Previously, we proposed a discrete fiber dispersion model [1] for soft biological tissues by using a triangular discretization of a unit sphere into a finite number of elementary areas. Over each elementary area, we define a representative fiber direction and an elementary fiber density based on the collagen fiber distribution in soft tissues. The strain energy of collagen fibers distributed in each elementary area is then approximated by the deformation of the representative fiber direction weighted by the corresponding elementary fiber density. A summation of fiber contributions of all elementary areas yields the resultant fiber strain energy. Due to the discrete treatment of fibers, any fibers under compression could be easily excluded from the total strain energy. In addition, fibers in a particular direction could be “deactivated” due to disease or other medical conditions while other fibers can still contribute to the total strain-energy function. Unlike other microstructurally-motivated constitutive models for fibrous tissues, this model does not depend on any particular numerical integration scheme over the sphere. The other advantage is that more realistic fiber dispersion data measured from biological tissues can be used with this model. However, in that study, we did not consider fiber recruitment, softening and damage. The goal of this study [2] is to incorporate these important properties of collagen fibers into the constitutive model. We first define a fiber recruitment stretch at which the fiber becomes straightened. Then, we adopt the continuum damage mechanics method for modeling fiber softening and damage. We implemented the proposed model in a finite element program and verified it with three representative examples including a uniaxial extension test of a dog-bone shaped specimen up to failure. As shown in the figure above, the areas with the highest stress correspond to the area of specimen tear/rupture observed in the experimental test. Read the full-text of the article for free with the link provided below [2].

[1] K. Li, R.W. Ogden and G.A. Holzapfel, A discrete fiber dispersion method for excluding fibers under compression in the modeling of fibrous tissues. Journal of the Royal Society Interface, 15:20170766, 2018.

[2] K. Li and G.A. Holzapfel, Multiscale modeling of fiber recruitment and damage with a discrete fiber dispersion method. Journal of the Mechanics and Physics of Solids, 126:226-244, 2019. Free access link: https://authors.elsevier.com/c/1Yehs_WmNnyGk