Skip to main content
SpringerLink
Log in

Beam model for the elastic properties of material with spherical voids

  • Special
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

The elastic properties of a material with spherical voids of equal volume are analysed using a new model, with particular attention paid to the hexagonal close-packed and the face-centred cubic arrangement of voids. Void fractions well above 74 % are considered, yielding overlapping voids as in an open-cell foam and hence a connected pore structure. The material is represented by a network of beams. The elastic behaviour of each beam is derived analytically from the material structure. By computing the linear elastic properties of the beam network, the Young’s moduli and Poisson ratios for different directions are evaluated. In the limit of rigidity loss, a power law is obtained, describing the relation between Young’s modulus and void fraction with an exponent of 5/2 for bending-dominated and 3/2 for stretching-dominated directions. The corresponding Poisson ratios vary between 0 and 1. With decreasing void fraction, these exponents become 2.3 and 1.3, respectively. The data obtained analytically and from the new beam model are compared to finite element simulations carried out in a companion study, and good agreement is found. The hexagonal close-packed void arrangement features anisotropic behaviour, comparable to that of fibre-reinforced materials. This may give rise to new applications of open-cell 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

Similar content being viewed by others

References

  1. Bakis, C., Bank, L.C., Brown, V., Cosenza, E., Davalos, J., Lesko, J., Machida, A., Rizkalla, S., Triantafillou, T.: Fiber-reinforced polymer composites for construction-state-of-the-art review. J. Compos. Constr. 6(2), 73–87 (2002)

    Article  Google Scholar 

  2. Chandra, R., Singh, S., Gupta, K.: Damping studies in fiber-reinforced composites—a review. Compos. Struct. 46(1), 41–51 (1999)

    Article  Google Scholar 

  3. Day, A., Snyder, K., Garboczi, E., Thorpe, M.: The elastic moduli of a sheet containing circular holes. J. Mech. Phys. Solids 40(5), 1031–1051 (1992)

    Article  Google Scholar 

  4. Gibson, L.J., Ashby, M.F.: Cellular Solids: Structure and Properties, 2nd edn. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  5. He, C., Liu, P., Griffin, A.C.: Toward negative poisson ratio polymers through molecular design. Macromolecules 31(9), 3145–3147 (1998)

    Article  Google Scholar 

  6. Heitkam, S., Drenckhan, W., Fröhlich, J.: Packing spheres tightly: influence of mechanical stability on close-packed sphere structures. Phys. Rev. Lett. 108, 148–302 (2012)

    Article  Google Scholar 

  7. Heitkam, S., Drenckhan, W., Titscher, T., Kreuter, D., Hajnal, D., Piechon, F., Fröhlich, J.: Elastic Properties of Material with Spherical Voids in Different Arrangements. Eur. J. Mech. A/Solids (submitted) (2015)

  8. Heitkam, S., Schwarz, S., Santarelli, C., Fröhlich, J.: Influence of an electromagnetic field on the formation of wet metal foam. Eur. Phys. J. Spec. Top. 220(1), 207–214 (2013)

    Article  Google Scholar 

  9. Jang, W.Y., Kraynik, A.M., Kyriakides, S.: On the microstructure of open-cell foams and its effect on elastic properties. Int. J. Solids Struct. 45(7), 1845–1875 (2008)

    Article  MATH  Google Scholar 

  10. Kikuchi, K., Ikeda, K., Okayasu, R., Takagi, K., Kawasaki, A.: Structural dependency of three-dimensionally periodic porous materials on elastic properties. Mater. Sci. Eng. A 528(28), 8292–8298 (2011)

    Article  Google Scholar 

  11. Kim, J.K., Mai, Yw: High strength, high fracture toughness fibre composites with interface control—a review. Compos. Sci. Technol. 41(4), 333–378 (1991)

    Article  Google Scholar 

  12. Lambert, J., Cantat, I., Delannay, R., Renault, A., Graner, F., Glazier, J.A., Veretennikov, I., Cloetens, P.: Extraction of relevant physical parameters from 3D images of foams obtained by X-ray tomography. Colloids Surf. A Physicochem. Eng. Asp. 263(1), 295–302 (2005)

    Article  Google Scholar 

  13. Lee, J., Kim, K., Ju, J., Kim, D.M.: Compliant cellular materials with elliptical holes for extremely high positive and negative Poisson’s ratios. J. Eng. Mater. Technol. 137(1), 011001 (2015)

    Article  Google Scholar 

  14. Mallick, P.K.: Fiber-Reinforced Composites: Materials, Manufacturing, and Design. CRC Press, Boca Raton (2007)

    Book  Google Scholar 

  15. Mbakogu, F., Pavlović, M.: Shallow shells under arbitrary loading: analysis by the two-surface truss model. Commun. Appl. Numer. Methods 3(3), 235–242 (1987)

    Article  MATH  Google Scholar 

  16. Mills, N.: Polymer Foams Handbook: Engineering and Biomechanics Applications and Design Guide. Butterworth-Heinemann, Oxford (2007)

    Google Scholar 

  17. Roberts, A., Garboczi, E.J.: Elastic properties of model random three-dimensional open-cell solids. J. Mech. Phys. Solids 50(1), 33–55 (2002)

    Article  MATH  Google Scholar 

  18. Salonen, E.M.: Triangular framework model for plate bending. J. Eng. Mech. Div. 97(1), 149–153 (1971)

    Google Scholar 

  19. Ting, T., Chen, T.: Poisson’s ratio for anisotropic elastic materials can have no bounds. Q. J. Mech. Appl. Math. 58(1), 73–82 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Warren, W., Kraynik, A.: Linear elastic behavior of a low-density Kelvin foam with open cells. J. Appl. Mech. 64(4), 787–794 (1997)

    Article  MATH  Google Scholar 

  21. Weaire, D., Aste, T.: The Pursuit of Perfect Packing. CRC Press, Boca Raton (2008)

    MATH  Google Scholar 

  22. Weaire, D., Phelan, R.: A counter-example to Kelvin’s conjecture on minimal surfaces. Philos. Mag. Lett. 69(2), 107–110 (1994)

    Article  Google Scholar 

  23. Yu, A., An, X., Zou, R., Yang, R., Kendall, K.: Self-assembly of particles for densest packing by mechanical vibration. Phys. Rev. Lett. 97(26), 265–501 (2006)

    Article  Google Scholar 

Download references

Acknowledgments

We gratefully acknowledge Frederic Piechon, Christophe Poulard, Thomas Titscher, Daniel Christopher Kreuter, and David Hajnal for fruitful discussions on the topic and during the work that originally motivated the present study [7]. Computation time was provided by the Center for Information Services and High Performance Computing (ZIH) at TU Dresden. We acknowledge support from the European Research Council (ERC) under the European Unions Seventh Framework Program (FP7/2007-2013) in form of an ERC Starting Grant, Agreement 307280-POMCAPS. We acknowledge support from the European Centre for Emerging Materials and Processes (ECEMP) at TU Dresden and the Helmholtz Alliance Liquid Metal Technologies (LIMTECH).

Author information

Authors and Affiliations

  1. Institute of Fluid Mechanics, Technische Universität Dresden, 01069, Dresden, Germany

    Sascha Heitkam & Jochen Fröhlich

  2. Laboratoire de Physique des Solides, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405, Orsay, France

    Wiebke Drenckhan

  3. School of Physics, Trinity College, Dublin 2, Ireland

    Denis Weaire

Authors
  1. Sascha Heitkam
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wiebke Drenckhan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Denis Weaire
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jochen Fröhlich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jochen Fröhlich.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heitkam, S., Drenckhan, W., Weaire, D. et al. Beam model for the elastic properties of material with spherical voids. Arch Appl Mech 86, 165–176 (2016). https://doi.org/10.1007/s00419-015-1102-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-015-1102-8

Keywords

Search

Navigation