Young's Modulus

Young's modulus of the four GRF models and three Boolean models is shown in figures 3 and 4, respectively. Each data point represents an average over five samples. About 10⁴ hours of CPU time, on various workstations, were used to generate the results presented in this paper. Rather than tabulating the data, the results are reported in terms of simple empirical structure-property relations. We found that the data of each of each model could be described by the form

$$\frac E{E_s}=\left(\frac{p-p_0}{1-p_0}\right)^m.$$ (4.1)

A non-linear least squares fitting program was used to determine p₀ and m. The fitting parameters, obtained for a solid Poisson's ratio of _s = 0.2, are reported in Table 2. The relative error for any data point is generally around a few percent or less. Note that the fitting parameter p₀ is not the percolation threshold p_c. For example, p₀ 0.26 for the single-cut GRF model, but p₀ 0.13 (Roberts & Teubner, 1995). Clearly, large errors would occur if equation (4.1) were used to extrapolate the data.

Figure 3. Finite element data and lines of best-fit to equation (4.1) for the four GRF models:
single-cut (), two-cut (), open-cell (), and closed-cell () (semi-log axes).
Data for the two-cut and closed-cell models are nearly conincident.

 

Figure 4: Finite element data and lines of best-fit to equation (4.1) for the three Boolean models:
spherical pores ($\Box$), oblate spheroidal pores ($\triangle$), and solid spheres ($\circ$).

Figure 5: Variation of Young's modulus with solid Poisson's ratio at all solid fractions. Data is shown at all volume fractions studied. Overlapping spheres and oblate spheroids ($\circ$), overlapping solid spheres ($\diamond$), two-cut and closed-cell GRF models ($\Box$), single-cut GRF model ($\triangle$), and open-cell GRF model ( $\bigtriangledown$). For comparison the results of the SCM at p = 0.9 (---) and p = 0.6 are shown (- - -).

By definition, the two-cut, open-cell and closed-cell GRF models have no percolation threshold (i.e. p_c = 0). A plot of E/E_s vs. p on bi-logarithmic axis (not shown) revealed a straight line for small p indicating that data for these models can be extrapolated using the equation

$$E/{E_s} \approx C p^n.$$ (4.2)

 

Table 2: Simple structure property relations for 7 different model porous materials considered in this paper. For p > pmin, we find the data can be described by E/Es = [(p - p0)/(1 - p0)]m to within a few percent. For the last three models, the data can be extrapolated using the power law E/Es = Cpn for p < pmax. The Poisson's ratio can be approximately described by relation (4.3) with parameters $\nu _1$1 and p1. In certain cases (*), formulae (4.4) and (4.5) should be used if more than a rough estimate is needed.
		p < pmax			p < pmin			$\nu$
Model	Fig.	n	C	pmax	m	p0	pmin	$\nu _1$ 1	p1
solid spheres*	1(a)				2.23	0.348	0.50	0.140	0.528
spherical pores	1(b)				1.65	0.182	0.50	0.221	0.160
oblate pores*	1(c)				2.25	0.202	0.50	0.166	0.396
single-cut GRF	2(a)				1.64	0.214	0.30	0.184	0.258
two-cut GRF	2(b)	1.58	0.717	0.50	2.09	-0.064	0.10	0.220	-0.045
open-cell GRF	2(c)	3.15	4.200	0.20	2.15	0.029	0.20	0.233	0.114
closed-cell GRF*	2(d)	1.54	0.694	0.40	2.30	-0.121	0.15	0.227	-0.029

 

The value of C and n for each model are given in table 2.

In figures 3 and 4, the slight variance observed at each volume fraction corresponds to varying the matrix Poisson's ratio. Evidently, as anticipated from the rigorous two dimensional results, the Young's modulus is nearly independent of the solid Poisson's ratio, i.e., E(p,_s) E(p). To quantify the weak dependence we plot E(p,_s) / E(p,_s = 0.2 vs. _s for the four GRF models and three Boolean models in figure 5. For 0 _s 0.4, the relative variance is less than 4%, and we presume that the variance will not be significantly larger for 0.4 < 0.5. Since virtually all solid materials have a solid Poisson's ratio in the range 0 _s 0.5, we conclude that the Young's modulus can practically be regarded as being independent of the solid Poisson's ratio. As $\nu _s$_s decreases, the maximum variance increases to 15 % at $\nu _s=-0.3$_s = -0.3. In the graph, we also show the SCM for spherical inclusions. The weak dependence of E(p,_s) on $\nu _s$_s in the data is qualitatively similar to the SCM theory.

Figure 6: 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 (- $\cdot$ -).

Figure 7: Hashin-Shtrikman (- -) upper bounds, 3-point (---) upper bounds, and Torquato's expansion ($\cdots$) versus the finite element data. The solid Poisson's ratio is $\nu _s=0.2$_s = 0.2. (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 figure 6, we make a quantitative comparison between data for overlapping spherical and oblate pores and the relevant effective medium theories. For p $p\geq0.9$ 0.9 (the dilute limit), all the theories give similar predictions and conform with the data. For both pore shapes the DEM performs significantly better than the SCM, providing a reasonable prediction for p $p\geq0.6$ 0.6. This might be anticipated from the close similarity between the definition of the models and the assumptions of the differential method. In both the models and DEM, the volume fraction is decreased by adding pores that are uncorrelated to the existing microstucture.

We compare the Hashin-Shtrikman bound, the 3-point bound and Torquato's expansion with the FEM data in Fig. 7. In all cases the data fall below the bounds. It has been argued (Torquato, 1991, §3.5) that the 3-point upper bound and expansion (Torquato, 1998) will provide a reasonable prediction if the pores are isolated. This is only true for the closed-cell model, and the data are well predicted by the expansion for p $p\geq 0.5$ 0.5. Even when the pores are interconnected, the expansion provides a reasonable prediction for p $p\geq0.6$ 0.6 in all but the case of overlapping solid spheres. Note that large relative differences between the expansion and data occur at lower volume fractions (these become more evident on bi-logarithmic plots).

Figure 8: Poisson's ratio (p,$\nu (p,\nu _s)$_s) of the single-cut GRF model plotted against p and $\nu _s$_s. In (a) the intercept of the data with the vertical axis at p = 1 corresponds to $\nu _s$_s. In (b) the dashed line shows the behaviour of a non-porous solid, the next line up (on the $\nu _s=-0.3$_s = 0.3 axis) is for p = 0.9 and so on up to p = 0.3. At p = 0.3 the Poisson's ratio is seen to be nearly independent of $\nu _s$_s. The symbols are data points and the lines are a best-fit to equation (4.3)

Figure 9: Generic flow pattern behaviour of the Poisson's ratio of porous models $\nu$ as a function of solid fraction p; (a) overlapping spherical pores, (b) overlapping solid spheres, (c) overlapping oblate spheroidal pores, (d) two-cut GRF, (e) open-cell GRF, (f) closed-cell GRF. The solid lines are numerical fits to equation (4.3). In cases (b), (c), and (f), a better fit is obtained with equation (4.4) (- -) or equation (4.5) ($\cdots$).