Journal of MATERIALS RESEARCH Welcome Comments Help Estimation of harness by nanoinentation of rough surfaces M. S. Bobji an S. K. Biswas Department of Mechanical Engineering, Inian Institute of Science, Bangalore, 560 02, Inia (Receive 22 July 997; accepte 0 February 998) The roughness of a surface influences the surface mechanical properties, estimate using nanoinentation ata. Assuming a relation between the penetration epth normalize with respect to a roughness scale parameter, an the effective raius encountere by the inenter, a first orer moel of roughness epenency of harness is propose. The practical usefulness of this moel is verifie by the numerical simulation of nanoinentation on a fractal surface. As the roughness of a surface is increase, the harness measure at epths comparable with the roughness scale eviates increasingly from the actual harness. Given the constants relate to inenter geometry, the present work provies a rationale an a metho for econvoluting the effect of roughness in arriving at real harness characteristics of the near surface region of a material. I. INTRODUCTION Inentation methos are use to evaluate the mechanical properties of surface layers an thin films. Traitionally, a har inenter is presse into the specimen surface with a known force, an the harness is estimate using the measure projecte area of the resulting impression. Alternatively, in epth sensing inentations, the penetration of the inenter as a function of applie loa is measure. From the resulting loa versus isplacement curves the material properties such as harness an elastic moulus can be estimate.,2 In nanoinentation, the penetration of the inenter is of the orer of nanometers. It has been foun that at low penetration epths, the harness is ifferent from the bulk harness an the scatter in the measurement is high. 3 Pollock et al. 4 have reviewe the relevant theory an the experimentation that escribe the behavior of materials in the 0 000 nm epth range. The variation in harness at low penetration epths may be attribute to surface chemical effects, 3,5 material property variation with epth, 3 an/or surface roughness of the specimen being inente. The variation is also influence by the metho of measurement: epth sensing or imaging an by the instrument errors in epth sensing measurement. The errors associate with nanoinentation measurement have been iscusse by Menčik an Swain 6 an the scatter has been stuie by Yost. 7 Here we are concerne with the effect of roughness on the surface mechanical property estimates mae using nanoinentation. We woul attempt in this paper to evolve a metho of econvoluting the genuine material property, from the results obtaine by the nanoinentation of a rough surface. It is not ifficult to visualize as Tabor 8 ha note many years ago that the effect of roughness on harness estimation is negligible if the inentation epths are much greater than the surface roughness. The selfaffine fractal nature of engineering surfaces has been emonstrate since then using STM an AFM. 9 The roughness wavelengths that affect a physical process are etermine by the length scale of the process. For nanoinentation the appropriate length scale is the size of the inent mae. The effect of asperities much smaller than the inent size is average out, as for example in the case of conventional harness measurements. Similarly the asperities that are much larger than the inent o not affect the measurements as they present almost a plane surface to the inenter. Polishing reuces the amplitue of roughness but at length scales greater than a certain value etermine by the size of the abrasive. The power spectrum of the rough surface at high frequency (low wavelength) is unaffecte by polishing. This means that except in the case of cleave, atomistically smooth surfaces, nanoinentations are invariably an effectively carrie out on rough surfaces. For an elastic-plastic inentation, the inentation pressure epens on the impose strain. If a cone inents a flat surface, the harness oes not change with penetration. For a spherical inenter, as the strain changes with penetration the harness also changes with penetration. Visualizing a rough surface as mae up of asperities of small raius riing on the back of asperities of larger raius, 0 penetration by inenter of any geometry brings asperities of small raius into play first (Fig. ). With increasing penetration asperities of larger raius are encountere. As the effective raius encountere by the inenter tip continues to change with penetration, the strain varies an thus the measure harness has to change with penetration. When nanoinentation is carrie out on a rough surface, the harness estimates are thus invariably epenent on the penetration epth. One of the key challenges here, given a escription of the roughness J. Mater. Res., Vol. 3, No., Nov 998 998 Materials Research Society 3227
FIG.. Schematic of an inenter on a fractal surface. For increasing penetration epth, effective raius of curvature of the asperity increases. an the penetration epth, is to arrive at an ab initio efinition an therefore an estimate of the effective raius as encountere by the inenter. Such an estimate is essential for econvoluting the surface mechanical properties from the measure nanoinentation ata. As the inenter is brought near the rough surface the contact is first establishe with a single asperity. When the loa is increase, this asperity eforms plastically an the neighboring asperities come into contact. The contact area thus consists of many tiny islans (see Fig. 3 inset). The loa versus isplacement graph measure is epenent on the way in which these islans are istribute. This in turn affects the scatter in the measure property. As the loa is increase, the contact islans increase both in size an number an a statistical averaging results. In this paper we evelop a roughness epenency moel base on single asperity contact. The harness estimates are experimentally valiate by inenting a spherical boy by a spherical inenter. The applicability of the moel for a real situation is teste by comparing the roughness epenency, as suggeste by the moel with that obtaine by numerical simulation of nanoinentation of a generate fractal surface. It is suggeste that the moel can be use to econvolute the real mechanical property of a surface from experimental ata by eliminating the roughness epenency. II. INDENTATION OF A SPHERE A. Theory Figure 2 shows a spherical surface being inente by a spherical inenter. It has been shown that the harness estimate from such an experiment is! R a cos 2 u H H s, () R i R a where H s is the harness of the smooth flat surface an R i an R a are the raii of the inenter an asperity, FIG. 2. Configuration of the inenter an the asperity use in the experiment. The inenter raius R i 2.5 mm. Asperity raius R a 25 mm, 2.5 mm, an 8 mm. u was varie from 0 ± to 30 ±. respectively. The angle u is small, because of the presence of neighboring asperities. It further has very little effect on measure harness. By expaning the term within the parentheses an neglecting the higher orer terms of R i R a (assuming R i! R a ), the equation can be simplifie to H 2 R i. H s R a R a varies continuously for an actual rough surface from zero to infinity as the inenter penetrates into the surface. The exact relation between R a an the penetration epth epens on the nature of the rough surface. A general form of such a relation may be written as! m R a K, (2) where r is a roughness parameter such as root mean square roughness, with respect to which the penetration epth can be normalize. K an m are the constants, epenent on the geometric nature of the rough surface. Thus, H 2 K 2 H m. (3) s r B. Experimental Inentation experiments were carrie out using a harene steel ball of raius (R i ) 2.5 mm as the inenter. Specimens as shown in Fig. 2, an of three ifferent raii (R a ), i.e., 25, 2.5, an 8 mm, were machine out of copper ros in a copying lathe. A fixture was use to position the specimen such that the istance between r 3228 J. Mater. Res., Vol. 3, No., Nov 998
the specimen an inenter axes (x) can be varie. Inentation was carrie out in a 0 ton ( 00 KN) universal testing machine, an loa-isplacement curves were recore. The experimental material stiffness was foun ( 50 KN m) to be two orers less than the machine stiffness of 6 GN m. Details of the experimental setup are given in a previous paper. 2 III. MULTIPLE ASPERITY CONTACT NUMERICAL SIMULATION A. Simulation The rough surface is simulate in a way similar to that outline by Majumar an Tien. 3 The height variation Z x of an isotropic an homogeneous rough surface in any arbitrary irection, along a straight line, can be represente by the Weierstrass Manelbrot function. 3 Z x G D2 `X n n l cos 2pg n x g 22D n ;, D, 2; g.. In this G is a scaling contact, D is the fractal imension of the profile, g n l is the frequency moe corresponing to the reciprocal of the wavelength (l) of roughness, an n l is the lower cutoff frequency of the profile which epens on the length of the sample L through the relation g n l L; g is chosen to be.5 for phase ranomization an high spectral ensity. This function has a power spectrum which can be approximate by a continuous spectrum 4 given by with 2 F D ln g 5 2 2D 7 2 2D p an g a p/2 Z 3 cos 522D u2cos 722D u u 0 2 3 2 0.5 g 22D nl2 g 422D 2 0.5 `X n n l! 0.5 cos 2pg n a g 22D n. A value of.5 is chosen for D, corresponing to brownian surface. 9 The summation to infinity is cut off at a higher inex. The inices are chosen to be 34 an 52, respectively, such that the roughness is simulate in the same length scale as the physical phenomena the inentation to be stuie. The surface is simulate by evaluating Eq. (4) over a gri of 28 3 28 uniformly space points. The geometry of the inenter use is shown in Fig. 3. The half cone angle (f) an the tip raius of curvature (R i ) are varie to get the ifferent area functions. The surface of the inenter Z i is generate over the same set of gri points as the simulate rough surface. The axis of the inenter is varie ranomly over the x-y plane, within the mm 3 mm simulate surface. P v G2 D2 2lng v 522D. The values of G an D can be obtaine from a profile measurement using this expression. For isotropic surfaces, Nayak 5 has establishe the relation between the spectrum of a surface an its profile along an arbitrary irection. The surface spectrum 3 is P s v 5 2 2D 7 2 2D p 3 G2 D2 2lng 2p Z 0 cos 522D u2cos 722D u u v 622D. The equation of the surface with this spectrum is Z x, y F D G D2 `X cos 2pg n x g y cos 2pg n y g x 3 ; g 22D n n n l, D, 2; g., (4) FIG. 3. Configuration of the conical inenter on a fractal surface, use in the numerical simulation. The inset shows the contact area for two penetration epths of 50 nm an 50 nm for the conical inenter with a tip raius of 0 nm. J. Mater. Res., Vol. 3, No., Nov 998 3229
Twenty-five such ranom inentations are carrie out for given inenter parameters u an R i. The smooth inenter is brought into contact with the simulate surface an the real contact area is obtaine. For this, first the sum surface 6 is foun using Z s Z i 2 Z. Z s gives the ifference in height between the inenter Z i an the rough surface Z (Fig. 3). The contact area for a particular penetration () of the inenter into the rough surface is the contour of the sum surface for the value of Z s equal to. The contours were obtaine using a stanar algorithm that uses linear interpolation for the Z s values in between the gri points. A typical real contact area is shown for two penetrations in the inset of Fig. 2. B. Contact moel The inentation of a soft, rough surface by a smooth an har inenter is equivalent to the penetration of a soft, smooth an flat surface by a set of har asperities. 8 From the volume an the area of a contact islan, the spherical cavity moel 7 is use to estimate the mean pressure acting over the contact islan. " #) p i 2 E tan b (2 3 Y 2Y 2 2 2n ln, (5) 3 2n where E is Young s moulus, Y is the yiel strength, an n is Poisson s ratio of the material being inente. Tanb is obtaine by equating the volume of the contact islan to that of a cone of attack angle b whose base area is equal to the contact area of the islan. It is assume that the inenter eformation is negligible. The upper limit to this mean pressure is set by the conition of fully plastic eformation. Thus the loa supporte by each islan is compute as ( P i p i 3 A i if p i, 3 3 3 Y 3 A i if p i > 3, (6) where A i is the contact area of the iniviual islans. The total loa is obtaine by summing up the iniviual loa supporte by all the islans for a given penetration. The harness is then obtaine by iviing this loa with the apparent area obtaine from the area function of the inenter. The penetration epth use in the area function can be obtaine in two ifferent ways, epening on the position of the reference plane. One way is to take the plane passing through the initial contact point an parallel to the mean plane of the rough surface. This simulates a epth sensing experiment with ieally infinite measurement resolution. The other way is to measure the penetration from the mean plane. This simulates the imaging type of nanoinentation experiments. 8 The area function for a given inenter geometry is compute using the same routine, but by letting the inentation be one on a smooth flat surface. The inentation was carrie out at 25 ranom locations on the simulate surface an the loa was foun out for ifferent penetration epths at a given location. The rms roughness (R rms ) of the inente surface is varie by varying the magnification constant G in Eq. (4). 9 Six ifferent rough surfaces with R rms ranging from 0.62 nm to 2.6 nm were generate for the inentation. To introuce the effect of the varying material property with the eformation volume or the penetration epth, the yiel strength Y use in Eq. (5) to calculate the mean pressure is allowe to vary as Y Y 0 c 0. (7) V n With n 3 this woul give, for a conical inenter, a flat surface harness variation of type H s H 0 c, (8) where H 0 is the bulk harness an c is a material constant. IV. RESULTS AND DISCUSSION The results of the numerical simulation of nanoinentation by a cone with a spherical tip on a fractal surface is summarize in Fig. 4. The figure clearly brings out the fact that even for a material the bulk an the surface mechanical properties of which are the same, the FIG. 4. Variation of mean harness with penetration epth, obtaine from numerical simulation for the conical inenter with a tip raius of 0 nm. 3230 J. Mater. Res., Vol. 3, No., Nov 998
harness changes with penetration as long as the surface is rough. The figure further elineates the influence of the metho of measurement an the actual property graient with epth on the measure harness. Given this, the following explores a simple way of econvoluting the measure ata to eliminate the effect of the roughness an arrive at (for a epth sensing instrument with zero measurement error) a first orer estimate of harness characteristic with penetration which reflects genuine property variation with epth. Figure 5 shows the experimental ata collecte at ifferent angular offset (u) an using ifferent specimen raii (R a ) to fall roughly on a single straight line when plotte as a function of R a cos 2 u R i R a. This valiates Eq. (). Figure 6 shows the results of inentation carrie out on two spherical surfaces of the same raius but of very ifferent heights h. The harness of the spherical asperity whose height is equal to the raius (h R a ) is inepenent of the penetration epth, except at very low penetration. The harness measure by inenting the asperity whose height is equal to onehalf of its raius (h R a 2), however, starts to increase as the penetration epth reaches a substantial proportion of the specimen height. The inenter beyon this stage may be visualize as encountering an effective raius which is a composite of the specimen an the flat surface raii (infinity). This simple experiment emonstrates that for a real rough surface as the inenter goes through from one level of asperity to the next layer of larger asperities, the effective raius an therefore the harness changes. FIG. 5. Plot of harness versus R a cos 2 u R i R a. The experimental points lie on the line preicte by Eq. (). FIG. 6. Effect of varying asperity raius with penetration on measure harness. Weiss 9 pointe out that the effect of roughness can be accounte for, by aing an error term e in isplacement. This woul give H r Af 6 e H 0 Af 6 e for spherical 6 2 e for conical/pyramial, where H 0 is the bulk harness an Af is the area function of the inenter. Comparing this with Eq. (3) it is clear that for the assume epenency of effective raius on penetration [Eq. (2)], e is relate to some roughness parameter an the inenter geometry. Accoringly we may write an H r H 0 e k r (9) 2 k n, (0) r where k an n are parameters epenent on inenter geometry; n for Weiss s analysis is for a spherical inenter an 2 for a conical inenter. The results of the numerical simulation fit Eq. (0) remarkably well for the values of k an n given in Table I. It is seen that n is an integer as preicte by Weiss, 9 an the value of k is relatively insensitive to inenter raius, both the parameters being primarily epenent on the J. Mater. Res., Vol. 3, No., Nov 998 323
TABLE I. The inenter relate parameters k an n obtaine from numerical simulation. H r H 0 2 k n / r Ref: Contact point Ref: Mean plane Inenter tip raius R i nm k n k n Remarks 500 25.5 22.2 Spherical 500 23.78 22.28 Almost spherical 0 4.07 2 24.92 2 Almost conical 0 3.29 2 25.66 2 Conical inenter shape. It may be note here that k an n are not moel (of the rough surface) specific an it is possible, given any measure real surface profile with proper resolution, to estimate the values of k an n by numerical simulation. Nanoinentation of rough surfaces belonging to a material of the type given by Eq. (7) was simulate for a range of roughness (rms values). Figure 7 shows the estimate harness points normalize with the rough surface harness [Eq. (0)] as a function of the penetration epth an roughness, for a sharp conical inenter. The continuous lines in the figure are rawn as per the equation, H v c H r c k r. () 2 It is seen that the variation of harness ue to the changing roughness an penetration epth is escribe well by this equation. (See Appenix for a physical FIG. 7. Variation of mean harness, normalize with the harness measure on the rough surface with no property variation, with penetration epth, for the inenter with zero tip raius. basis behin this equation.) Using Eq. (0) the harness measure on a rough surface with a material property variation can be written as H v! 2 k n! c r c k r H 2 0. (2) The first term in this equation gives the harness variation ue to roughness alone when there is no property variation with volume/epth. The secon term expresses the effect of property graient in its interaction with roughness on harness. This term comes about because the eformation volume in an asperity, for a given penetration epth, changes with roughness. This results in a change in the aggregate strength of the asperity. The roughness thus alters the asperity-wise istribution of strength an geometric constraint. Harness, which is a prouct of strength an constraint summe over the whole contact omain, changes with roughness. When there is no property variation with volume, c is zero an the harness reuces to Eq. (0). When the roughness, on the other han, is zero ( r 0) Eq. (2) reuces to the smooth surface material property profile [Eq. (8)]. Given the measure harness, the inenter geometry relate constants k an n an the bulk harness Eq. (2) can be use to etermine the material constant c which gives the graient of property with epth. Although the present simulation has been one for a particular type of property profile [Eq. (8)], it is suggeste that a more general profile may be etermine using the approach evelope here. The scatter in the harness measurement arising ue to the roughness of the surface is foun to be inepenent on the material property variation, but epens on the metho of measurement (Fig. 8). The magnitue of the scatter can be quantifie by a nonimensional parameter S s M where s is the stanar eviation an M is the mean of the values of harness for a given penetration epth. The scatter obtaine from the simulations of the imaging type of experiments is foun to be less than that obtaine from the simulations of the epth sensing experiment. 3232 J. Mater. Res., Vol. 3, No., Nov 998
FIG. 8. Variation of the stanar eviation of the measure harness values normalize with the respective mean values, with the penetration epth normalize with rms roughness, for the inenter with 0 nm tip raius. V. CONCLUSIONS () The roughness of a surface affects the harness estimate by nanoinentation, irrespective of whether the bulk an surface mechanical properties are the same. (2) The effective raius of the inente fractal surface increases in irect proportion to penetration an in inverse proportion to a roughness parameter. (3) Knowing the inenter geometry an given the roughness an penetration epth, it is possible to econvolute the effect of roughness on measure harness using a simple algebraic equation, to etermine the genuine mechanical property profile of the surface region. REFERENCES. M. F. Doerner an W. D. Nix, J. Mater. Res., 60 (986). 2. J. L. Loubet, J. M. Georges, an G. Meille, in Microinentation Techniques in Materials Science an Engineering, ASTM STP 889, eite by P. J. Blau an B. R. Lawn (American Society for Testing an Materials, Philaelphia, PA, 986), p. 72. 3. W. C. Oliver, R. Hutchings, an J. B. Pethica, in Microinentation Techniques in Materials Science an Engineering, ASTM STP 889, eite by P. J. Blau an B. R. Lawn (American Society for Testing an Materials, Philaelphia, PA, 986), p. 90. 4. H. M. Pollock, D. Maugis, an M. Barquins, in Microinentation Techniques in Materials Science an Engineering, ASTM STP 889, eite by P. J. Blau an B. R. Lawn (American Society for Testing an Materials, Philaelphia, PA, 986), p. 47. 5. W. W. Gerberich, S. K. Venkataramanan, H. Huang, S. E. Harrey, an D. L. Kohlstet, Acta Metall. Mater. 43, 569 (995). 6. J. Menčik an M. V. Swain, J. Mater. Res. 0, 49 (995). 7. F. G. Yost, Metall. Trans. 4A, 947 (983). 8. D. Tabor, The Harness of Metals (Oxfor University Press, Glasgow, 95). 9. A. Mujamar an B. Bhusan, Trans. ASME, J. Tribology 2, 205 (990). 0. J. F. Archar, Proc. Roy. Soc. Lonon A243, 90 (957).. M. S. Bobji, M. Fahim, an S. K. Biswas, Trib. Lett. 2, 38 (996). 2. M. S. Bobji an S. K. Biswas, Philos. Mag. A 73, 399 (996). 3. A. Majumar an C. L. Tien, Wear 36, 33 (990). 4. M. V. Berry an Z. V. Lewis, Proc. Roy. Soc. Lonon A370, 459 (980). 5. P. R. Nayak, Wear 26, 65 (973). 6. H. A. Fransis, Wear 45, 22 (977). 7. K. L. Johnson, Contact Mechanics (Cambrige University Press, Cambrige, 985). 8. B. Bhusan, V. N. Koinkar, an J-A. Ruan, Proc. Instn. Mech. Engrs. J, 208, 7 (994). 9. H. J. Weiss, Phys. Status Solius A29, 67 (992). APPENDIX For a rough surface with a material property variation of type given by Eq. (7), the effect of roughness can be introuce by aing an error term e to the penetration epth. Thus, from Eq. (8), for a sharp conical inenter, H v H r c. 2 e Substituting for e from Eq. (9), this can be written in a series form for. e as H v H r c c k r c! k r 2. 2 3 Equation (2) is obtaine from the above equation by neglecting higher orer terms an by substituting for H r from Eq. (0). J. Mater. Res., Vol. 3, No., Nov 998 3233