Quantification of the Effect of Cross-shear on the Wear of Conventional and Highly Cross-linked UHMWPE

Abstract

A computational model has been developed to quantify the degree of cross-shear of a polyethylene pin articulating against a metallic plate, based on the direct simulation of a multidirectional pin-on-plate wear machine. The principal molecular orientation (PMO) was determined for each polymer site. The frictional work in the direction perpendicular to the PMO was assumed to produce the greatest orientation softening (Wang et al., 1997). The cross-shear ratio (CS) was defined as the frictional work perpendicular to the PMO direction, divided by the total frictional work. Cross-shear on the pin contact surface was location-specific, and of continuously changing magnitude because the direction of frictional force continuously changed due to pin rotation. The polymer pin motion was varied from a purely linear track (CS=0) up to a maximum rotation of ±55° (CS=0.254). The relationship between wear factors (K) measured experimentally and theoretically predicted CS was defined using logarithmic functions for both conventional and highly cross-linked UHMWPE. Cross-shear increased the apparent wear factor for both polyethylenes by more than 5-fold compared to unidirectional wear.

1 Introduction

Wear debris generated from ultra high molecular weight polyethylene (UHMWPE) hip joint replacements has been identified as the major cause of osteolysis and loosening, leading to eventual revision surgery (Ingham and Fisher, 2000). In order to reduce the quantity and generation rate of wear debris and therefore the occurrence of aseptic loosening, many factors have been investigated in laboratory and clinical retrieval studies. One of these factors is the so called “cross-shear” effect, which refers to local counterface motion transverse to the direction of strain hardening of the polyethylene created by the joint articulation during sliding (Wang et al., 1997). Under linear tracking motion, the molecules of UHMWPE are stretched along the direction of sliding and then orient in that direction, leading to a significant degree of strain hardening in that direction, which results in an increase of wear resistance in that direction (Wang, 2001). Under conditions of multidirectional cross-shear motion, surface molecules align preferentially in the principal direction of sliding. Strengthening in one particular direction leads to weakening in the transverse direction (Wang et al., 1997). This weakening is known as orientation softening, which accelerates the generation of wear debris. These theoretical explanations are supported by many experimental findings (e.g., Bragdon et al., 1996; Saikko and Ahlroos, 1999). The wear factors obtained from simple tests of conventional polyethylene against polished cobalt-chrome alloy under linear tracking motions typically are of the order of 10⁻⁸ mm³ N⁻¹m⁻¹, two orders of magnitude less than the average wear factors inferred clinically (Bragdon et al., 1996). Contemporary hip simulators, on the other hand, produce cross-shear motion and therefore generate wear factors which match more closely to average clinical wear factors (Saikko and Ahlroos, 1999).

Several previous studies have suggested that the sliding track, frictional work and slip velocity angle might affect the amount of cross-shear and hence wear rate (Davey et al., 2005; Saikko and Calonius, 2002; Schwenke et al., 2006). In a pin-on-plate simulator with rectangular motion pathways, the volumetric wear increased significantly for a more square wear path, relative to a more elongated rectangular wear path (Turell et al., 2003). The wear rate was found to increase with the average sliding distance, with the inverse of the average aspect ratio of the motion loci, and with the product of both (Davey et al., 2005). A theoretical model based on the concept of frictional work indicated that the wear factor dramatically increased with the increase of maximum cross-shear angle; this was also verified experimentally (Wang 1996; Wang, 2001). In another simulator study, a continually changing direction of sliding was considered as another essential factor affecting wear, in addition to the shape and size of sliding track (Saikko, 1998). A linear relationship between the directional change of the velocity vector and the amount of wear was reported in a simulator study of total hip replacement (Saikko and Calonius, 2002). Thus, the importance of cross-shear to the wear mechanisms of UHMWPE is now well documented, with the wear factor and wear rate of UHMWPE wear increasing greatly with the increase of the amount of cross-shear (Kang et al., 2006; Knight et al., 2005; Turell et al., 2003).

There have been several attempts to further quantify cross-shear and to correlate it with the wear factor, using either computational or experimental approaches (Kang et al., 2006; Knight et al., 2005). A statistical formulation was adopted to estimate the most probable direction of polymer orientation for total knee arthroplasty (TKA) UHMWPE tibial insert surfaces under different in vivo kinematic conditions. The so-called crossing intensity had maximum values of 0.04 for gait and 0.09 for stair activities, respectively (Hamilton et al., 2005). There is no doubt that cross-shear has become one of the important factors to be considered in the design of multidirectional wear testers. The wear factors from in vitro simulator studies will be more comparable with those apparent in vivo if more similar cross-shear conditions can be applied.

A generic theoretical model (Wang et al., 1996; Wang 2001) has demonstrated that cross-shear is an overall effect of multidirectional sliding, frictional work and cross-linking intensity. Barbour et al. (1999) firstly showed that cross-shear should not be disregarded even under linear sliding conditions if polymer rotation occurs. Previous experimental studies investigated the relationship between the wear factor and aspect ratio (AR) of the sliding tracks alone, but the information obtained was not sufficient to quantify cross-shear (Turell et al., 2003; Saikko et al., 2004).

In a recent study, CS was defined as the ratio between the frictional work performed in the direction perpendicular to the PMO to the total frictional work (Kang et al., 2006). The PMO direction was determined as the average direction of the local sliding track or, equivalently, the average direction of frictional force. This calculation was dependent upon the direction of local motion trajectories, rather than upon the global kinematics applied to generate the sliding track in the hip or knee.

The present study extends investigation of the effect of cross-shear on the wear of conventional and highly cross-linked UHMWPE, by establishing relationships between CS and the wear factor. The CSs were calculated mathematically and the wear factors were determined experimentally, based on direct simulation of a simple pin-on-plate set up. A number of issues were specifically addressed: i) is the shape of the sliding track (described as AR in most previous studies) sufficient to interpret the amount of cross-shear? ii) can the PMO and CS be efficiently quantified? iii) what type of statistical relationship describes wear factor as a function of CS?. To elucidate the type of regression curve that best interpolates the experimental data, several curves formulations were considered.

2 Materials and Methods

2.1 Computational Method to Quantify Cross-shear

A computational method was developed to quantify cross-shear, based on the unified theory of wear and frictional work proposed by Wang (2001). The motion of a pin relative to a plate was reproduced analytically, and the sliding tracks of individual sites (discretization element centroids) on the pin were determined with respect to a reference coordinate system attached to the plate. A detailed numerical analysis of CS is presented in Appendix A.

For the present experiment, it was postulated that the direction of principal molecular orientation (PMO) for each element on the pin was given by the average direction of the corresponding sliding track (or, equivalently, the average direction of frictional force, corresponding to γ ≠ 0 in Appendix A), which indicated the overall direction of frictional force across each duty cycle. There is another simple way to potentially define the PMO, i.e., as the direction of plate sliding, which is the primary sliding direction (corresponding to γ = 0 in Appendix A). To compare the difference between these two definitions of PMO (Appendix A), a selection of five typical elements were made (Fig. A 3). Another factor, polymer rotation, was also isolated to investigate its effect on the value of CS. In the final numerical solution of CS, the definition of PMO as the average direction of the corresponding sliding track was applied, as well as the inclusion of polymer rotation.

Fig. A.3.

Illustrative elements on the pin surface, for detailed cross-shear ratio analysis under different PMO definitions.

The direction the greatest strain softening was taken as perpendicular to the PMO direction. The frictional work, defined as the duty-cycle integral of the frictional force times incremental sliding distance, was separated into two components: one perpendicular to and one along the PMO direction. The CS was defined as the frictional work component perpendicular to the PMO direction (W_cross-shear), divided by the total frictional work (W_total).

$$CS=\frac{W_{\mathit{cross}-\mathit{shear}}}{W_{\mathit{total}}}$$ (1)

The contact surface was discretized into a polar array of 29×29 elements (Fig. 1), and analysis was performed at each element’s centroid. To assess the effective overall cross-shear of a given set of input conditions, the CS was averaged over the whole surface. The numbers of discrete increments per duty cycle utilized for computing the PMO and CS were 41 and 401, respectively, based on convergence determinations. The numerical analyses were carried out using Matlab7.0 (The Mathworks, Inc, Massachusetts, USA).

Fig. 1.

Discretization of pin surface for the numeric cross-shear analysis (29×29 elements, based on discretization of polar coordinates)

2.2 Experiments

Multidirectional wear tests were conducted with a six-station pin-on-plate machine, each station consisting of a metallic plate reciprocating back and forth, and a polyethylene pin oscillating around its own axis. A gear mechanism was used to transmit the reciprocating motion of the rectangular bath (to which the plate was fixed) to the pin holder, therefore making the pin rotate (Figs 2 a and b). The angle of pin rotation depended on and varied with the amplitude of the reciprocating motion in this configuration. Both motions were in phase, having a common frequency of 1 Hz. To achieve a larger pin rotation angle, the gear ratio was reduced. The input kinematics for the polymer pin varied from purely linear tracking, up to a rotation of ±55°; the stroke length varied between_12 to 38 mm. A compressive load of 160 N was applied, resulting in a nominal contact pressure of 3.18 MPa for an 8 mm diameter pin contact surface (Fig. 2 c). This is within the physiological range of contact pressure in human articular joints (Saikko, 2006). The pins were machined from ram-extruded non-cross-linked (0 MRad) and highly cross-linked (10 MRad) GUR 1050 UHMWPE. The CoCr metallic plates were polished to an average surface roughness Ra ~0.01 μm. Each trial required two weeks of testing time.

Fig. 2.

(a) Pin holder with rack and pinion gear mechanism used to provide rotation; (b) schematic of the motion in the multidirectional pin-on-plate machine; (c) dimensions of specimen pin.

Eight combinations of pin rotation and plate sliding were utilized for 0 MRad UHMWPE: (0°, 20mm), (±10°, 10mm), (±15°, 12mm), (±20°, 20mm), (±20°, 12mm), (±30°, 28mm), (±45°, 26mm) and (±55°, 38mm). Among these eight combinations, two conditions: (±20°, 20mm) and (±20°, 12mm) were designed to compare the effect of different stroke length on wear for the same pin rotation. The input kinematics for 10 MRad UHMWPE were reduced to four combinations: (±10°, 10mm), (±30°, 28mm), (±45°, 26mm) and (±55°, 38mm). Between three to nine replicates were used for each kinematical condition, depending on statistical dispersion.

The pins were pre-soaked in sterile water for at least 250 hours prior to testing, to stabilize their moisture content. After each week of testing, the pins and metallic plates were ultrasonically cleaned in isopropanol for 10 minutes. The surface profiles of the metallic plates were measured using a surface profilometer (Taylor Hobson, Leicester, UK), with a high pass Gaussian filter at a cut-off 0.8 mm. The pins were replaced into their holders with the same orientation with respect to the sliding direction. The holders were meticulously cleaned to ensure that no third body wear occurred. The lubricant consisted of 25% (v/v) bovine serum in sterile water, with 0.1% (w/v) sodium azide to inhibit the growth of bacteria during testing, corresponding to an equivalent protein content of 15.46 mg/ml. For each test station, 40 ml of lubricant diluted as described above was used. The lubricant was changed each week, after approximately 330,000 cycles. Unloaded control pins were soaked in fluid for the duration of the test. They were used as a reference for mass change alone due to the water uptake of polyethylene. They were kept inside the test rig so that they were subjected to the same conditions of temperature as the test pins.

Before weighing, the pins were cleaned and then left for 48 hrs in an atmosphere of controlled temperature and humidity, to stabilize the polymer. Each pin was weighed until four measurements within a ±10 μg range were obtained. The mass losses of the pins were adjusted to account the mass change due to the moisture absorption determined by the control pins. The volume loss (W) for a specific sliding distance (S) was calculated gravimetrically, and the wear factor (K) was worked out as the volume loss per unit load per unit sliding distance as shown in eq. 2 (Galvin et al., 2006).

$$K=\frac{W}{LS}$$ (2)

where L is the compressive load.

2.3 Statistics

Statistical analysis was conducted using ANOVA with Fisher’s protected least significant difference (PLSD) post hoc test, in which a P-value of less than 0.05 was used to define significance. In order to best-fit the wear factor as a function of average CS, various types of regression methods contained in Microsoft Excel 2002 (Microsoft Corporation, USA) were tested. The expressions which gave the highest correlation coefficients were accepted in this study. Three regression methods, among which the linear growth and power law previously used by Turell et al. (2003) and Knight et al. (2005), were specifically examined. To compare the differences between these three regression methods, a null hypothesis was formed that there is no difference between these methods, by assuming the curve fit data were obtained from independent regression methods. The P-value was calculated in SPSS 13.0 (SPSS Inc, 2004).

3. Results

3.1 Cross-shear Analysis

Once the PMO was determined from the average orientation of the sliding track (or, equivalently, the average direction of frictional force), the CS was calculated. To examine the effect of the PMO direction on the CS calculation, a different PMO direction was postulated to be along the sliding direction of the plate. The comparisons of CS based on these two different definitions of PMO direction for five typical elements (Fig. A 3) are shown in Table 1, as well as the CS values with and without consideration of polymer rotation, which is another essential factor. For elements 2 and 4 in Fig. A 3, the sliding tracks were symmetric, and the average directions of the corresponding sliding tracks were in the sliding direction of the plate. Therefore, their PMO directions were the same under the two different PMO definitions. Any differences between their CSs presumably come from different sliding distances. Despite the range of CSs for the five individual illustrative elements, however, the average CSs for the whole pin were very close for the two alternative PMO definitions (Table 1). The PMO directions for element 5 were always in the horizontal direction under both PMO definitions, but the CS was nonzero (up to 0.086) when the pin’s rotation (polymer rotation) was considered. This demonstrated that the inclusion of polymer rotation plays a role in the analysis of CS. Fig 3 shows the CSs of all elements over the pin surface. The effective overall cross-shear of a given set of input kinetic conditions was averaged over the whole surface. Fig. 4 shows the spatial mean CS versus range of pin rotation.

Table 1.

Cross-shear ratios for the five elements in Fig. 2 for rotations of ±30°, under two different assumptions of PMO direction: a) γ ≠ 0, PMO determined by sliding track (or equivalently, frictional force) average; b) γ = 0, PMO in the sliding direction of the plate). Both PMO definitions are compared with and without consideration of polymer rotation.

Element	Cross-shear ratio (±30°)
	γ ≠ 0		γ = 0
	With rotation	Without rotation	With rotation	Without rotation
1	0.097	0.0004	0.114	0.018
2	0.068	0.001	0.072	0.005
3	0.074	0.0006	0.087	0.014
4	0.110	0.002	0.117	0.008
5	0.086	0	0.086	0

Fig. 3.

Distribution of cross-shear ratio over the pin surface at mid-cycle under rotation of ±30° (d=28mm).

Fig. 4.

Spatial mean cross-shear ratio as a function of the range of pin rotation, β₀.

3.2 Experimental Wear Factor

The volume losses of pins at various stages of testing cycles are shown in Figs 5 a & b, respectively for 0 MRad and 10 MRad UHMWPE. For the same number of testing cycles, the pin volume loss under a higher degree of rotation was greater than in a smaller degree of rotation. For 10 MRad UHMWPE, the pin volume loss under the uniaxial condition was nearly undetectable. Compared to the 0 MRad UHMWPE, the wear volume at the maximum cross-shear condition was reduced more than 5 times for 10 MRad UHMWPE. The experimental apparent wear factors for various average CS values for these two materials are shown in Figs 6 a & b. For 0 MRad UHMPWE, the mean wear factors were 4.7±0.37×10⁻⁷ and 5.5±1.7×10⁻⁷ mm³m⁻¹N⁻¹ for kinematical combinations of (±20°, 20mm) and (±20°, 12mm), respectively, although the computational analysis showed a consistent CS of 0.039 for these two stroke lengths combined with the same pin rotation. For all kinematical inputs involving rotation, the mean wear factors were at least an order of magnitude higher than those under unidirectional motion. The wear factor increased rapidly with the increase of CS up to 0.039, corresponding to the rotation angle of ±20°, beyond which point it changed only slowly. Three types of regression methods were used to best-fit wear factor as a function of average CS. The power function K = 8.0×10⁻⁷CS ^0.2071 had a best statistical R-squared value of 0.92. This high R-squared value reflected a good fit of the overall dataset, despite the dispersion of the replicate trials at each individual CS setting. Alternatively, piece-wise linear regressions were also employed, with best fits being, K = 1.0×10⁻⁵CS+1.0×10⁻⁷ (for CS<0.04, the upslope phase) & K = 6.0×10⁻⁷CS+5.0×10⁻⁷ (for CS ≥ 0.04, the plateau phase), with an average R-squared value of 0.9 for these two phases. The logarithmic expression was K=1.0×10⁻⁷ ln(CS)+ 7.0×10⁻⁷, with an R-squared value of 0.80. The p-values between different regression methods were greater than 0.05, supporting the null hypothesis that there was no significant difference of curve fit accuracy between these three methods. The logarithmic expression showed a slower growth than the power law, which may reflect the change of wear factor against various CSs more reasonably for 10 MRad UHMWPE (Fig. 6b). The function was K=1.0×10⁻¹⁰ ln(CS)+ 1.0×10⁻⁷(R²=0.72) and the mean wear factor under the maximum CS of 0.254 was 1.03±0.17×10⁻⁷ mm³m⁻¹N⁻¹ for 10 MRad UHMWPE.

Fig. 5.

Volume loss at various stages of cycles (dispersion bar: mean (n≥3) ± 95% CI) for (a) 0 MRad conventional UHMWPE and (b) 10 MRad highly cross-linked UHMWPE.

Fig. 6.

Wear factor versus average cross-shear ratio (dispersion bar: mean (n≥3) ± 95% CI) for (a) 0 MRad conventional UHMWPE and (b) 10 MRad highly cross-linked UHMWPE.

4. Discussion

Davey et al. (2004) reported the percentage orientation of UHMWPE measured in the direction of PMO in virgin material (a-axis), and in the cross-shear direction (b-axis), versus different ARs of the sliding track. They demonstrated that the percentage molecule orientation in the a-axis direction was strongly correlated with the AR of the sliding track. An increasing percentage of molecular orientation in the b-axis direction occurred when the sliding track had less elongation. This informed the definition of PMO in this study. Table 1 shows how alternative definitions of PMO could affect the computational average CSs. Previous work has assumed that the PMO direction coincides with the primary sliding direction (Sambasivan et al., 2004). The sliding tracks in the present study had high ARs, so the PMO deviated relatively little from the horizontal direction, and hence the average CSs under the two alternative PMO definitions were close to each other. The high CS value for element 5, which had a linear sliding track, demonstrated that CS was not correlated with AR alone, but depended also on polymer rotation. The PMO axis has to always be attached to and rotate with the polymer pin. It is clear that polymer rotation plays a large role in cross-shear even at the central point on the pin surface (element 5). Despite that fact that the friction force on element 5 remains constant in the horizontal direction, the direction of PMO is continuously changed as the pin rotates. Therefore, the direction of frictional force with respect to the direction of the PMO changes, which in turn generates appreciable cross-shear. Once the additional pin rotation is added into Turell’s pin-on-disc set-up, it can be postulated that both the mathematically determined CS and experimental wear factor would be increased.

Fig. 6 a, which shows the change of wear factor as a function of CS for conventional non-cross-linked UHMWPE, indicates that the wear factor increased strongly with the increase of CSs up to 0.04 (corresponding to a pin rotation of ±30°) and changed only slowly afterwards. This relationship was well fit by a power function, logarithmic expression or piece-wise linear regression. This is substantially different from the purely linear relation between the wear factor and $\frac{A}{A+B}$ ratio proposed by Turell et al. (2003), where A and B represent the lengths in cross-shear and primary directions in a square motion pattern. The CSs in the present study in line with the $\frac{A}{A+B}$ ratio in Turell et al.’s test, were both calculated from the concept of frictional work by Wang (2001), although the former seemingly would be more accurate due to using both an element-wise and a surface-wise definition. In the present study, the mean wear factors were 3.94×10⁻⁷ and 6.8×10⁻⁷ mm³N⁻¹m⁻¹ for the CSs of 0.087 and 0.18, respectively. The average wear factors were approximately 4.5×10⁻⁷ and 7.5×10⁻⁷ mm³N⁻¹m⁻¹ in Turell et al.’s test, corresponding to $\frac{A}{A+B}$ ratios of 0.1 and 0.2. The mean wear factors between these two studies were very close for close values of CS versus $\frac{A}{A+B}$. This coincidence might be due to use of the same theoretical model to quantify CS and the same contact pressure (approximately 3 MPa) applied on the wear screening devices in both of the studies. For the range of CS from 0 to 0.254, the present study used a larger dataset and described the experimental relationship between wear factor and CS more directly than did Turell et al.’s test. Beyond 0.254, Turell et al.’s data would be more applicable, although a purely linear relation seemingly is not sufficient to describe the relationship between wear factor and CS. The reason why the wear factor is so highly sensitive to cross-shear when the CS is less than 0.04 might be explained from the PMO and crystallinity in UHMWPE, which plausibly quickly change when the test condition changes from unidirectional to ±20° rotation. This hypothesis would need to be confirmed by future physical measurements of PMO and crystallinity of the worn pin surface.

The maximum experimental wear factor for conventional non-cross-linked UHMWPE in the present study was 6.8×10⁻⁷ mm³N⁻¹m⁻¹, which is one order of magnitude less than the maximum wear factor obtained in in Saikko et al.’s test for a circle path, or from in vivo and in vitro studies (Davey et al., 2005; Bragdon et al., 1996). This difference might be caused by the respective differences in sliding tracks, contact pressure and other non-mechanical reasons such as polyethylene quality and lubricant issues. The sliding tracks in the present experiments deviated relatively modestly from the sliding direction of plate, in contrast to the extreme deviations of directions in the work of Saikko et al. (2004). In addition, the sliding tracks in Saikko et al.’s test were enclosed ellipses, which meant that the polymer pin continuously rotated through up to a full 360 degrees, motion certainly resulting in a larger cross-shear and a larger wear factor. However, by extrapolating the power law shown in Figs 6 a & b, the wear factors may not be increased significantly even when large CSs (e.g. beyond the maximum CS of 0.25 in the present study) were applied.

Another reason for the difference of wear factor between Saikko et al.’s test and present study could be the different contact pressures applied in the tests. The contact pressure in Saikko et al’s test was 1 MPa, relatively lower than 3 MPa applied in present study. The wear factors have been found increased with the decrease in nominal contact stress only if the stress was within the limit to cause low cycle fatigue and rapid wear as a consequence of the structural failure (Jin and Fisher 2001). This can be explained by the asperity contact theory for polyethylene-on-metal bearings. The real contact area in a polyethylene-on-metal bearing is just a portion of the nominal contact area. An increase in load leads to an increase in both the real contact area and the nominal contact stress. Based on Hertizan theory (Johnson 1987), the real contact area shows a nonlinear increase with the increase in load, so, the wear factor would be decreased. In another study by Saikko (2006), it was suggested that the pressure value of 2 MPa should not be exceeded in the pin-on-disc wear tests, to achieve clinically comparable wear of UHMWPE cups. Therefore, further tests need to be carried out using a lower contact pressure, within this suggested limit, to reproduce higher wear factors under a range of CSs.

The cross-shear occurring in the square or looping circle motion paths (equivalent to a CS of 0.5) in the above-mentioned literature, or the more complicated sliding tracks in hip joints under normal gait cycles or under simulator test conditions, are larger than the maximum CS of 0.254 obtained from the present study (a maximum pin rotation of ±55°).

5. Conclusions

The effect of cross-shear on the wear of UHMWPE in a multi-directional pin-on-plate machine was modeled in the present study. The experimental wear factors for conventional and highly cross-linked UHMWPE were correlated with the spatial mean cross-shear ratios, using logarithmic expressions. Once cross-shear is introduced into the wear prediction, cross-shear dependent wear formulation of this type will presumably be applicable to the more complex kinematic and kinetic conditions prevailing in laboratory simulations of wear in total joint replacements.

Appendix A. Determination of Cross-shear Ratio

To quantify the sliding track, two moving coordinate systems x,y and X,Y were considered, respectively fixed to the pin and the plate. At the starting instant of each motion cycle (Fig. A.1a), X,Y and x,y were mutually parallel, with their origins offset by a distance of $\frac{d}{2}$ (d being the stroke length of the plate). For plate motion initially from right to left, the pin rotated clockwise in the first half cycle and anticlockwise in the second half cycle. For a given point p on the contact surface of the pin, the position vector P in x,y is written as:

Fig. A.1.

The positions of coordinate systems (C.S.) xy and XY, fixed to pin and plate respectively, (a) at the start point, and (b) any instant i over each motion cycle (the new position of p in XY coordinate system is represented by q).

$$\left\{P(r,\theta )\right\}=\left\{\begin{array}{c}rcos\theta \\ rsin\theta \end{array}\right\}$$ (A.1)

where r is the radius and θ indicates the angular position of p with respect to the x axis.

At any subsequent instant i, the new position vector for p, expressed as q, is mapped into X,Y (Fig. A.1b):

$$\{q\}=\left\{\begin{array}{c}X\\ Y\end{array}\right\}=\left\{\begin{array}{cc}cos\phi & sin\phi \\ -sin\phi & cos\phi \end{array}\right\}\left\{P(r,\theta )\right\}-\left\{\begin{array}{c}{X}_i(t)\\ 0\end{array}\right\}$$ (A.2)

where φ is the rotation angle of the pin and X_i(t) is the plate displacement in the X direction.

For a reciprocating plate undergoing simple harmonic motion, the displacement X_i(t) is:

$$X_i(t)=\frac{d}{2}cos\left(\frac{2\pi }{T}t\right)$$ (A.3)

where T is the oscillation period and t is the time. Because the rotation of the pin is also simple harmonic motion with the same oscillation period, the pin’s angular displacement β(t) is:

$$\beta (t)=-{\beta }_{0}cos\left(\frac{2\pi }{T}t\right)$$ (A.4)

where β₀ is the amplitude of maximum rotation. Therefore, the rotation angle φ is:

$$\phi (t)=\beta (t)+{\beta }_0$$ (A.5)

Combining eqs. (A.4), (A.5) and (A.6) with eq. (A.3), the instantaneous position vector q was calculated. Sliding tracks were defined as the locus of those instantaneous positions.

A separate local coordinate system (x′, y′) was also specified (Fig. A.2a), attached to each element’s centroidal point. The x′ axis was taken to be along the PMO at the datum instant of duty cycle initiation. As the element rotates with the pin, the attached axes x′ and y′ are rotated relative to their datum orientation. The instantaneous frictional force F and slip velocity V_total are always directed tangentially to the sliding track, and therefore in general change directions with respect to the x′ axis.

Fig. A.2.

(a) Schematics of polymer rotation in one motion cycle; velocity analysis for a single element with respect to the (b) oscillation plate and (c) principal molecular orientation (PMO) (x′ axis) at any instant.

Denoting P_X, P_Y as the coordinates on a given element’s centroid’s sliding track in the XY plane, the slope of the sliding track at instant i is:

$${\tau }_i=\frac{P_{Y(i+1)}-P_{Y(i)}}{P_{X(i+1)}-P_{X(i)}}$$ (A.6)

Dividing the motion cycle into a finite number (e) of equal temporal increments, the temporal average slope is:

$$\overline{\tau }=\frac{{\displaystyle \sum _{i=1}^e}{\tau }_i}{e}$$ (A.7)

The PMO was defined as the temporal average slope of the sliding track, or equivalently, as the temporal average direction of frictional force. The temporal average direction of the frictional force being the primary determinant of PMO, and that the average direction of the motion path happens to be equivalent when coefficient of friction and contact pressure remain unchanged during each motion cycle, both of these alternative definitions are equivalent.

The angle γ between the PMO and the horizontal axis X at the datum position (at initiation of the duty cycle) is:

$$\begin{array}{l}•\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\gamma ={tan}^{-1}\overline{\tau }\hspace{0.17em}(\text{in\hspace{0.17em}the\hspace{0.17em}first\hspace{0.17em}half\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}motion\hspace{0.17em}cycle)}\\ •\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\gamma =\pi +{tan}^{-1}\overline{\tau }\hspace{0.17em}(\text{in\hspace{0.17em}the\hspace{0.17em}second\hspace{0.17em}half\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}motion\hspace{0.17em}cycle)}\end{array}$$ (A.8)

From eqs. (A.4) and (A.5), the magnitude of sliding velocity V_slide of the plate and the angular velocity ω_rot of the pin can be expressed as:

$$\begin{array}{l}{V}_{\mathit{slide}}(t)={\stackrel{.}{X}}_i(t)=-\frac{2\pi }{T}\frac{d}{2}sin\left(\frac{2\pi }{T}t\right)\\ {\omega }_{\mathit{rot}}(t)=\beta (t)=\frac{2\pi }{T}{\beta }_{0}sin\left(\frac{2\pi }{T}t\right)\end{array}$$ (A.9)

Therefore, the speed at position p due to pin rotation in a fixed lab coordinate system is:

$$V_{\mathit{rot}}={\omega }_{\mathit{rot}}r$$ (A.10)

The magnitudes of X and Y components of V_rot are:

$$\begin{array}{l}{V}_{\mathit{rot}X}=V_{\mathit{rot}}sin(\theta -\phi )\\ V_{\mathit{rot}Y}=-V_{\mathit{rot}}cos(\theta -\phi )\end{array}$$ (A.11)

where φ is the rotation angle given in eq. (A.5).

The total velocity V_total of p is the resultant −V_slide and V_rot (Fig. A.2b). The X-directional magnitude of velocity V_totalX and the Y-directional magnitude of_velocity V_totalY are:

$$\begin{array}{l}{V}_{\mathit{total}X}=-V_{\mathit{slide}}+V_{\mathit{rot}X}\\ V_{\mathit{total}Y}=V_{\mathit{rot}Y}\end{array}$$ (A.12)

Therefore the x′-directional magnitude of velocity V_totalx′ and the y′-directional magnitude of velocity V_totaly′ are:

$$\begin{array}{l}{V}_{\mathit{total}{x}^{\prime }}=V_{\mathit{total}X}cos(\beta -\gamma )-V_{\mathit{total}Y}sin(\beta -\gamma )\\ V_{\mathit{total}{y}^{\prime }}=V_{\mathit{total}X}sin(\beta -\gamma )+V_{\mathit{total}Y}cos(\beta -\gamma )\end{array}$$ (A.13)

The angle α of the total velocity vector with respect to x′ axis is:

$$\alpha ={tan}^{-1}\left(\frac{V_{\mathit{total}{y}^{\prime }}}{V_{\mathit{total}{x}^{\prime }}}\right)$$ (A.14)

The total friction force (F) is in the opposite direction to the resultant velocity vector V_total.

The x′ and y′ components of F are:

$$\begin{array}{l}{F}_{x^{\prime }}=-Fcos\alpha \\ F_{y^{\prime }}=-Fsin\alpha \end{array}$$ (A.15)

The infinitesimal frictional work (dE) components in the x′ and y′ directions during an infinitesimal time interval (dt) are:

$$\begin{array}{l}{dE}_{x^{\prime }}=F_{x^{\prime }}{ds}_{x^{\prime }}=F_{x^{\prime }}{V}_{\mathit{total}{x}^{\prime }}dt\\ {dE}_{y^{\prime }}=F_{y^{\prime }}{ds}_{y^{\prime }}=F_{y^{\prime }}{V}_{\mathit{total}{y}^{\prime }}dt\end{array}$$ (A.16)

Therefore, the frictional work components over the entire duty cycle are:

$$\begin{array}{l}{E}_{x^{\prime }}=\underset{0}{\overset{T}{\int }}-Fcos\alpha \hspace{0.17em}{V}_{\mathit{total}{x}^{\prime }}dt\\ E_{y^{\prime }}=\underset{0}{\overset{T}{\int }}-Fsin\alpha \hspace{0.17em}{V}_{\mathit{total}{y}^{\prime }}dt\end{array}$$ (A.17)

Replacing F by f_μ σ_i, the cross-shear ratio CS is calculated:

$$CS=\frac{\underset{0}{\overset{T}{\int }}f{\sigma }_{i}sin\alpha \hspace{0.17em}{V}_{\mathit{total}{y}^{\prime }}dt}{\underset{0}{\overset{T}{\int }}f{\sigma }_i\sqrt{{V_{\mathit{total}{x}^{\prime }}}^2+{V_{\mathit{total}{y}^{\prime }}}^2}dt}$$ (A.18)

where f is the frictional coefficient and σ_i is instantaneous normal contact stress.

Dividing the motion cycle into n equal time increments Δt and eliminating f, CS is calculated as:

$$CS=\frac{{\displaystyle \sum _{i=1}^n}{\sigma }_{i}sin\alpha \hspace{0.17em}{V}_{\mathit{total}{y}^{\prime }}\mathrm{\Delta }t}{{\displaystyle \sum _{i=1}^n}{\sigma }_i\sqrt{{V_{\mathit{total}{x}^{\prime }}}^2+{V_{\mathit{total}{y}^{\prime }}}^2}\mathrm{\Delta }t}$$ (A.19)

Alternative forms of this equation, but with the same physical meaning, are given in Galvin et al. (2006) and in Wang (2001).

There is another assumption to define PMO as the direction of plate sliding (γ=0 in e.q.A.8), which is the primary sliding direction. Fig. A.3 shows the illustrative elements on the pin surface, for detailed CS analysis under two different assumptions.