Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2010, Article ID 631437, 6 pagesdoi:10.1155/2010/631437

Research Article

Cutting Force in Stone Machining by Diamond Disk

S. Turchetta

Dipartimento di Ingegneria Industriale, Università Degli Studi di Cassino, Via G. di Biasio 43, 03043 Cassino, Italy

Correspondence should be addressed to S. Turchetta, turchetta@unicas.it

Received 7 June 2010; Revised 1 September 2010; Accepted 6 October 2010

Academic Editor: João Paulo Davim

Copyright © 2010 S. Turchetta. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Stone machining by diamond disk is a widespread process to manufacture standard products, such as tiles, slabs, and kerbs.Cutting force and energy may be used to monitor stone machining. Empirical models are required to guide the selection of cuttingconditions. In this paper, the effects of cutting conditions on cutting force and cutting energy are related to the shape of theidealized chip thickness. The empirical models developed in this paper can be used to predict the variation of the cutting energy.Therefore these models can be used to guide the selection of cutting conditions. The chip generation and removal process has beenquantified with the intention of assisting both the toolmaker and the stonemason in optimising the tool composition and cuttingprocess parameters, respectively.

1. Introduction

Cutting force and energy are important parameters tobetter understand the machining process, since theyare directly related to tool wear, cutting temperatures,and surface integrity. In stone machining abrasive gritspass on machined surface by removing stone mineralconstituents. To understand the prevailing mechanism ofabrasive-workpiece interactions during stone machining is anecessary step in order to efficiently use the cutting process.The understanding of the cutting phenomena leads tomodels that voice the relationship between cutting behaviourand control parameters. In order to achieve better controlof a cutting process, a model is required to demonstrate therelationship between cutting and control parameters.

Very few researches exist in the literature on stonecutting. Jerro et al. [1] showed a mathematical approach todefine and derive theoretical chipping geometries. From theknowledge of the theoretical chipping geometries, chip areaand mean chip thickness relations were obtained. The rela-tionship between tangential cutting force and obtained chipthickness is empirically investigated. Brach et al. [2] studiedthe problem to convert dynamometer readings of specificcutting energy into power consumed. Asche et al. [3] showedthe empirical results of the influence of process parameterson tool wear. Tönshoff and Warnecke [4] developed a model

on stone cutting by disc-like diamond tools that is widelyused even if it is not completely tested. The model showsthe mechanical interaction of tool and workpiece as causedby the elastic and plastic workpiece deformation of thecutting grits, the friction between stone and diamonds, stoneand matrix, and swarf and matrix. Konstanty [5] presenteda theoretical model of natural stone sawing by means ofdiamond impregnated tools for both circular and framesawing. These models seem not to have been tested by meansof experiments. Pai et al. [6] collected and observed chipsamples under a scanning microscope and related them tothe specific grinding energy. Wang and Clausen [7] simulatedthe cutting process of diamond grit by single-point cuttingtools and demonstrated that the tendency of cutting forcesand characters of grooves are similar. Di Ilio and Togna[8] showed an analytical model to foresee the maximumwear rates of grains and matrix which provide evidence ofthe wear mechanism. It is based on the assumption thattool wear rate depends on two basic factors: the formerbeing only a matrix characteristic and the latter only a graincharacteristic, both to be determined experimentally. Ersoyet al. [9] established a relationship among the cutting specificenergy, the wear of the diamond saw, and the rock propertiesfor different rock types. These investigations do not try togive an organic comprehension of phenomena that happenat the tool-workpiece interface during stone cutting.

2 Advances in Materials Science and Engineering

The literature offers many works on grinding of ductileor brittle materials. Malkin and Hwang [10] proposed aninteresting model of the relationship between the grindingpower and the rate of plowed surface area generated byabrasive cutting grits that interact with the workpiece inceramics. He deepened the model by taking into accountthe influence of the rounding at the tip of the triangular-shaped grit on the specific grinding energy [11]. The work onmetal grinding shows many approaches to model grindingforce [12]. They are based on empirical [13] or physicalconsiderations [14].

The present work models the widespread process of stonemachining by electroplated diamond disk. The diamondgrits on the tool surface remove material through thecracking of the stone volume. The cut is mainly influenced bythe physical-mechanical stone material properties, like gritsize and strength: different minerals have different effectson the cutting mechanism, on the binder abrasion, and onthe diamond wear. Other factors influencing this processare the forces between diamonds and material, the stressdistribution in the rock, the temperatures in the tool-workpiece interface.

This work aims to investigate the relationship among thecutting force and energy and the relevant cutting parameters,such as the depth of cut and the feed rate. In particular,the machining conditions that are most interesting froman industrial point of view have been investigated. Thecutting force and energy have been modelled as a functionof equivalent chip thickness and material removal rate(MRR) by simple and general power function. The obtainedmodels have been tested for different values of processparameters.

In the following the models developed for cutting forceversus equivalent chip thickness and for specific cuttingenergy versus equivalent chip thickness or material removalrate are presented. Then, they have been tested for differentprocess conditions.

2. Stone Machining by a Diamond Disk:Cutting Force and Specific Cutting Energy

The cutting force may be measured by a dynamometer placedunder the workpiece during the stone machining, as shownin Figure 1. A dynamometer may measure the componentsof the cutting force, which acts on the workpiece, along thefeed rate direction and along the perpendicular to the feedrate direction, F f and Ffn, respectively; see Figure 2(a). Theresultant R of the F f and Ffn components may be calculatedas

R =√F2f + F

2fn. (1)

The resultant R forms an angle β with the component F f

β = tan−1(FfnF f

). (2)

Work piece

Dynamometer

Va

Figure 1: Stone machining test.

The angle of contact between disk and workpiece is given by

θ = cos−1(

1− 2dpd

), (3)

where d and dp are disk diameter and depth of cutThe tangential Fc and radial Ft components of the cutting

force may be calculated by the resultant R (see Figure 2(b))

Fc = R sin δ,Ft = R cos δ,

(4)

where

δ = β − Z · ϑ. (5)

The Z parameter in (5) depends on the location of theapplication point of the resultant force R on the arc of contactAC between disk and workpiece. Thus,

Z = ABAC

. (6)

Before obtaining the components Ft and Fc by the mea-surements of F f and Ffn values, some way of estimating thevalue of Z must be found. If the depth of cut has a smallvalue, the tangential Fc and the Ffn components of the cuttingforce roughly coincide (see Figure 2(b)). This is true whenthe ratio between the depth of cut and the tool diameteris smaller than 0.025 mm. A more careful estimate of Zparameter may be found by empirical investigations, such asthose found in [14]. A value of 0.5 may be adopted in thiswork, for a depth of cut up to 0.5 mm.

2.1. Cutting Force and Equivalent Chip Thickness. The tan-gential Fc and radial Ft cutting forces are proportional to theequivalent chip thickness heq by means of a power function

Ft = Kt · hυteq, (7)

Fc = Kc · hυceq, (8)

Advances in Materials Science and Engineering 3

Stone

Dynamometer

Va βF fR

A

Disk

d

Vt

BCFfn

dp

(a)

Ffn

β

Fc

δR

Ft

Zϑ

F f

(b)

Figure 2: (a) F f and Ffn measurements by dynamometer; (b) Ft and Fc force components.

where Kt and Kc are the cutting force coefficients, υt and υcare constants.

The equivalent chip thickness is equal to

heq =dp · vavt

. (9)

It is determined by depth of cut dp, the feed and the cuttingspeed, va and vt, respectively.

Since the cutting power is the product of the tangentialforce and the cutting speed, a nonlinear relationship alsoexists for cutting power. The specific cutting power can beexpressed as

Ec = Fc · vtva · dp · b

, (10)

where vt and b are the cutting speed and width of cut.The numerator is the time rate of power consumption,

while the denominator is the time rate of stone volumeremoval. It tends to be constant for a given work material,mill specification, and undeformed chip thickness just asthe fracture stress tends to have a characteristic value fora given material and type of loading. It varies significantlywith chip thickness as well as with the condition of thedisk face due to dressing technique and grit wear. Specificenergy is a convenient quantity to use in estimating cuttingforces. Substituting (8) and (9) in (10), the following result isobtained:

Ec = Kcb· hυc−1eq = Ke · hυeeq, (11)

where Ke = Kc/b and υe = υc − 1.The material removal rate (MRR) takes into account the

most important process parameters, and it is given by

MRR = va · dp · b. (12)

Substituting (9) in (12),

MRR = heq · vt · b, (13)

and therefore

heq = MRRvt · b

. (14)

Substituting (14) in (11) results in

Ec = Kcbυc · vυc−1t

·MRRυc−1 = Km ·MRRυe , (15)

where Km = Kc/bυc · vυc−1t .It can concluded that by defining 4 parameters

(Kc,Kt, υc, υt) it is possible to model both the cutting force(Fc,Ft) and the specific cutting energy (Ec) by means of(7), (8), (10), and (15). Those equations are general, theyare valid for different values of cutting parameters, as wedemonstrate in the following section.

2.2. Cutting Force and Maximum Chip Thickness. Cuttingforces play an important role in all stone machining pro-cesses. They are a function of the maximum chip thicknessand the geometry of the idealised sawing chip; see Malkinand Hwang [10], Milton [14]. Different chip shapes lead todifferent sawing behaviour. The shape of the idealised chip isoften characterised by the maximum thickness (hc) and thelength of the chip (lc).

The maximum chip thickness is equal to

hc =√

6 · dp ·VaC · r · vt · lc

, (16)

where C is the density of the active grain distribution on thetool surface, lc is the chip length, r = bc/hc is the chip shaperatio, and bc is the average width of the chips.

The density of the active grain distribution on the millsurface, C, and the with chip shape ratio, r, have beencalculated in previous study by Carrino et al. [15].

The cutting force is proportional to the maximum chipthickness hc by means of a power function

Fc = Hc · hεcc ,Ft = Ht · hεtc ,

(17)

where Fc and Ft are the tangential and radial force,Hc andHtare the cutting force coefficients, while εc and εt are constants.

3. Experimental Test

Experiments were undertaken on a Brembana CNC machin-ing centre. An electroplated diamond disk is commonly used

4 Advances in Materials Science and Engineering

Table 1: Tool properties.

Tool properties Diamond disk

Diamond mesh (#) 45/50

Diamond concentration (grain/mm2) 2.2

Tool diameter (mm) 180

Tool thickness (mm) 5

Table 2: Mechanical properties of Coreno Perlato Royal and WhiteCarrara marbles.

Material properties Coreno Perlato RoyalWhite Carrara

Carrara

Density (kg/m3) 2740 2705

Water absorption (%) 4.0 0.06

Compressive strength (MPa) 163 131

Young modulus (MPa) 72000 75000

Flexural strength (MPa) 12.8 16.9

Abrasion resistence 0.95 0.52

Impact resistance (cm) 32 61

Knoop hardness (MPa) 2001 1463

Table 3: Experimental plan.

Factors no. Levels Levels

Cutting depth (mm) 140.01-0.02-0.03-0.04-0.05-0.06-

0.07-0.08-0.09-0.1-0.2-0.3-0.4-0.5

Feed speed (mm/min) 3 200-400-600

Cutting speed (rpm) 1 2000

Replications 3

Total cuts 126

to cut ornamental stone. Its tool properties are shown inTable 1.

The workpiece material was Coreno Perlato Royal mar-ble. It mainly consists of CaCO3 with inclusions of seaweedand fossils that produce light and dark spots appreciatedfrom an aesthetic point of view. Its mechanical properties areshown in Table 2 in comparison with those of well-knownWhite Carrara marble.

Three feed speed values and fourteen cutting depthvalues were taken into account; they were chosen in orderto reproduce the commonly used industrial range of processvariables. Each cut was replicated three times, yielding a totalof 126 measured forces. The experimental plan is shownin Table 3. The cutting conditions were represented by theequivalent chip thickness heq. The experimental cuts wereperformed in a random sequence, in order to reduce theeffect of any possible systematic error. The cutting forcesFfn and F f have been measured by a Kistler piezoelectricplatform dynamometer (Type 9257 BA).

4. Result Discussion

ANOVA analysis underlined that both feed rate and depthof cut significantly influence the force components Ffn and

23456789

Ft(N

)

10111213141516171819

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003

heq (mm)

200 mm/min400 mm/min

600 mm/min

Ft = 61.3 · h0.232eqFt = 77.7 · h0.233eqFt = 121 · h0.236eq

400 mm/min

600 mm/min

Figure 3: Comparison of model and experimental data of Ft versusheq and va.

23456789

Fc(

N)

1011121314151617181920

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003

heq (mm)

200 mm/min400 mm/min

600 mm/min

Fc = 69.5 · h0.213eqFc = 82.1 · h0.215eqFc = 111 · h0.217eq

400 mm/min

600 mm/min

Figure 4: Comparison among model and experimental data of Fcversus heq and va.

F f , even if the depth of cut seems to be the most significantvariable. An increase of both the depth of cut and the feedspeed causes an increase of both force components.

Regression analysis of the experimental data was carriedout to the constant values in (7), (8), (11), (15), (17). Thecoefficients of determination were higher than 98%, whilethe hypotheses (normality and homogeneity of variance)about the residuals were satisfied. The radial cutting forceFt versus the increase of the equivalent chip thickness andthe feed speed is reported in Figure 3. It increases withthe increase of the feed speed from a maximum of 8 N at200 mm/min to 18 N at 600 mm/min. The tangential cuttingforce Fc versus the increase of the equivalent chip thicknessand the feed speed is shown in Figure 4. It increases with theincrease of the equivalent chip thickness from a maximum of12 N at 200 mm/min to 18 N at 600 mm/min. The values ofthe tangential cutting force Fc are comparable with those ofthe radial cutting force Ft .

Advances in Materials Science and Engineering 5

050

100

150

200Ec

(kJ/

m3)

250

300

350

400

450

500

0 0.00005 0.0001 0.00015 0.0002 0.00025 0.0003

heq (mm)

200 mm/min400 mm/min

600 mm/min

Ec = 13.9 · h−0.787eqEc = 16.5 · h−0.786eqEc = 22.1 · h−0.783eq

400 mm/min

600 mm/min

Figure 5: Comparison among model and experimental data of Ecversus heq and va.

0

50

100

150

200

Ec

(kJ/

m3)

250

300

350

400

450

500

0 200 400 600 800 12001000 1400 1600

MRR (mm3/min)

200 mm/min400 mm/min

600 mm/min

Ec = 149161 ·MRR−0.4871Ec = 30443 ·MRR−0.2851Ec = 26458 ·MRR−0.2804

400 mm/min

600 mm/min

Figure 6: Comparison among model and experimental data of Ecversus MRR and va.

The specific cutting energy Ec versus the increase of theequivalent chip thickness heq and the feed speed is shown inFigure 5. It decreases with the increase of the equivalent chipthickness from a maximum of 480 kJ/m3 at 200 mm/min to320 kJ/m3 at 600 mm/min. Finally, the relationship betweenthe specific cutting energy Ec and the MRR appears to be wellexplained by (15). An increase of MRR involves a decrease ofEc specific cutting energy, as shown in Figure 6 for the threeconsidered values of the feed speed.

The radial cutting force Ft versus the increase of the max-imum chip thickness is reported in Figure 7. The tangentialcutting force Fc versus the increase of the maximum chipthickness is shown in Figure 8.

5. Conclusion

This work shows the models to foresee the cutting forceand energy as a function of the process parameters in

02468F t

(N)

101214161820

0 0.0004 0.0008 0.0012 0.0016 0.002

hc (mm)

Ft = 38337 · h1.208eq

Figure 7: Comparison of model and experimental data of Ft .

02468F

c(N

)101214161820

0 0.0004 0.0008 0.0012 0.0016 0.002

hc (mm)

Fc = 88107 · h1.316c

Figure 8: Comparison of model and experimental data of Fc.

the widespread stone machining by electroplated diamonddisk. ANOVA demonstrated that the cutting force stronglydepends on both the depth of cut and the feed speed. Theinfluence of those two parameters has been syntheticallyconsidered by means of the equivalent chip thickness orthe material removal rate (MRR). The relationship betweencutting force (or specific cutting energy) and equivalent chipthickness (or MRR) was modelled by power function forthree different feed speeds, 200 mm/min, 400 mm/min, and600 mm/min.

Also, the cutting force components depend on themaximum thickness of chip removed by single diamond.

The obtained models are effective, simple, and general.They represent a first step towards the optimisation of thestone machining diamond disk.

Nomenclature

b: Width of cut (mm)dp: Depth of cut (mm)hc: Maximum chip thickness (mm)vt: Cutting speed (m/min)f : Feed rate (mm/root)C: Density of active diamond gritsr: Shape of chip sectionlc: Chip length (mm)

6 Advances in Materials Science and Engineering

va: Feed speed (m/min)F f : Cutting force component along feed

direction (N)Ffn: Cutting force component along the

perpendicular to the feed direction (N)Ft : Tangential cutting force (N)Fc: Radial cutting force (N)R: Resultant of F f and Ffn (N)β: Angle between R and F f (◦)δ: Angle between Ft and Fc (◦)θ: Angle at center of mill corresponding of arc

of mill work contact (◦)Ec: Specific cutting energy (J)MRR: Material removal rate (mm3/min).

Acknowledgment

This work has been carried out with the support of the ItalianMIUR (Ministry of Instruction, University and Research).

References

[1] H. D. Jerro, S. S. Pang, C. Yang, and R. A. Mirshams,“Kinematics analysis of the chipping process using the circulardiamond saw blade,” Journal of Manufacturing Science andEngineering, Transactions of the ASME, vol. 121, no. 2, pp. 257–264, 1999.

[2] K. Brach, D. M. Pai, E. Ratterman, and M. C. Shaw, “Grindingforces and energy,” Journal of Engineering for Industry, vol. 110,no. 1, pp. 25–31, 1988.

[3] J. Asche, H. K. Tönshoff, and T. Friemuth, “Cutting Principles,wear and applications of diamond tools in the stone andcivil engineering industry,” in Proceedings of Diamond ToolsConference, pp. 151–157, 1999.

[4] H. K. Tönshoff and G. Warnecke, “Research on stone sawing,”in Advances in Ultrahard Materials Applications Technology, P.Daniel, Ed., vol. 1, pp. 36–49, Hornbeam, London, UK, 1982.

[5] J. Konstanty, “Theoretical analysis of stone sawing withdiamonds,” Journal of Materials Processing Technology, vol.123, no. 1, pp. 146–154, 2002.

[6] D. M. Pai, E. Ratterman, and M. C. Shaw, “Grinding swarf,”Wear, vol. 131, no. 2, pp. 329–339, 1989.

[7] C. Y. Wang and R. Clausen, “Marble cutting with single pointcutting tool and diamond segments,” International Journal ofMachine Tools and Manufacture, vol. 42, no. 9, pp. 1045–1054,2002.

[8] A. Di Ilio and A. Togna, “A theoretical wear model fordiamond tools in stone cutting,” International Journal ofMachine Tools and Manufacture, vol. 43, no. 11, pp. 1171–1177, 2003.

[9] A. Ersoy, S. Buyuksagic, and U. Atici, “Wear characteristics ofcircular diamond saws in the cutting of different hard abrasiverocks,” Wear, vol. 258, no. 9, pp. 1422–1436, 2005.

[10] T. W. Hwang and S. Malkin, “Grinding mechanisms andenergy balance for ceramics,” Journal of Manufacturing Scienceand Engineering, Transactions of the ASME, vol. 121, no. 4, pp.623–631, 1999.

[11] T. W. Hwang and S. Malkin, “Upper bound analysis forspecific energy in grinding of ceramics,” Wear, vol. 231, no.2, pp. 161–171, 1999.

[12] H. K. Tönshoff, J. Peters, I. Inasaki, and T. Paul, “Modellingand simulation of grinding processes,” Annals of the CIRP, vol.41, no. 2, pp. 677–688, 1992.

[13] X. Chen, W. B. Rowe, D. R. Allanson, and B. Mills, “Agrinding power model for selection of dressing and grindingconditions,” Journal of Manufacturing Science and Engineering,Transactions of the ASME, vol. 121, no. 4, pp. 632–637, 1999.

[14] M. C. Shaw, Principles of Abrasive Processing, Oxford Science,Oxford, UK, 1996.

[15] L. Carrino, W. Polini, and S. Turchetta, “Stone chippingvolume using a diamont mill,” in Proceedings of the 16thInternational Conference on Production Research (ICPR ’01),p. 113, Prague, Czech Republic, July-August 2001, Summariesno. 5.

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