Poisson's Ratio

In figures 8(a) and (b) we plot ν(p, ν_s) for the single-cut GRF model as a function of p and ν_s, respectively. Two striking features are evident. In (a), a flow diagram is observed, with ν(p, ν_s) converging from ν(1, ν_s) = ν_s to a fixed point ν>₁ ≈ 0.2 as p decreases. The behaviour is very similar to the rigorous behavior in two-dimensions (Day et al. , 1992). (Thorpe and Jasiuk 1992) showed that though the flow behavior is rigorous, the actual value of the fixed point depends on the microstructure, specifically on how material is removed to achieve the percolation threshold. In (b), the lines, which correspond to fixed p, appear to rotate about a 'critical' point, which we call ν*. On the flow diagram, this point would be represented by a horizontal line, indicating that for ν_s = ν* Poisson's ratio of the porous material does not depend on the solid fraction, or ν(p, ν*) ≈ ν*. Since both the behaviors in (a) and (b) exist, evidently ν* and ν₁ must be identical.

However, the behavior observed in (b) is not a simple consequence of that in (a). A flow diagram can exist without all the flow lines, when plotted vs. matrix Poisson's ratio, passing through a single point. When this point exists and is found to be on the ν = ν_s line, it means that for a given value of matrix Poisson's ratio, the porous body Poisson's ratio is the same. As was mentioned in a previous section, this kind of behavior, along with the flow to a fixed point, is seen in both the SCM and DEM effective medium theories, as well as in the spherical pore dilute limit. The present numerical results indicate that this behavior appears in a realistic porous microstructure as well.

It is important to note at this point that the value of ν* is the same for each value of p, including the dilute limit (p ≈ 1). For values of p in this range, the value of ν* is simply predicted from the dilute limit. This remarkable result implies that the dilute limit can tell us something about the elastic behavior at the critical point! This is non-intuitive behavior, and requires thought. If the result only held true for models generated by iteratively using a single defect shape, like a spheroid, one might guess that this shape controlled the microstructure even at porosities approaching the critical point. But this result is true, as is shown below, for most of the Gaussian models as well, which are not made from a given shape.

To a reasonable approximation it appears that ν(p, ν_s) is a linear function of both p and ν_s. The above considerations indicate that the data can be described by the relation

$$\nu=\nu_s+\frac{1-p}{1-p_1}(\nu_1-\nu_s).$$ (4.3)

A least-squares procedure was used to determine p₁ = 0.258 and ν₁ = 0.184. The function, which is shown on figures 8, is seen to provide a reasonable fit.

In figure 9, we show that Poisson's ratio of the other six models have qualitatively similar flow-diagrams. Five of the models show a critical point (not shown) just as in figure 8 (b) in plots of ν vs. ν₂. In figure 10, it is seen that the closed-cell GRF model, in contrast, does not show a critical point. However, for all six models there appeared to be a linear relationship between ν and ν_s. The fitting parameters ν₁ and p₁ for each model are given in table 2. In three cases (b, c, and f), some non-linearity in the parameter p is evident, and equation (4.3) only provides a rough fit. To obtain a better fit we tested several generalisations of equation (4.3). The form,

$$\nu=\nu_s+\left(\frac{1-p}{1-p_1}\right)^k(\nu_1-\nu_s).$$ (4.4)

was able to provide a reasonable fit of the data for overlapping oblate pores (ν = 0.161, p₁ = 0.041, and k = 1.91) and overlapping solid spheres (ν₁ = 0.140, p₁ = 0.14, and k = 1.22). In the case of overlapping solid spheres, the error bars (not shown) are significantly larger than in the other models, and on the order of 50% at the largest porosity shown. The results are shown as a dashed line in figures 9(b) and (c). For the closed-cell GRF model, the simplest form able to fit the data was

$$\nu= A(1-p)+\nu_s \left( p+ B (1-p) + C (1-p)^2\right)$$ (4.5)

with A = 0.221, B = -0.210, and C = 0.342. The results are shown in figures 9(f) and  10. In contrast to equations (4.3) and  (4.4), this form does not have a critical point, and extrapolation to p = 0 gives ν = A + ν_s (B + C), i.e. Poisson's ratio does not become independent of ν_s in this formula. We would expect, however, that real data, if it could be accurately generated in this low pregime, would indeed flow to a fixed point. Again we emphasise that the flow to a fixed point is probably more generic behaviour than is the 'rotation' about a critical point, and indeed there is no reason that flow behaviour need imply the critical point behaviour.

Figure 10:The Poisson's ratio of the closed-cell model does not show a critical point.

In figure 11 we compare the SCM and DEM results to data for overlapping spherical and oblate pores. As for Young's modulus, all the formulas work well for p $p\geq0.9$ 0.9, the dilute limit, since they are all based on the exact expression in the dilute limit. The DEM performs reasonably well for p $p\geq 0.7$ 0.7, outside the dilute limit. The SCM theory, however, badly misrepresents the shape of the Poisson's ratio flow diagram for most values of p, at least in part due to the incorrect percolation threshold built into the SCM for spherical pores.

Figure 11: Comparison of various theories to FEM data for overlapping spherical pores> (a) and oblate spheroidal pores (b): SCM ($\cdots$), DEM (--), Christensen's SCM (- -), and Wu's SCM (- $\cdot$ -).

Torquato's truncated expansion is compared to the numerical data in figure 12. This expansion is truncated at the third order because of the difficulty of evaluating the correlation functions needed for the higher order terms. Apart from the case of solid spheres, the predictions are generally very good for p $p\geq 0.7$ 0.7. Since the expansion is not built explicitly upon any dilute limit, it makes sense to use it to compare to the Gaussian random field models. For the closed-cell GRF model, the predictions are very good for a larger range of p ( $p\geq 0.3$ 0.3). Note that the closed-cell model has isolated pores by definition. This supports Torquato's hypothesis that the truncated expansion is likely to work for dispersions of isolated pores. However, even when the pores are connected, the stress is still carried by the solid phase, which might be the reason that the expansion seems to work well even for systems with connected pores  (Torquato, 2001). At high solid fractions Torquato's expansion is able to quantitatively reproduce quite subtle non-linearities in the Poisson ratio plots of the different models, which are a much more stringent test of the theory. In addition to confirming Torquato's results, this provides strong evidence that our FEM, model simulations, and method of calculating the microstructural correlation functions and parameters (ζ and η) are quite accurate.

Figure 12: Torquato's expansion (--) versus the finite element data. (a) overlapping spherical pores, (b) overlapping solid spheres, (c) single-cut GRF, (d) two-cut GRF, (e) open-cell intersection set GRF, and (f) closed-cell union set GRF.

In figures 13 and 14, we show that all but one of the theoretical results exhibits the critical point behaviour we empirically observed in the data. Christensen's SCM, which is thought to model closed-cell materials when the inclusions are pores (Christensen, 1998), does not exbibit a critical point. In this respect it is similar to our data for the closed-cell GRF model. For spherical pores the SCM and DEM have exact critical points at ν_s = 0.2 (Garboczi & Day, 1995). For oblate pores Wu's generalisation of the SCM and the DEM have an approximate critical point at ν ≈ 0.14. Berryman's generalisation of the SCM has an approximate critical point at ν ≈ 0.17 in good agreement with the measured value of overlapping oblate pores (ν₁ = 0.166).

Figure 13: Critical point behaviour of various effective medium theories. Spherical pores; (a) SCM, (b) Differential method (c) Christensen's SCM. Oblate pores; (d) SCM, (e) Differential method (f) Wu's SCM.

Figure 14: Torquato's expansion has an approximate critical point at ν₁ ≈ 0.2. (a) overlapping spherical pores (ν₁ = 0.22), (b) overlapping solid spheres (ν₁ = 0.19), (c) single-cut GRF (ν₁ = 0.21), (d) two-cut GRF (ν₁ = 0.23), (e) open-cell intersection set GRF (ν₁ = 0.23), and (f) closed-cell union set GRF (ν₁ = 0.23).

Torquato's expansion revealed an approximate critical point for all six models near ν = 0.2. For overlapping spheres, and the two-cut, open-cell and closed-cell GRF models the prediction was accurate to within ±0.01. The form (2.13) in which we have reported Torquato's result shows the mathematical origin of the critical point. At ν_s = 0.2, ( ν − 0.2) $\nu-0.2) \propto (\zeta-\eta)$ (ζ − η) which is generally quite small (see table 1). It is instructive to linearise Torquato's result ν = g_T(p, ν_s) in terms of the variables µ = ν_s − 1/5 and θ = ζ − η which gives

$$\frac{\nu-\nu_s}{1-p}= \frac{9\theta-12(\zeta+4)\mu}{50\zeta(p\zeta+2-2p)} +O(\theta^2)+O(\mu^2) +O(\mu\theta).$$ (4.6)

This form predicts ν ≈ ν_s at µ* = 3θ / (4ζ + 16) which is weakly dependent on p via ζ and θ. Surprisingly, the absolute difference between ν = 1/5 + µ* and exact numerical solution of ν − g_T(ν, p) = 0, was less than 0.0015 for all models. For | θ | | µ | $\vert\theta\vert \ll \vert\mu\vert \ll 1$ 1 equation (4.6) predicts ν − ν_s (ν_s − 1/5) which corresponds to the cross-over behaviour in figure 14.