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An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites

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Abstract

In a nanostructured material, the interface-to- volume ratio is so high that the interface energy, which is usually negligible with respect to the bulk energy in solid mechanics, can no longer be neglected. The interfaces in a number of nanomaterials can be appropriately characterized by the coherent interface model. According to the latter, the displacement vector field is continuous across an interface in a medium while the traction vector field across the same interface is discontinuous and must satisfy the Laplace–Young equation. The present work aims to elaborate an efficient numerical approach to dealing with the interface effects described by the coherent interface model and to determining the size-dependent effective elastic moduli of nanocomposites. To achieve this twofold objective, a computational technique combining the level set method and the extended finite element method is developed and implemented. The numerical results obtained by the developed computational technique in the two-dimensional (2D) context are compared and discussed with respect to the relevant exact analytical solutions used as benchmarks. The computational technique elaborated in the present work is expected to be an efficient tool for evaluating the overall size-dependent elastic behaviour of nanomaterials and nano-sized structures.

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References

  1. Miller RE and Shenevoy VB (2000). Size-dependent elastic properties of nanosized structural elements. Nanotechnology 11(3): 139–147

    Article  Google Scholar 

  2. Haile JM (1992). Molecular dynamics simulation. Wiley, New York

    Google Scholar 

  3. Hoover WG (1986). Molecular dynamics. Springer, Berlin

    Google Scholar 

  4. Rapaport DC (2004). The art of molecular dynamics simulation. Cambridge University Press, New York

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Hadal R, Yuan Q, Jog JP and Misra RDK (2006). On stress whitening during surface deformation in clay-containing polymer nanocomposites: a microstructural approach. Mater Sci Eng A 418: 268–281

    Article  Google Scholar 

  7. Benveniste Y (2006). A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J Mech Phys Solids 54: 708–734

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  9. Sharma P and Ganti S (2006). Size-dependent Eshelby’s tensor for embedded nano-inclusions incorporating surface/interface energies. J Appl Mech Trans ASME 71(5): 663–671

    Article  Google Scholar 

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

    Article  Google Scholar 

  11. Chen T, Dvorak GJ and Yu CC (2007). Solids containing spherical nano-inclusions with interface stresses: Effective properties and thermal-mechanical connections. Int J Solids Struct 44((3-4)): 941–955

    Article  MATH  Google Scholar 

  12. Duan HL, Wang J, Huang ZP and 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  MATH  MathSciNet  Google Scholar 

  13. 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 Trans ASME 72:628

    Google Scholar 

  14. Sharma P, Ganti S, Bathe N (2006) Erratum: “Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities (82:535, 2003). Appl Phys Lett 89(4), No. 049901

    Google Scholar 

  15. Le Quang H, He Q-C. Estimation of the imperfect thermoelastic moduli of fibrous nanocomposites with cylindrically anisotropic phases. Arch Appl Mech (submitted)

  16. Le Quang H and He Q-C (2007). Size-dependent effective thermoelastic properties of nanocomposites with spherically anisotropic phases. J Mech Phys Solids 55(9): 1889–1921

    MathSciNet  Google Scholar 

  17. Dingreville R, Qu J and Cherkaoui M (2005). Surface free energy and its effect on the elastic behavior of nano-sized particles. J Mech Phys Solids 53(8): 1827–1854

    Article  MATH  MathSciNet  Google Scholar 

  18. Gao W, Yu S and Huang G (2006). Finite element characterization of the size-dependent mechanical behaviour in nanosystems. Nanotechnology 17: 1118–1122

    Article  Google Scholar 

  19. Osher S and Sethian JA (1998). Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79(1): 12–49

    Article  MathSciNet  Google Scholar 

  20. Belytschko T and Black T (1999). Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45(5): 601–620

    Article  MATH  MathSciNet  Google Scholar 

  21. Moës N, Dolbow J and Belytschko T (1999). A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1): 131–156

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  23. Sukumar N, Huang ZY, Prevost J-H and Suo Z (2004). Partition of unity enrichment for bimaterial interface cracks. Int J Numer Methods Eng 59: 1075–1102

    Article  MATH  Google Scholar 

  24. Do Carmo MP (1976). Differential geometry of curves and surfaces. Prentice-Hall, New Jersey

    MATH  Google Scholar 

  25. Thorpe JA (1979). Elementary topics in differential geometry. Springer, Berlin

    MATH  Google Scholar 

  26. Cammarata RC (1994). Surface and interface stress effects in thin films. Progr Surface Sci 46: 1–38

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  28. Belytchko T, Parimi C, Moës N, Sukumar N and Usui S (2003). Structured extended finite element method for solids defined by implicit surfaces. Int J Numer Methods Eng 56: 609–635

    Article  Google Scholar 

  29. Moës N, Cloirec M, Cartraud P and Remacle J-F (2003). A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192: 3163–3177

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  31. Barrett R, Berry M, Chan TF, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C, van der Vorst H (1994) Templates for the solution of linear systems: building blocks for iterative methods. SIAM, Philadelphia

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  1. Université Paris-Est, 5 Bd Descartes, 77454, Marne-la-Vallée Cedex 2, France

    J. Yvonnet, H. Le. Quang & Q. -C. He

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  1. J. Yvonnet
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  2. H. Le. Quang
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  3. Q. -C. He
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Yvonnet, J., Quang, H.L. & He, Q.C. An XFEM/level set approach to modelling surface/interface effects and to computing the size-dependent effective properties of nanocomposites. Comput Mech 42, 119–131 (2008). https://doi.org/10.1007/s00466-008-0241-y

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  • DOI: https://doi.org/10.1007/s00466-008-0241-y

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