School of Mathematics, University of Manchester, Alan Turing Building, Manchester M13 9PL, UK.
Biomech Model Mechanobiol. 2012 Jul;11(6):883-901. doi: 10.1007/s10237-011-0359-2. Epub 2011 Nov 23.
We compare theoretical predictions of the effective elastic moduli of cortical bone at both the meso- and macroscales. We consider the efficacy of three alternative approaches: the method of asymptotic homogenization, the Mori-Tanaka scheme and the Hashin-Rosen bounds. The methods concur for specific engineering moduli such as the axial Young's modulus but can vary for others. In a past study, the effect of porosity alone on mesoscopic properties of cortical bone was considered, taking the matrix to be isotropic. Here, we consider the additional influence of the transverse isotropy of the matrix. We make the point that micromechanical approaches can be used in two alternative ways to predict either the macroscopic (size of cortical bone sample) or mesoscopic (in between micro- and macroscales) effective moduli, depending upon the choice of representative volume element size. It is widely accepted that the mesoscale behaviour is an important aspect of the mechanical behaviour of bone but models incorporating its effect have started to appear only relatively recently. Before this only macroscopic behaviour was addressed. Comparisons are drawn with experimental data and simulations from the literature for macroscale predictions with particularly good agreement in the case of dry bone. Finally, we show how predictions of the effective mesoscopic elastic moduli can be made which retain dependence on the well-known porosity gradient across the thickness of cortical bone.
我们比较了皮质骨在细观和宏观尺度上的有效弹性模量的理论预测。我们考虑了三种替代方法的效果：渐近均匀化方法、 Mori-Tanaka 方案和 Hashin-Rosen 界限。这些方法在特定的工程模量（如轴向杨氏模量）上是一致的，但在其他模量上可能会有所不同。在过去的研究中，仅考虑了孔隙率对皮质骨细观性质的影响，将基质视为各向同性的。在这里，我们考虑了基质横向各向异性的额外影响。我们指出，微观力学方法可以通过两种替代方式来预测宏观（皮质骨样本的尺寸）或细观（微观和宏观之间）有效模量，具体取决于代表性体积元素尺寸的选择。人们普遍认为，细观行为是骨骼力学行为的一个重要方面，但包含其影响的模型直到最近才开始出现。在此之前，只研究了宏观行为。我们将宏观预测的实验数据和文献中的模拟进行了比较，在干燥骨骼的情况下，结果特别吻合。最后，我们展示了如何进行有效的细观弹性模量预测，这些预测仍然依赖于皮质骨厚度上众所周知的孔隙率梯度。
