Overlapping Solid Spheres

Realizations of the overlapping solid sphere model [14,21] are generated by placing solid spheres at random points in the unit cell. This produces a set of overlapping grains that mimic the microstructure of sintered ceramic composites (see Fig. 1a). The space outside the solid grains is the pore space, with porosity $\phi$. The pore phase is macroscopically connected above porosities of $\phi\approx 0.03$    0.03 and the solid phase remains connected for values of $\phi$ below $\approx 0.70$ 0.70 [14]. Above $\phi$ = 0.7, the solid phase is composed of isolated solid particles. So between $\phi = 0.03$  = 0.03 and $\phi = 0.70$ = 0.70, the overlapping solid sphere model is bi-continuous. In ceramics the porosity is generally less then 0.40, in this bi-continuous regime. We therefore consider the elastic properties for 0.1 $0.1 \leq \phi \leq 0.50$ 0.50, where the solid Poisson's ratio, $\nu _s$_s , varied over the range 0.1 $0.1 \leq \nu_s \leq 0.4$_s 0.4.

Figure 1: Showing pieces of the various models studied: (a) overlapping solid spheres, (b) overlapping spherical pores, and (c) overlapping ellipsoidal pores.

To generate the microstructure we chose solid spheres of radius r=1 µm. Note that the elastic properties are length scale invariant so the results apply to spheres of any radius for which the continuum assumption holds. A preliminary study showed that finite size errors were acceptably small for cubic samples with edge length T=12 µm. To study the discretization errors we generated one realization of the model with porosity $\phi$ = 0.5 at M = 48 $\dots$128. The elastic properties depend quite strongly on resolution. We found that the variation of Young's modulus with M could be described by the relation [22]

where E₀ can be identified as the continuum value (corresponding to infinitely large M). The same is true for Poisson's ratio. Even at M=128 the finite element code overestimates the 'exact' result for the Young's modulus by 30 %. Therefore, for the overlapping sphere model it is necessary to measure the elastic moduli at three different values of M and extrapolate the results to M $M\to\infty$ . We chose N_s=5 samples at each resolution and porosity, except at $\phi$ = 0.5 where large statistical variations implied a larger number of samples was necessary (N_s=10). Thus 30 different realizations of the models were considered, each at 3 different discretizations, for a total of 90 models.

The statistical variation in Young's modulus and Poisson's ratio for the case $\nu _s$ _s = 0.2 are shown in Table 1. The error bars shown in the table are equal to twice the standard error (S.E.= $\sigma/\sqrt{N_s}$ with $\sigma$ the standard deviation). Therefore there is a 95% chance that the "true" result lies between the indicated error bars. The results are accurate to within 20% at $\phi$ = 0.5; the error decreasing with porosity to less than 10% for $\phi \leq 0.30$ 0.30. The expected Gaussian distribution of the measured averages implies that the results are actually more accurate than this. For example, the anticipated relative errors are halved if a 68% likelihood threshold is used (i.e., ± one standard error).

Table 1: Elastic properties of the three models ($\nu _s$_s = 0.2).

	Overlapping solid spheres		Overlapping spherical pores		Overlapping ellipsoidal pores
$\phi$	E/Es	$\nu$	E/Es	$\nu$	E/Es	$\nu$
0.1	0.71 ± 1%	0.19 ± 1%	0.80 ± 1%	0.20 ± 1%	0.73 ± 2%	0.19 ± 3%
0.2	0.47 ± 2%	0.18 ± 4%	0.62 ± 2%	0.20 ± 2%	0.52 ± 3%	0.18 ± 4%
0.3	0.25 ± 6%	0.17 ± 9%	0.46 ± 3%	0.21 ± 3%	0.34 ± 4%	0.18 ± 6%
0.4	0.12 ± 13%	0.15 ± 25%	0.33 ± 4%	0.21 ± 4%	0.20 ± 3%	0.18 ± 4%
0.5	0.039 ± 22%	0.15 ± 21%	0.21 ± 8%	0.22 ± 9%	0.11 ± 4%	0.18 ± 6%

In addition to the above results we also computed the elastic moduli of the 90 model microstructures at solid Poisson's ratios $\nu _s$_s = 0.1, 0.3 and 0.4. The statistical variation was not significantly different from the case $\nu _s$_s = 0.2. Combined with the data for $\nu _s$_s = 0.2 this covers most commonly occuring solids. The scaled Young's modulus for each value of $\nu _s$ is plotted against porosity in Fig. 2. Remarkably, the scaled Young's modulus of the porous material appears to be practically independent of $\nu _s$_s . This result has been proven to be exact in 2-D [23,24] and appears to hold to a very good approximation in 3-D. We found that the Young's modulus data are well described by an equation of the form

$$\frac{E}{E_s}=\left( 1- \frac \phi\phi_0 \right)^n$$ (2)

with n=2.23 and $\phi _0$₀ = 0.652 and 0 $0 \leq \phi \leq 0.5$ 0.5. Note that n and $\phi _0$ ₀ are empirical correlation parameters and should not be interpreted as the percolation exponent and threshold, respectively. Percolation concepts are generally valid closer to the threshold _c 0.7 (for this model) and a higher value of n is expected. The computational cost of accurately measuring the elastic properties increases greatly as the percolation threshold is approached.

Figure 2: The Young's modulus of the three microstructure models. The solid lines are empirical fits to the equation E / E_s = (1 - / ₀ ) ⁿ. Data is shown for overlapping solid spheres $(\circ , n=2.23, \phi _0=0.652)$, spherical pores $(\Box , n=1.65, \phi _0=0.818)$ and ellipsoidal pores $(\triangle , n=2.25, \phi _0=0.798)$ for $\nu _s=-0.1,\dots ,0.4$_s =  -0.1,...,0.4. Note that the E is practically independent of the solid Poisson's ratio in each case [the different values of E ($E(\nu _s)$_s) at each porosity are almost indistinguishable].

The Poisson's ratio of the porous material is shown in Fig. 3 as a function of $\phi$ and $\nu _s$_s, and appears to be a flow diagram [23], where the Poisson's ratio asymptotically approaches a fixed point, independently of the value of the solid Poisson's ratio. This flow diagram has been analytically proven to hold in 2-D, when a percolation threshold exists at which the Young's modulus goes to zero [23,24]. This flow diagram also appears to be valid in 3-D as well, within numerical uncertainty. The Poisson's ratio data shown in Fig. 3 can be roughly described by the simple linear relation,

$$\nu =\nu_s+\frac \phi\phi_0 (\nu_0-\nu_s) = \nu_0 + \left(1- \frac \phi\phi_0\right)(\nu_s-\nu_0)$$ (3)

with two fitting parameters $\nu _0$₀ = 0.140 and $\phi _0$ ₀ = 0.472. A more accurate fit is obtained with the three parameter relation,

$$\nu= \nu_0 + \left(1- \frac \phi\phi_0\right)^m(\nu_s-\nu_0).$$ (4)

with $\nu _0$₀ = 0.140, $\phi _0$₀ = 0.500 and m=1.22.

Figure 3: The Poisson's ratio of the overlapping solid sphere model as a function of porosity for $\nu _s$_s = 0.1-0.4. The dashed lines are an empirical fit to Eq. (3). The solid lines correspond to the three parameter relation given in Eq. (4), with the value of all parameters given in the text. The intercepts of the lines at zero porosity correspond to the solid Poisson's ratio.