Effective medium theory for elastic composites

OTHER EFFECTIVE MEDIUM THEORIES

A great variety of effective medium theories exist for studies of the elastic properties of composites. Of these theories, the scattering theory presented by Zeller and Dederichs (1973), Korringa (1973), and Gubernatis and Krumhansl (1975) have the most in common with the scattering-theory approach presented here. However, the present approach appears to be unique among the self-consistent scattering-theory variety, being dynamic (i.e., frequency dependent), while all the others are based on static or quasi-static derivations. This difference becomes a very useful advantage if we want to generalize the approach to finite (nonzero) frequencies, as is required for viscoelastic media. The bounding arguments presented here do not carry over directly to the frequency dependent case, but they actually can be generalized -- as shown by Gibianksy and Milton (1993), Milton and Berryman (1997), and Gibiansky et al. (1999).

Another class of effective medium theories studied by Hill (1965), Budiansky (1965), Wu (1966), Walpole (1969), and Boucher (1974) does not yield the same results as the present one, except for the case of spherical inclusions. It has been shown elsewhere [Berryman (1980b)] how the derivation of the approach of Hill, Budiansky, and others can be kinds of symmetrized to yield the symmetrical results as presented here that I prefer. Since the CPA class of effective medium theories gives results equivalent to the Hashin-Shtrikman [Hashin and Shtrikman (1963,1961,1962)] bounds when the inclusions are disk-shaped, I conclude that these results are preferred - since they do satisfy these bounding constraints, while the alternatives do not. The numerical results show general satisfaction of the bounds.

To elucidate somewhat further the relationship between the static and dynamic derivations of the effective medium results, I will outline the static derivation next. The integral equations for the static strain field are given by

$$\epsilon_{ij}(\vec{x}) = \epsilon_{ij}^0(\vec{x}) + \int d^3... ...},\vec{x}') \Delta c_{klmn}(\vec{x}')\epsilon_{mn}(\vec{x}'),$$ (43)

where Green's function is

$$G_{ijkl}(\vec{x},\vec{x}') = \frac{1}{2}\left(g_{ik,jl}^0 + g_{jk,il}^0\right),$$ (44)

with the Kelvin solution given by

$$g_{pq}(\vec{x},\vec{x}') = \frac{1}{4\pi\mu_m}\left[\frac{\d... ...-\nu_m)}\frac{\partial^2 r}{\partial x_p\partial x_q}\right],$$ (45)

where $r = \vert\vec{x}-\vec{x}'\vert$ and $\mu_m$ and $\nu_m$ are, respectively, the shear modulus and Poisson's ratio of the matrix material. Equation (43) may be rewritten formally as

$$\epsilon = \epsilon^0 + G\Delta c \epsilon,$$ (46)

where $G$ is now an integral operator defined by

$$Gf = \int d^3x' G(\vec{x},\vec{x}') f(\vec{x}').$$ (47)

Iterating Equation (46), I obtain the well-known Born series

$$\epsilon = \epsilon^0 + G\Delta c \epsilon^0 + G\Delta c G\Delta c \epsilon^0 + \ldots,$$ (48)

and then summing the Born series formally yields

$$\epsilon = \left(I+Gt\right)\epsilon^0 = \left(I-G\Delta c\right)^{-1}\epsilon^0,$$ (49)

where the so-called $t$-matrix is defined by

$$t = \Delta c \left(I-G\Delta c\right)^{-1} = \Delta c\left(I + Gt\right).$$ (50)

Taking the ensemble average of Equation (49), I have

$$\left<\epsilon\right> = \left(I+G\left<t\right>\right)\epsilon^0 = \left<\left(I-G\Delta c\right)^{-1}\right>\epsilon^0.$$ (51)

For a single scatterer, Equation (49) is equivalent to Equation (10). Therefore, it is worth noting that Wu's (1966) tensor $T$ is formally related to the $t$-matrix by

$$T = I + Gt = \left(I-G\Delta c\right)^{-1}.$$ (52)

Equation (51) is now in a convenient form for use in determining the effective elastic tensor $c^*$ of a composite defined by

$$\left<\sigma\right> = \left<c \epsilon\right> \equiv c^*\left<\epsilon\right>,$$ (53)

where the averages in Equation (53) are again ensemble averages over possible composites having similar physical and statistical properties. Using the standard definition $c = c^m + \Delta c$, I find that

$$\left<c \epsilon\right> = c^m\left<\epsilon\right> + \left<... ...right> = c^m\left<\epsilon\right> + \left<t\right>\epsilon^0.$$ (54)

From Equation (54), it follows easily that the effective elastic tensor is given by

$$c^* = c^m + \left<t\right>\left(I + G\left<t\right>\right)^{-1}.$$ (55)

The choice of matrix elastic tensor $c^m$ is still completely free since the decomposition $c = c^m + \Delta c$ is not unique. Thus, I am free to choose, for example, $c^m = c^*$ (i.e., the matrix material has now exactly the properties of the equivalent composite material), which implies:

$$\left<t\right> \equiv 0.$$ (56)

Equation (56) is an implicit formula determining the effective elastic tensor $c^*$, and says that the effective scattering $t$-matrix averages to zero.

In principle, Equation (56) provides an exact solution for the effective moduli. However, the total $t$-matrix itself is generally too difficult to calculate. It turns out to be more reasonable and more effective [Velicky et al. (1968)] to rearrange the terms of the total $t$-matrix into a series of terms with repeated scattering from individual scatterers ($t_i$). Then, by setting the ensemble average of the individual $t$ matrices to zero

$$\left<t_i\right> = \sum_{i=1}^N f_i\Delta c_i\left(I - G\Delta c_i\right)^{-1} = 0,$$ (57)

and neglecting terms corresponding to fluctuations in the scattered wave [Velicky et al. (1968)], a tractable approximation for the estimate of the elastic moduli is obtained.

When the constituents and the composite as a whole are all relatively homogeneous and isotropic, the tensor Equation (57) reduces to two coupled equations:

$$\sum_{i=1}^N f_i(K_i - K^*)P^{*i} = 0,$$ (58)

and

$$\sum_{i=1}^N f_i (\mu_i - \mu^*)Q^{*i} = 0,$$ (59)

where Equations (13), (14), and (52) were used to simplify Equation (57). Note that Equations (58) and (59) are identical to Equations (18) and (19), thereby establishing the equivalence of these two approaches in the isotropic case.

Effective medium theory for elastic composites