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Analysis of factors influencing the friction coefficients of shoe sole materials
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Safety Science 46 (2008) 822–832 www.elsevier.com/locate/ssci
Analysis of factors influencing the friction coefficients of shoe sole materials S. Derler *, F. Kausch, R. Huber Empa – Materials Science and Technology, Lerchenfeldstrasse 5, CH-9014 St. Gallen, Switzerland Received 31 May 2006; received in revised form 21 December 2006; accepted 8 January 2007
Abstract The slip resistance of pedestrian surfaces is usually characterized by friction coefficients measured with samples of shoe sole materials. The factors that influence the results of friction measurements were investigated for two elastomer materials and leather using the portable tribometer Floor Slide Control 2000, which was operated under varying conditions in a climate chamber on two reference surfaces. The friction behaviour observed was in accordance with viscoelastic material properties and depended on the softness of the shoe sole materials. Temperature and mechanical abrasion of the sole materials during the measurements were the factors with the greatest systematic effects on the dynamic coefficients of friction. In some cases, the coefficients of friction also depended on the direction in which a test material was slid along the same track, indicating an influence of local floor surface variations. For dry leather, the influence of relative air humidity on the measured friction coefficients strongly depended on the floor surface. Due to varying systematic influences, the standard deviations of the friction coefficients in series of 20 repetitive measurements (repeatability standard deviation) differed for the individual material combinations. Linear regression analysis of the data revealed that pure random contributions to the repeatability standard deviation were similar and around 0.005 for all investigated materials. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Friction coefficient; Sole material; Slip resistance; Floor surface; Portable tribometer
1. Introduction The friction of shoes on floor coverings is generally considered important in connection with the occurrence of slips and falls (Chang et al., 2003a). For this reason, the slip resistance or, conversely, the slipperiness of pedestrian surfaces is normally assessed on the basis of friction coefficients measured with shoe sole materials such as elastomers or leather. A variety of tribometric measurement devices is available (Chang et al., 2001b), but up to now there is no international standard concerning the slip resistance of floor coverings. Due to its advantages in ease of use and accordance with scientific requirements (Skiba et al., 1994a), the portable device Floor Slide Control 2000 (FSC 2000) is widely used in Switzerland for determining the slip resistance of floor *
Corresponding author. Tel.: +41 71 274 77 66; fax: +41 71 274 77 62. E-mail address: [email protected] (S. Derler).
0925-7535/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssci.2007.01.010
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surfaces in situ by measuring dynamic coefficients of friction. In the present study, systematic and random variations in the measurement results of this tribometer were examined in detail, since the reliable assessment of slip resistance strongly depends on the precision that can be reached in friction measurements. Important factors influencing measurement uncertainty in pedestrian slip resistance metrology comprise the surface conditions of the sole materials and floor coverings in contact, the ambient conditions as well as measurement conditions related to the devices, test parameters, procedures and operators (Marpet, 2002). Due to the abrasability of a relatively soft elastomer when rubbed against a hard floor surface, the results of repetitive friction measurements might be systematically influenced by successive frictional contacts. The results of Fendley et al. (1999) indicated that the preparation of test materials using different abrasive-paper grit sizes had a significant effect not only on the measured coefficients of friction but also on their variability under dry conditions. Based on a large number of experimental results on footwear solings, also Wilson (1996) concluded that the state of abrasion or wear, i.e., the morphological surface state of solings can have a considerable effect on slip resistance. In a study on the slip resistance of different footwear on ice, Gao et al. (2003) found that sole abrasion increased the slip resistance on hard ice, while no significant effect was found on melting ice. Kim et al. (2001) investigated the microscopic wear progression and surface alterations for three sole materials, with which repetitive sliding-friction measurements were carried out on a dry surface. The surface modifications of the elastomers, being initiated in an early stage of sliding, were attributed to mechanisms such as abrasive wear, ploughing and fatigue wear. Both surface wear and its effect on the friction characteristics were found to be material-dependent. The surface roughness of floor coverings is known to have an important influence on the slip resistance (Chang, 1999; Manning and Jones, 2001; Manning et al., 1998). Therefore, local variations in the floor surface roughness are expected to increase the scatter in measured coefficients of friction. In field studies, considerable local variations were found in the slip resistance of floors (Chang et al., 2003b; Chang et al., 2004; Leclercq and Saulnier, 2002), and Derler et al. (2005) discussed the influence of wear patterns and maintenance on the local variations in floor surfaces in use. In laboratory tests, in which the friction of three soling materials on different surfaces was investigated under dry conditions, Kim and Smith (2000) found that frictional contacts changed the surface topography of the floors. The surface changes were explained by wear processes and the transfer of polymeric material from the shoe sole onto the floor. Due to viscoelastic properties, the frictional behaviour of shoe sole materials is time and temperaturedependent. Seasonal changes of climatic parameters and weather conditions can cause considerable variations in the skid resistance of road pavements (Anderson et al., 1986; Oliver et al., 1988). Recently, temperaturedependent seasonal variations in the slip resistance have also been reported for a resilient indoor floor covering (Derler et al., 2005). If friction measurements are repeated after a certain time period, changes in the ambient conditions are important in connection with measurement uncertainty. In addition, Chang et al. reported on effects of time on friction coefficients measured on the same day (Chang et al., 2004) and at intervals of several weeks or months (Chang, 2002), respectively. According to Smith and Bowden (2005), coefficients of friction between a sole material and a floor surface increase with contact time as a result of a creep mechanism that involves increasing interaction of surface asperities. This effect, however, mainly seems to be relevant for the measurement of static coefficients of friction and can only explain short-term variations in measurement results. For long-term variations, time-effects might be negligible compared to variations in the ambient and material conditions (Derler et al., 2005; Leclercq et al., 1997). Skiba et al. (1994a,b) discussed the technical characteristics and the measurement uncertainty of the specific device FSC 2000 in comparison with other portable devices. In the present study, the device FSC 2000 was used to investigate the factors that influence the friction coefficients between shoe sole materials and reference surfaces in more detail. In order to study the effects of elastomer abrasion, sliding direction, temperature and air humidity, dry and wet friction experiments were carried out under well defined conditions in a climate chamber. Linear regression analysis was applied to analyse the measured data regarding systematic and random contributions to the measurement uncertainty, given by the repeatability standard deviation in series of 20 successive measurements. A quantitative understanding of the factors that influence friction coefficients of shoe sole materials and measurement uncertainty improves the interpretation of tribometric measurement results and, thus, provides a more reliable assessment of slip resistance. On this basis, test materials can be optimized and test parameters
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can be adjusted for future test procedures. Although this study focused on the portable tribometer FSC 2000, the results are also relevant for other friction measurement devices that use similar elastomer pads and test parameters. 2. Methods 2.1. Friction measurements The portable tribometer FSC 2000 was discussed by Chang et al. (2001a). The device was operated over a distance of 0.6 m at a velocity of 0.2 m/s, measuring the dynamic coefficient of friction between a standard material (rubber, plastic or leather according to Table 1) and the underlying surface. Samples (1.1 m 0.3 m) of PVC flooring and of terrazzo tiles were used as reference surfaces. In profilometric measurements using a Perthometer M4P-150, the surface roughness parameters of terrazzo were about twice as high than for PVC and showed greater variation coefficients. For example, values of 7.8 ± 1.9 lm (terrazzo) and 4.2 ± 0.6 lm (PVC) were measured for the parameter Rz, characterizing the average height difference between highest peaks and lowest valleys in a surface profile. With the device FSC 2000 the measurement of friction coefficients is limited to values between 0 and 1. The dry friction coefficients of the materials rubber and plastic exceeded this range on both reference surfaces and were therefore used in combination with an aqueous lubricant (about 1 mm thick film of deionised water containing 0.5% sodium lauryl sulphate). The test materials were immersed in the lubricant for at least 5 min prior to the start of friction measurements. The case of dry friction was investigated using leather, which is not practicable in the wet condition due to swelling and slider modification. All friction experiments were carried out in a climate chamber in which the temperature was varied between 15 °C, 20 °C, 25 °C and 30 °C and could be controlled with an accuracy of ± 1 °C. The relative humidity in the climate chamber could be regulated at a constant level of 65 ± 5% for the three higher temperatures, but it drifted to a higher value of about 80% during the experiments at a temperature of 15 °C. To investigate the influence of changes in relative air humidity on the friction of dry leather, the temperature was held at 20 ± 1 °C, while the relative humidity was varied between 40 ± 5%, 60 ± 5%, 80 ± 5% and 95 ± 5%. Prior to the friction measurements, the elastomer materials and reference surfaces were conditioned during 2 h to a specific climate. For each experimental condition, 20 successive friction measurements were carried out, in which an elastomer was slid in alternating (opposite) lengthwise directions on a reference surface. It was ensured that the friction tester was always operated exactly along the same line on each surface. Before each series of 20 measurements, the elastomer surface was prepared using 320-grit abrasive-paper according to a standardized procedure. 2.2. Hardness measurements Hardness, Shore A, was measured according to DIN 53505 (2000) for all investigated shoe sole materials at the temperatures 15 °C, 20 °C, 25 °C and 30 °C and at a relative humidity of 65%. For each material, 10 measurements were carried out per temperature.
Table 1 Characterization of standard materials used with the device FSC 2000 Slider material
Hardness Shore A (DIN 53505, 2000)
Temperature dependence of hardness Shore A
Tg (°C)
Rubber Plastic Leather
70.5 ± 1.2 85.7 ± 1.7 91.2 ± 1.0
0.17 per °C 0.28 per °C 0.05 per °C
46 35 Not measured
For the hardness, Shore A, overall mean values and standard deviations of 40 measurements, carried out at four different temperatures (15 °C, 20 °C, 25 °C, 30 °C) are reported. The temperature dependence of the hardness was determined by linear regression (slope) of the mean values found for the different temperatures. The glass transition temperature Tg was measured by dynamic differential calorimetry.
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2.3. Data analysis Data analyses were carried out using the software MathematicaÒ (Wolfram, 1996) and statistical methods described in Sachs (2004). The repeatability standard deviation Dl for a series of n = 20 successive friction measurements (l1, l2 . . . , ln) was assumed to be a combination of a random and a deterministic contribution according to the equation Dl2 ¼ Dl2random þ Dl2trend ;
ð1Þ
where the systematic and random contributions can be calculated by linear regression and be expressed as pffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ Dltrend ¼ r Dl and Dlrandom ¼ 1 r2 Dl: The correlation coefficient r describes the linear relationship between the measurement results and their sequence (numbers from 1 to n). r is related to the slope m of the linear function fitted to the data by qffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðnþ1Þ ¼ r Dl, where the factor n ðn þ 1Þ=12 corresponds to the standard deviation of the integers m 12 from 1 to n. For r = ±1, the data represent a pure linear trend, and the corresponding repeatability standard deviation becomes rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ðn þ 1Þ jln l1 j n ðn þ 1Þ ¼ : ð3Þ Dl ¼ Dltrend ¼ jmj 12 n1 12 Periodic contributions in the series of measurement results, e.g. due to alternating sliding directions, additionally influence the data variation according to Dl2 ¼ Dl2random þ Dl2trend þ Dl2periodic :
ð4Þ
If a is the amplitude of a periodic component contributing with alternating signs to the measured data, the corresponding standard deviation can be estimated as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dlperiodic ¼ n=ðn 1Þ a: ð5Þ Eq. (5) directly follows from the standard deviation of a series of n numbers with equal absolute value and alternating signs (+a, a, +a, . . .). In the case of friction measurements alternately carried out in opposite directions, a is given as the half difference between the mean values for the two directions. 3. Results 3.1. Overview Results of the data analysis are summarised in Table 2. The mean repeatability standard deviation (Dl) of friction coefficients (COF) found in series of 20 successive measurements ranged from 0.004 to 0.029, strongly depending on the material combination. Systematic and random contributions to the repeatability standard deviation were estimated according to the data analysis described in Section 2.3. In order to confirm the results of the analysis for series of 20 measurements, equivalent analyses were carried out using sets of 10 friction measurements in the same direction. Linear trends in series of friction coefficients, characterized by the parameters m and r, were attributed to systematic effects due to progressive mechanical abrasion of the sole materials in frictional contacts (Section 3.2). Differences in friction coefficients between opposite sliding directions, described by the parameter a, were associated with local variations in the reference surfaces (Section 3.3). Both the results for linear trends and for the influence of opposite sliding directions were used to estimate the random contributions (Dlrandom) to the observed repeatability standard deviations (Section 3.4). The influence of temperature on friction coefficients was analysed by linear regression of the mean values for four different temperatures (Section 3.5). The same method was used to investigate the influence of air humidity on the friction of leather (Section 3.6) and to determine the influence of temperature on the hardness of the shoe sole materials (Section 3.7).
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Table 2 Results of the data analyses for the friction coefficients (COF) between three sole materials and two reference surfaces. The measurements at all temperatures were taken into account to calculate mean values for the listed parameters (according to Section 2.3). ‘‘SD” was used as an abbreviation for standard deviation PVC
Analysis of 20 measurements Range of mean COF (l) Mean repeatability SD (Dl) Mean linear trend (19 m) Mean correlation coefficient (r) Mean systematic contribution to the repeatability SD (Dltrend) Visible surface change Difference in mean COF for opposite sliding directions (2 a) Mean periodic contribution to the repeatability SD (Dlperiodic) Mean random contribution to the repeatability SD (Dlrandom)
Terrazzo
Rubber (wet)
Plastic (wet)
Leather (dry)*
Rubber (wet)
Plastic (wet)
Leather (dry)
0.364–0.437 0.007 0.014 0.66 0.005
0.351–0.369 0.004 0.009 0.52 0.003
0.320–0.361 0.019 +0.057 0.93 0.018
0.523–0.582 0.029 0.076 0.83 0.024
0.479–0.539 0.009 +0.008 0.30 0.003
0.271–0.305 0.005 0.000 0.14 0.003
Smoothing 0.004
Smoothing 0.000
Smoothing +0.012
Smoothing 0.032
Roughening 0.012
None 0.001
0.002
0.000
0.006
0.016
0.006
0.001
0.004
0.003
0.001
0.001
0.006
0.003
0.004 0.009 0.53 0.003
0.018 +0.053 0.97 0.018
0.024 0.068 0.96 0.023
0.007 +0.009 0.50 0.004
0.005 0.000 0.14 0.003
0.003
0.004
0.006
0.005
0.003
<0.001 –
+0.003 <0.001
+0.004 –
+0.004 –
+0.001 +0.004
Analysis of 2 10 measurements in the same direction Mean repeatability SD (Dl) 0.006 Mean linear trend (9 m) 0.013 Mean correlation coefficient (r) 0.65 Mean systematic contribution to the 0.004 repeatability SD (Dltrend) Mean random contribution to the 0.004 repeatability SD (Dlrandom) Linear regression of mean coefficients of friction Effect of temperature (1/°C) +0.005 Effect of relative humidity (1%) – *
For the material combination leather/PVC, the measurements at 15 °C showed two inconsistent tendencies within the same series of 20 friction coefficients and were therefore not taken into account in the analysis of systematic and random contributions to the repeatability standard deviation.
3.2. Effects of mechanical abrasion of the sole materials Fig. 1 shows results of friction measurements carried out on terrazzo. For leather, practically constant coefficients of friction were found in 20 successive measurements. In contrast, the results of plastic showed
0.6 Rubber, wet
COF
0.5 Plastic, wet
0.4 0.3
Leather , dry
5 10 15 No. of friction measurement
20
Fig. 1. Coefficients of friction of three materials on terrazzo, measured in alternating directions (at a temperature of 20 °C and a relative humidity of 65%). The data for rubber and plastic illustrate effects of mechanical abrasion as well as influences of the sliding direction.
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a slightly increasing trend and those of rubber a considerable decrease, respectively. Visual inspection of the elastomer surfaces indicated that the observed trends were caused by mechanical wear. For all but one material combination (leather/terrazzo) the surfaces of the sole materials were visibly modified during the friction measurements (Table 2). The surface of the plastic slider was roughened on terrazzo and, as a consequence, the friction coefficients showed an average increase of 0.008 within 20 measurements (r = 0.30). On PVC, plastic was polished and the friction coefficients were reduced by 0.009 on average (r = 0.52). Rubber was smoothed on both reference surfaces and the mean reductions in the measured coefficients of friction were 0.014 on PVC (r = 0.66) and 0.076 on terrazzo (r = 0.83), respectively. Whereas dry leather showed constant coefficients of friction on terrazzo (r = 0.14), the values increased by 0.057 due to surface polishing on PVC (r = 0.93). Correlation coefficients with absolute values above 0.44 indicated statistically significant linear trends at the 95% confidence level, those with absolute values above 0.56 at the 99% level. 3.3. Influence of the sliding direction Fig. 1 shows measurement results, in which alternating higher and lower COF values indicated a systematic influence of the direction of the friction experiments. On terrazzo, the average differences between the coefficients of friction for opposite sliding directions were 0.032 for rubber and 0.012 for plastic, whereas no effect was found for leather (Table 2). For PVC, on the other hand, the greatest influence of the sliding direction was observed with leather (+0.012), while practically no effects were found with rubber and plastic. 3.4. Repeatability of measured coefficients of friction The results in Table 2 illustrate considerable material-dependent effects of wear and sliding direction on the repeatability of friction measurements. The average repeatability standard deviations of friction coefficients (Dl) ranged from 0.004 (plastic on PVC) to 0.029 (rubber on terrazzo). The mean contributions of systematic trends (Dltrend) lay between 0.003 and 0.024, the mean periodic contributions (Dlperiodic) between 0.000 and 0.016, and the mean random contributions (Dlrandom) between 0.001 and 0.006 (Table 2). 3.5. Influence of temperature Fig. 2 shows that the mean coefficients of friction increased with temperature (0.003 per 1 °C on average). Linear regression analysis revealed clear differences between the specific material combinations. On both reference surfaces, the most significant temperature-dependence was found with rubber (Tg = 46 °C), for which the mean coefficients of friction increased by up to 0.005 per °C. For plastic (Tg = 35 °C), on the other hand, only those coefficients of friction measured on terrazzo depended on temperature. The results of leather depended on the temperature for both reference surfaces, showing a more significant effect on PVC. 3.6. Influence of relative air humidity on the friction of leather As Fig. 3 illustrates, the coefficients of friction of leather on terrazzo showed a marked increase with relative humidity (0.004 per % relative humidity) while practically constant coefficients of friction were measured on the relatively smooth PVC surface (<0.001 per % relative humidity). 3.7. Temperature dependence of material hardness Among the investigated shoe sole materials, rubber was the softest material while leather was the hardest (Table 1). The hardness of leather did only show a slight temperature dependence between 15 °C and 30 °C. Rubber and plastic clearly softened with increasing temperature (Fig. 4). The softening was stronger for the plastic material, for which the higher glass transition temperature was measured (Table 1).
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PVC 0.7 0.6
COF
0.5 0.4 0.3 0.2
Slope [1/°C] Rubber 0.0048 Plastic 0.0005 Leather 0.0034
0.1 15
17.5
20
22.5
25
27.5
30
27. 5
30
Temperature °C Terrazzo 0. 7 0. 6
COF
0. 5 0. 4 0. 3 0. 2
Slope [1/°C] Rubber 0.0040 Plastic 0.0036 Leather 0.0012
0. 1 15
17. 5
20
22. 5
25
Temperature °C Fig. 2. Temperature-dependence of coefficients of friction of three elastomers on PVC and terrazzo. Points and bars denote mean values ± one standard deviation for series of 20 measurements.
1
COF
0.8
Terrazzo
0.6 0.4 0.2
PVC
40
50
60
70
80
90
100
Relative air humidity [%] Fig. 3. Influence of the relative air humidity (rh) on mean coefficients of friction of leather on dry floor surfaces (measurements at 20 °C). The slopes of the calculated regression lines were 0.0040 per % rh (terrazzo) and 0.0004 per % rh (PVC), respectively. Points and bars denote mean values ± one standard deviation for series of 20 measurements.
4. Discussion The friction experiments carried out with three shoe sole materials under controlled climatic conditions showed that abrasion of the materials in frictional contacts, local variations in the roughness of the reference surfaces, ambient temperature and, in the case of dry leather also relative air humidity, are factors that sys-
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95
Hardness Shore A
90 85 Slope [1/°C] Leather - 0.050 Plastic - 0.278 Rubber - 0.172
80 75 70 15
17.5
20
22.5
25
27.5
30
Temperature [°C] Fig. 4. Temperature-dependence of the hardness, Shore A, for the three sole materials investigated. Points and bars denote mean values ± one standard deviation of 10 measurements.
tematically influence friction coefficients and are therefore important in connection with the uncertainty of friction measurements. For four out of six material combinations investigated, the mean repeatability standard deviations of friction coefficients were below 0.01 in series of 20 measurements. In those two cases, in which considerable systematic effects of mechanical abrasion were observed, the mean repeatability standard deviations were 0.019 and 0.029, respectively. Based on laboratory experiments using 18 test combinations of different sole materials (NBR, PUR and leather), floor surfaces (steel, tiles and PVC) and lubricants (oil, water, glycerine and air), Skiba et al. (1994b) reported repeatability standard deviations of up to 0.04 for coefficients of friction measured with the device FSC 2000. In field tests on various floor coverings using the FSC 2000 with the standard materials rubber, plastic and leather, Derler et al. (2005) found a mean repeatability standard deviation of 0.02 for series of 12 successive measurements. They varied the sliding track of an elastomer for each single measurement and considered the influence of local variations in the floor surfaces as important. In the present study, this influence was suppressed by constantly using the same sliding track on each reference surface. For certain material combinations, however, the effects of alternating sliding directions still indicated the importance of local variations in the reference surfaces. Although inhomogeneous surface properties of the soling materials – either due to the initial preparation or produced during the sliding friction contacts – could be a contributing factor, slightly different contact areas in combination with local variations in structure and roughness of the reference surface are probably the main reasons for the observed differences in the measurement results for opposite sliding directions. This interpretation is supported by two facts: (1) Surface roughness and surface roughness variations are greater for terrazzo than for PVC (Section 2.1). As surface roughness is an important factor for friction under wet conditions, it can be expected that the greater local variations in the surface roughness of terrazzo have a stronger effect on measured coefficients of friction compared to the relatively smooth PVC floor. (2) Whereas rubber was smoothed in friction measurements on terrazzo, plastic was roughened (Table 2). Despite this difference, both materials showed qualitatively similar alternating patterns in the friction coefficients measured on terrazzo, implying that the differences for opposite sliding directions were mainly determined by the properties of the floor surface. Linear regression analysis of series of 20 friction measurements showed that pure random contributions to the repeatability standard deviation were similar and around 0.005 for all investigated materials. The corresponding random variations can be attributed to the random nature of the measurement process itself and, in particular, of frictional contacts on the microscopic scale. Since the surface of a relatively soft material is continuously modified in a sliding-friction contact against a harder floor material, even the results of two successive friction measurements on an identical surface can be slightly different due to varying contact areas and stick-slip phenomena. For three material combinations, mechanical abrasion of the elastomers represented the most important contribution to the repeatability standard deviation, while periodic contributions due to opposite directions of the friction experiments were predominant in one case. For two material combinations, systematic and
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random effects were of the same order of magnitude. In those cases, in which mechanical abrasions of the sole materials led to distinct trends in the friction coefficients of measurement series, linear functions provided a good approximation to the data. As possible effects of wear of shoe soles, both roughening and polishing of the surface have been reported in the literature (Leclercq et al., 1995; Manning et al., 1990; Wilson, 1996). Despite a limited number of cases investigated, a variety of tribological phenomena was also observed in the present study. The friction behaviour of the sole materials was generally in accordance with viscoelastic material properties, and in line with others (Kim et al., 2001; Rowland et al., 1996) we found that the wear and friction characteristics depended on the individual material combination. A limitation of our study was that changes in the surface roughness of the slider materials were not monitored during the friction measurements. The friction of a viscoelastic material sliding on a rigid surface is mainly determined by adhesion due to molecular bonding of surface atoms in both contacting materials and hysteresis, resulting from the delayed recovery of the viscoelastic material after deformation by the surface asperities of the rigid material (Moore, 1972). According to Persson et al. (2005) the friction of rubber on smooth surfaces primarily depends on adhesion, while hysteresis becomes increasingly important for rough surfaces. For a tire sliding on a road surface, dry friction was found to be entirely due to the hysteresis contribution, whereas the reduced friction in the wet condition was explained by a sealing effect of rubber, which leads to the entrapment of water in pools of the rough surface, associated with an effective reduction of surface roughness. For the slip resistance of shoe soles on floor surfaces covered by a liquid film, the drainage capability of the shoe-floor contact surface, the draping of the sole material about floor surface asperities as well as the true contact area between the surfaces are considered as key factors (Chang et al., 2001b; Gro¨nqvist, 1995; Strandberg, 1985). Rubber is the softest among the FSC 2000 standard materials. Because a soft material tends to a higher effective contact area and more pronounced microscopic deformations when mechanically interacting with the surface asperities of a rigid material, greater friction coefficients can be expected for rubber than for plastic. This was found in the friction measurements under wet conditions. In addition, mechanical abrasions and floor surface inhomogeneities had a stronger influence for rubber. For both rubber and plastic, generally higher repeatability standard deviations of friction coefficients as well as greater systematic and random contributions were found on the rough reference surface (terrazzo). The dry friction of leather contrasted to the wet friction of rubber and plastic. In particular, lower coefficients of friction, smaller systematic trends as well as a weaker effect of the sliding direction were observed on the rough instead of the smooth surface. Among the viscoelastic materials investigated, leather was the hardest and the one with the smoothest surface. The dry friction of leather seemed mainly determined by adhesion, which generally increases with the effective contact area (Gro¨nqvist, 1995). This was indicated by the measurement series on smooth PVC, in which polishing of the leather surface took place, leading to an increase in the effective contact area and the resulting coefficients of friction. Because the true contact area of leather is expected to be smaller on a rough compared to a smooth surface, the fact that leather showed lower friction coefficients on terrazzo (rough) than on PVC (smooth) additionally supports the interpretation that the dry friction of leather was dominated by adhesion. For all investigated viscoelastic materials, the measured coefficients of friction increased with temperature. The most significant effects were found for rubber (+0.004 to 0.005 per °C), while the behaviour of plastic strongly depended on the substrate. The friction measurement results of leather in the dry condition showed influences of both temperature and relative humidity. The observation of friction coefficients increasing with temperature is in line with the results of Derler et al. (2005) who found the slip resistance of an indoor floor covering to be relatively high in summers and relatively low in winters. For viscoelastic materials, however, the opposite behaviour would be typical, because both the adhesion and the hysteresis contribution to friction are proportional to the dynamic loss tangent which decreases with increasing temperature above the glass transition (Moore, 1972). Accordingly, the coefficients of friction between rubber and road pavements are normally found to decrease with increasing temperature (Bazlamit and Reza, 2005). The increase in frictional coefficients with temperature found for sole materials in the present study can be explained by a gradual softening of the materials within the temperature range investigated (Table 1). For softer materials, a more effective draping about floor surface asperities is expected, leading to an increased effective contact area and higher friction forces.
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Leather is known to be highly hygroscopic and thus sensitive to air humidity (Tuckermann et al., 2001). In particular, the volume and the softness of leather are increased by moisture absorption and swelling. As a consequence, draping effects become increasingly important for the friction between moist leather and a rough surface. Such a mechanism would explain the considerable increase in the coefficients of friction measured with leather on terrazzo. Su and Rowlands (2000) stated that due to the viscoelastic material properties and the complex structure of leather, mechanical contacts cause a mixture of plastic and elastic deformations so that, in addition to the surface properties, leather softness has an influence on the frictional behaviour. 5. Conclusions The coefficients of friction between shoe sole materials and floor coverings are influenced by a variety of factors which are important to be analysed in order to quantify and improve measurement uncertainty in connection with slip resistance. Depending on the material combination, mechanical abrasion of the sole material was found to cause considerable linear trends in series of successive friction measurements. As a result, the repeatability standard deviation of the measured friction coefficients was systematically increased. Linear regression analysis allowed to separate systematic and random contributions to the repeatability standard deviation and, therefore, was found to be a useful tool to analyze the results of friction measurement series. For all material combinations investigated, the mean coefficients of friction increased with temperature. This effect was attributed to a gradual softening of the shoe sole materials, resulting in a more effective draping about floor surface asperities in combination with an increased contact area and higher friction forces. In this study, however, the detailed relationship to the friction behaviour of viscoelastic materials as well as the influence of the floor covering as a function of temperature remained unclear. Therefore, the temperature-dependence of elastomer friction needs to be further investigated for a wider range of shoe sole materials. Due to a high sensitivity to mechanical abrasion and temperature, rubber seemed to be the least appropriate standard material for friction measurements with the device FSC 2000. Substantial influences of abrasions and temperature changes would reduce the repeatability and the reproducibility of measurement results. A problem is also that in cases where elastomer abrasion causes systematically decreasing coefficients of friction, the slip resistance of a floor covering could be overestimated, if a too small number of measurements would be carried out. The application of linear regression analysis for the interpretation of friction measurement results would not only help quantify systematic effects such as the abrasion of test materials, but also allow the optimization of test materials and procedures and, thus, generally improve the assessment of slip resistance. Acknowledgements Parts of this research were financially supported by the Swiss Council for Accident Prevention (bfu) and the Swiss Accident Injurance Company (Suva). We thank Valentin Stra¨ssle for measuring the hardness of sole materials as a function of the temperature. References Anderson, D.A., Meyer, W.E., Rosenberger, J.L., 1986. Development of a procedure for correcting skid-resistance measurements to a standard end-of-season value. In: Crump, E.T. (Ed.), Pavement roughness and skid resistance. Transportation Research Board. National Research Council, Washington DC, pp. 40–48. Bazlamit, S.M., Reza, F., 2005. Changes in asphalt pavement friction components and adjustment of skid number for temperature. Journal of Transportation Engineering-ASCE 131, 470–476. Chang, W.R., 1999. The effect of surface roughness on the measurement of slip resistance. International Journal of Industrial Ergonomics 24, 299–313. Chang, W.R., 2002. The effects of slip criterion and time on friction measurements. Safety Science 40, 593–611. Chang, W.R., Gronqvist, R., Leclercq, S., Brungraber, R.J., Mattke, U., Strandberg, L., Thorpe, S.C., Myung, R., Makkonen, L., Courtney, T.K., 2001a. The role of friction in the measurement of slipperiness, Part 2: survey of friction measurement devices. Ergonomics 44, 1233–1261. Chang, W.R., Gronqvist, R., Leclercq, S., Myung, R., Makkonen, L., Strandberg, L., Brungraber, R.J., Mattke, U., Thorpe, S.C., 2001b. The role of friction in the measurement of slipperiness, Part 1: friction mechanisms and definition of test conditions. Ergonomics 44, 1217–1232.
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Chang, W.-R., Courtney, T.C., Gro¨nqvist, R., Redfern, M., 2003a. Measuring slipperiness – human locomotion and surface factors. Taylor & Francis, London. Chang, W.R., Cotnam, J.P., Matz, S., 2003b. Field evaluation of two commonly used slipmeters. Applied Ergonomics 34, 51–60. Chang, W.R., Li, K.W., Huang, Y.H., Filiaggi, A., Courtney, T.K., 2004. Assessing floor slipperiness in fast-food restaurants in Taiwan using objective and subjective measures. Applied Ergonomics 35, 401–408. Derler, S., Kausch, F., Huber, R., 2005. Systematic patterns and random fluctuations in time series of coefficients of friction measured on floor surfaces. Safety Science 43, 751–770. DIN 53505, 2000. Pru¨fung von Kautschuk und Elastomeren – Ha¨rtepru¨fung nach Shore A und Shore D. Deutsche Norm. Fendley, A., Marpet, M.I., Medoff, H., Schutter, D., 1999. Repeatability and reproducibility in walkway-safety tribometry: abrasive-grit size in test-foot preparation. Journal of Testing and Evaluation 27, 76–82. Gao, C., Abeysekera, J., Hirvonen, M., Aschan, C., 2003. The effect of footwear sole abrasion on the coefficient of friction on melting and hard ice. International Journal of Industrial Ergonomics 31, 323–330. Gro¨nqvist, R., 1995. Mechanisms of friction and assessment of slip resistance of new and used footwear soles on contaminated floors. Ergonomics 38, 224–241. Kim, I.J., Smith, R., 2000. Observation of the floor surface topography changes in pedestrian slip resistance measurements. International Journal of Industrial Ergonomics 26, 581–601. Kim, I.J., Smith, R., Nagata, H., 2001. Microscopic observations of the progressive wear on shoe surfaces that affect the slip resistance characteristics. International Journal of Industrial Ergonomics 28, 17–29. Leclercq, S., Saulnier, H., 2002. Floor slip resistance changes in food sector workshops: prevailing role played by ‘‘fouling”. Safety Science 40, 659–673. Leclercq, S., Tisserand, M., Saulnier, H., 1995. Assessment of slipping resistance of footwear and floor surfaces. Influence of manufacture and utilization of the products. Ergonomics 38, 209–219. Leclercq, S., Tisserand, M., Saulnier, H., 1997. Analysis of measurements of slip resistance of soiled surfaces on site. Applied Ergonomics 28, 283–294. Manning, D.P., Jones, C., 2001. The effect of roughness, floor polish, water, oil and ice on underfoot friction: current safety footwear solings are less slip resistant than microcellular polyurethane. Applied Ergonomics 32, 185–196. Manning, D.P., Jones, C., Bruce, M., 1990. Proof of shoe slip-resistance by a walking traction test. Journal of Occupational Accidents 12, 255–270. Manning, D.P., Jones, C., Rowland, F.J., Roff, M., 1998. The surface roughness of a rubber soling material determines the coefficient of friction on water-lubricated surfaces. Journal of Safety Research 29, 275–283. Marpet, M.I., 2002. Improved characterization of tribometric test results. Safety Science 40, 705–714. Moore, D.F., 1972. The friction and lubrication of elastomers. Pergamon Press, Oxford. Oliver, J.W.H., Tredrea, P.F., Pratt, D.N., 1988. Seasonal variation of skid resistance in Australia. Australian Road Research Board, Special Report No. 37. Persson, B.N.J., Tartaglino, U., Albohr, O., Tosatti, E., 2005. Rubber friction on wet and dry road surfaces: the sealing effect. Physical Review B 71, 035428. Rowland, F.J., Jones, C., Manning, D.P., 1996. Surface roughness of footwear soling materials: relevance to slip resistance. Journal of Testing and Evaluation 24, 368–376. Sachs, L., 2004. Angewandte Statistik. Springer, Berlin. Skiba, R., Scheil, M., Windhovel, U., 1994a. Comparative research of unsteady measurement of friction coefficient on floors. Kautschuk Gummi Kunststoffe 47, 513–518. Skiba, R., Scheil, M., Windho¨vel, U., 1994b. Vergleichsuntersuchung zur instationa¨ren Reibzahlmessung auf Fussbo¨den. Schriftenreihe der Bundesanstalt fu¨r Arbeitsschutz, Dortmund. Forschungsbericht Fb 701. Smith, R.H., Bowden, 2005. Test foot contact time effects in pedestrian slip-resistance metrology. Journal of Testing and Evaluation 33, 39–43. Strandberg, L., 1985. The effect of conditions underfoot on falling and overexertion accidents. Ergonomics 28, 131–147. Su, Z.W., Rowlands, H., 2000. Measurement and analysis of leather frictional forces. Journal of the Society of Leather Technologists and Chemists 84, 117–121. Tuckermann, M., Mertig, M., Pompe, W., 2001. In situ observation of moisture-induced swelling of leather in an environmental scanning electron microscope. Review of Scientific Instruments 72, 1449–1452. Wilson, M., 1996. Slip resistance characteristics of footwear solings assessed using the SATRA friction tester. Journal of Testing and Evaluation 24, 377–385. Wolfram, S., 1996. Mathematica. Wolfram Media, Champaign.
Safety Science 46 (2008) 822–832 www.elsevier.com/locate/ssci
Analysis of factors influencing the friction coefficients of shoe sole materials S. Derler *, F. Kausch, R. Huber Empa – Materials Science and Technology, Lerchenfeldstrasse 5, CH-9014 St. Gallen, Switzerland Received 31 May 2006; received in revised form 21 December 2006; accepted 8 January 2007
Abstract The slip resistance of pedestrian surfaces is usually characterized by friction coefficients measured with samples of shoe sole materials. The factors that influence the results of friction measurements were investigated for two elastomer materials and leather using the portable tribometer Floor Slide Control 2000, which was operated under varying conditions in a climate chamber on two reference surfaces. The friction behaviour observed was in accordance with viscoelastic material properties and depended on the softness of the shoe sole materials. Temperature and mechanical abrasion of the sole materials during the measurements were the factors with the greatest systematic effects on the dynamic coefficients of friction. In some cases, the coefficients of friction also depended on the direction in which a test material was slid along the same track, indicating an influence of local floor surface variations. For dry leather, the influence of relative air humidity on the measured friction coefficients strongly depended on the floor surface. Due to varying systematic influences, the standard deviations of the friction coefficients in series of 20 repetitive measurements (repeatability standard deviation) differed for the individual material combinations. Linear regression analysis of the data revealed that pure random contributions to the repeatability standard deviation were similar and around 0.005 for all investigated materials. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Friction coefficient; Sole material; Slip resistance; Floor surface; Portable tribometer
1. Introduction The friction of shoes on floor coverings is generally considered important in connection with the occurrence of slips and falls (Chang et al., 2003a). For this reason, the slip resistance or, conversely, the slipperiness of pedestrian surfaces is normally assessed on the basis of friction coefficients measured with shoe sole materials such as elastomers or leather. A variety of tribometric measurement devices is available (Chang et al., 2001b), but up to now there is no international standard concerning the slip resistance of floor coverings. Due to its advantages in ease of use and accordance with scientific requirements (Skiba et al., 1994a), the portable device Floor Slide Control 2000 (FSC 2000) is widely used in Switzerland for determining the slip resistance of floor *
Corresponding author. Tel.: +41 71 274 77 66; fax: +41 71 274 77 62. E-mail address: [email protected] (S. Derler).
0925-7535/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ssci.2007.01.010
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surfaces in situ by measuring dynamic coefficients of friction. In the present study, systematic and random variations in the measurement results of this tribometer were examined in detail, since the reliable assessment of slip resistance strongly depends on the precision that can be reached in friction measurements. Important factors influencing measurement uncertainty in pedestrian slip resistance metrology comprise the surface conditions of the sole materials and floor coverings in contact, the ambient conditions as well as measurement conditions related to the devices, test parameters, procedures and operators (Marpet, 2002). Due to the abrasability of a relatively soft elastomer when rubbed against a hard floor surface, the results of repetitive friction measurements might be systematically influenced by successive frictional contacts. The results of Fendley et al. (1999) indicated that the preparation of test materials using different abrasive-paper grit sizes had a significant effect not only on the measured coefficients of friction but also on their variability under dry conditions. Based on a large number of experimental results on footwear solings, also Wilson (1996) concluded that the state of abrasion or wear, i.e., the morphological surface state of solings can have a considerable effect on slip resistance. In a study on the slip resistance of different footwear on ice, Gao et al. (2003) found that sole abrasion increased the slip resistance on hard ice, while no significant effect was found on melting ice. Kim et al. (2001) investigated the microscopic wear progression and surface alterations for three sole materials, with which repetitive sliding-friction measurements were carried out on a dry surface. The surface modifications of the elastomers, being initiated in an early stage of sliding, were attributed to mechanisms such as abrasive wear, ploughing and fatigue wear. Both surface wear and its effect on the friction characteristics were found to be material-dependent. The surface roughness of floor coverings is known to have an important influence on the slip resistance (Chang, 1999; Manning and Jones, 2001; Manning et al., 1998). Therefore, local variations in the floor surface roughness are expected to increase the scatter in measured coefficients of friction. In field studies, considerable local variations were found in the slip resistance of floors (Chang et al., 2003b; Chang et al., 2004; Leclercq and Saulnier, 2002), and Derler et al. (2005) discussed the influence of wear patterns and maintenance on the local variations in floor surfaces in use. In laboratory tests, in which the friction of three soling materials on different surfaces was investigated under dry conditions, Kim and Smith (2000) found that frictional contacts changed the surface topography of the floors. The surface changes were explained by wear processes and the transfer of polymeric material from the shoe sole onto the floor. Due to viscoelastic properties, the frictional behaviour of shoe sole materials is time and temperaturedependent. Seasonal changes of climatic parameters and weather conditions can cause considerable variations in the skid resistance of road pavements (Anderson et al., 1986; Oliver et al., 1988). Recently, temperaturedependent seasonal variations in the slip resistance have also been reported for a resilient indoor floor covering (Derler et al., 2005). If friction measurements are repeated after a certain time period, changes in the ambient conditions are important in connection with measurement uncertainty. In addition, Chang et al. reported on effects of time on friction coefficients measured on the same day (Chang et al., 2004) and at intervals of several weeks or months (Chang, 2002), respectively. According to Smith and Bowden (2005), coefficients of friction between a sole material and a floor surface increase with contact time as a result of a creep mechanism that involves increasing interaction of surface asperities. This effect, however, mainly seems to be relevant for the measurement of static coefficients of friction and can only explain short-term variations in measurement results. For long-term variations, time-effects might be negligible compared to variations in the ambient and material conditions (Derler et al., 2005; Leclercq et al., 1997). Skiba et al. (1994a,b) discussed the technical characteristics and the measurement uncertainty of the specific device FSC 2000 in comparison with other portable devices. In the present study, the device FSC 2000 was used to investigate the factors that influence the friction coefficients between shoe sole materials and reference surfaces in more detail. In order to study the effects of elastomer abrasion, sliding direction, temperature and air humidity, dry and wet friction experiments were carried out under well defined conditions in a climate chamber. Linear regression analysis was applied to analyse the measured data regarding systematic and random contributions to the measurement uncertainty, given by the repeatability standard deviation in series of 20 successive measurements. A quantitative understanding of the factors that influence friction coefficients of shoe sole materials and measurement uncertainty improves the interpretation of tribometric measurement results and, thus, provides a more reliable assessment of slip resistance. On this basis, test materials can be optimized and test parameters
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can be adjusted for future test procedures. Although this study focused on the portable tribometer FSC 2000, the results are also relevant for other friction measurement devices that use similar elastomer pads and test parameters. 2. Methods 2.1. Friction measurements The portable tribometer FSC 2000 was discussed by Chang et al. (2001a). The device was operated over a distance of 0.6 m at a velocity of 0.2 m/s, measuring the dynamic coefficient of friction between a standard material (rubber, plastic or leather according to Table 1) and the underlying surface. Samples (1.1 m 0.3 m) of PVC flooring and of terrazzo tiles were used as reference surfaces. In profilometric measurements using a Perthometer M4P-150, the surface roughness parameters of terrazzo were about twice as high than for PVC and showed greater variation coefficients. For example, values of 7.8 ± 1.9 lm (terrazzo) and 4.2 ± 0.6 lm (PVC) were measured for the parameter Rz, characterizing the average height difference between highest peaks and lowest valleys in a surface profile. With the device FSC 2000 the measurement of friction coefficients is limited to values between 0 and 1. The dry friction coefficients of the materials rubber and plastic exceeded this range on both reference surfaces and were therefore used in combination with an aqueous lubricant (about 1 mm thick film of deionised water containing 0.5% sodium lauryl sulphate). The test materials were immersed in the lubricant for at least 5 min prior to the start of friction measurements. The case of dry friction was investigated using leather, which is not practicable in the wet condition due to swelling and slider modification. All friction experiments were carried out in a climate chamber in which the temperature was varied between 15 °C, 20 °C, 25 °C and 30 °C and could be controlled with an accuracy of ± 1 °C. The relative humidity in the climate chamber could be regulated at a constant level of 65 ± 5% for the three higher temperatures, but it drifted to a higher value of about 80% during the experiments at a temperature of 15 °C. To investigate the influence of changes in relative air humidity on the friction of dry leather, the temperature was held at 20 ± 1 °C, while the relative humidity was varied between 40 ± 5%, 60 ± 5%, 80 ± 5% and 95 ± 5%. Prior to the friction measurements, the elastomer materials and reference surfaces were conditioned during 2 h to a specific climate. For each experimental condition, 20 successive friction measurements were carried out, in which an elastomer was slid in alternating (opposite) lengthwise directions on a reference surface. It was ensured that the friction tester was always operated exactly along the same line on each surface. Before each series of 20 measurements, the elastomer surface was prepared using 320-grit abrasive-paper according to a standardized procedure. 2.2. Hardness measurements Hardness, Shore A, was measured according to DIN 53505 (2000) for all investigated shoe sole materials at the temperatures 15 °C, 20 °C, 25 °C and 30 °C and at a relative humidity of 65%. For each material, 10 measurements were carried out per temperature.
Table 1 Characterization of standard materials used with the device FSC 2000 Slider material
Hardness Shore A (DIN 53505, 2000)
Temperature dependence of hardness Shore A
Tg (°C)
Rubber Plastic Leather
70.5 ± 1.2 85.7 ± 1.7 91.2 ± 1.0
0.17 per °C 0.28 per °C 0.05 per °C
46 35 Not measured
For the hardness, Shore A, overall mean values and standard deviations of 40 measurements, carried out at four different temperatures (15 °C, 20 °C, 25 °C, 30 °C) are reported. The temperature dependence of the hardness was determined by linear regression (slope) of the mean values found for the different temperatures. The glass transition temperature Tg was measured by dynamic differential calorimetry.
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2.3. Data analysis Data analyses were carried out using the software MathematicaÒ (Wolfram, 1996) and statistical methods described in Sachs (2004). The repeatability standard deviation Dl for a series of n = 20 successive friction measurements (l1, l2 . . . , ln) was assumed to be a combination of a random and a deterministic contribution according to the equation Dl2 ¼ Dl2random þ Dl2trend ;
ð1Þ
where the systematic and random contributions can be calculated by linear regression and be expressed as pffiffiffiffiffiffiffiffiffiffiffiffi ð2Þ Dltrend ¼ r Dl and Dlrandom ¼ 1 r2 Dl: The correlation coefficient r describes the linear relationship between the measurement results and their sequence (numbers from 1 to n). r is related to the slope m of the linear function fitted to the data by qffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðnþ1Þ ¼ r Dl, where the factor n ðn þ 1Þ=12 corresponds to the standard deviation of the integers m 12 from 1 to n. For r = ±1, the data represent a pure linear trend, and the corresponding repeatability standard deviation becomes rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ðn þ 1Þ jln l1 j n ðn þ 1Þ ¼ : ð3Þ Dl ¼ Dltrend ¼ jmj 12 n1 12 Periodic contributions in the series of measurement results, e.g. due to alternating sliding directions, additionally influence the data variation according to Dl2 ¼ Dl2random þ Dl2trend þ Dl2periodic :
ð4Þ
If a is the amplitude of a periodic component contributing with alternating signs to the measured data, the corresponding standard deviation can be estimated as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dlperiodic ¼ n=ðn 1Þ a: ð5Þ Eq. (5) directly follows from the standard deviation of a series of n numbers with equal absolute value and alternating signs (+a, a, +a, . . .). In the case of friction measurements alternately carried out in opposite directions, a is given as the half difference between the mean values for the two directions. 3. Results 3.1. Overview Results of the data analysis are summarised in Table 2. The mean repeatability standard deviation (Dl) of friction coefficients (COF) found in series of 20 successive measurements ranged from 0.004 to 0.029, strongly depending on the material combination. Systematic and random contributions to the repeatability standard deviation were estimated according to the data analysis described in Section 2.3. In order to confirm the results of the analysis for series of 20 measurements, equivalent analyses were carried out using sets of 10 friction measurements in the same direction. Linear trends in series of friction coefficients, characterized by the parameters m and r, were attributed to systematic effects due to progressive mechanical abrasion of the sole materials in frictional contacts (Section 3.2). Differences in friction coefficients between opposite sliding directions, described by the parameter a, were associated with local variations in the reference surfaces (Section 3.3). Both the results for linear trends and for the influence of opposite sliding directions were used to estimate the random contributions (Dlrandom) to the observed repeatability standard deviations (Section 3.4). The influence of temperature on friction coefficients was analysed by linear regression of the mean values for four different temperatures (Section 3.5). The same method was used to investigate the influence of air humidity on the friction of leather (Section 3.6) and to determine the influence of temperature on the hardness of the shoe sole materials (Section 3.7).
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Table 2 Results of the data analyses for the friction coefficients (COF) between three sole materials and two reference surfaces. The measurements at all temperatures were taken into account to calculate mean values for the listed parameters (according to Section 2.3). ‘‘SD” was used as an abbreviation for standard deviation PVC
Analysis of 20 measurements Range of mean COF (l) Mean repeatability SD (Dl) Mean linear trend (19 m) Mean correlation coefficient (r) Mean systematic contribution to the repeatability SD (Dltrend) Visible surface change Difference in mean COF for opposite sliding directions (2 a) Mean periodic contribution to the repeatability SD (Dlperiodic) Mean random contribution to the repeatability SD (Dlrandom)
Terrazzo
Rubber (wet)
Plastic (wet)
Leather (dry)*
Rubber (wet)
Plastic (wet)
Leather (dry)
0.364–0.437 0.007 0.014 0.66 0.005
0.351–0.369 0.004 0.009 0.52 0.003
0.320–0.361 0.019 +0.057 0.93 0.018
0.523–0.582 0.029 0.076 0.83 0.024
0.479–0.539 0.009 +0.008 0.30 0.003
0.271–0.305 0.005 0.000 0.14 0.003
Smoothing 0.004
Smoothing 0.000
Smoothing +0.012
Smoothing 0.032
Roughening 0.012
None 0.001
0.002
0.000
0.006
0.016
0.006
0.001
0.004
0.003
0.001
0.001
0.006
0.003
0.004 0.009 0.53 0.003
0.018 +0.053 0.97 0.018
0.024 0.068 0.96 0.023
0.007 +0.009 0.50 0.004
0.005 0.000 0.14 0.003
0.003
0.004
0.006
0.005
0.003
<0.001 –
+0.003 <0.001
+0.004 –
+0.004 –
+0.001 +0.004
Analysis of 2 10 measurements in the same direction Mean repeatability SD (Dl) 0.006 Mean linear trend (9 m) 0.013 Mean correlation coefficient (r) 0.65 Mean systematic contribution to the 0.004 repeatability SD (Dltrend) Mean random contribution to the 0.004 repeatability SD (Dlrandom) Linear regression of mean coefficients of friction Effect of temperature (1/°C) +0.005 Effect of relative humidity (1%) – *
For the material combination leather/PVC, the measurements at 15 °C showed two inconsistent tendencies within the same series of 20 friction coefficients and were therefore not taken into account in the analysis of systematic and random contributions to the repeatability standard deviation.
3.2. Effects of mechanical abrasion of the sole materials Fig. 1 shows results of friction measurements carried out on terrazzo. For leather, practically constant coefficients of friction were found in 20 successive measurements. In contrast, the results of plastic showed
0.6 Rubber, wet
COF
0.5 Plastic, wet
0.4 0.3
Leather , dry
5 10 15 No. of friction measurement
20
Fig. 1. Coefficients of friction of three materials on terrazzo, measured in alternating directions (at a temperature of 20 °C and a relative humidity of 65%). The data for rubber and plastic illustrate effects of mechanical abrasion as well as influences of the sliding direction.
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a slightly increasing trend and those of rubber a considerable decrease, respectively. Visual inspection of the elastomer surfaces indicated that the observed trends were caused by mechanical wear. For all but one material combination (leather/terrazzo) the surfaces of the sole materials were visibly modified during the friction measurements (Table 2). The surface of the plastic slider was roughened on terrazzo and, as a consequence, the friction coefficients showed an average increase of 0.008 within 20 measurements (r = 0.30). On PVC, plastic was polished and the friction coefficients were reduced by 0.009 on average (r = 0.52). Rubber was smoothed on both reference surfaces and the mean reductions in the measured coefficients of friction were 0.014 on PVC (r = 0.66) and 0.076 on terrazzo (r = 0.83), respectively. Whereas dry leather showed constant coefficients of friction on terrazzo (r = 0.14), the values increased by 0.057 due to surface polishing on PVC (r = 0.93). Correlation coefficients with absolute values above 0.44 indicated statistically significant linear trends at the 95% confidence level, those with absolute values above 0.56 at the 99% level. 3.3. Influence of the sliding direction Fig. 1 shows measurement results, in which alternating higher and lower COF values indicated a systematic influence of the direction of the friction experiments. On terrazzo, the average differences between the coefficients of friction for opposite sliding directions were 0.032 for rubber and 0.012 for plastic, whereas no effect was found for leather (Table 2). For PVC, on the other hand, the greatest influence of the sliding direction was observed with leather (+0.012), while practically no effects were found with rubber and plastic. 3.4. Repeatability of measured coefficients of friction The results in Table 2 illustrate considerable material-dependent effects of wear and sliding direction on the repeatability of friction measurements. The average repeatability standard deviations of friction coefficients (Dl) ranged from 0.004 (plastic on PVC) to 0.029 (rubber on terrazzo). The mean contributions of systematic trends (Dltrend) lay between 0.003 and 0.024, the mean periodic contributions (Dlperiodic) between 0.000 and 0.016, and the mean random contributions (Dlrandom) between 0.001 and 0.006 (Table 2). 3.5. Influence of temperature Fig. 2 shows that the mean coefficients of friction increased with temperature (0.003 per 1 °C on average). Linear regression analysis revealed clear differences between the specific material combinations. On both reference surfaces, the most significant temperature-dependence was found with rubber (Tg = 46 °C), for which the mean coefficients of friction increased by up to 0.005 per °C. For plastic (Tg = 35 °C), on the other hand, only those coefficients of friction measured on terrazzo depended on temperature. The results of leather depended on the temperature for both reference surfaces, showing a more significant effect on PVC. 3.6. Influence of relative air humidity on the friction of leather As Fig. 3 illustrates, the coefficients of friction of leather on terrazzo showed a marked increase with relative humidity (0.004 per % relative humidity) while practically constant coefficients of friction were measured on the relatively smooth PVC surface (<0.001 per % relative humidity). 3.7. Temperature dependence of material hardness Among the investigated shoe sole materials, rubber was the softest material while leather was the hardest (Table 1). The hardness of leather did only show a slight temperature dependence between 15 °C and 30 °C. Rubber and plastic clearly softened with increasing temperature (Fig. 4). The softening was stronger for the plastic material, for which the higher glass transition temperature was measured (Table 1).
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PVC 0.7 0.6
COF
0.5 0.4 0.3 0.2
Slope [1/°C] Rubber 0.0048 Plastic 0.0005 Leather 0.0034
0.1 15
17.5
20
22.5
25
27.5
30
27. 5
30
Temperature °C Terrazzo 0. 7 0. 6
COF
0. 5 0. 4 0. 3 0. 2
Slope [1/°C] Rubber 0.0040 Plastic 0.0036 Leather 0.0012
0. 1 15
17. 5
20
22. 5
25
Temperature °C Fig. 2. Temperature-dependence of coefficients of friction of three elastomers on PVC and terrazzo. Points and bars denote mean values ± one standard deviation for series of 20 measurements.
1
COF
0.8
Terrazzo
0.6 0.4 0.2
PVC
40
50
60
70
80
90
100
Relative air humidity [%] Fig. 3. Influence of the relative air humidity (rh) on mean coefficients of friction of leather on dry floor surfaces (measurements at 20 °C). The slopes of the calculated regression lines were 0.0040 per % rh (terrazzo) and 0.0004 per % rh (PVC), respectively. Points and bars denote mean values ± one standard deviation for series of 20 measurements.
4. Discussion The friction experiments carried out with three shoe sole materials under controlled climatic conditions showed that abrasion of the materials in frictional contacts, local variations in the roughness of the reference surfaces, ambient temperature and, in the case of dry leather also relative air humidity, are factors that sys-
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95
Hardness Shore A
90 85 Slope [1/°C] Leather - 0.050 Plastic - 0.278 Rubber - 0.172
80 75 70 15
17.5
20
22.5
25
27.5
30
Temperature [°C] Fig. 4. Temperature-dependence of the hardness, Shore A, for the three sole materials investigated. Points and bars denote mean values ± one standard deviation of 10 measurements.
tematically influence friction coefficients and are therefore important in connection with the uncertainty of friction measurements. For four out of six material combinations investigated, the mean repeatability standard deviations of friction coefficients were below 0.01 in series of 20 measurements. In those two cases, in which considerable systematic effects of mechanical abrasion were observed, the mean repeatability standard deviations were 0.019 and 0.029, respectively. Based on laboratory experiments using 18 test combinations of different sole materials (NBR, PUR and leather), floor surfaces (steel, tiles and PVC) and lubricants (oil, water, glycerine and air), Skiba et al. (1994b) reported repeatability standard deviations of up to 0.04 for coefficients of friction measured with the device FSC 2000. In field tests on various floor coverings using the FSC 2000 with the standard materials rubber, plastic and leather, Derler et al. (2005) found a mean repeatability standard deviation of 0.02 for series of 12 successive measurements. They varied the sliding track of an elastomer for each single measurement and considered the influence of local variations in the floor surfaces as important. In the present study, this influence was suppressed by constantly using the same sliding track on each reference surface. For certain material combinations, however, the effects of alternating sliding directions still indicated the importance of local variations in the reference surfaces. Although inhomogeneous surface properties of the soling materials – either due to the initial preparation or produced during the sliding friction contacts – could be a contributing factor, slightly different contact areas in combination with local variations in structure and roughness of the reference surface are probably the main reasons for the observed differences in the measurement results for opposite sliding directions. This interpretation is supported by two facts: (1) Surface roughness and surface roughness variations are greater for terrazzo than for PVC (Section 2.1). As surface roughness is an important factor for friction under wet conditions, it can be expected that the greater local variations in the surface roughness of terrazzo have a stronger effect on measured coefficients of friction compared to the relatively smooth PVC floor. (2) Whereas rubber was smoothed in friction measurements on terrazzo, plastic was roughened (Table 2). Despite this difference, both materials showed qualitatively similar alternating patterns in the friction coefficients measured on terrazzo, implying that the differences for opposite sliding directions were mainly determined by the properties of the floor surface. Linear regression analysis of series of 20 friction measurements showed that pure random contributions to the repeatability standard deviation were similar and around 0.005 for all investigated materials. The corresponding random variations can be attributed to the random nature of the measurement process itself and, in particular, of frictional contacts on the microscopic scale. Since the surface of a relatively soft material is continuously modified in a sliding-friction contact against a harder floor material, even the results of two successive friction measurements on an identical surface can be slightly different due to varying contact areas and stick-slip phenomena. For three material combinations, mechanical abrasion of the elastomers represented the most important contribution to the repeatability standard deviation, while periodic contributions due to opposite directions of the friction experiments were predominant in one case. For two material combinations, systematic and
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random effects were of the same order of magnitude. In those cases, in which mechanical abrasions of the sole materials led to distinct trends in the friction coefficients of measurement series, linear functions provided a good approximation to the data. As possible effects of wear of shoe soles, both roughening and polishing of the surface have been reported in the literature (Leclercq et al., 1995; Manning et al., 1990; Wilson, 1996). Despite a limited number of cases investigated, a variety of tribological phenomena was also observed in the present study. The friction behaviour of the sole materials was generally in accordance with viscoelastic material properties, and in line with others (Kim et al., 2001; Rowland et al., 1996) we found that the wear and friction characteristics depended on the individual material combination. A limitation of our study was that changes in the surface roughness of the slider materials were not monitored during the friction measurements. The friction of a viscoelastic material sliding on a rigid surface is mainly determined by adhesion due to molecular bonding of surface atoms in both contacting materials and hysteresis, resulting from the delayed recovery of the viscoelastic material after deformation by the surface asperities of the rigid material (Moore, 1972). According to Persson et al. (2005) the friction of rubber on smooth surfaces primarily depends on adhesion, while hysteresis becomes increasingly important for rough surfaces. For a tire sliding on a road surface, dry friction was found to be entirely due to the hysteresis contribution, whereas the reduced friction in the wet condition was explained by a sealing effect of rubber, which leads to the entrapment of water in pools of the rough surface, associated with an effective reduction of surface roughness. For the slip resistance of shoe soles on floor surfaces covered by a liquid film, the drainage capability of the shoe-floor contact surface, the draping of the sole material about floor surface asperities as well as the true contact area between the surfaces are considered as key factors (Chang et al., 2001b; Gro¨nqvist, 1995; Strandberg, 1985). Rubber is the softest among the FSC 2000 standard materials. Because a soft material tends to a higher effective contact area and more pronounced microscopic deformations when mechanically interacting with the surface asperities of a rigid material, greater friction coefficients can be expected for rubber than for plastic. This was found in the friction measurements under wet conditions. In addition, mechanical abrasions and floor surface inhomogeneities had a stronger influence for rubber. For both rubber and plastic, generally higher repeatability standard deviations of friction coefficients as well as greater systematic and random contributions were found on the rough reference surface (terrazzo). The dry friction of leather contrasted to the wet friction of rubber and plastic. In particular, lower coefficients of friction, smaller systematic trends as well as a weaker effect of the sliding direction were observed on the rough instead of the smooth surface. Among the viscoelastic materials investigated, leather was the hardest and the one with the smoothest surface. The dry friction of leather seemed mainly determined by adhesion, which generally increases with the effective contact area (Gro¨nqvist, 1995). This was indicated by the measurement series on smooth PVC, in which polishing of the leather surface took place, leading to an increase in the effective contact area and the resulting coefficients of friction. Because the true contact area of leather is expected to be smaller on a rough compared to a smooth surface, the fact that leather showed lower friction coefficients on terrazzo (rough) than on PVC (smooth) additionally supports the interpretation that the dry friction of leather was dominated by adhesion. For all investigated viscoelastic materials, the measured coefficients of friction increased with temperature. The most significant effects were found for rubber (+0.004 to 0.005 per °C), while the behaviour of plastic strongly depended on the substrate. The friction measurement results of leather in the dry condition showed influences of both temperature and relative humidity. The observation of friction coefficients increasing with temperature is in line with the results of Derler et al. (2005) who found the slip resistance of an indoor floor covering to be relatively high in summers and relatively low in winters. For viscoelastic materials, however, the opposite behaviour would be typical, because both the adhesion and the hysteresis contribution to friction are proportional to the dynamic loss tangent which decreases with increasing temperature above the glass transition (Moore, 1972). Accordingly, the coefficients of friction between rubber and road pavements are normally found to decrease with increasing temperature (Bazlamit and Reza, 2005). The increase in frictional coefficients with temperature found for sole materials in the present study can be explained by a gradual softening of the materials within the temperature range investigated (Table 1). For softer materials, a more effective draping about floor surface asperities is expected, leading to an increased effective contact area and higher friction forces.
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Leather is known to be highly hygroscopic and thus sensitive to air humidity (Tuckermann et al., 2001). In particular, the volume and the softness of leather are increased by moisture absorption and swelling. As a consequence, draping effects become increasingly important for the friction between moist leather and a rough surface. Such a mechanism would explain the considerable increase in the coefficients of friction measured with leather on terrazzo. Su and Rowlands (2000) stated that due to the viscoelastic material properties and the complex structure of leather, mechanical contacts cause a mixture of plastic and elastic deformations so that, in addition to the surface properties, leather softness has an influence on the frictional behaviour. 5. Conclusions The coefficients of friction between shoe sole materials and floor coverings are influenced by a variety of factors which are important to be analysed in order to quantify and improve measurement uncertainty in connection with slip resistance. Depending on the material combination, mechanical abrasion of the sole material was found to cause considerable linear trends in series of successive friction measurements. As a result, the repeatability standard deviation of the measured friction coefficients was systematically increased. Linear regression analysis allowed to separate systematic and random contributions to the repeatability standard deviation and, therefore, was found to be a useful tool to analyze the results of friction measurement series. For all material combinations investigated, the mean coefficients of friction increased with temperature. This effect was attributed to a gradual softening of the shoe sole materials, resulting in a more effective draping about floor surface asperities in combination with an increased contact area and higher friction forces. In this study, however, the detailed relationship to the friction behaviour of viscoelastic materials as well as the influence of the floor covering as a function of temperature remained unclear. Therefore, the temperature-dependence of elastomer friction needs to be further investigated for a wider range of shoe sole materials. Due to a high sensitivity to mechanical abrasion and temperature, rubber seemed to be the least appropriate standard material for friction measurements with the device FSC 2000. Substantial influences of abrasions and temperature changes would reduce the repeatability and the reproducibility of measurement results. A problem is also that in cases where elastomer abrasion causes systematically decreasing coefficients of friction, the slip resistance of a floor covering could be overestimated, if a too small number of measurements would be carried out. The application of linear regression analysis for the interpretation of friction measurement results would not only help quantify systematic effects such as the abrasion of test materials, but also allow the optimization of test materials and procedures and, thus, generally improve the assessment of slip resistance. Acknowledgements Parts of this research were financially supported by the Swiss Council for Accident Prevention (bfu) and the Swiss Accident Injurance Company (Suva). We thank Valentin Stra¨ssle for measuring the hardness of sole materials as a function of the temperature. References Anderson, D.A., Meyer, W.E., Rosenberger, J.L., 1986. Development of a procedure for correcting skid-resistance measurements to a standard end-of-season value. In: Crump, E.T. (Ed.), Pavement roughness and skid resistance. Transportation Research Board. National Research Council, Washington DC, pp. 40–48. Bazlamit, S.M., Reza, F., 2005. Changes in asphalt pavement friction components and adjustment of skid number for temperature. Journal of Transportation Engineering-ASCE 131, 470–476. Chang, W.R., 1999. The effect of surface roughness on the measurement of slip resistance. International Journal of Industrial Ergonomics 24, 299–313. Chang, W.R., 2002. The effects of slip criterion and time on friction measurements. Safety Science 40, 593–611. Chang, W.R., Gronqvist, R., Leclercq, S., Brungraber, R.J., Mattke, U., Strandberg, L., Thorpe, S.C., Myung, R., Makkonen, L., Courtney, T.K., 2001a. The role of friction in the measurement of slipperiness, Part 2: survey of friction measurement devices. Ergonomics 44, 1233–1261. Chang, W.R., Gronqvist, R., Leclercq, S., Myung, R., Makkonen, L., Strandberg, L., Brungraber, R.J., Mattke, U., Thorpe, S.C., 2001b. The role of friction in the measurement of slipperiness, Part 1: friction mechanisms and definition of test conditions. Ergonomics 44, 1217–1232.
832
S. Derler et al. / Safety Science 46 (2008) 822–832
Chang, W.-R., Courtney, T.C., Gro¨nqvist, R., Redfern, M., 2003a. Measuring slipperiness – human locomotion and surface factors. Taylor & Francis, London. Chang, W.R., Cotnam, J.P., Matz, S., 2003b. Field evaluation of two commonly used slipmeters. Applied Ergonomics 34, 51–60. Chang, W.R., Li, K.W., Huang, Y.H., Filiaggi, A., Courtney, T.K., 2004. Assessing floor slipperiness in fast-food restaurants in Taiwan using objective and subjective measures. Applied Ergonomics 35, 401–408. Derler, S., Kausch, F., Huber, R., 2005. Systematic patterns and random fluctuations in time series of coefficients of friction measured on floor surfaces. Safety Science 43, 751–770. DIN 53505, 2000. Pru¨fung von Kautschuk und Elastomeren – Ha¨rtepru¨fung nach Shore A und Shore D. Deutsche Norm. Fendley, A., Marpet, M.I., Medoff, H., Schutter, D., 1999. Repeatability and reproducibility in walkway-safety tribometry: abrasive-grit size in test-foot preparation. Journal of Testing and Evaluation 27, 76–82. Gao, C., Abeysekera, J., Hirvonen, M., Aschan, C., 2003. The effect of footwear sole abrasion on the coefficient of friction on melting and hard ice. International Journal of Industrial Ergonomics 31, 323–330. Gro¨nqvist, R., 1995. Mechanisms of friction and assessment of slip resistance of new and used footwear soles on contaminated floors. Ergonomics 38, 224–241. Kim, I.J., Smith, R., 2000. Observation of the floor surface topography changes in pedestrian slip resistance measurements. International Journal of Industrial Ergonomics 26, 581–601. Kim, I.J., Smith, R., Nagata, H., 2001. Microscopic observations of the progressive wear on shoe surfaces that affect the slip resistance characteristics. International Journal of Industrial Ergonomics 28, 17–29. Leclercq, S., Saulnier, H., 2002. Floor slip resistance changes in food sector workshops: prevailing role played by ‘‘fouling”. Safety Science 40, 659–673. Leclercq, S., Tisserand, M., Saulnier, H., 1995. Assessment of slipping resistance of footwear and floor surfaces. Influence of manufacture and utilization of the products. Ergonomics 38, 209–219. Leclercq, S., Tisserand, M., Saulnier, H., 1997. Analysis of measurements of slip resistance of soiled surfaces on site. Applied Ergonomics 28, 283–294. Manning, D.P., Jones, C., 2001. The effect of roughness, floor polish, water, oil and ice on underfoot friction: current safety footwear solings are less slip resistant than microcellular polyurethane. Applied Ergonomics 32, 185–196. Manning, D.P., Jones, C., Bruce, M., 1990. Proof of shoe slip-resistance by a walking traction test. Journal of Occupational Accidents 12, 255–270. Manning, D.P., Jones, C., Rowland, F.J., Roff, M., 1998. The surface roughness of a rubber soling material determines the coefficient of friction on water-lubricated surfaces. Journal of Safety Research 29, 275–283. Marpet, M.I., 2002. Improved characterization of tribometric test results. Safety Science 40, 705–714. Moore, D.F., 1972. The friction and lubrication of elastomers. Pergamon Press, Oxford. Oliver, J.W.H., Tredrea, P.F., Pratt, D.N., 1988. Seasonal variation of skid resistance in Australia. Australian Road Research Board, Special Report No. 37. Persson, B.N.J., Tartaglino, U., Albohr, O., Tosatti, E., 2005. Rubber friction on wet and dry road surfaces: the sealing effect. Physical Review B 71, 035428. Rowland, F.J., Jones, C., Manning, D.P., 1996. Surface roughness of footwear soling materials: relevance to slip resistance. Journal of Testing and Evaluation 24, 368–376. Sachs, L., 2004. Angewandte Statistik. Springer, Berlin. Skiba, R., Scheil, M., Windhovel, U., 1994a. Comparative research of unsteady measurement of friction coefficient on floors. Kautschuk Gummi Kunststoffe 47, 513–518. Skiba, R., Scheil, M., Windho¨vel, U., 1994b. Vergleichsuntersuchung zur instationa¨ren Reibzahlmessung auf Fussbo¨den. Schriftenreihe der Bundesanstalt fu¨r Arbeitsschutz, Dortmund. Forschungsbericht Fb 701. Smith, R.H., Bowden, 2005. Test foot contact time effects in pedestrian slip-resistance metrology. Journal of Testing and Evaluation 33, 39–43. Strandberg, L., 1985. The effect of conditions underfoot on falling and overexertion accidents. Ergonomics 28, 131–147. Su, Z.W., Rowlands, H., 2000. Measurement and analysis of leather frictional forces. Journal of the Society of Leather Technologists and Chemists 84, 117–121. Tuckermann, M., Mertig, M., Pompe, W., 2001. In situ observation of moisture-induced swelling of leather in an environmental scanning electron microscope. Review of Scientific Instruments 72, 1449–1452. Wilson, M., 1996. Slip resistance characteristics of footwear solings assessed using the SATRA friction tester. Journal of Testing and Evaluation 24, 377–385. Wolfram, S., 1996. Mathematica. Wolfram Media, Champaign.