Canonical functions

To make progress towards our present goals, it will prove helpful to take advantage of some observations made earlier about both rigorous bounds and many of the known estimates for moduli of elastic composites (Berryman, 1982; 1995; Milton, 1987; 2002). In particular, it is known (Berryman, 1982) that if we introduce certain functionals -- similar in analytical structure to Hill's formula for the overall bulk modulus K^*, which is  

$$K^*= \left[\sum_{i=1}^{J} \frac{v_i}{K_i+4\mu/3}\right]^{-1} - 4\mu/3,$$ (1)

valid when the shear modulus $\mu$ is a uniform constant throughout the medium. Here K_i is the bulk modulus of the ith constituent out of J constituents, and v_i is the corresponding volume fraction, with the constraint that $\sum_{i=1}^J v_i = 1$.This form is also similar to the form of the Hashin-Shtrikman bounds (Hashin and Shtrikman, 1962; 1963) for both bulk and shear moduli -- many of the known formulas for composites can be expressed simply in terms of these functionals. Specifically, for analysis of effective bulk modulus K^*, we introduce  

$$\Lambda(\beta) \equiv \left[\sum_{i=1}^J \frac{v_i}{K_i + \beta}\right]^{-1} - \beta,$$ (2)

while, for the effective shear modulus $\mu^*$, we have  

$$\Gamma(\theta) \equiv \left[\sum_{i=1}^J \frac{v_i}{\mu_i + \theta}\right]^{-1} - \theta.$$ (3)

Here $\mu_i$ is the shear modulus of the ith constituent out of J isotropic constituents. The arguments $\beta$ and $\theta$ have dimensions of GPa, and are always nonnegative. Both functions increase monotonically as their arguments increase. Furthermore, when the argument of each functional vanishes, the result is the volume weighted harmonic mean (or Reuss average) of the corresponding physical property. Similarly, an analysis of the series expansion for each functional at large arguments shows that, in the limit when the arguments go to infinity, the functionals approach the volume weighted mean (or Voigt average) of the corresponding physical property. We call these expressions the ``canonical functions'' for elasticity, as results expressible in these terms appear repeatedly in the literature -- although published results are not necessarily manipulated into these canonical forms by all authors. The arguments $\beta$ and $\theta$ are called the ``transform parameters.''

TABLE 1. Various bounds on bulk and shear modulus can be expressed in terms of the canonical functions $\Lambda(\beta)$ and $\Gamma(\theta)$. Subscripts $\pm$ for $\beta$ and $\theta$are for upper/lower (+/-) bounds. Subscripts $\pm$ for the elastic constants imply the highest/lowest (+/-) values of the quantity present in the composite. $\Theta$, X, $\Xi$, and the averages $\left<\cdot\right\gt$ and $\left<\cdot\right\gt _\zeta$ are all defined in the text. $K_R = \left<K^{-1}\right\gt^{-1}$,$\mu_R = \left<\mu^{-1}\right\gt^{-1}$,$K_V = \left<K\right\gt$, and $\mu_V = \left<\mu\right\gt$are the Reuss and Voigt averages of the respective moduli.

$$\beta_- \beta_+ \theta_- \theta_+$$ 2.00 Bound