Effect of including damage at the tissue level in the nonlinear homogenisation of trabecular bone

Francesc Levrero-Florencio,Lee Margetts,Pankaj Pankaj,Krishnagoud Manda

doi:10.1007/s10237-017-0913-7

Francesc Levrero-Florencio, Lee Margetts + Show 2 more

Open Access

https://doi.org/10.1007/s10237-017-0913-7

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Abstract

Being able to predict bone fracture or implant stability needs a proper constitutive model of trabecular bone at the macroscale in multiaxial, non-monotonic loading modes. Its macroscopic damage behaviour has been investigated experimentally in the past, mostly with the restriction of uniaxial cyclic loading experiments for different samples, which does not allow for the investigation of several load cases in the same sample as damage in one direction may affect the behaviour in other directions. Homogenised finite element models of whole bones have the potential to assess complicated scenarios and thus improve clinical predictions. The aim of this study is to use a homogenisation-based multiscale procedure to upscale the damage behaviour of bone from an assumed solid phase constitutive law and investigate its multiaxial behaviour for the first time. Twelve cubic specimens were each submitted to nine proportional strain histories by using a parallel code developed in-house. Evolution of post-elastic properties for trabecular bone was assessed for a small range of macroscopic plastic strains in these nine load cases. Damage evolution was found to be non-isotropic, and both damage and hardening were found to depend on the loading mode (tensile, compression or shear); both were characterised by linear laws with relatively high coefficients of determination. It is expected that the knowledge of the macroscopic behaviour of trabecular bone gained in this study will help in creating more precise continuum FE models of whole bones that improve clinical predictions.

Highlights

Bone is a hierarchical biomaterial, exhibiting complex postyield properties at each of its scales (Schwiedrzik et al 2013)
It is interesting to note that in the case of a strain-controlled uniaxial normal load case, the shear stiffness components corresponding to the shear planes containing the loaded normal component reduce more than the other one
It can be seen from this figure that damage development under strain-controlled uniaxial tension and strain-controlled uniaxial compression can be reasonably well predicted by a linear relationship with respect to the macroscopic strain norm, but not so well for shear, as the coefficients of determination suggest

Summary

Introduction

Bone is a hierarchical biomaterial, exhibiting complex postyield properties at each of its scales (Schwiedrzik et al 2013). The macroscopic stiffness tensor of trabecular bone has been evaluated by using micro-CT (μCT) and homogenisation-based multiscale finite element (FE) models, the socalled micro-FE (μFE) models (Hollister et al 1994; van Rietbergen et al 1995) This method consists of picking a cubic region and converting the binarised μCT scans to high resolution FE meshes, which include detailed geometry of bone microstructure. The solid phase is usually assigned isotropic elastic properties, and the cubic specimen is subjected to six virtual load cases, stress or strain (Hollister and Kikuchi 1992) The response from these tests enables the evaluation of the macroscopic elastic stiffness tensor by using a standard mechanics approach (van Rietbergen et al 1996)

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