Modeling and prediction of surface roughness in ceramic grinding

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

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International Journal of Machine Tools & Manufacture

0890-69

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijmactool

Modeling and prediction of surface roughness in ceramic grinding

Sanjay Agarwal a,n, P. Venkateswara Rao b

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

a r t i c l e i n f o

Article history:

Received 25 January 2010

Received in revised form

21 August 2010

Accepted 27 August 2010Available online 7 September 2010

Keywords:

Analytical model

Surface roughness

Ceramic grinding

Chip thickness

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016/j.ijmachtools.2010.08.009

esponding author. Tel.: +91 51 0232 0349; fa

ail address: [email protected] (S. Ag

a b s t r a c t

Surface quality of workpiece during ceramic grinding is an ever-increasing concern in industries now-a-

days. 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.

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1. Introduction

Over last two decades, interest in grinding of advancedceramics has grown substantially with the widespread use ofceramic components in many engineering applications. Theadvantage of ceramics over other materials includes high hard-ness 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 nowbeing increasingly used in valves, packing (sealing) elements,bearings, pistons, rotors and other applications where closedimensional tolerances and good surface finish are required.However, the benefits mentioned above are accompanied bydifficulties associated with machining in general and withgrinding in particular because of the high values of hardnessand stiffness of the ceramics and very low fracture toughness ascompared to metallic materials and alloys. Precision ceramiccomponents require strict adherence to close tolerances andsurface finish as the performance and reliability of thesecomponents are greatly influenced by the accuracy and surfacefinish produced during the grinding process.

Surface roughness is one of the most important factors inassessing the quality of a ground component. However, there is nocomprehensive model that can predict roughness over a widerange of operating conditions; and after many decades ofresearch, this is an area that still relies on the experience andskills of the machine tool operators. The reason stems from the

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arwal).

fact that many variables are affecting the process. Many of thesevariables are nonlinear, interdependent, or difficult to quantify.Therefore, the models available so far are not fully feasible andexperimental investigations can be very exhaustive but withlimited applicability [2]. So, an attempt has been made to developa theoretical model for the prediction of surface roughness for thegrinding of silicon carbide with diamond abrasive.

Despite various research efforts in ceramic grinding over lasttwo decades, much needs to be established to standardize thetheoretical models for the prediction of surface roughness forimproving product quality and increasing productivity to reducemachining cost. A ground surface is produced by the action oflarge number of cutting edges on the surface of the grindingwheel which are randomly distributed all over the wheel surface.The groove produced on the workpiece surface by an individualgrain closely reflects the geometry of the grain tip with no sideflow of the work material. Thus, it is possible to evaluate thesurface roughness from the considerations of the grain tipgeometry and its location on the wheel surface under a givenset of grinding conditions. The size and location of these cuttingedges on the wheel surface are random in nature. Thus, thesurface roughness produced during ceramic grinding cannot bepredicted in a deterministic manner. Because of this randomness,a probabilistic approach for the evaluation of surface roughness ismore appropriate and hence any attempt to estimate surfaceroughness should be probabilistic in nature.

Extensive research has been carried out to predict the surfaceroughness of the workpiece manufactured by grinding. On thebasis of information available in the literature, theoreticalmethods of surface roughness evaluation can be classified intoempirical and analytical methods. In the empirical method,

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Nomenclature

A shaded area, mm2

ae wheel depth of cut, mmE expected valueFt tangential grinding force, N/mmerf error functionJ Jacobianp probabilityRa arithmetic mean surface roughness, mm

t undeformed chip thickness, mmtm maximum undeformed chip thickness by new model,

mmVs wheel speed, m/secVw table feed rate, m/minycl centre-line distance, mma width of grinding wheel, mmb parameter of probability density functionf overlap factors standard deviation

S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 50 (2010) 1065–10761066

surface roughness models are normally developed as a function ofkinematic conditions [3]. The empirical model, developed by Sutoand Sata [4], relates surface finish to the number of active cuttingedges using the experimental data and it has been found to behaving a logarithmic relationship. Although empirical modelshave the advantages that they require minimum efforts todevelop and are used in all fields of grinding technology but theinherent problem associated with this method is that the modeldeveloped under one grinding condition, cannot be used forsurface roughness prediction at other conditions i.e. it can be usedfor accurate description of the process within the limited range ofchosen parameters only. Hence the scope is limited.

To overcome the problem, the analytical method of developingmodels has been tried out to predict surface roughness in ceramicgrinding. The analytical models are always preferred to empiricalmodels as these models are deductively derived from funda-mental principles. With specific objective in mind, relevantfundamental approach is selected on the basis of process knowl-edge and experience, and a qualitative model is worked out. Then,an analytical model is established, based on the conformity tofundamental laws, using a mathematical formulation of thequalitative model. Thus, the main advantage of the analyticalmodel is that the results can easily be transferred to othergrinding conditions and other grinding processes. Hence, theseresults can be made applicable to a wide range of processconditions. The analytical surface roughness models have alwaysbeen characterized by the description of the microstructure of thegrinding wheel, in one-dimensional form, taking the graindistance, the width of cutting edge and the grain diameter intoaccount [5] and in two-dimensional form by considering the graincount and the ratio of width of cut to depth of cut [6]. However,these models did not consider the differing height of cutting edgesand assumed that the distance between the cutting edges wasuniform. Lal and Shaw [7] used similar approach to describe thesurface roughness based on chip thickness model. This model ismore successful in industry as it does not need the effort of wheelcharacterization. Tonshoff et al. [8] described the state of art inthe modeling and simulation of grinding processes comparingdifferent approaches to modeling. Furthermore, the benefits aswell as the limitations of the model applications and simulationwere discussed. This work identified one simple basic modelwhere all the parameters such as wheel topography, materialproperties, etc. were lumped into the empirical constant. Modelsdeveloped for the grinding process in the surface roughnessanalysis [5,9,10] assumed an orderly arrangement of the abrasivegrains on the grinding wheel. Zhou and Xi [11] used aconventional method to determine the surface roughness basedon the model using the mean value of the grain protrusionheights. However, the predicted value of the surface roughness,based on traditional method, was found to be less than themeasured value. To overcome this problem, proposed methodtakes into consideration the random distribution of the grainprotrusion heights.

Several analytical models, based on stochastic nature ofgrinding process, were proposed [12–15] to simulate the surfaceprofile generated during grinding. In these models, the abrasivegrains on the grinding wheel were taken as a number of smallcutting points distributed randomly over the wheel surface.Assuming a particular probability distribution of these randomcutting points, output surface profiles were generated for knowninput surface profile and input grinding conditions. To simulatethe relative cutting path of grains, Steffens [16] performed aclosed loop simulation, presupposing that thermo-mechanicalequilibrium had been established during the grinding process. Theinput for this simulation program was the quantities like grindingwheel topography, physical quantities of the system, set-upparameters of the machine tool. Simulations could closelyreproduce the ground surface using probabilistic analysis; how-ever, the applicability of this program was limited since thesimulation program was based on the measurement of micro-structure of grinding wheel. This method was time-consuming.Although many analytical models have been developed based onthe stochastic nature of the grinding process but Basuray et al.[17] proposed a simple model for evaluating surface roughness infine grinding based on probabilistic approach. Results of theapproximate analysis yielded values that agree reasonably wellwith the experimental results. However many parameters andproperties of materials were lumped into the empirical constantsin this analysis. Hecker and Liang [18] developed an analyticalmodel for the prediction of the surface roughness based on theprobabilistic undeformed chip thickness model, which wasverified using experimental data from cylindrical grinding;however, the geometrical analysis of the grooves left on thesurface has been carried out considering the ideal conic shape ofgrains which may not be true. Experiments conducted by Lal andShaw [19] with single abrasive grain under fine grindingconditions indicates that the grain tip could be better approxi-mated by circular arc. Therefore, it is evident that the grooveproduced by an individual grain can be better approximated by anarc of a circle. Based on this concept, Agarwal and Rao [20]developed an analytical model for the prediction of surfaceroughness in ceramic grinding. This model had been validated bythe experimental results of silicon carbide grinding.

In most of the models developed so far, the transverse shape ofthe grooves produced has been assumed to be triangular orcircular in shape. More realistic results may be obtained if it canbe assumed that the grain tip is to be of parabolic shape. Thus, thegroove generated by an individual grain tip would be of parabolicshape. Based on the above assumption, an analytical model [21]has been developed to predict the surface roughness based onprobabilistic approach to represent the stochastic nature of thegrinding process considering the grooves to be parabolic in shape.This model has enhanced the effectiveness of the existing surfaceroughness model. The new model proposed for predicting surfaceroughness during ceramic grinding appears to yield better resultsas compared to the model developed with groove to be circular in

Y

x2

+

Grain

Bond

T1

T2

Grinding Wheel

x1 T1

X

αT2

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

S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 50 (2010) 1065–1076 1067

shape. However this model was developed without consideringthe overlapping of parabolic shaped grooves. More realistic resultscould be obtained by incorporating the effect of overlapping ofparabolic grooves generated during the grinding operation.

In this paper, an analytical model has been envisaged,incorporating the effect of overlapping of parabolic shapedgrooves, to evaluate surface roughness from the chip thicknessprobability density function. The material properties, the wheelmicrostructure, the kinematic grinding conditions etc. have alsobeen included in the model through chip thickness model. Asimple relationship between the surface roughness and theundeformed chip thickness has been found, with the chipthickness as random variable, which can be used as a timeefficient solution for the reliable prediction of surface roughnessof ground workpieces.

Vs

Worksurface

m

z

m

x

Y

Groove traced by grain

Profile of groove

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

2. Model development

A complete description of analytical model for the predictionof surface roughness involves the parameters of abrasive wheeland kinematic conditions for a specified process. The wheel ischaracterized by its nominal grain size, nominal grain density,distribution of grains, etc. The kinematic conditions include thewheel peripheral speed, table speed, and wheel depth of cut. Thetable speed and wheel depth of cut are considered as theoperating variables. Fig. 1 illustrates the coordinate systemdefining the random relative positions of two neighboring graintips T1 and T2. It is assumed that the grain tips are parabolic inshape and are radially oriented with respect to the centre ofgrinding wheel.

Coordinate X defines the random position of a grain tip alongthe width of the wheel. Grains are randomly positioned on thewheel so that the X generally has a uniform distribution [15]. Thatis, the probability density function of X is given by

f ðxÞ ¼1

a for 0rxra ð1Þ

where a is the width of the wheel.Further, since the radial positions of grain tips are random, a

probability density function is required to describe the surface

roughness incorporating all the grains engaged. Thus, theundeformed chip thickness t can be described by Rayleigh’sprobability density function proposed by Younis and Alawi [22].Therefore, the spectrum of chip thickness generated can beassumed to have the same mathematical distribution. TheRayleigh p.d.f., f(t) is given by

f ðtÞ ¼ðt=b2

Þe�ðt2=2b2

Þ for tZ0

0 for to0

(ð2Þ

where b is a parameter that completely defines the probabilitydensity function and it depends upon the cutting conditions,microstructure of grinding wheel, the properties of workpiecematerial, etc. The expected value and a standard deviation of theabove function can be expressed as

EðtÞ ¼ ðffiffiffiffiffiffiffiffiffip=2

pÞb ð3Þ

sðtÞ ¼ fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4�pÞ=2

pgb ð4Þ

It is important to remark that what here is referred to as theundeformed chip thickness t is actually the depth of engagementof each individual active grain which are participating inremoving material. This function has a similar shape to thelogarithmic standard distribution suggested by Konig and Lortz[23] to describe the chip thickness distribution. However, theRayleigh distribution has the advantage of being uniquely definedby only one parameter, that is, b.

A schematic diagram showing the interaction of the grain tipto the workpiece is given in Fig. 2. The relative motion of thecutting grains with respect to the workpiece surface generates aremoved chip with a curved longitudinal shape, as shown in Fig. 3.This chip has an increasing chip thickness from zero to amaximum value tm with a cross-section determined by the graingeometry.

At any transverse section m–m, the profile of a groovegenerated by any grain is as shown in Fig. 4. Let xi and xi + 1 bethe positions of two successive grooves produced by any grain onthe ith and (i+1)th columns as shown in Fig. 5. As the grains areinteracting independently of each other through the workpiecesurface, xi and xi + 1 are independent random variables. So, for anysection ‘oh’ along the axial direction, a probability densityfunction f(x) will be obtained by replacing a with h in Eq. (1).

In order to incorporate the effect of overlapping on the surfaceroughness, it is necessary to calculate the distance between twoneighboring grains. Since it has been assumed that a groove canbe overlapped on either side of it, by the succeeding groove, so, asshown in Fig. 5, the centre-to-centre distance, in axial direction,between two successive grains c is given by

c¼ 9xiþ1�xi9 ð5Þ

Max Undeformed

Chip Thickness tm

Cross Sectional Area Ac

Widthb

Cutting Lengthlc

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

Profile of groove generated

X

Z

t

Work surface

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

xi+1

Xi

t

Z

X h 0

h

�

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

1 Equation number using superscript A refers to Appendix.


Let h1¼xi +1�xi and h2¼xi. It can also be written as

xi ¼ h2 and xiþ1 ¼ h1þh2 ð6Þ

A joint probability density function is required for thecalculation of the expected value of c. So after using the abovetransformation, it can be written as

f ðh1,h2Þ ¼ f ðx1ðh1,h2Þ,x2ðh1,h2ÞÞ9J9 ð7Þ

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

negative:

f ðh1Þ ¼

Z h

h1

f ðh1,h2Þdh2 ð8Þ

and

f ðh1Þ ¼

Z h

�h1

f ðh1,h2Þdh2 ð9Þ

Since c¼9h19, so the total probability to satisfy this conditionwill be

Pð9h19rcÞ ¼ Pðh1r�cÞþPðh1r�cÞ

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

fcðcÞ ¼ fh1ðcÞþ fh1

ðcÞ ð10Þ

So, the expected value of the centre-to-centre distance in axialdirection, E(c) between two successive grains can be obtained byusing Eq. (50)A1 as

EðcÞ ¼Z h

0c f ðcÞdc ð11Þ

where f(c) is the probability density function c. Since the grooveshape has been assumed to be parabolic, the maximum width ofcut, h will be equal to 4t as it has been assumed that cutting depth(uncut chip thickness t) corresponds to the distance from the tipof the parabola to its focus. Substituting the value of f(c) from Eq.(50)A, the expected value E(c), after limit evaluation, will beobtained as

EðcÞ ¼ ð4t=3Þ ð12Þ

So, the expected value of centre distance between two interact-ing grains along the X, will be 4/3 times the groove depth of cut.

The surface roughness, Ra, is defined as the arithmetic averageof the absolute values of the deviations of the surface profileheight from the mean line within the sampling length l. Therefore,the surface roughness Ra can be expressed as

Ra ¼1

l

Z l

0y�ycl dl

�� ð13Þ

where ycl denotes the distance of the centre line, drawn in such away that the areas above and below it are equal (Fig. 6). It can alsobe expressed statistically as

Ra ¼1

l

Z ymax

ymin

9y�ycl9pðyÞdy ð14Þ

where ymax and ymin are the lowest and highest peak height ofthe surface profile and p(y) is the probability that height of grainhas a particular value y.

The surface roughness, Ra, can be calculated using probabilitydensity function defined in Eq. (2). The complete description ofsurface generated is very difficult due to the complex behavior ofdifferent grains producing grooves because of the random grain–work interaction. Thus, certain assumptions have to be madewhile predicting the surface roughness. The assumptions aregiven below:

(1)
An individual grain has many tiny cutting points in its surface,therefore, for simplicity, the grain tips are approximated as

y cl

t 2

A2

4t2

t 1

A1

upperA2

4t1

lower

4/3 t2 4/3 t1

Fig. 6. Profile of grooves generated.


paraboliod in shape, randomly distributed throughout thewheel volume.

(2)
The profile of the grooves generated is same and completelydefined by the depth of engagement or undeformed chipthickness t.
(3)
Grooves will overlap each other on either side only once, withsame types of grooves and at a distance of 4/3 times theundeformed chip thickness.
(4)
On an average, the expected area of interference of grain tipand workpiece surface is up to focal point of parabola.
Under these assumptions, the profile generated by the grain isas shown in Fig. 5.

As per definition of surface roughness, the area above andbelow the centre line must be equal. Hence the total expectedarea could be written as

EfAðtÞg ¼ 0 ð15Þ

The above equation can be represented in terms of theprobability density function f(t) asZ 1

0AðtÞf ðtÞdt¼ 0 ð16Þ

During the grain–work interaction in the grinding, two types ofgrooves are generated depending upon their depth of engagementis either less or greater than centre line ycl and it is assumed thatoverlapping takes place with the same type of grooves. Definingthe overlap factor f as the ratio of area lost (due to overlapping),from the area contributing to surface roughness without over-lapping, to the area contributing to surface roughness when thereis no overlapping. So, for the groove with the depth ofengagement less than ycl, the expected value of area A01contributing to roughness, after overlapping, can be expressed as

EfAuðt1Þg ¼

Z ycl

0Auðt1Þf ðtÞdt¼

Z ycl

0Au1f ðtÞdt¼ ð1�fÞ

Z ycl

0A1f ðtÞdt

ð17Þ

where A1 is the intercepted area between grain and the centre linecontributing to surface roughness (Ra) before overlapping, asshown in Fig. 6.

Similarly, for the groove with depth of engagement greaterthan ycl, it can be expressed as

EfA00ðt2Þg ¼

Z 1ycl

A00ðt2Þf ðtÞdt¼

Z 1ycl

ðAupper200 �Alower

200 Þf ðtÞdt

¼ ð1�fÞZ 1

ycl

ðAupper2 �Alower

2 Þf ðtÞdt ð18Þ

Alower2 and Aupper

2 are the areas below and above the centre linebefore overlapping. Substituting the values from Eqs. (17) and

(18), in Eq. (16), equation becomesZ ycl

0A1f ðtÞdtþ

Z 1ycl

ðAupper2 �Alower

2 Þf ðtÞdt¼ 0 ð19Þ

or

p1EðA1Þþp2fEðAupper2 Þ�EðAlower

2 Þg ¼ 0 ð20Þ

where p1 and p2 are the probabilities defined in terms of the chipthickness probability density function f(t) and are given by

p1 ¼

Z ycl

0f ðtÞdt for toycl ð21Þ

p2 ¼

Z ycl

0f ðtÞdt for t4ycl ð22Þ

The expected value of area, contributing to surface roughnessRa, after introducing overlapping, for the groove with depth lessthan ycl, can be obtained as

EðAu1Þ ¼ ð1�fÞ 4yclEðt1Þ�8

3Eðt2

1Þ

� �ð23Þ

Similarly, the expected value of area for the groove with depthgreater than the centre line contributing to surface roughness (Ra)will be given as

EðAupper200 Þ ¼ ð1�fÞ 4yclEðt2Þ�

8

3Eðt2

2Þþ8

3E

ffiffiffiffit2

pðt2�yclÞ

3=2n o� �

ð24Þ

EðAlower200 Þ ¼ ð1�fÞ

8

3E

ffiffiffiffit2

pðt2�yclÞ

3=2n o

ð25Þ

where Alower200 and Aupper

200 are the areas below and above the centre

line as shown in Fig. 6. Rewriting Eq. (20) after substituting theexpected values from Eqs. (23), (24) and (25) as

2ycl p1Eðt1Þþp2Eðt2Þ� �

¼p2

p1Eðt21Þþp2Eðt2

2Þ� �

ð26Þ

Calculation of the expected values in the above equationrequires another probability density function for the cases wherethe chip thickness is smaller and greater than the centre linedistance ycl. Therefore, for the grains having depth of engagementlying between 0 and ycl, the probability density function of thechip thickness will be given by the conditional probability densityfunction f1(t) as

f1ðtÞ ¼ f1ðt90rtoyclÞ ¼f ðtÞR ycl

0 f ðtÞdtð27Þ

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


given by

f2ðtÞ ¼ f2ðt9yclrto1Þ¼f ðtÞR1

yclf ðtÞdt

ð28Þ

Substitute Eqs. (17), (18), (27) and (28) in Eq. (26) to find ycl.After simplification the expression for centre line can beexpressed as

ycl ¼2

3

� Eðt2Þ

EðtÞ

�ð29Þ

Substituting the expected values from Eq., and simplifying theequation, the value of the centre line will be given by

ycl ¼4

3

ffiffiffiffi2

p

rb ð30Þ

For the calculation of surface roughness, two types of groovesare considered. Since the contribution of the two types of groovesconsidered is different, thus, the total expected value of surfaceroughness can be calculated as

EðRaÞ ¼ p1EðRa1Þþp2EðRa2Þ ð31Þ

where E(Ra1) and E(Ra2) are the expected values of the surfaceroughness for depth of engagement smaller or greater than ycl andthese values can be calculated by the definition of the surfaceroughness. As per this, the surface roughness can be calculated byadding the area between the profile and the centre line and divideit by the total profile length. Hence from Fig. 5, the values can bewritten as

EðRa1Þ ¼ EAu14t1

� ð32Þ

EðRa2Þ ¼ EAupper

200 þAlower200

4t2

!ð33Þ

Rewriting Eqs. (32) and (33) after substituting the expressionsof A1, Aupper

2 and Alower2 from Eqs. (23)–(25) as

EðRa1Þ ¼ ð1�fÞ ycl�2

3

� Eðt1Þ

� ð34Þ

EðRa2Þ ¼ ð1�fÞ ycl�2

3

� Eðt2Þþ

4

3

� E t2 1�

ycl

t2

� 3=2 !( )

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 roughnesscan be expressed as

EðRaÞ ¼ 0:499ð1�fÞb ð36Þ

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

b¼ 0:795EðtÞ

Substituting this value in Eq. (36), expression becomes

EðRaÞ ¼ 0:396ð1�fÞEðtÞ ð37Þ

Eq. (37) shows a proportional relationship between the surfaceroughness and the chip thickness expected value under theassumption that the profile of groove generated by an individualgrain to be parabolic in shape with overlapping, without plowingand back transferring of material.

3. Chip thickness modeling

The chip-thickness model plays a major role in predicting thesurface quality. The chip-thickness models, proposed by Reich-enbach et al. [24] and others [25,26] were based on speed ratio,depth of cut, the equivalent diameter of the wheel, etc. But none ofthese models took the deformation due to the elasticity of thegrinding wheel and workpiece system into account. Geometrically,the contact deflections can influence both the surface finish of theworkpiece and the accuracy of size of ground components.According to Saini [27], the contact deflection in grinding can beviewed microscopically and macroscopically. Microscopically, awheel grain is deflected by the normal force exerted on it duringgrinding and the workpiece is plastically deformed in the grindingzone. Since a grain may have many tiny cutting points [3], suchanalysis is extremely cumbersome and the physical interpretationis difficult. Macroscopically, the grinding wheel may be consideredas a thick circular plate pressed against a curved surface fromwhich the material is ground. Macroscopic approach has beenadopted, in the present study because the core material (alminiumis the core material) is 94% by volume while the amount of theabrasive layer on the core is only 6% by volume, in the diamond-grinding wheel. Hence the diamond-grinding wheel can beconsidered as a thick circular plate and the modulus of elasticityof the wheel may be taken as the modulus of elasticity of its corematerial (alminium), in the ceramic grinding process. Therefore, anew chip-thickness model has been envisaged, based on macro-scopic approach, by incorporating the elasticity of the grindingwheel and the workpiece in the existing chip-thickness model.

During the process of grinding, the normal forces generated tendto elastically deform the wheel, which results in a decrease in theeffective diameter of the wheel in the wheel-work contact zone. Thiswould reduce the effective depth of cut and consequently reducesthe maximum chip thickness. Wheels with low modulus of elasticitywould have greater deformation than those with a high modulus ofelasticity. Hence, the reduction in the maximum chip thicknesswould be more for wheels with low modulus of elasticity. Similarly,workpiece with higher modulus of elasticity would deflect thegrinding wheel more. Hence, the reduction in the chip thicknesswould be more for grinding workpieces with higher modulus ofelasticity. Combining both the above effects, the maximum chipthickness can be expressed as [28]

tmpE1

E2

where E1 is the modulus of elasticity of the wheel and E2 is themodulus of elasticity of the workpiece. The modulus of elasticity ofthe diamond-grinding wheel (E1) is assumed to be the modulus ofelasticity of the core material of the wheel itself. This is because theamount of the abrasive layer on the core is only 4 mm thick (6% byvolume) and the core material is of 242 mm diameter (94% byvolume) in a grinding wheel of 250 mm diameter. Alminium is thecore material used in the diamond-grinding wheels and thus themodulus of elasticity of the wheel is taken as the modulus ofelasticity of alminium, which is 70 GPa. Hence E1 is taken as 70 GPain the present study. The value of modulus of elasticity of theworkpiece (E2) is taken as 410 GPa, which is provided by themanufacturer of the silicon carbide workpiece.

A well-known equation for estimating the maximum chipthickness in ceramic grinding [3] is as follows:

tmax ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4

Cr

Vw

Vs

� ffiffiffiffiffiffiffiae

deq

svuutwhere r is the chip width-to-thickness ratio, C is the number ofactive grits per unit area, Vw is the work velocity, Vs is the wheel


velocity, ae is the work engagement and deq is the equivalentwheel diameter. The value of r is equal to 4, in the present study,as the groove shape is assumed to be a parabolic. The equivalentwheel diameter in surface grinding is the wheel diameter itself.

Using a simple geometric relationship, the value of C derivedby Xu et al. [29] is as follows:

C ¼ 4f=fd2g ð4p=3uÞ2=3

g ð38Þ

where dg is the equivalent spherical diameter of diamond particle,v the volume fraction of diamond in the grinding wheel, and f thefraction of diamond particles that actively cut in grinding. Thegrinding wheel used in the present study has a density of 100, orin other words, volume fraction v is 0.25 [3]. To obtain the valueof C, it is assumed that only one-half of the diamond particles onthe wheel surface are actively engaged in cutting [29], or f¼0.5.The equivalent spherical diameter of diamond grit (dg) is given [3]as

dg ¼ 15:2M�1 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 dueto the elastic deformation, by incorporating a parameter related tothe ratio of elasticities of the workpiece and wheel. The newmodel for chip thickness can be expressed as

tm ¼E1

E2

� n

tm ð40Þ

where the exponent (n) is proportionality constant, to take care ofthe effects of linear and nonlinear deflections of the workpieceand grinding wheel.

Table 1Exponent (n) at various feeds and depths of cut.

Vw (m/min) ae (mm) Ft (N/mm) n

5 5 6.14 0.397

10 5 10.43 0.449

15 5 11.23 0.554

5 10 7.48 0.555

10 10 10.67 0.661

15 10 13.13 0.724

5 15 9.1 0.620

10 15 11.75 0.759

15 15 15.09 0.808

naverage¼0.614

4. Evaluation of the exponent (n)

In order to use the model effectively for reliable determinationof chip thickness, the value of exponent n must be known.Because a well-defined method is lacking, it has to be determinedby an approach that follows the fundamental principle, applicableto the grinding process. In this work, the concept of energybalance, i.e. the energy supplied by the grinding wheel is equal tothe amount of the energy required to remove the material, hasbeen used. So the value of exponent n obtained in this way can bereliably used for chip thickness determination.

The energy required to remove the material is supplied by thegrinding wheel. At slow removal rates, the specific cutting energy(Es) is extremely high, but it decreases at faster removal ratestending toward a minimum value for silicon carbide. As theexperiments during the present work are conducted at fastermaterial removal rates, the value of minimum specific energy hasbeen taken as 9 J/mm3 [3]. So the energy given by the grindingwheel is equal to the amount of the energy required to remove thematerial and it can be written as

FtVs ¼ Specific energy ðEsÞ � volume of material removed=unit time,

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

¼ ðCbsVsÞVc ð42Þ

where Vc is the volume of each undeformed chip produced and bs

is the grinding wheel width. Assuming a chip with paraboliccross-section, Vc can be approximated as one half times theproduct of the maximum cross-sectional area ð2rt2

m=3Þ and the

length lc from the following formula:

Vc ¼ rt

2m

3

ffiffiffiffiffiffiffiffiffiffiffiaedeq

qð_lc ¼


qÞ ð43Þ

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

FtVs ¼ EsðCbsVsÞVc

or

FtVs ¼ 9ðCbsVsÞrt

2m

3


q

Substituting the value of r in the above equation,

FtVs ¼ 12ðCbsVsÞt2m


q

or,

FtVs ¼ 12ðCbsVsÞE1

E2

� n

tm

� �2 ffiffiffiffiffiffiffiffiffiffiffiaedeq

qð44Þ

By measuring the tangential force during grinding using sixcomponent dynamometer, n can be evaluated. The experimentshave been carried out on a horizontal surface grinding machine.The diamond-grinding wheels of ASD 240 R100 B2 have been usedin the present experimental study. Silicon carbide with a modulusof elasticity of 410 GPa was ground at a speed of 2200 m/minwithout cutting fluid. The feed and depth of cut are varied duringexperimentation. The average value of exponent (n) was found tobe 0.614 and the results are shown in Table 1.

The new chip thickness model can thus be written as

tm ¼E1

E2

� 0:614

tm ð45Þ

The elastic properties of both the wheel and workpiece cause aconsiderable deflection in the grinding wheel, resulting in areduction in its effective diameter and thereby decreasing theactual depth of cut. In the present work, low modulus of elasticityof the wheel and high modulus of elasticity of the work materialcaused the wheel to deflect more and consequently resulted in asignificant reduction in the maximum chip thickness estimated bythe new model compared with that of the existing model, asobserved in the results shown in Table 2. These results strengthenthe representation of deflections of the work and wheel in termsof its elastic properties.

Table 2Undeformed chip thickness by existing and new chip-thickness model.

ae (mm) Vw (m/min) tm (mm) tm (mm)

5 5 2.299 0.776

5 10 3.251 1.098

5 15 3.981 1.344

10 5 2.734 0.923

10 10 3.866 1.305

10 15 4.734 1.599

15 5 3.025 1.021

15 10 4.279 1.445

15 15 5.240 1.769

Table 3Experimental values of surface roughness at different values of kinematic

parameters.

Exp. no. ae (mm) Vs/Vw Ra (mm) Average

value of Ra

1 2 3 4 5

1 5 440 0.161 0.166 0.162 0.168 0.163 0.164

2 5 293 0.229 0.227 0.220 0.228 0.213 0.229

3 5 220 0.268 0.269 0.271 0.273 0.274 0.271

4 5 176 0.310 0.311 0.315 0.310 0.309 0.311

5 5 146 0.328 0.329 0.327 0.330 0.331 0.329

6 15 440 0.201 0.208 0.207 0.209 0.205 0.206

7 15 293 0.215 0.248 0.246 0.245 0.246 0.244

8 15 220 0.303 0.299 0.292 0.293 0.298 0.297

9 15 176 0.336 0.338 0.340 0.340 0.341 0.339

10 15 146 0.363 0.362 0.371 0.369 0.370 0.367

11 25 440 0.209 0.215 0.210 0.215 0.216 0.213

12 25 293 0.271 0.271 0.270 0.270 0.273 0.271

13 25 220 0.331 0.332 0.330 0.333 0.329 0.331

14 25 176 0.372 0.367 0.366 0.375 0.365 0.369

15 25 146 0.402 0.393 0.399 0.405 0.401 0.400

16 35 440 0.259 0.249 0.249 0.252 0.256 0.253

17 35 293 0.321 0.326 0.330 0.322 0.321 0.324

18 35 220 0.355 0.355 0.348 0.347 0.350 0.351

19 35 176 0.396 0.391 0.398 0.387 0.398 0.394

20 35 146 0.429 0.424 0.429 0.425 0.423 0.426

21 45 440 0.314 0.305 0.310 0.316 0.315 0.312

22 45 293 0.357 0.352 0.345 0.356 0.355 0.353

23 45 220 0.397 0.400 0.395 0.389 0.399 0.396

24 45 176 0.432 0.431 0.433 0.435 0.429 0.432

25 45 146 0.497 0.489 0.499 0.491 0.489 0.4930

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

Surf

ace

roug

hnes

s (μ

m)

Speed ratio 100

Speed ratio 220

Speed ratio 550

Depth of cut (μm)10 20 30 40 50

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

0

0.1

0.2

0.3

0.4

0.5

0

Surf

ace

roug

hnes

s (μ

m)

Wheel 2 (ASD240R100B2)

Wheel 1 (ASD500R100B2)

Depth of cut (μm)5 10 15 20 25 30 35 40 45 50

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


5. Evaluation of overlap factor /

A comprehensive model for the quantitative prediction ofsurface roughness has been presented, as given by Eq. (37). Inorder to use the model effectively for the reliable prediction ofsurface 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 experi-ments were performed by grinding silicon carbide workpiece bydiamond-grinding wheel. An ‘ELLIOTT 8-18’ hydraulic surface-grinding machine was used to grind the sintered silicon carbidepieces with diamond-grinding wheels. The properties of SiCworkpiece material used for experimentation in this work are:density¼3.17 gm/cm3, hardness (HV)¼2700 kg/mm2, fracturetoughness (KlC)¼4.55 MPa m1/2, modulus of elasticity¼410 GPaand thermal conductivity¼145 W m�1 K�1. The workpiece ma-terial was supplied by H.C. Starck Ceramics GmbH & Co. KG,Germany. The tool was diamond-grinding wheel (ASD240R100B2) (Norton make) with modulus of elasticity of 70 GPa andalminium as core material. The size of the workpiece is20 mm�20 mm�5 mm. The other conditions taken for theexperimentation were as follows: wheel speed¼36.6 m/s, wheeldiameter¼250 mm, wheel width¼19 mm. The main kinematic

parameters for each experiment are depth of cut ae and the speedratio (Vs/Vw) where Vw is the feed rate and Vs is the wheel speed,along with the experimental value of surface roughness as shownin Table 3. Surface roughness measurements were made usingTalysurf-VI (cut-off length was 0.8 mm) at five different places onthe 20�5 mm2 cross-section of the workpiece after grinding andthe arithmetic mean of the values of the measurements has beenreported in the experimental results as shown in Table 3. Theexperiments are replicated five times (as shown in Table 3) tomask the variability of the process. The resolution of surfaceroughness-measuring instrument is 0.8 nm. This means that the0.8 nm is the minimum value of surface roughness that can bemeasured by the surface roughness-measuring instrument. How-ever the differences in the readings of surface roughness (Table 3)are much higher than the resolution of surface roughness-measuring instrument. So this instrument will be able todistinguish between the values clearly and hence measurementsmade by this instrument could be considered accurate enough forthe present study. Apart from this, there are sources of error inany measuring system. The term ‘precision’ is often used in thisconnection. Perfect precision means that the measurements willbe made with no random variability in the measured values orstandard deviation of the measuring system is zero. So, in order to


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

0.05

0.15

0.25

0.35

0.45

0depth of cut (μm)

surf

ace

roug

hnes

s (μ

m)

0.1

0.2

0.3

0.4

0.5

depth of cut (μm)

surf

ace

roug

hnes

s (μ

m)

0.15

0.25

0.35

0.45

0.55

depth of cut (μm)

surf

ace

roug

hnes

s (μ

m)

0.2

0.3

0.4

0.5

0.6

depth of cut (μm)

surf

ace

roug

hnes

s (μ

m)

0.25

0.35

0.45

0.55

0.65

depth of cut (μm)

surf

ace

roug

hnes

s (μ

m)

Vw = 5m/min

Vw = 7.5m/min

Vw = 12.5m/min

Vw = 10m/min

Vw = 15m/min

Ra, circular groove Ra, parabolic grrove

10 20 30 40 50

0 10 20 30 40 50

0 10 20 30 40 50

0 10 20 30 40 50

0 10 20 30 40 50

Fig. 9. Surface roughness presented at var

The expected value of chip thickness was calculated, for eachexperiment, 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 ofexpected value of chip thickness and surface roughness valueobtained experimentally (Table 3), using Eq. (37) and it was foundto be approximately 9.6%.

0.1

0.2

0.3

0.4

0.5

feed (m/min)

surf

ace

roug

hnes

s (μ

m)

0.1

0.2

0.3

0.4

0.5

feed (m/min)

surf

ace

roug

hnes

s (μ

m)

0.2

0.3

0.4

0.5

0.6

feed (m/min)

surf

ace

roug

hnes

s (μ

m)

0.25

0.35

0.45

0.55

0.65

4feed (m/min)

surf

ace

roug

hnes

s (μ

m)

d = 10μm

d = 15μm

d = 30μm

d = 45μm

0.05

0.15

0.25

0.35

0.45

feed (m/mi)

surf

ace

roug

hnes

s (μ

m)

d = 5μm

Ra, groove overlap Ra, experimental

7 10 13 16

4 7 10 13 16

4 7 10 13 16

4 7 10 13 16

4 7 10 13 16

ious values of depth of cut and feed.


6. Surface roughness prediction

The present work focuses on the prediction of surface roughnessbased on the chip thickness model. The chip thickness models play amajor role in predicting the surface quality. The chip thickness is avariable often used to describe the quality of ground surfaces as wellas to evaluate the competitiveness of the overall grinding system.The various parameters of the grinding process have been includedin the model through chip thickness model. Therefore, this modelcan be used to predict surface roughness under different conditionsof the parameters. The depth of cut and speed ratio are the twoimportant parameters that can be varied on the grinding machine.Fig. 7 shows the arithmetic mean value of surface roughness forvarious values of depth of cut for three different speed ratios. It canbe observed that at higher speed ratio, the surface finish is better.This is because, at higher speed ratio, i.e. at lower workpiecevelocity, more grains will be involved in removing a given volume ofmaterial, and thus depth of engagement will be low. Hence thesurface finish is better. Similarly, the wheel microstructure plays amajor role in the quality of the ground surfaces. Wheel 1 is made upof fine abrasive with average grit size of about 32 mm and wheel 2 isa coarser wheel with an average abrasive size of about 67 mm. It canbe observed from Fig. 8, the coarser wheel produces rougher surfaceas compared to that of fine wheel. Although a finer wheel producesbetter surface finish but it will cause higher forces and higher powerdue to the higher specific energy governed by a smaller expectedvalue of the chip thickness. This is a well-established fact, whichreinforces the correctness of the model.

7. Comparison with the existing models

The centre line average value of surface roughness (Ra) has beencompared with results, obtained experimentally and from theexisting surface roughness models [20,21] and new surface rough-ness model {Eq. (37)}. The deviation of the surface roughnesscalculated with the new model, from the existing model andexperimental values, for various values of feed and depth of cut, isshown in Fig. 9. It could be seen from this figure that the surfaceroughness increased with an increase in depth of cut and feed. Thisestablished behavior could be explained by observing the variationof maximum chip thickness with the grinding parameters. Increasein depth of cut causes the maximum chip thickness to increase andthereby resulting in a poor surface quality. It could also be seen fromFig. 9 that the surface roughness decreased with decrease in feed.This is as expected since the depth of engagement would be low atlow feed rate and hence the reduction in surface roughness could beobserved with the decrease in feed rate. Also at higher speed ratiosthe surface produced is smoother. This is because at lowerworkpiece 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 surfaceroughness shows a good agreement with the experimental dataobtained from different grinding conditions in surface grinding.Apart from this, the surface roughness values computed by newsurface roughness model are closer, to actual values obtainedexperimentally, as compared to that of the existing models and thuspredicting the performance of the process more accurately.

8. Conclusion

In this paper, an analytical model for surface roughnessprediction of ground ceramics, based on the analysis of thegrooves left by the grains that interact with the workpiece, which

is characterized by the undeformed chip thickness, has beendeveloped. The wheel microstructure, the kinematic and dynamicgrinding conditions, and the material properties were included inthe model through undeformed chip thickness model. The modelincorporates the overlapping effect of grooves left by the grains,apart from other grinding parameters. By incorporating theoverlapping effect, the model has been made more realistic, notonly to estimate the surface roughness more precisely, but also tomake the ceramic grinding reproducible. The model is capable ofhandling a wide variety of work and wheel speeds and is flexibleenough to incorporate the effects of other parameters. Hence thenew model can be reliably used to predict the surface roughnessin 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 thejoint probability density function of the transformation Eq. (6),which can be written as

f ðh1,h2Þ ¼ f ðx1ðh1,h2Þ,x2ðh1,h2ÞÞ9J9 ð46Þ

where J is the Jacobian. Substituting the value of probabilitydensity function for x1 and x2 from Eq. (1) in the above equation,the value of above function becomes

f ðh1,h2Þ ¼1

h

1

h1¼

1

h2ð47Þ

Thus using Eq. (47), the probability density function of h1 forthe selected length ‘0h’ as shown in Fig. 5, can be obtained as thesum of following two possibilities (whether h1 is positive ornegative) as

f ðh1Þ ¼

Z h

h1

f ðh1,h2Þdh2 ¼h�h1

h2

� ð48Þ

f ðh1Þ ¼

Z h

�h1

f ðh1,h2Þdh2 ¼hþh1

h2

� ð49Þ

Substituting the values from Eqs. (48) and (49), in Eq. (10),fc(c) becomes

fcðcÞ ¼ 2h�c

h2

� ð50Þ

A.2. Calculation of centre line position ycl (Eq. (29))

The calculation of the centre line value, ycl, Eq. (29) requiresthe calculation of the value of the undeformed chip thickness t

and square of the undeformed chip thickness t2. These expectedvalues can be calculated as

EðtÞ ¼

Z 10

tf ðtÞdt¼ �te�ðt2=2b2

Þ þ

ffiffiffiffip2

rberf

tffiffiffiffiffiffiffiffi2b2

q0B@

1CA

264

3751

0

ð51Þ

That gives the value of E(t) as

EðtÞ ¼

ffiffiffiffip2

rb¼ 1:257b ð52Þ


Similarly,

Eðt2Þ ¼

Z 10

t2f ðtÞdt ¼

Z 10

t3

b2e�ðt

2=2b2Þdt ð53Þ

It will give the value as

Eðt2Þ ¼ ½e�ðt2=2b2

Þð�2b2�t2Þ�10 ¼ 2b2

ð54Þ

where b is a parameter that completely defines the probabilitydensity 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 issmaller than the centre line value, ycl, can be calculated as

p1 ¼

Z ycl

0f ðtÞdt¼ 1�e�ðy

2cl=2b2Þ ð55Þ

Thus, as per definition of probability density function, theprobability of an undeformed chip thickness to be greater thancentre line value, ycl, will be

p2 ¼ 1�p1 ¼ e�ðy2cl=2b2Þ ð56Þ

The expected value of chip thickness smaller than ycl can becalculated by using the conditional probability density function(Eq. (27)), giving the expected value in this region as

Eðt1Þ ¼1

p1

Z ycl

0tf ðtÞdt ð57Þ

After solving the above integration and limit evaluation, it canbe expressed as

Eðt1Þ ¼1

1�e�ðy2cl=2b2Þ�ycle

�ðy2cl=2b2Þ þ

ffiffiffiffip2

rb erf

yclffiffiffiffiffiffiffiffi2b2

q0B@

1CA

264

375

After simplification, it can be written as

Eðt1Þ ¼ 0:713b ð58Þ

In the same way, the expected value of chip thickness greaterthan centre line value, ycl, can be calculated as

Eðt2Þ ¼1

p2

Z 1ycl

tf ðtÞdt ð59Þ

Using Eq. (28), the above integration after limit evaluation canbe expressed as

Eðt2Þ ¼1

e�ðy2cl=2b2Þ

ffiffiffiffip2

rþe�ðy

2cl=2b2Þycl�

ffiffiffiffip2

rerf

yclffiffiffiffiffiffiffiffi2b2

q0B@

1CA

264

375

or

Eðt2Þ ¼ 1:72b ð60Þ

One more expected value is required to be calculated tocompute the surface roughness for the chips whose chip thicknessvalue is more than ycl, as given by Eq. (35).

Expected value,

E t2 1�ycl

t2

� 3=2 !

,

can be calculated as

E Fðt2Þ� �

¼1

p2

Z 1ycl

FðtÞf ðtÞdt ð61Þ

where

Fðt2Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

y2cl

t22

s

After transformation of Eq. (61) in terms of z, which is afunction of t, the equation becomes

EfFðt2Þg ¼

Z 10

e�zf ðzÞdz ð62Þ

where

z¼t2

2b2

and

f ðzÞ ¼ e�ð8=3ffiffiffiffiffiffiz=pp

Þ ðffiffiffiffiffi2zp�ð4=3Þ

ffiffiffiffiffiffiffiffiffiffiffiffið2=pÞ

p 2Þ2ffiffiffiffiffi

2zp 1�

1

ð1þð3=4Þffiffiffiffiffiffipzp Þ

�3=2

Solving the above integration gives the value asR10: e�z f ðzÞdz¼ 0:3875.

Therefore, the expected value is

E t2 1�ycl

t2

� 3=2 !

¼ 0:3875b ð63Þ

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Modeling and prediction of surface roughness in ceramic grinding