Appendix B: Crack-influence Decomposition Method

Sayers and Kachanov (1991) present a useful method for decomposing the elastic potential of a cracked system into parts due to the (assumed) homogeneous and isotropic elastic background material, and those due to the presence of cracks up to moderate densities. The fundamental idea is that the elastic potential function is composed of just nine terms, representing all combinations of stress invariants of such a system. These invariants depend on the stress tensor $\sigma$and the crack density tensor $\alpha$. In particular, the tensor $\alpha$ is defined in three dimensions by  

$$\alpha = \frac{1}{V}\sum_c a_c^3 \hat{n}_c \hat{n}_c^T,$$ (37)

where V is the averaging volume, $\hat{n}_c$ is the unit normal of penny-shaped crack c having radius a_c. We use the notation $\hat{n}_c \hat{n}_c^T$, where T is the transpose, to express the outer product ($\hat{n}_c\otimes\hat{n}_c$)of two vectors; this notation is consistent with that commonly used to express the singular value decomposition of an arbitrary matrix in terms of its singular vectors. Another common, and entirely equivalent, form of notation for the same quantity that is often used in the mechanics literature is the dyadic form $\boldsymbol{n}_c \boldsymbol{n}_c$.

The elastic potential $\Phi(\sigma,\alpha)$ then takes the form  

$$\begin{array} {rclclcl} \Phi(\sigma,\alpha) & = & \Phi^{(0)}... ..._7 [\hbox{Tr}(\sigma\cdot\alpha\cdot\alpha)]^2, & &\end{array}$$ (38)

where $\hbox{Tr}$ is the trace operation, and the dot notation indicates a contraction over one set of indices. (Note that the significance of crack-influence parameters $\eta_4$, $\eta_6$, and $\eta_7$ have been changed from the definitions made by Kachanov (1980), Kachanov and Sevostianov (2005), and Sayers and Kachanov (1991), so that here $\eta_4$ is the coefficient of a contribution second order in $\alpha$,$\eta_6$ third order in $\alpha$, and $\eta_7$ fourth order in $\alpha$.) The coefficients pertinent to the isotropic background elastic medium are given by $\Phi^{(0)}_1 = (1+\nu_0)/2E_0$ and $\Phi^{(0)}_2 = -\nu_0/2E_0$,where E₀ is Young's modulus, and $\nu_0$ is Poisson's ratio.

Now, to illustrate the meaning of (38), we will reduce this to component form in two cases. For the cases of interest, we can assume the crack density tensor itself reduces to the form  

$$\alpha = \sum_{i=1}^3 \rho_i \hat{n}_i \hat{n}_i^T,$$ (39)

where $\hat{n}_i$, for i = 1,2,3, correspond to spatial directions x, y, z, respectively. Furthermore, $\hbox{Tr}\alpha = \rho_1 + \rho_2 + \rho_3 \equiv \rho = na^3$ is the scalar crack density defined in the main text.

Horizontal cracks, $\rho = \rho_3$

If all the cracks in the system have the same axis of symmetry (which we will take to be the z-axis), then $\rho = \rho_3$ and (38) reduces to the following expression:  

$$\begin{array} {rcl} \Phi(\sigma,\alpha) = \Phi^{(0)}_1(\hbox... ...\rho^2 + \eta_6\rho^3 + \eta_7\rho^4)\sigma_{33}^2,\end{array}$$ (40)

where the repeated index j is summed. At low crack densities $\rho$, we see that only the terms proportional to $\eta_1$ and $\eta_2$ are important in the crack-influence decomposition. As the crack density increases, the terms proportional to $\eta_3$, $\eta_4$, and $\eta_5$ start to contribute. Then, at the highest crack densities considered, all seven of these coefficients can come into play. Although we may imagine for example that $\eta_2(\rho)$ is actually a function of crack density $\rho$,it is clear from the form of (40) that such corrections would be indistinguishable from corrections due to $\eta_5(0)$.So, for our present purposes, we do not need to consider any coefficients except $\eta_1(0)$ and $\eta_2(0)$ at low crack densities, and we also do not need to consider any coefficients except $\eta_1(0)$ through $\eta_5(0)$, when we want to fit quadratic corrections for the moderate crack density results.

Typical values of $\rho$ of interest in many applications are around $\rho = 0.1$. So as long as the $\eta$'s for higher order corrections are of approximately the same order of magnitude as those for $\eta_1(0)$ and $\eta_2(0)$, we see that neglect of terms like $\eta_6\rho^3 + \eta_7\rho^4$ is entirely appropriate.

Now it is also easy to see how (40) gives rise to the low density result (4).

Vertical cracks, $\rho = \rho_1 + \rho_2$

It is also straightforward now to repeat the previous exercise by considering other types of crack density distribution. An interesting case is the one with all vertical cracks, having their crack normals in the xy-plane. Then, $\rho = \rho_1 + \rho_2$. A special case of this type is when the crack normals are completely randomly distributed so that $\rho_1 = \rho_2 = \rho/2$. Then, we get simplified formulas for all the terms in the elastic potential analogous to the previous example. The results are:  

$$\begin{array} {rcl} \Phi(\sigma,\alpha) = \Phi^{(0)}_1(\hbox... ...^3/8 + \eta_7\rho^4/16)(\sigma_{11}+\sigma_{22})^2,\end{array}$$ (41)

where again the repeated index j is summed.

The basic conclusions reached in the previous example clearly apply again. For small to moderate crack densities $\rho$, we do not need to consider $\rho$ dependence of crack-influence parameters $\eta_1(\rho)$ or $\eta_2(\rho)$,as such dependence cannot be distinguished from the low order contributions from $\eta_4(0)$ and $\eta_5(0)$, respectively. Similarly, $\eta_3(0)$ comes into play whenever $\eta_4(0)$ and $\eta_5(0)$ are important, while $\eta_6(0)$ and $\eta_7(0)$ can presumably be neglected in many low to moderate crack density applications.

TABLE 2.Examples of Sayers and Kachanov (1991) crack-influence parameters $\eta_1(\rho)$ and $\eta_2(\rho)$ when crack density $\rho << 1$for penny-shaped cracks. Four choices of effective medium theory are considered: NI (non-inteacting), DS, (differential scheme), CPA (coherent potential approximation), and SC (the Budiansky and O'Connell self-consistent scheme). Note that crack density is defined here as $\rho = Nr^3/V$, where N/V is number density of cracks, and $A = \pi r^2$ is the area of the circular crack face.

$$0.15in] \begin{tabular} {\vert c\vert cc\vert} \hline\hline... ...8 & 0.0290 \ SC & -0.0000207 & 0.0290 \hline\hline\end{tabular}$$