Interface study

One aim of this study was to establish a standard approach to defining the interface between the two bodies in subsequent studies. The second aim was to identify the impact that the interface conditions had on the construct. This helped to define the simulation strategy but also provided an insight into the difference in behaviour between the experimental disease model where a friction interface existed and the clinical scenario where a continuum between the two bodies exists.

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In the experimental disease model there was an interference fit between the plug and native bone. This held the plug of bone inside the femoral head securely but the coefficient of friction was not quantified experimentally. Therefore it was necessary to evaluate a range of friction coefficients to understand the sensitivity of the simulation to this parameter.

Once the effect of mesh density had been established, the model was adapted to include a plug of material with properties defined independently from the head component. Further mesh convergence was not required as the strains at the boundary were less than the strains in the contact region. As the model was loaded under displacement control, the strains in the contact region were the same as those evaluated in the original mesh convergence study.

The same boundary conditions were used as in the previous study (Section 3.2.2) when the design of the interface between the plug and head was investigated. Nodal alignment and the difference between using a tied interface and a friction interface were evaluated. The former was representative of the continuum between healthy bone and the necrotic lesion in patients’

femoral heads. The latter was more representative of the experimental disease model where an interference fit existed between the native and substituted bone. Two material property combinations were evaluated: one where the head and plug had the same modulus and one where the materials were defined separately.

The first three cases in this study isolated the effect of the interface on the outcome measures by maintaining homogeneity between the head and plug bodies while varying the interface conditions as described in Table 18. The hypothesis that a tied interface would give the same outcome as a solid model and that a reduction in friction coefficient at the interface between the head and plug would reduce the overall stiffness was posed. The remaining cases varied the friction coefficient to gauge its influence on outcome.

This study identified the influence that the characteristics of the interface had on the outcome.

This was investigated further in the parametric study (Section 3.3).

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Table 18. Case list for the interface study. Four models were generated during the preliminary interface study.

Sixteen models with varying head modulus, plug modulus, and interface conditions were generated

Study Case name Head

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The design for the disease model and control are shown in Figure 40. The head and platen were modelled in the same way as described in Section 3.2.2 including assignment of boundary conditions and loading under displacement control. The plug was modelled with a diameter of 10mm and a domed superior end with a radius of 8.5mm. The head and plug were both assigned a modulus of 300MPa in the initial phase of the study. This allowed direct comparison to the results reported in the mesh optimisation study (Section 3.2.2). The plug modulus was then reduced to 150MPa to make the construct more representative of the experimental disease model.

Figure 40. Left: Disease model design. The dotted line represents the boundary between the plug and head. This was either represented as a friction interface or by a tied constraint. To evaluate the effect of the boundary condition, both the head and plug were assigned a modulus of 300MPa. Right: Control model design.

A 0.5mm mesh was used in both models and the inferior edge was pinned to represent the experimental model and allow direct comparison to the results reported previously. For the preliminary interface study, the tied constraint model and friction model were constructed with two mesh orientations: one with nodes aligned at the top of the plug and one with nodes misaligned as shown in Figure 41. In the friction model, tangential movement at the boundary

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was governed by penalty friction with a co-efficient of 1.0; normal behaviour was defined as hard contact.

Figure 41. Mesh detail at the interface between the plug and head at the top of the plug. The arrows point to the boundary. Left: the edge was seeded to align the nodes along the contact surface; Right, the nodes were misaligned.

Results

The results are summarised in Table 20 and presented qualitatively in Figure 42. The tied model closely matched the results of the solid model regardless of mesh alignment at the interface. Comparing the tied interface model to the solid model, the reaction force and modulus increased by 1.8%; peak von Mises stress increased by 2.5%. Small discontinuities caused by nodal misalignment in the stress field were evident in the superior surface of the friction interface.

Changing the interface conditions from tied to friction resulted in a 13% reduction in reaction force and elastic modulus compared to a tied interface model. The peak VM stress increased by 4.3% in the friction model compared to the tied model.

Altering the coefficient of friction made no difference to the elastic modulus or peak VM stress therefore only one value for the friction interface is reported.

A friction constraint allowed separation of the two surfaces in contact caused by bending of the material above the plug. This provided an indication of where both tensile (splitting the interface) and compressive failure may occur. This splitting effect along the boundary extended approximately 80% down the edge of the plug (Figure 43). The actual magnitude, which was a function of the applied Poisson’s ratio, was small. The absolute separation of nodes along the interface was 0.1mm or less.

The reduction in elastic modulus was attributed to the fact that the friction interface did not allow generation of tensile stress at the plug-head interface: the surfaces were allowed to separate after contact. Deformation of the material superior to the plug also reduced the width of the plug in direct contact with the head, reducing the width over which load was transferred from 10mm to approximately 7mm.

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Perhaps the most relevant outcome of this study was identification of a (compressive) stress concentration and large stress gradient at the superior corners of the interface in the friction constraint model. The area of peak compression in the head corresponded to the location of cracks found in gross sections of the in vitro disease models (Figure 34). The outer surface of the interface was in compression with a VM stress of around 8.5MPa. The inner surface was under negligible stress (Figure 44).

Table 19. The effect that the boundary conditions between the head and plug had on the stress and elastic modulus.

In this table, a solid, homogenous model with E=300MPa was compared to a disease model where the head and plug both had E=300MPa. The tied constraint resulted in a very close approximation to the solid model. Replacing the tied constraint with a friction condition significantly reduced boundary reaction force and modulus. The peak von Mises stress increases slightly.

Model Reaction

force (N)

von Mises peak stress (MPa)

Elastic modulus (MPa)

Solid 74.6 15.6 264

Tied / Aligned nodes 75.9 16.1 269

Tied / Misaligned nodes 76.0 16.1 269

Friction interface / Misaligned nodes 66.0 16.3 233

Figure 42. Qualitative comparison of the effect of altering the boundary condition between the plug and head. Top left: A solid homogenous model used as a benchmark; Top right: A head-plug assembly with tied contact and an aligned mesh. Bottom left: a plug assembly with tied contact and a misaligned mesh; Bottom right: A head-plug assembly with a friction interface and misaligned mesh.

A

C D

B

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Figure 43. Example of a simplified disease model with a friction interface in its deformed state (Case four in the interface study, Table 18). The black arrows indicate separation of the interface; the white arrows indicate areas that could fail in compression. The splitting effect extended along approximately 80% of the length of the plug.

Figure 44. Close up view of the superior corner of the head-plug boundary with a friction-based interaction after deformation. The element edge length is approximately 0.5mm and it is clear that approximately three elements (1.5mm) on each side of the superior surface are not in contact, reducing the effective width of the plug to 7mm (applying symmetry). This helped to explain the reduction in elastic modulus. The von Mises stress at the element indicated by the black arrow was approximately 8.5MPa; at the white arrow it was less than 0.1MPa.

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A tied boundary was expected to give similar results to the solid model as the nodes at the boundary were constrained to move together and this was shown to be the case. A friction interface allowed the head and plug components to move independently: One surface deformed more than the other and also allowed the two surfaces to separate as demonstrated in Figure 45.

Thus the elastic modulus of a construct that used a friction constraint was lower than the solid model.

Figure 45. Plot of horizontal displacement caused by vertical compression of the head with a flat platen. The blue colour indicates displacement towards the left of the image; the red colour indicates displacement to the right of the image. Displacement was symmetrical and the head was displaced to a greater extent than the plug.

The difference in outcome measures between the tied models with aligned and misaligned meshes was negligible. However, some discontinuities were visible at the superior interface in the friction interface model with the misaligned mesh so an aligned mesh was used in all future models at the lesion-femur interface and platen-head interface. It was hypothesised that there would be no difference between a homogenous solid model and a homogenous head-plug assembly with a tied boundary constraint. The difference in elastic modulus was between the tied and solid simulations was 1.8% compared to 11.7% when a friction interface was compared to the solid model.

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3.2.4 Definition of measures and outputs