Scratching of Elastic/Plastic Materials With Hard Spherical Indenters

Transcription

1 Shane E. Flores Michael G. Pontin Frank W. Zok Materials Department, University of California, Santa Barbara, CA Scratching of Elastic/Plastic Materials With Hard Spherical Indenters A mechanistic framework has been developed for interpreting scratch tests performed with spherical indenters on elastic/plastic materials. The pertinent scaling relations have been identified through a plastic analysis and the model has been subsequently calibrated by finite element calculations. The results show that the ratio of scratch force to normal force (or apparent friction coefficient) can be partitioned into two additive components: one due to interfacial friction and another associated with plastic deformation. The plastic component scales parabolically with the normal force and depends only weakly on the true (elastic) friction coefficient. A simple formula for the scratch force, based on the plastic analysis and the numerical results, has been derived. Finally, experimental measurements on two material standards commonly used for nanoindenter calibration have been used to verify the theoretical results. DOI: / Introduction The advent of instrumented nanoindenters some two decades ago has enabled an unprecedented capability for probing the mechanical properties of materials over a wide range of length scales from nm to mm and forces from N ton. In addition to their now-routine use in measuring material stiffness and hardness, nanoindenters allow studies of creep, dynamic loading, thin film behavior, fracture, and adhesion. Good summaries of the test methods and the underlying mechanics are presented in textbooks by Bhushan 1 and Fischer-Cripps 2 as well as a recent review article by Gouldstone et al. 3. A comparatively recent advancement in the field has been the development of instrumented indenter probes that can be displaced in a precise manner both normal and tangential to the sample surface. These probes allow measurement of tribological properties those involving friction, abrasion, and wear at length scales and force ranges typical of normal indentation Despite the technological advancements, analysis protocols for extracting fundamental material properties from such tests have not reached maturity levels comparable to those used to ascertain modulus and hardness from indentation tests. The principal goal of the present article is to outline a mechanistic framework for interpreting measurements from scratch tests on elastic/plastic materials with spherical indenters. The latter shape selection is motivated by the fact that, at low force levels, stresses beneath a spherical indenter are below the elastic limits and hence the tribological properties can be ascertained in the absence of plasticity; yet, at higher force levels, responses in the transitional elastic/plastic and the fully plastic regimes can also be probed. In contrast, with sharp-tipped indenters such as the cubecorner, Berkovich, and cone, the accessible behavioral domains are far more restricted. That is, because of the self-similar deformation fields associated with sharp tips, the strain level is fixed independent of normal force and dictated by indenter shape Probing material properties over a range of strains requires use of indenters of varying shapes. Even then, if the tips are very sharp, measurements cannot be made in the elastic domain. Selection of the spherical indenter is further motivated by the recognition that the asperities that make contact during sliding of Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 22, 2008; final manuscript received June 30, 2008; published online August 22, Review conducted by Zhigang Suo. surfaces are more closely represented by protuberances with a constant finite curvature rather than ones with infinitely sharp points. As a prelude to forthcoming results and to provide perspective, the test conditions of interest and the dominant behavioral domains are illustrated in Fig. 1. Here a rigid sphere is pressed into contact with a flat slab of plastically deformable material with normal force F N and subsequently slid across the slab surface with lateral force F L. At sufficiently low levels of F N, wherein the contact is elastic, sliding occurs subject to Coulomb s law, with friction coefficient F L /F N. In contrast, at high force levels, both initial normal contact and subsequent sliding involve plastic deformation. In this domain, the normalized scratch force or apparent friction coefficient, F L /F N, increases approximately parabolically with F N and exceeds the true elastic friction coefficient. An intermediate force range exists within which deformation involves comparable amounts of elastic and plastic strains and the curves in Fig. 1 transition accordingly. The high load domain is the main focus of the present article. The remainder of this article consists of three parts. In the first, an approximate analytical model of scratching of a rigid, perfectly plastic material based on a virtual work analysis is presented. The model is used to identify the scaling relation between the scratch force and the geometric and material properties resulting in the nondimensional parameters of the coordinate axes in Fig. 1. Next, finite element calculations are used to investigate the effects of normal force and friction coefficient on scratch force, with the goal of ascertaining the key nondimensional parameters. Finally, experimental measurements on two material standards are presented and compared with the model predictions. 2 Analytical Model A lower-bound estimate of the scratch force is obtained using established theorems of classical plasticity. The geometry to be analyzed is depicted by the schematic in the top right corner of Fig. 1. Scratching proceeds in two steps. First, a rigid spherical indenter of radius R is pushed into a flat semi-infinite slab of rigid, perfectly plastic material with normal force, F N. The radius, a, of the resulting indentation is given by 19 a = C1 Ru 0 = F N C 2 y where u 0 is the maximum penetration depth, y is the material yield strength, C 1 =2.7, and C 2 =3.0. The indenter is then moved 1 Journal of Applied Mechanics Copyright 2008 by ASME NOVEMBER 2008, Vol. 75 /

2 Fig. 1 Overview of the scratch configuration, the dominant behavioral domains, and the trends in scratch force with normal force and friction coefficient laterally in the x-direction while maintaining constant normal force. Three additional assumptions are invoked. i Sliding at the interface between the two bodies obeys Coulomb s law. ii The scratch depth u s is equal to the initial indentation depth u 0 verified by subsequent finite element calculations. iii The forces at the indenter/material interface remain below those needed to produce sticking friction. In the steady-state domain, the rate of work done by scratching, dw/dx=f L, can be partitioned into two components: one, dw p /dx, due to plastic deformation beneath the indenter tip, and another, dw f /dx, from frictional sliding. The rate of dissipation is obtained from an analysis of the following virtual sequence of operations illustrated in Fig. 2. i A thin slab of thickness dx perpendicular to the scratch direction and upstream from the scratch tip is removed from the sample. ii The slab is indented by a cylindrical roller of radius R under plane strain conditions to produce a cylindrical divot of width 2a and depth u s identical to those in the scratch wake. iii The indenter is slid across the surface of the slab a distance dx. iv The deformed slab is pasted onto the opposite face, downstream from the scratch tip, thereby advancing the indenter tip by a distance dx. The work done during this sequence in Steps ii and iii in particular is u dw 0 dx = F L f N u du + F N 2 = 0 This relation identifies the two pertinent nondimensional parameters: the normalized scratch force, F L /F N, and the normalized normal force, F N /R 2 y. These represent the parameters on the coordinate axes in Fig. 1 and are utilized in the presentation of subsequent numerical and experimental results. 3 Numerical Analysis 3.1 Finite Element Model. Calculations of scratch response were performed using the commercial finite element code where f N u is the force per unit length of cylinder, given by f N u =2a y C 3 3 where C Combining Eqs. 1 3 and integrating yields where F L F N = + k 0 F N R 2 y 4 k 0 4C 3 3/ C 1 C 2 5 Fig. 2 Schematic of virtual cutting, indenting, and pasting operations used to model steady-state scratching / Vol. 75, NOVEMBER 2008 Transactions of the ASME

3 Fig. 3 Finite element mesh used for scratch and indentation simulations ABAQUS/EXPLICITV6.4. For consistency with the experiments described in the next section, the indenter is modeled as a rigid axisymmetric cone with a full apical angle 2 =60 deg and a spherical tip of radius R. All other length scales are subsequently normalized by R so the absolute value of R is arbitrary. The indenter is meshed with four-noded 3D rigid elements. The material being indented is represented by a biased mesh refined near the indentation surface and coarser toward its base using eight-noded 3D brick elements with reduced integration Fig. 3. Although all results are presented in a nondimensional form, the absolute material properties for most calculations were selected to be close to those of typical engineering polymers y =60 MPa, E=3 GPa, and =0.3. To assess the effects of yield strain, several simulations were performed with the same values of y and but with a higher modulus: E=300 GPa. To ensure numerical stability, the hardening rate subsequent to yielding was taken to be 3 MPa finite but small. Surface sliding was allowed to occur in accordance with Coulomb s law, with friction coefficients =0, 0.125, or The bottom surface of the specimen was fixed while symmetry boundary conditions were applied to its sides. The calculations were performed in two steps, consistent with those described in Sec. 2. That is, a normal load was applied to the indenter, up to peak values in the range F N /R 2 y 1 and peak normal displacements u 0 /R 0.2, and the indenter then displaced laterally up to a displacement of w/r Indentation. An initial assessment of the numerical results was made by comparing the indentation response with existing analytical and experimental results. To facilitate the comparisons, the forces and displacements have been normalized by their corresponding values at the onset of yield, F y and u y, given by 1,20 F y = 21.2 R2 3 y 6 and Ē 2 u y = 6.3 R 2 y 7 Ē 2 where Ē is the plane strain modulus. Two limiting behavioral domains exist. When F/F y is not much greater than unity, the spatial extent of plasticity and the magnitude of the plastic strains are small and hence the indentation response is given to a good approximation by the elastic Hertzian solution 1,20 F F y = u 3/2 u y At the other limit, where F/F y 1, the force-displacement response asymptotically approaches that for a rigid, perfectly plastic material, given by 1,20 F = F y 5.5 u u y Comparisons of the numerical results and the analytical solutions are presented in Fig. 4. Also shown are experimental results for steel from a previous study as well as those for polymethylmethacrylate PMMA from the present study, described below. Good agreement is obtained over the entire loading range. Additionally, friction has a minimal effect over the range of values examined here, consistent with previous numerical investigations 19,21. Parenthetically, the indentation response over the entire loading range can be adequately described by a simple formula that combines the results in Eqs. 8 and 9. Here the total displacement at a prescribed force is taken as the sum of those for purely elastic and purely plastic indentations, namely, 8 9 Journal of Applied Mechanics NOVEMBER 2008, Vol. 75 /

4 Fig. 4 Indentation of elastic-plastic materials with a rigid spherical indenter. Analytical solutions: Eq. 8 for elastic contact, Eq. 9 for plastic contact, and Eq. 10 for elastic/plastic contact. Data for steel adapted from Johnson 20. Finite element results and experimental measurements on PMMA are from the present study. u u y = F 2/3 F y F F y 10 This relation reduces to that in Eq. 8 as F/F y 0 and that in Eq. 9 when F/F y 1. Upon comparison with the numerical results Fig. 4, it appears to be reasonably accurate over the intermediate force range as well. 3.3 Scratching. Representative results from the scratch simulations are presented in Fig. 5. For normal forces above that needed to initiate yield, both the normal displacement u/ R and the scratch force F L /F N initially increase slightly with w/r a consequence of the loss of contact between the indenter and the material in the scratch wake, reach a peak, and then decrease and saturate at constant values, independent of scratch displacement. One manifestation of the steady state is the development of a uniform plastic strain field that translates with the indenter tip during sliding, as illustrated in Fig. 6. An additional notable result is that, for the case where =0, the scratch depth at steady state is essentially identical to the initial indentation depth following application of the normal force, consistent with the underlying assumption of the model in Sec. 2. For nonzero values of, the steady-state scratch depth falls somewhat below the indentation depth Fig. 5 d. Based on dimensional analysis, the critical scratch displacement, w c /R, needed to achieve steady state is expected to scale with the indent size, a/ R. Recognizing that, for small indents, a/r u/r, it follows that the corresponding normal displacement u c /R w c /R 2. An inspection of the numerical results reveals that the onset of steady state can be adequately described by u c /R 0.1 w c /R 2 Fig. 5 b. The effects of the normal force and the friction coefficient on the scratch force are summarized in Fig. 7. When plotted as F L /F N versus FN /R 2 y, the results are linear for a fixed value of, consistent with Eq. 4. Moreover, upon extrapolation to F N /R 2 y =0 where the material response is purely elastic, F L /F N, as required. However, in apparent contradiction to Eq. 4, the slopes of the lines in Fig. 7 are not constant but rather exhibit a weak nearly linear dependence on, characterized by Fig. 5 Results of finite element calculations, showing the effects of normal force and friction coefficient on scratch force and scratch depth. The open circles in b denote the approximate points at which steady state is attained / Vol. 75, NOVEMBER 2008 Transactions of the ASME

5 Fig. 8 Indentation hardness of PMMA, measured over a wide force range using both spheroconical and cube-corner indenters 4 Experimental Measurements Fig. 6 Development of plastic strain beneath the indenter during a typical scratch simulation for scratch displacements, w/r, of a 0, b 0.25, and c 2.5 for F N /R 2 y 1/2 =0.76, =0.125 k 0 = k 1 1+k 2 11 where k 1 =0.184 and k 2 =1.75. This result is depicted by the solid lines in Figs. 1 and 7. For friction coefficients , k 0 falls in the rather narrow range Furthermore, these values are considerably higher than the analytical prediction k , consistent with the lower-bound nature of the model. The effects of the elastic modulus appear to be small. For the case where =0, the scratch forces increase by only about 10% as E is increased by two orders of magnitude from 3 GPa to 300 GPa. Fig. 7 Effects of normal force and friction coefficient on scratch force. The solid lines calculated using the formula shown with k 1 =0.184 and k 2 =1.75. Filled symbols: E=3 GPa. Open symbols: E=300 GPa. 4.1 Materials and Test Methods. Scratch tests were performed on two material standards commonly used for indenter calibration: PMMA and a % pure 100 Al crystal. Scratching was performed using a 60 deg conical diamond indenter with a1 m tip radius. The test protocol consisted of i applying a normal load in the range mn for reasons described below over a period of 5 s, ii holding at the peak load for 5 s, iii displacing the indenter tip laterally over a distance of 10 m ata rate of 0.33 m/s, and iv holding for an additional 5 s before unloading. Material response was characterized by the normalized scratch force, F L /F N, and the scratch depth, u/r. Remnant scratches were imaged by scanning probe microscopy SPM. The normal forces used for the preceding scratch tests were selected on the basis of two criteria: i that deformation be well into the plastic domain, i.e., F/F y 1, and ii that the scratch depth remain below R, to prevent contact of the conical surface of the indenter with the test sample. To this end, preliminary indentation tests were performed using a cube-corner tip to ascertain hardness and modulus. Combining these property values with Eq. 6 and taking the lower limit on the allowable force to be 10F y, the first criterion can be expressed as FN /R 2 y 15 y 12 where y = y /E. Yield strains obtained from the cube-corner indentations are y =0.02 and for PMMA and Al, respectively, and the corresponding critical forces FN /R 2 y 0.3 and A further assessment was made by a series of tests with the spheroconical indenter over a wide force range, to confirm that the deformation was indeed well into the plastic domain, as manifested in a constant value of hardness independent of peak force. Indentation results of this type are plotted on Fig. 8. The maximum allowable normal force to satisfy the second criterion was estimated from the measured hardness and modulus coupled with the result in Eq. 12 and the condition u/r 1. The key property values are summarized in Table 1. The true elastic friction coefficient was measured by scratch tests performed using an indenter with a 50 m radius tip. Strictly, the forces for such tests should remain below that for yield, i.e., FN /R 2 y 5 y 0.1 for PMMA and 0.01 for Al. However, as demonstrated below, this criterion is overly stringent, since F L /F N remains essentially unchanged to significantly higher force levels. 4.2 Scratch Measurements. Typical scratch measurements are presented in Fig. 9. For PMMA, the results closely resemble those obtained from the finite element analysis. Notably, both F L /F N and u/r initially increase with w/r, reach a peak, and then fall back to steady-state values. The predicted onset of steady Journal of Applied Mechanics NOVEMBER 2008, Vol. 75 /

6 Table 1 Summary of indentation properties PMMA Al Hardness, H MPa Modulus, Ē GPa Yield stress, y MPa a Yield force, F y N b Yield displacement, u y nm b a Taken to be H/3. b Corresponds to R=1 m. state, given by u c /R 0.1 w c /R 2, agrees well with the measurements. Once at steady state, F L /F N and u/r exhibit minimal fluctuations. Although similar features are obtained with the Al sample, the reductions in F L /F N and u/r from their peaks to their steady-state values are considerably greater. Furthermore, both F L /F N and u/r exhibit periodic fluctuations with w/r, with wavelengths that increase with F N. The differences in steady-state response of the two materials appear to correlate with the scratch shapes, ascertained from SPM images Fig. 10. In PMMA, the scratches are remarkably uniform along their length, consistent with the constancy of F L /F N and u/ R. In contrast, in Al, the scratches exhibit scalloped edges, with characteristic wavelengths that mimic the oscillations in F L /F N and u/r. It is surmised that this behavior is due to a stick-slip phenomenon. Scratch force measurements from about 100 tests with the 1 m radius indenter are summarized in Fig. 11 a. When plotted as F L /F N versus FN /R 2 y assuming y =H/3, the results are linear for both materials and exhibit similar slopes: k 0 =0.17 and 0.26 for PMMA and Al, respectively. These values are in reasonable agreement with those obtained from the finite element calculations: 0.18 k for The results of scratch tests performed with the 50 m radius tip are plotted on Fig. 11 b. The apparent friction coefficients obtained from the latter tests initially decrease with increasing normal force likely due to fine-scale surface roughness 1 but then reach plateau levels and remain constant with further increases in force. For PMMA, yielding initiates at FN /R 2 y 0.09 somewhat above the values in Fig. 10 SPM images of scratches in PMMA top and Al bottom at various levels of normal force Fig. 11 b and thus the plateau value, F L /F N = , is deemed to be the intrinsic friction coefficient. For Al, yielding is predicted to occur at a lower force, FN /R 2 y 0.01, near the transition. Although the plateau is seemingly in the postyielding domain, the effect of plasticity via Eq. 4, plotted as a dashed line in Fig. 11 b is negligible over the force range of interest. As a result, the average plateau value, F L /F N = , is taken as the friction coefficient for this system. Both friction coefficients obtained in this manner are virtually identical to those inferred from extrapolations of the data in Fig. 11 a to FN /R 2 y =0. 5 Concluding Remarks A mechanistic framework for interpreting scratch tests on plastically deformable materials has been presented. Three behavioral Fig. 9 Experimental measurements of scratch force and normal displacement for a and b PMMA and c and d Al. The open circles in b and d denote approximate points at which steady state is attained / Vol. 75, NOVEMBER 2008 Transactions of the ASME

7 Strictly, the present numerical results are applicable to materials that exhibit time-independent, essentially perfectly plastic behavior subsequent to yielding. The effects of viscoplasticity pertinent to polymers such as PMMA and strain hardening intrinsic to pure Al have yet to be probed. Such effects may account for the slight differences in values of k 0 obtained for the two materials as well as discrepancies between the experimental values and those from the finite element calculations. Acknowledgment The authors gratefully acknowledge financial support from the National Institute of Health NIHR01DE Fig. 11 Summary of scratch force measurements. Tests performed with a 1 m and b 50 m tip radius indenters. The error bars represent standard deviations. The values of F L /F N at F N /R 2 y =0 in a were obtained from the plateau values in b. The dashed line in b represents the predicted dependence on FN /R 2 y through Eqs. 4 and 11 for =0.2. domains have been identified. For smooth surfaces, the scratch force F L /F N is constant when the normal force is below that needed to initiate yield FN /R 2 y 5 y and exhibits a linear dependence on FN /R 2 y in the high force domain FN /R 2 y 15 y. Within the transition 5 y FN /R 2 y 15 y, the effects of plasticity are small and, thus, to a good approximation, F L /F N. Although not explicitly addressed in this study, a fourth domain may arise at low force levels. For most real surfaces ones that are not atomically smooth, contact initially occurs at discrete asperities 1. If the number density of asperities per unit nominal contact area remains constant and the asperities deform elastically, the true contact area would scale as A t a 2 F N 2/3. Assuming, at the simplest level, that the lateral force needed for sliding is proportional to A t, it follows that the friction coefficient should scale as F L /F N F N 1/3. This prediction is qualitatively consistent with the reduction in friction coefficient with increasing F N for both PMMA and Al at the lowest force levels. References 1 Bhushan, B., 1999, Principles and Applications of Tribology, Wiley, New York. 2 Fisher-Cripps, A. C., 2004, Nanoindentation, Springer, New York. 3 Gouldstone, A., Chollacoop, N., Dao, M., Li, J., Minor, A. M., and Shen, Y. L., 2007, Indentation Across Size Scales and Disciplines: Recent Developments in Experimentation and Modeling, Acta Mater., 55, pp Pontin, M. G., Moses, D. N., Waite, J. H., and Zok, F. W., 2007, A Nonmineralized Approach to Abrasion Resistant Biomaterials, Proc. Natl. Acad. Sci. U.S.A., , pp Lafaye, S., and Troyon, M., 2006, On the Friction Behavior in Nanoscratch Testing, Wear, , pp Krupička, A., Johansson, M., and Hult, A., 2003, Use and Interpretation of Scratch Tests on Ductile Polymer Coatings, Prog. Org. Coat., 46 1, pp Bucaille, J. 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