Hood [1] reported permanent deformation in components of ... · Hood [1] reported permanent...

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Contact stress analysis of tibial components of prosthetic knee implants experimental and finite element analysis T. McGloughlin, J.Monaghan Department of Mechanical and Aeronautical Engineering, University of Limerick, Limerick, Ireland, and Department of Mechanical and Manufacturing Engineering, University of Dublin, Trinity College, Dublin 2, Ireland ABSTRACT Following the clinical success of total hip joint replacement in the 1960s, total knee joint replacement has become widespread. Coupled with these developments has been a concurrent development of engineering analysis of the implant to bone system. Of particular significance in total knee joint replacement have been the high levels of contact stresses which occur between the metallic femoral implant and the polymeric tibial component. These high contact stresses have led to the failure of some components and thus remain the subject of widespread research. The present paper will describe an experimental and finite element study of this contact condition. Using modelling theory and previously developed techniques, an experimental model of the femoral and tibial interface has been designed and constructed. The model made from Araldite resin (CT200) contains embedded strain gauges in the contact region which enable measurements of strains to be recorded for various loads. Stresses are subsequently evaluated for both normal loading and normal loading with sliding. In addition to experimental analysis, a Finite Element model of both the metal to CT200 interface and the metal to UHMWPE interface has been developed. The contact condition has been modelled using MARC software which has extensive capabilities for modelling contact. The FE results and the experimental results were benchmarked with known Hertzian solutions for the normal contact case for loads in the elastic range. Subsequent analysis examined the behaviour of the components in the elastic and the elasto- plastic range for both normal and sliding loads and results are compared to previously published theoretical solutions. 1 Introduction Modem knee implant designs have evolved in recent years from unrestrained highly con- forming designs to unrestrained highly non-conforming devices. In almost all cases the ultra-high molecular weight polyethylene (UHMWPE) tibial component has had a metallic tray to support and reinforce it. These design changes have significant engineering implica- tions for the long term life of Total Knee joint replacement components and particularly for the polyethylene tibial component. Recently the medical literature has reported a significant level of failure in knee implants and in particular failure of UHMWPE components. Transactions on Modelling and Simulation vol 10, © 1995 WIT Press, www.witpress.com, ISSN 1743-355X

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Contact stress analysis of tibial components

of prosthetic knee implants — experimental

and finite element analysis

T. McGloughlin, J. Monaghan

Department of Mechanical and Aeronautical Engineering,

University of Limerick, Limerick, Ireland, and Department

of Mechanical and Manufacturing Engineering, University

of Dublin, Trinity College, Dublin 2, Ireland

ABSTRACT

Following the clinical success of total hip joint replacement in the 1960s, total knee jointreplacement has become widespread. Coupled with these developments has been aconcurrent development of engineering analysis of the implant to bone system. Of particularsignificance in total knee joint replacement have been the high levels of contact stresseswhich occur between the metallic femoral implant and the polymeric tibial component.These high contact stresses have led to the failure of some components and thus remain thesubject of widespread research. The present paper will describe an experimental and finiteelement study of this contact condition. Using modelling theory and previously developedtechniques, an experimental model of the femoral and tibial interface has been designed andconstructed. The model made from Araldite resin (CT200) contains embedded strain gaugesin the contact region which enable measurements of strains to be recorded for various loads.Stresses are subsequently evaluated for both normal loading and normal loading withsliding. In addition to experimental analysis, a Finite Element model of both the metal toCT200 interface and the metal to UHMWPE interface has been developed. The contactcondition has been modelled using MARC software which has extensive capabilities formodelling contact. The FE results and the experimental results were benchmarked withknown Hertzian solutions for the normal contact case for loads in the elastic range.Subsequent analysis examined the behaviour of the components in the elastic and the elasto-plastic range for both normal and sliding loads and results are compared to previouslypublished theoretical solutions.

1 Introduction

Modem knee implant designs have evolved in recent years from unrestrained highly con-forming designs to unrestrained highly non-conforming devices. In almost all cases theultra-high molecular weight polyethylene (UHMWPE) tibial component has had a metallictray to support and reinforce it. These design changes have significant engineering implica-tions for the long term life of Total Knee joint replacement components and particularly forthe polyethylene tibial component. Recently the medical literature has reported a significantlevel of failure in knee implants and in particular failure of UHMWPE components.

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728 Computational Methods and Experimental Measurements

Hood [1] reported permanent deformation in components of less than 10 mm thickness andalso the collapse of support while Landy et al [2] reported very high contact stresses, exten-sive surface damage, fatigue and delamination of polyethylene and high sub-surface stressescoupled with wear rates higher than that in total hips. Jones et al [3] reported a clinicalfailure rate of 5% and found catastrophic levels of wear in low conforming tibial componentsof knees implants. They suggested that high contact stresses were a major contributor to thefailures. The low conforming knee implants were further indicted by Engh et al, and Engh[4,5]. Significant levels of wear due to high contact stresses set up by low conformity or dueto implant misalignment especially in thin implants, were cited as probable causes of failure.Nolan et al [6] also reported similar clinical problems with a low conforming knee implantand in unicondylar knee implants with low conformity Christensen et al [7] and Lindstrandet al [8] also reported fracture of the UHMWPE. Goodfellow [9] also drew attention to thehigh contact stresses and associated high wear rates being reported in non-conforming kneeimplants. Tsao et al [10] in 1993 reported on an extensive study of total knee replacementsurgery in which severe wear including cracking and delamination was present in 7% ofcases. Clearly the high contact stresses caused by the low conformity between the metalfemoral component and the polyethylene component contributed to these reported failures.

Wear and Contact StressesEarly work on wear rates by Rostoker and Galante [11] predicted accelerated wear rates onUHMWPE when the material was subjected to high contact stresses as is clearly the case inmany modem knee implants. They attempted to develop a predictive equation and sug-gested that for a doubling of contact stress, wear rate increases five fold. Rose et al [12]recommended caution in the use of UHMWPE in semi-constrained knees and they con-cluded that low conformity total knee prostheses joint replacements would require highermolecular weights than the high conformity hip replacements prostheses because of thehigher contact stresses caused by the non-conformity. Connolly et al [13] concluded thatminor changes in stress intensity in UHMWPE as might occur in low conformity kneearthroplasties could generate a rapid crack growth situation in a short space of time.Wrightand Bartel [14] found very high local contact stresses in a study of a wide range of prostheticgeometries. They concluded that these high contact stresses gave rise to a fatigue mechanismin the UHMWPE and thus contributed to failure. B artel et al [15] in a further study foundthat as thickness of the polyethylene component decreases even with metal backing that thecontact stress rose diamatically and further suggested that the maximum shear stress occursbelow the surface and that this may be an important indicator of subsurface cracks. Smith andLiu [16] considered the influence of both normal and tangential loads on contactstresses .Their work suggested that,when both normal and tangentail loads are present, themaximum shear stress moves to the surface and is 43% larger than the maximum shear stresswhich occurs below the surface when only a normal stress is active. Significant shear stressesmay well occur at the prothetic tibial-femoral interface and their results would suggest thatthe location and magnitude of the maximum shear stress are indeed a cause for concern.More recently Johnson [17,18],Hamilton et al [19] and Hamilton [20], have examined thestress distributions in sliding contacts in considerable detail and they found that in a slidingcontact the maximum tensile stress moves to the edge of the contact. With the presence offriction there is a compressive stress which occurs at the front edge and a tensile stress at thetrailing edge.In the context of tibial components of knee implants subjected to sliding loadsof an oscillating nature this has clear implications for the fatigue life of the plastic tibialcomponent described earlier. In relation to Finite Element Analysis (FEA) of contact sur-faces, noteworthy studies have been undertaken by Su and Youn [21] and Tian and Saka [22].Su and Youn examined the stress and strain distributions in the subsurface of High DensityPolyethylene and generated a wear model for HDPE taking account of frictional effects. Tianand Saka, conducted FEA by examining a two layer half-space without friction under normalload. Normal and tangential loads were then applied incrementally on the surface.

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Computational Methods and Experimental Measurements 729

Experimental Analysis

BackgroundDuring the normal walking cycle the components undergo cyclic compressive loading due tothe weight of the patient and in addition there is strong evidence to suggest that there arealso tangential loads caused by sliding of the components and by the tendency of the musclepulls to rotate the joint about the long axis of the bone. These combined normal and slidingloads are oscillatory in nature and thus induce cyclic tension and compression in the poly-meric materials. It is well known that such a loading condition can produce early fatiguetype failures in both metallic and polymeric materials. The contact conditions in manyprosthetic knee implants can be adequately described by reducing the geometries to theirsimplest form using the descriptions developed by Timoshenko [23] and others .

Figure 1: Cylinder in contact with flat surface

Model DesignIt was decided to begin the study with an investigation of a 2-D contact condition sincethere were published theoretical solutions for certain contact geometries.There is not anappropriate method for gauging polymeric materials such as UHMWPE and thus the meth-ods of embedded strain gauges developed by Bazergui and Little [24,25] and others werefelt to be an appropriate technique for the measurement of internal strains in the solids incontact with one another.The technique of embedded strain gauging requires the use of anappropriate castable material in which the gauges can be mounted. The castable materialmost widely used for this purpose is Araldite CT200 (Ciba-Geigy) and since it is not practi-cal to position strain gauges in components which have the same dimensions as the pros-thetic components an appropriately scaled model is needed. The model in this case wasbased on the dimensions of currently used implant components and the size of the modelwas determined using model analysis as described by Monch [26].The initial model chosenwas a cylindrical indentor coming into contact with a flat surface as shown in Figure 1. Thedimensions of the indentor and half-space were obtained from the modelling theory previ-ously described.

Strain Gauge Selection and LocationThe strain gauges were positioned in a vertical plane as shown in Figure 2 and this allowedthe experimental analysis to be simplified to a Plane Strain analysis. In order to measurestrains in the contact region, very small strain gauges were needed so that the gauges couldbe positioned as near as possible to the contact surface. Gauges of 2mm diameter wereselected for initial testing as they were readily available with suitable long lead wires. Thesetests demonstrated the suitability of using embedded strain gauges as a technique for themeasurement of the contact behaviour of solids. In all, seven 2mm diameter rosettes wereused in the tests and prior to mounting the gauges on the CT200 block, the resistances of thegauges were checked. The test blocks were manufactured in two halves and the gauges werepositioned on the vertical face of one half. When the two halves of the CT200 block wereglued together, the surface of the block was machined to an accurate finish level. The precisepositions of the strain gauges were needed to enable experimental results to be compared withtheoretical results and also to enable values obtained from the experimental data to be com-pared with nodal data obtained from FE analysis. The positions of the gauges were measuredaccurately using a travelling microscope and the locations are also shown in Figure 2,

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730 Computational Methods and Experimental Measurements

STRAIN GAUGE LAYOUT

Figure 2: Locations of embedded strain gauges within CT200 Block

Model ManufactureThe model material used in this study consisted of a suitable castable epoxy resin (AralditeCT200). The dimensions of the block were 140mm by 45mm by 65mm which ensured thatthe block was large enough to accommodate a reasonable number of strain gauges. Thegauges were selected with long lead wires to enable them to be connected to the bridgecircuit with a minimum number of junctions. The indentors were manufactured to theselected 150mm radius using stainless steel and the radius was ground and polished to amirror finish. The line contact produced when the single 150mm radius of the indentor cameinto contact with the flat surface of the CT200 block generated a contact condition whichwas comparable with the contact conditions occurring in some currently used knee implantpros theses.

Test Rig DesignThe requirements for loading the model were as follows1 A loading mechanism capable of applying static normal loads to the surface of a CT200model carrying embedded strain gauges.2 A suitable means of applying a horizontal load and thereby simulating sliding. Themagnitude of the sliding load was determined using the Coulomb Friction Law F = jiN.3 A suitable means of recording strain data.A loading arrangement was designed using the bed of a milling machine to provide rigidityand stability. The loading arm was designed to produce normal loads ranging from IkN to3kN at the indentor to CT200 interface. In addition, a cabling arrangement and rollerbearings enabled the horizontal load to be applied as shown in Figure 3.In order to establishthe magnitude of the normal loads being applied using the lever mechanism shown, the rigwas calibrated using a load cell.

Experimental SetupThe strain bridge consisted of an active arm (in the model block of CT200), a passive arm(additional gauges positioned in a further block of CT200), and precision wound 120Hresistors in the other two arms. The passive arm in the circuit ensured that thermal fluctua-tions in the model were balanced in the dummy ann.The bridge was connected to a TA880Multimonitor which enabled strain readings to be recorded. The Multimonitor was in turnconnected to a microcomputer for storing the strain data.

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Computational Methods and Experimental Measurements 731

PoUhedMetalcInd enter

CabteforapplyhgHoraontelLoad

\Arm andmass hangarfor applyingNormal Load

Figure 3: Schematic diagram of experimental test rig - Normal loads and Normal andSliding loads

Testing Procedure

(a) Normal loadsThe model under test was mounted in the holding fixtures prior to testing and checked forflatness. In addition the axis was positioned to coincide with the axis of the metallicindentor. Following the application of the load, strain readings were taken for all sevengauges and the load was removed. Additional weights were added to the arm in steps of 5kgand the loading sequence was repeated.

(b) Tangential loads and Normal loadsThe experimental procedure was as described in (a) but on this occasion a horizontal loadwas added. This horizontal load was applied using a horizontal cable attached to the carry-ing fixture and friction between the lower surface of the fixture and the table was minimisedby the introduction of suitable roller bearings also shown in Figure 3The horizontal loads were applied in such a fashion as to produce an incipient slidingcondition between the metal indentor and the CT200 block. Strain gauge readings wererecorded as before and appropriate stresses were calculated.

Finite Element Analysis

The contact condition which was examined experimentally using embedded strain gaugeswas also examined using Finite Element Analysis. The CT200 block was modelled using1064, 4-noded linear quadrilateral elements using Mental 2.1 and MarcK6 software.TheMARC software has extensive contact capabilities and the problem was analysed using arigid indentor coining into contact with the CT200 block. Figure 4 shows the meshgeometry and boundary conditions for normal loading while Figure 5 shows the meshgeometry and boundary conditions for normal and sliding loads. The loads were appliedincrementally using the time stepping capabilities of the software and the analysis wasconducted using plane strain analysis, treating the material as an elastic solid. Threethicknesses of block were examined with the application of normal loads and the resultswere compared to the Hertzian values for the same loads. In addition nodal stresses from theFE analysis were compared to the nodal stresses obtained from the strain gauge data.

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732 Computational Methods and Experimental Measurements

Figure 4 Mesh Geometry Normal Loading -- inset showing mesh in contact region

To further validate the FE results Fuji Presensor pressure sensitive film was used on the testrig to obtain contact area values for different loads.These were compared with both thetheoretically predicted contact area using Hertzian theory and to the values obtained fromthe FE output. (Table 1).

Light springs toprevent Rigid Bodydisplacement.

Horizontal Loads

Figure 5 Mesh Geometry Normal and Sliding Loads

Once satisfactory results were obtained from the FE anaysis for normal loads, additionalanalysis was conducted where the CT200 block was subjected to normal loads and slidingloads simultaneously. In addition, the indentor was modified and a rolling motion combinedwith a normal and sliding load was studied. This was felt to be necessary in order to try tomimic more accurately the motion of the prosthetic knee components.

TABLE 1CONTACT AREAS

CT200 BLOCK UHMWPE

LOAD (N/mm)58.6573.789.95105.451122

Hertz Analysis(lanun)3.463.884.244.644.79

Fuji Pfesenscr Rim(2amm)3.453.8567

FE Analysis(2a mm)

3.4 (Lmd 55 N/mm)

5 (Load 8 2V /nun)

5.08 (Lend 1 30N/min)

1-fcrtz A nalysis(2amm)8.689.7410.7611.6412

Fuji Presensor Rim(2amm)7891012

FEAnalyss(2amm)9.2(Load6Q6N/mni)

10.5<Load86.2N/mni)

13.4<Lcnd 128N/mm)

Results

Table 2 shows me results of the strain gauge analysis at the points within the block of CT200where the strains were experimentally measured. These positions were also located on the FEmodel of the CT200 block and the values are displayed for comparison.

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Computational Methods and Experimental Measurements 733

These values were all taken for a block thickness of 45mm since the effect of reducingthickness was not studied experimentally. One gauge wire on strain gauge 1 was damagedduring assembly and thus did not yield data and during subsequent testing gauge 7 failed toproduce strain readings. Considering that both of these gauges were remote from the contactzone, it was felt to be reasonable to neglect the values at those locations.In addition toobtaining the stress values at nodal positions in the CT200 block both in the FE analysis andin the experimental analysis, a comparison was made between the stress distribution ob-tained from the FE analysis and the coresponding Hertzian solution for Normal loading.Results are plotted in Figure 6. Further analysis was conducted using the MARC FE softwareto investigate the effects of sliding loads on the stress levels in the CT200 model and contourplots of the maximum shear stress distribution are shown in Figure 7. In addition an FEmodel was developed to investigate whether the rolling motion which occurs in the kneejoint would influence the stress distribution. Figure 8 shows contour plots of the stresses inthe y-direction and also shows the variation in shear stress in the block when rolling ispresent.

Table 2

NODAL STRESSESEXPERIMENTAL AND FECT200 BLOCK Normal Load

GaugeNa23456

(Tmax(N/mm')467-1.6(56(1051400.688

(Train(N/nm')-9.383-4.9885-1.7250.218-1.036

FENode

12741298474SI3)2

CTniax(N/mrf)1.4381.816-0.01280.42820.243

(Tmin(N/mnf )-9.95-7.836-1.194-5.914-3.73

> o' Q' +

0l>^ #***,

C> °k^

/-\O

^ B40

V*% o0

• FEA t=2<$, FEA 1=45O FEAt-7n(g, HmiZ

25 3 35CONTACT HALF-WIDTH 'a' nini

FE Analysis Vs Hertzian for varyingthickness of CT200

Figure 6: Plot of Normalised stress versus contact area,FE Vs Hertzian Analysis

Discussion

The results obtained from the FE model compare favourably with known Hertzian solutionsas shown in Figure 6.The results for the Normalised stress in the y-direction for the FEanalysis are within 10% of the values obtained from the theoretical treatment of a flatsurface loaded by a rigid cylindrical indentor. These results compare favourably also

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734 Computational Methods and Experimental Measurements

(a) (b)Figure 7: Contour plot of Maximum Shear Stress for (a) Normal Loads and (b) Normal andSliding loads

(a) (b)Figure 8: Contour Plots of (a) o in the contact region and (b) Tsubjeced to Normal, Rolling and Sliding Loads

when the block is

with the work of Jarvis and Kelly [27]. The very thin component deviates, however, quitesignificantly from the Hertzian stress distribution and since the magnitude of the contactarea is approaching the thickness of the CT200 block, this deviation is to be expected. Thevalues for 2a, the length of the contact, obtained using the Presensor film and the FEanalysis compared very favourably with the results obtained from Hertzian analysis fornormally applied loads as shown in Table 2 . The values for stresses at various nodal loca-tions when compared to the stresses obtained from the experimentally evaluated strainvalues only correspond moderately well with one another as shown in Table 3. There are anumber of possible reasons for the relatively poor outcome from the strain gauges. Theseare

(i) There may have been transverse strains present in the block which were neglectedin the analysis, since the problem was treated as a Plane Strain problem.

(ii) The gauges and the nodal points did not fully coincide with one another and thiscould also help explain the discrepancies between the nodal values from the FE model andthe stresses found from the strain data.

(iii) Furthermore, the 2mm diameter strain gauges may well be too large to record thestrain changes which are occuring in the contact zone, particularly as it is clear from the FEanalysis that the strain gradients in the contact zone are severe.The magnitude of the maximum shear stress in the FE analysis for normal loading is 9.26N/mnf, which compares favourably with the corresponding value of the theoretical of T of8.64 N/mnf , an error of only 7%.

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Computational Methods and Experimental Measurements 735

The location of the maximum shear stress was found to lie 1.75mm below thesurface of the CT200 block compared with 1.96mm predicted by theory (Johnson [17]) anerror of only ll%.The magnitude of the maximum shear stress does not rise significantlywith the introduction of sliding loads,although the location of the maximum value does shifttowrds the surface as suggested by Smith and Liu [16] and as shown in Figure 7.

The results obtained from the FE analysis when rolling was introduced show some veryinteresting trends. In particular the appearance of relatively sharp tensile stresses, as shownin Figure 8, at the extremities of the contact zone suggest that rolling motion in the pros-thetic components may be more influential than previously considered. In addition the peakshear stress now appears at the surface, a more severe condition than that which B artel et al[15] indicated could significantly effect the life of the UHMWPE implant. Further work isrequired on this aspect of the study and in addition work is in progress to examine thecontact behaviour of the polymeric material (UHMWPE), which unlike the CT200 material,is time dependent. In addition the investigation will proceed to a 3-D analysis.

Conclusions

1. The results obtained using embedded strain gauges for measurement of strains andsubsequent evaluation of stress in the contact zone were of limited value. Very mush largermodels and loads would be needed to ensure that the local strain gradients were not toosevere for the gauges.2. The results obtained from the FE analysis look promising and suggest that further workon the effect of sliding and rolling on the stress distribution in this type of contact problemwill give useful results for design of prosthetic knee components.

REFERENCES

[1] Hood R.W. et al, Contact Area and Pressure Distribution in Contemporary TotalKnee Designs.l9%\ ASME Biomechanics Symposium AMD Vol.43 pp233-236[2] Landy, M., Walker, P.S. Wear in condylar replacement knee-a 10 year follow.Proceedings, 31st Annual ORS, Las Vegas, Jan. 1985. p 96.[3] Jones, S.M.G., Pinder, I.M., Moraii, C.G. and Malcolm, A.J.Polyethylene wear inuncemented knee replacements JBJS Vol. 74-B N°l Jan 1992.pp 18-22.[4] Engh, G.A.Failure of the polyethylene bearing surface of a total knee replace-ment within four years. JBJS Vol.70-A, N°7. August 1988. pp 1093 - 1096.[5] Engh, G.A., Dwyer, K. & Hanes, C.Polyethylene wear of metal backed tibialcomponents in total and unicompartmental knee prostheses.J&AS Vol 74-B N°l Jan1992. pp 9-17.[6] Nolan, J.F. and Bucknill, T.M. Aggressive granulomatosis from Polyethylenefailure in an uncemented knee replacement. JBJS Vol. 74-B N l.Jan.1992.pp 23-24.[7] Christensen, O.M., Christiansen, T.G. and Johansen, T. Polyethylene failure in aPCA unicompartmental knee prosthesis.Acta Orthop. Scand. 1990; 61(6) pp. 578-79[8] Lindstrand, A., Ryd, L. and Stenstrom, A. Polyethylene failure in two total knees.Acta Orthop. Scand 1990; 61(6) pp 575-77.[9] Goodfellow, J., Knee prostheses - one step forward, two steps back. Editorial, JBJS Vol.74-B NOl Jan. 1992. pp 1-2.[10] Tsao, A., Mintz, L., McRae, C., Stulberg, D. and Wright, T.M. Failure of the Porous-Coated Anatomic Prosthesis in Total Knee Arthroplosty due to severe Polyethylene Wear.JBJS Vol. 75-ANO 1 January 1993. pp 19-25.[11] Rostoker, W. and Galante, J.O. Contact pressure dependence of wear rates of ultra highmolecular weight polyethylene. Journal of biomedical materials research. Vol. 13, 1979. pp957-964.

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736 Computational Methods and Experimental Measurements

[12] Rose. R.M., Ries, M.D., Paul, I.L., Crugnola, A.and Ellis, E. On the true wear rate ofultra high molecular weight polyethylene in the total knee prosthesis. Journal of BiotnedicalMaterials Research, Vol. 18, 1984.pp 207-224.[13] Connelly, G.M., Rimnac, C.M., Wright, T.M., Hertzberg, R.W. and MansonJ.A.Fatigue crack propagation behaviour of ultrahigh molecular weight polyethylene. Journal ofOrthop. Res. Vol. 2 N°2, 1984. pp 119-125.[14] Wright, T.M. and Bartel, D.L.Surface damage in polyethylene joint components inThe Changing Role of Engineering in Orthopaedic5.1 Mech E, London 1989. pp 187-192.[15] B artel, D.L., Burstein, A.M., Toda, H.H., and Edwards, D.L. The effect of conformityand plastic thickess on contact stresses in metal-backed plastic implants. Journal ofBiomechanical Engineering, August 1985, Vol 107, pp!93-199.[16] Smith, J.L. and Liu, C.K. Stresses due to tangential and normal loads on an elasticsolid with application to some contact stress problems. Journal ofAppi Mech. Vol. 20(1953) pp 157-166.[17] Johnson, K.L. Contact Mechanics, Cambridge University Press, Cambridge, 1985.[18] Johnson, K.L. One hundred years of Hertz contact. Proceedings I.Mech.E, Vol 196,1982. pp 363-378.[19] Hamilton.G.M. and Goodman, L.E. The stress field created by a circular slidingcontact. Journal Appl. Mech. Vol. 33 (1966) pp. 371-477.[20] Hamilton, G.M. Explicit equations for the stresses beneath a sliding spherical contact.Proceedings I.Mech.E. Vol. 197 C, 1983. pp 53-59.[21] Ching-Lo, Su. and Youn, Jae R., An elastic-plastic stress analysis of a polymericsubsurface with a thin layer under normal and tangential loading. WEAR, Vol. 123 (1988)pp 355-367.[22] Tian, H. and Saka, N.Finite element analysis of an elastic-plastic two-layer half space;sliding contact. WEAR, Vol. 148 (1991) pp 261-281.[23] Timohenko S. and Goodier J. Theory of Elasticity McGraw-Hill, 1970[24] Bazergui, A. and Meyer, M.L.Em bedded strain gauges for the measurement of strainsin rolling contact. Experimental Mechanics Vol. 8, October 1968. pp 433-441.[25] Little, E.G. Strain gauge measurement in Strain Measurements in Biomechanics. Miles,A.W. and Tanner, K.E. (eds), Chapman & Hall 1992.[26] Monch E. Similarity and model laws in photoelastic experiments.Experimental Mechanics Vol. 4 1964 pp 141-150.[27] Jarvis, J.L. and Kelly, J.A. Some posible tests for modelling contact using the finiteelement method. Proceedings, NAFEMS conference, Brighton, UK, May 26-29, 1993.

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