Modeling and prediction of surface roughness in ceramic grinding

International Journal of Machine Tools & Manufacture 50 (2010) 1065–1076

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Modeling and prediction of surface roughness in ceramic grinding Sanjay Agarwal a,n, P. Venkateswara Rao b a b

Department of Mechanical Engineering, Bundelkhand Institute of Engineering & Technology, Jhansi-284 128, India Department of Mechanical Engineering, Indian Institute of Technology, New Delhi-110 016, India

a r t i c l e in fo

abstract

Article history: Received 25 January 2010 Received in revised form 21 August 2010 Accepted 27 August 2010 Available online 7 September 2010

Surface quality of workpiece during ceramic grinding is an ever-increasing concern in industries now-adays. Every industry cares to produce products with supposedly better surface finish. The importance of the surface finish of a product depends upon its functional requirements. Since surface finish is governed by many factors, its experimental determination is laborious and time consuming. So the establishment of a model for the reliable prediction of surface roughness is still a key issue for ceramic grinding. In this study, a new analytical surface roughness model is developed on the basis of stochastic nature of the grinding process, governed mainly by the random geometry and the random distribution of cutting edges on the wheel surface having random grain protrusion heights. A simple relationship between the surface roughness and the chip thickness was obtained, which was validated by the experimental results of silicon carbide grinding. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Analytical model Surface roughness Ceramic grinding Chip thickness

1. Introduction Over last two decades, interest in grinding of advanced ceramics has grown substantially with the widespread use of ceramic components in many engineering applications. The advantage of ceramics over other materials includes high hardness and strength at elevated temperatures, chemical stability, attractive high temperature wear resistance and low density [1]. Structural ceramics such as silicon nitride, silicon carbide are now being increasingly used in valves, packing (sealing) elements, bearings, pistons, rotors and other applications where close dimensional tolerances and good surface finish are required. However, the benefits mentioned above are accompanied by difficulties associated with machining in general and with grinding in particular because of the high values of hardness and stiffness of the ceramics and very low fracture toughness as compared to metallic materials and alloys. Precision ceramic components require strict adherence to close tolerances and surface finish as the performance and reliability of these components are greatly influenced by the accuracy and surface finish produced during the grinding process. Surface roughness is one of the most important factors in assessing the quality of a ground component. However, there is no comprehensive model that can predict roughness over a wide range of operating conditions; and after many decades of research, this is an area that still relies on the experience and skills of the machine tool operators. The reason stems from the

n

Corresponding author. Tel.: + 91 51 0232 0349; fax: + 91 51 0232 0312. E-mail address: [email protected] (S. Agarwal).

0890-6955/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmachtools.2010.08.009

fact that many variables are affecting the process. Many of these variables are nonlinear, interdependent, or difficult to quantify. Therefore, the models available so far are not fully feasible and experimental investigations can be very exhaustive but with limited applicability [2]. So, an attempt has been made to develop a theoretical model for the prediction of surface roughness for the grinding of silicon carbide with diamond abrasive. Despite various research efforts in ceramic grinding over last two decades, much needs to be established to standardize the theoretical models for the prediction of surface roughness for improving product quality and increasing productivity to reduce machining cost. A ground surface is produced by the action of large number of cutting edges on the surface of the grinding wheel which are randomly distributed all over the wheel surface. The groove produced on the workpiece surface by an individual grain closely reflects the geometry of the grain tip with no side flow of the work material. Thus, it is possible to evaluate the surface roughness from the considerations of the grain tip geometry and its location on the wheel surface under a given set of grinding conditions. The size and location of these cutting edges on the wheel surface are random in nature. Thus, the surface roughness produced during ceramic grinding cannot be predicted in a deterministic manner. Because of this randomness, a probabilistic approach for the evaluation of surface roughness is more appropriate and hence any attempt to estimate surface roughness should be probabilistic in nature. Extensive research has been carried out to predict the surface roughness of the workpiece manufactured by grinding. On the basis of information available in the literature, theoretical methods of surface roughness evaluation can be classified into empirical and analytical methods. In the empirical method,

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Nomenclature A ae E Ft erf J p Ra

shaded area, mm2 wheel depth of cut, mm expected value tangential grinding force, N/mm error function Jacobian probability arithmetic mean surface roughness, mm

surface roughness models are normally developed as a function of kinematic conditions [3]. The empirical model, developed by Suto and Sata [4], relates surface finish to the number of active cutting edges using the experimental data and it has been found to be having a logarithmic relationship. Although empirical models have the advantages that they require minimum efforts to develop and are used in all fields of grinding technology but the inherent problem associated with this method is that the model developed under one grinding condition, cannot be used for surface roughness prediction at other conditions i.e. it can be used for accurate description of the process within the limited range of chosen parameters only. Hence the scope is limited. To overcome the problem, the analytical method of developing models has been tried out to predict surface roughness in ceramic grinding. The analytical models are always preferred to empirical models as these models are deductively derived from fundamental principles. With specific objective in mind, relevant fundamental approach is selected on the basis of process knowledge and experience, and a qualitative model is worked out. Then, an analytical model is established, based on the conformity to fundamental laws, using a mathematical formulation of the qualitative model. Thus, the main advantage of the analytical model is that the results can easily be transferred to other grinding conditions and other grinding processes. Hence, these results can be made applicable to a wide range of process conditions. The analytical surface roughness models have always been characterized by the description of the microstructure of the grinding wheel, in one-dimensional form, taking the grain distance, the width of cutting edge and the grain diameter into account [5] and in two-dimensional form by considering the grain count and the ratio of width of cut to depth of cut [6]. However, these models did not consider the differing height of cutting edges and assumed that the distance between the cutting edges was uniform. Lal and Shaw [7] used similar approach to describe the surface roughness based on chip thickness model. This model is more successful in industry as it does not need the effort of wheel characterization. Tonshoff et al. [8] described the state of art in the modeling and simulation of grinding processes comparing different approaches to modeling. Furthermore, the benefits as well as the limitations of the model applications and simulation were discussed. This work identified one simple basic model where all the parameters such as wheel topography, material properties, etc. were lumped into the empirical constant. Models developed for the grinding process in the surface roughness analysis [5,9,10] assumed an orderly arrangement of the abrasive grains on the grinding wheel. Zhou and Xi [11] used a conventional method to determine the surface roughness based on the model using the mean value of the grain protrusion heights. However, the predicted value of the surface roughness, based on traditional method, was found to be less than the measured value. To overcome this problem, proposed method takes into consideration the random distribution of the grain protrusion heights.

t tm Vs Vw ycl

a b f

s

undeformed chip thickness, mm maximum undeformed chip thickness by new model, mm wheel speed, m/sec table feed rate, m/min centre-line distance, mm width of grinding wheel, mm parameter of probability density function overlap factor standard deviation

Several analytical models, based on stochastic nature of grinding process, were proposed [12–15] to simulate the surface profile generated during grinding. In these models, the abrasive grains on the grinding wheel were taken as a number of small cutting points distributed randomly over the wheel surface. Assuming a particular probability distribution of these random cutting points, output surface profiles were generated for known input surface profile and input grinding conditions. To simulate the relative cutting path of grains, Steffens [16] performed a closed loop simulation, presupposing that thermo-mechanical equilibrium had been established during the grinding process. The input for this simulation program was the quantities like grinding wheel topography, physical quantities of the system, set-up parameters of the machine tool. Simulations could closely reproduce the ground surface using probabilistic analysis; however, the applicability of this program was limited since the simulation program was based on the measurement of microstructure of grinding wheel. This method was time-consuming. Although many analytical models have been developed based on the stochastic nature of the grinding process but Basuray et al. [17] proposed a simple model for evaluating surface roughness in fine grinding based on probabilistic approach. Results of the approximate analysis yielded values that agree reasonably well with the experimental results. However many parameters and properties of materials were lumped into the empirical constants in this analysis. Hecker and Liang [18] developed an analytical model for the prediction of the surface roughness based on the probabilistic undeformed chip thickness model, which was verified using experimental data from cylindrical grinding; however, the geometrical analysis of the grooves left on the surface has been carried out considering the ideal conic shape of grains which may not be true. Experiments conducted by Lal and Shaw [19] with single abrasive grain under fine grinding conditions indicates that the grain tip could be better approximated by circular arc. Therefore, it is evident that the groove produced by an individual grain can be better approximated by an arc of a circle. Based on this concept, Agarwal and Rao [20] developed an analytical model for the prediction of surface roughness in ceramic grinding. This model had been validated by the experimental results of silicon carbide grinding. In most of the models developed so far, the transverse shape of the grooves produced has been assumed to be triangular or circular in shape. More realistic results may be obtained if it can be assumed that the grain tip is to be of parabolic shape. Thus, the groove generated by an individual grain tip would be of parabolic shape. Based on the above assumption, an analytical model [21] has been developed to predict the surface roughness based on probabilistic approach to represent the stochastic nature of the grinding process considering the grooves to be parabolic in shape. This model has enhanced the effectiveness of the existing surface roughness model. The new model proposed for predicting surface roughness during ceramic grinding appears to yield better results as compared to the model developed with groove to be circular in

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roughness incorporating all the grains engaged. Thus, the undeformed chip thickness t can be described by Rayleigh’s probability density function proposed by Younis and Alawi [22]. Therefore, the spectrum of chip thickness generated can be assumed to have the same mathematical distribution. The Rayleigh p.d.f., f(t) is given by ( 2 2 2 ðt=b Þeðt =2b Þ for t Z0 f ðtÞ ¼ ð2Þ 0 for t o0

Grinding Wheel

+ Bond

where b is a parameter that completely defines the probability density function and it depends upon the cutting conditions, microstructure of grinding wheel, the properties of workpiece material, etc. The expected value and a standard deviation of the above function can be expressed as pffiffiffiffiffiffiffiffiffi EðtÞ ¼ ð p=2Þb ð3Þ

T2 X

T1

Grain

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

α x1

T2 T1

sðtÞ ¼ f ð4pÞ=2gb

Fig. 1. Distribution of grain tips in x–y plane.

shape. However this model was developed without considering the overlapping of parabolic shaped grooves. More realistic results could be obtained by incorporating the effect of overlapping of parabolic grooves generated during the grinding operation. In this paper, an analytical model has been envisaged, incorporating the effect of overlapping of parabolic shaped grooves, to evaluate surface roughness from the chip thickness probability density function. The material properties, the wheel microstructure, the kinematic grinding conditions etc. have also been included in the model through chip thickness model. A simple relationship between the surface roughness and the undeformed chip thickness has been found, with the chip thickness as random variable, which can be used as a time efficient solution for the reliable prediction of surface roughness of ground workpieces.

2. Model development A complete description of analytical model for the prediction of surface roughness involves the parameters of abrasive wheel and kinematic conditions for a specified process. The wheel is characterized by its nominal grain size, nominal grain density, distribution of grains, etc. The kinematic conditions include the wheel peripheral speed, table speed, and wheel depth of cut. The table speed and wheel depth of cut are considered as the operating variables. Fig. 1 illustrates the coordinate system defining the random relative positions of two neighboring grain tips T1 and T2. It is assumed that the grain tips are parabolic in shape and are radially oriented with respect to the centre of grinding wheel. Coordinate X defines the random position of a grain tip along the width of the wheel. Grains are randomly positioned on the wheel so that the X generally has a uniform distribution [15]. That is, the probability density function of X is given by 1

for 0 r x r a a where a is the width of the wheel.

ð4Þ

x2 Y

f ðxÞ ¼

1067

It is important to remark that what here is referred to as the undeformed chip thickness t is actually the depth of engagement of each individual active grain which are participating in removing material. This function has a similar shape to the ¨ logarithmic standard distribution suggested by Konig and Lortz [23] to describe the chip thickness distribution. However, the Rayleigh distribution has the advantage of being uniquely defined by only one parameter, that is, b. A schematic diagram showing the interaction of the grain tip to the workpiece is given in Fig. 2. The relative motion of the cutting grains with respect to the workpiece surface generates a removed chip with a curved longitudinal shape, as shown in Fig. 3. This chip has an increasing chip thickness from zero to a maximum value tm with a cross-section determined by the grain geometry. At any transverse section m–m, the profile of a groove generated by any grain is as shown in Fig. 4. Let xi and xi + 1 be the positions of two successive grooves produced by any grain on the ith and (i+ 1)th columns as shown in Fig. 5. As the grains are interacting independently of each other through the workpiece surface, xi and xi + 1 are independent random variables. So, for any section ‘oh’ along the axial direction, a probability density function f(x) will be obtained by replacing a with h in Eq. (1). In order to incorporate the effect of overlapping on the surface roughness, it is necessary to calculate the distance between two neighboring grains. Since it has been assumed that a groove can be overlapped on either side of it, by the succeeding groove, so, as shown in Fig. 5, the centre-to-centre distance, in axial direction, between two successive grains c is given by

c ¼ 9xi þ 1 xi 9

ð5Þ z

Worksurface

Groove traced by grain m Vs

x

Profile of groove

Y

ð1Þ

Further, since the radial positions of grain tips are random, a probability density function is required to describe the surface

m Fig. 2. Schematic view of the workpiece in cartesian coordinate system.

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negative: Z h f ðh1 Þ ¼ f ðh1 ,h2 Þdh2

Width b

Max Undeformed Chip Thickness tm

ð8Þ

h1

Cross Sectional Area Ac

and f ðh1 Þ ¼

Z

h

f ðh1 ,h2 Þdh2

ð9Þ

h1

Since c ¼ 9h19, so the total probability to satisfy this condition will be Pð9h1 9 r cÞ ¼ Pðh1 rcÞ þ Pðh1 r cÞ

lc Cutting Length

So, the probability density function of centre-to-centre distance, in axial direction, will be

Fig. 3. The 3D shape of an undeformed chip.

fc ðcÞ ¼ fh1 ðcÞ þ fh1 ðcÞ

Z Profile of groove generated

ð10Þ

So, the expected value of the centre-to-centre distance in axial direction, E(c) between two successive grains can be obtained by using Eq. (50)A1 as Z h c f ðcÞdc ð11Þ EðcÞ ¼

Work surface

0

t X Fig. 4. Sectional view showing the shape of groove generated.

where f(c) is the probability density function c. Since the groove shape has been assumed to be parabolic, the maximum width of cut, h will be equal to 4t as it has been assumed that cutting depth (uncut chip thickness t) corresponds to the distance from the tip of the parabola to its focus. Substituting the value of f(c) from Eq. (50)A, the expected value E(c), after limit evaluation, will be obtained as EðcÞ ¼ ð4t=3Þ

ð12Þ

Z So, the expected value of centre distance between two interacting grains along the X, will be 4/3 times the groove depth of cut. The surface roughness, Ra, is defined as the arithmetic average of the absolute values of the deviations of the surface profile height from the mean line within the sampling length l. Therefore, the surface roughness Ra can be expressed as Z  1 l  yycl dl ð13Þ Ra ¼ l 0

xi+1 

Xi

h

t

0

X h

Fig. 5. Sectional view showing the axial distance between the successive grooves.

Let h1 ¼ xi + 1  xi and h2 ¼xi. It can also be written as xi ¼ h2 and xi þ 1 ¼ h1 þ h2

ð6Þ

A joint probability density function is required for the calculation of the expected value of c. So after using the above transformation, it can be written as f ðh1 ,h2 Þ ¼ f ðx1 ðh1 ,h2 Þ,x2 ðh1 ,h2 ÞÞ9J9

ð7Þ

where J is the Jacobian determinant. Thus using Eq. (1), the probability density function of h1, for the section ‘0h’, will be given by the following two equations depending on the relative position of the overlapping grooves i.e. whether h1 is positive or

where ycl denotes the distance of the centre line, drawn in such a way that the areas above and below it are equal (Fig. 6). It can also be expressed statistically as Z max 1 y 9yycl 9pðyÞ dy ð14Þ Ra ¼ l ymin where ymax and ymin are the lowest and highest peak height of the surface profile and p(y) is the probability that height of grain has a particular value y. The surface roughness, Ra, can be calculated using probability density function defined in Eq. (2). The complete description of surface generated is very difficult due to the complex behavior of different grains producing grooves because of the random grain– work interaction. Thus, certain assumptions have to be made while predicting the surface roughness. The assumptions are given below: (1) An individual grain has many tiny cutting points in its surface, therefore, for simplicity, the grain tips are approximated as 1

Equation number using superscript A refers to Appendix.

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4/3 t1

1069

4/3 t2

4t1

4t2

t1

upper

t2

ycl

A2

A1

lower

A2 Fig. 6. Profile of grooves generated.

paraboliod in shape, randomly distributed throughout the wheel volume. (2) The profile of the grooves generated is same and completely defined by the depth of engagement or undeformed chip thickness t. (3) Grooves will overlap each other on either side only once, with same types of grooves and at a distance of 4/3 times the undeformed chip thickness. (4) On an average, the expected area of interference of grain tip and workpiece surface is up to focal point of parabola. Under these assumptions, the profile generated by the grain is as shown in Fig. 5. As per definition of surface roughness, the area above and below the centre line must be equal. Hence the total expected area could be written as EfAðtÞg ¼ 0

ð15Þ

The above equation can be represented in terms of the probability density function f(t) as Z 1 AðtÞf ðtÞdt ¼ 0 ð16Þ 0

During the grain–work interaction in the grinding, two types of grooves are generated depending upon their depth of engagement is either less or greater than centre line ycl and it is assumed that overlapping takes place with the same type of grooves. Defining the overlap factor f as the ratio of area lost (due to overlapping), from the area contributing to surface roughness without overlapping, to the area contributing to surface roughness when there is no overlapping. So, for the groove with the depth of engagement less than ycl, the expected value of area A0 1 contributing to roughness, after overlapping, can be expressed as Z ycl Z ycl Z ycl Auðt1 Þf ðtÞdt ¼ Au1 f ðtÞdt ¼ ð1fÞ A1 f ðtÞdt EfAuðt1 Þg ¼ 0

0

0

ð17Þ where A1 is the intercepted area between grain and the centre line contributing to surface roughness (Ra) before overlapping, as shown in Fig. 6. Similarly, for the groove with depth of engagement greater than ycl, it can be expressed as Z 1 Z 1 A00 ðt2 Þf ðtÞdt ¼ ðAupper Alower Þf ðtÞdt EfA00 ðt2 Þg ¼ 200 200 ycl ycl Z 1 ðAupper Alower Þf ðtÞdt ð18Þ ¼ ð1fÞ 2 2 ycl

Alower and Aupper are the areas below and above the centre line 2 2 before overlapping. Substituting the values from Eqs. (17) and

(18), in Eq. (16), equation becomes Z ycl Z 1 A1 f ðtÞdt þ ðAupper Alower Þf ðtÞdt ¼ 0 2 2 0

ð19Þ

ycl

or ÞEðAlower Þg ¼ 0 p1 EðA1 Þ þ p2 fEðAupper 2 2

ð20Þ

where p1 and p2 are the probabilities defined in terms of the chip thickness probability density function f(t) and are given by Z ycl p1 ¼ f ðtÞdt for t oycl ð21Þ 0

p2 ¼

Z

ycl

f ðtÞdt

for

0

t 4ycl

ð22Þ

The expected value of area, contributing to surface roughness Ra, after introducing overlapping, for the groove with depth less than ycl, can be obtained as   8 EðAu1 Þ ¼ ð1fÞ 4ycl Eðt1 Þ Eðt12 Þ ð23Þ 3 Similarly, the expected value of area for the groove with depth greater than the centre line contributing to surface roughness (Ra) will be given as  o 8 8 npffiffiffiffi t2 ðt2 ycl Þ3=2 ð24Þ EðAupper Þ ¼ ð1fÞ 4ycl Eðt2 Þ Eðt22 Þ þ E 200 3 3 o 8 npffiffiffiffi t2 ðt2 ycl Þ3=2 Þ ¼ ð1fÞ E EðAlower 200 3

ð25Þ

where Alower and Aupper are the areas below and above the centre 00 00 2

2

line as shown in Fig. 6. Rewriting Eq. (20) after substituting the expected values from Eqs. (23), (24) and (25) as   p  p1 Eðt12 Þ þp2 Eðt22 Þ 2ycl p1 Eðt1 Þ þ p2 Eðt2 Þ ¼ 2

ð26Þ

Calculation of the expected values in the above equation requires another probability density function for the cases where the chip thickness is smaller and greater than the centre line distance ycl. Therefore, for the grains having depth of engagement lying between 0 and ycl, the probability density function of the chip thickness will be given by the conditional probability density function f1(t) as f ðtÞ f1 ðtÞ ¼ f1 ðt90 r t oycl Þ ¼ R ycl 0 f ðtÞdt

ð27Þ

and for rest of the chip thickness i.e. for the grains lying above ycl, the conditional probability density function f2(t) will be

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3. Chip thickness modeling

given by f ðtÞ f2 ðtÞ ¼ f2 ðt9ycl r t o1Þ ¼ R 1 ycl f ðtÞdt

ð28Þ

Substitute Eqs. (17), (18), (27) and (28) in Eq. (26) to find ycl. After simplification the expression for centre line can be expressed as  2  2 Eðt Þ ð29Þ ycl ¼ 3 EðtÞ Substituting the expected values from Eq., and simplifying the equation, the value of the centre line will be given by rffiffiffiffi 4 2 ycl ¼ b ð30Þ 3 p For the calculation of surface roughness, two types of grooves are considered. Since the contribution of the two types of grooves considered is different, thus, the total expected value of surface roughness can be calculated as EðRa Þ ¼ p1 EðRa1 Þ þ p2 EðRa2 Þ

ð31Þ

where E(Ra1) and E(Ra2) are the expected values of the surface roughness for depth of engagement smaller or greater than ycl and these values can be calculated by the definition of the surface roughness. As per this, the surface roughness can be calculated by adding the area between the profile and the centre line and divide it by the total profile length. Hence from Fig. 5, the values can be written as  Au1 EðRa1 Þ ¼ E ð32Þ 4t1 Aupper þ Alower 00 200 EðRa2 Þ ¼ E 2 4t2

! ð33Þ

Rewriting Eqs. (32) and (33) after substituting the expressions and Alower from Eqs. (23)–(25) as of A1 , Aupper 2 2   2 Eðt1 Þ EðRa1 Þ ¼ ð1fÞ ycl  ð34Þ 3 (    !) 2 4 y 3=2 Eðt2 Þ þ E t2 1 cl EðRa2 Þ ¼ ð1fÞ ycl  3 3 t2

ð35Þ

Substituting the expected values of E(t1), E(t2) and E(t2(1  (ycl/t2))3/2) from Eqs. (58)A, (60)A and (63)A in Eqs. (34) and (35) and then, from Eq. (31), the expected value of surface roughness can be expressed as EðRa Þ ¼ 0:499ð1fÞb

ð36Þ

It can also be expressed in terms of the expected value of chip thickness E(t) by replacing b in terms of E(t) from Eq. (52)A as

b ¼ 0:795EðtÞ Substituting this value in Eq. (36), expression becomes EðRa Þ ¼ 0:396ð1fÞEðtÞ

ð37Þ

Eq. (37) shows a proportional relationship between the surface roughness and the chip thickness expected value under the assumption that the profile of groove generated by an individual grain to be parabolic in shape with overlapping, without plowing and back transferring of material.

The chip-thickness model plays a major role in predicting the surface quality. The chip-thickness models, proposed by Reichenbach et al. [24] and others [25,26] were based on speed ratio, depth of cut, the equivalent diameter of the wheel, etc. But none of these models took the deformation due to the elasticity of the grinding wheel and workpiece system into account. Geometrically, the contact deflections can influence both the surface finish of the workpiece and the accuracy of size of ground components. According to Saini [27], the contact deflection in grinding can be viewed microscopically and macroscopically. Microscopically, a wheel grain is deflected by the normal force exerted on it during grinding and the workpiece is plastically deformed in the grinding zone. Since a grain may have many tiny cutting points [3], such analysis is extremely cumbersome and the physical interpretation is difficult. Macroscopically, the grinding wheel may be considered as a thick circular plate pressed against a curved surface from which the material is ground. Macroscopic approach has been adopted, in the present study because the core material (alminium is the core material) is 94% by volume while the amount of the abrasive layer on the core is only 6% by volume, in the diamondgrinding wheel. Hence the diamond-grinding wheel can be considered as a thick circular plate and the modulus of elasticity of the wheel may be taken as the modulus of elasticity of its core material (alminium), in the ceramic grinding process. Therefore, a new chip-thickness model has been envisaged, based on macroscopic approach, by incorporating the elasticity of the grinding wheel and the workpiece in the existing chip-thickness model. During the process of grinding, the normal forces generated tend to elastically deform the wheel, which results in a decrease in the effective diameter of the wheel in the wheel-work contact zone. This would reduce the effective depth of cut and consequently reduces the maximum chip thickness. Wheels with low modulus of elasticity would have greater deformation than those with a high modulus of elasticity. Hence, the reduction in the maximum chip thickness would be more for wheels with low modulus of elasticity. Similarly, workpiece with higher modulus of elasticity would deflect the grinding wheel more. Hence, the reduction in the chip thickness would be more for grinding workpieces with higher modulus of elasticity. Combining both the above effects, the maximum chip thickness can be expressed as [28] tm p

E1 E2

where E1 is the modulus of elasticity of the wheel and E2 is the modulus of elasticity of the workpiece. The modulus of elasticity of the diamond-grinding wheel (E1) is assumed to be the modulus of elasticity of the core material of the wheel itself. This is because the amount of the abrasive layer on the core is only 4 mm thick (6% by volume) and the core material is of 242 mm diameter (94% by volume) in a grinding wheel of 250 mm diameter. Alminium is the core material used in the diamond-grinding wheels and thus the modulus of elasticity of the wheel is taken as the modulus of elasticity of alminium, which is 70 GPa. Hence E1 is taken as 70 GPa in the present study. The value of modulus of elasticity of the workpiece (E2) is taken as 410 GPa, which is provided by the manufacturer of the silicon carbide workpiece. A well-known equation for estimating the maximum chip thickness in ceramic grinding [3] is as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u  sffiffiffiffiffiffiffi u 4 Vw ae tmax ¼ t Cr Vs deq where r is the chip width-to-thickness ratio, C is the number of active grits per unit area, Vw is the work velocity, Vs is the wheel

S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 50 (2010) 1065–1076

velocity, ae is the work engagement and deq is the equivalent wheel diameter. The value of r is equal to 4, in the present study, as the groove shape is assumed to be a parabolic. The equivalent wheel diameter in surface grinding is the wheel diameter itself. Using a simple geometric relationship, the value of C derived by Xu et al. [29] is as follows: C ¼ 4f =fd2g ð4p=3uÞ2=3 g

length lc from the following formula: 2

Vc ¼ r

tm 3

qffiffiffiffiffiffiffiffiffiffiffi ae deq

ð_lc ¼

qffiffiffiffiffiffiffiffiffiffiffi ae deq Þ

ð43Þ

Therefore, by substituting Eqs. (42) and (43), Eq. (41) can be rewritten as

ð38Þ

where dg is the equivalent spherical diameter of diamond particle, v the volume fraction of diamond in the grinding wheel, and f the fraction of diamond particles that actively cut in grinding. The grinding wheel used in the present study has a density of 100, or in other words, volume fraction v is 0.25 [3]. To obtain the value of C, it is assumed that only one-half of the diamond particles on the wheel surface are actively engaged in cutting [29], or f¼ 0.5. The equivalent spherical diameter of diamond grit (dg) is given [3] as dg ¼ 15:2 M 1

1071

ð39Þ

where M is the mesh size used in the grading sieve. The existing chip-thickness model can be modified, to take into account the effect of deflections of the workpiece and wheel due to the elastic deformation, by incorporating a parameter related to the ratio of elasticities of the workpiece and wheel. The new model for chip thickness can be expressed as  n E1 tm ¼ tm ð40Þ E2 where the exponent (n) is proportionality constant, to take care of the effects of linear and nonlinear deflections of the workpiece and grinding wheel.

4. Evaluation of the exponent (n) In order to use the model effectively for reliable determination of chip thickness, the value of exponent n must be known. Because a well-defined method is lacking, it has to be determined by an approach that follows the fundamental principle, applicable to the grinding process. In this work, the concept of energy balance, i.e. the energy supplied by the grinding wheel is equal to the amount of the energy required to remove the material, has been used. So the value of exponent n obtained in this way can be reliably used for chip thickness determination. The energy required to remove the material is supplied by the grinding wheel. At slow removal rates, the specific cutting energy (Es) is extremely high, but it decreases at faster removal rates tending toward a minimum value for silicon carbide. As the experiments during the present work are conducted at faster material removal rates, the value of minimum specific energy has been taken as 9 J/mm3 [3]. So the energy given by the grinding wheel is equal to the amount of the energy required to remove the material and it can be written as Ft Vs ¼ Specific energy ðEs Þ  volume of material removed=unit time,

Ft Vs ¼ Es ðCbs Vs ÞVc or 2

Ft Vs ¼ 9ðCbs Vs Þr

tm 3

qffiffiffiffiffiffiffiffiffiffiffi ae deq

Substituting the value of r in the above equation, 2

Ft Vs ¼ 12ðCbs Vs Þt m

qffiffiffiffiffiffiffiffiffiffiffi ae deq

or, Ft Vs ¼ 12ðCbs Vs Þ

 n 2 qffiffiffiffiffiffiffiffiffiffiffi E1 tm ae deq E2

ð44Þ

By measuring the tangential force during grinding using six component dynamometer, n can be evaluated. The experiments have been carried out on a horizontal surface grinding machine. The diamond-grinding wheels of ASD 240 R100 B2 have been used in the present experimental study. Silicon carbide with a modulus of elasticity of 410 GPa was ground at a speed of 2200 m/min without cutting fluid. The feed and depth of cut are varied during experimentation. The average value of exponent (n) was found to be 0.614 and the results are shown in Table 1. The new chip thickness model can thus be written as tm ¼



E1 E2

0:614

tm

ð45Þ

The elastic properties of both the wheel and workpiece cause a considerable deflection in the grinding wheel, resulting in a reduction in its effective diameter and thereby decreasing the actual depth of cut. In the present work, low modulus of elasticity of the wheel and high modulus of elasticity of the work material caused the wheel to deflect more and consequently resulted in a significant reduction in the maximum chip thickness estimated by the new model compared with that of the existing model, as observed in the results shown in Table 2. These results strengthen the representation of deflections of the work and wheel in terms of its elastic properties. Table 1 Exponent (n) at various feeds and depths of cut.

ð41Þ where Ft is the tangential force on the grinding wheel (N). Volume of the material removed=unit time ¼ no: of chips produced=unit time  Volume of each chip ð42Þ ¼ ðCbs Vs ÞVc where Vc is the volume of each undeformed chip produced and bs is the grinding wheel width. Assuming a chip with parabolic cross-section, Vc can be approximated as one half times the 2 product of the maximum cross-sectional area ð2rtm =3Þ and the

Vw (m/min)

ae (mm)

Ft (N/mm)

n

5 10 15 5 10 15 5 10 15 naverage ¼ 0.614

5 5 5 10 10 10 15 15 15

6.14 10.43 11.23 7.48 10.67 13.13 9.1 11.75 15.09

0.397 0.449 0.554 0.555 0.661 0.724 0.620 0.759 0.808

S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 50 (2010) 1065–1076

Table 2 Undeformed chip thickness by existing and new chip-thickness model. ae (mm)

Vw (m/min)

tm (mm)

t m (mm)

5 5 5 10 10 10 15 15 15

5 10 15 5 10 15 5 10 15

2.299 3.251 3.981 2.734 3.866 4.734 3.025 4.279 5.240

0.776 1.098 1.344 0.923 1.305 1.599 1.021 1.445 1.769

Table 3 Experimental values of surface roughness at different values of kinematic parameters. Exp. no.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

ae (mm)

5 5 5 5 5 15 15 15 15 15 25 25 25 25 25 35 35 35 35 35 45 45 45 45 45

Vs/Vw

440 293 220 176 146 440 293 220 176 146 440 293 220 176 146 440 293 220 176 146 440 293 220 176 146

Ra (mm)

Average value of Ra

1

2

3

4

5

0.161 0.229 0.268 0.310 0.328 0.201 0.215 0.303 0.336 0.363 0.209 0.271 0.331 0.372 0.402 0.259 0.321 0.355 0.396 0.429 0.314 0.357 0.397 0.432 0.497

0.166 0.227 0.269 0.311 0.329 0.208 0.248 0.299 0.338 0.362 0.215 0.271 0.332 0.367 0.393 0.249 0.326 0.355 0.391 0.424 0.305 0.352 0.400 0.431 0.489

0.162 0.220 0.271 0.315 0.327 0.207 0.246 0.292 0.340 0.371 0.210 0.270 0.330 0.366 0.399 0.249 0.330 0.348 0.398 0.429 0.310 0.345 0.395 0.433 0.499

0.168 0.228 0.273 0.310 0.330 0.209 0.245 0.293 0.340 0.369 0.215 0.270 0.333 0.375 0.405 0.252 0.322 0.347 0.387 0.425 0.316 0.356 0.389 0.435 0.491

0.163 0.213 0.274 0.309 0.331 0.205 0.246 0.298 0.341 0.370 0.216 0.273 0.329 0.365 0.401 0.256 0.321 0.350 0.398 0.423 0.315 0.355 0.399 0.429 0.489

0.164 0.229 0.271 0.311 0.329 0.206 0.244 0.297 0.339 0.367 0.213 0.271 0.331 0.369 0.400 0.253 0.324 0.351 0.394 0.426 0.312 0.353 0.396 0.432 0.493

parameters for each experiment are depth of cut ae and the speed ratio (Vs/Vw) where Vw is the feed rate and Vs is the wheel speed, along with the experimental value of surface roughness as shown in Table 3. Surface roughness measurements were made using Talysurf-VI (cut-off length was 0.8 mm) at five different places on the 20  5 mm2 cross-section of the workpiece after grinding and the arithmetic mean of the values of the measurements has been reported in the experimental results as shown in Table 3. The experiments are replicated five times (as shown in Table 3) to mask the variability of the process. The resolution of surface roughness-measuring instrument is 0.8 nm. This means that the 0.8 nm is the minimum value of surface roughness that can be measured by the surface roughness-measuring instrument. However the differences in the readings of surface roughness (Table 3) are much higher than the resolution of surface roughnessmeasuring instrument. So this instrument will be able to distinguish between the values clearly and hence measurements made by this instrument could be considered accurate enough for the present study. Apart from this, there are sources of error in any measuring system. The term ‘precision’ is often used in this connection. Perfect precision means that the measurements will be made with no random variability in the measured values or standard deviation of the measuring system is zero. So, in order to

0.4 Speed ratio 100 0.35 Surface roughness (μm)

1072

0.3

Speed ratio 220

0.25 0.2

Speed ratio 550

0.15 0.1 0.05 0 0

30 20 Depth of cut (μm)

10

5. Evaluation of overlap factor /

40

50

Fig. 7. Surface roughness vs. depth of cut at three different speed ratios.

0.5 Wheel 2 (ASD240R100B2)

0.4 Surface roughness (μm)

A comprehensive model for the quantitative prediction of surface roughness has been presented, as given by Eq. (37). In order to use the model effectively for the reliable prediction of surface roughness, the value of overlap factor f must be known. Due to lack of a well-defined measuring method of overlap factor, it has to be determined experimentally. So, a series of experiments were performed by grinding silicon carbide workpiece by diamond-grinding wheel. An ‘ELLIOTT 8-18’ hydraulic surfacegrinding machine was used to grind the sintered silicon carbide pieces with diamond-grinding wheels. The properties of SiC workpiece material used for experimentation in this work are: density¼3.17 gm/cm3, hardness (HV)¼2700 kg/mm2, fracture toughness (KlC)¼4.55 MPa m1/2, modulus of elasticity¼410 GPa and thermal conductivity¼145 W m  1 K  1. The workpiece material was supplied by H.C. Starck Ceramics GmbH & Co. KG, Germany. The tool was diamond-grinding wheel (ASD240R100 B2) (Norton make) with modulus of elasticity of 70 GPa and alminium as core material. The size of the workpiece is 20 mm  20 mm  5 mm. The other conditions taken for the experimentation were as follows: wheel speed ¼36.6 m/s, wheel diameter¼250 mm, wheel width¼19 mm. The main kinematic

0.3

0.2

Wheel 1 (ASD500R100B2)

0.1

0 0

5

10

15

20 25 30 Depth of cut (μm)

35

40

45

Fig. 8. Surface roughness vs. depth of cut for two wheel conditions.

50

S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 50 (2010) 1065–1076

measure the reproducibility of the measurements, standard deviations were calculated for each set of measurements of the surface roughness. It was found that some values of surface roughness are inside one standard deviation from the experimental mean value while some other values need two standard deviations to fit the data. This means that the measured values of surface roughness (Table 3) have good degree of precision.

The expected value of chip thickness was calculated, for each experiment, by making use of new chip thickness model (Eq. (45)), after substituting all the parameters, as shown in Table 3. The factor for overlapping f was calculated, with the help of expected value of chip thickness and surface roughness value obtained experimentally (Table 3), using Eq. (37) and it was found to be approximately 9.6%.

surface roughness (μm)

surface roughness (μm)

0.45 Vw = 5m/min

0.35 0.25 0.15 0.05

10

20 30 40 depth of cut (μm)

Vw = 7.5m/min 0.4 0.3 0.2 0.1 10

20 30 40 depth of cut (μm)

surface roughness (μm)

surface roughness (μm)

0.45 0.35 0.25 0.15 50

0.6 0.5 0.4 0.3 0.2 20 30 depth of cut (μm)

40

Vw = 15m/min 0.55 0.45 0.35 0.25 10

20 30 40 depth of cut (μm)

Ra, circular groove

16

10 13 feed (m/min)

16

10 13 feed (m/min)

16

10 13 feed (m/min)

16

10 13 feed (m/min)

16

0.5 d = 10μm 0.4 0.3 0.2 0.1 7

0.5 d = 15μm 0.4 0.3 0.2 0.1 7

50

Ra, parabolic grrove

d = 30μm 0.5 0.4 0.3

4

0.65

0

10 13 feed (m/mi)

0.2

50

surface roughness (μm)

surface roughness (μm)

10

7

0.6

Vw = 12.5m/min

0

0.05

4

surface roughness (μm)

surface roughness (μm)

20 30 40 depth of cut (μm)

0.15

4

Vw = 10m/min

10

0.25

50

0.55

0

d = 5μm 0.35

4

0.5

0

0.45

50 surface roughness (μm)

surface roughness (μm)

0

1073

7

0.65 d = 45μm 0.55 0.45 0.35 0.25 4

7

Ra, groove overlap

Ra, experimental

Fig. 9. Surface roughness presented at various values of depth of cut and feed.

1074

S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 50 (2010) 1065–1076

6. Surface roughness prediction The present work focuses on the prediction of surface roughness based on the chip thickness model. The chip thickness models play a major role in predicting the surface quality. The chip thickness is a variable often used to describe the quality of ground surfaces as well as to evaluate the competitiveness of the overall grinding system. The various parameters of the grinding process have been included in the model through chip thickness model. Therefore, this model can be used to predict surface roughness under different conditions of the parameters. The depth of cut and speed ratio are the two important parameters that can be varied on the grinding machine. Fig. 7 shows the arithmetic mean value of surface roughness for various values of depth of cut for three different speed ratios. It can be observed that at higher speed ratio, the surface finish is better. This is because, at higher speed ratio, i.e. at lower workpiece velocity, more grains will be involved in removing a given volume of material, and thus depth of engagement will be low. Hence the surface finish is better. Similarly, the wheel microstructure plays a major role in the quality of the ground surfaces. Wheel 1 is made up of fine abrasive with average grit size of about 32 mm and wheel 2 is a coarser wheel with an average abrasive size of about 67 mm. It can be observed from Fig. 8, the coarser wheel produces rougher surface as compared to that of fine wheel. Although a finer wheel produces better surface finish but it will cause higher forces and higher power due to the higher specific energy governed by a smaller expected value of the chip thickness. This is a well-established fact, which reinforces the correctness of the model.

7. Comparison with the existing models The centre line average value of surface roughness (Ra) has been compared with results, obtained experimentally and from the existing surface roughness models [20,21] and new surface roughness model {Eq. (37)}. The deviation of the surface roughness calculated with the new model, from the existing model and experimental values, for various values of feed and depth of cut, is shown in Fig. 9. It could be seen from this figure that the surface roughness increased with an increase in depth of cut and feed. This established behavior could be explained by observing the variation of maximum chip thickness with the grinding parameters. Increase in depth of cut causes the maximum chip thickness to increase and thereby resulting in a poor surface quality. It could also be seen from Fig. 9 that the surface roughness decreased with decrease in feed. This is as expected since the depth of engagement would be low at low feed rate and hence the reduction in surface roughness could be observed with the decrease in feed rate. Also at higher speed ratios the surface produced is smoother. This is because at lower workpiece velocity (as wheel velocity is fixed in the present study), more grains participate in removing a given volume of material; hence the depth of engagement is lower, producing smooth surfaces. Further it has been observed from Fig. 9 that the predicted surface roughness shows a good agreement with the experimental data obtained from different grinding conditions in surface grinding. Apart from this, the surface roughness values computed by new surface roughness model are closer, to actual values obtained experimentally, as compared to that of the existing models and thus predicting the performance of the process more accurately.

is characterized by the undeformed chip thickness, has been developed. The wheel microstructure, the kinematic and dynamic grinding conditions, and the material properties were included in the model through undeformed chip thickness model. The model incorporates the overlapping effect of grooves left by the grains, apart from other grinding parameters. By incorporating the overlapping effect, the model has been made more realistic, not only to estimate the surface roughness more precisely, but also to make the ceramic grinding reproducible. The model is capable of handling a wide variety of work and wheel speeds and is flexible enough to incorporate the effects of other parameters. Hence the new model can be reliably used to predict the surface roughness in the surface grinding of silicon carbide ceramics.

Appendix A. Mathematical calculation of expected values A.1. Calculation of axial distance between the two successive overlapping grooves, c (Eq. (12)) The calculation of expected value of axial distance requires the joint probability density function of the transformation Eq. (6), which can be written as f ðh1 ,h2 Þ ¼ f ðx1 ðh1 ,h2 Þ,x2 ðh1 ,h2 ÞÞ9J9

ð46Þ

where J is the Jacobian. Substituting the value of probability density function for x1 and x2 from Eq. (1) in the above equation, the value of above function becomes f ðh1 ,h2 Þ ¼

1 1 1 1¼ 2 h h h

ð47Þ

Thus using Eq. (47), the probability density function of h1 for the selected length ‘0h’ as shown in Fig. 5, can be obtained as the sum of following two possibilities (whether h1 is positive or negative) as  Z h hh1 f ðh1 ,h2 Þdh2 ¼ f ðh1 Þ ¼ ð48Þ h2 h1 f ðh1 Þ ¼

Z

h

f ðh1 ,h2 Þdh2 ¼

h1



hþ h1 h2

ð49Þ

Substituting the values from Eqs. (48) and (49), in Eq. (10), fc(c) becomes  hc ð50Þ fc ðcÞ ¼ 2 h2

A.2. Calculation of centre line position ycl (Eq. (29)) The calculation of the centre line value, ycl, Eq. (29) requires the calculation of the value of the undeformed chip thickness t and square of the undeformed chip thickness t2. These expected values can be calculated as 2 0 131 rffiffiffiffi Z 1 2 p t 2 6 B 7 EðtÞ ¼ tf ðtÞdt ¼ 4teðt =2b Þ þ b erf @qffiffiffiffiffiffiffiffiC ð51Þ A5 2 2 0 2b 0

8. Conclusion In this paper, an analytical model for surface roughness prediction of ground ceramics, based on the analysis of the grooves left by the grains that interact with the workpiece, which

That gives the value of E(t) as rffiffiffiffi EðtÞ ¼

p 2

b ¼ 1:257b

ð52Þ

S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 50 (2010) 1065–1076

Similarly, Z 1 Z t 2 f ðtÞdt ¼ Eðt 2 Þ ¼ 0

1 0

t

3

b2

eðt

2

2

=2b Þ

dt

ð53Þ

1075

can be calculated as Z   1 1 E Fðt2 Þ ¼ FðtÞf ðtÞdt p2 ycl

ð61Þ

where It will give the value as ðt2 =2b2 Þ

2

Eðt Þ ¼ ½e

2

ð2b

t2 Þ1 0

¼ 2b

2

ð54Þ

where b is a parameter that completely defines the probability density function as in Eq. (2). A.3. Calculation of surface roughness, RA (Eqs. (34) and (35) The probability that an undeformed chip thickness value t is smaller than the centre line value, ycl, can be calculated as Z ycl 2 2 p1 ¼ f ðtÞdt ¼ 1eðycl =2b Þ ð55Þ

Fðt2 Þ ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffi y2 1 2cl t2

After transformation of Eq. (61) in terms of z, which is a function of t, the equation becomes Z 1 EfFðt2 Þg ¼ ez f ðzÞdz ð62Þ 0

where z¼

0

t2 2b

2

and Thus, as per definition of probability density function, the probability of an undeformed chip thickness to be greater than centre line value, ycl, will be 2

2

p2 ¼ 1p1 ¼ eðycl =2b

Þ

f ðzÞ ¼ eð8=3

pffiffiffiffiffiffiffiffiffiffiffiffi2

3=2 pffiffiffiffiffiffi ðpffiffiffiffiffi 2zð4=3Þ ð2=pÞ Þ2 1 z=pÞ pffiffiffiffiffi pffiffiffiffiffiffiÞ 1 ð1 þð3=4Þ pz 2z

ð56Þ Solving the above integration ez f ðzÞdz ¼ 0:3875. Therefore, the expected value is  ! ycl 3=2 E t2 1 ¼ 0:3875b t2

R1 The expected value of chip thickness smaller than ycl can be calculated by using the conditional probability density function (Eq. (27)), giving the expected value in this region as Z 1 ycl Eðt1 Þ ¼ tf ðtÞdt ð57Þ p1 0 After solving the above integration and limit evaluation, it can be expressed as 2 0 13 rffiffiffiffi 1 p 6 B ycl C7 ðy2cl =2b2 Þ ycl e þ b erf @qffiffiffiffiffiffiffiffiA5 Eðt1 Þ ¼ 2 4 2 2 2 1eðycl =2b Þ 2b After simplification, it can be written as Eðt1 Þ ¼ 0:713b

ð58Þ

In the same way, the expected value of chip thickness greater than centre line value, ycl, can be calculated as Z 1 1 Eðt2 Þ ¼ tf ðtÞdt ð59Þ p2 ycl Using Eq. (28), the above integration after limit evaluation can be expressed as 2 0 13 rffiffiffiffi rffiffiffiffi 1 p 6 p B ycl C7 ðy2cl =2b2 Þ ycl  Eðt2 Þ ¼ þe erf @qffiffiffiffiffiffiffiffiA5 2 4 2 2 2 2 eðycl =2b Þ 2b or Eðt2 Þ ¼ 1:72b

ð60Þ

One more expected value is required to be calculated to compute the surface roughness for the chips whose chip thickness value is more than ycl, as given by Eq. (35). Expected value,  ! y 3=2 E t2 1 cl , t2

gives

the

value

as

0:

ð63Þ

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