Bounds on size effects in composites via homogenization accounting for general interfaces

  • Soheil Firooz
  • George Chatzigeorgiou
  • Fodil Meraghni
  • Ali JaviliEmail author
Original Article


This manuscript provides novel bounds and estimates, for the first time, on size-dependent properties of composites accounting for generalized interfaces in their microstructure, via analytical homogenization verified by computational analysis. We extend both the composite cylinder assemblage and Mori–Tanaka approaches to account for the general interface model. Our proposed strategy does not only determine the overall response of composites, but also it provides information about the local fields for each phase of the medium including the interface. We present a comprehensive study on a broad range of interface parameters, stiffness ratios and sizes. Our analytical solutions are in excellent agreement with the computational results using the finite element method. Based on the observations throughout our investigations, two notions of size-dependent bounds and ultimate bounds on the effective response of composites are introduced which yield a significant insight into the size effects, particularly important for the design of nano-composites.


General interface Size effects Ultimate bounds Size-dependent bounds Homogenization Composites 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Javili, A.: Variational formulation of generalized interfaces for finite deformation elasticity. Math. Mech. Solids 23, 303–322 (2017)MathSciNetGoogle Scholar
  2. 2.
    Sanchez-Palencia, E.: Comportement limite d’un probleme de transmissiona travers une plaque faiblement conductrice. Comptes Rendus Mathematique Academie des Sciences 270, 1026–1028 (1970)zbMATHGoogle Scholar
  3. 3.
    Pham Huy, H., Sanchez-Palencia, E.: Phénomènes de transmission à travers des couches minces de conductivitéélevée. J. Math. Anal. Appl. 47, 284–309 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Hashin, Z.: Thin interphase/imperfect interface in conduction. J. Appl. Phys. 89, 2261–2267 (2001)ADSCrossRefGoogle Scholar
  5. 5.
    Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)ADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proc. R. Soc. A 326, 131–147 (1972)ADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Ogden, R.W.: On the overall moduli of non-linear elastic composite materials. J. Mech. Phys. Solids 22, 541–553 (1974)ADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Moeckel, G.P.: Thermodynamics of an interface. Arch. Ration. Mech. Anal. 57, 255–280 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. Q. J. Mech. Appl. Math. 29, 245–275 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Daher, N., Maugin, G.A.: The method of virtual power in continuum mechanics application to media presenting singular surfaces and interfaces. Acta Mechanica 60, 217–240 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    dell’Isola, F., Romano, A.: On the derivation of thermomechanical balance equations for continuous system with a nonmaterial interface. Int. J. Eng. Sci. 25, 1459–1468 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fried, E., Gurtin, M.E.: Thermomechanics of the interface between a body and its environment. Contin. Mech. Thermodyn. 19, 253–271 (2007)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Klarbring, A.: Derivation of a model of adhesively bonded joints by the asymptotic expansion method. Int. J. Eng. Sci. 29, 493–512 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Klarbring, A., Movchan, A.B.: Asymptotic modelling of adhesive joints. Mech. Mater. 28, 137–145 (1998)CrossRefGoogle Scholar
  16. 16.
    Chen, T., Chiu, M.S., Weng, C.N.: Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids. J. Appl. Phys. 100, 074308 (2006)ADSCrossRefGoogle Scholar
  17. 17.
    Javili, A., McBride, A., Steinmann, P.: Thermomechanics of solids with lower-dimensional energetics: on the importance of surface, interface, and curve structures at the nanoscale. A unifying review. Appl. Mech. Rev. 65, 010802 (2013a)ADSCrossRefGoogle Scholar
  18. 18.
    Javili, A.: A note on traction continuity across an interface in a geometrically non-linear framework. Math. Mech. Solids.
  19. 19.
    Huang, Z.P., Sun, L.: Size-dependent effective properties of a heterogeneous material with interface energy effect: from finite deformation theory to infinitesimal strain analysis. Acta Mechanica 190, 151–163 (2007)zbMATHCrossRefGoogle Scholar
  20. 20.
    Huang, Z.P., Wang, J.: A theory of hyperelasticity of multi-phase media with surface/interface energy effect. Acta Mechanica 182, 195–210 (2006)zbMATHCrossRefGoogle Scholar
  21. 21.
    Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49, 1294–1301 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Chhapadia, P., Mohammadi, P., Sharma, P.: Curvature-dependent surface energy and implications for nanostructures. J. Mech. Phys. Solids 59, 2103–2115 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Cordero, N.M., Forest, S., Busso, E.P.: Second strain gradient elasticity of nano-objects. J. Mech. Phys. Solids 97, 92–124 (2016)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Dingreville, R., Hallil, A., Berbenni, S.: From coherent to incoherent mismatched interfaces: a generalized continuum formulation of surface stresses. J. Mech. Phys. Solids 72, 40–60 (2014)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Dingreville, R., Qu, J.: Interfacial excess energy, excess stress and excess strain in elastic solids: Planar interfaces. J. Mech. Phys. Solids 56, 1944–1954 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Duan, H.L., Wang, J., Karihaloo, B.L.: Theory of elasticity at the nanoscale. Adv. Appl. Mech. 42, 1–68 (2009)CrossRefGoogle Scholar
  27. 27.
    Fried, E., Todres, R.E.: Mind the gap: the shape of the free surface of a rubber-like material in proximity to a rigid contactor. J. Elast. 80, 97–151 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Gurtin, M.E., Weissmüller, J., Larché, F.: A general theory of curved deformable interfaces in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)ADSCrossRefGoogle Scholar
  29. 29.
    Javili, A., Mcbride, A., Mergheim, J., Steinmann, P., Schmidt, U.: Micro-to-macro transitions for continua with surface structure at the microscale. Int. J. Solids Struct. 50, 2561–2572 (2013b)CrossRefGoogle Scholar
  30. 30.
    Liu, L., Yu, M., Lin, H., Foty, R.: Deformation and relaxation of an incompressible viscoelastic body with surface viscoelasticity. J. Mech. Phys. Solids 98, 309–329 (2017)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Steigmann, D.J., Ogden, R.W.: Elastic surface–substrate interactions. Proc. R. Soc. Lond. A 455, 437–474 (1999)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Wang, Z.Q., Zhao, Y.P., Huang, Z.P.: The effects of surface tension on the elastic properties of nano structures. Int. J. Eng. Sci. 48, 140–150 (2010)CrossRefGoogle Scholar
  33. 33.
    Zhong, Z., Meguid, S.A.: On the elastic field of a spherical inhomogeneity with an imperfectly bonded interface. J. Elast. 46, 91–113 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Han, Z., Mogilevskaya, S.G., Schillinger, D.: Local fields and overall transverse properties of unidirectional composite materials with multiple nanofibers and SteigmannOgden interfaces. Int. J. Solids Struct. 147, 166–182 (2018)CrossRefGoogle Scholar
  35. 35.
    Fedotov, A.: Interface model of homogenization for analysing the influence of inclusion size on the elastic properties of composites. Compos. B Eng. 152, 241–247 (2018)CrossRefGoogle Scholar
  36. 36.
    Barenblatt, G.I.: The formation of equilibrium cracks during brittle fracture. General ideas and hypotheses. Axially-symmetric cracks. J. Appl. Math. Mech. 23, 622–636 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Barenblatt, G.I.: The mathematical theory of equilibrium of crack in brittle fracture. Adv. Appl. Mech. 7, 55–129 (1962)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Dugdale, D.S.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960)ADSCrossRefGoogle Scholar
  39. 39.
    Needleman, A.: A continuum model for void nucleation by inclusion debonding. J. Appl. Mech. 54, 525–531 (1987)ADSzbMATHCrossRefGoogle Scholar
  40. 40.
    van den Bosch, M.J., Schreurs, P.J.G., Geers, M.G.D.: An improved description of the exponential Xu and Needleman cohesive zone law for mixed-mode decohesion. Eng. Fract. Mech. 73, 1220–1234 (2006)CrossRefGoogle Scholar
  41. 41.
    Wells, G.N., Sluys, L.J.: A new method for modelling cohesive cracks using finite elements. Int. J. Numer. Methods Eng. 50, 2667–2682 (2001)zbMATHCrossRefGoogle Scholar
  42. 42.
    Remmers, J.J.C., de Borst, R., Needleman, A.: The simulation of dynamic crack propagation using the cohesive segments method. J. Mech. Phys. Solids 56, 70–92 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Moës, N., Belytschko, T.: Extended finite element method for cohesive crack growth. Eng. Fract. Mech. 69, 813–833 (2002)CrossRefGoogle Scholar
  44. 44.
    Alfano, G., Crisfield, M.A.: Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. Int. J. Numer. Methods Eng. 50, 1701–1736 (2001)zbMATHCrossRefGoogle Scholar
  45. 45.
    Charlotte, M., Laverne, J., Marigo, J.J.: Initiation of cracks with cohesive force models: a variational approach. Eur. J. Mech. A/Solids 25, 649–669 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Despringre, N., Chemisky, Y., Bonnay, K., Meraghni, F.: Micromechanical modeling of damage and load transfer in particulate composites with partially debonded interface. Compos. Struct. 155, 77–88 (2016)CrossRefGoogle Scholar
  47. 47.
    Dimitri, R., Trullo, M., De Lorenzis, L., Zavarise, G.: Coupled cohesive zone models for mixed-mode fracture: a comparative study. Eng. Fract. Mech. 148, 145–179 (2015)CrossRefGoogle Scholar
  48. 48.
    Fagerström, M., Larsson, R.: Theory and numerics for finite deformation fracture modelling using strong discontinuities. Int. J. Numer. Methods Eng. 66, 911–948 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Gasser, T.C., Holzapfel, G.A.: Geometrically non-linear and consistently linearized embedded strong discontinuity models for 3D problems with an application to the dissection analysis of soft biological tissues. Comput. Methods Appl. Mech. Eng. 192, 5059–5098 (2003)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Ortiz, M., Pandolfi, A.: Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation analysis. Int. J. Numer. Methods Eng. 1282, 1267–1282 (1999)zbMATHCrossRefGoogle Scholar
  51. 51.
    Park, K., Paulino, G.H.: Cohesive zone models: a critical review of traction-separation relationships across fracture surfaces. Appl. Mech. Rev. 64, 060802 (2011)CrossRefGoogle Scholar
  52. 52.
    Park, K., Paulino, G.H., Roesler, J.R.: A unified potential-based cohesive model of mixed-mode fracture. J. Mech. Phys. Solids 57, 891–908 (2009)ADSCrossRefGoogle Scholar
  53. 53.
    Tijssens, M.G.A., Sluys, B.L.J., Van der Giessen, E.: Numerical simulation of quasi-brittle fracture using damaging cohesive surfaces. Eur. J. Mech. A/Solids 19, 761–779 (2000)ADSzbMATHCrossRefGoogle Scholar
  54. 54.
    Wu, C., Gowrishankar, S., Huang, R., Liechti, K.M.: On determining mixed-mode traction–separation relations for interfaces. Int. J. Fract. 202, 1–19 (2016)CrossRefGoogle Scholar
  55. 55.
    Qian, J., Lin, J., Xu, G.K., Lin, Y., Gao, H.: Thermally assisted peeling of an elastic strip in adhesion with a substrate via molecular bonds. J. Mech. Phys. Solids 101, 197–208 (2017)ADSMathSciNetCrossRefGoogle Scholar
  56. 56.
    Wang, J., Duan, H.L., Zhang, Z., Huang, Z.P.: An anti-interpenetration model and connections between interphase and interface models in particle-reinforced composites. Int. J. Mech. Sci. 47, 701–718 (2005)zbMATHCrossRefGoogle Scholar
  57. 57.
    Mosler, J., Scheider, I.: A thermodynamically and variationally consistent class of damage-type cohesive models. J. Mech. Phys. Solids 59, 1647–1668 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Hashin, Z.: Thin interphase/imperfect interface in elasticity with application to coated fiber composites. J. Mech. Phys. Solids 50, 2509–2537 (2002)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Benveniste, Y.: A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J. Mech. Phys. Solids 54, 708–734 (2006)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Benveniste, Y.: Models of thin interphases with variable moduli in plane-strain elasticity. Math. Mech. Solids 18, 119–134 (2013)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Benveniste, Y., Miloh, T.: Imperfect soft and stiff interfaces in two-dimensional elasticity. Mech. Mater. 33, 309–323 (2001)CrossRefGoogle Scholar
  62. 62.
    Gu, S.T., He, Q.C.: Interfacial discontinuity relations for coupled multifield phenomena and their application to the modeling of thin interphases as imperfect interfaces. J. Mech. Phys. Solids 59, 1413–1426 (2011)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Gu, S.T., Monteiro, E., He, Q.C.: Coordinate-free derivation and weak formulation of a general imperfect interface model for thermal conduction in composites. Compos. Sci. Technol. 71, 1209–1216 (2011)CrossRefGoogle Scholar
  64. 64.
    Monchiet, V., Bonnet, G.: Interfacial models in viscoplastic composites materials. Int. J. Eng. Sci. 48, 1762–1768 (2010)zbMATHCrossRefGoogle Scholar
  65. 65.
    Chatzigeorgiou, G., Meraghni, F., Javili, A.: Generalized interfacial energy and size effects in composites. J. Mech. Phys. Solids 106, 257–282 (2017)ADSMathSciNetCrossRefGoogle Scholar
  66. 66.
    Gu, S.T., Liu, J.T., He, Q.C.: Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities. Int. J. Solids Struct. 51, 2283–2296 (2014)CrossRefGoogle Scholar
  67. 67.
    Koutsawa, Y., Karatrantos, A., Yu, W., Ruch, D.: A micromechanics approach for the effective thermal conductivity of composite materials with general linear imperfect interfaces. Compos. Struct. 200, 747–756 (2018)CrossRefGoogle Scholar
  68. 68.
    Firooz, S., Javili, A.: Understanding the role of general interfaces in the overall behavior of composites and size effects. Comput. Mater. Sci. 162, 245–254 (2019)CrossRefGoogle Scholar
  69. 69.
    Brisard, S., Dormieux, L., Kondo, D.: Hashin–Shtrikman bounds on the shear modulus of a nanocomposite with spherical inclusions and interface effects. Comput. Mater. Sci. 50, 403–410 (2010)CrossRefGoogle Scholar
  70. 70.
    Chatzigeorgiou, G., Javili, A., Steinmann, P.: Multiscale modelling for composites with energetic interfaces at the micro- or nanoscale. Math. Mech. Solids 20, 1130–1145 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Duan, H.L., Karihaloo, B.L.: Effective thermal conductivities of heterogeneous media containing multiple imperfectly bonded inclusions. Phys. Rev. B 75, 064206 (2007)ADSCrossRefGoogle Scholar
  72. 72.
    Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L.: Size-dependent effective elastic constants of solids containing nano-inhomogeneities with interface stress. J. Mech. Phys. Solids 53, 1574–1596 (2005)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Lim, C.W., Li, Z.R., He, L.H.: Size dependent, non-uniform elastic field inside a nano-scale spherical inclusion due to interface stress. Int. J. Solids Struct. 43, 5055–5065 (2006)zbMATHCrossRefGoogle Scholar
  74. 74.
    Mogilevskaya, S.G., Crouch, S.L., Stolarski, H.K.: Multiple interacting circular nano-inhomogeneities with surface/interface effects. J. Mech. Phys. Solids 56, 2298–2327 (2008)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Nazarenko, L., Bargmann, S., Stolarski, H.: Closed-form formulas for the effective properties of random particulate nanocomposites with complete Gurtin–Murdoch model of material surfaces. Contin. Mech. Thermodyn. 29, 77–96 (2017)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Sharma, P.: Size-dependent elastic fields of embedded inclusions in isotropic chiral solids. Int. J. Solids Struct. 41, 6317–6333 (2004)zbMATHCrossRefGoogle Scholar
  77. 77.
    Sharma, P., Ganti, S., Bhate, N.: Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl. Phys. Lett. 82, 535–537 (2003)ADSCrossRefGoogle Scholar
  78. 78.
    Sharma, P., Wheeler, L.T.: Size-dependent elastic state of ellipsoidal nano-inclusions incorporating surface/interface tension. J. Appl. Mech. 74, 447–454 (2007)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Tian, L., Rajapakse, R.K.N.D.: Analytical solution for size-dependent elastic field of a nanoscale circular inhomogeneity. J. Appl. Mech. 74, 568–574 (2007)ADSzbMATHCrossRefGoogle Scholar
  80. 80.
    Fritzen, F., Leuschner, M.: Nonlinear reduced order homogenization of materials including cohesive interfaces. Comput. Mech. 56, 131–151 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    Javili, A., dell’Isola, F., Steinmann, P.: Geometrically nonlinear higher-gradient elasticity with energetic boundaries. J. Mech. Phys. Solids 61, 2381–2401 (2013c)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  82. 82.
    Monteiro, E., He, Q.C., Yvonnet, J.: Hyperelastic large deformations of two-phase composites with membrane-type interface. Int. J. Eng. Sci. 49, 985–1000 (2011)zbMATHCrossRefGoogle Scholar
  83. 83.
    Tu, W., Pindera, M.J.: Cohesive zone-based damage evolution in periodic materials via finite-volume homogenization. J. Appl. Mech. 81, 101005 (2014)ADSCrossRefGoogle Scholar
  84. 84.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Davydov, D., Javili, A., Steinmann, P.: On molecular statics and surface-enhanced continuum modeling of nano-structures. Comput. Mater. Sci. 69, 510–519 (2013)CrossRefGoogle Scholar
  86. 86.
    Elsner, B.A.M., Müller, S., Bargmann, S., Weissmüller, J.: Surface excess elasticity of gold: Ab initio coefficients and impact on the effective elastic response of nanowires. Acta Materialia 124, 468–477 (2017)CrossRefGoogle Scholar
  87. 87.
    He, J., Lilley, C.M.: Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 8, 1798–1802 (2008)ADSCrossRefGoogle Scholar
  88. 88.
    Levitas, V.I., Samani, K.: Size and mechanics effects in surface-induced melting of nanoparticles. Nature Commun. 2, 284–286 (2011)ADSCrossRefGoogle Scholar
  89. 89.
    Olsson, P.A.T., Park, H.S.: On the importance of surface elastic contributions to the flexural rigidity of nanowires. J. Mech. Phys. Solids 60, 2064–2083 (2012)ADSMathSciNetCrossRefGoogle Scholar
  90. 90.
    Park, H.S., Klein, P.A.: Surface stress effects on the resonant properties of metal nanowires: the importance of finite deformation kinematics and the impact of the residual surface stress. J. Mech. Phys. Solids 56, 3144–3166 (2008)ADSzbMATHCrossRefGoogle Scholar
  91. 91.
    Javili, A., Steinmann, P., Mosler, J.: Micro-to-macro transition accounting for general imperfect interfaces. Comput. Methods Appl. Mech. Eng. 317, 274–317 (2017)ADSMathSciNetCrossRefGoogle Scholar
  92. 92.
    McBride, A., Mergheim, J., Javili, A., Steinmann, P., Bargmann, S.: Micro-to-macro transitions for heterogeneous material layers accounting for in-plane stretch. J. Mech. Phys. Solids 60, 1221–1239 (2012)ADSMathSciNetCrossRefGoogle Scholar
  93. 93.
    Saeb, S., Steinmann, P., Javili, A.: Aspects of computational homogenization at finite deformations: a unifying review from Reuss’ to Voigt’s Bound. Appl. Mech. Rev. 68, 050801 (2016)ADSCrossRefGoogle Scholar
  94. 94.
    Kanouté, P., Boso, D.P., Chaboche, J.L., Schrefler, B.A.: Multiscale methods for composites: a review. Arch. Comput. Methods Eng. 16, 31–75 (2009)zbMATHCrossRefGoogle Scholar
  95. 95.
    Charalambakis, N., Chatzigeorgiou, G., Chemisky, Y., Meraghni, F.: Mathematical homogenization of inelastic dissipative materials: a survey and recent progress. Contin. Mech. Thermodyn. 30, 1–51 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Matouš, K., Geers, M.G.D., Kouznetsova, V.G., Gillman, A.: A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials. J. Comput. Phys. 330, 192–220 (2017)ADSMathSciNetCrossRefGoogle Scholar
  97. 97.
    Chen, Q., Wang, G., Pindera, M.J.: Homogenization and localization of nanoporous composites—A critical review and new developments. Compos. B Eng. 155, 329–368 (2018)CrossRefGoogle Scholar
  98. 98.
    Pindera, M.J., Khatam, H., Drago, A.S., Bansal, Y.: Micromechanics of spatially uniform heterogeneous media: a critical review and emerging approaches. Compos. B Eng. 40, 349–378 (2009)CrossRefGoogle Scholar
  99. 99.
    Khisaeva, Z.F., Ostoja-Starzewski, M.: On the size of RVE in finite elasticity of random composites. J. Elast. 85, 153–173 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  100. 100.
    Temizer, I., Zohdi, T.I.: A numerical method for homogenization in non-linear elasticity. Comput. Mech. 40, 281–298 (2007)zbMATHCrossRefGoogle Scholar
  101. 101.
    Gitman, I.M., Askes, H., Aifantis, E.C.: The representative volume size in static and dynamic micro-macro transitions. Int. J. Fract. 135, 3–9 (2005)zbMATHCrossRefGoogle Scholar
  102. 102.
    Kanit, T., Forest, S., Galliet, I., Mounoury, V., Jeulin, D.: Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int. J. Solids Struct. 40, 3647–3679 (2003)zbMATHCrossRefGoogle Scholar
  103. 103.
    Firooz, S., Saeb, S., Chatzigeorgiou, G., Meraghni, F., Steinmann, P., Javili, A.: Systematic study of homogenization and the utility of circular simplified representative volume element. Math. Mech. Solids.
  104. 104.
    Chatzigeorgiou, G., Seidel, G.D., Lagoudas, D.C.: Effective mechanical properties of ”fuzzy fiber” composites. Compos. B 43, 2577–2593 (2012)CrossRefGoogle Scholar
  105. 105.
    Dinzart, F., Sabar, H.: New micromechanical modeling of the elastic behavior of composite materials with ellipsoidal reinforcements and imperfect interfaces. Int. J. Solids Struct. 108, 254–262 (2017)CrossRefGoogle Scholar
  106. 106.
    Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330 (1979)ADSzbMATHCrossRefGoogle Scholar
  107. 107.
    Hashin, Z., Rosen, B.W.: The elastic moduli of fiber-reinforced materials. J. Appl. Mech. 31, 223–232 (1964)ADSCrossRefGoogle Scholar
  108. 108.
    Duan, H.L., Yi, X., Huang, Z.P., Wang, J.: A unified scheme for prediction of effective moduli of multiphase composites with interface effects. Part I: Theoretical framework. Mech. Mater. 39, 81–93 (2007)CrossRefGoogle Scholar
  109. 109.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. A 241, 376–396 (1957)ADSMathSciNetzbMATHGoogle Scholar
  110. 110.
    Benveniste, Y., Dvorak, G.J., Chen, T.: Stress fields in composites with coated inclusions. Mech. Mater. 7, 305–317 (1989)CrossRefGoogle Scholar
  111. 111.
    Wang, Z., Oelkers, R.J., Lee, K.C., Fisher, F.T.: Annular coated inclusion model and applications for polymer nanocomposites-Part I: Spherical inclusions. Mech. Mater. 101, 170–184 (2016a)CrossRefGoogle Scholar
  112. 112.
    Wang, Z., Oelkers, R.J., Lee, K.C., Fisher, F.T.: Annular coated inclusion model and applications for polymer nanocomposites-Part II: Cylindrical inclusions. Mech. Mater. 101, 50–60 (2016b)CrossRefGoogle Scholar
  113. 113.
    Benveniste, Y., Dvorak, G.J., Chen, T.: On diagonal and elastic symmetry of the approximate effective stiffness tensor of heterogeneous media. J. Mech. Phys. Solids 39(7), 927–946 (1991)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  114. 114.
    Hashin, Z.: Thermoelastic properties of fiber composites with imperfect interface. Mech. Mater. 8, 333–348 (1990)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringBilkent UniversityAnkaraTurkey
  2. 2.LEM3-UMR 7239 CNRS, Arts et Metiers ParisTech MetzMetzFrance

Personalised recommendations