  MathSciNet  Google Scholar 

  21. Fritzen F, Kunc O (2017) Two-stage data-driven homogenization for nonlinear solids using a reduced order model. Eur J Mech A Solids 69:201–220

    Article  MathSciNet  MATH  Google Scholar 

  22. Ghaboussi J, Garrett J Jr, Wu X (1991) Knowledge-based modeling of material behavior with neural networks. J Eng Mech 117(1):132–153

    Article  Google Scholar 

  23. Furukawa T, Yagawa G (1998) Implicit constitutive modelling for viscoplasticity using neural networks. Int J Numer Methods Eng 43(2):195–219

    Article  MATH  Google Scholar 

  24. Lefik M, Schrefler B (2003) Artificial neural network as an incremental non-linear constitutive model for a finite element code. Comput Methods Appl Mech Eng 192(28–30):3265–3283

    Article  MATH  Google Scholar 

  25. Yun GJ, Ghaboussi J, Elnashai AS (2008) A new neural network-based model for hysteretic behavior of materials. Int J Numer Methods Eng 73(4):447–469

    Article  MathSciNet  MATH  Google Scholar 

  26. Peherstorfer B, Willcox K (2015) Dynamic data-driven reduced-order models. Comput Methods Appl Mech Eng 291:21–41

    Article  MathSciNet  MATH  Google Scholar 

  27. Kirchdoerfer T, Ortiz M (2016) Data-driven computational mechanics. Comput Methods Appl Mech Eng 304:81–101

    Article  MathSciNet  MATH  Google Scholar 

  28. Versino D, Tonda A, Bronkhorst CA (2017) Data driven modeling of plastic deformation. Comput Methods Appl Mech Eng 318:981–1004

    Article  Google Scholar 

  29. Kirchdoerfer T, Ortiz M (2017) Data driven computing with noisy material data sets. Comput Methods Appl Mech Eng 326:622–641

    Article  MathSciNet  Google Scholar 

  30. Ibañez R, Abisset-Chavanne E, Aguado JV, Gonzalez D, Cueto E, Chinesta F (2018) A manifold learning approach to data-driven computational elasticity and inelasticity. Arch Comput Methods Eng 25(1):47–57

    Article  MathSciNet  MATH  Google Scholar 

  31. Nguyen LTK, Keip MA (2018) A data-driven approach to nonlinear elasticity. Comput Struct 194:97–115

    Article  Google Scholar 

  32. Waszczyszyn Z, Ziemiański L (2001) Neural networks in mechanics of structures and materials—new results and prospects of applications. Comput Struct 79(22–25):2261–2276

    Article  Google Scholar 

  33. Zhao H, Bai J (2015) Highly sensitive piezo-resistive graphite nanoplatelet-carbon nanotube hybrids/polydimethylsilicone composites with improved conductive network construction. ACS Appl Mater Interfaces 7(18):9652–9659

    Article  Google Scholar 

  34. Park OK, Jeevananda T, Kim NH, Kim S, Lee JH (2009) Effects of surface modification on the dispersion and electrical conductivity of carbon nanotube/polyaniline composites. Scr Mater 60(7):551–554

    Article  Google Scholar 

  35. Jia J, Sun X, Lin X, Shen X, Mai YW, Kim JK (2014) Exceptional electrical conductivity and fracture resistance of 3d interconnected graphene foam/epoxy composites. ACS Nano 8(6):5774–5783

    Article  Google Scholar 

  36. Wang T, Liang G, Yuan L, Gu A (2014) Unique hybridized graphene and its high dielectric constant composites with enhanced frequency stability, low dielectric loss and percolation threshold. Carbon 77:920–932

    Article  Google Scholar 

  37. Wei T, Luo G, Fan Z, Zheng Yan J, Yao C, Li W, Zhang C (2009) Preparation of graphene nanosheet/polymer composites using in situ reduction-extractive dispersion. Carbon 47(9):2296–2299

    Article  Google Scholar 

  38. Pang H, Chen T, Zhang G, Zeng B, Li ZM (2010) An electrically conducting polymer/graphene composite with a very low percolation threshold. Mater Lett 64(20):2226–2229

    Article  Google Scholar 

  39. Wang J, Yu S, Luo S, Chu B, Sun R, Wong CP (2016) Investigation of nonlinear I–V behavior of CNTs filled polymer composites. Mater Sci Eng B 206:55–60

    Article  Google Scholar 

  40. Martin-Gallego M, Bernal MM, Hernandez M, Verdejo R, López-Manchado MA (2013) Comparison of filler percolation and mechanical properties in graphene and carbon nanotubes filled epoxy nanocomposites. Eur Polym J 49(6):1347–1353

    Article  Google Scholar 

  41. Zeng X, Xu X, Kovalev PMSE, Baudot C, Mathews N, Zhao Y (2011) Characteristics of the electrical percolation in carbon nanotubes/polymer nanocomposites. J Phys Chem C 115(44):21685–21690

    Article  Google Scholar 

  42. Lu X, Yvonnet J, Detrez F, Bai J (2017) Multiscale modeling of nonlinear electric conductivity in graphene-reinforced nanocomposites taking into account tunnelling effect. J Comput Phys 337:116–131

    Article  MathSciNet  MATH  Google Scholar 

  43. Yvonnet J, He QC, Toulemonde C (2008) Numerical modelling of the effective conductivities of composites with arbitrarily shaped inclusions and highly conducting interface. Compos Sci Technol 68(13):2818–2825

    Article  Google Scholar 

  44. Lu X, Yvonnet J, Detrez F, Bai J (2018) Low electrical percolation thresholds and nonlinear effects in graphene-reinforced nanocomposites: a numerical analysis. J Compos Mater 52(20):2767–2775

    Article  Google Scholar 

  45. Simmons JG (1963) Electric tunnel effect between dissimilar electrodes separated by a thin insulating film. J Appl Phys 34(9):2581–2590

    Article  MATH  Google Scholar 

  46. Gu S, He QC (2011) Interfacial discontinuity relations for coupled multifield phenomena and their application to the modeling of thin interphases as imperfect interfaces. J Mech Phys Solids 59(7):1413–1426

    Article  MathSciNet  MATH  Google Scholar 

  47. Eglajs V, Audze P (1977) New approach to the design of multifactor experiments. Probl Dyn Strengths 35(1):104–107

    Google Scholar 

  48. Stankovich S, Dikin DA, Dommett GHB, Kohlhaas KM, Zimney EJ, Stach EA, Piner RD, Nguyen ST, Ruoff RS (2006) Graphene-based composite materials. Nature 442(7100):282–286

    Article  Google Scholar 

  49. Cybenko G (1989) Approximation by superpositions of a sigmoidal function. Math Control Signals Syst 2(4):303–314

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The financial support from Institut Universitaire de France (IUF) is gratefully acknowledged for JY.

Author information

Authors and Affiliations

  1. Laboratoire Modélisation et Simulation Multi Échelle, MSME UMR 8208 CNRS, Université Paris-Est, 5 bd Descartes, 77454, Marne-la-Vallée, France

    Xiaoxin Lu, Julien Yvonnet & Fabrice Detrez

  2. Department of Civil Engineering, Johns Hopkins University, Baltimore, USA

    Dimitris G. Giovanis

  3. Department of Civil Engineering, National Technical University of Athens, Athens, Greece

    Vissarion Papadopoulos

  4. Laboratoire de Mécanique des Sols, Structures et Matériaux, UMR 8579 CNRS, Université Paris Saclay, Paris, France

    Jinbo Bai

Authors
  1. Xiaoxin Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitris G. Giovanis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Julien Yvonnet
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Vissarion Papadopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Fabrice Detrez
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Jinbo Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julien Yvonnet.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lu, X., Giovanis, D.G., Yvonnet, J. et al. A data-driven computational homogenization method based on neural networks for the nonlinear anisotropic electrical response of graphene/polymer nanocomposites. Comput Mech 64, 307–321 (2019). https://doi.org/10.1007/s00466-018-1643-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-018-1643-0

Keywords

Search

Navigation