On the tension-compression switch of the Gasser-Ogden-Holzapfel model: Analysis and a new pre-integrated proposal

Many biological soft tissues are structurally composed of a mostly isotropic matrix (elastin) and fibers (collagen). These fibers are not perfectly aligned but dispersed around some referential, preferred directions. In order to account for the dispersion of the fibers, a probability distribution is assumed. The Generalized Structure Tensor (GST) models perform a pre-integration of the distribution in order to achieve improved computational efficiency. The best known model of this kind is the Gasser-Ogden-Holzapfel (GOH) model. However, in these models no singular treatment of fibers is made. Whenever they suffer compression it is usual to consider that fibers should not contribute to the overall stiffness. At this point, a switch criterion is employed. This switch criterion is important because it changes the model predictions and may also result in unphysical stress predictions or strain ranges at which no compatible equilibrium solution is found. We perform an analysis of different tension-compression switch criteria from the literature for the GOH model and show relevant physical and computational drawbacks when using these criteria. In order to overcome these drawbacks, we make a new proposal which yields continuous stress solutions. In our proposal, pre-integrated expressions given in terms of the usual set of invariants take into account an average amount of fibers working either in tension or in compression for a given deformation gradient and fiber family. Two distinct switches naturally emerge from our procedure. Furthermore, we keep the appealing GST pre-integrated approach for any proposed stored energy, including that of the GOH model.