Aging in the cortical bone: a constitutive law and its application

Abstract

It is well known that the material behavior of human cortical bone changes from ductile to more brittle due to aging. This process is accompanied by a decrease of the maximum specific deformation energy. Numerous mechanical tests of specimens have shown a relationship between the mechanical behavior, age and microstructure, especially the porosity, mineralization and fraction of the secondary osteonal area. But up to now, this relationship is not explicitly considered in a constitutive law. Measured stress–strain curves, taken from the literature, from one-dimensional mechanical experiments in tension (McCalden et al. in J Bone Joint Surg Am 75(8):1193–1205, 1993) have been characterized by Young’s modulus, elastic, plastic and fracture energy, fracture stress and strain. The specimens have been harvested from the femora of 46 deceased individuals. Based on this data, we set up a system of equations taking into account the microstructure of the bone material by analogy to common procedures in fracture and damage mechanics. Solving this system for all measured experimental data leads to the determination of the independent damage parameters for each individual person. It turned out that some characteristic mechanical values and one independent damage parameter are statistically significant dependent on age and microstructure. We receive a constitutive law, which describes the mechanical behavior up to fracture by measurable parameters for the microstructure and the individual age and gender only. In turn, we calculate the individual tolerable load for bending, using a nonlinear stress–strain curve, and postulate an age-dependent fracture load for healthy bone by means of the statistical regression. Deviation from the standard is an indication for a bone disease in particular for osteoporosis.