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Proceedings of the 6th International Conference on Mechanics and Materials in Design,
Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015
-893-
PAPER REF: 5630
PLASTICITY EFFECTS DURING THE 4-POINT BENDING OF
INTRAMEDULLARY LEG LENGTHENING IMPLANTS WITH
TELESCOPIC STRUCTURES
Mikko Kanerva1,2(*)
, Zahra Besharat3, Ryan Livingston
4, Harri Hallila
4, Mark Rutland
3
1Invalidisaatio ORTON, FI-00280 Helsinki, Finland
2Aalto University, Department of Applied Mechanics, FI-00076 Aalto, Finland
3Royal Inst. of Technology, Surface and Corrosion Sci. & Material Physics (ICT), Stockholm, Sweden
4Synoste Ltd, FI-02130, Espoo, Finland
(*)Email: [email protected]
ABSTRACT
A telescopic intramedullary leg lengthening implant during standard (ASTM F1264) four-
point bend testing is analysed in this study. The structure of a telescopic implant is simulated
using different finite element models in order to understand the ultimate bending behaviour.
The surface morphology on the internal contact surfaces is also analysed using scanning
electron microscopy and atomic force microscopy. The results show that the simulation of
correct non-linear behaviour necessitates plastic material models and representative contacts
between nested parts. Based on the microscopy analysis, the damage on the contact surfaces
inside the locking mechanism of a real tested implant is not found critical in terms of probable
crack nucleation sites.
Keywords: biomechanics, finite element analysis, atomic force microscopy.
INTRODUCTION
Human leg lengthening due to both inborn and accidental causes has been a challenging
procedure for medicine over 100 years (Hasler and Krieg, 2012). The leg lengthening
procedure is not only challenging from the surgery point of view but also from the mechanical
design point of view. The recent implant concepts for the femoral leg lengthening are based
on the intramedullary fixation, where the implant is fixed inside the femur bone cavity
(Baumgart et al., 2005; Thonse et al., 2005). The diameter of the bone cavity sets a space
limitation, which makes the mechanical strength an obvious optimization criterion for the
implant design. Moreover, due to the requirement, that an implant is not allowed to fracture
into pieces inside the bone cavity, overly fragile metal alloys are not favored.
Prior to the clinical test phase, an implant design faces experimental validation through
mechanical tests. The four-point bending test described in the ASTM F1264 standard is
basically the only procedure applied to the flexural performance validation of leg lengthening
implants. However, a telescopic structure places several problematic issues on the four-point
bending testing of leg lengthening implants. For example, the strength by the standard is
based on the assumption of linear (classical) beam bending. In this study, we focus on the
following issues of implant design and testing: (1) finite element analysis of the yield onset
and related effects on the bending of nested telescope parts; (2) consideration of the element
type and part interaction features in order to save computation time; (3) experimental surface
analysis of the Co-Cr alloy used in the internal parts of the implant.
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NUMERICAL PROBLEM FORMULATION
Finite element (FE) modelling
The part geometry, mesh generation, and problem solution were carried out using a
commercial code Abaqus® (version 6.13-3). The most time consuming simulations were
calculated by the services provided by CSC-IT Center for Science. 3-D models of a telescopic
intramedullary lengthening nail implant were generated to simulate the behaviour during a
four-point bending (4PB) test. The modelled system consists of three to five nested parts
depending on the simulation case studied. Either frictionless contacts or rigid tie constraints
between the separate parts were considered.
The boundary conditions were selected to meet the 4PB test setup. The representation of the
main parts and related boundary conditions as well as contacts is shown in Fig. 1. In all of the
simulation cases, analytical rigid loading points were used to model the 4PB loading fixtures
of the real-life test machine.
Material models were based on experiments (ASTM E8 -09) and a simulated FE model of a
real specimen. The material model verification and the implant models presented in this paper
were based on elements with an approximate volume of 1 mm3 (1.5 mm
2 for shell elements).
For the plastic deformation, we used a plasticity model assuming the basic strain rate
decomposition, dε = dεe + dε
p, and a yield condition based on the deviatoric part of stress (σ):
S = σ + pI
where p = -(trace[σ])/3. The requirement for plastic strain was assumed to satisfy uniaxial-
stress plastic-strain relationship, and a Mises-equivalent, scalar stress was exploited. The
scalar stress values and respective plastic strains were fitted using four (Co-Cr alloy) or three
(stainless steel) discrete points, as given in Table 1.
Simulation case I
The first simulation case was calculated using an implant model consisting of two nested
tubes by shell elements (quadrilateral S4R) with three integration points in the thickness
direction. The solid end of the outer tube and a loading elongation part were built of linear
tetrahedrons (C3D4). The contact between the nested tubes was made a frictionless, separable
contact. All the connections with analytical rigid loading points were made using tie
constraints, where one or two elements (for shells and tetras, respectively) per connection
were tied (see Fig. 2). The connection between the loading elongation and the solid outer tube
end was similarly a tie constraint (node to surface enforcement). Three different simulations
were run: first using linear material models and subsequently using the plastic material
models. Additionally, the effect of the number of shell integration points was studied.
Simulation case II
The second simulation case was created to analyse the effect of the contact type between the
tubes (by S4R shell elements, with three integration points in the thickness direction) and the
analytical rigid loading points. The contact between the nested tubes was made a frictionless,
separable contact, such as in the case I. The connection between the loading elongation and
the solid outer tube end was a frictionless, separable contact (node to surface). The solid end
of the outer tube and the loading elongation were built of linear elements (C3D4) as in the
case I. Two different simulations were run: first using linear material models and
subsequently using the plastic material models.
Proceedings of the 6th International Conference on Mechanics and Materials in Design,
Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015
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Simulation case III
The third simulation case was created to analyse the effect of the internal structure of the
implant during 4PB test. Estimative model of the internal structure was created using linear
hexahedrons (C3D8R) and linear tetrahedrons (C3D4), as illustrated in Fig. 3. The contact
between the rod and locking mechanism of the internal structure was a frictionless, separable
contact. The contact type between the inner tube and the lower analytical rigid loading point,
as well as the contact between the nested tubes, was a frictionless, separable contact. The
connection between the loading extension and the solid outer tube end was a frictionless,
separable contact (node to surface), as in the case II. The solid end of the outer tube and the
loading elongation were built of linear elements (C3D4). Three different simulations were
run: first using shell elements (S4R shell elements, with three integration points in the
thickness direction) for the tubes, second mostly using linear tetrahedrons (C3D4) for the
tubes, and finally using full-integrated quadratic elements (C3D20 / C3D10) throughout the
model. For all of the case III simulations, the plastic material models were applied.
Table 1 - Mises-equivalent scalar stress/strain points applied in material models for 4PB simulation
Material Poisson’s
ratio, -
Young’s
modulus,
GPa
1st yield /
plastic strain,
GPa (m/m)
2nd yield /
plastic strain,
GPa (m/m)
3rd yield /
plastic strain,
GPa (m/m)
4rd yield /
plastic strain,
GPa (m/m)
Co-Cr alloy 0.3 218 0.45 (0) 1.5 (6.7E-5) 1.8 (0.0017) 1.9 (0.0052)
Stainless steel 0.3 200 0.25 (0) 0.3 (0.0011) 0.35 (0.008) (0.4 (0.046))
Fig. 1 - Illustration of the main parts, boundary conditions and contacts included in the 4PB
model of a telescopic implant per simulation case.
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Fig. 2 - The contact used between an analytical rigid surface and deformable tube bodies;
a single element is tied using a node-to-surface enforcement.
Fig. 3 - Illustration of the implant’s internal structure model and contacts to the other parts
of the system per simulation case.
EXPERIMENTS
Atomic force microscopy (AFM)
The internal contact surfaces of the structure were analyzed in order to understand micro- and
nano-level characteristics of the alloy used. Morphology studies were carried out using the
Bruker Dimension Icon. The measurements were performed in a tapping mode using a single
cantilever probe (model µmasch NSC15/AlBS, force constant 40 N/m, tip radius 8 nm).
SEM/EDS (Scanning electron microscopy and energy dispersive spectroscopy)
The surface imaging analysis was performed using a FEI-XL 30 Series instrument, equipped
with an EDS X-Max SDD (Silicon Drift Detector) 50 mm2 detector from Oxford Instruments.
All images (secondary electrons) were obtained using an accelerating voltage of 15 kV.
Surfaces were imaged directly after the disassembling of a 4PB-tested implant.
Proceedings of the 6th International Conference on Mechanics and Materials in Design,
Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015
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RESULTS AND ANALYSIS
The effects of contact at loading points and plastic deformation on the bending stiffness
The results of the case I simulations are shown in Fig. 4. First of all, it can be seen that all the
case I models are overly rigid. The typical 4PB test bending stiffness values for telescopic
intramedullary implants in the existing literature vary between 30 Nm2 and 89 Nm
2 (Guichet
and Casar, 1997; Cole et al., 2001). For the case I simulations, the slope (4055-2028 N/mm)
gives a range of 950-1900 Nm2. The simulated rigidity follows from the axially restrained
nodal tie (in the longitudinal direction of both tubes) between the lower and upper loading
points. In order the implant to bend down, the tied tube ends are required to elongate axially.
This is clearly not true for a real 4PB test fixture. When using the plastic material models, the
system non-linearity starts almost immediately, approximately at a displacement of
-0.25 mm. The number of integration points (three and five) in the shell elements’ thickness
direction did not have effect on the overall flexural behaviour. Three integration points in the
thickness direction was selected to be used for shell elements in subsequent simulations.
Fig. 4 - Simulated bending results using the case I modelling inputs.
The effects of contact definition on bending stiffness
The results of the case II simulations are shown in Fig. 5. The graph illustrates the significant
effect of the type of contacts modelled. The allowance of the upper and lower loading point
per side (for the contacts of inner and outer tube against the loading points) to withdraw by a
frictionless contact decreases the simulated bending stiffness roughly 390%. Similarly as for
the case I simulations, the linear material models clearly incur too a high stiffness. What is
important to note, is that the release of the axial constraints changes the non-linear behaviour
of the system drastically. The onset of the non-linearity in the force-displacement curve starts
after a long essentially linear region. In other words, the release of the axial restraint is
required for both sides. The non-linearity due to material yielding starts far beyond the case I
non-linearity limit. For the case with only one side released (case II-A) the simulated non-
linearity limit is ≈ -1.5 mm of displacement, and for the simulation with both sides released
by frictionless contacts (case II-B) the simulated non-linearity limit is ≈ -9 mm of
displacement.
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Fig. 5 - Simulated bending results using the case II modelling inputs.
The effects of the implant’s internal structure on bending stiffness
The real telescopic implant contains an internal structure, which affects the bending stiffness.
However, the modelling of the internal structure is not straightforward due to its complexity.
An important part of the real internal structure is the locking
axial movement between the inner and outer tubes, i.e., the distal and proximal ends of the
implant when fixed to a bone. Different locking mechanisms have been applied in commercial
telescopic implants (e.g. Cole et al., 2001
represents a roll-based locking mechanism (e.g.
the modelling perspective, this contact is neither fully frictionless nor fully rigid (tied).
The results of simulations with either frictionless or tied contact between the internal parts are
shown in Fig. 7. It can be seen that the model with a tied internal structure is overly rigid.
Also, the element type in the tubes is expected to affect the behaviour when th
structure collides with the tubes’ internal surface and, therefore, solid inner and outer tubes
were analyzed. It can be seen that the solid elements (C3D4) in the tubes result in slightly
more flexible implant simulation compared to the FE mode
Fig. 6 - Roll-based locking mechanism constructed between the implant’s internal parts.
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Simulated bending results using the case II modelling inputs.
The effects of the implant’s internal structure on bending stiffness
The real telescopic implant contains an internal structure, which affects the bending stiffness.
However, the modelling of the internal structure is not straightforward due to its complexity.
An important part of the real internal structure is the locking mechanism, which controls the
axial movement between the inner and outer tubes, i.e., the distal and proximal ends of the
implant when fixed to a bone. Different locking mechanisms have been applied in commercial
telescopic implants (e.g. Cole et al., 2001). Here, the contact between the two internal parts
based locking mechanism (e.g. Engdahl, 2000), as described in Fig. 6. From
the modelling perspective, this contact is neither fully frictionless nor fully rigid (tied).
mulations with either frictionless or tied contact between the internal parts are
shown in Fig. 7. It can be seen that the model with a tied internal structure is overly rigid.
Also, the element type in the tubes is expected to affect the behaviour when th
structure collides with the tubes’ internal surface and, therefore, solid inner and outer tubes
were analyzed. It can be seen that the solid elements (C3D4) in the tubes result in slightly
more flexible implant simulation compared to the FE model with tubes of shells (S4R).
based locking mechanism constructed between the implant’s internal parts.
Simulated bending results using the case II modelling inputs.
The real telescopic implant contains an internal structure, which affects the bending stiffness.
However, the modelling of the internal structure is not straightforward due to its complexity.
mechanism, which controls the
axial movement between the inner and outer tubes, i.e., the distal and proximal ends of the
implant when fixed to a bone. Different locking mechanisms have been applied in commercial
). Here, the contact between the two internal parts
2000), as described in Fig. 6. From
the modelling perspective, this contact is neither fully frictionless nor fully rigid (tied).
mulations with either frictionless or tied contact between the internal parts are
shown in Fig. 7. It can be seen that the model with a tied internal structure is overly rigid.
Also, the element type in the tubes is expected to affect the behaviour when the internal
structure collides with the tubes’ internal surface and, therefore, solid inner and outer tubes
were analyzed. It can be seen that the solid elements (C3D4) in the tubes result in slightly
l with tubes of shells (S4R).
based locking mechanism constructed between the implant’s internal parts.
Proceedings of the 6th International Conference on Mechanics and Materials in Design,
Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015
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Fig. 7 - Simulated bending results using the case III modelling inputs.
Fig. 8 - Von Mises stress distributions for Case I and Case III models. Deformation scale = 1.
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The first non-linearity limit for the fully solid model appears at a displacement of
This local decrease in the force
thus, is related to more accurately simulated plastic deformation at th
(against the inner tube). The stress distribution of the internal structure (see cross
Fig. 9) shows that the internal structure does not yield during the simulated test. Anyhow,
essentially linear behaviour continues
than -8 mm. A general comparison between
A telescopic intramedullary lengthening nail implant has been 4PB tested by EndoLab
GmbH (Germany) and a comparison with the simulated data shows that the FE model with
the solid elements, the plastic material models and frictionless contacts between the internal
parts results in a reasonable estimate of the real bending behaviour (see Fig. 10). Especially
the non-linear behaviour is properly simulated. A notch appears also in the experimental data
(≈ -5.9 mm of displacement). The higher simulated bending stiffness is partly due to the
analytical rigid test fixture; some flexibility by the real test fixture is p
experimental data.
Fig. 9 - Cross-sectional von Mises stress distributions for Case III models where internal
contacts were modelled frictionless. Deformation scale = 1.
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linearity limit for the fully solid model appears at a displacement of
This local decrease in the force-displacement curve does not exist in the shell model and,
thus, is related to more accurately simulated plastic deformation at the left side loading points
(against the inner tube). The stress distribution of the internal structure (see cross
Fig. 9) shows that the internal structure does not yield during the simulated test. Anyhow,
essentially linear behaviour continues until permanent non-linearity at displacements larger
8 mm. A general comparison between case I and case III simulations is shown in Fig. 8.
A telescopic intramedullary lengthening nail implant has been 4PB tested by EndoLab
mparison with the simulated data shows that the FE model with
the solid elements, the plastic material models and frictionless contacts between the internal
parts results in a reasonable estimate of the real bending behaviour (see Fig. 10). Especially
linear behaviour is properly simulated. A notch appears also in the experimental data
5.9 mm of displacement). The higher simulated bending stiffness is partly due to the
analytical rigid test fixture; some flexibility by the real test fixture is probably included in the
sectional von Mises stress distributions for Case III models where internal
contacts were modelled frictionless. Deformation scale = 1.
linearity limit for the fully solid model appears at a displacement of -3.5 mm.
displacement curve does not exist in the shell model and,
e left side loading points
(against the inner tube). The stress distribution of the internal structure (see cross-sections in
Fig. 9) shows that the internal structure does not yield during the simulated test. Anyhow,
linearity at displacements larger
simulations is shown in Fig. 8.
A telescopic intramedullary lengthening nail implant has been 4PB tested by EndoLab®
mparison with the simulated data shows that the FE model with
the solid elements, the plastic material models and frictionless contacts between the internal
parts results in a reasonable estimate of the real bending behaviour (see Fig. 10). Especially
linear behaviour is properly simulated. A notch appears also in the experimental data
5.9 mm of displacement). The higher simulated bending stiffness is partly due to the
robably included in the
sectional von Mises stress distributions for Case III models where internal
Proceedings of the 6th International Conference on Mechanics and Materials in Design,
Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015
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Fig. 10 - Simulated bending results using the case III modelling inputs.
The size and usability of the FE models is illustrated in Fig. 11. It can be seen that the
efficiency in terms of CPU time is highly sensitive to the element type (integration scheme)
and contact definitions. The case III models with linear elements are a convenient
compromise between acceptable computational effort and accuracy for industrial-level design.
a)
b)
Fig. 11 - The size and usability of the FE models used in this study: (a) number of variables; (b)
CPU time required for simulations in this study. The simulation using the model with full-
integrated quadratic elements (case III) was suspended after one week of computation.
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AFM and SEM analysis of the surface damage of the internal part
The contact between the internal parts
was further studied in order to understand the nature of the contact. Any other than
frictionless contact is expected to leave microscopic damage on the contact surface.
SEM imaging of the surface at the
of the rolls, as shown in Fig. 12. However, local damage, in the form of longitudinal
scratches, occurred and indicated sliding of the rolls or contact to the surrounding static parts
(e.g. the cage of the rolls, see Fig. 6) during either assembly or testing.
The AFM height data measured at the roller dents and virgin surface shows that the dents are
smooth on a nano-scale (Fig. 13). Due to the smoothness of the dents (e.g. compared to the
ditches on virgin surface due to manufacture), similar damage in operative implants is not
expected to result in severe fatigue damage, i.e. crack nucleation.
a)
b)
Fig. 12 - Scanning electron microscopy of an internal surface of the locking mechanism after
testing of an implant: (a) general view; (b) detail imaging of the surface damage.
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AFM and SEM analysis of the surface damage of the internal part
The contact between the internal parts of the implant, representing the locking mechanism,
was further studied in order to understand the nature of the contact. Any other than
frictionless contact is expected to leave microscopic damage on the contact surface.
SEM imaging of the surface at the locking mechanism revealed the dents due to the gripping
of the rolls, as shown in Fig. 12. However, local damage, in the form of longitudinal
scratches, occurred and indicated sliding of the rolls or contact to the surrounding static parts
of the rolls, see Fig. 6) during either assembly or testing.
measured at the roller dents and virgin surface shows that the dents are
scale (Fig. 13). Due to the smoothness of the dents (e.g. compared to the
virgin surface due to manufacture), similar damage in operative implants is not
expected to result in severe fatigue damage, i.e. crack nucleation.
Scanning electron microscopy of an internal surface of the locking mechanism after
testing of an implant: (a) general view; (b) detail imaging of the surface damage.
of the implant, representing the locking mechanism,
was further studied in order to understand the nature of the contact. Any other than
frictionless contact is expected to leave microscopic damage on the contact surface.
locking mechanism revealed the dents due to the gripping
of the rolls, as shown in Fig. 12. However, local damage, in the form of longitudinal
scratches, occurred and indicated sliding of the rolls or contact to the surrounding static parts
measured at the roller dents and virgin surface shows that the dents are
scale (Fig. 13). Due to the smoothness of the dents (e.g. compared to the
virgin surface due to manufacture), similar damage in operative implants is not
Scanning electron microscopy of an internal surface of the locking mechanism after 4PB
testing of an implant: (a) general view; (b) detail imaging of the surface damage.
Proceedings of the 6th International Conference on Mechanics and Materials in Design,
Editors: J.F. Silva Gomes & S.A. Meguid, P.Delgada/Azores, 26-30 July 2015
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Fig. 13 - Atomic force microscopy analysis of an internal surface of the locking mechanism
after 4PB testing of an implant. Height data imaging of a dent due to the gripping of rolls.
CONCLUSIONS
A telescopic intramedullary implant during a standard (ASTM F1264) four-point bending test
was studied in this work. The implant structure was simulated using different finite element
models in order to understand the overall deformation and non-linearity.
The simulation results showed that the telescopic structure is locally stressed at the loading
points instead of regular bending. Also, in order to reach reasonable simulation accuracy,
plastic deformation as well as (frictionless) sliding contacts must be correctly modelled
between the nested parts of the implant. The nature of the contact inside a roll-based locking
mechanism is closer to a frictionless sliding (than rigid tie) during the four-point bending.
Scanning electron microscopy and atomic force microscopy of the contact surfaces inside the
locking mechanism of a real tested implant showed that the surface morphology at damage
sites is relatively smooth and is not expected to incur crack nucleation during operational use.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the funding by Orton Oy, Finland, under grant 9310/448.
The authors also acknowledge the computation services by CSC-IT Center for Science.
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