Advertisement

Doubly periodic array of coated cylindrical inclusions model and applications for nanocomposites

  • Yaoling XuEmail author
  • Qiao Tian
  • Junhua Xiao
Original Paper
  • 14 Downloads

Abstract

An analytical method is proposed to solve the problem of an infinite elastic matrix containing a doubly periodic array of coated cylindrical inclusions under antiplane shear. The elastic fields in the inclusions, the coatings/interphases and the matrix are derived, which are used to investigate the stresses and the effective stiffness coefficients of the nanofiber composites. Numerical examples demonstrate the size dependence of the stress and the effective stiffness coefficient, and the effects of the interphase thickness and stiffness and array configurations of the inclusions on the effective stiffness coefficient. A finite element analysis is used to benchmark the effective stiffness coefficient predicted by the proposed model, in which excellent agreement is observed. When letting the interphase be thin enough, the proposed coated inclusions model can be used to simulate the zero-thickness interface model, which is validated by the results comparisons of the two models. Instabilities of the stress fields are observed under certain conditions in simulating the zero-thickness interface model.

Notes

References

  1. 1.
    Paliwal, B., Cherkaoui, M.: Estimation of anisotropic elastic properties of nanocomposites using atomistic–continuum interphase model. Int. J. Solids Struct. 49(18), 2424–2438 (2012)CrossRefGoogle Scholar
  2. 2.
    Wong, E.W., Sheehan, P.E., Lieber, C.M.: Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science 277(26), 1971–1975 (1997)CrossRefGoogle Scholar
  3. 3.
    Mogilevskaya, S.G., Crouch, S.L., Stolarski, H.K.: Multiple interacting circular nano-inhomogeneities with surface/interface effects. J. Mech. Phys. Solids 56(6), 2298–2327 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Li, Y., Waas, A.M., Arruda, E.M.: A closed-form, hierarchical, multi-interphase model for composites—derivation, verification and application to nanocomposites. J. Mech. Phys. Solids 59(1), 43–63 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dingreville, R., Qiu, J.M., Cherkaoui, M.: Surface free energy and its effect on the elastic behavior of nano-sized particles, wires and films. J. Mech. Phys. Solids 53(8), 1827–1854 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Paliwal, B., Cherkaoui, M., Fassi-fehri, O.: Effective elastic properties of nanocomposites using a novel atomistic–continuum interphase model. C. R. Mec. 340(4–5), 296–306 (2012)CrossRefGoogle Scholar
  7. 7.
    Odegard, G.M., Clancy, T.C., Gates, T.S.: Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer 46(2), 553–562 (2005)CrossRefGoogle Scholar
  8. 8.
    Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57(4), 291–323 (1975)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gurtin, M.E., Murdoch, A.I.: Surface stress in solids. Int. J. Solids Struct. 14(6), 431–440 (1978)CrossRefGoogle Scholar
  10. 10.
    Tian, L., Rajapakse, R.K.N.D.: Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity. Int. J. Solids Struct. 44(24), 7988–8005 (2007)CrossRefGoogle Scholar
  11. 11.
    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(7), 1574–1596 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, T.Y., Dvorak, G.J., Yu, C.C.: Solids containing spherical nano-inclusions with interface stresses: effective properties and thermal–mechanical connections. Int. J. Solids Struct. 44(3–4), 941–955 (2007)CrossRefGoogle Scholar
  13. 13.
    Tian, L., Rajapakse, R.K.N.D.: Finite element modelling of nanoscale inhomogeneities in an elastic matrix. Comput. Mater. Sci. 41(1), 44–53 (2004)CrossRefGoogle Scholar
  14. 14.
    Dong, C.Y., Pan, E.: Boundary element analysis of nanoinhomogeneities of arbitrary shapes with surface and interface effects. Eng. Anal. Bound. Elem. 35(8), 996–1002 (2011)CrossRefGoogle Scholar
  15. 15.
    Parvanova, L., Vasilev, G.P., Dineva, P.S., Manolis, G.D.: Dynamic analysis of nano-heterogeneities in a finite-size solid by boundary and finite element methods. Int. J. Solids Struct. 80(1), 1–18 (2016)CrossRefGoogle Scholar
  16. 16.
    Paliwal, B., Cherkaoui, M.: Atomistic–continuum interphase model for effective properties of composite materials containing nano-inhomogeneities. Philos. Mag. 91(30), 3905–3930 (2011)CrossRefGoogle Scholar
  17. 17.
    Majewski, M., Kursa, M., Holobut, P., Kowalczyk-Gajewska, K.: Micromechanical and numerical analysis of packing and size effects in elastic particulate composites. Compos. Part B Eng. 124, 158–174 (2017)CrossRefGoogle Scholar
  18. 18.
    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 (2016)CrossRefGoogle Scholar
  19. 19.
    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 (2016)CrossRefGoogle Scholar
  20. 20.
    Xia, Z.H., Zhang, Y.F., Ellyin, F.: A unified periodical boundary condition for representative volume elements of composites and applications. Int. J. Solids Struct. 40(8), 1907–1921 (2003)CrossRefGoogle Scholar
  21. 21.
    Andrianov, I.V., Danishevs’kyy, V.V., Kalamkarov, A.L.: Micromechanical analysis of fiber-reinforced composites on account of influence of fiber coatings. Compos. Part B Eng. 39, 874–881 (2008)CrossRefGoogle Scholar
  22. 22.
    Espinosa-Almeyda, Y., Rodríguez-Ramos, R., Guinovart-Díaz, R., Bravo-Castillero, J., López-Realpozo, J.C., Camacho-Montes, H., Sabina, F.J., Lebon, F.: Antiplane magneto-electro-elastic effective properties of three-phase fiber composites. Int. J. Solids Struct. 51(21–22), 3508–3521 (2014)CrossRefGoogle Scholar
  23. 23.
    Trias, D., Costa, J., Mayugo, J.A., Hurtado, J.E.: Random models versus periodic models for fiber reinforced composites. Comput. Mater. Sci. 38(2), 316–324 (2006)CrossRefGoogle Scholar
  24. 24.
    Bouaoune, L., Brunet, Y., El Moumen, A., Kanit, T., Mazouz, H.: Random versus periodic microstructures for elasticity of fibers reinforced composites. Compos. Part B Eng. 103, 68–73 (2016)CrossRefGoogle Scholar
  25. 25.
    Lee, L., Boyd IV, J.G., Lagoudas, D.C.: Effective properties of three-phase electro-magneto-elastic composites. Int. J. Eng. Sci. 43(10), 790–825 (2005)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Masuda, H., Fukuda, K.: Ordered metal nanohole arrays made by a two-step replication of honeycomb structures of anodic alumina. Science 268, 1466–1468 (1995)CrossRefGoogle Scholar
  27. 27.
    Boyd, J.G., Lagoudas, D.C., Seo, C.S.: Arrays of micro-electrodes and electromagnets for processing of electro-magneto-elastic multifunctional composite materials. In: SPIE 10th Annual International Symposium on Smart Structures, San Diego, USA, vol. 5055, pp. 268-277 (2003)Google Scholar
  28. 28.
    Jiang, C.P., Cheung, Y.K.: A fiber/matrix/composite model with a combined confocal elliptical cylinder unit cell for predicting the effective longitudinal shear modulus. Int. J. Solids Struct. 35(30), 3977–3987 (1998)CrossRefGoogle Scholar
  29. 29.
    Li, X.: Application of Doubly Quasi-periodic Boundary Value Problems in Elasticity Theory. Berlin University, Berlin (1999)Google Scholar
  30. 30.
    Lu, J.K.: Boundary Value Problems for Analytic Function. World Scientific, Singapore (1993)Google Scholar
  31. 31.
    Xu, Y.L., Du, S.S., Xiao, J.H., Zhao, Q.X.: Evaluation of the effective elastic properties of long fiber reinforced composites with interphases. Comput. Mater. Sci. 61, 34–41 (2012)CrossRefGoogle Scholar
  32. 32.
    Liu, Y.W., Fang, Q.H.: Analysis of a screw dislocation inside an inhomogeneity with interface stress. Mater. Sci. Eng. A 464(1–2), 117–123 (2007)CrossRefGoogle Scholar
  33. 33.
    Xiao, J.H., Xu, Y.L.: Study on the effective anti-plane shear modulus of nano inhomogeneity composite materials. Chin. J. Solid Mech. 32(3), 287–292 (2011)Google Scholar
  34. 34.
    Chen, T.Y., Dvorak, G.J., Yu, C.C.: Size-dependent elastic properties of unidirectional nano-composites with interface stresses. Acta Mech. 188(1–2), 39–54 (2007)CrossRefGoogle Scholar
  35. 35.
    Luo, J., Wang, X.: On the anti-plane shear of an elliptic nano inhomogeneity. Eur. J. Mech. A Solids 28(5), 926–934 (2009)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Key Laboratory of Mechanical Reliability for Heavy Equipments and Large Structures of Hebei ProvinceYanshan UniversityQinhuangdaoChina

Personalised recommendations