Frame J C, Wheel M A, Riches P E
Department of Biomedical Engineering, University of Strathclyde, Glasgow, G4 0NW, UK.
Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow, G1 1XJ, UK.
Int J Numer Method Biomed Eng. 2018 Jan;34(1). doi: 10.1002/cnm.2903. Epub 2017 Jul 19.
In this paper, we present 2- and 3-dimensional finite element-based numerical models of loaded bovine cortical bone that explicitly incorporate the dominant microstructural feature: the vascular channel or Haversian canal system. The finite element models along with the representation of the microstructure within them are relatively simple: 2-dimensional models, consisting of a structured mesh of linear elastic planar elements punctuated by a periodic distribution of circular voids, are used to represent beam samples of cortical bone in which the channels are orientated perpendicular to the sample major axis, while 3-dimensional models, using a corresponding mesh of equivalent solid elements, represent those samples in which the canals are aligned with the axis. However, these models are exploited in an entirely novel approach involving the representation of material samples of different sizes and surface morphology. The numerical results obtained for the virtual material samples when loaded in bending indicate that they exhibit size effects not forecast by either classical (Cauchy) or more generalized elasticity theories. However, these effects are qualitatively consistent with those that we observed in a series of carefully conducted experiments involving the flexural testing of bone samples of different sizes. Encouraged by this qualitative agreement, we have identified appropriate model parameters, primarily void volume fraction but also void separation and matrix modulus by matching the computed size effects to those we observed experimentally. Interestingly, the parameter choices that provide the most suitable match of these effects broadly concur with those we actually observed in cortical bone.
在本文中，我们展示了基于二维和三维有限元的加载牛皮质骨数值模型，这些模型明确纳入了主要的微观结构特征：血管通道或哈弗斯管系统。有限元模型及其内部微观结构的表示相对简单：二维模型由线性弹性平面单元的结构化网格组成，网格中周期性分布着圆形空隙，用于表示皮质骨的梁状样本，其中通道垂直于样本主轴方向；三维模型则使用等效实体单元的相应网格，用于表示通道与轴对齐的样本。然而，这些模型采用了一种全新的方法，涉及对不同尺寸和表面形态的材料样本进行表示。对虚拟材料样本进行弯曲加载时获得的数值结果表明，它们表现出经典（柯西）弹性理论或更广义弹性理论均未预测到的尺寸效应。不过，这些效应在定性上与我们在一系列精心开展的、涉及不同尺寸骨样本弯曲测试的实验中所观察到的效应一致。受这种定性一致性的鼓舞，我们通过将计算得到的尺寸效应与实验观察结果相匹配，确定了合适的模型参数，主要是空隙体积分数，还有空隙间距和基体模量。有趣的是，能提供这些效应最适宜匹配的参数选择与我们在皮质骨中实际观察到的情况大致相符。
