Consider an RVE with N fibers, with direction unit vectors d(1) ,...,d(N) . The components of the orientation tensor A are defined as
Note: The sign of the direction unit vectors d(k) can be chosen arbitrarily, as it does not influence the orientation tensor.
The orientation tensor is a symmetric tensor and its trace (the sum of the diagonal elements) is always 1. The diagonal entries fall into the range [0,1], whereas the off-diagonal entries fall into [-1/2,1/2].
Material Designer strives to create an RVE that has (approximately) a diagonal orientation tensor; in other words, we try to generate an RVE where the principal axes of the orientation tensor match the global coordinate system.
Then we are left with the three diagonal entries: a11, a22, a33. The values specify how closely the fibers are aligned with the corresponding direction. For instance, if the orientation tensor is
then all the fibers are aligned in X direction. If the fiber orientations are uniform in all directions, then the orientation tensor is
If a33 is zero, for instance for
then the fibers are oriented parallel to the XY plane.
Since the sum of a11, a22, and a33 is always 1, you can specify only the target values for a11 and a22; a33 is computed automatically.