I

VM

a

ARR1A

KAFMR

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Wear 267 (2009) 71–80

Contents lists available at ScienceDirect

Wear

journa l homepage: www.e lsev ier .com/ locate /wear

nvestigations into abrasive flow finishing of complex workpieces using FEM

.K. Jain ∗, Rajani Kumar, P.M. Dixit, Ajay Sidparaechanical Engineering Department, Indian Institute of Technology, Kanpur 208016, India

r t i c l e i n f o

rticle history:eceived 3 August 2008eceived in revised form

a b s t r a c t

Abrasive flow machining (AFM) process provides a high level of surface finish and close tolerances withan economically acceptable rate of surface generation for a wide range of industrial components. It isattempted to analyze the AFM process using finite element method (FEM) for finishing of external sur-

4 November 2008ccepted 14 November 2008

eywords:FMEM

faces. To study the material removal mechanism of AFM, finite element model of forces acting on a singlegrain has been developed. Response surface method (RSM) is used to carry out an experimental researchto analyze the effect of extrusion pressure and number of cycles on material removal and surface fin-ish. Results obtained from finite element analysis for material removal have been compared with theexperimental data obtained during AFM.

aterial removal mechanismSM

. Introduction

Finishing operations represent a critical and expensive phasef overall production processes. Abrasive flow machining (AFM)s a nontraditional finishing process of finish machining (polish-ng, radiusing, deburring, recast layer removal, etc.) of surfacesnd edges [1,2]. In this process, a semi-solid medium consists ofpolymer of special properties, a small quantity of mineral oil, andbrasive particles mixed in a definite proportion. The medium isxtruded under pressure through or across the workpiece surfaceo be finished. Fig. 1(a) shows a schematic diagram of the experi-

ental setup [3,4] in which workpiece is placed in between the twopposite piston cylinder arrangement. When extrusion pressure ispplied through the hydraulic system to the medium by the pistonin the medium cylinder), two types of forces are generated, i.e.,xial force and radial force (Fig. 1(b)).

The AFM process is not completely understood as there is aack of quantitative relationships between process input param-ters (extrusion pressure, abrasive grain size, concentration) anderformance evaluation parameters (surface finish and materialemoval rate), specially for external surface finishing. The processodeling and analysis help in understanding the influence of vari-

us process parameters on the surface generating mechanism and

aterial removal. This is also helpful in developing an effective

ontrol mechanism for automation of a finishing process.Rhoades [1,5,6] studied the basic principle of AFM and reported

hat the depth of cut primarily depends upon abrasive grain size,

∗ Corresponding author. Tel.: +91 512 2597916; fax: +91 512 2597408.E-mail addresses: vkjain@iitk.ac.in (V.K. Jain), pmd@iitk.ac.in (P.M. Dixit),

jays@iitk.ac.in (A. Sidpara).

043-1648/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.wear.2008.11.005

© 2009 Elsevier B.V. All rights reserved.

relative hardness and sharpness, and extrusion pressure. Przylenk[7] concluded that with a small bore diameter of workpiece,more grains come in contact with the wall and material removalincreases. Kohut [8] presented some fundamentals of the processand reported that for any working pressure the amount of abrasionthat occurs is directly related to the slug length of flow. Williams andRajurkar [9,10] analyzed the effects of several machining parame-ters on key performance measures and concluded that the materialremoval and surface finish depend on medium viscosity, extrusionpressure and workpiece material. Jain et al. [11] reported aboutcomputationally intensive FEM-based model to predict materialremoval and surface roughness generated during AFM of axisym-metric internal shapes.

Jain et al. [12] used multi-variable regression analysis and neuralnetworks for the empirical modeling of AFM process. Jain and Jain[13] used neural networks for selecting the optimum values of fourimportant AFM process parameters namely concentration of abra-sive grains (% w/w), abrasive mesh size, number of AFM cycles, andflow velocity of AFM medium. Walia et al. [14] explored the appli-cation of centrifugal force for the productivity enhancement of theAFM process. They developed a finite element model of a polymer-based non-Newtonian viscoelastic fluid used in the process usingANSYS software and the same was used to evaluate the resultantpressure, velocity, and radial stresses during a working cycle.

Since, the fixture requirement for finishing of external surfaceis comparatively complex, the earlier research work in the field ofAFM is constrained in the finishing of internal surfaces including

fine holes. For internal and uniform surfaces, FEM results reportedby the researchers [11] are in agreement with the experimentalresults. Hence, an attempt is made to study AFM of tapered andcylindrical surfaces on a single workpiece. It is attempted to ana-

72 V.K. Jain et al. / Wear 26

Nomenclature

Ae area of the element[B] spatial derivatives of shape functions matrix used in

strain rateC weight of abrasives/weight of medium{Frp} global RHS vectorf coefficient of frictionFn normal force acting on the grain[K] global coefficient matrix for deformation analysis{m} constant factorNBER number of boundary elements in r direction on � 2NBEZ number of boundary elements in z direction on � 1NEM number of elements in mesh{Nb} one-dimensional matrix of biquadratic shape func-

tions{Np} column vector of bilinear shape functions for the

approximation of pressure[Nv] biquadratic shape functions matrix for velocity

approximationp hydrostatic pressure, N/m2, extrusion pressure, barRa surface roughness value{tb} traction vector at boundariestn normal traction componentsts tangential traction componentsV velocity componentsVa volume of the abrasive around the workpiece{W} global vector of weight functions in deformation

analysis{X} global values of pressure and velocities� boundary of the domainε̇ij strain rate tensor� viscosity of the medium� proportionality constant�a density of the medium around the workpiece�m density of the medium�y yield stress of the medium�ij stress tensor

lebmobamet

2

rvassca

The Galarkin weighted residual method [16] is used to reduce

�rr normal stress acting on the grain˝ region of deformation

yze the AFM process using finite element method for finishing ofxternal surfaces. The effort is made to determine the relationshipetween the flow characteristics of the medium and the results ofachining using FEM. To study the material removal mechanism

f AFM, finite element model of forces acting on a single grain haseen developed. An experimental research has been carried out tonalyze the effect of extrusion pressure and number of cycles onaterial removal and surface finish. Results obtained from finite

lement analysis for material removal have been compared withhe experimental data obtained during AFM.

. Finite element analysis

The viscoplastic or Bingham plastic material behaves as an idealigid solid when subjected to shear stress less than the yield stressalue. Above the yield stress, the viscoplastic material behaves asNewtonian fluid with shear stress being the linear function of

hear rate. This type of behavior is found in materials like pastes,

uspensions, paints, putty, etc. The medium used in AFM processan be idealized as viscoplastic material when the elastic strainsre negligible.

The following assumptions are made in analyzing the process:

7 (2009) 71–80

• The medium behaves like viscoplastic material.• Coulomb friction condition is assumed between the medium and

workpiece surface.• Process is considered isothermal as the velocities, and therefore

the temperature rise, are small.• The process is idealized as a steady-state axisymmetric process.

2.1. Rigid plastic formulation of AFM process

Under the assumption of small elastic strains, the materialbehavior beyond the yield stress is essentially as that of a fluid,i.e., material is incapable of sustaining any deviatoric stress with-out motion. Deformation of such materials can be convenientlyanalyzed by Eulerian or flow formulation. In general, the materialbehavior under deforming condition is governed by the equationsof conservation of mass, balance of momentum, and conservationof energy. Since, the process can be considered to be isothermal, theequation of conservation of energy need not be considered duringthe analysis. The Eulerian reference frame, the momentum equa-tions for an axisymmetric steady-state process (neglecting the bodyforces) have the following form:

�

(Vr

∂Vr

∂r+ Vz

∂Vr

∂z

)= −∂p

∂r+ 1

r

∂

∂r(r� ′

rr) − � ′

r+ ∂� ′

rz

∂z(1)

�

(Vr

∂Vz

∂r+ Vz

∂Vz

∂z

)= −∂p

∂z+ 1

r

∂

∂r(r� ′

rz) + ∂� ′zz

r(2)

where �: density of the medium; Vr and Vz: velocity componentswith respect to r and z axes; p: hydrostatic pressure, and � ′

ij: devi-

atoric part of the stress tensor �ij.In plastic deformation, volume remains constant. Therefore, the

continuity equation takes the form of

ε̇rr + ε̇zz + ε̇ = 0 (3)

where ε̇ij: strain rate tensor.In the constitutive equation, the deviatoric stress � ′

ijfor the vis-

coplastic/rigid plastic material is related to the strain rate by

� ′ij = 2�ε̇ij (4)

From Eq. (4), the generalized stress �̄ and generalized strain rate ¯̇εare given by the following equation:

�̄ = 3� ¯̇ε (5)

where �̄ =√

(3/2)� ′ij� ′

ijand ¯̇ε =

√(2/3)ε̇ijε̇ij .

According to Von-mises yield criterion, the expression for � forviscoplastic material [15] is given by

� = �y

3¯̇ε+ � (6)

Substituting in Eq. (5), we get

�̄ = 3(

�y

3¯̇ε+ �

)¯̇ε ⇒ �̄ = �y + 3� ¯̇ε (7)

2.2. Finite element formulation of AFM process

Eight noded rectangular isoparametric elements are chosen todescritize the domain. The domain and finite element mesh config-uration (Fig. 2(a)) with sample element are shown in Fig. 2(b).

the governing differential equations of the problem into algebraicequations by using appropriate weighting function for the momen-tum and continuity equations. Let Vz, Vr, and p be the functionswhich satisfy all the essential boundary conditions exactly. Then

V.K. Jain et al. / Wear 267 (2009) 71–80 73

etup a

ti

Hs˝t∫

w

I

I

I

ε

Fig. 1. (a) Schematic diagram of experimental s

he Vz, Vr, and p constitute a weak solution if the following integrals satisfied:∫

˝

[{ε̇rr + ε̇zz + ε̇}Wp +

{�

(Vr

∂Vr

∂r+ Vz

∂Vr

∂z

)

+∂p

∂r− 1

r

∂

∂r(r� ′

rr) − � ′

r− ∂� ′

rz

∂z

}Wr

+{

�

(Vr

∂Vz

∂r+ Vz

∂Vz

∂z

)+ ∂p

∂z− 1

r

∂

∂r(r� ′

rz) − ∂� ′zz

∂z

}Wz

]

× 2r dr dz = 0 (8)

ere, Wp, Wr, and Wz are the appropriate weight functions whichatisfy the homogeneous versions of the boundary conditions and

represents the area of the domain. After performing the integra-ion by parts, we get

˝

[I1 + I2]2r dr dz −∫

�1

I32r d�1 −∫

�2

I42r d�2 = 0 (9)

here

1 = {ε̇rr + ε̇zz + ε̇}Wp,

2 = �

{(Vr

∂Vr

∂r+ Vz

∂Vr

∂z

)Wr +

(Vr

∂Vz

∂r+ Vz

∂Vz

∂z

)Wz

}

− p{ε̇rr(W) + ε̇zz(W) + ε̇(W)} + 2�{ε̇ε̇(W)

+ ε̇rr ε̇rr(W) + ε̇zzε̇zz(W) + 2ε̇rzε̇rz(W)},

3 = tzWz, I4 = trWr,

˙ rr(W) = ∂Wr

∂r,

Fig. 2. (a) The domain and finite element mesh for axisymmetric fl

nd (b) types of forces acting on an abrasive [4].

ε̇(W) = W

r,

ε̇zz(W) = ∂Wz

∂z,

and

ε̇rz(W) = 12

(∂Wr

∂z+ ∂Wz

∂r

).

In the present work, Vz, and Vr are chosen to be biquadraticfunction of z and r, respectively and p is chosen to be a bilinear func-tion. This approximation certainly satisfies both the convergence:completeness and compatibility. The final finite element equationis obtained by assembling the elemental coefficient matrices andright side vectors into global coefficient matrix and right side vector,respectively. The elemental coefficient matrices [Ke] are calculatedby evaluating the integrals by Gauss quadrature integration tech-nique. The global finite element equation after application of all theboundary conditions is as follows:

NEM∑e=1

{We}T [Ke]{Xe} =NBER∑b=1

{Wbr }T {f b

r } +NBEZ∑b=1

{Wbz }T {f b

z } (10)

Along the boundary, the coordinates (z, r) are approximated using1D quadratic shape functions. All the elemental matrices are evalu-ated using 2 × 2 Gauss quadrature. Similarly, the elemental vectorsof the boundary are evaluated using two point Gauss quadrature.

The assembly of elemental coefficient matrices into the global coef-ficient matrix is done by transferring the elements correspondingto a local degree of freedom to positions of corresponding globaldegrees of freedom in the global coefficient matrix. Similar proce-dure is followed for the assembly of global boundary matrix. The

ow of medium and (b) sample element in natural coordinate.

74 V.K. Jain et al. / Wear 267 (2009) 71–80

a

[

To

2

ci(dcpitiii

3

ttltba

•

•

•

•

•

TB

B

ABFEA

Fig. 5) can be calculated as

V(1) = V(3) = L2dg

(D1 + D2

2

)(17)

Fig. 3. Boundaries for external surfaces.

ssembled finite element equation from Eq. (10) can be written as

K]{X} = {Frp}. (11)

he above Eq. (11) is solved using FORTRAN-77 after substitutionf the boundary conditions.

.3. Boundary conditions

The boundary conditions specified on any boundary can belassified into three categories: velocity boundary conditions (spec-fication of velocity components), stress boundary conditionsspecification of traction components) and mixed boundary con-itions (i.e. combination of one velocity and one stress boundaryondition). The domain along with the coordinate system for AFMrocess is shown in Fig. 3. Table 1 describes boundary conditions

mposed on different surfaces. After imposing boundary condi-ions as discussed in the above section, the final matrix Eq. (11)s solved by Householder Method [17]. The Householder Methods used because the resulting matrix of the mixed formulation isll-conditioned.

. Model for material removal

The material removal in AFM process is assumed by indenta-ion (caused due to normal force) of the abrasive particle (A) inhe workpiece surface followed by its linear movement along theength of the workpiece. This action of abrasive particle is analyzedo determine the volume of the material removed in the given num-er of cycles. In the modeling of material removal, the followingssumptions are made:

In each stroke the material removed by each abrasive particle isassumed to be same and constant.For the simplification of the problem, all the abrasives are con-sidered to be spherical in shape having the same size. In reality,these abrasives are irregular in shape, and size varies in a range.The normal force acting on each abrasive particle is assumed tobe same and equal. It is determined using finite element analysis.It is assumed that all the abrasive particles are evenly distributed

in the medium.The rise in temperature of the medium during machining isassumed to be negligible.

able 1oundary conditions.

oundaries Conditions

B and CD Vz = V0 (velocity of the piston) and Vr = 0C and GH |tz| = f|tr| and Vr = 0 (friction condition prevails)G and HI Vn = 0 ⇒ Vz tan ˛ + Vr = 0 and |ts| = f|tn| (inclined boundaries)F and IJ Vz = 0 and Vr = 0E and JD Vr = 0 and tz = 0

Fig. 4. Schematic diagram of a spherical abrasive and workpiece interaction.

The volume of the material removed (Vg) due to indentation by eachgrain is given by

Vg = Rh2 (12)

From geometry (in Fig. 4)

(dg

2

)2

=(

dg

2− h

)2

+ r2 (13)

As the penetration depth ‘h’ in AFM process is very small, h2 termcan be neglected. Simplifying the above equation, we get

r =√

dgh (14)

3.1. Volume of the abrasive around the workpiece

The total number of abrasive particles in contact with theworkpiece can be calculated as follows. It is assumed that the layer-thickness of the medium around workpiece is equal to the diameterof the grit. Fig. 5 shows a schematic diagram of different work-piece surfaces and layer of abrasive medium over the workpiece. S(cylindrical), T1 (tapered left side) and T2 (tapered right side) are dif-ferent surfaces of the workpiece. From Fig. 5, volume of the mediumaround the cylindrical part is

V(2) =

4((D1 + 2dg)2 − D2

1)L1 = (d2g + D1dg)L1 (15)

As the grain diameter dg is very small compared to D1, we canneglect the higher order term (d2

g ). Then,

V(2) = L1D1dg (16)

Volume of the medium around the tapered portions (1 and 3 in

Fig. 5. Schematic diagram of workpiece and layer of the abrasive media over theworkpiece (L1 = 15 mm, L2 = 7.5 mm, D1 = 40 mm, and D2 = 30 mm).

ear 26

w

V

3

fosf

C

Sa

V

3

tcItsct

t

w

l

t

ww

L

Ni

N

w

N

V.K. Jain et al. / W

From the geometry, total volume of the medium around theorkpiece can be calculated

m = 2V(1) + V(2) ⇒ Vm = dg(D1L1 + (D1 + D2)L2) (18)

.2. Number of abrasives in contact with the workpiece

The total number of abrasive particles around the workpiece sur-ace can be calculated by calculating the ratio between total volumef abrasive particles and volume of each grain. Total volume of abra-ive particles around the workpiece surface can be calculated asollows:

oncentration (C) = �aVa

�mVm⇒ Va = C�mVm

�a

ubstituting the value of Vm, we get the volume of the abrasive (Va)round the workpiece as

a = C�mdg(D1L1 + (D1 + D2)L2)�a

(19)

.3. Slug length

In AFM process, the medium extruded after abrading thoroughhe workpiece is called “Slug”. The numbers of abrasive particles inontact with the workpiece per stroke depend upon the slug length.f the slug length is more, then the abrasive particles in contact withhe workpiece surface is more and vice-versa. For calculating thelug length, it is assumed that the medium is filled in the mediumylinder and the workpiece is tightly held (up to AA in Fig. 6). Thenime taken for the piston to travel one stroke length is

= Ls

Vp(20)

here Ls: stroke length and Vp: piston velocity.In the same time (t), the medium travels the length equal to slug

ength through the workpiece. Hence,

= Lslug

Vf(21)

here Lslug: slug length and Vf: velocity of the medium near theorkpiece.

Equating Eqs. (20) and (21), we get slug length as

slug = LsVf

Vp(22)

o. of abrasives in contact with the workpiece in one stroke length,s given by

a = volume of abrasive around workpiecevolume of each abrasive

× Lslug

Lw(23)

here Lw: workpiece length (L1 + 2L2).Substituting Eq. (19) in Eq. (23) and simplifying, we get

a = 6C�mLsVf

�ad2gVpLw

(D1L1 + (D1 + D2)L2) (24)

Fig. 6. Slug length for AFM of external surfaces.

7 (2009) 71–80 75

But in actual practice, out of this no. of abrasive particles all donot participate actively in removing the material. Let us assume K1percent of the particles are active, then Eq. (24) becomes

Na = K16C�mLsVf

�ad2gVpLw

(D1L1 + (D1 + D2)L2) (25)

3.4. Penetration depth

The maximum stress developed on the work surface due to nor-mal force exerted by the abrasive grain is given by

�w = Fn

area of contact(26)

where area of contact = r2 = dgh. Hence

�w = Fn

dgh(27)

Normal force acting on a grain is given by

Fn = �rr ×

4d2

g (28)

The flow stress (�w) and Brinell hardness (Hw) of the workpiece arerelated by,

�w = K2Hw (29)

where K2 = 1 for brittle materials and K2 > 1 for ductile materials (forsteel, K2 = 3.1).

h = Fn

dgK2Hw(30)

Substituting the value of ‘h’ from Eq. (30) in Eq. (12), the volume ofmaterial removed per grain will be as follows:

Vg =

2dg

[Fn

dgK2Hw

]2

(31)

Assuming that the axial force acting on each grain is high enoughto remove the material in the form of chips produced due to thedepth of penetration equal to ‘h’.

Total volume of material removal (Vt) in one stroke:

Vt = VgNa (32)

Total material removal after ‘n’ number of cycles (2n strokes):

V =

2dg

[Fn

dgK2Hw

]2

× K16C�mLsVf

�ad2gVpLw

(D1L1 + (D1 + D2)L2)(2n)

(33)

Simplifying Eq. (33), we finally get the total volumetric materialremoval after ‘n’ number of cycles:

V = 6KnC�mLsVf (D1L1 + (D1 + D2)L2)

�ad3gVpLw

(Fn

Hw

)2(34)

where K = K1K−22

The value of ‘K’ is determined in next Section 5.3. Normal stress(�rr) acting on a grain is obtained from the finite element analysisfrom which the normal force acting on each grain (Fn) is calculated(Eq. (28)).

4. Experimentation

Response surface methodology (RSM) is applied to predict theresponse parameters with a given set of conditions. Workpiecematerial is brass (62% Cu, 2% Pb and balanced Zn) having hardnessVPN 110-130. Workpieces are initially taper turned from the cylin-drical bars and then finished by taper turning process itself. Silicon

76 V.K. Jain et al. / Wear 267 (2009) 71–80

Table 2Coded levels and corresponding actual values of process parameters.

Parameter Levels

PN

cstatcaas[bll

5

aolremaors

5

wiiv

Y

5

a

Fig. 7. Effect of extrusion pressure on material removal.

TP

R

−1.414 −1 0 1 1.414

ressure (bar) 20 25 40 55 60o. of cycles (N) 5 8 15 22 25

arbide (average particle size = 330 �m, 56% w/w) is used as an abra-ive, and it is mixed with polyborosilixane carrier and oil to makehe medium for AFM. By using RSM, two independent parametersre chosen viz. extrusion pressure (P), and number of cycles (N) andheir selected levels are shown in Table 2. −1.414 to +1.414 indicatesoded values of the parameters whose absolute values (all positive)re given in the table. Table 3 shows the randomized experimentsnd levels of each independent parameter. This randomized testequence is necessary to prevent the effects of unknown variables18]. When the upper cylinder completes its movement from top toottom and reaches again at the top is referred as one cycle. Stroke

ength is 65 mm and it is little more than the twice of the workpieceength. So medium passes over the workpiece completely.

. Results and discussion

Based on the response surface model obtained after regressionnalysis, the results in terms of the effect of pressure and numberf cycles on material removal and surface finish have been calcu-ated and discussed based on both experimental and theoreticalesults. Surface roughness is measured at four different places onach surface (cylindrical and taper) and then averaged. The paperainly deals with modeling of the process, and the experiments

re conducted to validate the proposed model. During comparisonf theoretical and experimental results, almost same initial surfaceoughness values were maintained to minimize the effect of initialurface roughness variability on the responses.

.1. Material removal

Material removal is calculated by measuring weight of theorkpiece before and after finishing operation. From the exper-

mental results (Table 3), following response equation (Eq. (35))s obtained for evaluating material removal (YMR) by analysis ofariance (ANOVA).

2 2

MR = −92.72 + 3.99P + 6.37N − 0.03P − 0.13N − 0.03PN (35)

.1.1. Effect of extrusion pressureFig. 7 shows the effect of extrusion pressure on material removal

t different number of cycles. It is clear from the graph that the effect

able 3lan of experiments and material removal results for external surfaces.

un no. Exp. no. Coded level

Pressure, P (bar) No. of cycles, N

1. 8 0 1.4142. 6 1.414 03. 5 −1.414 04. 4 1 15. 2 1 −16. 3 −1 17. 7 0 −1.4148. 1 −1 −19. 13 0 0

10. 11 0 011. 9 0 012. 10 0 013. 12 0 0

Fig. 8. Effect of number of cycles on material removal.

of pressure is significant on material removal. The normal force act-ing on the abrasive particle is responsible that has direct influenceon depth of cut and hence material removal. As the extrusion pres-sure increases the normal force acting on the abrasive particle alsoincreases so material removal increases.

5.1.2. Effect of number of cyclesFrom Fig. 8, it is evident that as the number of cycles increases

the material removal increases. In the initial cycles of machining theincrease in material removal is high because the workpiece surface

Actual level Material removal, YMR (mg)

Pressure, P (bar) No. of cycles, N

40 25 73.0060 15 81.6620 15 42.3355 22 83.0055 8 73.3325 22 42.6640 5 46.6625 8 19.6740 15 70.0040 15 66.3340 15 75.6740 15 65.3340 15 72.00

V.K. Jain et al. / Wear 267 (2009) 71–80 77

Table 4Surface finish results for external surfaces (see Fig. 5).

S. no. P (bar) No. of cycles Ra on surface ‘S’ (�m) Ra on surface ‘T1’ (�m) Ra on surface ‘T2’ (�m)

Before mach. After mach. �Ra Before mach. After mach. �Ra Before mach. After mach. �Ra

1 40 25 4.9375 3.4475 1.4900 2.4400 1.3275 1.1125 1.8225 1.0350 0.78752 66 15 5.4975 3.3350 2.1625 2.8525 1.4300 1.4225 2.4675 1.4175 1.05003 20 15 4.9250 3.8025 1.1225 2.0625 1.1075 0.9550 2.5075 1.9675 0.54004 55 22 3.8550 1.4325 2.4225 2.4250 0.9275 1.4975 2.9575 0.9925 1.96505 55 8 1.8325 0.7425 1.0900 2.3300 1.0850 1.2450 1.9200 1.0300 0.89006 25 22 2.6350 1.4975 1.1375 2.0350 1.0650 0.9700 2.6925 2.0300 0.66257 40 5 1.9200 0.7675 1.1525 2.8450 1.8000 1.0450 2.3900 1.6575 0.73258 25 8 2.4150 1.4350 0.9800 4.0750 3.8725 0.2025 1.4550 1.2200 0.23509 40 15 2.6950 0.7825 1.9125 3.4500 2.3575 1.0925 2.0775 1.3600 0.7175

3.953.94

1 4.404.28

iambpsi

5

fgc

Y

10 40 15 3.0350 1.3625 1.672511 40 15 2.8625 1.0300 1.83252 40 15 3.0600 1.2875 1.7725

13 40 15 3.2700 1.4150 1.8550

s having more in number and sharp irregularities. As a result thebrasives are able to easily abrade the peaks of the surface. Afterachining for a certain number of cycles (say, 20 or so), the surface

ecomes more uniform than in the beginning (i.e. height of theeaks reduces). Now, the abrasive particles may just flow over theurface through the newly created path, if the axial force requireds more than the one acting on an abrasive particle [19].

.2. Surface finish

For surfaces S, T1, T2, as shown in Fig. 5, the model equationsor �Ra have been derived by RSM using the experimental results

iven in Table 4. Using Eqs. (36–38), parametric analysis has beenarried out.

�Ra (S) = −0.0059 + 0.023P + 0.079N − 0.0005P2

−0.0052N2 + 0.0028PN (36)

Fig. 9. (a) Effect of pressure on �Ra for surface ‘S’, (b) effect of pressure on

25 2.9125 1.0400 1.9950 1.1750 0.820050 2.8150 1.1300 2.0200 1.1825 0.837550 3.3825 1.0225 3.6125 2.7825 0.830000 3.2500 1.0300 2.3625 1.6475 0.7150

Y�Ra (T1) = −0.73 + 0.03P + 0.09N + 0.00008P2

−0.0006N2 − 0.0012PN (37)

Y�Ra (T2) = 0.87 − 0.02P − 0.05N

+0.0002P2 + 0.0005N2 + 0.0015PN (38)

5.2.1. Effect of extrusion pressureFrom the graphs shown in Fig. 9, it is evident that as the pres-

sure increases the change in Ra (�Ra) also increases. This is becausewhen the pressure increases the normal force acting on each grain

also increases which results in deeper indentation on the workpiecesurface. If the axial force on an abrasive is more than the resistanceoffered by the workpiece material, the removal of the peaks overthe surface of the workpiece takes place leading to the improvedsurface finish. At high extrusion pressure, time required to finish a

�Ra for surface ‘T1’ and (c) effect of pressure on �Ra for surface ‘T2’.

78 V.K. Jain et al. / Wear 267 (2009) 71–80

ycles o

ci

if

•

•

5

c

Fig. 10. (a) Effect of no. of cycles on �Ra for surface ‘S’, (b) effect of no. of c

omponent is reduced because medium provides more and deeperndentations on the workpiece surface.

Comparing Fig. 9(a)–(c) for the surfaces S, T1, and T2, the increasen �Ra is more for cylindrical surface (S) compared to tapered sur-aces (T1 and T2). This is because of the following reasons:

There is a possibility for the medium to flow either withouttouching the tapered surface or with insufficient force to removematerial from certain regions of workpiece during reverse stroke.The restriction passage for the cylindrical surface is higher com-pared to tapered surfaces, hence higher force acting on an abrasive

particle interacting with cylindrical surface.

.2.2. Effect of number of cyclesFig. 10(a)–(c) shows that as the number of cycles increases, the

hange in �Ra also increases. This is because major part of the ini-

Fig. 11. Comparison of theoretical and experimental m

n �Ra for surface ‘T1’ and (c) effect of no. of cycles on �Ra for surface ‘T2’.

tial roughness of the workpiece is removed in a few cycles in thebeginning of AFM. It can be observed that at higher pressure therate of improvement in surface finish is higher. At higher pressure,less number of cycles is required to attain a particular surface finishvalue as compared to the low pressure. At low pressure, indentationforce exerted on the workpiece surface is low which results in lowfinishing rate.

5.3. Comparison of theoretical and experimental material removal

From the finite element analysis the stress exerted on each abra-

sive grain (i.e. the stress in r-direction) is obtained. The normalforce applied on each abrasive grain is calculated by multiply-ing the cross sectional area of the grain. The volumetric materialremoval is calculated using Eq. (34). The value of constant K inEq. (34) is calculated using the experimental value of the mate-

aterial removal at (a) 10 cycles and (b) 20 cycles.

V.K. Jain et al. / Wear 267 (2009) 71–80 79

Fig. 12. Comparison of theoretical and experimental material

Table 5Values of constant K at different conditions.

S. no. Pressure (bar) No. of cycles K

1 40 25 0.040592 55 8 0.044823

K

rT

5

irir

••

•

5

ihtoaloweidr

6

omii

[

[

60 15 0.01706

avg 0.03415

ial removal. Values of K for different conditions are given inable 5.

.3.1. Effect of extrusion pressureFrom Fig. 11(a) and (b), it is evident that the trend of the theoret-

cal material removal is in agreement with experimental materialemoval. As the pressure is increasing the material removal isncreasing. The discrepancy between theoretical and experimentalesults may be due to following reasons:

It may be due to fixture design.It may be due to simplified assumptions made in the theoreticalmodel.The value of constant K is determined by back calculation, whichmay give erroneous value.

.3.2. Effect of number of cyclesFrom Fig. 12(a) and (b), Theoretical MR is linearly varying with

ncrease in no. of cycles. In AFM process, the material removal isigher in initial stage of machining due to highly irregular peaks onhe surface. After a few no. of cycles, the surface becomes morer less uniform and the increase in material removal becomeslmost negligible. But in theoretical model, to simplify the prob-em, it is assumed that equal material is removed in each numberf cycles, and abrasive particle is also assumed as a sphericalhile in real case it is irregular shape. As pressure increases, sharp

dges of the abrasive provide more cutting action which resultsn higher material removal. This may be some of the reasons foriscrepancy in the experimental and theoretical values of materialemoval.

. Conclusions

In the present study, finite element model is applied for AFMf external surfaces by considering the medium as a viscoplasticaterial. Model for material removal is developed by considering

ndentation of abrasive grain on the workpiece surface. The exper-ments are carried out on brass material to study the effects of

[

[

removal at (a) 20 bar pressure and (b) 50 bar pressure.

pressure and number of cycles on material removal and surfacefinish. The following conclusions have been derived:

• The extrusion pressure affects the material removal significantly.As the extrusion pressure increases the material removal alsoincreases.

• The effect of number of cycles on material removal is higherin initial stage. As the number of cycles increases (say, 20 orso) the experimental increase in material removal becomesnegligible.

• For the same extrusion pressure and number of cycles the increasein �Ra for conical surface is less as compared to cylindrical sur-face.

• Theoretical material removal calculated from the model devel-oped is compared with the experimental results. They are foundto have the same trend but quantitatively there are deviationsbetween the two.

Acknowledgement

The authors acknowledge the financial support provided bythe Aeronautics R&D Board, New Delhi, for Project No. 471 DARO/08/1161387/M/I.

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