For non-isotropic materials, an appropriate coordinate system is given by the so-called principal material axes. While the Cartesian coordinates are the direction of computation (and experiment e.g., taking CT and applying loads), the structure material has its own symmetries e.g., the plane perpendicular to the mean direction of the fibers. These intrinsic symmetries may not be aligned with the Cartesian coordinates, but they define another orthonormal coordinate system. The estimation of principal axes is a means to find these intrinsic symmetries.

The principal material axes are fitted as in Rutka et al., 2006 who followed Browaeys and Chevrot, 2004. They are shown in the results file and are used for approximations of the elasticity tensors.

Here, once again, if the orientation of a structure is not known, the PMAs might give a hint: the PMA of the isotropic structure are more uniformly distributed than the transversely isotropic structure. The letter has one dominant PMA (value) for each coordinate of the original coordinate system (e.g. the first column: 0.99 for the x-axis and -0.03 and 0.02 for y-and z-axis) whereas the discrepancy in magnitude between the values for the isotropic structure are not high (e.g. first column: 0.64, 0.48 and 0.58).

If we denote by U the matrix with the new coordinate vectors:

Then, the coordinates of the elasticity tensor in the new coordinate system are

See the section about the elasticity theory for the relationship between the elasticity tensor as 4th order tensor and the matrix notation.

The elasticity tensor for different assumptions like anisotropy or orthotropy, e.g., are displayed behind the estimated PMAs, as described in the following section.