Elastic properties of model cellular solids

The finite element method uses a variational formulation of the linear elastic equations, and finds the solution by minimising the elastic energy via a fast conjugate gradient method. The digital image is assumed to have periodic boundary conditions. Details of the theory and copies of the actual programs used are reported in the papers of Garboczi & Day [Garboczi & Day, 1995] and Garboczi [Garboczi, 1998].

Given a digital microstructure, the FEM provides a numerical solution of the elasticity equations. The accuracy is only limited by the finite number of pixels which can be used (around 10⁶ in this study). We consider continuum models with a fixed length scale, such as cell size, and measure the properties of a T x T x T µm region, divided into M ³ cubic pixels. Here T is much greater than the cell size. If the foam were regular and periodic, just one unit cell would be sufficient. In this section we discuss the sources of error and how they can be minimised.

Discretisation errors occur in the FEM when there are insufficient pixels in a solid region to correctly model continuum elasticity. To check the effect of resolution for the FEM we measured the Young's modulus of the simple cell model shown in Figure 1(b) (with L=6, w=2 and d=1 µm) at finer and finer resolutions M = 7,14...77. Here, and in subsequent calculations, we use E_s=1 GPa and a solid Poisson's ratio of ν =0.2. The results are shown in Figure 2(a). An empirical fit of the form E _FEM  ≈ E_exact  + aM ⁻¹ is used to determine the 'exact' modulus; the linear nature of the graph (Figure 2a) for M > 21 confirms the ansatz. The error is less than 10 % for M 28, which corresponds to a strut thickness of 4 pixels. As the square beams have stress concentrations at the corners, we expect the code will perform better for model foams with rounded edges. The extrapolated numerical value E_exact  = 0.16 is close to the theoretical value E₁₀₀ =0.13 [(equation 3)], the difference being attributed to the finite density of the model (ρ / ρ _s = 0.14) and the assumption of clamped ends.

To check the density dependence of (equation 3) we measured the Young's modulus and Poisson's ratio of the simple cell model. We used the parameters L=60, w=20 µm and varied d over the range 1-15 µm at a resolution of 1 µm /pixel (giving a density range of ρ / ρ_s  =0.0024−.024). The results for E₁₀₀ are shown in Figure 2(b) and confirm that (equation 3) with C=2/3 is a reasonable approximation.

Figure 2: The Young's modulus of the cell model ( Figure 1b). (a) As a function of resolution. The computational results (symbols) approach the extrapolated 'exact' value as M increases. (b) As a function of density. The line is (equation 3) with C=2/3 (corresponding to L/w=3).

We also measured the properties of the open-cell tetrakaidecahedral model (Figure 1d) with cylindrical struts. We employed a unit cell size of M=80 pixels with side length T=80 µm and varied the cylinder radius in the range r=1-8 µm. The reduced density is given by the formula $\rho/\rho_s=12\pi r^2/ \sqrt2 T^2$. The results, shown in Figure 3(a), agree with the theoretical results [(equation 4) with C _Z =0.9] in the low density limit. In Figure 3(b) we show ν₁₂ as a function of density, as well as ν ₁₂ for the simple cell model (Figure 1b), which does not exhibit incompressible behaviour.

Figure 3: The properties of the tetrakaidecahedral cell model shown in Figure 1(d) as a function of density. (a) The Young's ($\circ$), bulk ($\Box$) and shear ($\triangle$) modulus. The lines correspond to (equation 4) with C_Z =0.9. (b) The Poisson ratio ν₁₂ ($\circ$) compared with theory (line) [(equation 4), C_Z =0.9]. The results ($\Box$) for the simple cell model shown in Figure 1(b) are also plotted.

For random foams, we also need to consider finite size effects. If there are too few cells in the computational cube, the estimates will not correspond to the properties of a macroscopic system (which may have many thousands of cells). Preliminary studies indicated that about 100 cells are necessary (roughly five cells in each direction) to keep the finite-size errors of the same order as the discretisation errors. Finally, one has to determine the number of samples N_s that need to be studied to ensure that the statistical variation of individual samples does not bias the results. For the sample size considered, we found five samples were sufficient. The resulting statistical errors are generally less than 10 %, but at the lowest densities we measure, can increase to 20 %.