predictive modeling of undeformed chip thickness in ceramic grinding

International Journal of Machine Tools & Manufacture 56 (2012) 59–68

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International Journal of Machine Tools & Manufacture

0890-69

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Predictive modeling of undeformed chip thickness in ceramic grinding

Sanjay Agarwal a,n, P. Venkateswara Rao b

a Department of Mechanical Engineering, Bundelkhand Institute of Engineering and Technology, Jhansi 284128, Indiab Department of Mechanical Engineering, Indian Institute of Technology, New Delhi 110016, India

a r t i c l e i n f o

Article history:

Received 21 April 2011

Received in revised form

23 December 2011

Accepted 3 January 2012Available online 13 January 2012

Keywords:

Undeformed chip thickness

Ceramic grinding

Silicon carbide

55/$ - see front matter & 2012 Elsevier Ltd. A

016/j.ijmachtools.2012.01.003

esponding author. Tel.: þ91 51 0232 0399; fa

ail address: [email protected] (S. Ag

a b s t r a c t

The quality of the surface produced during ceramic grinding is important as it influences the

performance of the finished part to great extent. The undeformed 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. Hence, the estimation of undeformed chip thickness can cater to the requirements of

performance evaluation. But, the undeformed chip thickness is governed by many factors and its

experimental determination is laborious and time consuming. So the establishment of a model for the

reliable prediction of undeformed chip thickness is still a key issue for ceramic grinding. In this study, a

new undeformed chip-thickness model is developed, for the reliable prediction of undeformed chip

thickness in ceramic grinding, on the basis of stochastic nature of the grinding process, governed

mainly by the random geometry and the random distribution of cutting edges. The model includes the

real contact length that results from combined contact length, due to wheel–workpiece contact zone

deflection and the local deflection due to the microscopic contact at the grain level and contact length

due to geometry of depth of cut. The mechanical properties of workpiece material and the grinding

parameters are also considered in the undeformed chip thickness model through normal grinding force

model. The new model has been validated by the experimental results of silicon carbide grinding,

taking the surface roughness as a parameter of evaluation.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Interest in grinding of advanced ceramics has grown substan-tially with the widespread use of ceramic components in variousengineering applications such as cutting tools, automobile valves,packing (sealing) elements, bearings, pistons, rotors, etc. The cera-mic components offer many advantages over metallic counterpartsin terms of improved performance and better efficiency [1].However, the favorable features are accompanied by difficultiesassociated with machining and mainly with grinding because ofthe high hardness and stiffness of these advanced ceramics anddemand for the required accuracy and surface quality of theground components. Grinding is an important process for shapingbrittle materials to obtain good surface finish and high dimen-sional accuracy. It is a complicated process, consisting of complexinteractions between a large number of variables, such as machinetool, grinding wheel, workpiece material and operating para-meters. Precision ceramic components require strict adherenceto close tolerances and surface finish as the performance andreliability of these components are greatly affected by the accu-racy and surface finish produced during the grinding process.

ll rights reserved.

x: þ91 51 0232 0312.

arwal).

There are various factors, which govern the dimensional accuracyand surface finish in grinding, and hence, the development ofanalytical or empirical models for reliable prediction of machiningperformance becomes a key issue [2].

A consistent modeling must begin from the basic physics ofthe process given by the interaction of individual grinding grainswith the workpiece. It must then be expanded to the behavior ofthe whole grinding wheel. The single grit–workpiece interactioncan be characterized by the undeformed chip thickness. Theundeformed chip thickness is a variable often used to describethe quality of ground surfaces as well as to evaluate the competi-tiveness of the overall grinding system. However, there is nocomprehensive model that can predict undeformed chip thick-ness over a wide range of operating conditions. The reason stemsfrom the fact that many variables affect the process. Many ofthese variables are non-linear, interdependent, or difficult toquantify. Therefore, the models available so far are not fullyfeasible and experimental investigations can be very exhaustivebut with limited applicability [3]. So, an attempt has been madeto develop a theoretical model for the prediction of undeformedchip thickness for the grinding of silicon carbide with diamondabrasive.

Despite various research efforts in ceramic grinding over lasttwo decades, much needs to be established to standardize thetheoretical models for the prediction of undeformed chip

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S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 56 (2012) 59–6860

thickness for improving product quality and increasing produc-tivity to reduce machining cost. Since a ground surface isproduced by the action of large number of cutting edges on thesurface of the grinding wheel, the groove produced on the work-piece surface by an individual grain closely reflects the geometryof the grain tip. Thus, it is possible to evaluate the undeformedchip thickness from the considerations of the grain tip geometry.Since the size of these cutting edges on the wheel surfaceis random in nature, the undeformed chip thickness cannotbe predicted in a deterministic manner. Because of this random-ness, a probabilistic approach for the evaluation of undeformedchip thickness is more appropriate and hence any attempt toestimate undeformed chip thickness should be probabilistic innature.

Further the nature of contact behavior between the grindingwheel and the workpiece during grinding process is one of theprincipal factors that contribute to the quality of the ground work-piece. The importance of local contact deflections in grinding hasbeen recognized both by researchers and industry practitioners.Geometrically, the local contact deflections can influence both thesurface finish of the workpiece and the accuracy of size of groundcomponents [4]. There is however still much use in industry of‘‘rules of thumb’’ or spark-out techniques in finish grinding opera-tions to produce components with good surface quality and closedimensional tolerances. These operations can be time-consumingand reduce equipment productivity. Therefore, the effect of localcontact deflections must also be taken into consideration whiledeveloping a new undeformed chip thickness model for reliableprediction of undeformed chip thickness in the grinding operation.

2. Literature review

Advanced engineering ceramics have been extensively used inindustrial applications during the last two decades [5]. However,the actual utilization of advanced ceramics has been quite limitedmainly because of the machining difficulties and associated highcost of machining these materials by grinding while ensuring theworkpiece quality. A technological basis to achieve more efficientutilization of the ceramic grinding process requires an under-standing of the interaction between the abrasive and the work-piece, which has direct bearing on the surface quality, produced.Although extensive research has been carried out to predict theperformance of the ground ceramic workpiece, a complete under-standing is yet to be achieved.

Many researchers have attempted to predict the performance ofgrinding processes by using undeformed chip-thickness models.The undeformed chip-thickness model, used to assess the perfor-mance of grinding processes, plays a major role in predicting thesurface quality. The chip-thickness models, proposed by Reich-enbach et al. [6] were based on speed ratio, depth of cut, theequivalent diameter of the wheel, etc. The undeformed chipthickness previously modeled was obtained from the continuityanalysis based on the balance between the volume of the chipsgenerated and the total material removal rate. In this case, as wasdescribed by Malkin [7], an idealized grinding wheel was assumedwith cutting points uniformly distributed over the wheel surface.Therefore, the value calculated corresponds to an average unde-formed chip thickness. A common model was derived by Tonshoffet al. [3] from the comparison of a number of chip thicknessmodels. These models include one, two and three dimensionaldescriptions of the wheel surface as well as each grain in thecontact area involved in the chip formation process. None of thesemodels take into account the deformation of the grinding contactzone, which can considerably increase the number of cutting edgesin contact with the workpiece, with a consequent reduction of the

chip thickness. Furthermore, the workpiece material is not expli-citly considered. Nakayama et al. [8] developed the relationshipbetween the force and elastic deflection of the wheel. Experimentswere conducted to measure the deflection associated with theindividual grains. It was shown that the deflection of the individualgrains was of the same order of magnitude as that of theundeformed chip thickness. Brown et al. [9] analyzed the influenceof elastic deflection on the contact length by separating the elasticdeflection into two parts, the wheel body and workpiece deflectionand the deflection between an active grain and the workpiece. Acontact length model was established using Hertz theory. How-ever, the contact length model did not consider the depth of cutwhich has a significant influence on the grinding contact length.Kumar and Shaw [10] analyzed the deflection of the wheel and theworkpiece separately, and approximated the contact to the loadingof a work roll in metal rolling and developed the relationship toinclude the elastic deflection and grinding depth of cut. However,this model uses extrapolated results to estimate the elastic deflec-tions of a smooth contact situation. Good agreement was claimedbetween theory and experiment although the contact lengthsfound were generally smaller than measured value. Recentresearch has mainly concentrated on methods to improve themeasurement of the grinding contact length [11–15]. The researchhas provided valuable evidence on the real contact length,although the theory to support this evidence remains unsatisfac-tory. Saini et al. [6] emphasized the importance of contact deflec-tion in grinding. The various components of local contactdeflections, including that due to grain rotation and their combinedinfluence on the ground workpiece from the point of view ofindustrial application, was described. Some possible approaches toimprove the production efficiency of the grinding operations wereconsidered. Chen and Rowe [16] gave a detailed analysis ofgrinding process modeling and simulation. The wheel surfacegenerated by dressing was simulated. The grinding forces wereanalyzed by simulating the force on each grain, which passedthrough the section of the workpiece.

Chip formation and material removal depends to a large extenton the microstructure of the grinding wheel, the quantities ofmotion and the geometric parameters. The chip-thickness modelby Snoeys and Peters [17] for determining the equivalent chipthickness was based on the equation of continuity. This char-acteristic quantity represents the sum of all individual chipthicknesses in the contact area between grinding wheel andworkpiece. The simple model, which was developed by Petersfor determining the equivalent chip thickness, offers advantagesin practical application as the characteristic quantities ofthe grinding wheel topography do not have to be determined.Reichenbach et al. [6] and Malkin [7] have considered the grindingwheel topography in two-dimensional form by determining thegrain count. The derivation of the undeformed chip-thicknessmodel given by Malkin elaborates the various aspects of grindingprocess and various parameters related to it. The topography ofthe wheel and its kinematic interaction with workpiece was alsodescribed. All these models take the main influencing quantitiesinto consideration. Among these are the speed ratio, the workingengagement and the equivalent diameter. Furthermore, thesemodels have in common that, according to the exponents, thespeed ratio has the most important influence on chip formation.The basic model derived from these models not only considers theset-up parameters for the machine tool and the geometric propor-tions but also includes a factor for describing the microstructure ofthe grinding wheel. Consequently, many measurements are neces-sary, similar to topography models. Thus, the development ofthese undeformed chip-thickness models is complicated and timeconsuming. So, the present work focuses on developing a newundeformed chip thickness model.

zprofile of groove

t

x

Fig. 2. Sectional view showing the shape of groove generated.

cross-sectional areaAch

bmax undeformedchip thickness tm

width

S. Agarwal, P. Venkateswara Rao / International Journal of Machine Tools & Manufacture 56 (2012) 59–68 61

The literature review above indicates that the undeformedchip thickness models developed so far were based on either thegeometric contact length alone or the contact length obtained dueto the combined effect of deflection due to geometric grindingcontact zone and the wheel–workpiece contact. However littlehas been done to develop a chip thickness model based on realcontact length that results from the combined effect of deflectioncaused by all the three components of the grinding contact zonei.e. the geometric grinding contact zone, the elastic deflection dueto contact between the wheel body and the workpiece and thedeflection due to the microscopic contact at the grain level. Inaddition to this, in most of the models developed so far, thetransverse shape of the grooves produced has been assumed to betriangular. A simple abrasive grain on the wheel surface generallyhas many tiny cutting points on its surface. Experiments con-ducted by Lal and Shaw [18] with single abrasive grain under finegrinding conditions indicate that the grain tip could be betterapproximated by circular arc. Therefore, it is evident that thegroove produced by an individual grain can be better approxi-mated by an arc of a circle. Based on the above analysis, it can beobserved that there is a need to develop an analytical model topredict the undeformed chip thickness, based on probabilisticapproach to represent the stochastic nature of the grindingprocess, considering the grooves to be a part of circular arc andincorporating the real contact length.

In the present work, a new undeformed chip-thickness modelhas been developed on the basis of stochastic nature of the grindingprocess, governed mainly by the random geometry and the randomdistribution of cutting edges and based on elastic deformation ofcontact length which takes account of the depth of cut and thecontact deflection which includes the elastic deflection between thewheel and the workpiece and the deflection due to the microscopiccontact at the grain level. The mechanical properties of the work-piece material and the grinding parameters are also considered inthe undeformed chip thickness prediction model through normalgrinding force model. As the measurement of undeformed chipthickness is difficult, the experimental validation has been carriedout with the help of surface-roughness model obtained from theabove undeformed chip-thickness model.

cutting lengthl

Fig. 3. Three-dimensional shape of an undeformed chip.

3. Model development

A schematic diagram showing the interaction of the grain tipto the work piece is given in Fig. 1. At any transverse sectionm–m, the profile of groove generated by any grain is as shown in

Vs

work surface z

m

Y

Fig. 1. Schematic view of the workpiec

Fig. 2. Since an individual grain has many tiny cutting points onits surface and the speed ratio is high, the groove produced by anindividual grain can be assumed to be an arc of a circle. Therelative motion of the cutting grains with respect to the work-piece surface generates a removed chip with a curved long-itudinal shape, as shown in Fig. 3. This chip has an increasingchip thickness from zero to a maximum value tm with a crosssection determined by the grain geometry.

It can be further assumed that the material is either plowedwith little or no side pile-up or removed in the form of chipswhenever grain–work piece interference occurs. Further, since theradial positions of grain tips are random, a probability densityfunction is required to describe the surface roughness incorpor-ating all the grains engaged. Thus, the undeformed chip thickness‘t’ can be described by Rayleigh’s probability density functionproposed by Younis and Alawi [19]. Therefore, the spectrum ofundeformed chip thickness generated can be assumed to have the

mx

groove traced by grain

profile of groove

e in Cartesian coordinate system.


same mathematical distribution. The Rayleigh p.d.f., f(t) is given by

f ðtÞ ¼ðt=b2

Þe�ðt2=2b2

Þ for tZ0

0 for to0

(ð1Þ

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 ð2Þ

sðtÞ ¼ fffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið4�pÞ=2

pgb ð3Þ

It is important to mention here that the undeformed chipthickness ‘t’ is actually the depth of engagement of each indivi-dual active grain which are participating in removing material.This function has a similar shape to the logarithmic standarddistribution suggested by Konig and Lortz [20] to describe theundeformed chip thickness distribution. However, the Rayleighdistribution has the advantage of being uniquely defined by onlyone parameter, that is, b.

The complete description of surface generated is very difficultdue to the complex behavior of different grains producing groovesbecause of the random grain–work interaction. Thus, certainassumptions have to be made while predicting the undeformedchip thickness [21]. The assumptions are given as [21]: (i) anindividual grain has many tiny cutting points in its surface, there-fore, for simplicity, the grain tips are approximated as hemi spheresof diameter dg (¼2t), randomly distributed throughout the wheelvolume, (ii) the profile of the grooves generated is same andcompletely defined by the depth of engagement or undeformedchip thickness ‘t’ and (iii) on an average, the expected area ofinterference of grain tip and workpiece surface is about half of thearea of a circle.

Under these assumptions, the expected chip cross section E

(Ach) is determined by the assumed to be an arc of a circle.Therefore, the expected chip cross section (as shown in Fig. 2) canbe expressed as

EðAchÞ ¼p2

Eðt2Þ ð4Þ

The expected value of the total area of engagement projectedonto a plane perpendicular to the grain direction of movement inthe grinding zone is the ensemble of all the areas engaged by eachindividual active grain, which can be expressed as

EðAtotalÞffiNdEðAchÞ ¼Ndp2

Eðt2Þ ð5Þ

where the instantaneous number of active cutting edges isNd ¼ blcC. Where lc is the wheel–workpiece contact length, b isthe wheel width of cut and C is the number of active grits per unitarea of the wheel periphery (grit surface density). By the con-servation of volume in plasticity, the total projected engaged areamultiplied by the wheel velocity, Vs, must equal the materialremoval rate (MRR), i.e. MRR¼baeVw, or

EðAtotalÞVs ¼ baeVw ð6Þ

where Vw is the workpiece tangential velocity, Vs is the grindingwheel surface speed and ae is the depth of cut. Now, the relationbetween the undeformed chip thickness and the main variablescan be found from Eqs. (4)–(6) as

Eðt2Þ ¼2aeVw

pCVs

1

lcð7Þ

The left side of Eq. (7) can be calculated as

Eðt2Þ ¼

Z 10

t2f ðtÞdt ð8Þ

Substituting the value of f(t)from Eq. (1) in Eq. (8), theequation can be written as

Eðt2Þ ¼

Z 10

t2f ðtÞdt¼

Z 10

t3

b2e�ðt

2=2b2Þdt

it will give the value as

Eðt2Þ ¼ ½e�ðt2=2b2

Þð�2b2�t2Þ�10 ¼ 2b2

ð9Þ

where b is the parameter that completely defines the prob-ability density function as in Eq. (1).

The expected value of the undeformed chip thickness ‘t’ can becalculated in terms of parameter b as

EðtÞ ¼

Z 10

tf ðtÞdt¼ �te�ðt2=2b2

Þ þ

ffiffiffiffip2

rberf

tffiffiffiffiffiffiffiffi2b2

q0B@

1CA

264

3751

0

ð10Þ

that gives the value of EðtÞ as

EðtÞ ¼

ffiffiffiffip2

rb

or,

b¼

ffiffiffiffi2

p

rEðtÞ ð11Þ

Substituting the value of b from Eq. (11), in Eq. (9), theexpected value of chip thickness E(t) can be expressed as

EðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaeVw

CrVs

1

lc

sð12Þ

where r is the chip width-to-thickness ratio, C is the number ofactive grains per unit area of wheel surface. The variables lc and C

must be analyzed in more detail due to their strong interactionwith the undeformed chip thickness. However, the variables lcand C are calculated from the effect resulting from the grains incontact with the workpiece.

The value of C, in Eq. (12), can be obtained by a simplegeometric relationship, derived by Xu et al. [22] as follows:

C ¼ 4f=fd2g ð4p=3nÞ2=3

g ð13Þ

where dg is the equivalent spherical diameter of diamond particle,v is the volume fraction of diamond in the grinding wheel and f isthe fraction 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 [7]. To obtain the valueof C, it is assumed that only one-half of the diamond particles onthe wheel surface are actively engaged in cutting [22], or thevalue of f is equal to 0.5. The equivalent spherical diameter ofdiamond grit (dg) is given [7] as

dg ¼ 15:2M�1

where M is the mesh size used in the grading sieve.

3.1. Wheel–workpiece contact length (lc)

Grinding efficiency and workpiece surface integrity are greatlyaffected by deflections that occur within the grinding contactzone. The contact length is therefore an important determinantand depends on contact deflections which modify the shape of thecontact zone. Geometrically, the contact deflections can influenceboth the surface finish of the workpiece and the accuracy of sizeof ground components. According to Saini [13], the contactdeflection in grinding can be viewed microscopically and macro-scopically. Microscopically, a wheel grain is deflected by thenormal force exerted on it during grinding and the workpiece isdeformed in the grinding zone. Macroscopically, the grinding


wheel may be considered as a thick circular plate pressed againsta curved surface from which the material is ground. Due to elasticdeformation, the wheel–workpiece contact length and thedeformed grinding wheel radius are increased. So the grindingcontact zone has three components, which are contributing tocontact length and that are (a) the geometric grinding contactzone, (b) the elastic deflection between the wheel body and theworkpiece and (c) the microscopic contact at the grain level, asshown in Fig. 4 (Fig. 4(i)–(iii)). The real contact length would bedue to the combined effect of all the three components. In orderto calculate the real contact length, first of all, the mutual elasticdeflection between the wheel body and the workpiece and thegrits and the workpiece are calculated from Hertz relations and

+

+Combined effect of 4 (i), 4 (ii) and 4 (iii)

a

l

undeformed workpiece contact curve

deformed workpiececontact curve

workpiece

grinding wheel

active grits

F+

(iii)

grinding wheel

wd

active g

+

(ii)

w+

grinding wheel

a′

(i)

grinding wheel

+

l

a

F

F

deflectedd

δ

Fig. 4. Combined effect of (i), (ii) and (iii), showing the elastic deflection of grinding co

due to the normal grinding force. (i) Geometric grinding contact zone. (ii) Elastic conta

workpiece (iv) Enlarge view of contact zone of (ii) showing the wheel deflection. (v) Enla

of (iv) and (v), showing the elastic deflection of a grinding, wheel due to the normal g

then effect of depth of cut is taken into consideration to get thereal contact length. Now the two modes of deflection (i.e. (b) and(c) as mentioned above) are separately considered. Firstly, theelastic deflection due to the grit–workpiece contact, assumingthat no elastic deflection of the wheel takes place (as shown inFig. 4(iii) and (v)) is considered. Secondly, the elastic deflectiondue to the contact between the wheel and the workpiece,assuming the individual grinding grits are undeformed (as shownin Fig. 4(ii) and (iv)) is considered. These two modes are thencombined to give the total contact deflection of the grindingwheel–workpiece, as shown in Fig. 4(vi).

In order to calculate the elastic deflection of the grit–workpiece contact (assuming no elastic deflection of the wheel),

F

+

(iv)

undistorted grains

deflected grindingwheel

a'

+

grains assumed to be circular in shape

(v)

l

δ

+

grains assumed to be circular in shape

grinding wheel remainundeflected under load

distortedgrains

orkpiece

rits

orkpiece

workpiece

l

+

deflected grinding wheel distorted grains

(vi)

grains assumed tobe circular in shape

d

F

F

workpiecesurface

workpiecesurface

workpiecesurface

deflection

F

ntact zone that results from combined effect of depth of cut and elastic deflection

ct between grinding wheel and workpiece. (iii) Elastic contact between grits and

rge view of contact zone of (iii) showing the grains deflection. (vi) combined effect

rinding force.


a grinding grit, although irregular in shape, will be assumedspherical with diameter dg. Considering a single grit is pressedagainst a workpiece surface under a normal load Fng; the Hertzrelations for a sphere [9,23] give the mutual approach of remotepoints in the grit and the workpiece surface as

d¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3p=2

ffiffiffi2pÞ2ðKwþKgÞ

2ð1=dgÞF

2ng

3

qð14Þ

where Fng is the normal load applied to the grit, dg is the meandiameter of the grit. The workpiece and grit elasticity can beestimated, respectively, by Ki ¼ ð1�n2

i Þ=pEi, i¼ gðgritÞ,wðworkpieceÞ, where Ei is the modulus of elasticity and ni isPoisson’s ratio for the grit and workpiece material. Similarly, inorder to calculate the elastic deflection of the wheel–workpiececontact, the elastic behavior of the wheel, although the wheel is acomposite structure of grits, bond and voids, can be approximatedby assuming it to be continuum. This will be a poor approximation ifregions of both tension and compression exist, because the elasticproperties will be different in the two regions. It will also be a poorapproximation if a very small element of the structure is considered.However, it is reasonable for the present contact calculation;because the stresses will be compressive everywhere and thestressed region will be large, relative to the size of the grain [9].Assuming the grinding to be continuum, the Hertz relations forcontact between a plate (i.e. grinding wheel) and a flat surface (i.e.workpiece surface) as shown in Fig. 4(ii) and (iv), can be applied. Thelength of the contact between grinding wheel and workpiece surfacewill be given by [9,24] as

a0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2:56ðpds=bÞðKwþKsÞFn

qð15Þ

where Fn is the normal load and ds is the diameter of the grindingwheel. The wheel and workpiece elasticity can be estimated,respectively by Ki ¼ ð1�n2

i Þ=pEi, i¼ sðwheelÞ, wðworkpieceÞ,where Ei is the modulus of elasticity and ni is Poisson’s ratio forthe grinding wheel and workpiece material.

In order to assess the combined local elastic deflection, itwould be essential to club local elastic deflection caused by boththe modes (i.e. the modes (b) and (c) as mentioned above and asshown in Fig. 4(ii) and (iii)). Now from Fig. 4(iii) and (v), the loadper grit Fng , in Eq. (9) depends on the normal wheel load Fn, andthe number of active grits contacting the work surface Nd. Thislatter factor is a function of the wheel composition and the elasticdeformation itself. The influence of deformation on grits contactsis indicated in Fig. 4(v). For deflection condition shown inFig. 4(vi), the number of grains contacting the workpiece will beNd ¼ blcC. Since the distribution of grains on the wheel is random,it will be reasonable to assume that all grains are equally loadedby the normal grinding force, or Fng ¼ Fn=Nd. Substituting thevalue of Nd, the load per grit, Fng ¼ Fn=blcC. Fig. 4(vi) shows thecondition of a grinding wheel under load and can be visualized intwo steps. First the grains alone deflect, as shown in Fig. 4(v); thisgives a wheel–work contact length, l0c. Following this, wheel andworkpiece deflect, which will cause the contact length to increase.Since the radius of the grinding wheel is large as compared to thecontact length, it will be a reasonable approximation that thetotal contact length will be given by the sum of the contactlengths from the two deflections i.e.

l00cffi l0cþa0 ð16Þ

where a0 and l0c are the contact lengths as indicated in Fig. 4(iv)and (v), a0 is given by Eq. (15), the length l0c is, l0cffi2

ffiffiffiffiffiffiffidsd

pwhere d

is the mutual approach of remote points in the grit and theworkpiece, given by Eq. (14). The total contact length l00c can beevaluated by substituting the value of l0c and a0 in Eq. (16).

The contact length given by Eq. (16) does not incorporate theeffect of depth of cut. It is important to mention here that the

depth of cut has a significant influence on the grinding contactlength. Rowe et al. [25–27] proposes to calculate the real contactlength as

lc ¼ ðl2f þ l2g Þ

0:5ð17Þ

where lf is the contact length due to elastic deflection as a resultof applied force and lg (geometric contact length) relates to thecontact length due to the depth of cut, given by lg ¼

ffiffiffiffiffiffiffiffiffidsae

p.

Substituting the value of lf (that is equal to l00c) from Eq. (16) andlg ¼


p, in Eq. (17), the value of real contact length lc can be

expressed as

lc ¼2

ffiffiffiffiffiffiffiffiffiFds

pffiffiffiffiffiffiffiffiffiblcC3

p ÞF1=3n þCF1=2

n

!2

þdsae

0@

1A

0:50B@ ð18Þ

where, F¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3p=2

ffiffiffi2pÞ2ðKwþKgÞ

2ð1=dgÞ

3

qand C¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2:56ðpds=bÞðKwþKsÞ

p.

If the elastic properties Kw, Ks and Kg are known, then Eq. (18)can be evaluated numerically to give the lc for a particular wheeland work combination at any normal grinding force, Fn.

The normal grinding force play an important role in grindingprocess as it has a strong influence on local contact deflection andthe nature of the contact deflection has an important effect on themechanism of material removal. This means that the normalgrinding force is an important quantitative indicator to character-ize the mode of material removal in ceramic grinding. So therelationship between the normal grinding force and the work-piece properties, as well as the grinding parameter, could beestablished depending on the mode of material removal. Theresults of the previous studies [28–32] have revealed that thematerial removal in the machining of advanced ceramics wasmainly due to grain dislodgement. For material removal by graindislodgement removal mode, a relationship for the materialremoval rate in scratching based on a simple fracture mechanicsanalysis has been established. Generally, the removal of materialin the machining of brittle material like ceramics is influenced byits microstructure [33–35], which dominates the mechanicalproperties of the material. The volume of material removed perunit sliding distance (i.e. the cross-sectional area of a singlegroove) can be expressed approximately as [36]

V ¼ AEw

4=5

H9=5

!p2

KIC

where KIC is the fracture toughness of work material; A the constantthat may depend on the indenter geometry, lubrication condition,sliding speed, etc.; p the normal load on the abrasive particle; Ew

the modulus of elasticity of workpiece material; H the Vickersharness. It is important to mention here that the normal scratchingforce p is equivalent to the normal force per grit in grinding. Thespecific normal grinding force can be expressed as [36]:

Fng ¼ b0K1=2

IC H9=10

Ew2=5

!Vw

Vs

� �3=4

a11=12e d1=12

s ð19Þ

where b0 is the constant that includes information related to thewheel topography. Eq. (19) could be simplified to the followingform by ignoring the parameters with small exponents and byrounding off some of the exponents [36]:

Fng ¼ b0K1=2

IC H

Ew2=5

!Vw

Vs

� �3=4

ae ð20Þ

This equation indicates that the specific normal grinding force isdependent on mechanical properties of the workpiece material andthe grinding parameters, the constant b0 can be approximately equalto 0.85 [36], for grinding conducted in a micro-fracture mode.


Substituting the value of specific normal grinding force fromEq. (20) in Eq. (18), the equation for lc can be expressed as

lc ¼ w2 ae

q3=4

� �2=3

þx2lcþ2wx ae

q4

� �1=4

l1=2c þdsae

!0:5

ð21Þ

where, w¼ 2b01=3ðK1=6IC H1=3=E2=15

w ÞffiffiffiffiffiffiffiffiffiFds

p, x¼C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib0ðK1=2

IC H=E2=5w ÞbC

qand q¼Vs/Vw.

If the kinematic conditions and material properties of thegrinding wheel and the workpiece are known, then equation canbe evaluated numerically to give the lc for a particular wheel andwork combination.

It is important to mention that lc is the only part of right handside of Eq. (12) affected by elastic deflection, lc can be found fromEq. (21). If no allowance is made for elastic deflection, the contactlength lc will be equal to the geometric contact length lg and givenby lg ¼


p. The maximum undeformed chip thickness, tm

under the influence of elastic deflection can be expressed as

tm ¼lglc

� �0:5

tm ð22Þ

where tm is maximum undeformed chip thickness when noallowance is made for elastic deflection (i.e. tm can be obtainedby substituting lc ¼ lg in Eq. (12)), lg is the geometric contactlength and lc is the real contact length that can be obtained fromEq. (21).

The model developed above considers only the deflections ofgrains and the wheel. However, the deflection of machine tool isnot considered, which may also affect the undeformed chipthickness. The reason for this is that, the machine tool is a veryrigid structure as compared to diamond grinding wheel (withaluminum as the core material), which is used in the presentinvestigation. The diamond grinding wheel can be considered as athick circular plate made up of aluminum as the amount of theabrasive layer on the core is only 6% by volume while the corematerial is 94% by volume, in a diamond grinding wheel. So thesubstantial deflection would take place to the grinding wheelwhile deflection of the machine tool would be very small andhence considered insignificant in the present investigation.

A model has to be evaluated to determine its validity. Thevalidity of the model is assessed through a comparison betweenthe predicted values and measured values of the undeformed chipthickness within the predefined range of parameters. As themeasurement of undeformed chip thickness is difficult, an experi-mental validation was carried out with the help of a surfaceroughness model developed based on the undeformed chip thick-ness. It is important to mention here that the surface roughness isnot only influenced by the grooves scratched by the grains but alsoby many other factors such as wheel wear, wheel dressing, coolantapplication, dynamic stability of the machine, etc. However thefactors like wheel wear, wheel dressing, coolant application,dynamic stability of the machine, etc. are considered insignificantin the present investigation, as explained below.

A resin bonded diamond-grinding wheel is used in the presentstudy. One of the earlier research finding [37] have shown thatthe use of a resin bonded diamond grinding wheel results in alower normal grinding force as compared to that obtained withother types of diamond grinding wheels such as metal bondeddiamond grinding wheel, under the same process conditions,during ceramic grinding. This result in less value of resultantgrinding force per unit contact area between the workpiece andthe grinding wheel (grinding stress) as well as the cumulativesliding length of the grain cutting edge and thus very low wheelwear, which may be considered insignificant for the presentinvestigation. As far as the wheel dressing is concerned, sincenew diamond grinding wheels are used for the experimentation,

it does not require any kind of dressing. So this factor is alsoconsidered insignificant for the present study.

Further, heat and friction are the byproducts of the grindingprocess that adversely affects the accuracy and surface finish ofthe workpiece and wheel wear. Since, the experiments areconducted under moderate operating conditions; it will resultinto less heat generation. This does not cause any thermal damageto the workpiece (silicon carbide) and wheel (diamond grindingwheel) as silicon carbide, being a ceramic material, can sustain avery high temperature and diamond (in the diamond grindingwheel) is a very good conductor of heat. So no coolant is requiredin the present investigation. Apart from this, the microscopicobservation of the grooves is very difficult, as a ground surface isproduced by the action of large number of cutting edges (grains)on the surface of the grinding wheel (i.e. surface is generated byoverlapping of grooves produced by interacting grains).

In view of the above mentioned facts, an experimental validationwas carried out with the help of a surface roughness model developedbased on the undeformed chip thickness. The surface-roughnessmodel written in terms of undeformed chip thickness [7] is

Ra ¼t2

m

ae

� �q

qþ1

� �2

ð23Þ

where Ra is the center line average value of surface roughness and q isthe ratio of wheel speed to work speed.

A series of experiments were performed, on an ‘ELLIOTT 8–18’hydraulic surface-grinding machine, by grinding silicon carbide work-piece with diamond grinding wheels. The properties of silicon carbideworkpiece material used for experimentation in this work are:density¼3.17 g/cm3, hardness (HV)¼2700 kg/mm2, fracture tough-ness (KlC)¼4.55 MPa m1/2, modulus of elasticity Ew¼410 GPa andthermal conductivity¼145 W m�1 K�1, Poisson’s ratio (nw)¼0.14.The workpiece material was supplied by H.C. Starck Ceramics GmbHand Co. KG, Germany. The tool was diamond-grinding wheel(ASD240R100B2) (Norton make) with modulus of elasticity(Es)¼70 GPa and aluminum as core material. The modulus ofelasticity (Eg) of diamond grit is taken as 925 GPa and Poisson’s ratioof aluminum (ns) and diamond grit (ng) are taken as 0.33 and 0.20,respectively. The size of the workpiece is 20 mm�20 mm�5 mm.The other conditions taken for the experimentation were as follows:wheel speed (Vs)¼36.6 m/s, wheel diameter¼250 mm, wheelwidth¼19 mm. The main kinematic parameters for each experimentare depth of cut (ae) and the speed ratio (Vs/Vw) where Vw is the feedrate and Vs is the wheel speed, along with the experimental value ofsurface roughness as shown in Table 1. Surface roughness measure-ments were made using Talysurf-VI (cut-off length was 0.8 mm) atfive different places on the 20�5 mm2 cross-section of the workpieceafter grinding and the arithmetic mean of the values of the measure-ments has been reported in the experimental results as shown inTable 1. The experiments are replicated five times (as shown inTable 1) to mask the variability of the process. The resolution ofsurface roughness measuring instrument is 0.8 nm. This means thatthe 0.8 nm is the minimum value of surface roughness that can bemeasured by the surface roughness-measuring instrument. Howeverthe differences in the readings of surface roughness (Table 1) aremuch higher that the resolution of surface roughness measuringinstrument. So this instrument will be able to distinguish betweenthe values clearly and hence measurements made by this instrumentcould be considered accurate enough for the present study. Apartfrom this, there are sources of error in any measuring system. Theterm ‘precision’ is often used in this connection. Perfect precisionmeans that the measurements will be made with no randomvariability in the measured values or standard deviation of themeasuring system is zero. So, in order to measure the reproducibilityof the measurements, standard deviations were calculated for eachset of measurements of the surface roughness. It was found that some

Table 1Experimental 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 0.289 0.297 0.240 0.268 0.253 0.269

3 5 220 0.283 0.284 0.271 0.273 0.274 0.277

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 10 440 0.231 0.228 0.207 0.229 0.205 0.221

7 10 293 0.205 0.248 0.226 0.245 0.226 0.234

8 10 220 0.283 0.289 0.292 0.273 0.298 0.287

9 10 176 0.306 0.338 0.310 0.340 0.311 0.321

10 10 146 0.373 0.362 0.391 0.409 0.410 0.389

11 15 440 0.209 0.215 0.210 0.215 0.216 0.213

12 15 293 0.301 0.271 0.270 0.311 0.303 0.291

13 15 220 0.321 0.332 0.330 0.333 0.324 0.328

14 15 176 0.342 0.367 0.366 0.375 0.355 0.361

15 15 146 0.372 0.393 0.384 0.395 0.401 0.389

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

ae (mm) Vw (m/min) t0m (mm) t m (mm)

5 5 2.817 1.204

5 7.5 3.450 1.458

5 10 3.981 1.515

5 12.5 4.455 1.683

5 15 4.881 1.818

10 5 3.350 1.675

10 7.5 4.103 2.032

10 10 4.734 2.071

10 12.5 5.297 2.300

10 15 5.803 2.501

15 5 3.708 2.002

15 7.5 4.541 2.468

15 10 5.239 2.604

15 12.5 5.863 2.889

15 15 6.422 3.138

0.35

0.4

0.45

0.5

0.55

ess

(µm

)

Speed ratio 100

Speed ratio 220


values of surface roughness are inside one standard deviation fromthe experimental mean value while some other values need twostandard deviations to fit the data. This means that the measuredvalues of surface roughness (Table 1) have good degree of precision.

0

0.05

0.1

0.15

0.2

0.250.3

0 10 20 30 40 50Depth of cut (µm)

Sur

face

roug

hn

Speed ratio 550

Fig. 5. Surface roughness vs. depth of cut at three different speed ratios.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30 35 40 45 50Depth of cut (µm)

Surf

ace

roug

hnes

s (µ

m)

Wheel 1(ASD500R100B2)

Wheel 2(ASD240R100B2)

Fig. 6. Surface roughness vs. depth of cut for two wheel conditions.

4. Results and discussion

Abrasive machining in general and grinding in particular areprocesses that increase their importance with high demands onaccuracy of surface finish. The principal relation between therelevant process parameters and the responses, which describethe microtopography of the grinding wheel and workpiece mate-rial, are commonly in agreement in grinding research. Majordifferences appear in the absolute values of derived data. Thisseems to be mainly due to the lack of specification of theworkpiece and the grinding wheel. The question of which modelis suitable for a given application can not be answered in general.The decision must take into account the required accuracy. But italso has to be mentioned that, with higher complexity, thenumber of parameters and the effort to determine them increases.

The undeformed chip-thickness models are used to predict theperformance of the grinding process as well as to develop othermodels, like force model, surface roughness model, etc. So accuracyof these models depends much on the validity of the undeformedchip-thickness model. The existing undeformed chip-thicknessmodels somewhat deviate from the real process, as almost in allof them, the transverse shape of the grooves produced has beenassumed to be triangular in shape and are not based on the realcontact length. The elastic properties of both the wheel and work-piece cause a considerable deflection in the contact zone, resultingin an increase in the contact length and thereby decreasing theactual depth of cut. With the model developed in the present work(Eq. (22)); there is a significant reduction in the maximum unde-formed chip thickness, compared with that obtained by the existingundeformed chip thickness model [7], as observed in the resultsshown in Table 2. These results strengthen the representation oflocal contact deflection of grinding contact zone in terms of theelastic properties of the work and wheel.

Fig. 5 shows the arithmetic mean value of surface roughnesscalculated based on the new undeformed chip thickness model, forvarious values of depth of cut for three different speed ratios. It can beobserved that at higher speed ratio, the surface finish is better. This isbecause, at higher speed ratio i.e. at lower workpiece velocity, more

grains will be involved in removing a given volume of material, andthus depth of engagement will be low. Hence the surface finish isbetter. Similarly, the wheel microstructure plays a major role in thequality of the ground surfaces. Wheel 1 is made up of fine abrasivewith average grit size of about 32 mm and wheel 2 is a coarser wheel

Ra, experimental Ra, new model Ra, circular groove

00.5

11.5

22.5

33.5

44.5

5

4 6 8 10 1412 16feed (m/min)

Ra

(mic

rons

)

0

0.5

1

1.5

2

2.5

3

3.5

4 6 8 10 12 14 16feed (m/min)

Ra

(mic

rons

)

0

0.5

1

1.5

2

2.5

3

4 6 8 10 12 14 16

Ra

(mic

rons

)

ae=5�m ae=15�mae=10�m

feed (m/min)

Ra, existing model

Fig. 7. Surface finish presented against feed at various depths of cut.


with an average abrasive size of about 67 mm. It can be observed fromFig. 6, the coarser wheel produces rougher surface as compared tothat of fine wheel. Although a finer wheel produces better surfacefinish but it will cause higher forces and higher power due to thehigher specific energy governed by a smaller expected value of theundeformed chip thickness. This is a well-established fact, whichreinforces the correctness of the new undeformed chipthickness model.

4.1. Comparison with the existing model

The deviation of the surface roughness calculated with thenew undeformed chip thickness model (Eq. (22)), from theexisting undeformed chip thickness model [7] and the experi-mental values, for various values of feed and depth of cut, isshown in Fig. 7. It could be seen from this figure that the surfaceroughness increased with an increase in depth of cut and feed.This established behavior could be explained by observing thevariation of maximum undeformed chip thickness with thegrinding parameters. Increase in depth of cut causes the max-imum undeformed chip thickness to increase and thereby result-ing in a poor surface quality. It could also be seen from Fig. 7 thatthe surface roughness decreased with decrease in feed. This is asexpected since the depth of engagement would be low at low feedrate and hence the reduction in surface roughness could beobserved with the decrease in feed rate. Also at higher speedratios the surface produced is smoother. This is because at lowerworkpiece velocity (as wheel velocity is fixed in the presentstudy), more grains participate in removing a given volume ofmaterial; hence the depth of engagement is lower, producingsmooth surfaces. Further it has been observed from Fig. 7 that thecomputed values of surface roughness by new undeformed chipthickness model shows a good agreement with the experimentaldata obtained from different grinding conditions in surfacegrinding. Apart from this, the surface roughness values computedby new undeformed chip thickness model are closer, to actualvalues obtained experimentally, as compared to that calculatedby the existing undeformed chip thickness model and thuspredicting the performance of the process more accurately.

5. Conclusion

In this paper, a new analytical model for undeformed chipthickness prediction, based on the random distribution of thegrain protrusion heights and by assuming the profile of groovegenerated by an individual grain to be an arc of a circle, has beendeveloped. The model incorporates the real contact length, whichis in terms of the elastic properties of the wheel and workpiece

material and the other traditional grinding parameters. Themechanical properties of the workpiece material and the grindingparameters are also considered in the model through normalgrinding force model. The proposed model results in a significantreduction in the undeformed chip thickness, compared with thatobtained by the existing undeformed chip thickness model, underthe same operating conditions. The elastic properties of both thewheel and workpiece cause a considerable deflection in thecontact zone, resulting in an increase in the contact length andthereby decreasing the actual depth of cut which causes reductionin the undeformed chip thickness. This information gives a newunderstanding into ceramic grinding that the local contact deflec-tion of grinding contact zone, in terms of the elastic properties ofthe work and wheel, has a strong influence on the undeformedchip thickness. In addition to this, the predicted surface rough-ness, based on this new undeformed chip thickness model, showsa good agreement with experimental data obtained from differentkinematic conditions, as compared to that calculated by theexisting undeformed chip thickness model, in silicon carbidegrinding. The effect on the surface roughness due to grindingparameters changes was simulated and discussed. The grindingparameters include changes in the depth of cut, speed ratio andwheel microstructure. The predicted surface roughness, based onthe proposed model, predicts that the low depth of cut, higherspeed ratio and a finer wheel gives better surface finish (producesmoother surface), due to reduction in the undeformed chipthickness. This information leads to an important conclusion thatundeformed chip thickness can be controlled by properly con-trolling the different grinding parameters. Hence, this model canbe reliably used for the performance evaluation of the ceramicgrinding process without conducting laborious experimentation.

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predictive modeling of undeformed chip thickness in ceramic grinding

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