To estabilsh a numerical formula between the orientation distribution function(ODF) and eigenvalues of the orientaion tensor( the second oder) such as e1,e2,e3, do not konw which kind of ODF can be used to do this. By now, a suitable ODF cannot be found to solve this problem. Maybe need to construct a new ODF by Fish or other distribution. Who would have a good idea about this?
Hello Yuan,
I tried some tests last year to see how the orientation tensor represented different sets of vectors. As you will know, the orientation tensor is the sum of the dyadic products of a set of unit vectors with themselves, divided by the number of unit vectors (like a population covariance matrix but for unit vectors). Basically, I believe it just gives the second-order tensor part of whatever orientation density function you give it. The tests that I did were for discrete numbers of unit vectors. For example, I could get an isotropic orientation tensor from poles to a cube, poles to an octahedron and poles to a trapezohedron. As these are all polymodal point distributions, it appears that there is no unique corresponding ODF.
David Durney
Hello David,
Thanks for your tests and idea. Now, I mainly focus on the ice (HCP)material and try to construct a general relation between the eigenvalues of the second-order tensor and principal strain(s )(rates ) with a single-modal assumption. This work had been done by P.R.Cobbold and D. Gapais in 1979(in a paper: specification of fabric shapes using an eigenvalue method:discussion ), after Woodcock(1977) gave a publication. But, they did not work out a general relation (λ1≠λ2≠λ3; λ indicates the principal strain axes)between both, just included relations about λ1＝λ2 and λ2＝λ3). From your perspective, if it is possible to figure out a numerical relation between both in single-modal distributions, such as single-maximum, equatorial girdle, isotropic distribution?
Yuan Li
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