Temperature field and wear prediction for UHMWPE acetabular cup with assumed rectangular surface texture

Materials & Design Materials and Design 28 (2007) 2402–2416 www.elsevier.com/locate/matdes

Temperature field and wear prediction for UHMWPE acetabular cup with assumed rectangular surface texture Guang-Neng Dong b

a,b

, Meng Hua

b,*

, Jian Li c, Kong Bieng Chuah

b

a Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi’an Jiaotong University, Xi’an 710049, PR China Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Kowloon, Hong Kong c Wuhan Research Institute of Materials Protection, 430030 Wuhan, PR China

Received 13 March 2006; accepted 21 September 2006 Available online 2 January 2007

Abstract Surface texture design of a tribo-pair affects its tribological behaviors and temperature distribution, subsequently its performance behaviors and servicing life. Numerical simulation and experiment to study the influence of rectangular texture, on UHMWPE acetabular cup reciprocally slid with steel ball was performed. Results demonstrated the occurrence of periodical and regular fluctuation of the surface temperatures in the cup and ball in accordance with the swing of the ball arm. Higher temperature and amplitude of fluctuation were observed in the ball for cup/steel-ball pairs. Predicted temperature for steel ball sliding in UHMWPE cup with rectangular texture is close and comparable with experimental data under steady state. Prediction reveals that: (i) increase in load and frequency increases the wear depths; (ii) wear of rectangular texture for the condition of short sliding time seems agreeable with experimental data; (iii) thermal effect changes the service performance of UHMWPE after a significant long period of operation that may be explained by the viscoelastic behavior of polymer; and (iv) thermal effect is likely to accelerate the degradation of polymer.  2006 Elsevier Ltd. All rights reserved. Keywords: UHMWPE; Thermoplastics; Creep; Wear; Simulation; Temperature rise; Artificial joint

1. Introduction Ultra-high molecular weight polyethylene (UHMWPE) are used as a bearing material in joint replacement prostheses in the last three decades because of its excellent properties of bio-compatibility, chemical stability, and effective impact load damping and low friction coefficient [1]. Currently available orthopaedic implants usually consist of metallic or ceramic component articulating against an UHMWPE part. Subsequently, the generation of UHMWPE debris is the growing concern in the use of conventional UHMWPE acetabular components since it is detrimental to tissue and likely to promote complications like tissue inflammation, bone loss osteolysis and

*

Corresponding author. Tel.: +852 2788 8443; fax: +852 2788 8423. E-mail address: [email protected] (M. Hua).

0261-3069/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2006.09.015

implant loosening. Although osteolysis does not usually occur in patients with low acetabular cup wear rates [2], failure of a pair of joint replacement is recognized as a result of the osteolysis and aseptic loosening caused by the wear of UHMWPE and by the induced tissue response to the worn debris. Alleviation of such detrimental effects of UHMWPE wear debris has aroused a great deal of research interest. Friction in a system always leads to temperature rise. Under certain condition, the temperature rise both in vivo/vitro can be very high that prompts to total failure of UHMWPE in the hip joint replacement. Knowledge of the temperature rise and the corresponding wear allows the successful application of the hip joint replacement. Measurement of the behavior of temperature rise at the contact of hip joint replacements in vivo [3] and in vitro [4] was therefore useful in joint implant design. Study [3] in total hip implants with aluminum oxide ceramic heads

G.-N. Dong et al. / Materials and Design 28 (2007) 2402–2416

and UHMWPE cups, walking condition under a steady state after an hour, showed the obtaining of a maximum peak temperature of 43.1 C. 2D and 3D finite element prediction on smooth surface [5,6] illustrated that the temperature of synovia fluid would be beyond 46 C. At such level of temperatures, the function of the synovia fluid was likely to degenerate and the surface layer would be experiencing dramatic temperature and stress change. However, available prediction seems not taking the surface roughness and texture of ultra-thin surface layer into consideration. Tribological damage is generally initiated from the surface layer of a mating system. A temperature rise in the order of 1–10 C would rapidly change the properties of the bearing materials and/or the lubricant, and subsequently leads to non-physiological wear mechanisms. As a result, the use of low friction, high wear resistant and good lubricating implant materials seems to be the best way to alleviate frictional heating [7]. Unfortunately, such materials are still awaited to be explored. In vivo, frictional heating generally elevates the temperature of the system [8,9], which raises the risk to increase wear, creep and degradation of the UHMWPE as well as harms the surrounding biological tissues. As the tribobehaviors of a system are significantly influenced by the contact nature within the system and also by the properties of its surface materials, the surface topography and materials, and their treatment processes, as well as the loading conditions of the surface, etc., likely vary the system contact nature. Knowledge of these topographical and material system parameters surely helps to understand the tribo-behaviors of the artificial hip joints. However, available literature in studying femoral components and artificial hip joints seldom dealt with the surface roughness, waviness and texture, and their thermal effect on wear. Since constructing an optimum surface profile of materials used in human body is a key importance, this paper analyzes the thermal dispersion characteristics in a known surface roughness of UHMWPE hip cup and its influence on wear phenomenon. It further discusses the influence of the sensitivity of frictional heating on the cup surface texture with an assumed rectangular pattern, the load capacitance and the active frequency. 2. Theoretical modeling As exactly modeling a femoral system is rather difficult, establishment of a theoretical model is thus based on the simplified simulation physical configuration as shown in Fig. 1. The model consists of a spherical femoral head of radius R1 (= 16 mm) which is loading and moving in harmonic oscillatory mode in a hemi-spherical UHMWPE acetabulum cup [10,11] of inner radius R2 (= 16.05 mm) with its base subtending an angle of a = 145 about its centre O. The physical model gives a clearance cR of 0.1 mm between the ball and the cup. The steel femoral head is made by material having thermal and mechanical properties as tabulated in Table 1, and is machined to an

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Fig. 1. Schematic dynamics of a modeled hip joint: (a) application of forces; (b) analysis of applied forces.

average surface roughness Ra ranging between 0.012 and 0.015 lm. 2.1. Motion equation of the femoral ball As the physical configuration of femoral head ball and hip joint cup (Fig. 1), under steady state, may be assumed to resume a relative dynamic motion approximately similar to the harmonic oscillatory mode of a single pendulum [5,16], the swing of the femoral joint with a swing angle h 0 is therefore described by a sine function h 0 = A sin(xt) [5] (where: 30 < h < 30 is the harmonic oscillatory angle; A is the amplitude; x is the frequency; and t is the time). Let the sum of the radial load (P) and the radial component of the weight of human body (mg, where m is the mass and g is the gravitational acceleration) be the total applied load Pt (=P + mg cos h 0 ) on the acetabular at the contact of the femoral ball, which is supported by a normal force N from the cup in the opposite direction at the contact. A tangential locomotion force (F) activating the relative movement of femoral ball and acetabular cup is applying at the contact where a tangential friction force (f) opposes the motion. Let the radius of swinging be equal to the cup radius R1, the equilibrium condition of both static and dynamic loadings on the system allows the tangential force components to be expressed as: F  lðP þ mg cos h0 Þ  mg sin h0 ¼ mR1 €h0 , where: l is the friction coefficient between the femoral ball and acetabular cup; and h 0 is the angle in radian as measured from x-axis. As the swing angle h 0 is practically small, linearization by taking sin h 0  h 0 and cos h 0  1 for the equilibrium expression of tangential force components gives a secondary ordinary differential equation as €h0 þ g h0 þ lg þ lP  F ¼ 0 R1 R1 R1 m

ð1Þ

When a person is walking in steps with equal swing, the balance of the external disturbance forces with the rubbing forces simplifies the mode of motion as a free oscillatory with F  lP = 0. With small l, the solution of h 0 and its corresponding angular velocity h_ 0 can thus be respectively approximated as

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Table 1 Mechanical and thermal properties of ceramics, steel and UHMWPE materials Steela (femoral head) Young’s modulus (GPa) Nano-Young’s modulus (GPa) Density (kg m3) Poisson’s ratio Nano-hardness (GPa) Hardness (MPa) Specific heat (50 C)(J (kg K)1) Expansion coefficient (20–100 C) (K1) Thermal conductivity (20–100 C) (W (m K)1) Natural convective heat-transfer coefficient (W m2 K1) h of water (W m2 K1) Friction coefficient a

E En c t Hn HV c k h l

210 7810 0.3 848 475 11.9 · 106 46.6

UHMWPE [5] (Acetabulum) 0.985 2.2 (3 mN) [13] 965 0.46 0.17 [13] 53.0 [14] 2300 9.0 · 105 0.5 30 [12] 4.26 [15] 0.06–0.16 [6]

www.MatWeb.com, AISI E 52100 Steel.

pffiffiffiffiffiffiffiffiffiffi h0 ¼ 0:5236 cosð g=R1 tÞ; h0 2 ½p=6; p=6; pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi h0 ¼ 1:6391= R1 sinð g=R1 tÞ

and ð2Þ

The discreterization of Eq. (2) gives the time-dependentparameters, h 0 (k) and h_ 0 ðkÞ, for the hip contact between the ball and the cup, computed at kth computing step, for the Nj number of perimeter nodes starting from the ball swing position j0 (defined as j0 = 85Nj/360) as pffiffiffiffiffiffiffiffiffiffi hðkÞ ¼ 0:5236½1  cosð g=R1 kdtÞ=ðp=N j Þ þ j0 ; pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi _ hðkÞ ¼ 1:6391= R1 sinð g=R1 kdtÞ

and ð3Þ

2.2. Constitution of regular surface texture for acetabulum Distribution of actual surface asperities on the acetabular cup is very complicated and difficult to model [17] although most of the rough machining processes like turning, milling, planning and so on produce largely similar asperities and approximately periodic surface morphology [18]. Hence, the peak and trough of surface asperities are approximately analyzed by simplifying them as identical, ideally and regularly distributed shape and height. When the surface of a hip cup is patterned in rectangular texture, its shape can be described mathematically by modulating the cosine function superimposed with Kroneck function d as qi ¼ cd½f ðxi Þ

ð4Þ

in which: f(xi) = cos(xi); xi = 2pi/L0 (i = 1,. . .,1461) and 0; jf ðxi Þj > c0 d½f ðxi Þ ¼ ; c is the height of rectangular 1; jf ðxi Þj < c0 wave, and c0 is the pitch coefficient of a rectangular texture, as shown in Fig. 2a. The assumed distribution of rectangular patterns on the surface of the cup for this specific study is shown in Fig. 2b. 2.3. Formulation of micro-contact When a hemispherical femoral head is brought to contact with a hip joint cup, the hemispherical femoral head

Fig. 2. Schematic of surface profiles for the acetabular cup: (a) schematic of rectangular texture; (b) detail view of rectangular textures.

may: (i) be simply contacting a single micro-hemispherical pattern, and/or (ii) be contacting several micro-spherical patterns on the inner surface of the hip cup. These two kinds of contacting modes can be treated approximately as plain-stress problem and analyzed by Hertzian contact theory. 2.3.1. Separation between two non-conformal cylinders Fig. 3, on which the solid line illustrates the contact deformation while the dash line represents the primary surface, shows the 2D view of a femoral ball with radius R2 contacting normally with the micro-hemispheres of radius r on the textured surface of the cup (Fig. 3a). By denoting the physical variants of the corresponding micro-balls with a prime, the magnitude of the line MN joining the two

G.-N. Dong et al. / Materials and Design 28 (2007) 2402–2416

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Fig. 3. Contact schemes of two mating cylinders: (a) contact without loading; (b) contact after application of load.

intersecting points M and N where are xu apart from the contact origin O on the line O01 O2 (Fig. 3a) can be expressed as MN ¼ y 01 þ y 2 . Using the condition of single contact, the

( y01

þ y2 ¼

pðxÞ ¼

E pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2  x 2 2R

ð8Þ

Integrating Eq. (7), after substituting of Eq. (8), leads to

1  2R ½x2  þ ;

jxja pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2R ½x2  sgnðxÞðx x2  a2  a2 ln jx þ x2  a2 j þ a2 ln jajÞ þ ;

separation y 01 between the points M and N on the unloaded surface can be related to a deformed radius R* (i.e., R* = 1/ (1/r + 1/R2)) (Fig. 3b) by y 01 þ y 2 ¼ ðx2 =2rÞ þ ðx2 =2R2 Þ ¼ ðx2 =2R Þ. Then, the contacting deformation of the femoral ball and the hemisphere pattern on an acetabular cup can be simulated as the deformation of the surface, over a stripe of width 2a (Fig. 3b), of the two contacting cylinders. Hence, the approach to the contact from the remote points d away must satisfy [19] the following condition: y 01 þ y 2 ¼ d 

x2 ; 2R

jxj 6 a

ð5Þ

As those points outside the contact zone, at where jxj > a, are practically not contacting, their corresponding displacement can be described as y 01 þ y 2 > d  x2 =ð2R Þ;

jxj > a

where the ‘‘sign’’ function, sgn(x), is defined as  1; x>0 sgnðxÞ ¼ 1; x < 0 Under practical loading condition, there are usually several micro-hemispherical contacts between two rough mating surfaces taking place. By neglecting the difference in radius of the cup and the ball, the lines O0i O2 ði ¼ 1; 2; . . . ; nÞ can thus be considered as a constitution of n number of individual pairs of contact points. Using the expression y 01 þ y 2 ¼ ðx2 =2rÞ þ ðx2 =2R2 Þ ¼ ðx2 =2R Þ to describe the domains of individual separation, their contribution to the overall separation y 0i of the contact point 1 can be obtained by superimposing its total i-numbers of constituting points as n X ðy 01;1 þ y 01;2 þ    þ y 01;i Þ y 01 ¼ i¼1

ð6Þ

When the contacting surface in the regions of a 6 x 6 a is being elastically loaded by a pressure distribution of p(x), its common gradient can be described by Z a oy 01 oy 2 2 pðsÞ ds ð7Þ þ ¼  pE a x  s ox ox Hence, the substitution of Eq. (5) leads to a Hertzian pressure distribution as described below:

jxj > a

¼

8  > > ½x1;i þð2iþ1Þa2 > > ; x 2 ½ð2i  2Þa; 2ia > 2r > > > > > > 2 > < ðx1;1 þaÞ ; x 2 ½2a; 0 2r

2

ðx1;1 aÞ > ; x 2 ½0; 2a > 2r > > > > > > > ðx1;i 2iaÞ2 > ; x 2 ½2ia; ð2i þ 2Þa > > 2r : 

ð9Þ

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in which y 0 is a piecewise function; and y 01;i is the separation of the ith-micro-hemispherical contact points on the textured surface of the cup. To facilitate the analysis, cylindrical coordinates are used to predict the contact phenomena of multi-micro-hemispheres, whilst cartesian coordinates are used for the expeditious analysis of the acetabular cup problem. When a hemispherical texture is flattened at the contacting incipience during deformation, it becomes rectangular texture. The texture on the concave surface of the cup can subsequently be treated as a micro-plane consisting of a series of micro-rectangular protruding peaks neighboring to the troughs. The relevant analysis can therefore be performed by simply substituting the radius R* with R2. The simulation also accounts for the possible plastic yielding, due to high stress at the contact tip of asperities, by setting those contact pressures exceeding the UHMWPE hardness either to a cut-off value or simply to a hardness value [20]. In the simulation of micro-contact, the mechanical properties, such as hardness and etc., of materials are their corresponding micro-values instead of their bulk ones [13,21]. Typical example is the use of the hardness values, after visco-elastic modification [13], from nano-indentation tests and the measured Young’s modulus of UHMWPE in the simulation studies. 2.4. Evaluation of contact temperature For formulating the transient state of contact temperature, following assumptions are specifically made. (1) Plastic deformation and wear of the UHMWPE are assumed independent of the heat flow behaviors so that the conservative law of energy is applicable in the temperature prediction. (2) Effect of flow lubricant is negligible and therefore completely excluded. (3) Friction coefficient remains constant and is temperature independent during contact. (4) Materials are homogeneous and their properties are temperature independent for both ball and cup. Furthermore, there is not any phase transformation throughout the process. (5) All friction energy is dissipated between the articulating surfaces and there is not any irradiation and convective heat transfer taking place. 2.4.1. Heat conduction equation The use of cylindrical coordinate system to analyze the heat conduction between the ball and the cup facilitates the governing equation for the non-steady state heat-conduction to be expressed as o2 t 1 ot 1 o2 t 1 ot þ 2 þ ¼ 2 2 or r or r o/ a os

ð10Þ

where t is the instantaneous temperature, r is the instantaneous radius at t, / is the corresponding perimeter variant,

a (=k/qc) is the heat diffusive coefficient, and s is the time. Eq. (10) can therefore be discretized as a large sparse matrix of    1 tiþ1;j;kþ1 ti;j;kþ1 ¼ ti;j;k =F m þ 1 þ 2ði  1Þ   1 1 þ 1 ti1;j;kþ1 þ 2 2 2ði  1Þ ði  1Þ ðD/Þ  ðti;jþ1;kþ1 þ ti;j1;kþ1 Þ =f2 þ 2=½ði  1Þ2 ðD/Þ2  þ 1=F m g

ð11Þ

2

in which 1/Fm = (Dr) /(amDs). 2.4.2. Boundary conditions Assuming that the base of the acetabular cup is being the chord of a circle subtending an open angle of 145 that swings from 30 to 30, the flow of rubbing heat is thus limited in the zone. The discretized heat flow boundaries in the zone are therefore the natural convection transfer mode and considered being constant. Let tf be the temperature of the convective media, h be the convective coefficient, and the subscript m = 1 be the labeling for the cup and m = 2 be for the ball, the boundary conditions can be expressed as tiþ1;j;kþ1  ti;j;kþ1 ðiÞ Constant heat flow km ¼ qm ðm ¼ 1;2Þ Dr ð12Þ t1;j;k þ 2F 0m ðt2;j;kþ1 þ Bi  tf Þ ðiiÞ Convective heat t1;j;kþ1 ¼ 1 þ 2Bi  F 0 þ 2F 0m ð13Þ  ðiiiÞ Non-contact zone ti ¼ 37 C ðHuman body temperatureÞ ð14Þ As r = 0 at the ball center leads the R.H.S of Eq. (10) to infinitive and unsolvable, the temperature t0 is thus taken as the average of the n neighboring point temperatures ti P (i.e. t0 = ti/n). 2.4.3. Evaluation of heat partition Expression of the Peclet number Pe at an ith contact point can be written as   v i ai Pe ¼ max ði ¼ 1; 2Þ ð15Þ 2ki =ðqi ci Þ where v is the sliding speed, a is the radius of the contact zones, kball and kcup are the thermal conductivity of the ball and the cup, respectively. Generally, the Peclet number under steady state heat conduction condition is Pe 6 0.5 for slow sliding mode and Pe > 10 for fast sliding mode. As the contact spot at slow sliding can be considered as a ‘slow heat source’ when either the sliding speed of the system is low or the system thermal diffusivity (which is a function of the thermal conductivity, density and specific heat) is high, the total heat Q generated, when the i number of contact points are individually under Hertz pressures pi with

G.-N. Dong et al. / Materials and Design 28 (2007) 2402–2416

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the ball moving relatively at a sliding speed of vi, can be expressed as the sum of its elemental heat qi by [22–24]: X X Q¼ qi ¼ ai li pi vi ði ¼ 1; 2Þ ð16Þ

When the test time is very short and approaches zero (i.e., t ! 0), the creep can be treated approximately linear with time (Eq. (21)).

in which the individual ai for Pe < 0.5 (i.e. slow sliding heat source) are written as

2.5.1. Effect of stress on creep By Boltzmann superposition principle, the response of UHMWPE to a given load is independent of any other loads. Since the deformation of a specimen at any instant is directly proportional to its applied stress, the overall strain in the linear viscoelastic region is thus expressed as Z t J ðt  sÞdrðtÞ ð22Þ eðtÞ ¼

a1 ¼ 1=ð1 þ k2 =k1 Þ a2 ¼ 1  a1 ¼ 1=ð1 þ k1 =k2 Þ and for Pe > 10 (i.e. fast sliding heat source) as rffiffiffiffiffiffiffi1  k2 Pe2 a1 ¼ 1 þ k1 Pe1

ð17Þ

ð18Þ

while for 0.5 < Pe < 10 (i.e. in the middle sliding heat source) as a1 ¼

0:282 0:282 þ 0:112v=Pe

ð19Þ

2.5. Creep Creep usually occurs in plastics under sufficient period of stressing and temperature loading, etc. and significantly influences the properties of viscoelastic materials. The effective utilization of UHMWPE as acetabular cup material for artificial joints needs to understand the creep damages of UHMWPE. Since creep of acetabular cup would recursively be the result of wear, idea of overall wear characteristics may help to analyze creep although it is difficult to differentiate clearly creep from the experiments of friction and wear. By using the linear visco-elastic Kelvin model, which consists of two Hooken springs and one Newtonian dashpot as shown in Fig. 4, to represent the creep (typically, the springs with Young’s modulus E to imitate elastic response; the dashpot with viscosity g2 to simulate the viscous and/or time dependent response to load), the governing equation of the creep e from the model can therefore be expressed as E i r0 r0 h  2t e¼ þ 1  e g2 ð20Þ E 1 E2 Eq. (20) can be simplified as  ðr0 =E1 Þ þ ðr0 =g2 Þt when t ! 0 e¼ ðr0 =E1 Þ þ ðr0 =E2 Þ when t ! 1

ð21Þ

1

in which compliance function J(t) = 1/E(t) is independent of stress at the particular time. Hence the overall strain e(t) can be related to time t by eðtÞ ¼ J ðt  s0 Þr0 þ J ðt  s1 Þðr1  r0 Þ ¼ e0 þ es

where e0 is a strain that can be expressed in the form of Eq. (24), es is the strain under the stress r0 and defined as Eq. (25) below:   r0 r0 1 1 e0 ¼ þ t ¼ r0 þ t ð24Þ E 1 g2 E 1 g2   r r 1 1 þ t¼r þ t ð25Þ es ¼ E 1 g2 E 1 g2 The expressions (24) and (25) allow the overall strain (Eq. (23)) to be re-written as below:  n  X 1 1 þ t r ð26Þ e¼ E 1 g2 i¼1 This can be written as following empirical formula with the empirical constants a, b to be determined empirically. e ¼ ða þ btÞr

ð27Þ

2.5.2. Determination of empirical constants Lee and Pienkowski [25] determined data of the overall compressive creep of UHMWPE by immersing specimens in a 36.5 ± 0.3 C bovine serum solution (with serum and solution volumetric ration of 2:1). To retard any possible bacterial growth, they diluted the bovine serum solution with 1% (w/v) sodium azide solution. Correlating their data with Eq. (27), it was found that, after transforming the time value to log form, it gave a = 2.93 · 104 and b = 7.996 · 104, respectively. Hence, the definition of the empirical creeping depth dcreep = e/h allows the amount of the strain e and stress r due to the deforming surface in the total time interval Dt of implantation [26] to be expressed in the form of " !# n X 4 4 e ¼ 2:93  10 þ 7:996  10 log i Dt rh; and r¼

Fig. 4. A Kelvin viscoelastic model.

ð23Þ

n X i¼1

, pci Dtci

n X i¼1

! Dtci

i¼1

ð28Þ

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in which the notations are as previously defined except that: (i) the subscript c denotes their occurrence at the instants i when under the non-zero contact pressure pi MPa at time t minutes; (ii) h is the initial thickness, in mm, of the UHMWPE cup used.

dent wear at any point on the surface of the cup obeys the Archard wear law with constant wear coefficient of kw(T). The small gradient of the contact temperature allows the wear coefficient kw(T) to take the form of [31]:

2.5.3. Effect of temperature on creep Mechanical properties of polymers are strongly affected by temperature. Creep phenomena of polymers, even simply due to the change of 1 C in temperature, effectively reduce service life [15]. A polymer under a short loading condition at high temperature may behave similarly to that being asserted at low temperature for a long duration. Therefore, they are correlated to the amount of horizontal shift by a frequently used shift factor aT [27] as follows:   Eac 1 1  log aT ¼ ð29Þ 2:3R T T R

where kw0 is the initial wear coefficient, b is a coefficient and defined as b = a1(1 + m1)/(1  m1), a1 is the coefficient of thermal expansion of the cup material, m1 is the Poisson’s ratio of the cup material, and kw = kw0 = kw1 = 3.9 · 105 mm3/N m.

where R (= 8.314 J/mol/K) is a molar gas constant, Eac is the activation energy, TR is an arbitrary reference temperature that is taken as the human body temperature at healthy state (normally as 37 C). By taking the active energy Eac for UHMWPE equal to 235.2 kJ/mol for high density polyethylene (HDPE) [15], Eq. (29) can therefore be simplified as log aT ¼

1:23  103  3:973 T

ð30Þ

When the temperature of the cup is higher than 37 C, the time used in the computation of creep and wear is either substituted by iDt/aT or being the running time iDt [28]. 2.6. Wear 2.6.1. Initial wear coefficient For a pair of UHMWPE and metal mating system, pinon-disk experiments [29,30] derived the relationship of wear coefficient kw in mm3/N m and surface roughness Ra in lm scale either in the form [29] of kw = 8.68 · 106Ra + 1.51 · 106, or in the form [30] of kw = 2.35 · 104Ra2.03. Although the aforementioned two expressions for kw give values almost in the same order of magnitude, the kw used in this simulation study is taken as the average of the two values evaluated by the two equations, which is

k w ðT Þ ¼ k 0 þ k 1 bT ðiDtÞ;

b ¼ a1 ð1 þ m1 Þ=ð1  m1 Þ

ð32Þ

2.6.3. Wear depth As a ball of radius r is loading onto the concave surface of a hemispherical cup whose radius R = br (Fig. 5) under a normal load W, its nominal pressure P (= r) is the load divided by the ‘‘foot-print’’ of the overall contact area. The normalized dimensionless wear depth d/r can geometrically be simply related to the angles of a0 and b0 [32,33] as (d/r) = [(1  cos b0)  b(1  cos a0)] where a0 = arcsin[(sinb0)/b]. When both b0 and a0 are very small and almost approach zero, the above two expressions can be approximately reduced to d ¼ 1=2b20 r. Subsequently it leads to the volumetric wear V to be approximated as V ¼ 145 prb 180 3 2 3 ½r2 ðb0  sin b0 cos b0 Þ  145 pbr  b , where: b  sin b0 0 180 3 0 cos b0  2=3b30 . When the radius of a ball is the same as the inner radius of hemi-spherical cup (i.e. R = r), b in the expressions for (d/r) and V is therefore equal to 1, the expression for V can be simplified as (V/r3) = (145/180)p(2/3)(b0)3. Equating this expression of V with the Archard volumetric wear equation (V/r3) = pkwPN allows the elimination of b0 and d to be written as  2 1 180  3k w PiDt 3 d¼ r ð33Þ 2 145  2 Although the interfacial pressure between the worn surface of the cup and the ball is not known, it can be determined by the consideration that: (i) the loading must satisfy both

k w ¼ 7:55  107 þ ð8:68  106 Ra þ 2:35  104 Ra2:03 Þ=2 ð31Þ Since the measured roughness of the UHMWPE cup was 1.6 lm, its corresponding wear coefficient has therefore being taken as kw = 3.9 · 105 mm3/N m (i.e. 3.9 · 1014 Pa1). 2.6.2. Temperature dependent wear coefficient When a metallic ball having radius r slides reciprocally against a UHMWPE cup under a normal load W, the sliding time iDt (see Section 2.5.3) and temperature T depen-

Fig. 5. Geometry of a wearing spherical bearing—the swing sphere of radius r carries a uniform and stationary load W (dimension D represents the depth by which the sphere has penetrated the softer bearing cup which is of radius R = br).

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the overall force equilibrium, and (ii) the wear at every point on the sliding interface obeying the Archard law. 3. Experiments 3.1. Set up Experimental wear phenomenal behaviors of UHMWPE cup slid against a steel ball with chromium plating have been evaluated by a Plint wear tester was performed. A hemispherical UHMWPE cup with squared backing block was bolted to the bottom plate of a pin, which was mounted into head-block of the tester. The mating steel ball was tightly fixed to the top tip of a pushing and pulling lever that was hole-pinned through by a bearing on a shaft (Fig. 6) which was rigidly mounted to a supporting frame. The bottom tip of the lever was fitted into a tapered hole in a holding block which was mounted to a reciprocating moving basin on the platform of Plint wear tester (Fig. 7). The reciprocating moving basin was activated by a double acting air cylinder controlled by regulators and solenoid valves, and it swung the ball within a 60 swing angle in its contacting cup with a radius of 16 mm about the bearing shaft under a set contact load on the head of the pin holding block of the Plint tester. Fig. 7 shows the complete configuration of the system on the Plint wear tester. In the process of testing, the nominal contact load was set at 100 N and the magnitude of the sliding velocity was 16.76 mm/s. Each test run was performed for 325 min with reciprocating frequency set at 0.2 Hz. Alcohol was intermittently dripped into the mating surface during tests so as to achieve a mild boundary lubrication condition.

Fig. 7. Arrangement of test rig.

3.2. Measurement of temperature rise For measuring temperature rise, a blind hole with 1 mm diameter was drilled at the back bottom centre of UHMWPE cup for the insertion of a thermocouple, which was consisted of a Ni–Cr wire arc-welded to a 0.6 mm diameter Cu wire and pre-calibrated in turbine oil. For stabilizing the signal of temperature from disordering fluctuation, the bottom of the blind-hole was approximately 1 mm away from the inner concaved cup surface (see Fig. 8). The thermocouple was properly adhered in the hole by silicon grease (i.e. a kind of heat conduction glue), and its output analog electric signal was amplified and then connected to a data acquisition card (DAC) that calculated the temperature according to a pre-calibration equation.

Fig. 8. Layout of thermocouple.

3.3. Wear observation At the end of running, the UHMWPE cup was further cleaned with alcohol bath and dried in a vacuum desiccator at 40 C for 1 h. Its worn surfaces were then analyzed by a 3D Form stylus Talysurf and SEM. The scanning area for the measurement of surface profile was 2 · 2 mm2 firstly about the centre of the cup, and then about the point at the locations 5 mm to the left and the right side of cup centre respectively, as illus-

Fig. 9. Locations where surface measurements being taken.

trated in Fig. 9. The corresponding surface profile enabled the distance between minimum and maximum heights to be evaluated. The comparison of the original and the worn profile at the scanning positio

4. Results and discussion Fig. 6. Schematic drawing of the cup and ball test apparatus for sliding wear test.

Relevant expressions presented in Section 2 above were used to predict the temperature field and wear on the sur-

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face with rectangular textures of a UHMWPE acetabular cup, according to the solution flow chart shown in Fig. 10. In the prediction, a 32 mm diameter steel ball was assumed to swing in an UHMWPE cup, having outer diameter 42 mm and inner diameter 32 mm, with radial clearance of 0.05 mm between them. The grids subdivision for the ball (Fig. 11) was 32 nodes in radial direction and 361 nodes in perimeter direction, whilst that for the cup (Fig. 11) was 21 nodes in the radial direction and 1461 nodes in the perimeter direction. Furthermore, the friction coefficient used for the simulation was taken as 0.16 and the body load of 100 N. 4.1. Influence of contact position on contact pressure 4.1.1. Prediction In an acetabular cup and femoral head ball system, contact pressure usually varies with body weight, applied load, and swing angle. The various contact positions generally have different values of swing angles and local contact pressures. Fig. 12 shows the contact pressures for the two locations as illustrated in Fig. 13. It shows that there exists a contact peak outside of the vicinity of contact centre points (typically, the position 426 and the position 893). The exact position of points 426 and 893 was shown in Fig. 2 – it was on the bottom of a valley of the

Fig. 10. Solution procedure.

Fig. 11. Schematic of numerical grids.

assumed rectangular texture and at the outside valley bottom of the pattern at the cup center, respectively. Trend of the results indicated that the pressure distribution on the textured surface differed from its smooth surface counterpart. When the spherical femoral head in harmonic oscillatory motion swung firstly from the central position to the left and then from left to right, the predicted distribution of the contact pressure for the smooth surface cup would follow the dashed semi-elliptical curve, as labeled with ‘‘1’’, in Fig. 12. The curve showed symmetry about the position at x/a = 0.0 (i.e., at the investigated node position), with a maximum value occurring at the position and the minimum value at either rim of the moving stroke. The symmetrically reciprocating movement of the spherical femoral head decreased the magnitude of normal loading from the central to either side towards its two rim-edges, hence giving the profile of the contact pressure distribution. Comparing the contact pressure of spherical femoral head being swung in smooth cup surface with that in contacting with rectangular roughness profile in Fig. 12, it was observed that there were two humps of pressure peaks stretching approximately in the range of 1.05 5 jx/aj 5 1.7 likely to give a maximum value at jx/aj  1.5. When the ball swung to the left position from the centre, the arm of the ball has larger inclination towards the right-hand-side that created larger contact zone than the one at the left (Fig. 12a). Generally, the existence of patterns reduced the PCD (pitch circle diameter) of the cup in relation to the diameter of the ball in the simulation. When the ball and its arm are both in vertical position, i.e. at the centre of the cup, the protrusion of the rectangular patterns at the swept angle of ±30 hindered the down moving of the ball if there was not any deformation of the two rectangular patterns at the terminus of the swept angle, either side of the vertical axis through the central position of the cup. Geometrically, the flat surface of these two rectangular patterns was likely to extend downwardly in certain depth. Subsequently, more material would be under compression as the ball was moving downwards and thus led to the pressure humps at these two contacts where 2 < jx/aj < 4 and its highest normalized pressure peak was about 4.5 at jx/aj  3 (Fig. 12b).

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Fig. 12. Contact pressure distributions at various positions (for smooth cylinder vs. rectangular cup surface and under a weight of 100 N): (a) at the position on the cup center; (b) at the right positions from the cup center.

Fig. 13. Schematic of boundary points on rectangular for the ball and cup.

4.1.2. Implications of the pressure distribution Comparison of the pressure distribution behaviors (Fig. 12), it can be seen that the high pressure humps occurs at the end-edge of the two swept angles when either for (i) the ball is at the right position or (ii) it is at the central position of the cup. Under sliding, this implies that the severe wear is likely to initiate firstly at these two positions and gradually propagate towards the central position of the cup. The higher accumulative normalized pressure at the right positions implies that the former positions would be worn off completely much quicker than the latter under dry condition. However, the smooth surface cup generally has highest wear from

the central position and slowly spreads out to its vicinities at either side. 4.2. Time dependent surface layer temperature of the cup 4.2.1. Prediction A prediction was performed for a 32 mm diameter steel ball loading a 100 N load in a UHMWPE acetabular cup (with outer diameter of 42 mm and inner diameter of 32 mm), both were subdivided as shown in Fig. 11. The initial environment temperature used in the simulation was taken as 37 C. The simulation related the frictional coefficient lBL under boundary lubrication condition with the

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frictional coefficient lBF of the boundary film, and the frictional coefficient lUHMWPE of UHMWPE mating with steel under dry friction condition as following: lBL ¼ alBF þ ð1  aÞlUHMWPE in which a is the overlay ratio of boundary film on the rubbing surface. Generally, a = 0 implies the dry frictional condition whilst a = 1 implies the occurrence of the boundary film. When a tribo-pair of steel/UHMWPE was loaded under 100 N and run at velocity of 0.2 m/s with frequency of 0.2 Hz for 50 s, the surface temperature rise when the coefficient of friction and convective heat transfer was taken as 0.16 and 5, respectively, is very low and not easily detectable because the rise is mainly limited within a bound of 1 mm [34]. The existence of patterns and the harmonic swinging of ball in relation to the cup varied the contact pressure with position (Fig. 12). Similar manner was likely to occur for the sliding friction as well, hence leading to almost the same characteristics of the temperature distribution (Fig. 14). At certain points on the surface layer, their temperature–time relation shows a periodical characteristic (Fig. 14) and their position of temperature peaks matches

with that of the contact pressure peaks well on the temperature vs. radian plot (Fig. 15). It is observed from Fig. 14 that the peak temperature of the cup surface is at 37.0003 C under the loading condition, whilst the corresponding steady peak temperature of node 871 is approximately 37.04 C (Fig. 15) and that of node 427 is 37.02 C. 4.2.2. Temperature rise measurement Under steady heat of the same operational conditions as described above, the simulated temperature rise is comparable with that of the experimental value for the position about 1 mm away from the cup inner surface of the cup centre (cf. Fig. 16 and Fig. 8). The simulation showed that it reached a heat steady condition quickly, only after heating about 50 s. However, the tested result suggested that it needed about 200 s to reach the heat steady state. Obviously, the time to reach a steady for the former is shorter than for the latter. This would be attributed to: (i) the slower response of thermocouple; (ii) that the resolution of the thermocouple to such low range of temperature rise was not high enough; and (iii) the interfacial resistance to heat between thermocouple and the cup substrate.

Fig. 14. Temperature field of cup surface layer (frequency 0.2 Hz, load, 100 N, friction coefficient, 0.16 convective heat transfer coefficient 5, velocity 0.2 m/s).

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Fig. 15. Cup surface temperature vs. radian at a time for various points in vicinities of contact centre on rectangular cup surface (frequency 0.2 Hz, convective heat transfer coefficient 5, velocity 0.2 m/s, weight 100 N, friction 0.16) (a) position 427 (b) position 871.

Fig. 16. Dependence of surface layer temperature for the centre of cup on time. Simulation and test conditions: frequency 0.2 Hz, convective heat transfer coefficient 5, velocity 0.2 m/s, weight 100 N, friction 0.16.

4.3. Temperature field of the ball For the tribo-system with steel ball slid at a speed of 0.2 m/s with 0.2 Hz frequency, convective heat transfer coefficient five and under loading of 100 N, the point temperatures of the ball varied with time as shown in Fig. 17. The transient temperature at node 73 increased with time and reached its steady state within 2 s, from then on the temperature fluctuating about the mean value between 37.45 C and 37.50 C because of the harmonic motion of the tribo-system. Since steel ball has good coefficient of heat conduction, the time (about 2 s) to reach its steady state is shorter than that (about 50 s – Fig. 16) of the polymer cup. As the node 73 (Fig. 13) was at the centre of hot source on the ball surface, it subsequently had higher temperature. Although the temperature (Fig. 17) for those points around the centre (typically the point 70) of the hot source was lower than the node 73, it was higher than that for those points (e.g. the node 80) far away from the centre of the

Fig. 17. Influence of position on temperature on the ball surface for rectangular pattern (frequency 0.2 Hz, convective heat transfer coefficient 5, velocity 0.2 m/s, Load 100 N – Refer Fig. 13 for the respective positions).

hot source. Plot of isothermal contours (Fig. 18) indicated that there was a region with temperature even higher than the maximum value on the cup surface. This was surely possible since the region would be regularly encountering sliding contact with the pattern-points on the cup surface, of which higher contact pressure would be exerted. Unlike those pattern-points on the cup surface which had longer

Fig. 18. Temperature field contour on the steel ball surface on the cup with rectangular pattern at position 73 (frequency 0.2 Hz, convective heat transfer coefficient 5, velocity 0.2 m/s, weight 100 N).

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non-contact intermittence for convective heat transfer, the regular sliding contact phenomenon thus led to higher temperature in the region. The temperatures of the ball surface vary with positions and nature of contact, the contact centre normally gives a maximum temperature comparing with its vicinity (cf, node 70 and node 80 – Fig. 17).

Table 2 Comparison between a calculation result and experimental result

Wear depth (lm) Conditions

4.4. Wear calculation Simulated results (Fig. 19) showed that the wear depths on the cup surface increased with load and swinging frequency. This implies that the body weight and violent sports may both be detrimental to implants. Comparing the slope of the curves in Fig. 19a and b suggests that wear is more susceptible to frequency than load. Practically, under the conditions of severely violent sporting and heavy body weight, wear damage of artificial joints is likely to shorten the service life of an artificial joint. The average value of the experimentally measured wear at the three measuring points as indicated in Fig. 9 was tabulated in Table 2 and plotted in Fig. 19a. Apparently, it was higher than the simulated result (Fig. 19a). This may be owing to the negligence of those factors like the further wear acceleration of wear debris and wear scatter, etc. However, the prediction enables the understanding of influence of surface textures on wear of UHMWPE cup surface. It also allows the identification of important factors susceptible to wear in a pair of artificial joints. 4.5. Phenomenal observation of wear surface As there was some difficulty for direct comparison of prediction with experimental measurements, observation of swung surface of UHMWPE acetabular cup for phenomenal wear condition was thus made (Section 3). Since high temperature normally softens the UHMWPE locally where is susceptible to wear under high squeezing pressure repetitively, the comparison of the change in the surface morphology of a UHMWPE cup before and after swing-

Load (N) Frequency (Hz) Friction coefficient

Steel ball/ UHMWPE

Test

0.4980

1.5

100.0 0.167 0.16

100.0 0.167 No record

slid by a steel ball may allow the verification of the distribution of localized pressure and temperature due to the effect of surface patterns. Experimental study (Figs. 8 and 9) using a Plint wear tester (Fig. 7) with setting conditions as described in Section 3.1 showed trace of conversion of the frictional heat into the temperature rise locally on the cup surface (Fig. 16), which subsequently softens relatively and locally the UHMWPE that prompts for the local wear on the surface of the UHMWPE cup. Although the comparison of the original surface (Fig. 20a) with its worn counterpart (Fig. 20b) after swung against by a steel ball for 320 min did not obviously show any severe wear scar, it was observed that some crests on the originally turned surface were locally flattened and widened. In addition, there was sign of rough and fiber-like peels appearing on the cup surface (see the circled region in Fig. 20b). Further magnified SEM micrographs of some texture peaks on the worn surface (Fig. 20c and d) of UHMWPE cup gave trace of peeled off hair-like fiber being drawn and squeezed to adhere in the swung track as circled in Fig. 20c. Such phenomenon of scattering of flattening scars clearly illustrated the influence of localized temperature rise and contact pressure, which could be considered as the raiser of local wear on the surface of the UHMWPE cup. Further measurement of the individual heights of the surface texture verified that their wear depth was very shallow that con-

Fig. 19. Wear depths depending on load and frequency: (a) load; (b) frequency.

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Fig. 20. SEM wear surface of UHMWPE cup: (a) original surface of the cup; (b) worn surface of the cup; (c) local magnification of worn surface of the white circle in (b); (d) local magnification of the white circle in (c).

firmed the occurrence of mild cup wear as predicted in Section 4.4 and tabulated in Table 2. However, research [35] found that those fibers (Fig. 20b and d) would form a brush-like surface structure that further improve the friction and wear properties of the cup. 5. Conclusions (1) Prediction by using the assumed rectangular surface textures showed trends of periodical and regular fluctuation of the local surface temperatures of the cup and ball. Furthermore, it was observed that the ball tended to give higher temperature and larger amplitude of fluctuation than those in the surface of mating cup. (2) Predicted temperature under steady state for a steel ball sliding over the concave surface of a UHMWPE cup with an assumed rectangular texture was comparable to the result of its experimental counterpart. (3) Results of wear simulation showed that, for a steelball/UHMWPE-cup artificial joint swinging system, increasing in load and frequency increased the wear depths on the surface of the UHMWPE cup. Among load and frequency, the latter is more susceptible to prompt for wear than the former, and thus has to be suitable control in order to prolong the service life of a steel/UHMWPE artificial joint.

(4) Experimental wear phenomenal observation by comparing the surface morphology before and after swinging showed sight of localized discontinuous original machined traces and localized scaling. This serves as an evidence of differential wear at different locations on the surface as the result of differential localized temperature and pressure due to the existence of surface on UHMWPE cup.

Acknowledgements The work described in this paper was supported by the grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU7001429]. The authors thank Mr. M.K. Chan, a graduate in 2005, MEEM Dept., City University of Hong Kong, for his help in collecting some experimental data. References [1] Dangsheng Xiong, Shirong Ge. Friction and wear properties of UHMWPE/Al2O3 ceramic under different lubricating conditions. Wear 2001;250:242–5. [2] Dowd JE, Sychterz CJ, Young AM, Charles A. Characterization of long-term femoral-head-penetration rates: Association with and prediction of osteolysis. J Bone Joint Surg Am 2000;82A(8): 1102–7.

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