MEASUREMENT OF NORMAL CONTACT STIFFNESS ON FRACTAL ROUGH SURFACES

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1 23 rd Australasian Conerence on the Mechanics o Structures and Materials (ACMSM23) Byron Bay, Australia, 9-2 December 204, S.T. Smith (Ed.) MEASUREMENT OF NORMAL CONTACT STIFFNESS ON FRACTAL ROUGH SURFACES Chongpu Zhai * School o Civil Engineering, The University o Sydney Sydney, NSW, 2006, Australia. chongpu.zhai@gmail.com (Corresponding author) Sébastien Bevand Charles Delaunay Institute, LASMIS, University o Technology o Troyes (UTT) UMR 628, CNRS, 2 Rue Marie Curie, 0000 Troyes, France Yixiang Gan, Dorian Hanaor, Gwénaëlle Proust School o Civil Engineering, The University o Sydney Sydney, NSW, 2006, Australia. Bruno Guelorget, Delphine Retraint Charles Delaunay Institute, LASMIS, University o Technology o Troyes (UTT) UMR 628, CNRS, 2 Rue Marie Curie, 0000 Troyes, France ABSTRACT We investigate the eects o roughness and ractality on the normal contact stiness o rough suraces. Samples o isotropically roughened aluium suraces are considered. The roughness and ractal dimension were altered through blasting using dierent sized particles. Subsequently, surace mechanical attrition treatment (SMAT) was applied to the suraces in order to modiy the surace at the micro-scale. The surace topology was characterised by intererometry-based proilometry. The normal contact stiness was measured through nano-indentation with a lat tip utilising the partial unloading method. We ocus on establishing the relationships between surace stiness and roughness, combined with the eects o ractal dimension. The experimental results, or a wide range o suraces, showed that the measured contact stiness depended very closely on suraces root mean squared (RMS) slope and their ractal dimension, with correlation coeicients o around 90%, whilst a relatively weak correlation coeicient o 57% was ound between the contact stiness and RMS roughness. KEYWORDS Contact stiness, Rough suraces, Fractal dimension, Nano-indentation. INTRODUCTION In contact mechanics, surace morphology and eatures play a considerable role in detering how solids will interact with one another in processes including riction, wear, lubrication, thermal and electrical conductance (Bowden et al. 986; Johnson 987). Rough suraces can be characterised using a variety o descriptors including surace roughness, e.g., root mean squared roughness, skewness, and ractal dimension. Surace ractal dimension is regarded as an eective roughness parameter since it represents three-dimensional surace eatures over a signiicant range o length scales (Go et al. 2006). In the past years, investigations have been done extensively regarding the normal interacial stiness o random rough suraces with varying ractal dimensions. Numerical simulations have been carried

2 out with the help o molecular dynamics (Akarapu et al. 20), boundary element methods (e.g., Pohrt and Popov 202, 203; Campana et al. 20), etc. Studies have ocused on interpreting parameters o contact area and contact stiness under dierent loading conditions and surace characteristics (Popov 200). Several related experiments have also been conducted on rough suraces to obtain interacial behaviour (e.g. Buzio et al. 2003; Lorenz et al. 2009; Hanaor et al. 203). In spite o recent technological advancements in nano-indentation and surace morphology characterisation, which acilitate the mechanical and morphological analysis o suraces at micrometer and nanometer scales, the structure-dependence o contact behaviour o suraces with random multiscale eatures remains largely unknown, and particularly the existing experimental results are limited. In this work, we present an experimental study to detere correlations between interacial contact stiness and surace morphology. METHOD Sample Preparation Thick round disks made o aluium alloy 20 were selected as testing samples due to the material s chemical stability and deormability. The suraces o these samples were irstly treated by standard sand blasting procedures. The sand particles used to blast the sample suraces were garnet and glass beads, the sizes o which are 0.2 mm and 0.3 mm, respectively. SMAT treatment utilising the vibration and shock o micro-scale particles was urther employed on certain samples in order to urther modiy surace eatures. The surace morphology o the testing samples is o undamental importance in governing physical properties and interacial phenomena. We described the sample suraces through descriptors o RMS roughness as well as RMS slope, which are two widely used parameters or rough suraces (Hanaor et al. 203). For the purpose o quantiying the surace roughness, the aluium suraces were scanned using an optical surace proilometer (NanoMap 000WLI) to obtain three-dimensional topographies o the samples. Then, the mean roughness parameters were detered rom the digitised surace data across multiple scans o mm 2 or each individual sample. As RMS roughness is inluenced primarily by highest level eatures while RMS slope values are inluenced primarily by lowest scale eatures, these parameters are rather insuicient to give an entire description o the surace over multiple length scales. The concept o ractal dimension provides a useul method or characterising rough suraces in terms o cross-scales analysis. A number o existing algorithms have been developed to detere surace ractal dimension rom digitised surace data. Here we adopted a triangulation method to calculate the ractal dimension o a given surace scan. The method works as shown in Fig.. A grid o unit dimension L is placed to mesh the surace into a number o triangles. For example, when L= X /4, with X being the total scan length, the surace is covered by 32 triangles o dierent areas inclined at various angles with respect to the projected plane. These triangles have an equal projected triangle size, although their real areas are dierent and larger than the projected area A. The areas o all triangles are calculated and summed to obtain an P approximation o the surace area A(L)depending S on L. The grid size is then decreased by a successive actor o 2, and the previously mentioned process continues until L corresponds to the distance between two adjacent pixel points, i.e., the highest resolution o the scan. The ractal dimension D can be calculated (Douketis et al. 995) as ollows: dlog(a / A ) dlog(l) S P D =- +2. () As evident rom Fig., the sample suraces exhibit sel-ainity only over a certain range o scales, ollowing the tendency o natural suraces to generally have complex structures with sel-ainity (statistical sel-similarity) ater anisotropic scaling (Zahn et al. 999). We used the anisotropic scaling to normalise three-dimensional scans to calculate D or characterising the sample suraces. ACMSM

3 0.4 log(a S /A P ) Sample Sample 2 Sample 3 Sample 4 Sample D Indentator size ,000 L: µm Figure. Schematics o the calculation o D rom dierent sample suraces. The size o lat tip used in nano-indentation is also shown. Sample details can be ound in Table. As shown in Table, ive typical sample suraces with distinct roughness parameters were prepared and characterised prior to nano-indentation tests. For each sample, the mean RMS roughness, RMS slope, and ractal dimension are reported with standard deviations as indicated in Table. It is ound that the RMS slope and ractal dimension reveal similar changing trends, i.e., with smaller particles used to modiy the suraces, larger RMS slope and ractal dimension values are achieved. As or the RMS roughness, results show that the parameter has no clear correlation with varying particle sizes. Table : Surace characterization with dierent treatments. RMS Fractal Suraces RMS slope roughness/µm dimension Sample : Surace blasted with ± ± ± garnet (particle size: 0.2 mm) Sample 2: Surace blasted with glass beads (particle size: 0.3 mm) Sample 3: Surace blasted with garnet + SMAT (particle size: 2 mm) Sample 4: Surace blasted with glass beads + SMAT (particle size: 2 mm) Sample 5: Polished surace with SMAT treatment (particle size: 2 mm) Nano-indentation 4.76 ± ± ± ± ± ± ± ± ± ± ± ± Surace contact stiness o aluium samples with dierent surace morphology was assessed through nano-indentation tests with a 54 µm-diameter lat tip (SYNTON-MDP FLT-D050), as shown ACMSM

4 in Fig. 2. The tip size has a length scale comparable with the surace correlation length, shown in Fig.. During the tests, the lat tip penetrated into random rough samples. We chose a lat tip so that the apparent contact area under the tip does not change with respect to the indentation depth, unlike spherical and Berkovich tips. All the values o stiness were gained by averaging over 0 indentation tests at dierent positions within each sample. Unloading processes at predetered loading levels were applied to eliate the inluence o plastic deormation while measuring contact stiness. For each indentation test, 0 partial unloading procedures were employed with 0% o the current loading level Nanoindentation into surace FLT-D050 Aluum sample Load on sample: mn Contact stiness Displacement into surace: nm Figure 2. Typical loading curve o nano-indentation tests on the sample suraces. Partial unloading paths were applied to extract surace contact stiness under dierent loading levels. RESULTS AND DISCUSSIONS Experimental Data Fig. 3(a) shows the contact stiness o ive selected suraces or dierent values o stress σ, which is calculated as loading orce divided by projected area o the tip. For all tested samples, the results agreed well with the existing theory that the contact stiness increases with the increment o loading orce. It is also ound that a high RMS slope and ractal dimension results in a low contact stiness value. To urther clariy the speciic dierences o contact stiness between distinct suraces, we converted the raw data to non-dimensional values by dividing the stiness and stress values ( k and σ, respectively) by their values at the irst unloading stage or each surace. It can be seen in the s low stress region (i.e., <00MPa), o Fig. 3(b) that the logarithm o dimensionless stiness is proportional to the logarithm o the dimensionless normal stress. That is to say, the stiness shows a power-law relationship with the normal orce, which can be described as k : F α, (2) where N N F is the normal orce acting on the surace and α is the exponent o the power unction. The stiness exponent α ranges rom to , changing as the ractal dimension D changes rom 2.26 to 2.69 and the RMS slope rom 0.07 to As a comparison, the typical value o Hertzian contact is /3. or ACMSM

5 The power-law relationship ound in these experimental results is consistent with previous theoretical predictions (Pohrt et al. 202) on a quantitative basis. The results obtained here also correspond well with results rom simulations (Geike et al. 2007) and experimental measurements carried out through AFM on thin ilms (Buzio et al. 2003). k: GPa Sample Sample 2 Sample 3 Sample 4 Sample 5 (a) 0 00 σ: MPa Low Stress H ig h Stress log(k/k ) Sample α = Sample 2 α = Sample 3 α = Sample 4 α = Sample 5 α = (b) α α2 α3 α4 α5 α = dlog( σ / σ ) log(σ/σ ) dlog(k / k ) Figure 3. Contact stiness o ive tested samples: (a) Raw data rom multiple indentation tests at dierent loading levels, and the stress is calculated by F / A where F is the normal orce and A denotes the projected area o the lat tip; (b) Curve itting or normalised stiness and stress or each individual sample, with k andσ being stiness and stress at the irst unloading stage, respectively. We emphasise here that the above observations are valid primarily or the low stress region. As the loading orce increases towards the high stress region, the exponent α exhibits a slight downward trend. The trend is more proound as the normal orce approaches the compressive strength o the aluium alloy. The non-uniorm pressure distributions within the real contact areas remain nonlinear and complex in the plasticity-doant regions. The impact o plastic deormation o individual asperities can be a possible explanation or this trend. (a) (b) (c) a = 0.066x a = x a = x c = c 2= c 3 = α RMS roughness/µm RMS slope Fractal Dimension Figure 4. Correlations between the stiness variation exponent α and roughness parameters: (a) RMS roughness; (b) RMS slope; (c) Fractal dimension. ACMSM

6 Correlation Analysis Fig. 4 illustrates the signiicance o RMS roughness, RMS slope and ractal dimension in governing contact stiness or rough suraces. Speciically, the stiness variation exponent α across dierent loading orces illustrates a relatively weak correlation to RMS surace roughness. While in contrast, the exponent correlates more closely with the RMS slope and the ractal dimension with the corresponding correlation coeicients around 90%. CONCLUSIONS We experimentally demonstrate the eects o surace roughness on the normal contact stiness o various rough suraces. The results in our experiments show that the contact stiness ollows a powerlaw unction with respect to the normal orce. The exponents o stiness variation over loading show strong dependence on the RMS slope and ractal dimension. Some urther research can be carried out with various tip sizes to explore cross-scale contact properties. ACKNOWLEDGMENTS Financial support or this research rom the Australian Research Council through grants DE and Civil Engineering Research Development Scheme (CERDS) in School o Civil Engineering at The University o Sydney is greatly appreciated. REFERENCES Akarapu, S., Sharp, T., & Robbins, M. O. (20). Stiness o contacts between rough suraces. Physical Review Letters, 06(20), Bowden, F. P. and Tabor, D. (986). The Friction and Lubrication o Solids, Clarendon Press, Oxord. Buzio, R., Boragno, C., Biscarini, F., De Mongeot, F. B., & Valbusa, U. (2003). The contact mechanics o ractal suraces. Nature Materials, 2(4), pp Campana, C., Persson, B. N. J., & Müser, M. H. (20). Transverse and normal interacial stiness o solids with randomly rough suraces. Journal o Physics: Condensed Matter, 23(8), Douketis, C., Wang, Z., Haslett, T. L., & Moskovits, M. (995). Fractal character o cold-deposited silver ilms detered by low-temperature scanning tunneling microscopy. Physical Review B, 5(6), 022. Greenwood, J. A., & Williamson, J. B. P. (966). Contact o noally lat suraces. Proceedings o the Royal Society o London. Series A. Mathematical and Physical Sciences, 295(442), Go, J. Y., & Pyun, S. I. (2006). Fractal approach to rough suraces and interaces in electrochemistry. Modern Aspects o Electrochemistry, Springer US, pp Geike, T., & Popov, V. L. (2007). Mapping o three-dimensional contact problems into one dimension. Phsical Review- E, 76(3), Hanaor, D. A., Gan, Y., & Einav, I. (203). Eects o surace structure deormation on static riction at ractal interaces. Géotechnique Letters 3(2), Johnson, K. L. (987). Contact Mechanics. Cambridge University Press. Lorenz, B., & Persson, B. N. J. (2009). Interacial separation between elastic solids with randomly rough suraces: comparison o experiment with theory. Journal o Physics: Condensed Matter, 2(), Popov, V. L. (Ed.). (200). Contact Mechanics and Friction. Springer Berlin Heidelberg. Pohrt, R., & Popov, V. L. (202). Normal contact stiness o elastic solids with ractal rough suraces. Physical Review Letters, 08(0), Pohrt, R., & Popov, V. L. (203). Contact stiness o randomly rough suraces. Scientiic Reports, 3. Zahn, W., & Zösch, A. (999). The dependence o ractal dimension on measuring conditions o scanning probe microscopy. Fresenius' Journal o Analytical Chemistry, 365(-3), ACMSM