The singular value decomposition (SVD), or equivalently the eigenvalue decomposition in the case of a real symmetric matrix, for (6) is relatively easy to perform. We can immediately write down four eigenvectors:
The ranges of values for are and (since ) .The interpretation of the solutions is simple for the isotropic limit where and , corresponding respectively to pure compression and pure shear modes. For all other cases, these two modes have mixed character, indicating that pure compression cannot be excited in the system, and must always be coupled to shear. Some types of pure shear modes can still be excited even in the nonisotropic cases, because the other four eigenvectors in (24) are not affected by this coupling, and they are all pure shear modes. Pure shear (in strain) and pure compressional (in stress) modes are obtained as linear combinations of these two mixed modes according to
To understand the behavior of in terms of the layer property fluctuations, it is first helpful to note that the pertinent functional is easily shown to be a monotonic function of its argument x. So it is sufficient to study the behavior of the argument x = (a+b-c)/f.
Exact results in terms of layer elasticity parameters
Combining results from Eqs.(11)-(14), we find after some work on rearranging the terms that
Our main conclusion is that the shear modulus fluctuations giving rise to the anisotropy due to layering are (as expected) the main source of deviations of (33) from unity. But, there are some other more subtle effects present having to do with the interplay between , correlations as well as the strength of the fluctuations that ultimately determine the magnitude of the deviations of (33) from unity.
A fifth effective shear modulus?
From what has gone before, we know that there are four eigenvalues of the system that are easily identified with effective shear moduli (two are l and two are m). The bulk modulus K of the simple system is well-defined, and the bounds on Keff for polycrystals are quite simple to apply and interpret. But we are still missing an important element of the overall picture of this system, and that is how the remaining degree of freedom is to be interpreted. It seems clear that it should be imterpreted as an effective (quasi-)shear mode, since we have already accounted for the bulk mode. It is also clear that analysis of this remaining degree of freedom is not so easy because it is never an eigenfunction of the elasticity/poroelasticity tensor except in the cases that are trivial and therefore of interest to us here only as the isotropic baseline for comparisons.
Although it seems problematic to define a new shear modulus arbitrarily, we will now proceed to enumerate a number of possibilities and then, by a process of elimination, arrive at what appears to be a useful result.