LAUSR

The interface between materials with mismatched elastic moduli is failure-prone due to singular stress concentrations that could develop at the interface. Yet most ligaments tear at mid-substance and not at the ligament-bone interface. Current physiological understanding of these interfaces centers around the graded material properties of the ligament as it nears the bone, with increasing mineralization of the ligament. Open questions remain on which features of the gradation, or more generally of the ligament-bone interface, significantly improve the robustness and eliminate the stress singularity. To address these questions, we develop a 2D elasticity model of the interface and analyze it using a combination of numerical simulations and asymptotics. Increasing size of the gradation zone yields diminishing returns once the zone gets longer than the width of the ligament. Moreover, we apply a previous result in elasticity and show that the frequently-observed fanned geometry of ligaments near the interface can also eliminate the singularity. Previously, the functional advantage of the fanned geometry was thought simply to increase the cross-sectional area. However, our analysis reveals a different and more significant functional benefit to this geometry. The contribution of this work is to identify scaling laws for the region with graded material properties and a fanned geometry. Based on the observation that neonatal ligament tears often occur at the ligament-bone interface, while injuries in adults primarily occur mid-substance, we propose that the graded interface develops in response to the stress state experienced at dissimilar material interfaces.