plasticity effects during the 4-point bending · pdf filethe modelled system consists of three...

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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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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Biomechanical Applications

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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.



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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.



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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



An important part of the real internal structure is the locking mechanism, which controls the



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


mulations with either frictionless or tied contact between the internal parts are


Also, the element type in the tubes is expected to affect the behaviour when th



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.



mechanism, which controls the



). Here, the contact between the two internal parts

2000), as described in Fig. 6. From


mulations with either frictionless or tied contact between the internal parts are


Also, the element type in the tubes is expected to affect the behaviour when the internal



l with tubes of shells (S4R).

based locking mechanism constructed between the implant’s internal parts.



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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


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



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


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



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



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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,



SEM imaging of the surface at the locking mechanism revealed the dents due to the gripping



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


of the implant, representing the locking mechanism,



locking mechanism revealed the dents due to the gripping



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




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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.

REFERENCES

[1]-Baumgart R, Bürklein D, Hinterwimmer, et al. The management of leg-length discrepancy

in Ollier’s disease with a fully implantable lengthening nail. J Bone Joint Surg (2005), 87,

p.1000–4.

[2]-Cole JD, Justin D, Kasparis T, DeVlught D, Knobloch C. The intramedullary skeletal

kinetic distractor (ISKD): first clinical results of a new intramedullary nail for lengthening of

the femur and tibia. Int J Care Injured (2001), 32, p.129–139.

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[3]-Engdahl, G. Handbook of giant magnetostrictive materials, Academic Press (2000), San

Diego, CA, USA.

[4]-Guichet J-M, Casar RS. Mechanical characterization of a totally intramedullary gradual

elongation nail. Clin Orthop Relat Res (1997), 337, p.281–290.

[5]-Hasler C, Krieg A. Current concepts of leg lengthening. J Child Orthop (2012), 6, p.89–

104.

[5]-Thonse R, Herzenberg JE, Standard SC, Paley D. Limb lengthening with a fully

implantable, telescopic, intramedullary nail. Operative Tech Orthopaedics (2005), 15, p.355–

362.

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