THE BACKGROUND

I have made a finite element model of the intervertebral disc of the human spine and used the Holzapfel-Gasser-Ogden (HGO) material model to model its mechanical behavior. I have used this material model because two families of fibers can be found in the intervertebral disc (see Figure 1) and the HGO strain energy function has a component to represent the behavior of these fibers.

Figure 1. Criss-cross pattern of the fibers of the intervertebral disc.

To define the direction of the fibers, I have defined a cylindrical coordinate system in the center of the intervertebral disc, as it follows:

Definition of the cylindrical coordinate system

*Orientation, name=CoordSystem, definition=coordinates, local directions=2, system=CYLINDRICAL
 412,  -492,    22,   412,  -492,    23
3, 0.
0, ,
0, -,

Definition of the material parameters

*Material, name=ANNULUS-FIBROSUS
*Anisotropic Hyperelastic, holzapfel, local direction=2
, , , ,

Material section assignment

** Section: Annulus_Fibrosus
*Solid Section, elset=Annulus_Fibrosus, orientation=CoordSystem, material=ANNULUS-FIBROSUS
,

The result of this simulation is shown in Figures 2 and 3.

Figure 2. Resultant direction of the 1st family of fibers (LOCALDIR1).

Figure 3. Circumferential component of the direction of the 1st family of fibers (LOCALDIR1).

THE PROBLEM

As you may have noticed, the vertical axis of the cylindrical coordinate system (represented in red in Figures 2 and 3) is aligned to the Z-axis of the model. How can you check this? The first line of *Orientation is used to define the direction of the axis of the cylindrical coordinate system by the coordinates of two points belonging to the axis. In this case, the first point is (412,  -492,    22) and the second one is (412,  -492,    23). However, the results are not ok if I do the same with another intervertebral disc whose axis is not aligned with one of the 3 Cartesian axis of the model (X, Y, Z). So the question is: is that related to the second line, i.e., "3, 0" rotation?

On the other hand, as you can see in Figures 2 and 3, the shape of the intervertebral disc is not perfectly rounded, so it makes the circumferential direction defined by the cylindrical coordinate system not to be perfectly tangent to every element, but tangent to the radial direction of the cylindrical coordinate system at the centroid of the element. I have defined this cylindrical coordinate system because the geometry of an intervertebral disc can be approximated as a cylinder, but what about if my geometry is more irregular? How can I define the HGO model with the local coordinate system of each element without creating one coordinate system per element? I guess that's the way the Living Heart Project is made, because its shape is irregular.

Thank you so much!!