Estimation schemes based on bounds for elasticity

One very famous approximation scheme for elastic composites is due to Hill (1952). The idea is to take the known Voigt and Reuss averages of the elastic system stiffnesses or compliances, and then make direct use of this information by computing either the arithmetic or geometric mean of these two limiting values. These formulas have been found to be very effective for fitting real data in a wide variety of circumstances (Simmons and Wang, 1971; Thomsen, 1972; Watt and Peselnick 1980). Clearly the same basic idea can be applied to any pairs of bounds for scalars, such as the Hashin-Shtrikman bounds; or, for complex constants, a similar idea based on finding the center-of-mass of a bounded region in the complex plane could be pursued (but to date apparently has not been). The advantage of such approaches is that they can provide the user with just one estimate per choice of volume fraction, while at the same time requiring no additional information over that contained in the bounds themselves.

Hill's concept clearly works just as well, and possibly somewhat better, if we apply it instead -- whenever we have an analytical function at our disposal as we do here in the canonical functions -- to the transform variables $\beta$ and $\theta$ rather than to the moduli K and $\mu$ directly. So one set of estimates we might test in our examples takes the form  

$$\beta_H \equiv \frac{1}{2}(\beta_- + \beta_+) \quad\hbox{and}\quad \theta_H \equiv \frac{1}{2}(\theta_- + \theta_+),$$ (10)

where the bounds on $\beta$ and $\theta$ were already given in TABLE 1, and the averages are just the arithmetic means. The subscript H is intended to reference Hill's contribution to this idea.

Another rather different approach (although still expected to give quite similar results) is to examine the forms of the $\beta$ and $\theta$ transform variables in order to determine if some other estimate that lies between the bounds might suggest itself. One useful tool we can introduce here is the weighted geometric mean. For example, if we define  

$$\mu_G^\zeta \equiv \mu_1^{\zeta_1}\mu_2^{\zeta_2},$$ (11)

it is well-known (Hardy et al., 1952) that this is a geometric mean and it always lies between (or on) the corresponding mean $\left<\mu\right\gt _\zeta$ and harmonic mean $\left<\mu^{-1}\right\gt^{-1}_\zeta$:  

$$\left<\mu^{-1}\right\gt^{-1}_\zeta \le \mu_1^{\zeta_1}\mu_2^{\zeta_2} \le \left<\mu\right\gt _\zeta.$$ (12)

So $\beta_G = \frac{4}{3}\mu_G^\zeta$ is one natural choice we could make for the bulk modulus transform parameter estimate. This approach has one clear advantage over the usual self-consistent estimates in that the microstructural information can easily be incorporated this way, whereas the means of doing so for self-consistent methods usually involves more complicated calculations via scattering theory (Gubernatis and Krumhansl, 1975; Berryman, 1980). This approach also provides a formula, rather than an implicit equation requiring an iteration procedure for its solution, thus eliminating another common criticism of implicit estimators.

Similar results are not as easy to find for the shear modulus bounds. The reason is that there are either two or three averages that come into play for shear, always including $\langle\cdot\rangle_\zeta$ and $\langle\cdot\rangle_\eta$,while the formulas (5) and (6) also depend on the usual volume averages $\langle\cdot\rangle$. Since it is known that the McCoy-Silnutzer bounds are never tighter than those of Milton and Phan-Thien (1982), we will consider only the Milton and Phan-Thien bounds from here on, since they have only two types of averages present.

In general $\zeta_i$ and $\eta_i$ differ. But in some cases (spheres and disks, for example) they are the same. Furthermore, it is easy to show that for any modulus M, we have the result (relevant in particular to needles) that  

$$\begin{array} {r} \left<M\right\gt _\eta - \left<M\right\gt ... ...ght\gt\right] \ = \frac{1}{12}(v_1-v_2)(M_1-M_2). \end{array}$$ (13)

Thus, the differences always vanish for 50-50 concentrations, and furthermore the factor of $\frac{1}{12}$ reduces the difference further by an order of magnitude. If we make the approximation that $\langle\cdot\rangle_\eta \simeq \langle\cdot\rangle_\zeta$, this is often a quite reasonable compromise. When this is so, we can then choose to make the further approximations that  

$$\left<M\right\gt _\zeta \simeq M_G^\zeta = M_1^{\zeta_1}M_2^{\zeta_2},$$ (14)

and also that  

$$\left<M^{-1}\right\gt _\zeta \simeq M_G^{-\zeta}.$$ (15)

Substituting these approximations into the Milton and Phan-Thien bounds (7) and (8), we find that both transform parameters for the upper and lower bounds are replaced by the same effective transform parameter:  

$$\theta_G^\zeta \equiv \Theta(K_G^\zeta,\mu_G^\zeta).$$ (16)

This result provides a unique estimate that will always lie between these bounds.

A somewhat better (i.e., more balanced) approximation is achieved for $\zeta_i \ne \eta_i$by defining $\epsilon_i \equiv \frac{1}{2}(\zeta_i+\eta_i)$. Then, all occurrences of $\langle\mu\rangle_\zeta$, $\langle\mu\rangle_\eta$,$\langle\mu^{-1}\rangle^{-1}_\zeta$, and $\langle\mu^{-1}\rangle^{-1}_\eta$ are replaced by $\mu_G^\epsilon$. The errors introduced now through differences $\eta_i-\epsilon_i$ are half those in (13). But new errors are introduced through the differences $\zeta_i-\epsilon_i$. The resulting geometric approximation turns out to be  

$$\theta_G^* = \Theta(K_G^\zeta,\mu_G^\epsilon),$$ (17)

which still reduces to (16) whenever $\eta_i = \zeta_i$.Also note that, if $\eta_i + \zeta_i = 1$, then $\mu_G^\epsilon = \sqrt{\mu_1\mu_2}$.

[Note: If $\zeta_i$ is known but $\eta_i$ is not known (either experimentally or theoretically), Berryman and Milton (1988) discuss how to use knowledge of $\zeta_i$ to constrain estimates of $\eta_i$.However, we will not pursue this option here.]

To maintain internal consistency of the approximation, we can choose to set  

$$\beta_G^* = \frac{4}{3}\mu_G^\zeta,$$ (18)

or we could choose instead to use $\beta_H$ from (10). However, we do not expect that these choices will differ by very much for the bulk modulus estimates.

 

laminated_poly_l12
Figure 1 Schematic illustrating the model of random polycrystals of laminates. Grains are assumed to fit tightly so there is no misfit porosity. But the shapes of the grains are not necessarily the same, and the symmetry axes of the grains (three examples are shown here) are randomly oriented so the overall polycrystal is equiaxed (statistically isotropic).