Effective medium theory for elastic composites

EFFECTIVE ELASTIC CONSTANTS

Mal and Knopoff (1967) derived an integral equation for the scattered displacement field from a single elastic scatterer. Let $\Omega_i$ symbolize the volume of the region occupied by a single inclusion $i$. Let the incident field be $\vec{u}^0(\vec{x})\exp(-i\omega t)$ and let $\vec{u}(\vec{x})\exp(-i\omega t)$ and $\vec{v}(\vec{x})\exp(-i\omega t)$ be the total field outside and inside the inclusion volume such that

$$\begin{array}{c} \vec{u}(\vec{x}) = \vec{u}^0(\vec{x}) + \ve... ...c{x}) \qquad\hbox{for}\qquad \vec{x} \in \Omega_i. \end{array}$$ (1)

The scattered fields are $\vec{u}^s$ and $\vec{v}^s$. Both $\vec{u}(\vec{x})$ and $\vec{u}^0(\vec{x})$ satisfy the same equation:

$$c^m_{\ell npq}\frac{\partial^2u_p}{\partial x_n\partial x_q} + \rho_m\omega^2 u_\ell = 0$$ (2)

outside the inclusion, while $\vec{v}(\vec{x})$ satisfies

$$c^i_{\ell npq}\frac{\partial^2v_p}{\partial x_n\partial x_q} + \rho_i\omega^2 v_\ell = 0$$ (3)

inside the inclusion. The indices $\ell$,$n$,$p$,$q$ take the values $1,2,3$ for the three spatial dimensions, and the Einstein summation convention applies in Equations (2) and (3), and also throughout this paper. The elastic tensor for the matrix and inclusion are respectively:

$$c^m_{\ell npq} = \lambda_m\delta_{\ell n}\delta_{pq} + \mu_... ...elta_{\ell p}\delta_{nq} + \delta_{np}\delta_{\ell q}\right),$$ (4)

$$c^i_{\ell npq} = \lambda_i\delta_{\ell n}\delta_{pq} + \mu_... ...\ell q}\right) \equiv c^m_{\ell npq} + \Delta c^i_{\ell npq},$$ (5)

and $\rho_m$ and $\rho_i (\equiv \rho_m + \Delta\rho_i)$ are the respective densities.

Green's function for a point source in an infinite, isotropic, homogeneous elastic medium of the matrix material is given by

$$g_{pq}(\vec{x},\vec{\zeta}) = \frac{1}{4\pi\rho_m\omega^2}\l... ...(\frac{\exp{(ikr)}}{r} - \frac{\exp{(isr)}}{r}\right)\right],$$ (6)

where $r = \vert\vec{x}-\vec{\zeta}\vert$, $k = \omega[\rho_m/(\lambda_m+2\mu_m)]^{1/2}$, and $s = \omega[\rho_m/\mu_m]^{1/2}$ -- with $k$ and $s$ being, respectively, the magnitudes of the wavevectors for compressional and shear waves in the matrix. Given the form of $g_{pq}$, Mal and Knopoff (1967) then derive an integral equation for $\vec{u}(\vec{x})$. Since the derivation follows standard lines of argument, I will not repeat it here. The result is

$$u_\ell(\vec{x}) = u^0_\ell(\vec{x}) +\int_{\Omega_i}d\vec{\z... ...tial}{\partial\zeta_j}\right]g_{\ell n}(\vec{x},\vec{\zeta}).$$ (7)

Equation (7) is an exact integral equation for the displacement field in the region exterior to the scatterer in terms of the displacement and strain fields inside the inclusion volume $\Omega_i$.

To evaluate the integral (7), estimates of the interior displacement and strain fields are required. Considering the first Born approximation from quantum scattering theory suggests the estimates for wave speed and strain at $\vec{\zeta} \in \Omega_i$:

$$\vec{v}(\vec{\zeta}) \simeq \vec{u}^0(\vec{\zeta}),$$ (8)

and

$$\epsilon_{pq}(\vec{\zeta}) \simeq \epsilon_{pq}^0(\vec{\zeta}).$$ (9)

By Equations (8) and (9), I mean to approximate $\vec{v}$ and $\epsilon$ by the values $\vec{u}^0$ and $\epsilon^0$ would have achieved at position $\vec{\zeta}$ if the matrix contained no scatterers. For scatterers with small volumes, it follows from (7) that $\vec{u}^s(\vec{x})$ and its derivatives are small quantities for $\vec{x}$ outside of $\Omega_i$. Since the displacement is continuous across the boundary, it follows that Equation (8) will be a good approximation to $\vec{v}(\vec{\zeta})$. However, this argument fails for Equation (9), because the strains are not continuous across the boundary. Equation (9) should therefore be replaced by the formula:

$$\epsilon_{pq} = T_{pqrs}\epsilon_{rs}^0,$$ (10)

where $T$ is Wu's tensor [Wu (1966)], relating $\epsilon_{pq}$ for an arbitrary ellipsoidal inclusion to the uniform strain at infinity $\epsilon^0_{pq}$. Now, if the wavelength of the incident waves is large compared to the size of the ellipsoid (i.e., $a/\bar{\lambda} « 1$, where $\bar{\lambda}$ is the wavelength), then the fields both near the ellipsoid and inside scatterer volume $\Omega_i$ will be essentially static and uniform [Eshelby (1957)]. Thus, to the lowest order of approximation, it is valid to make the substitutions (8) and (10). When the ellipsoid is centered at $\zeta_i$, it follows easily that

$$u_{\ell}^s(\vec{x}) = \Omega_i\left[\Delta\rho_i\omega^2u_n^... ...t) \epsilon_{rs}^0g_{\ell n,j}(\vec{x},\vec{\zeta}_i)\right],$$ (11)

where the symmetry properties of $T$ have been used in simplifying the expression. A comma preceding a subscript indicates a derivative with respect to the as-labelled component.

Equation (11) gives the first order estimate of the scattered wave from an ellipsoidal inclusion whose principal axes are aligned with the coordinate axes. When the ellipsoid is oriented arbitrarily with respect to the coordinate axes, Equation (11) must be changed by replacing $T_{pqrs}$ everywhere with

$$U_{pqrs} = \ell_{p\alpha}\ell_{q\beta}\ell_{r\gamma}\ell_{s\delta}T_{\alpha\beta\gamma\delta},$$ (12)

where $\ell_{\alpha\beta}$ are the appropriate direction cosines. For homogeneous, isotropic composites with randomly oriented ellipsoidal inclusions, the general form of the average tensor as given by Wu (1966) is

$$\bar{U}_{pqrs} = \frac{1}{3}(P-Q)\delta_{pq}\delta_{rs} + \frac{1}{2}Q(\delta_{pr}\delta_{qs} + \delta_{ps}\delta_{qr}),$$ (13)

where

$$P = \frac{1}{3}T_{ppqq} \qquad\hbox{and}\qquad Q = \frac{1}{5}\left(T_{pqpq}-T_{ppqq}\right).$$ (14)

Finally, suppose $N$ inclusions are contained in a small volume of radius $a$ centered at $\vec{\zeta}_0$. Assume that the effects of multiple scattering may be neglected at sufficiently low frequencies (i.e., long wavelengths appropriate for seismology) to the lowest order. Then, to the same degree of approximation used in Equation (11) (i.e., $a/\bar{\lambda} « 1$), the scattered wave has the form:

$$\left<u_\ell^s(\vec{x})\right>^m \simeq \sum_{i=1}^N \Omega_... ...ht)\epsilon_{rs}^0g_{\ell n,j}(\vec{x},\vec{\zeta}_0)\right],$$ (15)

where the superscripts $m$ and $i$ again refer to matrix and inclusion properties, respectively. Note especially that distinct superscripts $i$ must be used in Equation (15) to specify both the inclusion material itself, and also the shape of each distinct type of inclusion.

To apply this thought experiment to the analytical problem of estimating elastic constants, consider replacing the true composite sphere with a sphere composed of matrix material identical to the imbedding material and of ellipsoidal inclusions of the same materials as those in the true composite, and also in the same proportions. Then, if multiple scattering effects may be (and are) neglected, the theoretical expression which determines the elastic constants is

$$\left<u_\ell^s(\vec{x})\right>^* = 0,$$ (16)

where the left hand side is given by Equation (15) with matrix-type $m=*$. Equation (16) states simply that the net (overall) scattering -- due to many scatterers - in the self-consistently determined medium vanishes to lowest order.

If the volume fraction of the $i$-th component is defined by $f_i = \Omega_i/\sum_{j=1}^N\Omega_j$, then Equation (16) implies the following formulas:

$$\sum_{i=1}^N f_i(\rho_i-\rho^*) = 0,$$ (17)

$$\sum_{i=1}^N f_i(K_i - K^*)P^{*i} = 0,$$ (18)

and

$$\sum_{i=1}^N f_i(\mu_i - \mu^*)Q^{*i} = 0.$$ (19)

Equation (17) states that the effective density $\rho^*$ is just the volume average density (which is what one might reasonably expect, but nevertheless is not always true for effective medium theories). Equations (18) and (19) provide implicit formulas for $K^*$ and $\mu^*$. Such implicit formulas are typically solved numerically by iteration [Berryman (1980b)]. This step is usually necessary because the factors $P^{*i}$ and $Q^{*i}$ are themselves both typically functions of both the unknown quantities $K^*$ and $\mu^*$. Experience has shown that such iterative methods often converge in a stable fashion, and usually after a small number of iterations (typically 10 or less).

The derivation given and final results attained here are very similar to methods discussed by Elliott et al. (1974) and Gubernatis and Krumhansl (1975). I will therefore refer to the resulting effective medium method as the ``coherent potential approximation'' (or CPA), as is typically done in the physics literature, since the early work of Soven (1967). Equations (18) and (19) were also obtained independently by Korringa et al. (1979), while using an entirely different method. In the following sections, I will compare the results obtained from this effective medium theory to the known rigorous bounds on elastic constants and also to the results of other effective medium theories.

Effective medium theory for elastic composites