Add material model - Modeling of non-symmetric fiber dispersion using the isoparametric discrete fiber dispersion (DFD) method

[mrolfpissarczyk]

References:
https://doi.org/10.1016/j.cma.2020.113511
https://doi.org/10.1098/rspa.2021.0592

and

https://doi.org/10.1098/rsif.2017.0766

Short description:
Collagen fibers in soft biological tissues provide the overall stiffness and strength of the material. The latest imaging techniques, such as second-harmonic generation, have enabled detailed visualization of the underlying microscopic constitution of biological tissues such as arterial walls and other tissues. The collagen fibers in these tissues may be dispersed randomly in space, in a certain pattern such as predominantly in a particular direction, as a rotationally symmetric dispersion about a mean direction, or as the recently observed non-symmetric dispersion in arterial walls, or other arrangements.

One approach for modeling the dispersed fiber distributions in a constitutive equation is the discrete fiber dispersion method. By using the DFD method, the fiber dispersion in the tissue can be incorporated into the strain-energy function by an integrable probability density function, which is defined over the unit hemisphere. This unit hemisphere is then discretized into a finite number of elementary areas, such as spherical triangles. Over each elementary area, we define a representative fiber direction and a discrete fiber density. Then, the strain energy of all the fibers distributed over each elementary area is approximated based on the deformation of the representative fiber direction weighted by the corresponding discrete fiber density. A summation of fiber contributions over all elementary areas then yields the resultant fiber strain energy. This treatment allows us to exclude fiber under compression in a discrete manner by evaluating the tension-compression status of the representative fiber directions only.

Implementation:
Coming soon.