Elastic energy and the crack density tensor

Sayers and Kachanov (1991) also introduce a very interesting and useful scheme in the same paper that permits the calculation of constants for anisotropic cracked media from estimates of the behavior (such as that predicted by DS) for the isotropic case. This approach is a tremendous simplification of an otherwise very difficult technical problem. The key idea they use is to introduce an elastic potential energy quadratic in the stress tensor that can be expressed in terms of invariants of the stress tensor in various combinations involving the ``crack density tensor.'' This approach results in a fairly complicated energy potential function involving nine distinct terms. But this function has the advantage that, upon linearization in the crack density, it reduces to only four terms. Two of these terms are the standard ones for the pure (uncracked) medium and the remaining two terms contain the linear contributions due to the cracks. Now it is not obvious that linearization is permissible in the crack density ranges of interest, but Sayers and Kachanov (1995) showed in later work that the remaining contributions from the fourth rank crack-density tensor are always small -- and therefore negligible in most situations of practical interest. The neglect of these terms nevertheless implies a certain amount of error in any calculation made based on their neglect, but -- if this error is of the size of our measurement error or less -- it should not be a serious impediment to studies and analysis of these systems.

To give one example, we find that the corrections to the compliance matrix S_ij due to the presence of an isotropic crack distribution take the form:  

$$\Delta S_{ij} = \rho\left(\begin{array} {cccccc} 2(\eta_1+\e... ... & 4\eta_2/3 & \cr & & & & & 4\eta_2/3 \cr\end{array}\right),$$ (1)

where $\eta_1$ and $\eta_2$ are the two coefficients appearing in the Sayers and Kachanov (1991) theory that depend on the presence of cracks, and $\rho = Nr^3/V$is the crack density (where N/V is the number desnity and r is the radius of the flat cracks when they are penny-shaped as assumed here). These two coefficients can be determined for any crack density by computing the bulk and shear moduli from the compliance matrix $S_{ij}^* = S_{ij} + \Delta S_{ij}$ and comparing the results one-to-one with the results from any effective medium theory one trusts. For these purposes, the differential scheme (DS) is the one that Sayers and Kachanov (1991) recommend, but I have shown elsewhere that another scheme -- a symmetric self-consistent scheme that is sometimes called the CPA (for coherent potential approximation) - gives very comparable results. The results can also be compared to rigorous bounds (this work is in progress) and, therefore, also used to obtain rigorous upper bounds on both $\vert\eta_1\vert$ and $\eta_2$. I have done some initial studies of this type and found that the value of $\vert\eta_1\vert$is generally much smaller in magnitude that of $\eta_2$.In particular, $\vert\eta_1/\eta_2\vert \le 0.01$is typical of the observed results for both DS and CPA.

The real advantage of this approach can now be shown very simply using a couple of examples.

First, consider the situation in which all the cracks in the system have the same vertical (z-)axis of symmetry. Then, the cracked/fractured system is not isotropic, and we have the compliance correction matrix  

$$\Delta S_{ij} = \rho\left(\begin{array} {cccccc} 0 & 0 & \et... ... \cr & & & & 2\eta_2 & \cr & & & & & 0 \cr\end{array}\right).$$ (2)

Now it is also not difficult to see that, if the cracks were oriented instead so that all their normals were pointed horizontally along the x-axis, then I would have one permutation of this matrix and, if instead they were all pointed horizontally along the y-axis, then I would have a third permutation of the matrix. If I then want to understand the isotropic correction matrix in (1), I can average these three permutations: just add the three $\Delta S$'s together and then divide by three. Having done that, I exactly recover the isotropic compliance corrections matrix displayed in (1). This construction shows in part both the power and the simplicity of the Sayers and Kachanov (1991) approach.

 

ranpc2vert Figure 1 Example of a vertical cross-section (xz-plane) through a medium having penny-shaped cracks with radius r/L = 0.05, where L = 1.0 is the length on each side of a cube in 3D. This image was produced by randomly placing 2000 crack centers in the box of volume = L3 (so crack density $\rho = 0.25$), and testing to see if the center is within a distance $r \le 0.05$ of the central square at y = 0.5. If so, then a random angle is chosen for the crack. If this crack orientation results in an intersection with the plane y=0.5, the line of intersection is plotted here. The resulting lines can have any length from 2r = 0.1 to zero. The number of intersections found for this realization was 114, whereas the expected value for any particular realization is approximately $(2/\pi)\times 200 \simeq 127$.

Next, consider the case when all cracks have their normals lying randomly in parallel planes. Then, if the parallel planes are taken to be horizontal, the cracks are all vertically aligned as in Figure 1. So, I immediately find the anisotropic (i.e., vertical transverse isotropy or VTI) result that  

$$\Delta S_{ij} = \rho\left(\begin{array} {cccccc} (\eta_1+\et... ...& & & & \eta_2 & \cr & & & & & 2\eta_2 \cr\end{array}\right).$$ (3)

The reader should check that adding two-thirds of (3) to one-third of (2) recovers (1), since this combination also represents an isotropic ensemble of fractures.

This same basic concept then works very well for any assumed symmetry that we might like to model. There is no additional work to be done once (i) the isotropic results are known (for some EMT) and (ii) the layout of the two $\eta$'s in the correction matrix $\Delta S$ have been determined once and for all for a given elastic symmetry resulting from a specific choice of crack orientation distribution. Sayers and Kachanov (1991) give a precise prescription for this. Although I make use of this prescription here, I will not show the details in order to avoid some of the mathematical complications inherent in their tensorial expressions.

There are interesting and important questions of uniqueness related to the inverse problem (i.e., deducing the $\eta$'s from seismic wave observations) since more than one type of distribution can give rise, for example, to vertical transverse isotropy (VTI). Then, the question is whether quantities such as the Thomsen parameters of anisotropy can help us to remove some of these possible ambiguities from the interpretations of field measurements.