Modelling of bone fracture using the fundamental functional unit – Osteon

Abstract

Limited bone sample size and number, coupled with hierarchical microstructures and non-linear damage, make credible assessment of bone quality difficult. This study presents a simple non-Linear Elastic Fracture Mechanics (non-LEFM) model, which can be easily adopted to measure bone strength and toughness using a limited number of small bone samples. The new closed-form model assumes that osteon, the fundamental functional unit of cortical bone, plays the dominant role in bulk bone properties. Different to Griffith theory [1] or LEFM [2] containing 0 microstructure, the new non-LEFM model contains 1 microstructure, the osteon diameter OD or the characteristic microstructure Cch. The “0 to 1” leap in microstructure modelling is critical for the evaluation of the non-linear damage prior to a transverse fracture. And it leads to the fracture relation Pmax = ft × Ae (Fracture Load = Strength × Area), in which the longitudinal tensile strength ft is the slope of the linear Pmax – Ae relation while the equivalent area Ae contains sample dimensions as well as Cch. Fracture measurements from samples with different sizes or initial cracks are on one straight line through the origin (0, 0), i.e. ft can be determined from any sample group. Fracture toughness KIC can then be transferred from ft and Cch. A statistical analysis based on normal distribution has been combined into the model so that the reliability band for bone strength and toughness can be specified. Bone data from literature are analyzed by the present model, and non-linear damage zones at Pmax in dry and wet bones are estimated and compared.