Explicit expressions for elastic constants of osteoporotic lamellar tissue and damage assessment using Hashin failure criterion

Abstract

Osteoporosis is one of the most prevalent diseases in the last decades. The ageing of the population has led to a large increase in the number of people who suffer this musculoskeletal disease. For this reason, an early diagnosis of osteoporosis in order to improve bone fracture risk assessment is essential. To this end, efforts to enhance the knowledge in the elastic, post-yielding and fracture properties of bone are required.In this work, we have derived explicit equations to estimate the orthotropic elastic constants of lamellar tissue as a function of the porosity at tissue level (microporosity) and the bone mineral density, following the multiscale approach presented in [1]. Microporosity has been explicitly mod-elled in a range from 1% to 25% [2]. Two types of pores, ellipsoid and sphere-shaped, have been modelled. On the other side, we have obtained the elastic constants of lamellar tissue in a range of bone mineral density comprised from 0,653 g/cm3 to 1,50 g/cm3 [3]. A non-linear multivariable regression by means of the least square fitting has been performed and the explicit expressions for the elastic constants of lamellar tissue have been provided as a function of the volumetric bone mineral density and porosity.Moreover, independent quasi-static load cases are numerically simulated. Bone failure onset has been modeled by Hashin’s orthotropic failure criterion and damage evolution has been assessed through the material property degradation (MPD) method. Strength limits of lamellar tissue have been inferred from Ascenzi and Bonucci [4] and Giner et al. [5]. Results reveal that failure onset is mainly due to the tension in the transversal direction of the mineralized collagen bundles.