Hysteretic pinching of human secondary osteons subjected to torsion

The mechanical behavior of bone tissue's ultra- and micro- structure is fundamental to assessment of macroscopic bone mechanics. This paper explores the ultra-structural characteristics of human femoral tissue responsible for energy absorption of secondary osteons under mechanical loading. A novel mathematical interpretation of single osteon mechanics elucidates the behavior of the collagen-apatite interface. Fully calcified single osteon specimens were mechanically tested quasi-statically under cyclic torsional loading about their longitudinal axis. On each hysteretic diagram, all cycles after the initial monotonic cycle appear pinched and share two points. Stiffness degradation and pinching degradation were investigated on the torque versus deflection-angle-per-unit-length diagrams as the number of cycles increases, in relation to the appearance of osteons in cross-section under circularly polarized light microscopy. Material science's Bauschinger effect, originally defined for metals and later extended to structures reinforced with metal bars, is adapted to describe pinching. Material science's prying effect, defined as amplification of eccentric tensile load through lever action, is employed to explain pinching. The presence of the two points shared by all complete cycles is analyzed in terms of the mathematical fixed point theorem. The results allow formulation of the following conjectures: (1) the prying of carbonated apatite crystallites at the interface with the 40 nm long bands of non-calcified collagen fibrils causes pinching; (2) the prying effect increases with the increasing percentage of collagen-apatite elements that form a larger angle with the osteon axis; and (3) micro-cracks increase more in number than in length as the number of cycles increases.