Wang Congyu, Feng Liang, Jasiuk Iwona
Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, H3G 1M8, Canada.
J Biomech Eng. 2009 Dec;131(12):121008. doi: 10.1115/1.4000192.
We study apparent elastic moduli of trabecular bone, which is represented, for simplicity, by a two- or three-dimensional periodic cellular network. The term "apparent" refers to the case when the region used in calculations (or specimen size) is smaller than a representative volume element and the moduli depend on the size of that region and boundary conditions. Both the bone tissue forming the network and the pores (represented by a very soft material) are assumed, for simplicity, as homogeneous, linear elastic, and isotropic. In order to investigate the effects of scale and boundary conditions on the moduli of these networks we vary the specimen size and apply four different boundary conditions: displacement, traction, mixed, and periodic. The analysis using periodic boundary conditions gives the effective moduli, while the displacement, traction, and mixed boundary conditions give apparent moduli. The apparent moduli calculated using displacement and traction boundary conditions bound the effective moduli from above and below, respectively. The larger is the size of the region used in our calculations, the closer are the bounds. Our choice of mixed boundary conditions gives results that are very close to those obtained using periodic boundary conditions. We conduct this analysis computationally using a finite element method. We also investigate the effect of mismatch in elastic moduli of bone tissue and soft fill, trabecular bone structure geometry, and bone tissue volume fraction on the apparent elastic moduli of idealized periodic models of trabecular bone. This study gives guidance on how the size of the specimen and boundary conditions (used in experiments or simulations) influence elastic moduli of cellular materials. This approach is applicable to heterogeneous materials in general.
我们研究了松质骨的表观弹性模量，为简单起见，松质骨由二维或三维周期性多孔网络表示。术语“表观”指的是计算中使用的区域（或样本尺寸）小于代表性体积单元，且模量取决于该区域的尺寸和边界条件的情况。为简单起见，假定构成网络的骨组织和孔隙（由非常柔软的材料表示）均为均匀、线性弹性且各向同性的。为了研究尺度和边界条件对这些网络模量的影响，我们改变样本尺寸并应用四种不同的边界条件：位移、牵引、混合和周期性边界条件。使用周期性边界条件进行分析可得到有效模量，而位移、牵引和混合边界条件则给出表观模量。使用位移和牵引边界条件计算得到的表观模量分别从上下限界定了有效模量。我们计算中使用的区域尺寸越大，界限就越接近。我们选择的混合边界条件给出的结果与使用周期性边界条件得到的结果非常接近。我们使用有限元方法进行了这项计算分析。我们还研究了骨组织和软填充物的弹性模量不匹配、松质骨结构几何形状以及骨组织体积分数对理想化周期性松质骨模型表观弹性模量的影响。这项研究为样本尺寸和边界条件（在实验或模拟中使用）如何影响多孔材料的弹性模量提供了指导。这种方法一般适用于非均质材料。
