Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods

Abstract

The key parameters determining the elastic properties of an unidirectional mineralized bone fibril-array decomposed in two further hierarchical levels are investigated using mean field methods. Modeling of the elastic properties of mineralized micro- and nanostructures requires accurate information about the underlying topology and the constituents' material properties. These input data are still afflicted by great uncertainties and their influence on computed elastic constants of a bone fibril-array remains unclear. In this work, mean field methods are applied to model mineralized fibrils, the extra-fibrillar matrix and the resulting fibril-array. The isotropic or transverse isotropic elastic constants of these constituents are computed as a function of degree of mineralization, mineral distribution between fibrils and extra-fibrillar matrix, collagen stiffness and fibril volume fraction. The linear sensitivity of the elastic constants was assessed at a default set of the above parameters. The strain ratios between the constituents as well as the axial and transverse indentation moduli of the fibril-array were calculated for comparison with experiments. Results indicate that the degree of mineralization and the collagen stiffness dominate fibril-array elasticity. Interestingly, the stiffness of the extra-fibrillar matrix has a strong influence on transverse and shear moduli of the fibril-array. The axial strain of the intra-fibrillar mineral platelets is 30-90% of the applied fibril strain, depending on mineralization and collagen stiffness. The fibril-to-fibril-array strain ratio is essentially ~1. This study provides an improved insight in the parameters, which govern the fibril-array stiffness of mineralized tissues such as bone.

Influence of the mineral staggering on the elastic properties of the mineralized collagen fibril in lamellar bone

“…In this case, a transverse isotropic behaviour is obtained (in agreement with Reisinger et al (2010)) that tends to be more anisotropic when volume fraction of mineral increases. Following the array proposed by Rubin et al (2003), results shown in this work reveal that the fibril assembly inside the fibre is stiffer in the axial direction regardless the presence of mineral overlapping in the individual fibrils.…”

“…The influence on the mechanical properties of bone, of both the constituents properties and their specific arrangement at every length scale, is nowadays an important investigation aim. Reisinger et al (2010) …”

X-ray diffraction as a promising tool to characterize bone nanocomposites

“…The diffracted intensity pattern from the polycrystalline aggregates hence can be useful in characterizing the spatial arrangement and orientation of the crystals. X-ray pole figure analysis is a popular tool in the texture characterization [14,[80][81][82] and the orientation patterns of collagen fibers in bone are well documented [83][84][85][86][87][88]. The texture analysis of mineral crystals using x-ray pole figure reveals that the mineral crystals are oriented not only parallel to the long axis of bone but also in other directions [80,82], and that the orientations of mineralized fibrils form a spiral around the central axis, exhibiting varying orientation angles in different lamellae [81,89].…”

Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models

“…The orientation of collagen fibrils changes in successive layers. Reisinger et al (2011) develop a detailed analysis of elastic properties of the microstructure using a finite element model in which the elastic constants of lamellar bone have been calculated using homogenization theories in a previous work (Reisinger et al, 2010). These authors analyze several collagen fibril orientation patterns.…”

“…Several works have been developed to calculate elastic constants of mineralized collagen fibrils. In Reisinger et al (2010), homogenization theory of Mori Tanaka is used to estimate elastic properties of mineralized collagen fibrils. However, they suppose that the crystals are spheroidal and randomly distributed.…”