Skip to main content
SpringerLink
Log in

Fast statistical homogenization procedure (FSHP) for particle random composites using virtual element method

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

Abstract

Mechanical behaviour of particle composite materials is growing of interest to engineering applications. A computational homogenization procedure in conjunction with a statistical approach have been successfully adopted for the definition of the representative volume element (RVE) size, that in random media is an unknown of the problem, and of the related equivalent elastic moduli. Drawback of such a statistical approach to homogenization is the high computational cost, which prevents the possibility to perform series of parametric analyses. In this work, we propose a so-called fast statistical homogenization procedure (FSHP) developed within an integrated framework that automates all the steps to perform. Furthermore within the FSHP, we adopt the numerical framework of the virtual element method for numerical simulations to reduce the computational burden. The computational strategies and the discretization adopted allow us to efficiently solve the series (hundreds) of simulations and to rapidly converge to the RVE size detection.

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.

Institutional subscriptions

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

Similar content being viewed by others

References

  1. Anthoine A (1995) Derivation of the in-plane elastic characteristics of masonry through homogenization theory. Int J Solids Struct 32(2):137–163

    Article  MATH  Google Scholar 

  2. Artioli E, Beirão Da Veiga L, Lovadina C, Sacco E (2017) Arbitrary order 2D virtual elements for polygonal meshes: Part I, elastic problem. Comput Mech 60(3):355–377

    Article  MathSciNet  MATH  Google Scholar 

  3. Beirão Da Veiga L, Brezzi F, Cangiani A, Manzini G, Marini LD, Russo A (2013) Basic principle of virtual element methods. Math Models Methods Appl Sci 23(01):199–214

    Article  MathSciNet  MATH  Google Scholar 

  4. Beirão Da Veiga L, Brezzi F, Marini L (2013) Virtual elements for linear elasticity problems. SIAM J Numer Anal 51(2):794–812

    Article  MathSciNet  MATH  Google Scholar 

  5. Beirão Da Veiga L, Brezzi F, Marini LD, Russo A (2014) The Hitchhiker’s guide to the virtual element method. Math Models Methods Appl Sci 24(08):1541–1573

    Article  MathSciNet  MATH  Google Scholar 

  6. Bouaoune L, Brunet Y, Moumen AE, Kanit T, Mazouz H (2016) Random versus periodic microstructures for elasticity of fibers reinforced composites. Compos Part B Eng 103:68–73

    Article  Google Scholar 

  7. Bouyge F, Jasiuk I, Ostoja-Starzewski M (2001) Micromechanically based couple-stress model of an elastic two-phase composite. Int J Solids Struct 38(10–13):1721–1735

    Article  MATH  Google Scholar 

  8. Capecchi D, Ruta G, Trovalusci P (2011) Voigt and poincaré’s mechanistic–energetic approaches to linear elasticity and suggestions for multiscale modelling. Arch Appl Mech 81(11):1573–1584

    Article  MATH  Google Scholar 

  9. Cecchi A, Sab K (2002) A multi-parameter homogeneization study for modelling elastic masonry. Eur J Mech A/Solids 21:249–268

    Article  MathSciNet  MATH  Google Scholar 

  10. De Bellis ML, Wriggers P, Hudobivnik B, Zavarise G (2018) Virtual element formulation for isotropic damage. Finite Elem Anal Des 144:38–48

    Article  MathSciNet  Google Scholar 

  11. Djebara Y, Moumen AE, Kanit T, Madani S, Imad A (2016) Modeling of the effect of particles size, particles distribution and particles number on mechanical properties of polymer-clay nano-composites: numerical homogenization versus experimental results. Compos Part B Eng 86:135–142

    Article  Google Scholar 

  12. Du X, Ostoja-Starzewski M (2006) On the size of representative volume element for darcy law in random media. Proc R Soc Lond A Math Phys Eng Sci 462(2074):2949–2963

    Article  MathSciNet  MATH  Google Scholar 

  13. Filipovic N (2016) Modeling the behavior of smart composite materials. In: Montemor M (ed) Smart composite coatings and membranes, Woodhead publishing series in composites science and engineering. Woodhead, Sawston, pp 61–82 (Chap. 3)

    Chapter  Google Scholar 

  14. Forest S, Sab K (1998) Cosserat overall modeling of heterogeneous materials. Mech Res Commun 25(4):449–454

    Article  MathSciNet  MATH  Google Scholar 

  15. Ghosh S, Kubair DV (2016) Exterior statistics based boundary conditions for representative volume elements of elastic composites. J Mech Phys Solids 95:1–24

    Article  MathSciNet  Google Scholar 

  16. Gitman IM, Askes H, Sluys LJ (2007) Representative volume: existence and size determination. Eng Fract Mech 74(16):2518–2534

    Article  Google Scholar 

  17. Hill R (1963) Elastic properties of reinforced solids: some theoretical principles. J Mech Phys Solids 11(5):357–372

    Article  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(13–14):3647–3679

    Article  MATH  Google Scholar 

  19. Khisaeva ZF, Ostoja-Starzewski M (2006) On the size of rve in finite elasticity of random composites. J Elast 85(2):153

    Article  MathSciNet  MATH  Google Scholar 

  20. Kubair D, Pinz M, Kollins K, Przybyla C, Ghosh S (2018) Role of exterior statistics-based boundary conditions for property-based statistically equivalent representative volume elements of polydispersed elastic composites. J Compos Mater 52(21):2919–2928

    Article  Google Scholar 

  21. Lourenço PB, Milani G, Tralli A, Zucchini A (2007) Analysis of masonry structures: review of and recent trends in homogenization techniques. Can J Civ Eng 34(11):1443–1457

    Article  Google Scholar 

  22. Matsuda T, Ohno N, Tanaka H, Shimizu T (2003) Effects of fiber distribution on elastic–viscoplastic behavior of long fiber-reinforced laminates. Int J Mech Sci 45(10):1583–1598 (6th Asia-Pacific symposium on advances in engineering plasticity and its applications)

    Article  MATH  Google Scholar 

  23. Ostoja-Starzewski M (1998) Random field models of heterogeneous materials. Int J Solids Struct 35(19):2429–2455

    Article  MATH  Google Scholar 

  24. Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21:112–132

    Article  Google Scholar 

  25. Ostoja-Starzewski M (2007) Microstructural randomness and scaling in mechanics of materials. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  26. Ranganathan SI, Ostoja-Starzewski M (2008) Scaling function, anisotropy and the size of RVE in elastic random polycrystals. J Mech Phys Solids 56:2773–2791

    Article  MathSciNet  MATH  Google Scholar 

  27. Ranganathan SI, Ostoja-Starzewski M (2009) Towards scaling laws in random polycrystals. Int J Eng Sci 47(11):1322–1330 (“Mechanics, Mathematics and Materials” a Special Issue in memory of A.J.M. Spencer FRS)

    Article  MathSciNet  MATH  Google Scholar 

  28. Reccia E, De Bellis ML, Trovalusci P, Masiani R (2018) Sensitivity to material contrast in homogenization of random particle composites as micropolar continua. Compos Part B Eng 136:39–45

    Article  Google Scholar 

  29. Sab K, Nedjar B (2005) Periodization of random media and representative volume element size for linear composites. C R Mec 333(2):187–195

    Article  MATH  Google Scholar 

  30. Sadowski T, Marsavina L (2011) Multiscale modelling of two-phase ceramic matrix composites. Comput Mater Sci 50(4):1336–1346

    Article  Google Scholar 

  31. Sadowski T, Pankowski B (2016) Numerical modelling of two-phase ceramic composite response under uniaxial loading. Compos Struct 143:388–394

    Article  Google Scholar 

  32. Salmi M, Auslender F, Bornert M, Fogli M (2012) Apparent and effective mechanical properties of linear matrix-inclusion random composites: improved bounds for the effective behavior. Int J Solids Struct 49(10):1195–1211

    Article  Google Scholar 

  33. Sansalone V, Trovalusci P, Cleri F (2006) Multiscale modelling of materials by a multifield approach: microscopic stress and strain distribution in fiber-matrix composites. Acta Mater 54(13):3485–3492

    Article  Google Scholar 

  34. Shewchuk JR (2002) Delaunay refinement algorithms for triangular mesh generation. Comput Geom 22(1):21–74 (16th ACM symposium on computational geometry)

    Article  MathSciNet  MATH  Google Scholar 

  35. Stefanou I, Sulem J, Vardoulakis I (2008) Three-dimensional Cosserat homogenization of masonry structures: elasticity. Acta Geotech 3:71–83

    Article  Google Scholar 

  36. Trovalusci P, Augusti G (1998) A continuum model with microstructure for materials with flaws and inclusions. J Phys IV Pr8:383–390

    Google Scholar 

  37. Trovalusci P, De Bellis ML, Ostoja-Starzewski M, Murrali A (2014) Particulate random composites homogenized as micropolar materials. Meccanica 49(11):2719–2727

    Article  MathSciNet  Google Scholar 

  38. Trovalusci P, Ostoja-Starzewski M, De Bellis ML, Murrali A (2015) Scale-dependent homogenization of random composites as micropolar continua. Eur J Mech A/Solids 49:396–407

    Article  Google Scholar 

  39. Trovalusci P, De Bellis ML, Ostoja-Starzewski M (2016) A statistically-based homogenization approach for particle random composites as micropolar continua. Adv Struct Mater 42:425–441

    Article  MathSciNet  Google Scholar 

  40. Trovalusci P, De Bellis ML, Masiani R (2017) A multiscale description of particle composites: from lattice microstructures to micropolar continua. Compos Part B Eng 128:164–173

    Article  Google Scholar 

  41. Wriggers P, Rust WT, Reddy BD (2016) A virtual element method for contact. Comput Mech 58(6):1039–1050

    Article  MathSciNet  MATH  Google Scholar 

  42. Wriggers P, Reddy BD, Rust W, Hudobivnik B (2017) Efficient virtual element formulations for compressible and incompressible finite deformations. Comput Mech 60(2):253–268

    Article  MathSciNet  MATH  Google Scholar 

  43. Zeman J, Sejnoha M (2007) From random microstructures to representative volume elements. Model Simul Mater Sci Eng 15(4):S325–S335

    Article  Google Scholar 

Download references

Funding

This work is supported by Italian Ministry of University and Research—P.R.I.N. 2015, Sapienza Research Unit, under Grant B86J16002300001, Project 2015JW9NJT “Advanced mechanical modeling of new materials and structures for the solution of 2020 Horizon challenges—and by Sapienza—University Grant 2016, B82F16005920005, and University Grant 2017, B83C17001440005.

Author information

Authors and Affiliations

  1. Department of Structural and Geotechnical Engineering, Sapienza University of Rome, via Gramsci 53, Rome, Italy

    M. Pingaro, E. Reccia, P. Trovalusci & R. Masiani

Authors
  1. M. Pingaro
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Reccia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Trovalusci
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Masiani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Pingaro.

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

Pingaro, M., Reccia, E., Trovalusci, P. et al. Fast statistical homogenization procedure (FSHP) for particle random composites using virtual element method. Comput Mech 64, 197–210 (2019). https://doi.org/10.1007/s00466-018-1665-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-018-1665-7

Keywords

Search

Navigation