Comparison with micro-mechanical and MSA formulae

In this section we compare a selection of well known theoretical results with the 'numerically exact' data computed in the previous section. These results include both analytically exact results (bounds, expansions, dilute limits, composite sphere assemblage), as well as approximate results (effective medium theories, minimum solid area models).

There are several kinds of exact bounds that have been derived for elastic materials [5,14]. These are equations involving the different phase moduli, the volume fractions of the different phases, and various correlation functions that define the geometry of the composite. The upper bound gives the maximum possible composite elastic moduli, and the lower bound gives the lowest possible composite elastic moduli. The bounds used in this paper are three- point bounds, which have been written out explicitly for overlapping solid spheres and overlapping spherical pores [14]. In the case where one phase has zero elastic moduli, as is true in this paper, the lower bound becomes zero as well, and so only the upper bound is meaningful.

An exact perturbation expansion also exists, where the elastic moduli of a two-phase material is expanded in terms of parameters involving the individual elastic moduli of each phase and geometrical quantities [27,28]. This expansion has been carried out to three terms explicitly, and it is this truncated form to which we will compare our numerical data.

Another exact result, which is used later in this section to build the various effective medium theories, is the case of dilute spherical pores for which the exact effective moduli are given by

K = K_m + c _i P^mi (K_i-K_m) (5)

G = G_m + c_i Q^mi (G_i-G_m) (6)

with

$$\frac{3K_m+4G_m}{3K_i+4G_m},\;\; Q^{mi}=\frac{G_m+F_m}{G_i+F_m},$$ P^mi =

$$\frac{G_m}6 \frac{9K_m+8G_m}{K_m+2G_m}.$$ F_m = (7)

Here c_i denotes the concentration (volume fraction) of inclusions and the subscripts i and m on the bulk K and shear modulus G denote the properties of the inclusion and matrix, respectively. The result is attributed to numerous authors [5]. For a porous matrix K_i=G_i=0 and the porosity is $\phi=c_i$ = c_i. The result is strictly valid for small concentrations of inclusions $\phi\ll 1$ 1 (in practice $\phi < 0.1$ < 0.1.) Cast in terms of the engineering constants for porous inclusions this result becomes

$$E_m - \frac32 \phi E_m \frac{9-4\nu_m-5\nu_m^2}{7-5\nu_m} + O(\phi^2)$$ E = (8)

$$\nu \nu_m - \frac32 \phi \frac{(5\nu_m-1)(1-\nu_m^2)}{7-5\nu_m} + O(\phi^2)$$ = (9)

Our prior statement[29] of Eq. (8) inadvertently omitted the factor of 3/2, although the correct result was used in the paper. A non zero quadratic term can be added (as an empirical correction) to ensure that E = 0 at $\phi$ = 1. This was suggested by Coble and Kingery [1] for MacKenzie's [30] result for spherical pores, which is equivalent to Eqs. (5)-(7) with K_i=G_i=0.

To adapt the dilute formulas to the case of finite porosity a number of proposals have been made. The approximate equations that result are usually called effective medium theories. The most common approximation is the so-called self consistent method (SCM) of Hill [9] and Budiansky [10]. In this model the equations of elasticity are solved for a spherical inclusion embedded in a medium of unknown effective moduli. The effective moduli K and G are then derived. In the dilute case the embedding medium is just the matrix. The Hill-Budiansky result can be stated as [12]

c_i P^*i(K_i-K_*) + c_m P^*m(K_m-K_*) = 0 (10)

c_i Q^*i(G_i-G_* ) + c_m Q^*m(G_m-G_*) = 0 (11)

where K_* and G_* denote the effective moduli and P^*m and Q ^*m are given in Eq. (7). The equations cannot be explicitly solved and numerical methods are necessary (see Hill [9] and Berryman [12] for details). In the case of porous inclusions, the moduli vanish at $\phi=\frac12$ = 1/2, which is a property not shared with most composites (e.g., the overlapping sphere model). To derive a more realistic result, Christensen and Lo [31] generalized the SCM (GSCM) to the case of a spherical shell embedded in a matrix of unknown moduli. The result is complicated and not reproduced here.

The differential method (reviewed by McLaughlin [8]) provides an alternative model using a similar philosophy. Suppose that the effective moduli of a composite medium are known to be K_* and G_*. Now if a small additional concentration of inclusions are added, the change in K_* and G_* is approximated to be that which would arise if a dilute concentration of inclusions were added to a uniform, homogeneous matrix with moduli K_* and G_*. This leads to a pair of coupled differential equations,

$$\frac{dK_*}{d c_i} P^{*i} \frac {K_i-K_*}{1-c_i};\; K_*(c_i=0)=K_m$$ = (12)

$$\frac{dG_*}{d c_i} Q^{*i} \frac {G_i-G_*}{1-c_i};\; G_*(c_i=0)=G_m.$$ = (13)

The dilute result, the self consistent result [12], and the differential method [8] can all be extended to the case of spheroidal inclusions. The general results [11] for P^mi and Q^mihave been given by Berryman [12]. In addition to these results Wu [11] derived a variant of the self consistent method, where K_* and G_*, the effective moduli, are found by implicitly solving the equations

K_* = K_m + c_i P^*i (K_i-K_m) (14)

G_* = G_m + c_i Q^*i (G_i-G_m). (15)

A different type of microstructure is provided by Hashin's [32,5] model of space-filling poly-disperse hollow spheres (the "composite-sphere assemblage"). Although a simple formula exists for the bulk modulus over the full porosity range [32], exact results for the Young's moduli are not available. Ramakrishnan and Arunachalam [33] recently derived the approximation

$$\frac{E}{E_s} \frac{(1-\phi)^2}{(1+2\phi-3\nu_s\phi)}$$ = (16)

$$\nu \frac{(4\nu_s+3\phi-7\nu_s\phi)}{4(1+2\phi-3\nu_s \phi)}.$$ = (17)

However, the derivation is not rigorous. In particular, the exact result for the bulk modulus of the model [32] is around twice that predicted by Eqs. (16-17) at $\phi=0.5$ = 0.5. Since Eq. (16) was found to provide reasonable agreement with experimental data for porous ceramics [17], we compare its predictions to our FEM data below.

The final class of results we consider is provided by the 'minimum solid area' (MSA) models [34] (which have been recently reviewed by Rice [3,13]). This approach is based on the assumption that the ratio of the effective moduli to the solid moduli is directly proportional to the minimum ratio of solid contact area to the total cross-sectional area of periodic structures. The approximation derived depends on the particular model considered. We consider two basic models most closely aligned with our FEM data: simple cubic arrays of solid and porous spheres. The latter case provides a particularly simple example of the type of result which can be derived. Suppose the repeat distance of the lattice is 2h and the sphere radius is r. The Young's modulus is assumed to be proportional to the area fraction, giving

$$\frac{E}{E_s}=\frac{(2h)^2-\pi r^2}{(2h)^2}=1-\frac\pi4 \left( \frac{6}{\pi} \right)^{\frac23} \phi^\frac23$$ (18)

since = 1/6 (r/h)³ . The form of the result changes for r>h (or $\phi>\pi/6$ > / 6 = 0.52) as the spheres begin to coalesce. Rice [3] has noted the results of many different periodic structures can be approximated by the form E / E_s = e ^-b over a range of porosities. For example, b $b\approx5$ 5 for the solid sphere model and b $b\approx3$ 3 for the porous sphere model. It is argued that for a given set of data, b can be compared with known values to assess the type of porosity. Often fractions of different types of porosity are assumed to match experimental data making the method an interpretive rather than a predictive tool. Since we have measured E for microstructures based on solid sphere contacts and porous spheres we should be able to ascertain the accuracy of the MSA formulae for these cases.

Fig. 6 shows the comparison between the exact three-point bounds for the overlapping solid sphere and spherical pore case, the truncated expansion for the overlapping spherical pore case, and the numerical results. Clearly the expansion does better than the three-point bound for the overlapping spherical pore case, though both are fairly close to the numerical results. The bound lies far away from the overlapping solid sphere numerical results, however. For this case, the exact expansion does not exist. Only the $\nu_s = 0.2$_s = 0.2 data is shown. Using the exact expansion, one can show that in 3-D, the Young's modulus is not exactly independent of the solid Poisson's ratio, but is rather a very good approximation, as was shown earlier in this paper.

Figure 6: A comparison of rigorous bounds and expansions to the FEM data for overlapping spherical pores ($\circ$) and overlapping solid spheres ($\Box$). The exact expansion (---) and the three-point upper bound (- - -) are shown for the spherical pore case. Only the three-point bound ($\cdots$) is shown for the solid sphere case. The three-point lower bound is zero for porous materials. The Poisson's ratio is 0.2 for all the results.

In Fig. 7, we compare the FEM data ($\nu_s = 0.2$_s = 0.2) for overlapping spherical pores with dilute and effective medium theory analytic results. At this Poisson's ratio the SCM and dilute results reduce to E / E_s = 1 - 2 $E/E_s=1-2\phi$ while the differential and dilute results with the Coble-Kingery correction reduce to E / E_s =  (1 - $E/E_s=(1-\phi)^2$)² . Since the analytic results are based on the case of dilute spherical pores they all match the FEM data at $\phi=0.1$ = 0.1. The dilute and SCM results under-estimate the FEM data at higher porosities because of the aphysical percolation threshold at $\phi=\frac12$ = 1/2. The generalized SCM over-estimates the data, while the differential method performs reasonably well over the entire porosity range. The latter observation might have been anticipated given the close association between the definition of the spherical pore model and the assumption of the differential method. At increasing porosities we are simply adding additional spherical pores to a porous matrix. The data for overlapping solid spheres is also shown in the figure, and seen to be quite different from any of the available results. This demonstrates that microstructure (the geometrical nature of the porosity) is an important factor besides the actual value of the porosity.

Figure 7: A comparison of different theories to the FEM data for overlapping spherical pores ($\circ$). The lines correspond to the dilute result and self consistent method [9,10] (or SCM) (- - -), the differential method [8] and dilute result with Coble-Kingery correction (---) and the generalized SCM [31] (-- -- --). Data for the overlapping solid sphere model (for which no rigorous theories exist) are also shown ($\Box$).

In Fig. 8, the minimum solid area models and the Ramakrishnan and Arunachalam results [33] are compared with the data. The MSA model for spherical pores performs reasonably well, although underestimating the FEM data for overlapping random spherical pores at low porosities $\phi <0.3$ < 0.3. The MSA model for solid spheres considerably underestimates these data for $\phi <0.3$ < 0.3. The Ramakrishnan and Arunachalam [33] approximation falls between the FEM data for $\phi>0.1$ > 0.1 indicating that it corresponds to neither of the microstructures.

Figure 8: A comparison of the minimum solid area (MSA) models [3] to the FEM data for overlapping spherical pores ($\circ$) and solid spheres ($\Box$). The MSA solid sphere model (---) and MSA porous sphere model (- - -) (in simple cubic packings) are seen to under-estimate the data for low porosities ($\phi <0.3$ < 0.3). The formula of Ramakrishnan and Arunachalam [33] E / E_s = (1 - )² / (1 + 1.4$E/E_s=(1-\phi )^2/(1+1.4\phi )$ ) is also shown ($\cdots$).

The FEM data for overlapping oblate ellipsoidal pores is compared with the available theories in Fig. 9. The SCM results of Wu [11] and Berryman [12] underestimate the porosity as a result of underestimating the physical percolation threshold. The Berryman result performs significantly better than does the Wu result. As for the case of spheres, the differential method matches the data quite closely because of the similarity between the assumptions of the theory and the definition of the model.

Figure 9: A comparison of different theories to the FEM data for overlapping oblate ellipsoidal pores ($\circ$). The lines correspond to the differential method [8] (---) and the self consistent methods of Wu [11] (- - -) and Berryman [12] ($\cdots$).

We have also compared the Poisson's ratio predicted by the various self-consistent and differential methods to the FEM data for overlapping spherical and ellipsoidal pores. The theoretical results converge to different fixed points (e.g. Fig. 3) in qualitative agreement with the data. But only the differential method provides reasonable agreement with the FEM data (with absolute error less than 0.02 for 0.4 and 0.1 _s 0.4 ).