Numerical simulation and experimental validation of fixed abrasive ...

Int J Adv Manuf Technol (2016) 83:1253–1264 DOI 10.1007/s00170-015-7643-8

ORIGINAL ARTICLE

Numerical simulation and experimental validation of fixed abrasive grinding pad topography Pengfei Liu 1 & Bin Lin 1 & Shuai Yan 1 & Yan Li 1 & Bo Wang 1

Received: 9 March 2015 / Accepted: 22 July 2015 / Published online: 8 August 2015 # Springer-Verlag London 2015

Abstract The topography of fixed abrasive grinding pad has a significant effect on the process of grinding analysis. A new numerical modeling technique has been proposed to generate the grinding pad topography with spherical grains in this paper. The simulation result was given by software. Five fixed abrasive grinding pads with different grain sizes were measured by using a confocal scanning laser microscope. Comparing the results of simulation and the experiment, it can be concluded that the simulated profile of the grinding pad is corresponding well with that of the actual pad. Keywords Numerical modeling . Grinding pad topography simulation . Experiment verification

1 Introduction Extensive studies have been proposed in modeling the topography of a grinding pad. Doman et al. [1] summarized the research of grinding wheel topography models and recommended the future directions for topography modeling. The methods reported in the literature can be classified into empirical methods and analytical methods. In the empirical method, Kolmogorov–Smirnov normality tests were carried out to

* Pengfei Liu [email protected] * Bin Lin [email protected] 1

Key Laboratory of Advanced Ceramics and Machining Technology, Tianjin University, Ministry of Education, Tianjin 300072, China

obtain distribution characteristics of abrasive grains [2]. Suto et al. [3] introduced empirical method into simulating the grinding process. The results of the simulation had a good agreement with experimental results; however, the size of grain was not considered into the simulation. Steffens et al. [4] also simulated the grinding process based on measurements of the wheel surface. The grain shape, distribution, and the wheel’s structure got from the 3D picture were considered in modeling the grinding wheel [5–7]. The advantage of empirical methods is that it is intuitionistic to achieve the data of the topography. While, on the other hand, the disadvantage for empirical method is obvious that the universal usage is limited. Empirical method is on a case-to-case basis; generally, the model established under a particular grinding pad is unusable for another situation and had no numerical model to support it. Besides, measuring the pad surface accurately with an appropriate instrument is another serious problem. Analytical methods can cover the shortages of empirical method. In the early stage, Monte-Carlo method was used [8–11]. It remained a big problem how to determine the cutting edge density function. Cao et al. [12] got the distance between two adjacent particles which could determine the edge length of the mesh. However, the steady particles stayed in the organism in a good line were not meeting the real situation. Zhou et al. [13] focused on the roughness of the workpiece, the sinusoidal model was used in simulating the topography of the pad, but the simulation result was not to be proved by experiments or some other method. Zitt et al. [14] have developed a geometric kinematic model of the grinding process in which three dimensional abrasive grains are created and the interaction of each individual grain is observed and considered.

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There were some other popular approaches adopted by many researchers meshing the grinding wheel surface with simple shape abrasive grains [15–19]. Those studies focused on the fact that the particles which are randomly distributed in the wheel surface with different sizes, shapes, orientations, and distribution density, but did not get the law of mesh. Shortage of the experiment verification was the deficiency of all those analytical methods. The uniformity between the simulation results and the real situation remained unknown. A new numerical simulation methodology was developed to generate the grinding pad surface topography in this paper. The simulation result was verified by experiments. First, the mathematical model of the pad topography was established based on a theory called “body cubic model,” followed by generation of the ground surface topography by calculation with math software. Five different grinding pads with different particle sizes produced by Saint-Gobain were selected as test samples, and the particles are CeO2. The surface topography was measured by using KEYENCE VHX-1000. The theory result was corresponding to the experimental result very well.

Int J Adv Manuf Technol (2016) 83:1253–1264

Fig. 2 Central position of the particles

Factors abovementioned are random for different pads from different process technic. The objective of this model is to obtain a universal method with some given processing technic parameters to ascertain these unknown elements appropriately.

2 Establishment of the mathematical model

2.2 Shape and size distribution of the grains

2.1 Basic framework of the model

Grain shapes on the pad tend to be cubic, cuboctahedron, and spherical. The cubic and cuboctahedron CeO2 grains produced by the regular method can easily scratch the workpiece and make the subsurface damage; the spherical grains produced by detonation technique show a better performance. For the universality of the method, all different type grains are simulated in this paper, but the sample from SaintGobain is spherical. Thus, representing other shapes by the spherical will not lose the generality. In fact, it is reasonable to make the simplification because the grains have a great negative rake when cutting. In the actual production, a particular level of CeO2 powder particle size can be got from the standard sieve’s screening. The particle size can be controlled in a certain range, but the specific size for a single particle is not definite; thus, the central-limit theorem of statistics is used in the prediction of

Generally, there are many uncertain factors, which decided by process technic of the pad, that affect the surface topography. In summary, the macro characteristics of the pad are decided by the following factors: (a) Shape of the grains (b) Height distribution of the exposed grains and the percentage of the projected area (c) Density of the grains on the surface statically and dynamically (d) Position and distance of the grains

Fig. 1 Particle size Gaussian distribution diagram

Fig. 3 Modification of the cubic model


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Fig. 4 Particle random offset schematic diagram

the particle size distribution. Several hypotheses are made as follows: the particles follow Gaussian distributions in which size was controlled by a standard sieve between interval [dg2 min, dgmax], and the expectancy was μg, the variance was σg. Equations 1 and 2 can be got from the property of Gaussian distributions: d gmin þ d gmax 2 d −d gmax gmin σ2g ¼ 6

μg ¼

μg σ2g dgmin dgmax

ð1Þ ð2Þ

Expectancy of the particles’ distributions Variance of the particles’ distributions Minimal size of the particles Maximal size of the particles

The distribution of the particle size could be shown in Fig. 1 [1] Probability density function of the particles can be got as Eq. 3: 1 f d g ¼ pffiffiffiffiffiffi e 2π

2.3 Central position of the particles Grain shape and the size have been solved based on the hypothesis. The position of the grains should be calculated based on the data of last two sections. The central position of the grains should be generated in a coordinate system. The method in this paper was based on cell structure proposed by Chen and Rowe [20]. Firstly, there was a simple cubic unit cell model in which the grains were arranged homogeneously; in other words, there was a single particle in every vertex of the cubic cell built as Fig. 2 [1]: It can be clearly seen that the Sg (length of the cubic edge) is the most important parameter which determines the density of particles. The particular position for one single particle is determined by the parameters in the randomization process, Sg can be got from Eqs. 4, 5, and 6 [1]: ¼ V simp g

1 3 πd 6 g

ð4Þ

3 V simp cell ¼ S g

ð5Þ

ðdg −μg Þ2 2σ2g

;

d gmin ≤ d g ≤ d gmax

f(dg) probability density function of the particles Fig. 5 Particle number schematic diagram

ð3Þ Vg ¼

V simp g V simp cell

3

¼

πd g 6S 3g

ð6Þ

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a

Grid points before randomize b

grig points after randomize c

simulation suface of the pad Fig. 6 Simulation procedure in MATLAB. a Grid points before randomization, b grid points after randomization, c simulation surface of the pad, d cylindrical coordinates sampling


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d

Cylindrical coordinates sampling Fig. 6 (continued)

Sg ¼

rffiffiffiffiffiffiffiffiffi π 3 *μ 6V g g

ð7Þ

2.4 Specific operations in simulation 2.4.1 Calculation of particle number of unit volume

Vg Vsimp cell Vsimp g dg Sg

Volume concentration of the pad Volume of the cell Total volume of the particles Average diameter of the particles Length of the cubic edge

This method is based on the particle number of the unit volume, and it will mesh the simulation space before generating the particles. Based on the uniform grid, particles will be defined on the corner and, after that, will randomize every position of the particle. The advantage of the method is that it can save a large storage space with the relative certainty of every particle’s central position.

Table 1

It can be concluded that particle number of unit volume deriving from the volume concentration of the pad is the foundation of modeling. Vg (volume concentration of the pad) of the sample from Saint-Gobain is about 15.52 %. The maximum Vg is about 52.36 % when Sg =dg, the sample’s Vg is less than the maximum theoretical calculated value. But the model is not universal for some other pad which, for example, the V g is more than 52.36 %. The model has been modified from a simple cubic cell model to body-centered cubic cell model shown in Fig. 3 for the universality. Equation 4 should be modified as the model has been changed.

Basic simulation parameters

No.

1

2

3

4

5

Particle grade

W1.5

W3.5

W7

W14

W28

Particle size range (μm) Abrasive volume fraction Vg (%) Cell edge Sg (μm) Radial grid spacing (μm) Circumference grid spacing (rad) Angle of the sector C (rad) Outer diameter R1 (mm) Inner diameter R2 (mm)

1.0∼1.5

2.5∼3.5

5.0∼7.0

10∼14

2.3620 0.5 5×10−5 0.005 20 5

5.6689 1.0 1×10−4 0.005

11.3378 2.0 2×10−4 0.005

22.6757 2.0 2×10−4 0.005

20∼28 15.52 45.3513 2.0 2×10−4 0.015

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Fig. 7 Part of the pad surface topography

Vg ¼

V cent g V cent cell

3

¼

πd g 3S 3g

1 3 1 3 V cent ¼ 2* πd g ¼ πd g g 6 3 simp 3 V cent cell ¼ V cell ¼ S g

Vcent g Vcent cell

ð8Þ

ð9Þ ð10Þ

Total volume of the particles in the cubic Volume of the cubic

Sg can be deduced as d g¼μg : Sg ¼

rffiffiffiffiffiffiffiffiffi π 3 *μ 3V g g

Fig. 8 Cylindrical coordinate sampling result

ð11Þ

In the polishing process, the surface of the pad is contacted with the workpiece directly, it is reasonable that simulating the surface of the pad instead of simulating the whole pad. The surface of the pad can be treated as a combination of several body-centered cubic cells. The particles on the corner were mutual for the adjacent cells. 2.4.2 Random processing for the particle central position In the real situation, the particles could not stay in the corners uniformly. For the purpose of approaching the real situation, the particles’ central positions need a random migration processing. The offsets for directions X, Y, Z are σx , σy, σz which should follow Gaussian distributions whose mathematical expectation is 0 and variance is S g /6 [1] (Fig. 4).


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Fig. 9 Simulation in different models

In order to describe every particle’s offset, numbering the particle which sequence number is i, j, k in directions X, Y, Z as ijk, and the central position is (xijk, yijk, zijk) named Gijk [1] (Fig. 5). Equation 12 is the arithmetic in simulation. 2

Gi jk

3 2 x 3 G000 þ i⋅S g þ σx xi jk y ¼ 4 yi jk 5 ¼ 4 G000 þ j⋅S g þ σy 5 zi jk Gz000 þ k⋅S g þ σz

xijk, yijk, zijk Gx000, Gy000, Gz000 σx, σy, σz

ð12Þ

Particles’ central positions Coordinate location parameters Offset value

Gx000, Gy000, and Gz000 could control the spatial position of the whole simulation space. Particles on the surface of the pad are Fig. 10 Simulation in different distributions

subjects of the study, and the k in Gijk could be assigned to be 0. Besides, some terms qualify the generation of the particles including the interference between abrasive grains, exposure to the air completely, and submerging in the pad body completely. Because of the particle sequence, we can verify the terms by Eq. 13: 8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 > d gijk þ d gi−1 jk > > xi jk −xi−1 jk þ yi jk −yi−1 jk þ zi jk −zi−1 jk ≥ > > 2 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r > > 2 > 2 2 d gijk þ d gi j−1k > > < xi jk −xi j−1k þ yi jk −yi j−1k þ zi jk −zi j−1k ≥ 2 ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > 2 2 d gijk þ d gi−1 j−1k > > xi jk −xi−1 j−1k þ yi jk −yi−1 j−1k þ zi jk −zi−1 j−1k ≥ > > > 2 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r > 2 > 2 2 d gijk þ d gi−1 jþ1k > > : xi jk −xi−1 jþ1k þ yi jk −yi−1 jþ1k þ zi jk −zi−1 jþ1k ≥ 2 ( d gijk ; ∀zi jk < H H −zi jk < 2 zi jk −H ≤ K*d gijk ; ∀zi jk > H

ð13Þ

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Fig. 11 Simulation in different K

dg H K

Particle diameter Surface height of binder Coefficient of exposure k 0 There are other two kinds of models named uniform distribution model [18] and sinusoidal distribution model [19]. All these models were not suitable for the body-centered cubic cell model. The blade height will not follow the real situation which were Gaussian distributions just like the experiment results if combining the grains which followed the Gaussian distributions and the offsets which followed other distributions, as shown in Fig. 10. The red line which was the Gaussian distribution corresponded to expectation distribution more closely comparing to other distributions. 3.1.3 Coefficient of exposure The coefficient of exposure (K) was a parameter which avoids the grains falling off from the organism. When K=0, the Fig. 17 Specific percentages and tendency

central position of the grain will be on the surface of the organism; in other words, half of the grain will be exposed to the air. When k>0, meaning more than half of the grain will be out there. Figure 11 shows the tendency of different K. The peak of the line was moving when the K was changing from −0.1 to 0.1. The experience point could be determined to be −0.1 because the blue line’s peak was the closest one to the expectation distribution.

4 Experiment verification In order to verify the correctness of this simulation method, five different pads with different particles size were observed under KEYENCE VHX-1000. In order to keep the contextual consistency, W14 pad was being taken for the sample. Figure 12 showed the real surface of the W14 pad, the spherical CeO2 particles were clearly revealed. The pad organism was made of polyurethane, and its color was close to the particles. For the convenience of handling and analyzing the


picture, dye processing was made to the organism with black ink, like Fig. 12. The light could not follow the maximum camera range, so it was dark around the center of the picture. So, part of the picture was captured to be analyzed as shown in Fig. 13. After being processed, the grayscale image and the binary image were gotten, and the grayscale image has more information than the binary image. We can get the proportion of the particles on the surface from the latter and the particle height information from the former (Fig. 14). We could get the percentage of abrasive particles on the surface of the polishing pad from Figs. 15 and 16. The white part could be regarded as particles, and the black part could be the organism. Comparing with the simulation data, the percentages on every size level were very close and the deviation was lower than 5 %. Figure 17 shows the specific data and tendency with the size changing. The percentages were all ranging between 15 and 18 %, being close to the abrasive volume fraction 15.52 %. The simulation results and the real situation both complied with our expectation. The percentages were rising with the particle size getting bigger. On one hand, the smaller particles are easily covered by the organism, so the uncovered percentage will be lower than the bigger particles. On the other hand, the particles could affect the organism around them that the marginal parts will upheave slightly with the particles. Figure 16 which is the grayscale image could give us the detailed height information. Gray level ranges from 0 to 255 and the color changes from black to white as the number gets bigger. Every single gray level could be corresponding to a certain height of the blade. Figure 18 showed the statistical result. Restricted by the microscope’s field of view, the sample size was smaller than the simulation result; even so, the result was still powerful for proving the validity of the simulation method. The red folding line was the statistical result. For the convenience of the comparison, the data was fitted by logarithmic function showed by the black line. Comparing with Fig. 11, the real situation met the simulation result perfectly, and it also revealed a strong Gaussian distribution feature. Besides, the mathematical expectation was almost the same blade height which powerfully demonstrates the method of simulation for the pad is correct.

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a new numerical modeling technique has been proposed to generate the grinding pad topography with spherical grains. In this model, the size of the particles was being considered to be following the Gaussian distribution; the distance between the particles was calculated in body-centered cubic cell model which was modified from a simple cubic cell model; and the central position of the particles was based on the randomized coordinate points which should meet the noninterference principle, no submerged principle and the no dropout principle. The simulation result matches the blade height map, which is based on the numeric statistics of W14 pad’s grayscale map, perfectly. The particle percentage on the pad surface is very close for the simulation result and the real situation, and both of them were close to the abrasive volume fraction. Besides, in the experiment result, the height distribution showed a strong Gaussian distribution feature just like our simulation result and the mathematical expectation was almost the same number. The consistency between the experiment result and the simulation result strongly demonstrated the validity and the practicability of the new numerical modeling technique proposed in this paper. Acknowledgments This work is mainly supported by the National Science and Technology Major Project of the Ministry of Science and Technology of China (Grant No. 2013ZX04001000-207)

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