Exact bounds and expansions

There are several kinds of exact bounds that have been derived for the elastic properties of composite materials (see the reviews of (Torquato 1991) and (Hashin 1983)). If the properties of each phase in a composite are not too dissimilar, the bounds can be quite restrictive. For porous materials, however, the bounds on Poisson's ratio are no more restrictive than the range guaranteed by the non-negativity of K and G for isotropic materials ( -1 $-1\leq \nu \leq 0.5$ 0.5), and the lower bound on E reduces to zero. The upper bound on E is sometimes found to provide a reasonable approximation of the actual property. The most commonly applied bounds for isotropic composites are due to (Hashin and Shtrikman 1963). The upper bound E_u is

$$\frac{E_u}{E_s} = \frac{p}{1+C(1-p)};\;\; C=\frac{(1+\nu_s)(13-15\nu_s)}{2(7-5\nu_s)}.$$ (2.8)

Thus the Hashin-Shtrikman bounds imply 0 < E < E_u and -1 < $-1< \nu <0.5$ < 0.5, and only depend on microstructure via the volume fraction.

It is possible to improve the bound if more statistical information, in the form of N-point correlation functions, is available for the composite. The 2-point correlation function p₂(r) represents the probability that two points a distance r apart will fall in the solid phase. The 3-point correlation function p⁽³⁾ (r, s, t) is equal to the probability that three points distances r, s and tapart all belong to the solid phase. Bounds that depend on this information are referred to as 3-point bounds. The form of the 3-point bounds (Beran & Molyneux, 1966), (Milton & Phan-Thien, 1982) is quite complex, but to show their qualitative behaviour we report the result for porous media where the solid matrix has Young's modulus E_sand Poisson's ratio $\nu _s=0.2$_s = 0.2. In this case the upper bound becomes

$$\frac{E_u}{E_s}=\frac{p}{1+C(1-p)};\;\; C=\frac{33\eta+7\zeta}{5\zeta(9\eta-\zeta)}.$$ (2.9)

The three-point bounds on the Poisson's ratio are -1 < $-1< \nu <0.5$ < 0.5. The 'microstructure parameters' $\zeta$ and $\eta$ are determined by (Milton & Phan-Thien, 1982)

$$\zeta \!\!\frac9{2pq}\int_0^\infty\!\!\frac{dr}{r} \int_0^\infty \!\! \... ...\!\!\! du P_2(u) \left( p^{(3)}(r,s,t)-\frac{p^{(2)} (r) p^{(2)}(s)}{p} \right)$$ = (2.10)

$$\eta \!\!\frac 5{21}\zeta+ \frac{150}{7pq}\int_0^\infty\!\!\frac{dr}{r... ...\!\!\! du P_4(u) \left( p^{(3)}(r,s,t)-\frac{p^{(2)}(r) p^{(2)}(s)}{p} \right).$$ = (2.11)

where t ² = r ² + s ² - 2rsu and P₂(u) = ¹/₂(3u² - 1) and P₄(u) = ¹/₈(35u⁴ - 30u² + 3) are Legendre polynomials. A useful numerical listing of these parameters for various systems is given in (Torquato, 1991).

(Torquato 1998) has recently derived predictive formulae for arbitrary composites in the form of exact expansions. For a porous medium, where the solid matrix has Poisson's ratio of $\nu _s=0.2$_s = 0.2, the results simplify to

$$\frac{E}{E_s}=\frac{p}{p+(1-p) C};\;\; C=\frac{74\zeta+6\eta}{5\zeta(5\zeta+3\eta)}$$ (2.12)

$$\nu=\frac15+ \frac{36(\zeta-\eta)} {25\zeta(5\zeta+3\eta)}\times\frac{1-p}{p+ C(1-p)}.$$ (2.13)

A clear advantage of the bounds and expansions is that they incorporate microstructural information, so can be applied to arbitrary composites. In principle, it should be possible to increase the accuracy of both methods by incorporating more statistical information. However, in practice, third-order information is only available for a restricted number of models, and it is very difficult to include fourth or higher-order information. In the truncated forms given above, the results are thought to be accurate for dispersed inclusions (Torquato, 1998),(Torquato, 1991, §3.5). Their accuracy for interpenetrating porous media is not known.

In this section we have described a range of well known theories for predicting the elastic properties of random porous materials. We have shown that for materials with interconnected pores, none of the theories can be confidently used to predict properties or interpret experimental structure-property relationships. Therefore, in order to apply the theories, it is necessary to check their range of validity. We do this by computationally studying the properties of several well known models that span a wide range of physically observed microstructure. These numerical results also reveal behaviour that, to a good approximation, is quite similar to rigorous 2-D behaviour. The next section introduces the kinds of microstructure considered.