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The Elasto-Plastic Properties of Trabecular Bone and Polyurethane Foam:
An Experimental and Computational Characterisation
Kelly, N. 1,2 , McGarry, J.P. 1,2
1 Department of Mechanical and Biomedical Engineering, National University of Ireland, Galway
2 National Centre for Biomedical Engineering Science, National University of Ireland, Galway
n.kelly3@nuigalway.ie
Abstract
During compressive loading trabecular bone can
undergo extensive inelastic deformation reaching
strains up to 60 % prior to ultimate failure [1] . Previous
bone plasticity studies have considered a number of
pressure dependent constitutive formulations including
the Drucker-Prager (DP) and Mohr-Coulomb (MC)
models [1-3] . The crushable foam (CF) plasticity model
has been used to model polymeric foams but has not
been investigated for trabecular bone.
1. Introduction
The current study entails a detailed experimental and
numerical investigation of the inelastic behaviour of
bovine trabecular bone (BTB) and a commercially
available trabecular bone analogue material (PU foam,
sawbones). Material behaviour under conditions of
unconfined and confined compression is determined and
a suitable inelastic constitutive formulation is identified.
Specifically the following constitutive formulations are
investigated: DP; MC; isotropic crushable foam (CFiso);
volumetric crushable foam (CFvol). Following this, we
investigate the surgical implantation of a tibial
component (Genesis II, Smith&Nephew) into a sawbone
tibia (composed of PU foam to replicate trabecular
bone) to identify if material yield occurs.
2. Materials and Methods
Experimental: 15 mm cubic specimens of PU foam
(ρ = 0.16 g/cm 3 ) and proximal tibial BTB were tested
destructively in unconfined and confined (custom rig)
uniaxial compression at a rate of 5 mm/min (Instron
4467, Instron Corp., USA).
Computational: 3D FE confined and unconfined
uniaxial compression tests of PU and BTB were
simulated (Abaqus v6.8). A 3D FE model was created
to simulate surgical tibial component implantation into a
sawbone tibia (#3402) (Fig.1B). Frictionless contact
was assumed between the components.
3. Results & Discussion
Experimental and computational results for PU foam
and BTB are shown in Fig.1A. Unconfined
experimental results for PU (E = 35 MPa, σy = 1.5 MPa)
and BTB (E = 364 MPa, σy = 10 MPa) are within the
reported range for human trabecular bone [6] . Simulation
of tibial component implantation results in a maximum
computed stress of ~14 MPa (Fig.1B-C).
60
(A)
Nominal Stress (MPa)
Nominal Stress (MPa)
Nominal Stress (MPa)
Nominal Stress (MPa)
3
2
1
0
5 53
0 4 4
2
3 3
0.1 0.2 0.2 0.3 0.3 0.4 0.4
Nominal
PU Nominal Strain
Foam Confined Strain
0.5 0.5 0.6 0.6
Nominal Stress (MPa)
2 2
1
1 1
PU Foam Unconfined
0 0
0 0
0.1 0.1 0.2
0.2 0.2 0.3
0.3 0.3 0.4
0.4 0.4 0.5
0.5 0.5
Nominal
Nominal Strain
Strain
Cfiso
Cfiso
Cfvol
Cfvol
MC
MC
DP
DP
Experimental
Experimental
0.6
0.6 0.6
Nominal Stress (MPa)
Nominal Stress (MPa)
Nominal Stress (MPa)
Nominal Stress (MPa)
Nominal Stress (MPa)
BTB Unconfined
0
0
5 0
100
100
4
80
80
3
60
60
40 2
40
20
20 1
0.1
0.1
0.2 0.3
0.2 0.3
Nominal
Nominal
BTB Confined Strain
Strain
0.4
0.4
0
0
0 0.1 0.2 0.3
0 0.1 0.1 0.2 0.3 0.2 0.4 0.3 0.5
Nominal Strain
Cfiso Cfvol Nominal
Nominal
MC DP Strain
Strain
Experimental
CFiso CFvol MC DP Experimental
CFiso CFvol MC DP Experimental
Mises/σy 0.6
(B) (C)
Figure 1 (A) Unconfined and confined experimental and
computational results for PU and BTB. (B) Schematic of
tibial component implantation into a sawbone tibia (PU
foam). (C) Tibial component implantation results.
In unconfined compression all 4 plasticity models are
calibrated to the experimental results (Fig.1A). Under
both loading conditions only the CFiso and CFvol models
demonstrate reasonable correlation with experimental
results. The pressure dependent yield criteria in both CF
models allow for accurate simulation of the inelastic
behaviour of PU and BTB under confined compression.
Our investigations suggest that for confined
compression the MC and DP models are not appropriate
for trabecular bone as the perfect plasticity type
behaviour cannot be captured post yield.
6. Conclusions
To the authors knowledge these constitutive
formulations have not previously been applied to
trabecular bone and this is also the first study to
investigate plastic deformation caused during tibial
component implantation. Simulation of tibial
component implantation results in trabecular stresses in
excess of the σy of PU, highlighting the importance of
modelling elasto-plastic material behaviour.
8. References and Acknowledgements
[1] Mercer (et al.), Acta Mater. 2:59, 2006. [2] Mullins (et
al.), J.M.B.B.M. 2:460, 2009. [3] Wang (et al.), Bone 43:775,
2008. [4] Desphande & Fleck 48:1253, 2000. [5] ABAQUS
Analysis Theory Manual, 2010. [6] Li & Aspden, J. Bone
Miner. Res. 12: 641, 1997.
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