A discrete fibre dispersion method for excluding fibres under compression in the modelling of fibrous tissues

Recently, micro-sphere-based methods derived from the angular integration approach have been used for excluding fibres under compression in the modelling of soft biological tissues. However, recent studies have revealed that many of the widely used numerical integration schemes over the unit sphere are inaccurate for large deformation problems even without excluding fibres under compression. Thus, in this study, we propose a discrete fibre dispersion model based on a systematic method for discretizing a unit hemisphere into a finite number of elementary areas, such as spherical triangles. Over each elementary area, we define a representative fibre direction and a discrete fibre density. Then, the strain energy of all the fibres distributed over each elementary area is approximated based on the deformation of the representative fibre direction weighted by the corresponding discrete fibre density. A summation of fibre contributions over all elementary areas then yields the resultant fibre strain energy. This treatment allows us to exclude fibres under compression in a discrete manner by evaluating the tension-compression status of the representative fibre directions only. We have implemented this model in a finite-element programme and illustrate it with three representative examples, including simple tension and simple shear of a unit cube, and non-homogeneous uniaxial extension of a rectangular strip. The results of all three examples are consistent and accurate compared with the previously developed continuous fibre dispersion model, and that is achieved with a substantial reduction of computational cost.