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Influence of kinematics and abrasive configuration on the grinding process of glass
Journal of Materials Processing Technology 213 (2013) 728–739
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Influence of kinematics and abrasive configuration on the grinding process of glass F.J.P. Sousa a,∗ , D.S. Hosse a,1 , I. Reichenbach a,1 , J.C. Aurich a,1 , J. Seewig b,2 a Department of Mechanical and Process Engineering, The Institute for Manufacturing Technology and Production Systems, Technical University of Kaiserslautern, D-67653 Kaiserslautern, Germany b Department of Mechanical and Process Engineering, The Institute for Measurement and Sensor-Technology, Technical University of Kaiserslautern, D-67663 Kaiserslautern, Germany
a r t i c l e
i n f o
Article history: Received 2 March 2012 Received in revised form 8 November 2012 Accepted 27 November 2012 Available online 7 December 2012 Keywords: Glass grinding Grinding kinematics Scratching intersections Abrasive configuration
a b s t r a c t The micro- and macro-kinematics performed by the abrasives play a key role during the grinding of brittle materials. The present work intends to evaluate the influence of the abrasive configuration and the trajectory of the abrasive tool on the grinding process of glass. For this purpose, two different arrangements of abrasive pins were tested under three different kinematic curves: two epitrochoidal and one hypotrochoid. Grinding tests were carried out on commercial glass tiles, using 1 h of grinding time and water as lubricant. Computational simulations were used to quantify the spatial distribution of abrasive contacts over the sample surface, including the scratching orientation of each abrasive contact. The amount of material removed from the surface due to the grinding was measured by contour profilometry. By assembling the results from both experiments and simulations the average removal rate of the grinding process were determined and mapped for the entire abraded surface. The effect of the abrasive configuration was found to be either minimized or maximized according to the kinematics chosen, and differences of up to 30% were detected. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The texture of machined products defines not only the tribomechanical properties of the surface but also glossiness and further optical properties expected for the product. For instance, in the floor tile industry, predominance of micro-scale scratches having different curvatures and directions is usually noticed even in polished regions exhibiting considerable glossiness, as result of the kinematics adopted for the machining process (Sousa et al., 2007). For traditional ceramic materials like stoneware floor tiles the material removal under grinding takes place mainly by generation and propagation of cracks around the trajectory of the abrasive contacts (Huang et al., 2002). According to the interactions of successive scratches, different values of removal rates can occur all over the surface under grinding and, as consequence, punctual variations of removal depth and surface qualities can be expected (Sousa et al., 2010).
∗ Corresponding author. Tel.: +49 631 205 3768; fax: +49 631 205 3238. E-mail addresses: [email protected] (F.J.P. Sousa), [email protected] (J. Seewig). 1 Tel.: +49 631 205 3768; fax: +49 631 205 3238. 2 Tel.: +49 631 205 3961; fax: +49 631 205 3963. 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.11.026
An attempt to investigate the effect of the tool path on the removal depth during polishing on aluminium plates was conducted by Hon-Yuen and Cheng (2010) using both simulations and experimental tests. In their tests four different types of tool path were selected and the corresponding profiles of removal depth were measured. It was revealed that the roughness parameters may be similar regardless the tool path, but the final textures differ. Nevertheless, the study of Hon-Yuen and Cheng (2010) was focused on the macro-kinematics and limited to orthogonal crossing paths. This work is focused on the micro-kinematic interactions of the abrasive particles while performing different trochoid curves. The main goal was to evaluate the influence of the geometric configuration of the abrasive tool under different trajectories on the removal rate during the grinding process of homogeneous brittle materials. Such geometric configuration is referred to the relative position of the abrasive pins inside a multiple-pin abrasive tool. The two abrasive configurations investigated are indicated in Fig. 1. It must be noticed that the only difference between abrasive tools is the radial phase angle of the three inner abrasive pins, shifted by 60◦ . They were accordingly named of (Delta) and Y (Ypsilon) configuration. In literature the abrasion volume is usually expressed in terms of sliding distance, mostly in order to check the adequacy with the Archard law of wear (Archard, 1953), in which the effects of several factors are computed altogether using a multiplicative wear coefficient calibrated for each particular tribo-system. This is
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Fig. 1. Geometric details of the multi-pin abrasive tools. (a) Delta “” and (b) epsilon “Y” configuration, and (c) the produced tools in epoxy matrix.
especially true for investigations focused only on the abrasive pin, which remains continuously exposed to the wear during the tests. In contrast, the present work is interested in the distribution of removal rate over the sample surface, inside which a particular region will suffer an intermittent abrasive action. Therefore, instead of the sliding distance, the intensity of the exposition to grinding was quantified by the number of abrasive contacts cumulated by each surface region during the entire process. Based on fruitful investigations of Belkhir et al. (2009) and Hutchings et al. (2004), addressing the polishing of optical glasses and porcelain stoneware tiles respectively, three very different kinematic curves were then selected for the grinding tests: two epitrochoids and one hypotrochoid. These curves were chosen to impose very different distributions of scratching conditions over the surface under grinding, such as scratching speed and radius, and also intersection angles between successive scratches. Details on each one of these curves are provided in the next topic, including the corresponding kinematic equations. 2. Theoretical considerations The three abovementioned kinematic curves are exhibit in Fig. 2, this time only considering the trajectory of a single abrasive particle. Kinematics K1 was adopted to provide scratches with a smooth curvature variation and also high-angle scratching intersections close to the tile centre (Fig. 2a). Kinematics K2 and K3 were selected to assure scratching intersections in different well-defined positions and in different quantities. Contrasting with kinematics K1 and K2, in kinematics K3 the scratching speed is the slowest at the periphery and the fastest at the tile centre. All the three kinematic curves considered in this work were achieved using the kinematic configuration presented in Fig. 3a,
where two rotation motions ω1 and ω2 can be observed. The axis of the primary rotation motion remains at the tile centre, whereas the axis of the secondary motion moves along the circumference of the primary circle. R1 and R2 represent the radii of the primary and secondary circular motion, respectively. RTC refers only to the tool centre, and not for any particular abrasive pin. While R1 is constant, the value of R2 varies slightly in length and direction according to the position of the abrasive pin taken into consideration. As consequence, each abrasive pin may perform a slightly different trajectory during the grinding process. An example of such trajectories is presented in Fig. 3b, admitting an abrasive tool with configuration and the epitrochoid curve K1. A different trajectory interaction is expected for each combination of abrasive configuration and kinematic curve. The final effect of all those interactions on the removal rate is the main subject of the present investigation. Considering the kinematics presented in Fig. 3a, with origin at the tile centre the unitary vectors i and j, the position of the abrasive can be analytically determined pin, given by vector displacement D, for each abrasive pin according to the following equation taken from literature (Sousa et al., 2010): D(t) = {R1 · cos(ω1 · t) + R2 · cos[(ω1 + ω2 ) · t]} · i + {R1 · sin(ω1 · t) + R2 · sin[(ω1 + ω2 ) · t]} · j
(1)
By using the equation above for each abrasive particle and during the entire grinding process, it is possible to access how often each region on the surface had suffered abrasive contacts. The assembling of all these results together furnishes the spatial distribution of grinding imposed onto the glass surface. Besides the position of the abrasives on time, the velocity vector V for a given
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Fig. 2. Kinematic curves adopted: (a) K1 – epitrochoid, (b) K2 – hypotrochoid, and (c) K3 – epitrochoid.
abrasive at each instant is directly achieved deriving Eq. (1) in time, namely: V (t) = − {ω1 R1 · sin(ω1 · t) + (ω1 + ω2 )R2 · sin[(ω1 + ω2 ) · t]} · i + {ω1 R1 · cos(ω1 · t) + (ω1 + ω2 )R2 · cos[(ω1 + ω2 ) · t]} · j (2) With further developments of Eq. (2) it is possible to determine the orientation towards which the scratch is being performed. Taking the positive direction of vector i as reference, the scratching angle for abrasive at each instant t can be given by: Tg() =
D˙ j D˙ i
of intersection and hence interaction between all the successive scratches occurred over that region. Thus, the scratching angle was determined for every single abrasive contact by means of Eq. (3). The corresponding historical data were registered for a fine mesh of points covering the entire tile surface and during the entire grinding process. The statistical dispersion of scratching angle was quantified by the standard deviation . Thus, final profiles of along the radial distance could be obtained and presented as theoretical results. The results achieved in this work are therefore divided into two parts: the first one presenting all the theoretical results provided by the simulations, and a second part containing all experimental results. 3. Experiment details
Vj −{ω1 R1 · cos(ω1 · t) + (ω1 + ω2 )R2 · cos[(ω1 + ω2 ) · t]} = Vi {ω1 R1 · sin(ω1 · t) + (ω1 + ω2 )R2 · sin[(ω1 + ω2 ) · t]} (3) The scratching angle belongs to the plane XY, which represents the tile surface, and it varies within = {−/2 ≤ ≤ /2}. The importance of this parameter rises when the process of texture generation over the glass surface is of interest. Regions scratched by abrasives coming from several different directions will consequently present much less preferential textures than those regions in which the scratches were mostly produced towards some particular direction. In grinding brittle materials, it would be reasonable to expect this variability of scratching angles to be associated with the removal rate of a given region. In other words, the statistical dispersion of scratching angle indirectly represents the degree
The difference of abrasive configuration was achieved by adopting two abrasive tools, each one composed of six identical cylindrical pins deliberately located at different places inside an epoxy resin matrix. Geometric details of the two different abrasive tools were given earlier in Fig. 1. The abrasive pins were fabricated by the company Dremel, under the commercial code 457. Each pin has 4 mm of diameter ( = 4 mm), consisting of a compressed non-woven nylon fibbers impregnated with aluminium oxide abrasive grains. The nominal contact area of the six pins together was 75.4 mm2 , and the first moment of inertia related to the tool centre was 1808 mm3 . It must be mentioned that these values do not change with the abrasive configurations, as the distances from the pins to the tool centre are the same. A constant nominal normal force of 40 N was applied upon the abrasive tool in all experiments. Abundant tap water (pH = 7.7)
Fig. 3. Kinematic configuration: (a) motion sources and (b) an example of the trajectory performed by the abrasive tool.
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grinding process, including probe tip compensations, bow error and linearity corrections. In order to represent an entire tile surface, a total of 36 radial roughness profiles for each tile were collected with a constant step of 10◦ , starting from the tile centre, as indicated in Fig. 4. Each contour roughness profile has a total length of 108 mm. To provide a reference level for the posterior evaluations of the removal depth, a segment of 3 mm from the flat (not abraded) surface was included at both sides of each profile. Quantitative comparisons between total removed volumes using Y and configured tools could then be accomplished by integrating all the measured depth profiles over their respective radial distance. 4. Simulation algorithm
Fig. 4. Measuring of the resulting surface.
was used as lubricant, and common float glass plates with 240 mm × 240 mm × 4 mm were used as samples, representing a homogeneous brittle material. The glass tiles were submitted to the grinding separately, during 60 min each. It was admitted four replications for each grinding condition, which was composed in turn by an abrasive configuration together with a kinematic curve. Inspired by results from Hutchings et al. (2004), Hutchings et al. (2005), and also by the experience from previous works, as in Sousa et al. (2010), a variable speed metallographic polishing machine (mark Buehler, model PowerPro 5000) was employed so that both hypotrochoids and the epitrochoids could be available for carrying out the grinding tests. These kinematic curves were presented earlier in Fig. 2. The sample holder of the polishing machine was slightly adapted to support the glass tiles instead of a sand paper. To obtain the three selected kinematic curves with the kinematic parameters presented in Fig. 3a, the following nominal values were taken: R1 = 40 mm, R2 = 52 mm, and R3 = 8 mm for the three inner abrasive pins and R4 = 16 mm for the outer pins. The first rotational level was ω1 = 150.38 rpm for kinematic K1 (epitrochoid), ω1 = −61.13 rpm for kinematic K2 (hypotrochoid), and ω1 = 50.75 rpm for kinematic K3 (epitrochoid). The rotation level ω2 was kept constant at 60.5 rpm. These rotation motions were experimentally measured with an optical revolution counter (Mark Hofman, model Uni-balancer), and all simulations were calculated computing the measured values. The corresponding kinematic curves considering a single abrasive pin were previously exposed in Fig. 1. A contour profilometre mark Hommel (model Waveline 120) was used to characterize the surface of the glass tiles after the
While the profiles of removal depth due to the grinding process could be directly measured by contour profilometry, the determination of the cumulative number of abrasive contact suffered by a given region required the development of a computational algorithm based on Eqs. (1)–(3) previously given. This algorithm was developed in LabVIEW v10 platform. In the algorithm developed, an abrasive contact is said to occur inside a given region of the tile surface every time its instantaneous coordinates (X and Y) cross inside the nominal contact area of any one of the six abrasive pins. Thus, using such computational algorithm, the cumulative number of scratching contacts was calculated for a complete mesh of points over the entire tile surface. As result, maps containing the spatial distribution of grinding intensity, given in terms of total number of abrasive contacts, could be quantitatively determined and presented as theoretical results. At the same time, the scratching angle associated to each single abrasive contact was also determined. The resulting histogram of scratching angles was then registered in terms of both average and standard deviation values, and for the entire tile surface. Colour intensity graphs were conveniently adopted to present the simulated results. In such graphs, each elementary small region on the real surface is directly and univocally associated with a pixel. For instance, the evolution of the grinding process in terms of accumulation of abrasive contacts is simulated and presented in Fig. 5, considering the kinematics K1, configuration, and at three different instants. A time increment of 1.5 ms (dt = 0.0015 s) and a lateral resolution of 1 mm2 for each pixel were admitted in all the simulations. The plane X–Y refers to the tile. Once the distribution of abrasive contacts was obtained, the removal rate for any region under grinding could be calculated simply by dividing the experimental result of removal depth by the corresponding number of abrasive contacts simulated for that region.
Fig. 5. Evolution of the grinding process. Simulation of the cumulative number of abrasive contacts: (a) right after starting, (b) after 0.5 s and after (c) 5.0 s.
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Fig. 6. Kinematics K1 – simulated distribution of abrasive contacts for (a) and (b) Y configurations. (c) Corresponding averaged radial profiles.
5. Theoretical results As a first theoretical result, simulated distributions of abrasive contacts cumulated during the entire grinding process are given in Figs. 6–8, for K1, K2, and K3, respectively. Results are presented
in both radial profiles and 3D-plots. The Z axis indicates the number of abrasive contacts furnished by the simulation algorithm. The 3D-plots for and Y configuration were presented vis-à-vis to provide a direct comparison. Taking profit of the radial symmetry of the grinding experiment, all the results from the same grinding
Fig. 7. Kinematics K2 – simulated distribution of abrasive contacts for (a) and (b) Y configurations. (c) Corresponding averaged radial profiles.
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Fig. 8. Kinematics K3 – simulated distribution of abrasive contacts for (a) and (b) Y configurations. (c) Corresponding averaged radial profiles.
condition were assembled together in order to become represented by a single profile along the radial distance. Further theoretical results are the dispersion regarding the orientation of the successive scratches punctually cumulated during grinding. As previously mentioned, for each pixel this dispersion was quantified by taking the standard deviation of the total population of scratches occurred inside this pixel. The values of were determined by means of Eq. (3). Noteworthy to recall is that the tendency for texture generation and the overlapping degree of successive scratches are higher where the standard deviation is small, as consequence of the higher concentration of scratches pointing towards similar directions. Radial profiles of standard deviation for both and Y configurations are given in Figs. 9–11, considering the kinematics K1, K2 and K3, respectively. To concentrate on the actual difference between geometric configurations, exceptionally in these graphs the simulation of the scratching angle referred to only two abrasive pins for each abrasive tool: one from the outer ring and another from the inner ring, according to the illustrated in the legend. The scratching speed was also determined for each abrasive contact, and no differences were observed between the abrasive
configurations. The mean values determined for the scratching speed were 0.87 m/s for K1, 0.80 m/s for K2 and 0.2 m/s for K3, and the respective minimum and maximum values were 0.50–1.31 m/s, 0.30–1.12 m/s, and 0.14–0.28 m/s. 6. Experimental results The surfaces of the glass tiles after the grinding process are presented in Fig. 12. The tiles are arranged in three lots according to the kinematics. The four lines inside each lot represent the replicated samples, whereas the two columns separate the abrasive configurations and Y. The radial profiles representing the removal depth experimentally measured for each abrasive configuration are presented in Figs. 13–15, considering the kinematic curves K1, K2 and K3, respectively. Graphs for and Y configurations are presented side by side for direct comparisons. Each one of the 36 plotted curves represents the contour of a different radial profile, varying from 10◦ to 360◦ , and calculated taking the average of four profilometry scans collected from each glass tile. As described before, the first and the last 3 mm of each profile were collected from an untouched area of
Fig. 9. Simulated profile of the standard deviation of scratching angle for K1 with and Y configurations.
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Fig. 10. Simulated profile of the standard deviation of scratching angle for K2 with and Y configurations.
Fig. 11. Simulated profile of the standard deviation of scratching angle for K3: (a) and (b) Y configurations.
Fig. 12. Glass tiles after the grinding tests. Resulting patterns for kinematics (a) K1, (b) K2 and (c) K3.
Fig. 13. Radial profiles of removal depth for kinematics K1: (a) and (b) Y configuration.
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Fig. 14. Radial profiles of removal depth for kinematics K2: (a) and (b) Y configuration.
Fig. 15. Radial profiles of removal depth for kinematics K3: (a) and (b) Y configuration.
Table 1 Removed volume for different conditions of kinematics and abrasive configuration. Removed volume (mm3 )
K1 K2 K3
Z-test
Y
Vol. difference
(Z0 = 1.64)
Mean
S.D.
Mean
S.D.
Mean
ZCalc
302.86 629.41 316.08
35.02 71.62 98.34
317.81 549.07 213.46
39.22 46.87 20.26
-14.95 80.34 102.62
0.57 1.88 2.04
the glass surface, so that a reference line for posterior evaluations of material removal could be provided. The total amount of removed volume (in mm3 ) calculated for each combination of kinematic curves and abrasive configurations are given in Table 1. A statistical analysis using Z-test was used in
order to evaluate the effects of abrasive configuration on the total volume removed from the surface, for each kinematics. The difference of removed volume found between Y and configuration and the corresponding comparison parameters (ZCalc ) were also included in the table, admitting a confidence level of 95% (Z0 = 1.64). A difference in volume removal is accepted as statistically significant when the criterion ZCalc > Z0 is satisfied. These values are written in bold. Once the results from experiments and simulations were achieved, further analyses became then available by combining these two kinds of results. In this sense, the ratio between the total removal depth for a given pixel and the corresponding number of abrasive contacts provides an estimative of the material removed per abrasive contact. Such specific removal rates were calculated for each pixel, and the resulting radial profiles are presented in
Fig. 16. Radial profiles of removal rate for both and Y configurations when performing kinematics K1.
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Fig. 17. Radial profiles of removal rate for both and Y configurations when performing kinematics K2.
Figs. 16–18, considering kinematics K1, K2 and K3, respectively, and for both abrasive configurations. The reference level of removal rate indicated in these figures was based on a model for the wear of brittle solids under fixed abrasive conditions proposed by Lawn (1975), discussed in the next section.
7. Discussion Visual inspections after the tests indicate that a stable and homogeneous grinding has taken place in all tests. As expected, the different positioning of the abrasive pins between and Y abrasive configurations caused different distributions of abrasive contacts for all kinematic curves. In a phenomenological point of view, the different abrasive distribution means variations in the degree of overlapping among successive scratches. This effect was probably responsible for the different profiles of material removal experimentally measured and presented in Figs. 13–15. The accumulation of abrasive contacts for the Y configuration started about 4 mm closer to the tile centre than that for the configuration, despite the distances from abrasive pins to tool centre to be the same in both configurations. This is explained focusing on the inner abrasives only. All grinding tests were set to start with the maximum reach of the abrasive pins as indicated in Fig. 19a. During the two rotation motions, the alignment of both axes R1 and R2 repeats periodically, defining the minimum and maximum abrasive reaches. In the Y configuration, the extra reach ı of the inner pins rises due to their alignment with the radius R2 , just like the outer abrasive pins. The inner pins have no influence on the maximum reach, but recalling that R2 > R1 , they overrun the tile centre and affect the minimum reach. As compared in Fig. 19b and c, this extra reach is given by ı = R4 ·cos[60◦ ] = 4 mm.
In average, the value of scratching speed was the highest for K1 (0.87 m/s), successively followed by K2 (0.80 m/s) and K3 (0.20 m/s). The same duration of 1 h was set in all tests and simulations, so that the higher is the level of scratching speed, the longer is the resulting sliding distances of the abrasive pins, along which the abrasive contacts occur. This may explain the minor occurrence of abrasive contacts seen in kinematics K3 when compared to the others. However, the higher concentrations of abrasive contacts invariably observed around the tile centre require further explanations. Besides, such concentration was surprisingly the highest for the kinematics K2 and the lowest for K3. The explanation found was based on the two rotational motions adopted. Within the same kinematics, the abrasive tool is driven towards both periphery (outer ring) and tile centre (inner ring) by the same number of times, regardless the abrasive configuration or the instantaneous scratching speed. Nevertheless, admitting both inner and outer ring to have the same constant width, the area exposed by the inner ring will be a smaller area than the outer ring. Taking into account the constant contact area of the abrasive pins, the accumulation of contacts inside the inner ring tends to be proportionally greater than the outer ring. However, the advantage of the inner ring in cumulating abrasive contacts still depends on the trajectory of the abrasive pin. An example of such dependence is given in Fig. 20, which presents the track left by a single abrasive while performing kinematics K1 (Fig. 20a) and K2 (Fig. 20b). In the latter, the overlapping of the abrasive trajectory was proportionally greater inside the inner ring than inside the outer ring. In this case, after successive passages of the abrasive pins the contact accumulation increases more rapidly around the tile centre than the periphery. On the other hand, the same did not happen for kinematics K1, where both inner and outer rings had similar overlapping degrees, causing the concentration of
Fig. 18. Radial profiles of removal rate for both and Y configurations when performing kinematics K3.
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Fig. 19. Reach distances during grinding: (a) recurrent alignments of the rotation axes and of the inner pins in (b) and (c) Y configurations.
Fig. 20. Proportions of abrasive track and both outer and inner grinding rings for (a) K1 and (b) K2.
abrasive contacts around the tile centre and the peripheral regions to be nearly the same (Fig. 6). In both cases, however, the abrasive pin remains a longer time running along the peripheral ring than in intermediary zones, so that the accumulation of abrasive contact was also favoured in this region. As seen in Fig. 2, the trajectory of kinematics K3 is scarcely coinciding with the peripheral ring, and hence no remarkable contact gaining was observed along this region. Still on the simulated distributions of abrasive contacts cumulated during grinding, all profiles have presented a steady behaviour along the intermediary grinding zones surrounded by few accumulation peaks in both inner and outer rings. Those peaks are explained by the geometrical features adopted for the grinding tests. As seen in Fig. 21, within a distance ε from the peripheral border there is a region exclusively abraded by the same abrasive pin, which is the same that defines the maximal grinding reach, namely (R1 + R2 + R3 + /2) = 110 mm. Analogously, in the vicinity of
the inner ring there is a region inside which the abrasive contacts also receive the contribution of extra abrasive pins transcending the tile centre. Therefore, there is an intermediary range inside which all abrasive pins are equally involved, so that no disturbing effects occur and a homogeneous grinding is expected. As observed in all those graphs, pronounced oscillations took place only outside these limits. The oscillation peaks differ markedly in intensity, but are the same in number and only shifted in their positions. According to the indicated in Fig. 19, this is explained considering that the limits of such regions depend on the position of the inner abrasive pin. For instance, the width ε is 8 mm for the Y configured tool and 12 mm for the · Taking the simplified sketch presented in Fig. 21, the limits of the region homogeneously abraded range from (R2 − R1 + R3 + /2) = 30 mm and (R1 + R2 − R3 − /2) = 74 mm of radial distance. Beyond these limits the number of active abrasive
Fig. 21. Regions reached for the abrasive pins.
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pins becomes inconstant. Towards the peripheral region, the sudden reduction in abrasive contacts right after the peaks indicates the loss of abrasive pins acting on that region. Besides, the peaks itself are probably caused by the longer overlapping between abrasive trajectories and peripheral regions when the abrasive pin approaches its maximum reach, as previously seen in Fig. 20. Regarding the removal caused by each abrasive configuration after grinding, a very similar behaviour was found for both and Y abrasive configurations while performing the same kinematics. Differences were mainly in the maxima values of removal depth rather than in qualitative terms. In all the kinematics tested, the Y configured tool has caused the deepest removals, as revealed in the experimental profiles of surface depth presented in Figs. 13–15. For the kinematic curves K2 and K3, the Z-test indicated that the abrasive tools with Y configuration have removed significantly more material than those with configuration. However, for the curve K1, no significant differences in removed volume were found between these configurations. This may suggest that for the trajectory K1, the difference in scratching interactions introduced by the change of the abrasive configuration was not big enough, in average, to affect the total removal volume. However, an appreciable difference in qualitative terms observed when comparing Fig. 13a and b. The trough on the removal profile using configuration was clearly narrower than that using Y configuration, but also considerably deeper, as if part of the removal was slightly shifted towards higher radial distances. In fact, such effect can be reasonably explained by the different contribution of the inner abrasive pins. As seen in Fig. 19, the abrasive pin in Y configuration reaches ca. 4 mm further than in D configuration (Fig. 19), displacing its contribution from 92 to 96 mm of the radial distance. This range is in total accordance with the removal profiles of Fig. 13. From this it follows that the effect of the abrasive configuration on the removal volume can be either qualitative or quantitative, depending on the macro-kinematics adopted. In case of kinematics K1 the removal rates were the same for both configurations, whereas for kinematics K2 and K3 a significant influence of the abrasive configuration was observed. Such influence was always in favour of abrasive tools using Y configuration, and it occurred for different ranges of scratching speed and rotation direction, since K2 represents a hypotrochoid and both K1 and K3 stand for epitrochoids. When analysing the results from kinematics K3, the importance of the scratching angle becomes evident. The higher predominance of scratching intersections observed along the scratching trajectory may explain the higher removal rates observed. It must be recalled that the only difference between the two abrasive configurations is the positioning of the inner abrasive pins, which in turn changes the resulting pattern of scratching intersections. The strongest effect of the abrasive configuration was detected under the myriad of scratching condition offered by the kinematics K3. The higher removal rates determined for the abrasive tool mounted in Y configuration are in total accordance with the results from the Z-test previously made. However, a comparison between the profiles of (Figs. 9–11) and removal rate (Figs. 16–18) reveals that the degree of overlapping between successive scratches cannot explain alone all the differences obtained on the removal rates. No further explanation could be found for such behaviours. The distribution of scratching angle was also influenced by the macroscopic trajectory of abrasive tool. In the kinematic K1, for instance, the variation of the scratching orientation in time, which is in turn related to the inverse of the instantaneous scratching radius, decreases steadily with the radial distance (Fig. 9). Thus, the dispersion of scratching angle also decreases with the radial distances. The same occurred in kinematics K2 (Fig. 11). Contrasting with these two epitrochoid curves, the scratching radius of K3 was the
smallest along the outer ring, and the greatest close to the inner ring, where high angle intersections were also taking place. As result, the lowest level of for this hypocycloid curve occurred at about the middle of the radial distance (Fig. 11). Focusing hereafter only on the homogeneously abraded region, i.e. from 30 mm to 74 mm of radial distance, a different behaviour of removal rates over the radial distance was obtained for each combination of abrasive configuration and kinematic curve investigated. None of them has presented a constant removal rate along the radial distance, as seen in Figs. 16–18. The deviation of the resulting profiles from a straight line reveals how strong the removal rate is affected by the scratching condition rather than the applied load. Within the same kinematics, the main difference of scratching condition is the degree of intersection between the successive scratches. To contextualize the absolute removal rates obtained, a model developed by Lawn (1975) for the determining the wear volume of brittle solids under fixed abrasive conditions was used as reference. Such model relates both macroscopic and microscopic parameters by the following equation: ˙ = · P · V · (˛ · · H)−1 W
(4)
˙ is the wear rate, P is the macroscopic load of the abrasive where W onto the tile surface, V is the scratching speed, H is the dynamic hardness, is a linear scaling factor and ˛ depends on the geometry of the scratching particle. After multiplying both sides by the same arbitrary period of time t, and also considering a sliding distance D of one pixel (1 mm), the resulting equation is: W = · P · (˛ · · H)−1
(5)
Taking P = 75.4 N, and the same values assumed by Lawn (1975), which are H ≈ 104 N/mm2 (soda-lime glass), ≈ 1 (chip areas directly given by micro-deformation tracks), and ˛ ≈ 1 (conical indenter), the total removed volume is W ≈ 24 × 10−4 mm3 . Finally, a theoretical average value of removal depth h¯ caused by each abrasive contact could be then estimated by: h¯ =
W
D · LW
(6)
where LW is the macroscopic width of the abrasion section, and it can be simply represented by the sum of the diameters of the six abrasive pins (LW = 6 × 4 mm = 24 mm). This leads to a value of removal depth of h¯ = 0.1 m, which is comparable to the range experimentally found. The higher removal rates outside the zone of homogeneous grinding could be explained considering the test procedure. Due to the rigidity of the grinding machine, during the grinding process the abrasive tool is forced to remain parallel to the sample surface. Therefore, regions outside the intermediary ring are forced to be equally removed. Such tendency may lead to temporarily increases in the normal load, so that the removal per abrasive contacts also increases. Knowledge on the micro-kinematics can provide support for several kinds of phenomenological investigations. In this sense, further researches on the scratching conditions, especially on the extension of damages produced underneath the surface by scratches under different conditions, may be recommended so that an optimized grinding process could be achieved. 8. Conclusions Based on the theoretical and experimental results obtained in this work the following conclusions can be drawn about the grinding process of glass:
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- The chosen kinematics has a major influence on the distribution of abrasive contacts over the surface under grinding, and as consequence, on the surface profiles. With favourable kinematic parameters the effect of the abrasive configuration can be either minimized or maximized, either qualitatively or quantitatively. - The effect of the abrasive configuration on the material removal was quantitatively significant for two of the three kinematics tested. The Y configured tool was capable to provide higher removal rates than the configured tool under both epicycloid and hypocycloid curves. The advantage of the Y configured tool was ascribed to the degree of intersection between the successive scratches. - The distribution of cumulative number of abrasive contacts occurred during the entire grinding process can be computationally simulated based on the kinematic equations derived from the experimental configuration. Pure geometric effects, such as the distribution of scratches (grinding uniformity), scratching speed, and especially the degree of overlapping between successive scratches, could be obtained and evaluated. - The greatest volume removal occurred close to both inner and outer peripheral ring, where according to the simulated results, the abrasive contacts were accumulated at most. Further researches on this subjected will be fruitful for the understanding of grinding processes of glassy materials, especially on the extension of damages produced by scratches under different conditions. Acknowledgements This work was carried out with the support of the following entities: Deutsche Forschungsgemeinschaft – DFG, the
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Coordination for the Improvement of High Education Personnel – CAPES, in the scope of the Brazilian–German Collaborative Research Initiative on Manufacturing Technology – BRAGECRIM, and the Collaborative Research Center 926 (SBF): Microscale Morphology of Component Surfaces (MICOS), project A4. References Archard, J.F., 1953. Contact and rubbing of flat surfaces. Journal of Applied Physics 24, 981–988. Belkhir, N., Bouzid, D., Herold, V., 2009. Wear behaviour of the abrasive grains in optical glass polishing. Journal of Materials Processing Technology 209 (20), 6140–6145. Huang, H., Li, Z., Shen, J.Y., Zhu, H.M., Xu, X.P., 2002. Micro-structure detection of a glossy granite surface machined by the grinding process. Journal of Materials Science Technology 129, 403–407. ˜ M.J., Quereda, M.F., Hutchings, I.M., Adachi, K., Xu, Y., Sánchez, E., Ibanez, 2004. Laboratory Simulation of the Industrial Ceramic Tile Polishing Process. Global Forum on Ceramic Tile – Qualicer, Castelón, Spain, pp. 19–30. ˜ Hutchings, I.M., Adachi, K., Xu, Y., Sánchez, E., Ibanez, M.J., Quereda, M.F., 2005. Analysis and laboratory simulation of an industrial polishing process for porcelain ceramic tiles. Journal of the European Ceramic Society 25, 3151–3156. Lawn, B.R., 1975. A model for the wear of brittle solids under fixed abrasive conditions. Wear 33, 369–372. Sousa, F.J.P., Aurich, J.C., Weingaertner, W.L., Alarcon, O.E., 2007. Kinematics of a single abrasive particle during the industrial polishing process of porcelain stoneware tiles. Journal of the European Ceramic Society 10, 3183–3190. Sousa, F.J.P., Aurich, J.C., Alarcon, O.E., Weingaertner, W.L., 2010. Influence of the trajectory of the abrasive pin on the grinding process of glassy ceramics. Journal of Materials Science and Engineering 4, 19–29. Hon-Yuen, T., Cheng, H., 2010. An investigation of the effects of the tool path on the removal of material in polishing. Journal of Materials Processing Technology 5, 807–818.
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Journal of Materials Processing Technology journal homepage: www.elsevier.com/locate/jmatprotec
Influence of kinematics and abrasive configuration on the grinding process of glass F.J.P. Sousa a,∗ , D.S. Hosse a,1 , I. Reichenbach a,1 , J.C. Aurich a,1 , J. Seewig b,2 a Department of Mechanical and Process Engineering, The Institute for Manufacturing Technology and Production Systems, Technical University of Kaiserslautern, D-67653 Kaiserslautern, Germany b Department of Mechanical and Process Engineering, The Institute for Measurement and Sensor-Technology, Technical University of Kaiserslautern, D-67663 Kaiserslautern, Germany
a r t i c l e
i n f o
Article history: Received 2 March 2012 Received in revised form 8 November 2012 Accepted 27 November 2012 Available online 7 December 2012 Keywords: Glass grinding Grinding kinematics Scratching intersections Abrasive configuration
a b s t r a c t The micro- and macro-kinematics performed by the abrasives play a key role during the grinding of brittle materials. The present work intends to evaluate the influence of the abrasive configuration and the trajectory of the abrasive tool on the grinding process of glass. For this purpose, two different arrangements of abrasive pins were tested under three different kinematic curves: two epitrochoidal and one hypotrochoid. Grinding tests were carried out on commercial glass tiles, using 1 h of grinding time and water as lubricant. Computational simulations were used to quantify the spatial distribution of abrasive contacts over the sample surface, including the scratching orientation of each abrasive contact. The amount of material removed from the surface due to the grinding was measured by contour profilometry. By assembling the results from both experiments and simulations the average removal rate of the grinding process were determined and mapped for the entire abraded surface. The effect of the abrasive configuration was found to be either minimized or maximized according to the kinematics chosen, and differences of up to 30% were detected. © 2012 Elsevier B.V. All rights reserved.
1. Introduction The texture of machined products defines not only the tribomechanical properties of the surface but also glossiness and further optical properties expected for the product. For instance, in the floor tile industry, predominance of micro-scale scratches having different curvatures and directions is usually noticed even in polished regions exhibiting considerable glossiness, as result of the kinematics adopted for the machining process (Sousa et al., 2007). For traditional ceramic materials like stoneware floor tiles the material removal under grinding takes place mainly by generation and propagation of cracks around the trajectory of the abrasive contacts (Huang et al., 2002). According to the interactions of successive scratches, different values of removal rates can occur all over the surface under grinding and, as consequence, punctual variations of removal depth and surface qualities can be expected (Sousa et al., 2010).
∗ Corresponding author. Tel.: +49 631 205 3768; fax: +49 631 205 3238. E-mail addresses: [email protected] (F.J.P. Sousa), [email protected] (J. Seewig). 1 Tel.: +49 631 205 3768; fax: +49 631 205 3238. 2 Tel.: +49 631 205 3961; fax: +49 631 205 3963. 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.11.026
An attempt to investigate the effect of the tool path on the removal depth during polishing on aluminium plates was conducted by Hon-Yuen and Cheng (2010) using both simulations and experimental tests. In their tests four different types of tool path were selected and the corresponding profiles of removal depth were measured. It was revealed that the roughness parameters may be similar regardless the tool path, but the final textures differ. Nevertheless, the study of Hon-Yuen and Cheng (2010) was focused on the macro-kinematics and limited to orthogonal crossing paths. This work is focused on the micro-kinematic interactions of the abrasive particles while performing different trochoid curves. The main goal was to evaluate the influence of the geometric configuration of the abrasive tool under different trajectories on the removal rate during the grinding process of homogeneous brittle materials. Such geometric configuration is referred to the relative position of the abrasive pins inside a multiple-pin abrasive tool. The two abrasive configurations investigated are indicated in Fig. 1. It must be noticed that the only difference between abrasive tools is the radial phase angle of the three inner abrasive pins, shifted by 60◦ . They were accordingly named of (Delta) and Y (Ypsilon) configuration. In literature the abrasion volume is usually expressed in terms of sliding distance, mostly in order to check the adequacy with the Archard law of wear (Archard, 1953), in which the effects of several factors are computed altogether using a multiplicative wear coefficient calibrated for each particular tribo-system. This is
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Fig. 1. Geometric details of the multi-pin abrasive tools. (a) Delta “” and (b) epsilon “Y” configuration, and (c) the produced tools in epoxy matrix.
especially true for investigations focused only on the abrasive pin, which remains continuously exposed to the wear during the tests. In contrast, the present work is interested in the distribution of removal rate over the sample surface, inside which a particular region will suffer an intermittent abrasive action. Therefore, instead of the sliding distance, the intensity of the exposition to grinding was quantified by the number of abrasive contacts cumulated by each surface region during the entire process. Based on fruitful investigations of Belkhir et al. (2009) and Hutchings et al. (2004), addressing the polishing of optical glasses and porcelain stoneware tiles respectively, three very different kinematic curves were then selected for the grinding tests: two epitrochoids and one hypotrochoid. These curves were chosen to impose very different distributions of scratching conditions over the surface under grinding, such as scratching speed and radius, and also intersection angles between successive scratches. Details on each one of these curves are provided in the next topic, including the corresponding kinematic equations. 2. Theoretical considerations The three abovementioned kinematic curves are exhibit in Fig. 2, this time only considering the trajectory of a single abrasive particle. Kinematics K1 was adopted to provide scratches with a smooth curvature variation and also high-angle scratching intersections close to the tile centre (Fig. 2a). Kinematics K2 and K3 were selected to assure scratching intersections in different well-defined positions and in different quantities. Contrasting with kinematics K1 and K2, in kinematics K3 the scratching speed is the slowest at the periphery and the fastest at the tile centre. All the three kinematic curves considered in this work were achieved using the kinematic configuration presented in Fig. 3a,
where two rotation motions ω1 and ω2 can be observed. The axis of the primary rotation motion remains at the tile centre, whereas the axis of the secondary motion moves along the circumference of the primary circle. R1 and R2 represent the radii of the primary and secondary circular motion, respectively. RTC refers only to the tool centre, and not for any particular abrasive pin. While R1 is constant, the value of R2 varies slightly in length and direction according to the position of the abrasive pin taken into consideration. As consequence, each abrasive pin may perform a slightly different trajectory during the grinding process. An example of such trajectories is presented in Fig. 3b, admitting an abrasive tool with configuration and the epitrochoid curve K1. A different trajectory interaction is expected for each combination of abrasive configuration and kinematic curve. The final effect of all those interactions on the removal rate is the main subject of the present investigation. Considering the kinematics presented in Fig. 3a, with origin at the tile centre the unitary vectors i and j, the position of the abrasive can be analytically determined pin, given by vector displacement D, for each abrasive pin according to the following equation taken from literature (Sousa et al., 2010): D(t) = {R1 · cos(ω1 · t) + R2 · cos[(ω1 + ω2 ) · t]} · i + {R1 · sin(ω1 · t) + R2 · sin[(ω1 + ω2 ) · t]} · j
(1)
By using the equation above for each abrasive particle and during the entire grinding process, it is possible to access how often each region on the surface had suffered abrasive contacts. The assembling of all these results together furnishes the spatial distribution of grinding imposed onto the glass surface. Besides the position of the abrasives on time, the velocity vector V for a given
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Fig. 2. Kinematic curves adopted: (a) K1 – epitrochoid, (b) K2 – hypotrochoid, and (c) K3 – epitrochoid.
abrasive at each instant is directly achieved deriving Eq. (1) in time, namely: V (t) = − {ω1 R1 · sin(ω1 · t) + (ω1 + ω2 )R2 · sin[(ω1 + ω2 ) · t]} · i + {ω1 R1 · cos(ω1 · t) + (ω1 + ω2 )R2 · cos[(ω1 + ω2 ) · t]} · j (2) With further developments of Eq. (2) it is possible to determine the orientation towards which the scratch is being performed. Taking the positive direction of vector i as reference, the scratching angle for abrasive at each instant t can be given by: Tg() =
D˙ j D˙ i
of intersection and hence interaction between all the successive scratches occurred over that region. Thus, the scratching angle was determined for every single abrasive contact by means of Eq. (3). The corresponding historical data were registered for a fine mesh of points covering the entire tile surface and during the entire grinding process. The statistical dispersion of scratching angle was quantified by the standard deviation . Thus, final profiles of along the radial distance could be obtained and presented as theoretical results. The results achieved in this work are therefore divided into two parts: the first one presenting all the theoretical results provided by the simulations, and a second part containing all experimental results. 3. Experiment details
Vj −{ω1 R1 · cos(ω1 · t) + (ω1 + ω2 )R2 · cos[(ω1 + ω2 ) · t]} = Vi {ω1 R1 · sin(ω1 · t) + (ω1 + ω2 )R2 · sin[(ω1 + ω2 ) · t]} (3) The scratching angle belongs to the plane XY, which represents the tile surface, and it varies within = {−/2 ≤ ≤ /2}. The importance of this parameter rises when the process of texture generation over the glass surface is of interest. Regions scratched by abrasives coming from several different directions will consequently present much less preferential textures than those regions in which the scratches were mostly produced towards some particular direction. In grinding brittle materials, it would be reasonable to expect this variability of scratching angles to be associated with the removal rate of a given region. In other words, the statistical dispersion of scratching angle indirectly represents the degree
The difference of abrasive configuration was achieved by adopting two abrasive tools, each one composed of six identical cylindrical pins deliberately located at different places inside an epoxy resin matrix. Geometric details of the two different abrasive tools were given earlier in Fig. 1. The abrasive pins were fabricated by the company Dremel, under the commercial code 457. Each pin has 4 mm of diameter ( = 4 mm), consisting of a compressed non-woven nylon fibbers impregnated with aluminium oxide abrasive grains. The nominal contact area of the six pins together was 75.4 mm2 , and the first moment of inertia related to the tool centre was 1808 mm3 . It must be mentioned that these values do not change with the abrasive configurations, as the distances from the pins to the tool centre are the same. A constant nominal normal force of 40 N was applied upon the abrasive tool in all experiments. Abundant tap water (pH = 7.7)
Fig. 3. Kinematic configuration: (a) motion sources and (b) an example of the trajectory performed by the abrasive tool.
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grinding process, including probe tip compensations, bow error and linearity corrections. In order to represent an entire tile surface, a total of 36 radial roughness profiles for each tile were collected with a constant step of 10◦ , starting from the tile centre, as indicated in Fig. 4. Each contour roughness profile has a total length of 108 mm. To provide a reference level for the posterior evaluations of the removal depth, a segment of 3 mm from the flat (not abraded) surface was included at both sides of each profile. Quantitative comparisons between total removed volumes using Y and configured tools could then be accomplished by integrating all the measured depth profiles over their respective radial distance. 4. Simulation algorithm
Fig. 4. Measuring of the resulting surface.
was used as lubricant, and common float glass plates with 240 mm × 240 mm × 4 mm were used as samples, representing a homogeneous brittle material. The glass tiles were submitted to the grinding separately, during 60 min each. It was admitted four replications for each grinding condition, which was composed in turn by an abrasive configuration together with a kinematic curve. Inspired by results from Hutchings et al. (2004), Hutchings et al. (2005), and also by the experience from previous works, as in Sousa et al. (2010), a variable speed metallographic polishing machine (mark Buehler, model PowerPro 5000) was employed so that both hypotrochoids and the epitrochoids could be available for carrying out the grinding tests. These kinematic curves were presented earlier in Fig. 2. The sample holder of the polishing machine was slightly adapted to support the glass tiles instead of a sand paper. To obtain the three selected kinematic curves with the kinematic parameters presented in Fig. 3a, the following nominal values were taken: R1 = 40 mm, R2 = 52 mm, and R3 = 8 mm for the three inner abrasive pins and R4 = 16 mm for the outer pins. The first rotational level was ω1 = 150.38 rpm for kinematic K1 (epitrochoid), ω1 = −61.13 rpm for kinematic K2 (hypotrochoid), and ω1 = 50.75 rpm for kinematic K3 (epitrochoid). The rotation level ω2 was kept constant at 60.5 rpm. These rotation motions were experimentally measured with an optical revolution counter (Mark Hofman, model Uni-balancer), and all simulations were calculated computing the measured values. The corresponding kinematic curves considering a single abrasive pin were previously exposed in Fig. 1. A contour profilometre mark Hommel (model Waveline 120) was used to characterize the surface of the glass tiles after the
While the profiles of removal depth due to the grinding process could be directly measured by contour profilometry, the determination of the cumulative number of abrasive contact suffered by a given region required the development of a computational algorithm based on Eqs. (1)–(3) previously given. This algorithm was developed in LabVIEW v10 platform. In the algorithm developed, an abrasive contact is said to occur inside a given region of the tile surface every time its instantaneous coordinates (X and Y) cross inside the nominal contact area of any one of the six abrasive pins. Thus, using such computational algorithm, the cumulative number of scratching contacts was calculated for a complete mesh of points over the entire tile surface. As result, maps containing the spatial distribution of grinding intensity, given in terms of total number of abrasive contacts, could be quantitatively determined and presented as theoretical results. At the same time, the scratching angle associated to each single abrasive contact was also determined. The resulting histogram of scratching angles was then registered in terms of both average and standard deviation values, and for the entire tile surface. Colour intensity graphs were conveniently adopted to present the simulated results. In such graphs, each elementary small region on the real surface is directly and univocally associated with a pixel. For instance, the evolution of the grinding process in terms of accumulation of abrasive contacts is simulated and presented in Fig. 5, considering the kinematics K1, configuration, and at three different instants. A time increment of 1.5 ms (dt = 0.0015 s) and a lateral resolution of 1 mm2 for each pixel were admitted in all the simulations. The plane X–Y refers to the tile. Once the distribution of abrasive contacts was obtained, the removal rate for any region under grinding could be calculated simply by dividing the experimental result of removal depth by the corresponding number of abrasive contacts simulated for that region.
Fig. 5. Evolution of the grinding process. Simulation of the cumulative number of abrasive contacts: (a) right after starting, (b) after 0.5 s and after (c) 5.0 s.
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Fig. 6. Kinematics K1 – simulated distribution of abrasive contacts for (a) and (b) Y configurations. (c) Corresponding averaged radial profiles.
5. Theoretical results As a first theoretical result, simulated distributions of abrasive contacts cumulated during the entire grinding process are given in Figs. 6–8, for K1, K2, and K3, respectively. Results are presented
in both radial profiles and 3D-plots. The Z axis indicates the number of abrasive contacts furnished by the simulation algorithm. The 3D-plots for and Y configuration were presented vis-à-vis to provide a direct comparison. Taking profit of the radial symmetry of the grinding experiment, all the results from the same grinding
Fig. 7. Kinematics K2 – simulated distribution of abrasive contacts for (a) and (b) Y configurations. (c) Corresponding averaged radial profiles.
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Fig. 8. Kinematics K3 – simulated distribution of abrasive contacts for (a) and (b) Y configurations. (c) Corresponding averaged radial profiles.
condition were assembled together in order to become represented by a single profile along the radial distance. Further theoretical results are the dispersion regarding the orientation of the successive scratches punctually cumulated during grinding. As previously mentioned, for each pixel this dispersion was quantified by taking the standard deviation of the total population of scratches occurred inside this pixel. The values of were determined by means of Eq. (3). Noteworthy to recall is that the tendency for texture generation and the overlapping degree of successive scratches are higher where the standard deviation is small, as consequence of the higher concentration of scratches pointing towards similar directions. Radial profiles of standard deviation for both and Y configurations are given in Figs. 9–11, considering the kinematics K1, K2 and K3, respectively. To concentrate on the actual difference between geometric configurations, exceptionally in these graphs the simulation of the scratching angle referred to only two abrasive pins for each abrasive tool: one from the outer ring and another from the inner ring, according to the illustrated in the legend. The scratching speed was also determined for each abrasive contact, and no differences were observed between the abrasive
configurations. The mean values determined for the scratching speed were 0.87 m/s for K1, 0.80 m/s for K2 and 0.2 m/s for K3, and the respective minimum and maximum values were 0.50–1.31 m/s, 0.30–1.12 m/s, and 0.14–0.28 m/s. 6. Experimental results The surfaces of the glass tiles after the grinding process are presented in Fig. 12. The tiles are arranged in three lots according to the kinematics. The four lines inside each lot represent the replicated samples, whereas the two columns separate the abrasive configurations and Y. The radial profiles representing the removal depth experimentally measured for each abrasive configuration are presented in Figs. 13–15, considering the kinematic curves K1, K2 and K3, respectively. Graphs for and Y configurations are presented side by side for direct comparisons. Each one of the 36 plotted curves represents the contour of a different radial profile, varying from 10◦ to 360◦ , and calculated taking the average of four profilometry scans collected from each glass tile. As described before, the first and the last 3 mm of each profile were collected from an untouched area of
Fig. 9. Simulated profile of the standard deviation of scratching angle for K1 with and Y configurations.
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Fig. 10. Simulated profile of the standard deviation of scratching angle for K2 with and Y configurations.
Fig. 11. Simulated profile of the standard deviation of scratching angle for K3: (a) and (b) Y configurations.
Fig. 12. Glass tiles after the grinding tests. Resulting patterns for kinematics (a) K1, (b) K2 and (c) K3.
Fig. 13. Radial profiles of removal depth for kinematics K1: (a) and (b) Y configuration.
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Fig. 14. Radial profiles of removal depth for kinematics K2: (a) and (b) Y configuration.
Fig. 15. Radial profiles of removal depth for kinematics K3: (a) and (b) Y configuration.
Table 1 Removed volume for different conditions of kinematics and abrasive configuration. Removed volume (mm3 )
K1 K2 K3
Z-test
Y
Vol. difference
(Z0 = 1.64)
Mean
S.D.
Mean
S.D.
Mean
ZCalc
302.86 629.41 316.08
35.02 71.62 98.34
317.81 549.07 213.46
39.22 46.87 20.26
-14.95 80.34 102.62
0.57 1.88 2.04
the glass surface, so that a reference line for posterior evaluations of material removal could be provided. The total amount of removed volume (in mm3 ) calculated for each combination of kinematic curves and abrasive configurations are given in Table 1. A statistical analysis using Z-test was used in
order to evaluate the effects of abrasive configuration on the total volume removed from the surface, for each kinematics. The difference of removed volume found between Y and configuration and the corresponding comparison parameters (ZCalc ) were also included in the table, admitting a confidence level of 95% (Z0 = 1.64). A difference in volume removal is accepted as statistically significant when the criterion ZCalc > Z0 is satisfied. These values are written in bold. Once the results from experiments and simulations were achieved, further analyses became then available by combining these two kinds of results. In this sense, the ratio between the total removal depth for a given pixel and the corresponding number of abrasive contacts provides an estimative of the material removed per abrasive contact. Such specific removal rates were calculated for each pixel, and the resulting radial profiles are presented in
Fig. 16. Radial profiles of removal rate for both and Y configurations when performing kinematics K1.
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Fig. 17. Radial profiles of removal rate for both and Y configurations when performing kinematics K2.
Figs. 16–18, considering kinematics K1, K2 and K3, respectively, and for both abrasive configurations. The reference level of removal rate indicated in these figures was based on a model for the wear of brittle solids under fixed abrasive conditions proposed by Lawn (1975), discussed in the next section.
7. Discussion Visual inspections after the tests indicate that a stable and homogeneous grinding has taken place in all tests. As expected, the different positioning of the abrasive pins between and Y abrasive configurations caused different distributions of abrasive contacts for all kinematic curves. In a phenomenological point of view, the different abrasive distribution means variations in the degree of overlapping among successive scratches. This effect was probably responsible for the different profiles of material removal experimentally measured and presented in Figs. 13–15. The accumulation of abrasive contacts for the Y configuration started about 4 mm closer to the tile centre than that for the configuration, despite the distances from abrasive pins to tool centre to be the same in both configurations. This is explained focusing on the inner abrasives only. All grinding tests were set to start with the maximum reach of the abrasive pins as indicated in Fig. 19a. During the two rotation motions, the alignment of both axes R1 and R2 repeats periodically, defining the minimum and maximum abrasive reaches. In the Y configuration, the extra reach ı of the inner pins rises due to their alignment with the radius R2 , just like the outer abrasive pins. The inner pins have no influence on the maximum reach, but recalling that R2 > R1 , they overrun the tile centre and affect the minimum reach. As compared in Fig. 19b and c, this extra reach is given by ı = R4 ·cos[60◦ ] = 4 mm.
In average, the value of scratching speed was the highest for K1 (0.87 m/s), successively followed by K2 (0.80 m/s) and K3 (0.20 m/s). The same duration of 1 h was set in all tests and simulations, so that the higher is the level of scratching speed, the longer is the resulting sliding distances of the abrasive pins, along which the abrasive contacts occur. This may explain the minor occurrence of abrasive contacts seen in kinematics K3 when compared to the others. However, the higher concentrations of abrasive contacts invariably observed around the tile centre require further explanations. Besides, such concentration was surprisingly the highest for the kinematics K2 and the lowest for K3. The explanation found was based on the two rotational motions adopted. Within the same kinematics, the abrasive tool is driven towards both periphery (outer ring) and tile centre (inner ring) by the same number of times, regardless the abrasive configuration or the instantaneous scratching speed. Nevertheless, admitting both inner and outer ring to have the same constant width, the area exposed by the inner ring will be a smaller area than the outer ring. Taking into account the constant contact area of the abrasive pins, the accumulation of contacts inside the inner ring tends to be proportionally greater than the outer ring. However, the advantage of the inner ring in cumulating abrasive contacts still depends on the trajectory of the abrasive pin. An example of such dependence is given in Fig. 20, which presents the track left by a single abrasive while performing kinematics K1 (Fig. 20a) and K2 (Fig. 20b). In the latter, the overlapping of the abrasive trajectory was proportionally greater inside the inner ring than inside the outer ring. In this case, after successive passages of the abrasive pins the contact accumulation increases more rapidly around the tile centre than the periphery. On the other hand, the same did not happen for kinematics K1, where both inner and outer rings had similar overlapping degrees, causing the concentration of
Fig. 18. Radial profiles of removal rate for both and Y configurations when performing kinematics K3.
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Fig. 19. Reach distances during grinding: (a) recurrent alignments of the rotation axes and of the inner pins in (b) and (c) Y configurations.
Fig. 20. Proportions of abrasive track and both outer and inner grinding rings for (a) K1 and (b) K2.
abrasive contacts around the tile centre and the peripheral regions to be nearly the same (Fig. 6). In both cases, however, the abrasive pin remains a longer time running along the peripheral ring than in intermediary zones, so that the accumulation of abrasive contact was also favoured in this region. As seen in Fig. 2, the trajectory of kinematics K3 is scarcely coinciding with the peripheral ring, and hence no remarkable contact gaining was observed along this region. Still on the simulated distributions of abrasive contacts cumulated during grinding, all profiles have presented a steady behaviour along the intermediary grinding zones surrounded by few accumulation peaks in both inner and outer rings. Those peaks are explained by the geometrical features adopted for the grinding tests. As seen in Fig. 21, within a distance ε from the peripheral border there is a region exclusively abraded by the same abrasive pin, which is the same that defines the maximal grinding reach, namely (R1 + R2 + R3 + /2) = 110 mm. Analogously, in the vicinity of
the inner ring there is a region inside which the abrasive contacts also receive the contribution of extra abrasive pins transcending the tile centre. Therefore, there is an intermediary range inside which all abrasive pins are equally involved, so that no disturbing effects occur and a homogeneous grinding is expected. As observed in all those graphs, pronounced oscillations took place only outside these limits. The oscillation peaks differ markedly in intensity, but are the same in number and only shifted in their positions. According to the indicated in Fig. 19, this is explained considering that the limits of such regions depend on the position of the inner abrasive pin. For instance, the width ε is 8 mm for the Y configured tool and 12 mm for the · Taking the simplified sketch presented in Fig. 21, the limits of the region homogeneously abraded range from (R2 − R1 + R3 + /2) = 30 mm and (R1 + R2 − R3 − /2) = 74 mm of radial distance. Beyond these limits the number of active abrasive
Fig. 21. Regions reached for the abrasive pins.
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pins becomes inconstant. Towards the peripheral region, the sudden reduction in abrasive contacts right after the peaks indicates the loss of abrasive pins acting on that region. Besides, the peaks itself are probably caused by the longer overlapping between abrasive trajectories and peripheral regions when the abrasive pin approaches its maximum reach, as previously seen in Fig. 20. Regarding the removal caused by each abrasive configuration after grinding, a very similar behaviour was found for both and Y abrasive configurations while performing the same kinematics. Differences were mainly in the maxima values of removal depth rather than in qualitative terms. In all the kinematics tested, the Y configured tool has caused the deepest removals, as revealed in the experimental profiles of surface depth presented in Figs. 13–15. For the kinematic curves K2 and K3, the Z-test indicated that the abrasive tools with Y configuration have removed significantly more material than those with configuration. However, for the curve K1, no significant differences in removed volume were found between these configurations. This may suggest that for the trajectory K1, the difference in scratching interactions introduced by the change of the abrasive configuration was not big enough, in average, to affect the total removal volume. However, an appreciable difference in qualitative terms observed when comparing Fig. 13a and b. The trough on the removal profile using configuration was clearly narrower than that using Y configuration, but also considerably deeper, as if part of the removal was slightly shifted towards higher radial distances. In fact, such effect can be reasonably explained by the different contribution of the inner abrasive pins. As seen in Fig. 19, the abrasive pin in Y configuration reaches ca. 4 mm further than in D configuration (Fig. 19), displacing its contribution from 92 to 96 mm of the radial distance. This range is in total accordance with the removal profiles of Fig. 13. From this it follows that the effect of the abrasive configuration on the removal volume can be either qualitative or quantitative, depending on the macro-kinematics adopted. In case of kinematics K1 the removal rates were the same for both configurations, whereas for kinematics K2 and K3 a significant influence of the abrasive configuration was observed. Such influence was always in favour of abrasive tools using Y configuration, and it occurred for different ranges of scratching speed and rotation direction, since K2 represents a hypotrochoid and both K1 and K3 stand for epitrochoids. When analysing the results from kinematics K3, the importance of the scratching angle becomes evident. The higher predominance of scratching intersections observed along the scratching trajectory may explain the higher removal rates observed. It must be recalled that the only difference between the two abrasive configurations is the positioning of the inner abrasive pins, which in turn changes the resulting pattern of scratching intersections. The strongest effect of the abrasive configuration was detected under the myriad of scratching condition offered by the kinematics K3. The higher removal rates determined for the abrasive tool mounted in Y configuration are in total accordance with the results from the Z-test previously made. However, a comparison between the profiles of (Figs. 9–11) and removal rate (Figs. 16–18) reveals that the degree of overlapping between successive scratches cannot explain alone all the differences obtained on the removal rates. No further explanation could be found for such behaviours. The distribution of scratching angle was also influenced by the macroscopic trajectory of abrasive tool. In the kinematic K1, for instance, the variation of the scratching orientation in time, which is in turn related to the inverse of the instantaneous scratching radius, decreases steadily with the radial distance (Fig. 9). Thus, the dispersion of scratching angle also decreases with the radial distances. The same occurred in kinematics K2 (Fig. 11). Contrasting with these two epitrochoid curves, the scratching radius of K3 was the
smallest along the outer ring, and the greatest close to the inner ring, where high angle intersections were also taking place. As result, the lowest level of for this hypocycloid curve occurred at about the middle of the radial distance (Fig. 11). Focusing hereafter only on the homogeneously abraded region, i.e. from 30 mm to 74 mm of radial distance, a different behaviour of removal rates over the radial distance was obtained for each combination of abrasive configuration and kinematic curve investigated. None of them has presented a constant removal rate along the radial distance, as seen in Figs. 16–18. The deviation of the resulting profiles from a straight line reveals how strong the removal rate is affected by the scratching condition rather than the applied load. Within the same kinematics, the main difference of scratching condition is the degree of intersection between the successive scratches. To contextualize the absolute removal rates obtained, a model developed by Lawn (1975) for the determining the wear volume of brittle solids under fixed abrasive conditions was used as reference. Such model relates both macroscopic and microscopic parameters by the following equation: ˙ = · P · V · (˛ · · H)−1 W
(4)
˙ is the wear rate, P is the macroscopic load of the abrasive where W onto the tile surface, V is the scratching speed, H is the dynamic hardness, is a linear scaling factor and ˛ depends on the geometry of the scratching particle. After multiplying both sides by the same arbitrary period of time t, and also considering a sliding distance D of one pixel (1 mm), the resulting equation is: W = · P · (˛ · · H)−1
(5)
Taking P = 75.4 N, and the same values assumed by Lawn (1975), which are H ≈ 104 N/mm2 (soda-lime glass), ≈ 1 (chip areas directly given by micro-deformation tracks), and ˛ ≈ 1 (conical indenter), the total removed volume is W ≈ 24 × 10−4 mm3 . Finally, a theoretical average value of removal depth h¯ caused by each abrasive contact could be then estimated by: h¯ =
W
D · LW
(6)
where LW is the macroscopic width of the abrasion section, and it can be simply represented by the sum of the diameters of the six abrasive pins (LW = 6 × 4 mm = 24 mm). This leads to a value of removal depth of h¯ = 0.1 m, which is comparable to the range experimentally found. The higher removal rates outside the zone of homogeneous grinding could be explained considering the test procedure. Due to the rigidity of the grinding machine, during the grinding process the abrasive tool is forced to remain parallel to the sample surface. Therefore, regions outside the intermediary ring are forced to be equally removed. Such tendency may lead to temporarily increases in the normal load, so that the removal per abrasive contacts also increases. Knowledge on the micro-kinematics can provide support for several kinds of phenomenological investigations. In this sense, further researches on the scratching conditions, especially on the extension of damages produced underneath the surface by scratches under different conditions, may be recommended so that an optimized grinding process could be achieved. 8. Conclusions Based on the theoretical and experimental results obtained in this work the following conclusions can be drawn about the grinding process of glass:
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- The chosen kinematics has a major influence on the distribution of abrasive contacts over the surface under grinding, and as consequence, on the surface profiles. With favourable kinematic parameters the effect of the abrasive configuration can be either minimized or maximized, either qualitatively or quantitatively. - The effect of the abrasive configuration on the material removal was quantitatively significant for two of the three kinematics tested. The Y configured tool was capable to provide higher removal rates than the configured tool under both epicycloid and hypocycloid curves. The advantage of the Y configured tool was ascribed to the degree of intersection between the successive scratches. - The distribution of cumulative number of abrasive contacts occurred during the entire grinding process can be computationally simulated based on the kinematic equations derived from the experimental configuration. Pure geometric effects, such as the distribution of scratches (grinding uniformity), scratching speed, and especially the degree of overlapping between successive scratches, could be obtained and evaluated. - The greatest volume removal occurred close to both inner and outer peripheral ring, where according to the simulated results, the abrasive contacts were accumulated at most. Further researches on this subjected will be fruitful for the understanding of grinding processes of glassy materials, especially on the extension of damages produced by scratches under different conditions. Acknowledgements This work was carried out with the support of the following entities: Deutsche Forschungsgemeinschaft – DFG, the
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Coordination for the Improvement of High Education Personnel – CAPES, in the scope of the Brazilian–German Collaborative Research Initiative on Manufacturing Technology – BRAGECRIM, and the Collaborative Research Center 926 (SBF): Microscale Morphology of Component Surfaces (MICOS), project A4. References Archard, J.F., 1953. Contact and rubbing of flat surfaces. Journal of Applied Physics 24, 981–988. Belkhir, N., Bouzid, D., Herold, V., 2009. Wear behaviour of the abrasive grains in optical glass polishing. Journal of Materials Processing Technology 209 (20), 6140–6145. Huang, H., Li, Z., Shen, J.Y., Zhu, H.M., Xu, X.P., 2002. Micro-structure detection of a glossy granite surface machined by the grinding process. Journal of Materials Science Technology 129, 403–407. ˜ M.J., Quereda, M.F., Hutchings, I.M., Adachi, K., Xu, Y., Sánchez, E., Ibanez, 2004. Laboratory Simulation of the Industrial Ceramic Tile Polishing Process. Global Forum on Ceramic Tile – Qualicer, Castelón, Spain, pp. 19–30. ˜ Hutchings, I.M., Adachi, K., Xu, Y., Sánchez, E., Ibanez, M.J., Quereda, M.F., 2005. Analysis and laboratory simulation of an industrial polishing process for porcelain ceramic tiles. Journal of the European Ceramic Society 25, 3151–3156. Lawn, B.R., 1975. A model for the wear of brittle solids under fixed abrasive conditions. Wear 33, 369–372. Sousa, F.J.P., Aurich, J.C., Weingaertner, W.L., Alarcon, O.E., 2007. Kinematics of a single abrasive particle during the industrial polishing process of porcelain stoneware tiles. Journal of the European Ceramic Society 10, 3183–3190. Sousa, F.J.P., Aurich, J.C., Alarcon, O.E., Weingaertner, W.L., 2010. Influence of the trajectory of the abrasive pin on the grinding process of glassy ceramics. Journal of Materials Science and Engineering 4, 19–29. Hon-Yuen, T., Cheng, H., 2010. An investigation of the effects of the tool path on the removal of material in polishing. Journal of Materials Processing Technology 5, 807–818.