A determination of the minimum sizes of representative volume elements for the prediction of cortical bone elastic properties

Abstract

At its highest level of microstructural organization-the mesoscale or millimeter scale-cortical bone exhibits a heterogeneous distribution of pores (Haversian canals, resorption cavities). Multi-scale mechanical models rely on the definition of a representative volume element (RVE). Analytical homogenization techniques are usually based on an idealized RVE microstructure, while finite element homogenization using high-resolution images is based on a realistic RVE of finite size. The objective of this paper was to quantify the size and content of possible cortical bone mesoscale RVEs. RVE size was defined as the minimum size: (1) for which the apparent (homogenized) stiffness tensor becomes independent of the applied boundary conditions or (2) for which the variance of elastic properties for a set of microstructure realizations is sufficiently small. The field of elastic coefficients and microstructure in RVEs was derived from one acoustic microscopy image of a human femur cortical bone sample with an overall porosity of 8.5%. The homogenized properties of RVEs were computed with a finite element technique. It was found that the size of the RVE representative of the overall tissue is about 1.5 mm. Smaller RVEs (~0.5 mm) can also be considered to estimate local mesoscopic properties that strongly depend on the local pores volume fraction. This result provides a sound basis for the application of homogenization techniques to model the heterogeneity of cortical microstructures. An application of the findings to estimate elastic properties in the case of a porosity gradient is briefly presented.

Stochastic multiscale modelling of cortical bone elasticity based on high-resolution imaging

“…It was found that the minimum RVE size should be about half a millimetre. This value is coherent with the findings of other authors (Grimal et al 2011a) who have addressed this issue using different approaches. Moreover, within the RVE, our model assumes homogeneous material properties around the pores at the tissue scale.…”

The asymptotic homogenization elasticity tensor properties for composites with material discontinuities

“…This volume should be large enough to statistically represent the whole structure and, at the same time, sufficiently small to enable a computationally feasible approach for the actual computation of the effective elasticity tensor. This can be achieved by performing elastic-type computations on a representative volume element (RVE), which, by definition, satisfies Hill's condition [17,18], i.e., the coarse scale energy (which involves the effective elastic properties) equals the average fine scale energy (see, e.g., [14] for a practical application of the RVE technique for cortical bone). Further average field techniques involve the well-known results by Eshelby [12], i.e.…”

Numerical Investigation of Thermal and Thermo-mechanical Effective Properties for Short Fibre Reinforced Composite

“…Schapery's based his approach on an iso-strain behaviour of fibres and matrix. Equation (15) and Equation (16) show the calculation of longitudinal and transverse coefficients of thermal expansion in Schaperys model:…”

“…RVE concept not only can be used in man-made composite material, natural material, e.g. bones are often described using this concept (see, among others [15,16,17,18]). …”