MS_anisotropy
MS_ANISOTROPY - Simple measures of anisotropy
// Part of MSAT - The Matlab Seismic Anisotropy Toolkit //
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
Shivakumar, 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