Archive of Applied Mechanics

, Volume 86, Issue 1–2, pp 165–176 | Cite as

Beam model for the elastic properties of material with spherical voids

  • Sascha Heitkam
  • Wiebke Drenckhan
  • Denis Weaire
  • Jochen Fröhlich


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.


Void material Open-cell foam Young’s modulus  Poisson ratio Void fraction 



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).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Sascha Heitkam
    • 1
  • Wiebke Drenckhan
    • 2
  • Denis Weaire
    • 3
  • Jochen Fröhlich
    • 1
  1. 1.Institute of Fluid MechanicsTechnische Universität DresdenDresdenGermany
  2. 2.Laboratoire de Physique des SolidesCNRS, Université Paris-Sud, Université Paris-SaclayOrsayFrance
  3. 3.School of PhysicsTrinity CollegeDublin 2Ireland

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