Skip to main content
SpringerLink
Log in

An embedded-FEM approach accounting for the size effect in nanocomposites

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

Abstract

A numerical approach based on the Embedded Finite Element Method (E-FEM) has been elaborated in order to model size effect in nanocomposites. For this purpose, the E-FEM standard Statically and Kinematically Optimal Nonsymmetric formulation is enhanced to incorporate a surface elasticity at the interface between the matrix and the inclusions. The results obtained with the proposed approach for a couple of problems have been compared to analytical solutions and other numerical approaches. This study has been carried out considering both linear and nonlinear behaviors of the materials. The developed approach is shown to be an efficient tool for the evaluation and prediction of the mechanical behavior of nanocomposite materials.

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
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Douce J, Boilot J-P, Biteau J, Scodellaro L, Jimenez A (2004) Effect of filler size and surface condition of nano-sized silica particles in polysiloxane coatings. Thin Solid Films 466(1–2):114–122

    Article  Google Scholar 

  2. Mishra S, Sonawane SH, Singh RP (2005) Studies on characterization of nano caco3 prepared by the in situ deposition technique and its application in pp-nano caco3 composites. J Polym Sci, Part B: Polym Phys 43(1):107–113

    Article  Google Scholar 

  3. Cho J, Joshi MS, Sun CT (2006) Effect of inclusion size on mechanical properties of polymeric composites with micro and nano particles. Compos Sci Technol 66(13):1941–1952

    Article  Google Scholar 

  4. Blivi AS, Benhui F, Bai J, Kondo D, Bédoui F (2016) Experimental evidence of size effect in nano-reinforced polymers: Case of silica reinforced pmma. Polym Test 56:337–343

    Article  Google Scholar 

  5. Blivi AS (2018) Effet de taille dans les polymères nano-renforcés: caractérisation multi-échelles et modélisation. PhD thesis

  6. Miller RE, Shenoy VB (2000) Size-dependent elastic properties of nanosized structural elements. Nanotechnol 11(3):139

    Article  Google Scholar 

  7. Duan HL, Wang J, Huang ZP, Karihaloo BL (2005) Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J Mech Phys Solids 53(7):1574–1596

    Article  MathSciNet  MATH  Google Scholar 

  8. Shenoy VB (2005) Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys Rev B 71:094104

    Article  Google Scholar 

  9. Mi C, Jun S, Kouris DA, Kim SY (2008) Atomistic calculations of interface elastic properties in noncoherent metallic bilayers. Phys Rev B 77:075425

    Article  Google Scholar 

  10. Brown D, Mele P, Marceau S, Alberola ND (2003) A molecular dynamics study of a model nanoparticle embedded in a polymer matrix. Macromol 36(4):1395–1406

    Article  Google Scholar 

  11. Brown D, Marcadon V, Mélé P, Albérola ND (2008) Effect of filler particle size on the properties of model nanocomposites. Macromol 41(4):1499–1511

    Article  Google Scholar 

  12. Marcadon V, Brown D, Hervé E, Mélé P, Albérola ND, Zaoui A (2013) Confrontation between molecular dynamics and micromechanical approaches to investigate particle size effects on the mechanical behaviour of polymer nanocomposites. Comput Mater Sci 79:495–505

    Article  Google Scholar 

  13. Paliwal B, Cherkaoui M, Fassi-Fehri O (2012) Effective elastic properties of nanocomposites using a novel atomistic-continuum interphase model. Comptes Rend Mec 340(4–5):296–306

    Article  Google Scholar 

  14. Le T-T, Guilleminot J, Soize C (2016) Stochastic continuum modeling of random interphases from atomistic simulations. application to a polymer nanocomposite. Comput Methods Appl Mech Eng 303:430–449

    Article  MathSciNet  MATH  Google Scholar 

  15. Sharma P, Ganti S, Bhate N (2003) Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl Phys Lett 82(4):535–537

    Article  Google Scholar 

  16. Sharma P, Ganti S (2004) Size-dependent eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies. Trans Am Soc Mech Eng j Appl Mech 71(5):663–671

    Article  MATH  Google Scholar 

  17. Sharma P, Ganti S (2005) Erratum: Size-Dependent Eshelby’s Tensor for Embedded Nano-Inclusions Incorporating Surface/Interface Energies. [J Appl Mech 71(5): 663–671 2004]. J Appl Mech 72(4):628–628

  18. Sharma P, Ganti S, Bhate N (2006) Erratum: effect of surfaces on the size-dependent elastic state of nano-inhomogeneities [Appl Phys Lett 82:535 (2003)]. Appl Phys Lett 89(4):535

  19. Chen T, Dvorak GJ, Yu CC (2007) Size-dependent elastic properties of unidirectional nano-composites with interface stresses. Acta Mech 188(1–2):39–54

    Article  MATH  Google Scholar 

  20. Le Quang H, He QC (2009) Estimation of the effective thermoelastic moduli of fibrous nanocomposites with cylindrically anisotropic phases. Arch Appl Mech 79(3):225–248

    Article  MATH  Google Scholar 

  21. Brisard S, Dormieux L, Kondo D (2010) Hashin-shtrikman bounds on the bulk modulus of a nanocomposite with spherical inclusions and interface effects. Comput Mater Sci 48(3):589–596

    Article  Google Scholar 

  22. Brisard S, Dormieux L, Kondo D (2010) Hashin-shtrikman bounds on the shear modulus of a nanocomposite with spherical inclusions and interface effects. Comput Mater Sci 50(2):403–410

    Article  Google Scholar 

  23. Benveniste Y (2013) Models of thin interphases and the effective medium approximation in composite media with curvilinearly anisotropic coated inclusions. Int J Eng Sci 72:140–154

    Article  MATH  Google Scholar 

  24. Marcadon V, Herve E, Zaoui A (2007) Micromechanical modeling of packing and size effects in particulate composites. Int J Solids Struct 44(25–26):8213–8228

    Article  MATH  Google Scholar 

  25. Zhang WX, Wang TJ (2007) Effect of surface energy on the yield strength of nanoporous materials. Appl Phys Lett 90(6):063104

    Article  Google Scholar 

  26. Dormieux L, Kondo D (2010) An extension of gurson model incorporating interface stresses effects. Int J Eng Sci 48(6):575–581

    Article  MathSciNet  MATH  Google Scholar 

  27. Dormieux L, Kondo D (2013) Non linear homogenization approach of strength of nanoporous materials with interface effects. Int J Eng Sci 71:102–110

    Article  MATH  Google Scholar 

  28. Antoine Lucchetta, Stella Brach, Djimédo Kondo (2021) Effects of particles size on the overall strength of nanocomposites: Molecular dynamics simulations and theoretical modeling. Mech Res Commun 114:103669

    Article  Google Scholar 

  29. Wei G, Shouwen YU, Ganyun H (2006) Finite element characterization of the size-dependent mechanical behaviour in nanosystems. Nanotechnol 17(4):1118

    Article  Google Scholar 

  30. Javili A, Chatzigeorgiou G, McBride AT, Steinmann P, Linder C (2015) Computational homogenization of nano-materials accounting for size effects via surface elasticity. GAMM-Mitt 38(2):285–312

    Article  MathSciNet  Google Scholar 

  31. Javili A, Steinmann P, Mosler J (2017) Micro-to-macro transition accounting for general imperfect interfaces. Comput Methods Appl Mech Eng 317(Supplement C):274–317

    Article  MathSciNet  MATH  Google Scholar 

  32. Yvonnet J, Le Quang H, He Q-C (2008) An xfem/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites. Comput Mech 42(1):119–131

    Article  MathSciNet  MATH  Google Scholar 

  33. Farsad M, Vernerey FJ, Park HS (2010) An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials. Int J Numer Meth Eng 84(12):1466–1489

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhu Q-Z, Gu S-T, Yvonnet J, Shao J-F, He Q-C (2011) Three-dimensional numerical modelling by xfem of spring-layer imperfect curved interfaces with applications to linearly elastic composite materials. Int J Numer Meth Eng 88(4):307–328

    Article  MathSciNet  MATH  Google Scholar 

  35. Hachi BE, Benkhechiba AE, Kired MR, Hachi D, Haboussi M (2020) Some investigations on 3d homogenization of nano-composite/nano-porous materials with surface effect by fem/xfem methods combined with level-set technique. Comput Methods Appl Mech Eng 371:113319

    Article  MathSciNet  MATH  Google Scholar 

  36. Bach DP, Brancherie D, Cauvin L (2019) Xfem/level set approach and interface element approach. Size effect in nanocomposites. Finite Elem Anal Des 165:41–51

    Article  MathSciNet  Google Scholar 

  37. Brancherie D, Ibrahimbegovic A (2009) Novel anisotropic continuum-discrete damage model capable of representing localized failure of massive structures: Part i: theoretical formulation and numerical implementation. Eng Comput 26(1/2):100–127

    Article  MATH  Google Scholar 

  38. Ibrahimbegovic A, Brancherie D (2003) Combined hardening and softening constitutive model of plasticity: precursor to shear slip line failure. Comput Mech 31(1–2):88–100

    Article  MATH  Google Scholar 

  39. Roubin E, Vallade A, Benkemoun N, Colliat J-B (2015) Multi-scale failure of heterogeneous materials: a double kinematics enhancement for embedded finite element method. Int J Solids Struct 52:180–196

  40. Benkemoun N, Hautefeuille M, Colliat J-B, Ibrahimbegovic A (2010) Failure of heterogeneous materials: 3d meso-scale fe models with embedded discontinuities. Int J Numer Meth Eng 82(13):1671–1688

  41. Jirásek M (2000) Comparative study on finite elements with embedded discontinuities. Comput Methods Appl Mech Eng 188(1–3):307–330

    Article  MATH  Google Scholar 

  42. Cazes F, Meschke G, Zhou M-M (2016) Strong discontinuity approaches: an algorithm for robust performance and comparative assessment of accuracy. Int J Solids Struct 96:355–379

    Article  Google Scholar 

  43. Gurtin ME, Ian Murdoch A (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57(4):291–323

    Article  MathSciNet  MATH  Google Scholar 

  44. Povstenko YZ (1993) Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. J Mech Phys Solids 41(9):1499–1514

    Article  MathSciNet  MATH  Google Scholar 

  45. Gurtin ME, Weissmüller J, Larche F (1998) A general theory of curved deformable interfaces in solids at equilibrium. Philos Mag A 78(5):1093–1109

    Article  Google Scholar 

  46. Simo JC, Hughes TJR (2006) Computational inelasticity, vol 7. Springer Science & Business Media, New York

    MATH  Google Scholar 

  47. Ibrahimbegovic A (2009) Nonlinear solid mechanics: theoretical formulations and finite element solution methods, vol 160. Springer Science & Business Media, New York

    Book  MATH  Google Scholar 

  48. Simo JC, Rifai MS (1990) A class of mixed assumed strain methods and the method of incompatible modes. Int J Numer Meth Eng 29(8):1595–1638

    Article  MathSciNet  MATH  Google Scholar 

  49. Wilson EL, Ibrahimbegovic A (1990) Use of incompatible displacement modes for the calculation of element stiffnesses or stresses. Finite Elem Anal Des 7(3):229–241

    Article  Google Scholar 

  50. Ibrahimbegovic A, Wilson EL (1991) A modified method of incompatible modes. Communications in applied numerical methods 7(3):187–194

    Article  MATH  Google Scholar 

  51. Chessa J, Belytschko T (2003) An enriched finite element method and level sets for axisymmetric two-phase flow with surface tension. Int J Numer Meth Eng 58(13):2041–2064

    Article  MathSciNet  MATH  Google Scholar 

  52. Wilson EL (1974) The static condensation algorithm. Int J Numer Meth Eng 8(1):198–203

    Article  Google Scholar 

  53. Sukumar N, Chopp DL, Moës N, Belytschko T (2001) Modeling holes and inclusions by level sets in the extended finite-element method. Comput Methods Appl Mech Eng 190(46–47):6183–6200

    Article  MathSciNet  MATH  Google Scholar 

  54. Doan T, Le-Quang H, To Q-D (2020) effective elastic stiffness of 2d materials containing nanovoids of arbitrary shape. Int J Eng Sci 150:103234

    Article  MathSciNet  MATH  Google Scholar 

  55. Wriggers P (2008) Nonlinear finite element methods. Springer Science & Business Media, New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire Roberval FRE UTC-CNRS 2012, Alliance Sorbonne Universités, Université de Technologie de Compiègne, Compiègne, France

    Dang Phong Bach, Delphine Brancherie & Ludovic Cauvin

Authors
  1. Dang Phong Bach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Delphine Brancherie
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ludovic Cauvin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dang Phong Bach.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Bach, D.P., Brancherie, D. & Cauvin, L. An embedded-FEM approach accounting for the size effect in nanocomposites. Comput Mech 70, 745–762 (2022). https://doi.org/10.1007/s00466-022-02194-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00466-022-02194-7

Keywords

Search

Navigation