Effect of geometrical structure variations on the viscoelastic and anisotropic behaviour of cortical bone using multi-scale finite element modelling

Micromechanical modelling of transverse fracture behaviour of lamellar bone using a phase-field damage model: The role of non-collagenous proteins and mineralised collagen fibrils

At the tissue-scale and above, there are now well-established structure-property relationships that provide good approximations of the biomechanical performance of bone through, for example, power-law relationships that relate tissue mineral density to elastic properties. However, below the tissue-level, the individual role of the constituents becomes prominent and these simple relationships tend to break down, with more detailed theoretical and computational models are required to describe the mechanical response. In this study, a two-dimensional micromechanics damage-based representative volume element (RVE) of lamellar bone was developed, which included a novel implementation of a phase-field damage model to describe the behaviour of non-collagenous proteins at mineral-mineral and mineral-fibril interface regions. It was found that, while the stiffness of the tissue was governed by the relative proportion of extra-fibrillar mineral and mineralised collagen fibrils, the strength and toughness of the tissue in transverse direction relied on the interactions occurring at mineral-mineral and mineral-fibril interfaces, highlighting the prominence of non-collagenous proteins in determine fracture-based processes at this scale. While fractures tended to initiate in mineral rich areas of the extra-fibrillar mineral matrix, it was found that the presence of mineralised collagen fibrils at low density did not provide a substantial contribution to crack propagation behaviour under transverse loading. However, at physiological volume fraction (VfMCF=50%), different scenarios could arise depending on the relative strength value of the interphase around the MCFs ( [Formula: see text] ) to the interphase between individual minerals ( [Formula: see text] ): (i) When [Formula: see text] , MCFs appear to facilitate crack propagation with MCF-mineral debonding being the dominant failure mode; (ii) once γ>1, the MCFs hinder the microcracks, leading to inhibition of crack propagation, which can be regarded as an energy dissipation mechanism. The effective fracture properties of the tissue also experience a sudden increase in fracture work density (J-integral) once the crack is arrested by MCFs or severely deflected. Collectively, the predicted behaviour of the model compared well to those reported through experimental and computational methods, highlighting its potential to provide further understanding into the mechanistic response of bone ultrastructure alterations related to the structural and compositional changes resulting from disease and aging.

Effective Governing Equations for Viscoelastic Composites

We derive the governing equations for the overall behaviour of linear viscoelastic composites comprising two families of elastic inclusions, subphases and/or fibres, and an incompressible Newtonian fluid interacting with the solid phases at the microscale. We assume that the distance between each of the subphases is very small in comparison to the length of the whole material (the macroscale). We can exploit this sharp scale separation and apply the asymptotic (periodic) homogenization method (AHM) which decouples spatial scales and leads to the derivation of the new homogenised model. It does this via upscaling the fluid-structure interaction problem that arises between the multiple elastic phases and the fluid. As we do not assume that the fluid flow is characterised by a parabolic profile, the new macroscale model, which consists of partial differential equations, is of Kelvin-Voigt viscoelastic type (rather than poroelastic). The novel model has coefficients that encode the properties of the microstructure and are to be computed by solving a single local differential fluid-structure interaction (FSI) problem where the solid and the fluid phases are all present and described by the one problem. The model reduces to the case described by Burridge and Keller (1981) when there is only one elastic phase in contact with the fluid. This model is applicable when the distance between adjacent phases is smaller than the average radius of the fluid flowing in the pores, which can be the case for various highly heterogeneous systems encountered in real-world (e.g., biological, or geological) scenarios of interest.

Effect of geometrical structure variations on strength and damage onset of cortical bone using multi-scale cohesive zone based finite element method

Three-dimensional multi-scale finite element models were designed to examine the effects of geometrical structure variations on the damage onset in cortical bone at multiple structural scales. A cohesive zone finite element approach, together with anisotropic damage initiation criteria, is used to predict the onset of damage. The finite element models are developed to account for the onset of microdamage from the microscopic length scales consisting of collagen fibres, to the macroscopic level consisting of osteons and the Haversian canals. Numerical results indicated that the yield strain at the initiation of microcracks is independent of variations in the local mineral volume fraction at each structural scale. Further, the yield strain and strength properties of cortical bone are dependent on its structural anisotropy and hierarchical structure. A positive correlation is observed between bone strength and mineral content at each length scale.

Closing cones create conical lamellae in secondary osteonal bone

Lamellae are sheets of mineralized collagen 1-20 µm thick, extending over hundreds of µm in bone tissue, occupying bone's structural hierarchy at a level above collagen fibres and osteocytes, and below osteons and trabeculae. Osteons are tubular arrangements of lamellae surrounding central neurovascular canals. Lamellae in osteons are usually described as concentric cylinders based on their annular appearance in transverse section. In this review, I provide a perspective on current understanding of the relationship between geometry of the bone formation front and the shape of lamellae produced at it, reaching the conclusion that the 'closing cone' bone formation front in secondary osteonal remodelling must necessarily result in cone-shaped lamellae in the mature secondary osteon. Secondary osteons replace primary osteons through a tunnelling process of bone turnover, meaning that conical lamellae may become more common in older and damaged bone which is at greatest risk of fracture. Visualization and measurement of three-dimensional lamellar shape over hundreds of microns is needed to provide data for accurate micromechanical simulations. Treating secondary osteonal lamellae as a 'stack of cones' rather than 'nested cylinders' may have important implications for our appreciation of bone's function as a load-bearing tissue and of its behaviour in fracture.

Damage analysis of human cortical bone under compressive and tensile loadings

Developing advanced fracture tools can increase the understanding of crack growth trajectories in human cortical bone. The present study investigates fracture micromechanics of human cortical bone under compressive and tensile loadings utilizing a phase field method. We construct two-dimensional finite element models from cortical microstructure of a human tibia cross section. We apply compression on the cortical bone models to create compressive microcracks. Then, we simulate the fracture of these models under tension to discover influential parameters on microcracks formation and post-yielding behavior. The results show that cement lines are susceptible sites to damage nucleation under compression rather than tension. The findings of this study also indicate a higher accumulation of initial damage (induced by compression) can lead to a lower microscopic stiffness as well as a less resistant material to damage initiation under tension. The simulations further indicate that the post-yielding properties (e.g., toughness) can be dependent on different variables such as morphological information of the osteons, the initial accumulation of microcracks, and the total length of cement lines.