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On the use of fractal geometry methods for the wear process characterization
Wear 258 (2005) 1462–1465
On the use of fractal geometry methods for the wear process characterization Miriam Kupkov´aa,∗ , Martin Kupkab , Em¨oke Rudnayov´aa , J´an Duszaa a b
Institute of Materials Research, SAS, Watsonova 47, Koˇsice, Slovakia Institute of Experimental Physics, SAS, Watsonova 47, Koˇsice, Slovakia
Received 8 April 2004; received in revised form 21 September 2004; accepted 23 September 2004 Available online 11 November 2004
Abstract Irregular fluctuations of friction coefficient around the mean value were analysed by means of fractal geometry methods. Particular data were recorded during the steady-state stage of dry sliding tests carried out in Si3 N4 ball – on – Si3 N4 /SiC nanocomposite disc configuration. It was proven that the set of points {sliding distance (time), friction coefficient}, considered as a geometric object in a two-dimensional space, has the property of a fractal curve. The fractal dimension of this curve increased with increasing wear rate gained in a particular wear test. This could indicate a possible correlation between the wear rate and the fractal dimension of friction coefficient as a function of sliding distance (time). © 2004 Elsevier B.V. All rights reserved. Keywords: Fractal analysis; Friction coefficient; Sliding wear; Si3 N4 /SiC nanocomposite
1. Introduction The wear of machine parts, that is, the progressive damage and material loss which occur on component surfaces as a result of the relative motion of adjacent structural parts, is an important problem related to the operating life of machines. And as such it has serious economic consequences. Therefore, considerable efforts have been expended on the development of deterministic models, which could enable engineers and designers to predict product life with confidence. Unfortunately, no simple and universal model is applicable to all situations [1,2], and the development of a reliable predictive theory of wear process is still a scientific and industrial challenge. Wear process complexity is linked to a great number of properties and variable parameters which affect this process. For example, the response of material to mechanical loading can be usually either ductile or brittle. But under the ∗
Corresponding author. Tel.: +421 55 633 8115; fax: +421 55 633 7108. E-mail address: [email protected] (M. Kupkov´a).
0043-1648/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2004.09.059
peculiar conditions generated around intensely loaded point or line contacts the materials which are usually classified as brittle (such as ceramics) can show significant plastic deformation and vice versa. In addition, jumps or nonlinearities in wear behaviour can occur with increasing severity of loading, when any pair of materials can suddenly extremely rise the wear rate, performing the transition from mild to severe wear. Then, recognizing all this complexity, every piece of knowledge concerning at least a partial tribological problem represents a valuable contribution. Investigation of friction coefficient is of peculiar importance with respect to the possible use of particular materials for disc brakes in automotive applications. The friction coefficient is defined as the ratio of the tangential or traction force to the normal loading for two solids under sliding [3]. During each wear test period, the evolution of friction coefficient is similar practically for all materials. Within a short running-in stage the friction coefficient increases and reaches a steady-state value. In particular, a steady-state stage is reached within the first 50–100 m of sliding and the steadystate friction coefficient is then maintained during the entire
M. Kupkov´a et al. / Wear 258 (2005) 1462–1465
test. However, very intense fluctuations of friction coefficient around the mean value may be present at the steady state. And just these fluctuations are object of interest in the contribution presented. There is a variety of wear behaviour controlling mechanisms. Practically all result in subsurface deformation damages and (micro) fragmentation of material. Fragmentation involves the initiation and propagation of fractures. Fracture propagation is highly nonlinear process. The interaction between successive wear events and/or their statistical combination generate observed final surface topographies and friction coefficient behaviour. The friction coefficient fluctuates during the fragment formation due to release of the stress accumulated by the relative motion of two sliding surfaces (this slightly resembles the earthquake scenario [4]). Due to above matters, it seems that the dependence friction coefficient versus sliding distance (time) can possess the fractal character and its analysis could provide an information on character of wear mechanisms. In recent years, many researchers have applied the fractal theory to the field of tribology. But almost exclusively, the fractal concepts are used for description of engineering surfaces and wear particles (e.g. [5,6]). In this contribution, the attention is focused upon different application of fractal theory. It is investigated whether the friction coefficient evolution reveals fractal character, and what kind of information on the wear process could be extracted from the subsequent “fractal geometry” analysis. In that context, some results of dry sliding tests carried out on a recently developed carbon derived Si3 N4 /SiC nanocomposite were analysed. Similar advanced ceramic materials are expected to be a kind of promising wearresistant tribo-materials due to better hardness, corrosion and high-temperature resistance than those of metallic materials.
2. Experimental The studied material (Si3 N4 matrix with introduced SiC nanoparticles) was prepared at the Institute of Inorganic Chemistry of the Slovak Academy of Sciences by hotpressing at 1750 ◦ C for 2 h under a specific heating regime, atmosphere, and mechanical pressure in the form of discs with diameter of 50 mm and thickness of 5 mm [7]. From a number of various experiments [8], the tests of interest were those carried out on the ground surfaces of discs by means of the pin-on-disc (ball-on-disc) ultra-high-vacuum tribometer (Austrian Research Centers, Seibersdorf) at room temperature, in air and without any lubricant. Initial surface finishing conditions for all discs prior to testing were characterized by the roughness Ra = 0.2 m. The ball of diameter of 6 mm was made of Si3 N4 . The applied load was 20 N, sliding distances 900 m. The wear volume on each flat specimen was calculated from the surface profile traces across the track and perpendicular to the sliding direction. These data were then
1463
used to calculate the wear coefficient (or wear rate) (the wear volume per sliding distance per normal load). The purpose of this orientation study was to analyse the irregular fluctuations superposed on the steady-state evolution of friction coefficient during the wear process of various intensity. Differently intense wear was caused by means of different sliding velocities (the values 0.1, 0.2 and 0.3 m/s were used). But the search for the source of the correlation between the sliding velocity and the wear process intensity was beyond the scope of this study.
3. Data processing Fluctuations of the friction coefficient reveal an irregular structure. With respect to their possible origin (as mentioned in Section 1), they can represent a non-stationary process. Such signals are quite difficult to process properly in a standard way (e.g. by means of Fourier analysis). Then a fractalbased study can be carried out. A variety of algorithms are available for the computation of fractal dimension. The Higuchi’s estimator [9] was chosen here as it is one of the most robust method to compute the fractal dimension of discrete time series and provides the most accurate estimates of this dimension [10]. This algorithm is briefly sketched below. Consider X(1), X(2), . . ., X(N) the time sequence to be k are analysed. From a given time series, k new time series Xm constructed, defined as follows: k Xm = {X(m), X(m + k), X(m + 2k), ...,
X(m + nli (N, m, k)k)} ,
for m = 1, 2, . . . , k.
where nli (N, m, k) stands for the lower integer part of (N − m)/k. m indicates the initial time value, and k represents the discrete time interval (delay) between points. For a time interval equal to k, one gets k sets of new time series. k , the length of the For each of the curves or time series Xm curve is defined as follows: N −1 Lm (k) = nli (N, m, k)k2 ×
nli (N,m,k)
|X(m + ik) − X(m + (i − 1)k)| .
i=1
The term (N − 1)/nli (N, m, k)k2 represents the normalization factor for the curve length of subset time series. The length
L(k) of the curve for the time delay k is defined as the average value of k lengths Lm (k). If L(k) ∝ k−D , then the curve analysed is fractal with the dimension D.
4. Results and discussion Observed behaviour of friction coefficient revealed obvious features (Fig. 1): a quick increase to the peak value,
1464
M. Kupkov´a et al. / Wear 258 (2005) 1462–1465
Fig. 1. Coefficient of friction as a function of sliding distance for various sliding speeds, and the results of application of Higuchi’s algorithm. In that context, k represents the discrete time interval between points constituting a particular subset of time sequence being analysed, and L(k) is the average value of the length of the friction coefficient trace defined on the time subsequence with time delay k. D represents the corresponding fractal dimension of friction coefficient trace.
followed by a transition to a nearly constant value with superposed fluctuation. The steady state stage of evolution of friction coefficient of Si3 N4 /SiC nanocomposite was observed after 150 m of sliding approximately. Therefore, the chosen sliding distance between 150 and 900 m was sufficient for investigating a real friction coefficient. In general, the measured fluctuations of friction coefficient consist of real fluctuations of frictional force and a “noise” introduced by a measuring device (due to rigidity of the apparatus, method of force measurement, etc.). It can be assumed that the “noise” contribution of a particular device is nearly the same for various materials being tested and for various experimental conditions. The magnitude of apparatus “noise” can be estimated by means of accuracy of measurement available for a given device, as the minimum measurable quantity is usually proportional to the magnitude of internal “noise”. In the case presented here, the measured amplitudes of fluctuations exceed the accuracy of measurements by some orders. Therefore, the measured fluctuations are mainly due to fluctuations of frictional force and can be used directly for qualitative characterization and comparison of various experimental results. Of course, for detailed quantitative analysis and comparison of results from different devices, the properties of measuring apparatus should be taken into account more carefully. Fluctuations of the friction coefficient around the mean value revealed an irregular structure. The fractal dimension D can be understood as a “measure of roughness” of the curve considered. The higher the fractal dimension, the more rough and irregular the curve is. A number of wear tests (friction coefficient traces) were analysed. The results of experiments carried out under the
same conditions coincide within a few percents. For the sake of illustration, some examples of friction coefficient behaviour for various sliding velocities as well as corresponding results of “fractal-geometry” analysis are presented in Fig. 1. Comparison of results of fractal-geometry analysis with the set of conventional wear test data indicates the possible correlation between the wear rate and the fractal dimension of friction coefficient versus sliding distance (time) dependence being considered a geometric figure. In particular, the fractal dimension increases with increasing wear rate (Fig. 2). This is in accordance with our working hypothesis that, at least for brittle materials, the friction coefficient fluctuations reflect the fluctuations in the stress–strain state of the subsurface material during the wear damage and fractal frag-
Fig. 2. Disc and ball wear rates for various sliding speeds, and fractal dimensions of corresponding friction coefficient vs. sliding distance (time) dependence. The data depicted are for experiments presented in Fig. 1.
M. Kupkov´a et al. / Wear 258 (2005) 1462–1465
mentation. The more intense the damage and fragmentation, the more intense the fluctuations in the stress field are. This results in intense fluctuations in surface traction force and consequently in friction coefficient determined by means of this force. To confirm or reject the correlation between the wear rate and fractal dimension of corresponding friction coefficient trace, it is necessary to analyse a wider set of various experimental data recorded under different experimental conditions and for different materials. It is also desirable to obtain certain quantitative relationships. This needs further investigation both theoretical and experimental. For example, it would be desirable to obtain the frequency-size distribution of wear debrits (experimental task), to extract the eventual frequency-size distribution of fracture events from the friction coefficient fluctuations (theoretical task), to compare these distribution, and so forth.
5. Conclusion In this orientation study, the possible fractal character of friction coefficient traces was investigated, and prospective correlations between fractal parameters and ordinary wear test data were searched for. For this reason, experimental data relevant to a tribological system consisting of Si3 N4 ball dry sliding against the Si3 N4 /SiC nanocomposite disc were analysed. The results of this preliminary study can be summarized as follows: (i) Application of the Higuchi’s estimator to the recorded friction coefficient data, considered as a time (sliding distance) sequence, confirmed that the friction coefficient trace is a fractal curve. (ii) Comparison of results of fractal-geometry analysis with the set of conventional wear test data indicates the possible correlation between the wear rate and the fractal dimension of friction coefficient versus sliding distance (time) dependence being considered a geometric figure. In particular, the fractal dimension increases with increasing wear rate.
1465
To confirm or reject the working hypothesis that the friction coefficient fluctuations reflect the intensity of nonstationary process of fractal fragmentation of (sub)surface material, a further analysis of a wider set of various experimental data recorded under different conditions and for different materials is necessary.
Acknowledgements The authors are grateful to the Slovak Grant Agency for Science (grant 2/3208/23) and NANOSMART, Center of Excellence of SAS for support of this work.
References [1] H.C. Meng, K.C. Ludema, Wear models and predictive equations: their form and content, Wear 181–183 (1995) 443–457. [2] J.A. Williams, Wear modelling: analytical, computational and mapping: a continuum mechanics approach, Wear 225–229 (1999) 1–17. [3] G. Straffelini, A simplified approach to the adhesive theory of friction, Wear 249 (2001) 79–85. [4] D.L. Turcotte, Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge, 1992. [5] H. Zhu, S. Ge, X. Huang, D. Zhang, J. Liu, Experimental study on the characterization of worn surface topography with characteristic roughness parameter, Wear 255 (2003) 309–314. [6] C.Q. Yuan, J. Li, X.P. Yan, Z. Peng, The use of the fractal description to characterize engineering surfaces and wear particles, Wear 255 (2003) 315–326. ˇ [7] P. Sajgal´ ık, M. Hnatko, Z. Lenˇce´ sˇ, P. Warbichler, F. Hofer, SiC–Si3 N4 nanocomposite prepared by the addition of SiO2 + C, Zeitschrift f¨ur Metallkunde 92 (2001) 937–941. ˇ [8] M. Kaˇsiarov´a, E. Rudnayov´a, J. Dusza, M. Hnatko, P. Sajgal´ ık, A. Merstallinger, L. Kuzsella, Some tribological properties of a carbonderived Si3N4/SiC nanocomposite, J. Eur. Ceram. Soc. 24 (2004) 3431–3435. [9] T. Higuchi, Approach to an irregular time series on the basis of the fractal theory, Physica D 31 (1988) 277–283. [10] P. Simard, E. La Tavernier, Fractal approach for signal processing and application to the diagnosis of cavitation, Mechan. Syst. Signal Processing 14 (2000) 459–469.
On the use of fractal geometry methods for the wear process characterization Miriam Kupkov´aa,∗ , Martin Kupkab , Em¨oke Rudnayov´aa , J´an Duszaa a b
Institute of Materials Research, SAS, Watsonova 47, Koˇsice, Slovakia Institute of Experimental Physics, SAS, Watsonova 47, Koˇsice, Slovakia
Received 8 April 2004; received in revised form 21 September 2004; accepted 23 September 2004 Available online 11 November 2004
Abstract Irregular fluctuations of friction coefficient around the mean value were analysed by means of fractal geometry methods. Particular data were recorded during the steady-state stage of dry sliding tests carried out in Si3 N4 ball – on – Si3 N4 /SiC nanocomposite disc configuration. It was proven that the set of points {sliding distance (time), friction coefficient}, considered as a geometric object in a two-dimensional space, has the property of a fractal curve. The fractal dimension of this curve increased with increasing wear rate gained in a particular wear test. This could indicate a possible correlation between the wear rate and the fractal dimension of friction coefficient as a function of sliding distance (time). © 2004 Elsevier B.V. All rights reserved. Keywords: Fractal analysis; Friction coefficient; Sliding wear; Si3 N4 /SiC nanocomposite
1. Introduction The wear of machine parts, that is, the progressive damage and material loss which occur on component surfaces as a result of the relative motion of adjacent structural parts, is an important problem related to the operating life of machines. And as such it has serious economic consequences. Therefore, considerable efforts have been expended on the development of deterministic models, which could enable engineers and designers to predict product life with confidence. Unfortunately, no simple and universal model is applicable to all situations [1,2], and the development of a reliable predictive theory of wear process is still a scientific and industrial challenge. Wear process complexity is linked to a great number of properties and variable parameters which affect this process. For example, the response of material to mechanical loading can be usually either ductile or brittle. But under the ∗
Corresponding author. Tel.: +421 55 633 8115; fax: +421 55 633 7108. E-mail address: [email protected] (M. Kupkov´a).
0043-1648/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2004.09.059
peculiar conditions generated around intensely loaded point or line contacts the materials which are usually classified as brittle (such as ceramics) can show significant plastic deformation and vice versa. In addition, jumps or nonlinearities in wear behaviour can occur with increasing severity of loading, when any pair of materials can suddenly extremely rise the wear rate, performing the transition from mild to severe wear. Then, recognizing all this complexity, every piece of knowledge concerning at least a partial tribological problem represents a valuable contribution. Investigation of friction coefficient is of peculiar importance with respect to the possible use of particular materials for disc brakes in automotive applications. The friction coefficient is defined as the ratio of the tangential or traction force to the normal loading for two solids under sliding [3]. During each wear test period, the evolution of friction coefficient is similar practically for all materials. Within a short running-in stage the friction coefficient increases and reaches a steady-state value. In particular, a steady-state stage is reached within the first 50–100 m of sliding and the steadystate friction coefficient is then maintained during the entire
M. Kupkov´a et al. / Wear 258 (2005) 1462–1465
test. However, very intense fluctuations of friction coefficient around the mean value may be present at the steady state. And just these fluctuations are object of interest in the contribution presented. There is a variety of wear behaviour controlling mechanisms. Practically all result in subsurface deformation damages and (micro) fragmentation of material. Fragmentation involves the initiation and propagation of fractures. Fracture propagation is highly nonlinear process. The interaction between successive wear events and/or their statistical combination generate observed final surface topographies and friction coefficient behaviour. The friction coefficient fluctuates during the fragment formation due to release of the stress accumulated by the relative motion of two sliding surfaces (this slightly resembles the earthquake scenario [4]). Due to above matters, it seems that the dependence friction coefficient versus sliding distance (time) can possess the fractal character and its analysis could provide an information on character of wear mechanisms. In recent years, many researchers have applied the fractal theory to the field of tribology. But almost exclusively, the fractal concepts are used for description of engineering surfaces and wear particles (e.g. [5,6]). In this contribution, the attention is focused upon different application of fractal theory. It is investigated whether the friction coefficient evolution reveals fractal character, and what kind of information on the wear process could be extracted from the subsequent “fractal geometry” analysis. In that context, some results of dry sliding tests carried out on a recently developed carbon derived Si3 N4 /SiC nanocomposite were analysed. Similar advanced ceramic materials are expected to be a kind of promising wearresistant tribo-materials due to better hardness, corrosion and high-temperature resistance than those of metallic materials.
2. Experimental The studied material (Si3 N4 matrix with introduced SiC nanoparticles) was prepared at the Institute of Inorganic Chemistry of the Slovak Academy of Sciences by hotpressing at 1750 ◦ C for 2 h under a specific heating regime, atmosphere, and mechanical pressure in the form of discs with diameter of 50 mm and thickness of 5 mm [7]. From a number of various experiments [8], the tests of interest were those carried out on the ground surfaces of discs by means of the pin-on-disc (ball-on-disc) ultra-high-vacuum tribometer (Austrian Research Centers, Seibersdorf) at room temperature, in air and without any lubricant. Initial surface finishing conditions for all discs prior to testing were characterized by the roughness Ra = 0.2 m. The ball of diameter of 6 mm was made of Si3 N4 . The applied load was 20 N, sliding distances 900 m. The wear volume on each flat specimen was calculated from the surface profile traces across the track and perpendicular to the sliding direction. These data were then
1463
used to calculate the wear coefficient (or wear rate) (the wear volume per sliding distance per normal load). The purpose of this orientation study was to analyse the irregular fluctuations superposed on the steady-state evolution of friction coefficient during the wear process of various intensity. Differently intense wear was caused by means of different sliding velocities (the values 0.1, 0.2 and 0.3 m/s were used). But the search for the source of the correlation between the sliding velocity and the wear process intensity was beyond the scope of this study.
3. Data processing Fluctuations of the friction coefficient reveal an irregular structure. With respect to their possible origin (as mentioned in Section 1), they can represent a non-stationary process. Such signals are quite difficult to process properly in a standard way (e.g. by means of Fourier analysis). Then a fractalbased study can be carried out. A variety of algorithms are available for the computation of fractal dimension. The Higuchi’s estimator [9] was chosen here as it is one of the most robust method to compute the fractal dimension of discrete time series and provides the most accurate estimates of this dimension [10]. This algorithm is briefly sketched below. Consider X(1), X(2), . . ., X(N) the time sequence to be k are analysed. From a given time series, k new time series Xm constructed, defined as follows: k Xm = {X(m), X(m + k), X(m + 2k), ...,
X(m + nli (N, m, k)k)} ,
for m = 1, 2, . . . , k.
where nli (N, m, k) stands for the lower integer part of (N − m)/k. m indicates the initial time value, and k represents the discrete time interval (delay) between points. For a time interval equal to k, one gets k sets of new time series. k , the length of the For each of the curves or time series Xm curve is defined as follows: N −1 Lm (k) = nli (N, m, k)k2 ×
nli (N,m,k)
|X(m + ik) − X(m + (i − 1)k)| .
i=1
The term (N − 1)/nli (N, m, k)k2 represents the normalization factor for the curve length of subset time series. The length
L(k) of the curve for the time delay k is defined as the average value of k lengths Lm (k). If L(k) ∝ k−D , then the curve analysed is fractal with the dimension D.
4. Results and discussion Observed behaviour of friction coefficient revealed obvious features (Fig. 1): a quick increase to the peak value,
1464
M. Kupkov´a et al. / Wear 258 (2005) 1462–1465
Fig. 1. Coefficient of friction as a function of sliding distance for various sliding speeds, and the results of application of Higuchi’s algorithm. In that context, k represents the discrete time interval between points constituting a particular subset of time sequence being analysed, and L(k) is the average value of the length of the friction coefficient trace defined on the time subsequence with time delay k. D represents the corresponding fractal dimension of friction coefficient trace.
followed by a transition to a nearly constant value with superposed fluctuation. The steady state stage of evolution of friction coefficient of Si3 N4 /SiC nanocomposite was observed after 150 m of sliding approximately. Therefore, the chosen sliding distance between 150 and 900 m was sufficient for investigating a real friction coefficient. In general, the measured fluctuations of friction coefficient consist of real fluctuations of frictional force and a “noise” introduced by a measuring device (due to rigidity of the apparatus, method of force measurement, etc.). It can be assumed that the “noise” contribution of a particular device is nearly the same for various materials being tested and for various experimental conditions. The magnitude of apparatus “noise” can be estimated by means of accuracy of measurement available for a given device, as the minimum measurable quantity is usually proportional to the magnitude of internal “noise”. In the case presented here, the measured amplitudes of fluctuations exceed the accuracy of measurements by some orders. Therefore, the measured fluctuations are mainly due to fluctuations of frictional force and can be used directly for qualitative characterization and comparison of various experimental results. Of course, for detailed quantitative analysis and comparison of results from different devices, the properties of measuring apparatus should be taken into account more carefully. Fluctuations of the friction coefficient around the mean value revealed an irregular structure. The fractal dimension D can be understood as a “measure of roughness” of the curve considered. The higher the fractal dimension, the more rough and irregular the curve is. A number of wear tests (friction coefficient traces) were analysed. The results of experiments carried out under the
same conditions coincide within a few percents. For the sake of illustration, some examples of friction coefficient behaviour for various sliding velocities as well as corresponding results of “fractal-geometry” analysis are presented in Fig. 1. Comparison of results of fractal-geometry analysis with the set of conventional wear test data indicates the possible correlation between the wear rate and the fractal dimension of friction coefficient versus sliding distance (time) dependence being considered a geometric figure. In particular, the fractal dimension increases with increasing wear rate (Fig. 2). This is in accordance with our working hypothesis that, at least for brittle materials, the friction coefficient fluctuations reflect the fluctuations in the stress–strain state of the subsurface material during the wear damage and fractal frag-
Fig. 2. Disc and ball wear rates for various sliding speeds, and fractal dimensions of corresponding friction coefficient vs. sliding distance (time) dependence. The data depicted are for experiments presented in Fig. 1.
M. Kupkov´a et al. / Wear 258 (2005) 1462–1465
mentation. The more intense the damage and fragmentation, the more intense the fluctuations in the stress field are. This results in intense fluctuations in surface traction force and consequently in friction coefficient determined by means of this force. To confirm or reject the correlation between the wear rate and fractal dimension of corresponding friction coefficient trace, it is necessary to analyse a wider set of various experimental data recorded under different experimental conditions and for different materials. It is also desirable to obtain certain quantitative relationships. This needs further investigation both theoretical and experimental. For example, it would be desirable to obtain the frequency-size distribution of wear debrits (experimental task), to extract the eventual frequency-size distribution of fracture events from the friction coefficient fluctuations (theoretical task), to compare these distribution, and so forth.
5. Conclusion In this orientation study, the possible fractal character of friction coefficient traces was investigated, and prospective correlations between fractal parameters and ordinary wear test data were searched for. For this reason, experimental data relevant to a tribological system consisting of Si3 N4 ball dry sliding against the Si3 N4 /SiC nanocomposite disc were analysed. The results of this preliminary study can be summarized as follows: (i) Application of the Higuchi’s estimator to the recorded friction coefficient data, considered as a time (sliding distance) sequence, confirmed that the friction coefficient trace is a fractal curve. (ii) Comparison of results of fractal-geometry analysis with the set of conventional wear test data indicates the possible correlation between the wear rate and the fractal dimension of friction coefficient versus sliding distance (time) dependence being considered a geometric figure. In particular, the fractal dimension increases with increasing wear rate.
1465
To confirm or reject the working hypothesis that the friction coefficient fluctuations reflect the intensity of nonstationary process of fractal fragmentation of (sub)surface material, a further analysis of a wider set of various experimental data recorded under different conditions and for different materials is necessary.
Acknowledgements The authors are grateful to the Slovak Grant Agency for Science (grant 2/3208/23) and NANOSMART, Center of Excellence of SAS for support of this work.
References [1] H.C. Meng, K.C. Ludema, Wear models and predictive equations: their form and content, Wear 181–183 (1995) 443–457. [2] J.A. Williams, Wear modelling: analytical, computational and mapping: a continuum mechanics approach, Wear 225–229 (1999) 1–17. [3] G. Straffelini, A simplified approach to the adhesive theory of friction, Wear 249 (2001) 79–85. [4] D.L. Turcotte, Fractals and Chaos in Geology and Geophysics, Cambridge University Press, Cambridge, 1992. [5] H. Zhu, S. Ge, X. Huang, D. Zhang, J. Liu, Experimental study on the characterization of worn surface topography with characteristic roughness parameter, Wear 255 (2003) 309–314. [6] C.Q. Yuan, J. Li, X.P. Yan, Z. Peng, The use of the fractal description to characterize engineering surfaces and wear particles, Wear 255 (2003) 315–326. ˇ [7] P. Sajgal´ ık, M. Hnatko, Z. Lenˇce´ sˇ, P. Warbichler, F. Hofer, SiC–Si3 N4 nanocomposite prepared by the addition of SiO2 + C, Zeitschrift f¨ur Metallkunde 92 (2001) 937–941. ˇ [8] M. Kaˇsiarov´a, E. Rudnayov´a, J. Dusza, M. Hnatko, P. Sajgal´ ık, A. Merstallinger, L. Kuzsella, Some tribological properties of a carbonderived Si3N4/SiC nanocomposite, J. Eur. Ceram. Soc. 24 (2004) 3431–3435. [9] T. Higuchi, Approach to an irregular time series on the basis of the fractal theory, Physica D 31 (1988) 277–283. [10] P. Simard, E. La Tavernier, Fractal approach for signal processing and application to the diagnosis of cavitation, Mechan. Syst. Signal Processing 14 (2000) 459–469.