MS_ANISOTROPY - Simple measures of anisotropy

Calculate the degree of anisotropy of an elasticity matrix.

[ uA, ... ] = MS_anisotropy( C, ... )

Usage:
    [ uA ] = MS_anisotropy( C )
        Return the Universal Elastic Anisotropy Index of Ranganathan
        and Ostoja-Starzewski (2008). Valid for any elasticity matrix,
        uA is zero for an isotropic case and increases for increasing
        anisotropy.

[ uA, lmA ] = MS_anisotropy( C )
    Also return the general measure of anisotropy proposed by
    Ledbetter and Miglion (2006). This the the ratio of the fastest
    and slowest squared shear wave velocity over all propogation
    and polarization directions. Equal to one in the isotropic case,
    increases with increasing anisotropy.

[ uA, lmA, zA ] = MS_anisotropy( C )
    Also return the Zenner (1948) measure of anisotropy. This is
    only valid for cubic crystals (NaN is returned if C does not
    represent a cubic crystal). zA is 1 for an isotropic case and
    increases or decreases with incresing anisotropy.

[ uA, lmA, zA, cbA ] = MS_anisotropy( C )
    Also return the Chung-Buessem (1967) anisotropy index. This
    is a single valued measure of anisotropy derived from zA. Like
    uA, this is zero for an isotropic case and increases for increasing
    anisotropy. Only valid for matricies representing cubic crystals.

[ uA, lmA, ... ] = MS_anisotropy( C, n )
    Set the number of random directions to sample for the calculation
    of lmA. Defaults to 1000, which seems to give results accurate to
    two decimal places. Ledbetter and Miglion (2006) use 10000 which
    gives results reproducable to three decimal places and a
    noticable slow down.

Notes:
    These measures of anisotropy are independent of orientation. However,
    the test for cubic symmetry assumes the matrix is in an ideal
    orention. Use MS_AXES to reorentate the imput matrix for the general
    case. MS_NORMS can be used to provide an alternate measure of
    anisotropy. Ledbetter and Miglion (2006) claim lmA is identcal to zA
    for cubic cases but Ranganathan and Ostoja-Starzewski (2008) point
    out cases where zA < 1 while lmA > 1 by construction.

References:
    Zenner, C. (1948) Elasticity and Anelasticiy of Metals. University
    of Chicago.

Chung, D. H. and W. R. Buessem (1967) Journal of Applied Physics
vol.38 p.5

Ledbetter, H. and A. Miglion (2006) "A general elastic-anisotropy
measure" Journal of Applied Physics vol.100 art.num.063516
http://dx.doi.org/10.1063/1.2338835

Ranganathan, S. I. and M. Ostoja-Starzewski (2008) "Universal Elastic
Anisotropy Index" Physical Review Letters vol.101 art.num.055504.
http://dx.doi.org/10.1103/PhysRevLett.101.055504

See also: MS_POLYAVERAGE, MS_NORMS, MS_AXES, MS_PHASEVELS