theoretical analysis of cutting fluid interaction in machining

Theoretical analysis of cutting fluid interaction in machining

T. Smith, Y. Naerheim and M.S. Lan*

A theoretical model of the interaction of cutting fluids in machining is proposed. Based on the capillary flow theory developed in this paper, the model states that fissures in the chip and along the tool-chip interface allow cutting fluid or vapour to be transported rapidly into the chip and along the tool-chip interface. This facilitates both the formation of the chip and its sliding along the rake face of the tool, thereby reducing the energy required for the cutting process. Mathematical expressions relating cutting force and penetrating properties of the cutting fluid or vapour using basic physical and chemical properties are derived. The expressions are applied using carbon tetrachloride as a model cutting fluid.

Keywords: cutting fluids, machining, interaction

Nota t ion

A m

b D D~ F

G g h k /eft

lg lg, f lg, Ii Is lv L rn M n N e. Pv O F

r c

l

t

te v V

Area per molecule Depth-of-cut Shape factor Diffusion coefficient Force resisting liquid penetration along tool-chip interface Viscous resistance to flow of liquid Gravitational constant Cutting fluid head Boltzmann's constant Effective chip length that contributes to coverage maintenance Tool vapour--chip contact length Free molecular flow penetration length Viscous vapour penetration length Liquid penetration length Metal-metal contact length Length for which vacuum exists Total contact length (l s + 11 -4- Ig m A- lg, f +ol 0 Grams per molecule Molecular weight Number of fissures per centimetre of chip Avogrado's number Atmospheric pressure Vapour pressure Flow rate Distance from tool rake face to chip undersurface Radius of curvature of liquid meniscus Penetration time " Uncut chip thickness (feed) Chip thickness Velocity through fissures Cutting velocity

*Rockwell International Science Center, 1049 Camino Dos Rios, Thousand Oaks, CA 91360, USA

7

A~

0

V

~ T , x

P , O"

T s

Chip velocity Rake angle Friction angle Cutting fluid surface tension Activation energy for viscous flow Enthalpy of vaporization Fluid viscosity Angle between rake face and chip as it parts from the rake face Slip velocity at y = r Liquid or gas velocity Slip coefficient at the chip surface Slip coefficient at the tool surface Cutting fluid density Molecular diameter Shear strength of workpiece material in primary shear zone Shear plane angle

In t roduct ion

Cutting fluids improve machining productivity and product quality by reducing cutting forces and vibrations, and by improving tool life, surface finish, and tolerances. This results from (1) cooling the tool, workpiece, and chip, (2) influencing the process so that chip formation is facilitated, and (3) reducing chip compression, built-up edge, and tool-chip seizure. However, the detailed mechanisms of the cutting fluid interaction during machining have been extensively researched and discussed without much consensus for almost 40 years. To elucidate detailed mechanisms, the authors have performed experimental studies. Acoustic emission studies were performed to detect changes in the cutting process caused by the cutting fluid interaction 1. Auger electron spectroscopy studies were performed to provide evidence of cutting fluid penetration along the tool and in the chip 2.

Fig. 1 illustrates the main parameters involved during machining. The chip is formed as the material passes

TRIBOLOGY international 0301-679X/88/0502394)9 $3.00 ~) 1988 Butterworth & Co (Publishers) Ltd 239

Smith et al--cutt ing fluid interaction

-I I v

F G

Primary shear zone

Rake face

Workpiece I

Vc t

c

Chip

Gas

Liquid

Flank face

s

~ ~ ~ L i ~ / i

Too I

Fig 1 Schematic diagram of the metal cutting process and cutting fluid penetration along the tool-chip interface

through the primary shear zone which is oriented at an angle, @, relative to the direction of the cutting motion. The uncut chip thickness (the feed) is t, and the chip thickness is 4. The force in the direction of the cutting motion, the main cutting force, Fp, can be expressed as 3

Fp = 2 t b r s cot@ (1)

where zs is the shear strength of the workpiece material in the primary shear zone and b is the depth of cut. The shear plane angle, @, depends to a large extent on the friction between the tool and chip at the rake face. It can be related to the friction angle, fl, according to the equation of least energy:

1 @ = ~ - ~ (fl - ~) (2)

Merchant 3 suggested that the cutting fluid penetrates between the chip and tool to form a film with lower coefficient of friction than that of the workpiece material in direct contact with the tool. Since the coefficient of friction, # = tan /3, a lowering of # and, therefore, /3 increases the shear angle @ and decreases the shear force (Eq (1)).

Rebinder 4 and others 5-8 attributed the reduction in cutting forces to the decrease in shear strength of the material caused by interaction of cutting fluid species with the near-surface dislocations to facilitate plastic

deformation. Furthermore, they suggested diffusion of cutting fluid through the chip to the tool-chip interface to form a soft low-friction film between the chip and the tool rake face.

The difficulty in visualizing penetration of cutting fluid all the way to the cutting edge, between the chip and tool, or the transport through the bulk chip, led Rowe 9 to hypothesize partial penetration along the tool-chip interface, reducing the length of metal contact with the tool. The equation for the main cutting force is expressed a s

tc/t Fp = 2 t b Ts - - - t a n ~ (3)

COS~

where the thickness ratio tUt is related to the contact length I s by

tUt = x/1 + (lJt) cos~ (4)

Penetration between the chip and tool reduces I s and thus Fp. Williams and Tabor 1° point out that the reduction in contact length affects the reduction in the cutting force in two ways: (1) the contact length affects chip-tool friction which dissipates energy, and (2) the contact length controls the inclination of the primary shear zone, where the greater part of the energy of cutting is consumed. Most investigators tend to support the idea that fluids,

240 October 88 Vol 21 No 5

most likely in the vapour phase, gain partial access to the tool-chip interface where chemical reaction occurs to form lubricating films ~1 -~3. There is less agreement on the question of cutting fluid vapour diffusion through the shear zone of the chip 1°' la, 14.

It is well documented that microcracks form in the primary shear zone when two-phase or single-phase materials with inclusions (ie most engineering materials) are machined ~s- 19. These cracks which have clean and reactive walls provide extremely fast capillary paths for cutting fluid and/or vapour penetration. Transport through such paths is much faster than diffusion through a distorted lattice. In fact the recent study by the authors provides evidence of cutting fluid penetration along the rake face of the tool at the tool-chip interface and along fissures in the chip generated during chip formation, rather than through the lattice 1. Although this in general supports the Rebinder 4 and Rowe 9 hypotheses of cutting fluid interaction in machining, the mechanism of penetration through the chip differs.

The purpose of this paper is to derive theoretical models based on basic chemical and physical properties that describe the experimentally observed penetration of cutting fluids along fissures between the tool and chip and within the chip itself. Such models would be desirable for guiding the development of improved cutting fluids or evaluation of existing ones. This would help reduce the costly and time consuming empirical machinability tests presently required for evaluating cutting fluids.

The models

The various regions and parameters of the cutting process used in the models are schematically shown in Fig 1. The effect of the tool-chip contact length on the main cutting force is obtained by combining Eqs (3) and (4):

f p = 2 t b Zs {[1 + (lJt)cosct] 1 / 2 / C O S 0 ~ - - tanct} (5)

It should be noted that z s, in Eq (5), will vary over the contact regions so that z, is considered to be an average effective value rather than the normal bulk value. Expres- sing F v in terms of the contact length, Is, one might consider l~ to be the total length, L, minus the cutting fluid liquid,/], and vapour, lg, penetration lengths.

These penetration lengths are designated l I for liquid, lg, for vapour in the viscous flow regime, lg, f for vapour in the free molecular flow regime, and Iv for regions in which no liquid or vapour has penetrated but for which separation has occurred and vacuum exists. The reduced contact length is expressed as

I s = L - (ll + lg,. + lg, f + Iv) (6)

where L is the total length for which capillary conditions exist. The reduction in Fp, arising from cutting fluid penetration, can be estimated by substituting the value of Is from Eq (6) into Eq (5).

The chip moves with velocity vo over the tool rake face. The stress distribution and temperature gradient within the chip cause it to curl away from the tool. The ability of the chip to pull away from the tool depends on the adhesion between them. This, in turn, depends upon the reaction between the freshly generated under-surface of


the chip, the tool surface, and the cutting fluid or vapour (ie the penetration of the fluid or vapour). The farther the penetration, the lower the adhesion, which allows greater curl, which, in turn, further enhances the penetration.

The chip formation in the primary shear zone and chip flow over the rake face of the tool are dynamic rather than static processes involving rapidly repeating slip and microcracking in the primary shear z o n e 1 5 - 2 2 a s well as chip curling and breaking. This gives rise to fissures in the chip (Figs 2(a) and 2(b)) and along the tool-chip contact length (Fig l) x. Hence, at any instant there will be a distance, ls, of intimate metal-to-metal contact with the tool. Further away from the cutting edge there may be a length, lg, in which vapour, but no liquid, has penetrated, and still further away there may be a length, Ii, that contains the liquid cutting fluid. In the next instant, this fissure may close, extruding the vapour and liquid. However, wherever the gas and liquid have penetrated, a reaction product layer will be left that will decrease the

i ....................... i I ! 500 iJrn

b

~ Fluid

Workpiece_ s

Fig 2 (a) micrograph showing the chip formation and oriented microstructure in the chip, and (b) schematic diagram of fissures in chip (medium carbon steel at 106 m/rain)

TRI BOLOGY international 241


Table 1 Expressions for fluid and vapour penetration

Models Equation Expression

Between chip and tool Liquid Vapour (viscous)

Vapour (free molecular flow)

Through chip Liquid

Vapour (viscous)


A14 A22

A28

A29

A30

A28

/, ~ (2r2/3r/Vc)(P- P~) /g.~ ~ (~/2r/fl) (P~ - Pg.~) + (ct k T Oo.~12~l f12) In

/g.f ~ 16Dx/2k Tm rz/9n2a 2 Qg.f

/1= z(~E-+ ~cr2)(P - Pv)/2n O~f

/9, . = z ( ~ + ¢~r2)(P~v-P~g.,)/4nkTO~f

Ig,f ~ 16Dx//2k Tm rz/gn2a2Qgj

Pv - (k T Og,,/fl) Pg,, - (k T Og.,/fl)

adhesion and the friction between the chip and tool L 2. The net result of this process is that the greater the penetration of the fluid or its vapour, the greater will be the effective length lg + 11, and the shorter will be the length l~. A similar model can be proposed for penetration from the top of the chip along fissures generated in the chip during chip formation and curl (see Fig 2(b)). Such fissures provide fast penetration paths by capillary action combined with reactions with clean reactive fissure walls.

In the Appendix the various models for penetration of cutting fluids and vapours are considered, both for the chip--tool interface.and through the chip. Table 1 gives the theoretical expressions that have been derived for the various models in the Appendix.

Estimates for various mechanisms

Cutting fluid penetration between chip and tool

Consider penetration of liquid between the tool and chip (Eq (A14). The driving pressure P is the sum of three terms:

P = P,t + ?/rc + p g h (7)

where Pat is the atmospheric pressure, y/r~ is the capillary pressure, and p g h is the liquid head pressure. The atmospheric pressure P ~ 103g cm- ~ s-2; at room temperature for CCL, ~ ~ 10 g s - 2, rc ~ r/2 cm, and the liquid head can be considered negligible. The vapour pressure o f CCla can be calculated as

P~ ~ 10 7 e -6600/2T (8)

which yields

Pv ~ 102g cm-1 s-2 '

at room temperature. The viscosity of CC14 can be expressed as

r/,~ 1 x e -25°°/2r (9)

o r

r / ~ 1 0 - 2 g c m 1 s - 1

at room temperature. If we use a chip velocity of V~ ~ 70 cm s- 1 from our previous study, the penetration depth can be estimated to be

I I ~ 20 r cm (10)

Hence, for r equal to 5 x 10-3 cm, the penetration will be 1 × 10- ~ cm. Raising the temperature of the CCI4 to its boiling point increases l~ by a factor of three.

Consider penetration by C C I 4 vapour at room temperature in the viscous regime (Eq (A22)). The minimum rate of gas flow is assumed to be just enough to cover the freshly formed chip under-surface:

Qg, n ~ v e z / A m (molecules s- 1) (11)

where A m is the area per molecule (ie about 10-14 cm2). The vapour pressure, Pv ~ 10Zg cm-1 s-2, the viscosity ~/~2 × 104gcm -1 s - l , and V~ ~ 7 0 c m s 1. If(~ >>rand Pg,'l << Pv, Eq (A22) reduces to

lg, n ~ 10Sr 2 + 6 x 103r In [1 -- 10r] (12)

It follows that r must be of the order of 3 × 10- 3 cm or greater for significant penetration to occur (ie l~ > 0.1 cm).

For penetration by CC14 vapour in the free molecular flow region, Eq (A28) yields

lg,f ~ 2 × 104 D r (13)

It follows that since D ~ 1, r must be of the order 5 × 10 -6 cm or greater, for significant penetration to occur .

Cutting fluid penetration through chip fissures

Consider penetration of liquid CCI4 through fissures in the chip. For ~c <<r, and y / r > > ( P a t + p g h - P v ) , ie y/r ~ ( P - Pv), Eq (7), Eq (A29) yields

1/Qcf ~ 2q tc/z r 2 7 (14)

since l~ = to. The time delay (tp) for fluid entering the chip from the top to exit into the chip-tool interface is

tp : tell; (15)


where v, the fluid velocity, is

v = O~f/z r cm s- a

From Eqs (A31) and (14) to (16)

ls .~ 2t~,7 VJr ~,

(16)

(17)

If the chip forms sufficient fissures and tc ~ 0.1 cm, q ~ 10-2 gcm -1 s -a, V¢ ~ 7 0 c m S -1, and 7 ~ 10g S -2, then

l, ~ 1.4 × l0 3/r (18)

If r is less than 10 -3, I s is unreasonably large (about 1.4 cm). It follows that r must be greater than 10-2 cm for a significant reduction in 12 to occur, but considering the chip compression, this value of r is unrealistic. In the case of ~ >>r, ~¢ must be greater than 10-3cm, and, therefore, r > 10 -4 cm for a significant reduction in 12 to occur.

For viscous flow of CCI 4 vapour through the chip

Qg,, = z r v P/k T molecules s- a (19)

and from Eqs (A30), (A31 ), and (19), if ~ << r and assuming P ~ Pv/2, Pv >> Pg. f then

12 ,~ 3t~ Vv q/Vv r2 cm (20)

For t c ~ 0 . 1 c m , V c ~ 7 0 c m s -a, r / , ~ 2 x 10-4g cm -a, and P~ ~ 102g cm -a s -2

Is ,,~ 0.4 x 10-5/r 2 (21)

It follows that r must be greater than about 3 x 10 -3 for a significant reduction in 12. In the case of ~¢ >> r, ~ r must be greater than about 10 -5 cm, or r > 10 -3 cm.

For free molecular flow through the chip,

Qg. f = z v r p = z v r Pm/k T g s- a (22)

where p is the vapour density. From Eqs (A28), (A31), (15), and (22)

12 ~ 3 Vc t 2 x / ~ m/2k T/aD r (23)

For V~ ~ 70 cm s- ~, t¢ ~ 0.1 cm, m = 2.5 x 10 -22 g molecule -a, k = l . 3 8 x 10a6gcm -2 K -1 molecule -a , and T ~ 300 K,

Is ~ 5 x l O - 5 / D r (24)

It follows that D r must be greater than about 10-. 6 cm, or if D ~ l, r > 10 . 6 cm, for a significant reduction in 12 to occur.

An estimate of the limiting of fissures per unit l~ngth of chip, n, can be made by using the above estimates of Is and the following approximation:

n l~ffQ~f ,~ z V~/Am (25)

In Eq (25), n len is the number of fissures in effective chip length laf, Qef is the flow rate through a single fissure, and z V JAm is the total flow sufficient to maintain a monolayer coverage on the chip undersurface. To obtain Q~f in units of molecules per second needed in Eq (25). for liquid flow in terms of Eq (14), 1/QcF must be multiplied by re~p, where m is grams per molecule and p is the liquid density. Eq (25) becomes

n/eft ~ 2 V¢ rl t¢ m/7 0 A~ r 2 (26)

Smith et a l - -cut t ing f luid interaction

which reduces to

/1/eft ~ 5 X 10- 11/r2 (fissures) (27)

If r > 10 -4 cm for significant reduction in Is, then one fissure is sufficient, since n/eft cannot be less than unity.

In the case of viscous vapour flow,

n/eft ~'~ 12 V¢ ~1 k Ttc/2r 3 p~ A m (28)

which reduces to

n/eft ~ 3 x 10- 7/r3 (29)

If for significant reduction in 12, r > 10-3cm, then n l~ff ~ 3 x 102, and if a reasonable value of lef t is about 10-1 cm then n ~ 3 x 103 fissures cm- a, a plausible value.

For free molecular flow

n/eft ~ 97z2 Vc 0"2 t J16D A m x / ~ T/m r (30)

which reduces to

~1 lef t ~ 2 x 10 -2 /D r (31)

I fa reasonable value of D is about 1 and/eft ~'~ 10-1, then for a significant reduction in 12, r > 1 0 - 6 c m , and n ~ 2 x l05 fissures cm- a, a plausible number.

Table 2 gives a comparison of the gap between chip and tool or fissure opening for penetration through the chip for the various models.

To demonstrate that penetration cannot be explained by a diffusion process through the solid or along grain boundaries, regardless of how distorted the lattice may be, consider the following analysis. Fick's first law of diffusion is stated as

J = - D~ OC/Ox ~ D C1/t~ (32)

where D~ is the diffusion coefficient and OC/Ox is the concentration gradient. Assume the gradient is linear through the chip thickness (t,---0.1 cm) and that the concentration of CC14 at the top is that in the liquid

Table 2 Comparison of gap or fissure opening for various penetration models for cutting fluids during machining

Model r is gap between chip and tool or fissure opening, n = fissures/cm

Between" chip and tool

Liquid

Vapour (viscous)


For significant penetration I r > 5 x 10-3cm r > 3 x 10-3cm

r> 5 x 10 -6c~

Through chip

Liquid Vapour (viscous) Vapour (free molecular flow)

For signif icant reduction in I s r > 10-4cm, n ~ l r > 1 0 - 3 c m , n ~ 1 0 3 r> 10-8cm, n ~ l O s


Smith et a l - -cut t ing f luid interaction

(C 1 ~ 1022 molecules c m - 3 ) . In order to provide a monolayer of reaction product on the chip underside, the flux J must be l0 ts Vc molecules cm- 1. For chip velocity Vc = 70 cm s- 1, and from Eq (25), the diffusion coefficient is calculated to be D c ~ 10 - 6 c m 2 S -1 . This is the order of magnitude for grain boundary diffusion at 1000°C. In order to provide sufficient CC14 at the interface, all of the chip would need to have average properties of grain boundaries at about 1000°C. Since the chip average temperature is far below 1000°C, and since only a small fraction of the chip has slip planes that act like grain boundaries, and since the C1 estimate is orders of magnitude too large, it is concluded that solid state diffusion is not a factor in transporting cutting fluid species to the chip-tool interface.

Summary and conclusions

Theoretical aspects of three models to describe the effect of cutting fluids on machining have been considered. The three models that have been reported in the literature are: the physico-chemical action theory 4, the soft film theory 3, and the contact reduction theory '9. Particular attention has been given to the penetration of cutting fluid as liquid, as viscous vapour, and as free molecular flow vapour, into fissures formed during machining. Two regions have been considered as possible sites for cutting fluid access: (1) fissure-like separation between the tool anal the chip; and (2) fissures in the chip. In either case, the fissures are expected to be dynamic in nature, opening and closing. However, the proposed models do not rely on the fissures being dynamic. If a fissure is exposed to cutting fluid vapour or liquid as it opens momentarily, the fluid will penetrate a distance that depends on many factors. The surfaces of the fissures will be coated, at least with a monolayer of reaction products between the cutting fluid and the fresh, clean metal. As the fissure closes, rewelding will not occur, because of the reaction product layer. This layer will reduce the length of metal-metal contact, l~, and thus the friction of machining. If penetration proceeds partially through the chip along fissures in the chip in the primary shear zone, the shear strength of chip materials will be affected. Complete penetration through the chip to the tool-chip interface will reduce the friction between the chip and tool. Theoretical relationships for capillary penetration between the chip and tool and along fissures in the chip have been developed.

With carbon tetrachloride as a model cutting fluid and using reasonable parameters and physical constants, the fissure separation, r, was estimated for liquid, viscous vapour, and molecular flow along the tool-chip interface and through the chip fissures. Significant reduction in ls was considered to be in the order of 0.1 cm. Table 2 gives estimated values for r and n for which significant penetration might occur.

Although analysis of solid state diffusion indicates the diffusion process to be negligible, it is concluded from Table 2 that for liquid and viscous vapour flow, r > 10 -3 cm is necessary for significant penetration between tool and chip to occur. It is concluded that for the fissure between the tool and the chip, although the liquid and viscous vapour act as a source of cutting fluid, only

free molecular flow can account for sufficient transport to reduce the metal-metal contact length.

In the case of transport through the chip, if the fissure gaps are of the order of lO-4cm, only one fissure is required for monolayer formation of reaction product on the chip underside, for liquid capillary flow. However, considering the compression in the chip, gaps of the order o f 10 -6 cm are more plausible. In this case, n ~ 105 fissures cm - 1 is required for a formation of a monolayer, by freemolecular flow.

The average time these fissures must remain open can be estimated from Eq(A31), the penetration time r ~ l~/V~. For I, to be reduced to a small value, eg 10- 3 cm, t o is of the order of microseconds, and for l~ to be 0.1 cm, tp must be of the order of milliseconds, if the chip speed is of the order of 70 cm s- 1.

It is concluded that because of the large separation or gap necessary for penetration of liquid or viscous vapour, they cannot significantly affect a reduction in the contact length Is or the adhesion or frictional drag beyond l s However, it is concluded that vapour penetration via free molecular flow, both through fissures in the chip and between the chip and tool, is feasible and might account for a reduction in l~ and a decrease in adhesion and fractional drag beyond 1,.

This analysis is consistent with carbon tetrachloride as an ideal cutting fluid. The small size of the carbon tetrachloride molecule, its high vapour pressure, and the reactive chemical attraction of chlorine for clean metal are properties to be looked for in cutting fluid. The expressions for penetration lengths in Table 2 can be used for evaluating the potential effectiveness of cutting fluids from their chemical and physical properties.

Acknowledgments

This work was supported by the Rockwell International Independent Research and Development Program.

References 1. Naerheim Y., Smith T., and Lan M.S. Experimental investigation of

cutting fluid interaction in machining. Trans. ASME,. Tribology, 1986, 108

2. Naerheim Y. and Lan M.S. Acoustic emission reveals new inform- ation about the metal cutting process and tool wear. Submitted.for presentation at NAMRC XVI, University t~f Illinois, Urbana- Champaign, Illinois, 25-28 May 1988

3. Merchant M.E. Fundamentals of cutting fluid action. Lubr. Eng., 1950 6, 163 181

4. Rebinder P.A. and Shreiner L.A. Doklady Akad. Nauk. CCCP, 1949, 64, 879

5. Westwond A.R.C. and Mills S.S. Application of chemicomechanical effects to fracture-dependent processes. In Surface effects in crystal plasticity, Nordoff, London, 1977, p 836

6. Pertsov N.V. Environmentally-assisted hard materials machining. ibid, 851

7. Cassia C. and Boothroyd G. Lubricating action of cutting fluids. J. Mech. Eng. Sci. 1965 7, 67

8. Arshinov V. and Alekseev G. Metal cutting theory and cutting tool design. Mir. Publishers, Moscow, 1970

9. Rowe G.W. Lubrication and lubricants, edited by E.R. Braithwaite, Elsevier, Amsterdam, 1967

10. Williams J.A. and Tabor D. The role of lubricants in machining. Wear, 1977, 43 275

11. Horne S.G., Doyle E.D., and Tabor D. Direct observation of contact and lubrication at the chip-tool interface. Proc, Int. Conf. on Lubrication, HT, Chicago, IL, 7-9 June 1978


12. Childs T.H.C. Rake face action of cutting lubricants. Proc, lnstn. Mech. Engrs., 1972, 186, 717

13. Sehey J.A, Tribology in metal working: friction, lubrication, and wear, ASM, 1983

14. Childs T.H.C. and Rowe G.W. Physics in metal cutting, Rep. Prog. Phys., 1973 36, 223

15. Walker T.J. and Shaw M.C. Proc. 10th Machine Tool Design and Research Conf. Pergamon. Oxford, 1970, 242

16. Komanduri R. and Brown R.H. Metals and Materials, 1972, 6, 531 17. lwata K. Proc. Intl. Conf. on Production Technology, Inst. Eng.,

Australia, 183 (1974). 18. Luong LH.S. Metals Technology, 1980, 7, 465 19. Komanduri R. and Brown R.H. Trans. ASME, J. Eng.for Industry,

1981, 103, 33 20. Lindberg B. and Lindstrom B. Measurement of the segmentation

frequency in the chip formation process. Ann. CIRP 1984, 32, 17 21. Sullivan K.F., Wright P.K., and Smith P.D. Metallurgical appraisal

of instabilities arising in machining. Metals Technology, 1978, 5, 181 22. Wright P.K., Home S.G., and Tabor D. Boundary conditions at the

chip-tool interface in machining: comparison between seizure and sliding friction. Wear, 1979, 54, 371

23. Tnrnbull H.H., Barton R.S. and Riviere J.C. An introduction to vacuum technology, John Wiley and Sons, New York, 1962, p 19

Appendix: Interaction and penetration of cutting fluid

Interaction at chip-tool interface

Liquid penetration

As the chip moves with velocity Vc over the tool and gradually separates from the rake face, it leaves capillary fissures under ultrahigh vacuum between the tool and chip. If the fissures are exposed to the cutting fluid, the fluid will be sucked into the fissures by three forces that relate to external atmospheric pressure, P,, the liquid head, p h g, and capillary pressure, y/r, where p is the liquid density, g is the gravitational constant, h is the liquid head, y is the liquid surface tension, and r is the radius of curvature at the liquid meniscus. The depth of liquid penetration in any fissure will depend on the fissure dimensions, the viscous flow properties of the fluid at the fissure temperature, and the time the fissure is open. The time the fissure is open will be dependent on the workpiece, tool material, and the machining conditions. The forces resisting liquid penetration are the viscous flow of the fluid and the viscous drag caused by the moving chip. Although the separation between the chip and tool has been grossly exaggerated in Fig 1 to facilitate identifica- tion of the parameters, we simplify the analysis by assuming that in the region of capillary flow the bottom of the chip is approximately parallel to the tool rake surface. Assuming that these forces are balanced within any element dx, the force balance can be expressed as

--/~ + F. +/~d = 0 (A1)

where F is the force due to the pressure drop across dx, F, is the viscous resistance to flow, and F d is the viscous drag of the moving chip. It follows from first principles (see Fig 3) that

F = zy dp (A2)

Fn = 2zrl dx dv/dy (A3)

Fd = 2zq dx Vc/a (A4)

Smith et al--cutt ing f luid interaction

ii ii i iiiiiiiiii iiiiiiiiiii!iii ! i iii i ii i i iiiiiiii iiiiiiiiiii iiiii i i ii iii iii i iiii i i i i! !i i "

Fig 3 S c h e m a t k re )resentation o f chip-tool configuration identifying the various parameters

where y is the height of the fluid element, z is the width (ie 2z dx. is the area of the top and bottom of the element), dp is the pressure drop across dx, ~ is the fluid viscosity, dv/dy is the fluid velocity gradient, and a is the distance from the tool surface to the chip surface at the position of the element dx, corrected for slip at these surfaces, That is,

a = r x + ~c,x + iv,x (A5)

where Ic,~ is the slip coefficient at the chip surface and IT, x is the slip coefficient at the tool surface.

From Eqs (A1), (A2), (A3), and (A4),

dP Y d x = 2rl dv/dy + 2rl Vc/a (A6)

Integrating with respect to y and inserting the boundary condition (v = #, the slip velocity, at y = r), we obtain

o r

v y y

f, ,f f,vj dv - 2rl dx y dy - a)dy # • •

1 dP (y2 r2"~ v - # = 2~ d~ \ 2 - - 2 J - (VdaXy - r)

The slip velocity is defined as

_ ,( v3 = _ e r

la = \ d y / , = , 2~I d x a

Therefore

v = B ~ ~¢r - ( V ¢ / a X y - r - ~ ¢ )

(A7)

(A8)

(A9)

(AIO)

where B = (1/2rl)dP/dx. The liquid volume rate of flow can be expressed as

r r

f ) Q l = z v d y = z B 2 ~cr dy

o o

- z i (~la)(Y - r - ~o)dy (All)


Smith et a l - -cut t ing f lu id interaction

Therefore

/2r3 ~cr2)-t - z(V~/a)(~+ ¢cr) ()~ = - z a ~ + (A12)

Integrating Eq (A12) with respect to dx and dP, over the lengths of liquid in the capillary (l 0, yields

{ (r2)} 2[,2 r , dx = 2q (A13)

where L' = I s + 1 s.

For the liquid at steady state, Ql = 0. Also, slip at the interface is zero, ie. ~x and ~ = 0. Thus, Eq (A13) yields the desired value of liquid penetration, l~, as a function of r, P, and Pv, r/and V:

l, ,~ (2r2/3q V~)(P - P,) (A14)

Eq (A14) shows that the cutting fluid penetration will increase with driving pressure P, fissure opening r, and it will decrease as the vapour pressure, Pv, liquid viscosity 7, and velocity V~ increase. The parameters Pv and t/are highly sensitive to temperature, both being exponentially related to temperature by the equations

Pv o - A H / R T = e~ e ~ (A15)

= qOe-An~/R T (A16)

where AHv is the enthalpy of vaporization, AH ~ is the activation energy for viscous flow, and pO and qo are pre-exponential factors related to the entropy changes for these processes.

Vapour penetration (viscous f low)

The region I s can be divided into three lengths: Is. ~, for viscous gas flow, lg.f for free molecular flow, and Iv, for a region of vacuum, where no vapour penetration has been reached. In region l,,, the gas pressure is high enough so that the molecular mean free path is small compared with the tool-chip separation dimensions (ie Knudsen number < 0.2). In region ls, f, the pressure is low enough so that the molecular mean free path is large with respect to the tool-chip separation dimensions (ie Knudsen number > 1). There is a region between these for which no theoretical analysis is available for gas flow. For the region Is, ~ of gaseous viscous flow, Eq (A13) is valid, except that ~ and Q are not zero. The flow rate Q is not zero because the vapour is being consumed by reacting with the fresh metal underside of the chip. Also, to obtain a molecular (rather than a volume) rate of flow, Eq (A 11) is changed to

Qs,. ~- z i (P/k T)v dy o

(A17)

where, from the ideal gas law (P/kT)v yields molecules per cm 2 s-~. To simplify the analysis, assume as before that in the region ls, ., the walls of the tool-chip separation

are approximately parallel (ie 0 ~ 0, r = re). Let

f 2r3 ) = z ~ T + ~cr 2 (A18)

fl= z(VJa)([~ + (¢r) (A19)

then from Eqs (A12) and (A17)

Qs,. = (P/k 7)( - Bct + fl) (A20)

Integration with respect to P and X yields

P~ (L" +l,.~)

f ~t PdP ; p,., 2tl k - Qs. ,1 L"

and

lg, . -~ (at/2t/flXP v - Pg,,) + (~ k TQg" d2tl flz)

Pv - k TQg.,/fl In Ps,,7 - k TQs .,]fl

(A21)

(A22)

The penetration distance lg, ~ for vapour with viscous flow increases with the tool-chip separation, r, the slip coefficient, ~c, and the vapour pressure, Pv. The penetration distance decreases with the gas viscosity, ~,, the chip speed, Vc, and the gas pressure, Ps.,"

Vapour penetration (free molecular f low)

In the region lg, f the gas pressure is low enough for collisions of molecules with each other to be negligible in comparison with their impacts on the walls. From the Knudsen equation 23, the penetration distance lg, f can be expressed as

lg.f ~ D(8/3) 2 M ~ M ~ T(A2/Br Q,.f)(AP) (A23)

where At is the cross-sectional area of the tool-chip separation, Bf is the perimeter, D is a factor that depends upon the shape of the tool-chip separation, T is the absolute temperature, and AP is the gas pressure drop in the free molecular flow region.

In the free molecular flow regime, the Knudsen number (ie the ratio of the mean free path to the effective tool-chip separation) is greater than unity. An approximate expression for this ratio is:

2t/) k T > 1 (A24) eZ; m

where Pm is the mean pressure. If it is assumed that the downstream pressure is small compared with the up- stream pressure, then Pm ~ AP/2 and

AP < ( 2 r / / r ) ~ (A25)

Since, from kinetic theory, the gas viscosity can be expressed as

rl = mx/Sk T/x m/3x/2 ~.2 (A26)


where a is the molecular diameter, the upper limit of Eq (A25) can be expressed as

AP ~ 4k T/37r3/2tr r 2 (A27)

Combining Eqs (A23) and (A27) yields

lg, f "~ 16Dx//~ - Tm r z/97zEo'EQg, f (A28)

Vacuum Region In the region lv none of the flowing gas has reacted with the clean metal chip prior to vibrational closing of the fissure. However, mating of the chip surface with the tool will be imperfect (ie asperity contact only) which will leave a fraction, f, of the area unbonded. This fraction will have a much lower friction than the fraction that rewelds. Consequently, there will be an effective vacuum penetration length fly, if vapour or liquid penetration has not been complete.

Cutting fluid penetration of chip

It has been shown that significant cutting fluid penetration occurs through the chip from the top 1 as represented in Fig 2(b), which shows enormously exaggerated dimensions of the fissures. The penetration down these fissures will be controlled by capillary flow, but will differ from the tool-chip interface in that one side of the fissure will not be moving with respect to the other side (ie V c = 0). For this situation, Eq (AI3) is valid and

I/2r3 ) 1, = z ~ - + ~cr 2 (e -- P,)/2tl Qce (A29)

where cf refers to chip-fissure.


For viscous gas flow, Eq (A21) is valid, but with fl = 0, which yields

{2r3 ) l,,, = z ~ - + ~¢r 2 (p2 _ p2,,)/ag k TQcf (A30)

Eq (A28), for free molecular flow, should remain un- changed, although the shape factor D will be different.

The relationship between Fp and the penetration of cutting fluid from the top of the chip will be quite different from that expessed by Eq (5) for penetration between tool and chip. Partial fluid penetration through the chip will reduce the energy needed to shear the chip in the primary shear zone and thereby facilitate the chip formation. Complete fluid penetration will also reduce friction between the tool and chip. The length of the contact zone Is, along the chip-tool interface, will depend on the time needed to penetrate the chip, and thus, also depend on chip velocity. That is,

I s ~ Vet p (A31)

where tp is the penetration time.

The time will depend on the limiting mechanism of liquid or vapour penetration and is calculated as 1/Qef from rearrangement of Eqs (A22) and (A28), with chip thickness, t c, replacing Is, ~ or Ig, f.

The total flow of cutting fluid through chip fissures will be leff r /Qef, where n is the number if fissures per unit length needed to maintain surface coverage between chip and tool at least a monolayer thick. /elf is the effective chip length that contributes to the coverage maintenance.


theoretical analysis of cutting fluid interaction in machining

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