Finite Element Analysis

Finite element wear simulation studies have been published for total joint replacements [103], [10] although the validity and accuracy of the results are under great scrutiny. The primary FEA software package used within this project was Abaqus (Dassault Systèmes Simulia Corp.) which is widely used. This software has been developed for over 30 years and is used to solve many complex engineering problems within industry and academic research. It is designed to interact efficiently with other computer aided engineering software, and this has been explored within the scope of this project. There are examples within the literature showing how Abaqus has been used as a tool to model the contact between joints in the body as well as predict mechanical wear [104].

The importance of verification and validation of the finite element method was discussed by Jones and Wilcox [105]. A method of assessing the validity of the finite element results for a hip implantation was completed and the deformation of the finite element model was within 7.1% difference of mechanical testing of the component. For contact area validation, a study by Jin et al. [106] was conducted to assess the displacement and contact area comparison between the finite element analysis results and experimental work with very good agreement reached between results. A method was described by D’Lima et al. [107] who used pressure transducers to compare experimental peak contact stress and contact areas against results obtained from finite element analysis.

2.3.1.1. Analysis Types and Options

The different types of finite element analysis are clearly and concisely summarised by Heisler [108], as the understanding of this is important to ensure the problem is modelled with

2.3. Computational and Numerical Methods

41

appropriate solvers. As a starting point, linear analysis assumes that the material properties and displacement responses are linear. This offers a chance to assess the model at a less computationally intensive level than considering non-linearity. In terms of hip implant contact problems, it may be reasonable to assume linearity if the predicted contact stresses are not large enough to cause plastic deformation [109]. The complexity of the analysis would be increased when including non-linear effects. Non-linearity occurs when the stiffness of a material changes as the force exerted on it increases (i.e. the material deforms). The types of non-linearity to consider in tribological problems are contact, geometry and material. Non- linearity becomes increasingly important to consider when modelling non-linear materials with large deformations [87].

2.3.1.2. Finite Element Analysis with Wear Simulations

A combination of Lagrangian and Eulerian formulations allows for adaptive meshing techniques to be used for user defined mesh customisation and this technique has been formulated to be used to model specific finite element analysis problems [110]. The Arbitrary Lagragian Eulerian adaptive meshing techniques can be used to deal with large deformations or material loses (i.e. material wear). This can be achieved by a technique allowing the mesh to move independently to the material, therefore moving only the nodes. There are limitations of using adaptive meshing, including the restrictions of element types and contact interaction properties, therefore compromises may be required when using these techniques. The use of adaptive meshing has become an important part of conducting wear simulations for orthopaedic devices, and this methodology has been applied and published in [111]. This technique should not to be confused with adaptive ‘remeshing’ techniques which are used to refine the mesh of a model in areas which require high accuracy.

The need to study artificial joints in terms of its long term performance has been stated [39], and the drive to minimise wear even at the nanoscale is an ongoing ambition of many

2.3. Computational and Numerical Methods

42

engineers. A method of a contact-coupled wear simulation is carried out without the use of the FEM in [109] and good agreement was obtained against results from using the FEM [112]. Jin et al. [109] have criticised the use of a finite element model to study wear due to the amount of time required to build and refine the models, they also argue that this problem can be overcome by using simple equation based methods. It should be noted that as research continues in this field, flexibility, customisation and solving time will continue to improve. By assuming an Archard abrasive wear model [54], the following relationship can be used to simulate the wear between two surfaces in contact using finite element analysis. The following formulation is adapted from [113] to obtain the wear depth, h,

ps

k

h

=

which is also used by Kurtz et al. [114], where the material and surface dependent dimensional wear coefficient kw has units of mm3/Nmm. The contact stress p and sliding distance s are

obtained directly from the finite element post-processing solver. From (2.21), the wear depth, h, is calculated, however within hip implant contact problems, both the sliding distance and contact pressures vary during cyclic loading and angular displacement. Therefore, an equation to cover one complete patient walking gait cycle has been provided [113] for the wear depth distribution matrix W(θ,φ) as an integral with respect to time t from the limits of heal strike hs to toe off to, where θ and φ are the polar angle and azimuthal angle respectively,

(

)

k

P(

t) (v

t)dt

W

,

,

,

,

,ϕ

θ

ϕ

θ

ϕ

θ

=∫

×

×

This equation is formulated to include the point location on the contact surface in terms of spherical co-ordinates (θ,φ), the distribution of normal stress components P(θ,φ,t) and the instantaneous sliding velocity at the contact surface v(θ,φ,t). In its temporally discretized form this equation becomes [113],

2.3. Computational and Numerical Methods 43

( )

k

P(

t)

s(

t)

W

,

,

,

,

,

ϕ

θ

ϕ

θ

ϕ

θ

∑

×

×

=

where Pi(θ,φ,t) is the contact stress distribution, si(θ,φ,t) is the incremental sliding distance and n is the number of increments during the gait cycle. Along with the inclusion of hip joint angular displacement, this is in a form that can be utilised in combination with the finite element method along with the inclusion of hip joint angular displacement.

A similar approach was taken in a separate study which was a 10 year wear simulation of a UHMWPE (ultra high molecular weight polyethylene) patellofemoral joint [103]. The wear rates are comparable to that observed for the hip joint; however, the wear model does not consider wear such as delamination or pitting which are known to be factors from these types of retrieved joints and polyethylene joint implant materials. Although abrasive wear is a major contributing factor to the wear, it is not the only type of wear to be considered (as will be discussed further in this chapter) and it may not be safe to assume a constant wear coefficient. Ong et al. [115] also used a very similar methodology, by applying Archard’s Law and using finite element analysis to determine the contact stress in combination with the sliding distance and wear coefficient. From this, the wear depth under component loading was obtained. Throughout this type of implicit finite element analysis, equilibrium requires establishment as demonstrated by Maxian et al. [116] where, once the contact is established, the displacement control can be replaced by load control.

To provide confidence and further validation of using an adaptively-meshed sliding-distance- coupled finite element model for predicting wear simulations for orthopaedic implants, Callaghan et al. [117] published volumetric wear results which compared within 4.1% of the gravimetric results from experimental testing. By extending the literature review outside of the wear of just orthopaedic device implants, wear simulations have been conducted based on more traditional applications. The work published by Kim et al. [100] presents a good agreement between the results obtained from finite element wear simulation and pin-on-disc