Skip to main content
SpringerLink
Log in

Numerical Investigation of Effective Moduli of Porous Elastic Material with Surface Stresses for Various Structures of Porous Cells

  • Chapter
  • First Online:
Wave Dynamics, Mechanics and Physics of Microstructured Metamaterials

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 109))

Abstract

The chapter deals with the model problem of finding the effective moduli of a nanoporous elastic material, in which the surface stresses are defined on the pore surface to reflect the size effect using the Gurtin–Murdoch model. One cell of a porous material in the form of a cube with one pore located in the center is considered. The objective of the study is to assess the influence of the pore shape and the magnitude of the scale factors on the effective moduli of the composite material. The homogenization problem is formulated within the framework of the effective moduli method, and to find its solution, the finite element method and the ANSYS software package are used. In the finite element model, the surface stresses are taken into account by membrane elements covering the pore surfaces and conformable with the finite element mesh of bulk elements. Numerical experiments carried out for pores of cubic and spherical shapes show the cumulative significant effect of pore geometry and scale factors on the effective elastic moduli.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  CAS  Google Scholar 

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

    Article  CAS  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. Chatzigeorgiou, G., Meraghni, F., Javili, A.: Generalized interfacial energy and size effects in composites. J. Mech. Phys. Solids. 106, 257–282 (2017)

    Article  Google Scholar 

  5. Chen, T., 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, 941–955 (2007)

    Article  Google Scholar 

  6. Duan, H.L., Wang, J., Huang, Z.P., Karihaloo, B.L.: Eshelby formalism for nano-inhomogeneities. Proc. R. Soc. A. 461, 3335–3353 (2005)

    Article  Google Scholar 

  7. 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)

    Article  Google Scholar 

  8. Duan, H.L., Wang, J., Huang, Z.P., Luo, Z.Y.: Stress concentration tensors of inhomogeneities with interface effects. Mech. Mater. 37, 723–736 (2005)

    Article  Google Scholar 

  9. Duan, H.L., Wang, J., Karihaloo, B.L., Huang, Z.P.: Nanoporous materials can be made stiffer than non-porous counterparts by surface modification. Acta Materialia. 54, 2983–2990 (2006)

    Article  CAS  Google Scholar 

  10. Eremeyev, V.A.: On effective properties of materials at the nano- and microscales considering surface effects. Acta Mech. 227, 29–42 (2016)

    Article  Google Scholar 

  11. Eremeyev, V., Morozov, N.: The effective stiffness of a nanoporous rod. Dokl. Physics. 55(6), 279–282 (2010)

    Article  CAS  Google Scholar 

  12. Gad, A.I., Mahmoud. F.F., Alshorbagy. A.E., Ali-Eldin. S.S.: Finite element modeling for elastic nano-indentation problems incorporating surface energy effect. Int. J. Mech. Sciences. 84, 158–170 (2014)

    Google Scholar 

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

    Article  CAS  Google Scholar 

  14. 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)

    Article  Google Scholar 

  15. Gurtin, M.E., Murdoch, A.I.: A continuum theory of elastic material surfaces. Arch. Rat. Mech. Analysis. 57(4), 291–323 (1975)

    Article  Google Scholar 

  16. Hamilton, J.C., Wolfer, W.G.: Theories of surface elasticity for nanoscale objects. Surface Sci. 603, 1284–1291 (2009)

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  18. Javili, A., McBride, A., Mergheima, J., Steinmann, P., Schmidt, U.: Micro-to-macro transitions for continua with surface structure at the microscale. Int. J. Solids Struct. 50, 2561–2572 (2013)

    Article  Google Scholar 

  19. 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, 010802-1–31 (2013)

    Google Scholar 

  20. Javili, A., Steinmann, P.: A finite element framework for continua with boundary energies. Part I: the two-dimensional case. Comput. Methods Appl. Mech. Engrg. 198, 2198–2208 (2009)

    Google Scholar 

  21. Javili, A., Steinmann, P.: A finite element framework for continua with boundary energies. Part II: The three-dimensional case. Comput. Methods Appl. Mech. Engrg. 199, 755–765 (2010)

    Google Scholar 

  22. Jeong, J., Cho, M., Choi, J.: Effective mechanical properties of micro/nano-scale porous materials considering surface effects. Interact. Multiscale Mech. 4(2), 107–122 (2011)

    Article  Google Scholar 

  23. Kushch, V.I., Mogilevskaya, S.G., Stolarski, H.K., Crouch, S.L.: Elastic fields and effective moduli of particulate nanocomposites with the Gurtin-Murdoch model of interfaces. Int. J. Solids Struct. 50, 1141–1153 (2013)

    Article  CAS  Google Scholar 

  24. Le Quang, H., He, Q.-C.: Variational principles and bounds for elastic inhomogeneous materials with coherent imperfect interfaces. Mech. Mater. 40, 865–884 (2008)

    Article  Google Scholar 

  25. Miller, R.E., Shenoy, V.B.: Size-dependent elastic properties of nanosized structural elements. Nanotechnology. 11, 139–147 (2000)

    Google Scholar 

  26. Nasedkin, A.V., Kornievsky, A.S.: Finite element modeling and computer design of anisotropic elastic porous composites with surface stresses. In: M.A. Sumbatyan (Ed.) Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials. Ser. Advanced Structured Materials, vol. 59, pp. 107–122. Springer, Singapore (2017)

    Google Scholar 

  27. Nasedkin, A.V., Kornievsky, A.S.: Finite element modeling of effective properties of elastic materials with random nanosized porosities. ycisl. meh. splos. sred – Computational Continuum Mechanics. 10(4), 375–387 (2017)

    Google Scholar 

  28. Nasedkin, A.V., Kornievsky, A.S.: Finite element homogenization of elastic materials with open porosity at different scale levels. AIP Conf. Proc. 2046, 020064 (2018)

    Google Scholar 

  29. Nasedkin, A.V., Nasedkina, A.A., Kornievsky, A.S.: Modeling of nanostructured porous thermoelastic composites with surface effects. AIP Conf. Proc. 1798, 020110 (2017)

    Google Scholar 

  30. Nasedkin, A.V., Nasedkina, A.A., Kornievsky, A.S.: Finite element modeling of effective properties of nanoporous thermoelastic composites with surface effects. In: Greece. M. Papadrakakis, E. Onate, B.A. Schrefler (eds.) Coupled Problems 2017 - Proceeding VII International Conference on Coupled Problems in Science and Engineering, 12–14 June 2017, pp. 1140–1151. Rhodes Island, CIMNE, Barcelona, Spain (2017)

    Google Scholar 

  31. Nazarenko, L., Bargmann, S., Stolarski, H.: Energy-equivalent inhomogeneity approach to analysis of effective properties of nanomaterials with stochastic structure. Int. J. Solids Struct. 59, 183–197 (2015)

    Article  Google Scholar 

  32. Riaz, U., Ashraf, S.M.: Application of Finite Element Method for the Design of Nanocomposites. In: Musa, S.M. (ed.), Computational Finite Element Methods in Nanotechnology, pp. 241–290. CRC Press (2012)

    Google Scholar 

  33. 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)

    Google Scholar 

  34. Tian, L., Rajapakse, R.K.N.D.: Finite element modelling of nanoscale inhomogeneities in an elastic matrix. Comp. Mater. Sci. 41, 44–53 (2007)

    Article  Google Scholar 

  35. Wang, J., Huang, Z., Duan, H., Yu, S., Feng, X., Wang, G., Zhang, W., Wang, T.: Surface stress effect in mechanics of nanostructured materials. Acta Mechanica Solida Sinica. 24(1), 52–82 (2011)

    Article  CAS  Google Scholar 

  36. Wang, K.F., Wang, B.L., Kitamura, T.: A review on the application of modified continuum models in modeling and simulation of nanostructures. Acta Mech. Sin. 32(1), 83–100 (2016)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Russian Science Foundation (grant number 15–19-10008-P).

Author information

Authors and Affiliations

  1. Institute of Mathematics, Mechanics and Computer Science, Southern Federal University, Milchakova Street 8a, Rostov-on-Don, 344090, Russia

    A. V. Nasedkin & A. S. Kornievsky

Authors
  1. A. V. Nasedkin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. S. Kornievsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Nasedkin .

Editor information

Editors and Affiliations

  1. Institute for Mathematics, Mechanics and Computer Science, Southern Federal University, Rostov-on-Don, Russia

    Mezhlum A. Sumbatyan

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Nasedkin, A.V., Kornievsky, A.S. (2019). Numerical Investigation of Effective Moduli of Porous Elastic Material with Surface Stresses for Various Structures of Porous Cells. In: Sumbatyan, M. (eds) Wave Dynamics, Mechanics and Physics of Microstructured Metamaterials. Advanced Structured Materials, vol 109. Springer, Cham. https://doi.org/10.1007/978-3-030-17470-5_15

Download citation

Publish with us

Policies and ethics

Search

Navigation