Keywords: cellular materials, open-cell foams, metallic foams, lattice modelling, perturbation method, Voronoi tessellation.

Materials, combining lightweight and high mechanical energy absorption, are of great interest for a variety of automotive, locomotive, marine, and aerospace applications. Recent advances in materials fabrication and processing have led to the emergence of stochastic metallic foams, which combine low density and high efficiency in absorbing external mechanical energy through its deformation. Thus, significant research effort has been focused on research of metallic foams and periodic cellular material characterisation and structural integrity [1].

Many analytical and experimental models have been developed for predicting mechanical properties of cellular solids based on idealized volume elements, i.e. unit cells [1]. A properly defined unit cell (repeating unit) can capture the essential microstructural features of an actual cellular material. The cubic, tetrahedral, dodecahedral and tetrakaidecahedral cells have been mostly used as repeating units for modelling of three-dimensional cellular solids [2].

Unit cell-based models can provide some insight into the behaviour of cellular structures, but they are limited only to regular structures and cannot account for microstructural imperfections which are present in most real cellular materials. Since cellular structures of metallic foams are typically non-periodic, non-uniform and disordered, more complex, statistical models are required to obtain improved predictions. A different approach to computational modelling of cellular structures is therefore required to account for the stochastic nature of cellular solids.

Aluminium open cell foams may be best represented by packed tetrakaidecahedral cells, since the tetrakaidecahedron is known to be the polyhedron that can pack with identical units to fill space by minimizing the surface energy [3].

A regular open-cell lattice model can thus be constructed by using the tetrakaidecahedral cells and replacing the edges of tetrakaidecahedral cells with the beam finite elements. The irregularity of such lattice models is introduced by controlled disorder of lattice's vertex coordinates [4], while the cross-sectional area of beams and their shape remain identical on all edges.

As a result of the similarity between the mathematical procedure of the Voronoi tessellation and the physics of foam production, the Voronoi tessellation technique is another choice for describing the structure of foams resulting from a bubble forming process [5].

Both methods presented have certain advantages. The Voronoi tessellation method produces lattice models with different cell topologies while the perturbation method preserves the cell topologies. At the same time, the perturbation method takes significantly less time for generation of larger lattice models and produces meshes with evenly distributed cell edge lengths.

Van der Burg et al. [6] showed that the effects of using orders of magnitude shorter beam finite elements for static simulations are negligible when observing the deformation behaviour of the entire model. However, for efficient use of presented lattice models for explicit dynamic simulations certain smoothing techniques should be introduced to assure proper stability and accuracy of the simulations.

1

L.J. Gibson, M.F. Ashby, "Cellular solids, Structure and Properties, Second Edition", Cambridge University Press, Cambridge, United Kingdom, 1999.

2

W.E. Warren, A.M. Kraynik, "Linear elastic behaviour of a low-density Kelvin foam with open cells", Journal of Applied Mechanics, 64, 787-794, 1997. doi:10.1115/1.2788983

3

A.P. Roberts, E.J. Garboczi, "Elastic properties of model random three-dimensional open-cell solids", Journal of the Mechanics and Physics of Solids, 50, 33-55, 2002. doi:10.1016/S0022-5096(01)00056-4

4

Z. Zheng, J. Yu, J. Li, "Dynamic crushing of 2D cellular structures: A finite element study", International Journal of Impact Engineering, 32, 650-664, 2005. doi:10.1016/j.ijimpeng.2005.05.007

5

J.S. Huang, L.J. Gibson, "Creep of open-cell Voronoi foams", Materials Science and Engineering, A339, 220-226, 2003. doi:10.1016/S0921-5093(02)00152-1

6

M.W.D. Van der Burg, V. Shulmeister, E. Van der Geissen, R. Marissen, "On the linear elastic properties of regular and random open-cell foam models", Journal of Cellular Plastics, 33, 31-54, 1997.

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