Department of Civil Engineering (Structural Engineering), Babol Noshirvani University of Technology, Babol, Iran.
ARTORG Centre for Biomedical Engineering Research, University of Bern, Bern, Switzerland.
Biomech Model Mechanobiol. 2020 Dec;19(6):2127-2147. doi: 10.1007/s10237-020-01328-1. Epub 2020 Apr 24.
In this study, 3D finite element analyses (FEA) are conducted to quantify the orthotropic elastic properties and investigate the load transfer mechanism of bone at the sub-lamellar level. Three finite element (FE) unit cells with periodic boundary conditions are presented to model a two-scale microstructure of bone including a mineralized collagen fibril (MCF), the extrafibrillar matrix (EFM) and the resulting fibril array (FAY) under arbitrary loading. The axial and transverse elastic properties of the FAY computed by FEA are calibrated with unique experimental results on ovine micro-samples showing a coherent fibril orientation. They are then systematically compared with those calculated using analytical methods including the basic Voigt, Reuss and shear-lag models, the Mori-Tanaka scheme and the upper and lower bounds by Hashin and Shtrikman. The predicted axial strain ratios between the two-scales are discussed with respect to a recent small-angle X-ray scattering and wide-angle X-ray diffraction study. Beyond apparent elastic properties, the FE models provide stress distributions at both hierarchical levels, confirm the shear lag mechanisms within the MCF and between MCF and EFM and identify potential damage sites under arbitrary loading conditions. A comprehensive sensitivity analysis shows that mineral volume fraction in the fibril array is the dominant parameter on the axial and transverse elastic moduli, while the MCF volume fraction in FAY is the most sensitive variable for the ratio of axial versus transverse elastic modulus followed by the elastic moduli of hydroxyapatite and collagen. The FE model of the FAY developed and calibrated in the current study represents an anatomically realistic, experimentally validated and computationally efficient basis for investigating the apparent yield, post-yield and failure behaviors of lamellar bone in future research.
在这项研究中，进行了三维有限元分析（FEA），以量化骨的各向异性弹性特性，并研究亚层水平的骨的载荷传递机制。提出了三个具有周期性边界条件的有限元（FE）单元，以模拟包括矿化胶原纤维（MCF）、细胞外基质（EFM）和由此产生的纤维束排列（FAY）的两尺度微观结构，在任意载荷下。通过 FEA 计算的 FAY 的轴向和横向弹性特性与绵羊微样本的独特实验结果进行了校准，这些结果显示出一致的纤维取向。然后，将它们与使用分析方法（包括基本的 Voigt、Reuss 和剪切滞后模型、Mori-Tanaka 方案以及 Hashin 和 Shtrikman 的上下限）计算的结果进行了系统比较。讨论了两尺度之间的预测轴向应变比与最近的小角 X 射线散射和广角 X 射线衍射研究的关系。除了明显的弹性特性之外，FE 模型还提供了在两个层次上的应力分布，证实了 MCF 内部和 MCF 与 EFM 之间的剪切滞后机制，并确定了在任意载荷条件下的潜在损伤部位。全面的敏感性分析表明，纤维束排列中的矿物质体积分数是轴向和横向弹性模量的主要参数，而 FAY 中的 MCF 体积分数是轴向与横向弹性模量之比的最敏感变量，其次是羟磷灰石和胶原的弹性模量。当前研究中开发和校准的 FAY 的 FE 模型代表了一种解剖学上真实、实验验证和计算有效的基础，可用于未来研究中研究层状骨的明显屈服、屈服后和失效行为。
