A microstructure-based study on compact human bones: hierarchical length scale parameter
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O R I G I NA L PA P E R
Delara Soltani · Majid Akbarzadeh Khorshidi
A microstructure-based study on compact human bones: hierarchical length scale parameter
Received: 5 March 2020 / Revised: 10 August 2020 / Accepted: 23 September 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract This paper presents a microstructure-based study on compact human bone using the modified couple stress theory and investigates some of its mechanical behaviors at the micro-scale. First of all, length scale parameters in torsion and bending are assessed for bone micro-samples on the basis of experimental data proposed by Yang and Lakes. Then, the effect of micro-rotations on the torsion and bending of single osteons is studied and compared with experimental results obtained by Ascenzi et al., and the length scale parameter of osteons is consequently obtained. According to the present study on osteons and using an approximate formulation of multi-layered tubes under torsion, a material length scale parameter of lamellae is acquired to capture the size effect. These achievements lead to introducing a hierarchical structure for the material length scale parameter of the modified couple stress theory. In addition, the buckling, static nonlinear response, torsional vibration, and flexural dispersion analyses of osteons are presented.
1 Introduction Compact or cortical bone is one of two main patterns in the bone tissue, which has special characteristics and appearance. It can have different interpretations at different structural levels, i.e., macrostructure, microstructure, sub-microstructure, nanostructure and sub-nanostructure [1]. At the macro-level, a hard and dense part of bones is considered as the compact bone while we can observe a system of osteons at the meso-scale. Each osteon (or Haversian system) is surrounded by the cement line [2]. In the microscopic view of osteons, it can be found that each osteon consists of concentric layers (lamellae) that surround a central canal (Haversian canal). The Haversian canal contains the bone’s blood supplies and may be enveloped by between 5 and 20 lamellae, approximately. By focusing more to reach the submicron level, each lamella layer can be seen which is 3–7µm thick [1]. These lamellae consist of a large number of collagen fibers which are oriented in different directions [1,3]. In fact, several rod-like nanofibrils with less than 1 µm diameter [4] (nanostructure) generate these collagen fibers. Finally, at the sub-nanoscale level, the mineral crystals and collagen molecules become significant. To understand more about the compact bones, some invaluable studies can be advantageous to refer to [1–5]. According to the studies carried out on the substructures of the compact bones, it can be inferred that this type of bone follows a hierarchical system in its structure especially in terms of geometry and architecture. The hierarchical patterns can be observed in many different structures specifically in natural materials [6–13]. D. Soltani Faculty of New Sciences a
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