Node-bond models

As mentioned above, cellular solids are not necessarily derived from a liquid foam. For example, metallic foams may be generated by burning out particulate inclusions or infiltrating a porous matrix which is later removed by leaching. Other cellular solids, such as bone and sponge, are generated by complex organic processes [Gibson & Ashby, 1988]. Since simulating the actual physics and chemistry of the development of cellular materials is beyond the scope of this paper, we instead consider three different statistical models that have features resembling those observed in real cellular solid materials. In this section, we consider node-bond models with variable coordination number (i.e. the number of bonds connected to each node) generated from random seed points.

There are several methods of generating open and closed-cell cellular models from seed-points. An example is provided by the Delaunay (or Voronoi-dual) tessellation [Stoyan et al., 1995]. Starting from a Voronoi tessellation, the cell edges are now defined by a rod placed between two points in space which share a common face. Since the cells of a Voronoi tessellation have approximately 14 faces [Oger et al., 1996], the coordination number of an open-cell Delaunay tessellation will also be around   14.

It is possible to define a more general model, called in this paper a 'node-bond' model, by placing a bond between each seed (or node) and its nearest neighbours. The coordination number, average bond-length, and bond-length distribution depend on the rules used for defining the nearest neighbours of a given node. For example, the coordination number zcan be fixed by connecting a node to its z nearest neighbours. The coordination number can also be allowed to fluctuate if nodes are only connected when node-node distance is smaller than some specified value. If the resulting average coordination number of the foam is around 14, we expect the model to be similar to an open-cell Delaunay tessellation.

To be specific, we connected the centres of an equilibrium hard-sphere (diameter d₀ ) distribution that were closer than the distance Fd₀ (F > 1). We employed the same five distributions of 122 points used for the Voronoi tessellation (c=0.511). To generate the microstructure for the elastic computations we placed a cylinder of radius r between each pair of connected points. Hemi-spherical caps were added at each end to avoid gaps occurring between cylinders that intersect at an angle. An illustration of the node-bond model is shown in Figure 4(b).

We first chose F =1.5, which yielded a high-coordination number foam of $\bar z=12.5$ = 12.5 with average bond-length of $\bar d/d_0=1.16$. This model is illustrated in Figure 4(c), which is a digital model actually used in the elastic computations. To simulate a low-coordination number foam we also studied the case F =1.1, which yielded $\bar z=5.5$ = 5.5 with an average bond-length of $\bar d/d_0=1.04$. Dangling branches in the model were avoided by deleting all nodes with less than two 'nearest' neighbours. The process was repeated until all nodes had two or more nearest neighbours. For F =1.1, only two or three nodes were deleted from each sample. In spring lattices, this iterative removing of low-connectivity nodes has been called 'trimming' [Feng et al., 1985].

The Young's modulus of the high-coordination number model can be described to within a 4 % relative error by,

$$\frac EE_s = 0.376 \left(\frac{\rho}{\rho_s} \right)^{1.29} \rm {for}\;\; 0.04 < \frac{\rho}{\rho_s} < 0.25,$$ (10)

and to within 4 % for 0.1 < / _s < 1 by ( equation 9) with p₀=-0.198 and m=2.80. The Young's modulus of the low-coordination number model can be described to within a 5 % relative error by

$$\frac EE_s = 0.535 \left(\frac{\rho}{\rho_s} \right)^{1.81} \rm {for}\;\; 0.026 < \frac\rho\rho_s < 0.35$$ (11)

and by equation (9) with p ₀=-0.445 and m=4.27 for 0.25 < / _s 1. The FEM data and (equations 10) and (11) are shown in Fig 6. The behaviour of the Poisson's ratio of this model will be discussed in § 4.

Figure 6: The Young's modulus of the non-foam-like cellular models;. high coordination number node-bond model ($\Box$), low coordination number node-bond ($\circ$) and Gaussian random field model ($\triangle$). The lines are the empirical fits to the data given in the text.