Modeling of bending and torsional stiffnesses of bone at sub-microscale: Effect of curved mineral lamellae

Recent transmission electron microscopy images of transverse sections of human cortical bone showed that mineral lamellae (polycrystalline sheets of apatite crystals) form arcuate multi-radius patterns around collagen fibrils. The 3-6 nm thick mineral lamellae are arranged in stacks of 3-20 layers and curve around individual fibrils, few fibrils, and higher numbers of collagen fibrils. We evaluate the effect of these stacked mineral lamellae with various radius of curvature patterns on the elastic bending and torsional responses of bone at the sub-microscale using a finite element method. We find that the curved multi-radius stack patterns increased the bending and torsional stiffnesses by 7% and 23%, respectively, compared to when the stacks of mineral lamellae only encircle individual fibrils for the idealized geometric models considered. This study provides new insights into the structure-property relations for the bone ultrastructure.