Prediction of specific cutting force and cutting force ratio in turning

Journal of Materials Processing Technology, 41 (1994) 71-82 71 Elsevier

Prediction of specific cutting force and cutting force ratio in turning

B.Y. Lee

Department of Mechanical Manufacture Engineering, National Yunlin Polytechnic Institute, Yunlin, Taiwan, 63208, ROC

Y.S. T a r n g

Department of Mechanical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan, 10672, ROC

(Received October 21, 1992; accepted March 30, 1993)

Industrial Summary

Presented in this paper is a model which enables the prediction of the specific cutting force and the cutting force ratio in metal cutting, based on a predictive machining theory using a shear-zone model of chip formation. An important feature of this model is that the variations of the flow-stress properties of the work material with strain, strain-rate, and temperature and of the thermal properties of the work material with temperature and chemical composition are taken into consideration. Good correlation between the predicted and measured specific cutting forces and cutting force ratios in the turning operation is demonstrated.

1. Introduction

I t is well k n o w n t h a t the d e t e r m i n a t i o n of the specific cu t t ing force and the cu t t ing force ra t io is ve ry i m p o r t a n t for the ana lys i s of cu t t ing dynamics in m a c h i n i n g [1]. The specific cu t t ing force and the cu t t ing force r a t io a re usua l ly ob t a ined f rom expe r imen t s based on a la rge a m o u n t of cu t t ing tes t da t a [2-8]. In real i ty , the ob ta in ing of re l iab le cu t t ing tes t da t a is ex t r eme ly cos t ly in t e rms of t ime and mate r ia l . F u r t h e r m o r e , a sys t ema t i ca l a p p r o a c h to the u n d e r s t a n d i n g of the v a r i a t i o n of the specific cu t t ing force and the cu t t ing force r a t io wi th v a r i a t i o n in the cu t t ing condi t ions is stil l no t avai lable . Therefore , i t is p roposed to develop a model for the p red ic t ion of the specific

Correspondence to: Dr. Y.S. Tarng, Department of Mechanical Engineering, National Taiwan Institute of Technology, 43, Keelung Road, Section 4, Taipei, Taiwan 10672, Republic of China.

0924-0136/94/$07.00 © 1994 Elsevier Science BV. All rights reserved. SSDI 0924-0136(93)E0052-I

72 B.Y. Lee and Y.S. Tarng / Prediction of specific cutting force

cutting force and the cutting force ratio in metal cutting without resorting to the empirical approach.

In order to be able to understand the variation of specific cutting force and the cutting force ratio with variation in the cutting conditions, a model of chip formation which takes into account the variation of the flow-stress properties of the work material with strain, strain-rate, and temperature and of the thermal properties of the work material with temperature and chemical com- positions is considered [9]. As a result, once knowledge of the workpiece material flow-stress, the thermal properties, and the cutting conditions, includ- ing the associated tool geometry, is given, the specific cutting force and the cutting force ratio can be predicted with reasonable accuracy.

In the following, a cutting-force model based on predictive machining theory is explained. Then, the specific cutting force and the cutting force ratio are calculated using this cutting-force model. The theoretical predictions show good agreement with the experimental results for a wide range of cutting conditions, except for low cutting speeds, due to the presence of a built-up edge. Finally, short Conclusions are drawn.

2. Specific cutt ing force and cutt ing force ratio in cutt ing

In a dynamic cutting process the tangential force F¢ and the radial force Ft are given by

Fc=kgbtl (1)

Ft=krF¢ (2)

where ks is the specific cutting force, kr is the cutting force ratio, tl is the undeformed chip thickness and b is the chip width.

Usually, the specific cutting force and the cutting force ratio are obtained from experiments based on a large amount of cutting test data. In order to predict the specific cutting force ks and the cutting force ratio k r , a model of chip formation is used in the analysis (Fig. 1). It is assumed that the chip is formed by plastic deformation (no fracture) with no built-up edge. The tangential force Fc and the radial force Ft can therefore be represented also by the equations:

F~ = R cos ()~ - a) (3)

F t = R sin (,~ - a) (4)

where ~ is the rake angle, ~ is the friction angle, and R is the resultant force which can be determined from the following equation:

Fs kab tl b R - - (5)

cos0 s i n ¢ c o s 0

in which Fs is the shear force, ¢ is the shear angle, and kab is the shear flow stress along AB.

B.Y. Lee and Y.S. Tarng/ Prediction of specific cutting force

(I

s ~ F,

tOO[ ~ I work

Fig. 1. Mode] of chip formation use in the orthogonal machining theory.

73

The resultant force can also be resolved into the friction force F and the normal force N at the plane of the tool/chip interface. That is

F = R sin ,~ (6)

N = R cos ~ (7)

In the foregoing discussion, it is shown that once the shear angle ~b, the shear flow stress kab, the friction angle ,~, and the angle 0, are known then the various components of force can be determined. As a result, the specific cutting force ks and the cutting force ratio kr can be predicted using eqns. (1) and (2). In the following, an orthogonal machining theory is presented for the estimation of the cutting force components so as to be able to predict the specific cutting force ks and the cutting force ratio kr in orthogonal cutting.

3. Orthogonal machining theory

The model of chip formation used in the theory is given in Fig. 1. The basis of the theory is to analyze the stress distributions along AB and the tool/chip interface in terms of the shear angle ¢, the work-material properties, etc. and then to select the proper shear angle ¢ so that the resultant forces transmitted by AB and the interface are in equilibrium, the tool being assumed to be perfectly sharp.

By starting at the free surface just ahead of A and applying the appropriate stress equilibrium equation along AB, it can be shown that for 0 < ¢ ~< ~/4 the

74 B.Y. Lee and Y.S. Tarng/ Prediction of specific cutting force

angle 0 made by the resultant R with AB is given by

tan 0= 1 + 2(~/4- ¢ ) - C n (8)

where C is the strain-rate constant in the empirical strain-rate relationship proposed by Stevenson and Oxley [10]:

c y , ~ab-- la b (9)

in which ?ab is the maximum shear strain rate at AB, V, (Fig. 1) is the shear velocity and lab is the length of AB, and n is the strain-hardening index in the empirical stress/strain relationship:

a = a l e n (10)

in which a and s are the uniaxial flow stress and strain and al and n are constants that define the stress/strain curve for given values of strain rate and temperature. From the geometry of Fig. 1, angle 0 can be expressed also in terms of other angles by the equation:

¢=0-~+~ (11)

The temperature at AB which is needed, together with the strain rate and strain at AB, to determine kab and n is found from the equation:

[ ' -P sCOS ] Tab = Tw + Lp- b cJss (12)

where Tw is the initial workpiece temperature, Fs is the shear force along AB, (0 < ~ ~< 1) is a factor which allows for not all of the plastic work of chip

formation occurring at AB, p and S are the density and specific heat of the work material respectively and fl is the proportion of the heat conducted into the workpiece, which is estimated from the following empirical equations based on a compilation of experimental data made by Boothroyd [11].

f l=O.5- -O.351og(RTtan¢) for 0.04 ~< RTtan~b ~< 10.0 (13)

f l=O.3--O.151og(Rwtan¢) for 10.0 ~< Rwtan¢ (14)

in which RT is a non-dimensional thermal number given by

p S Uti RT -- - - (15)

K

where K is the thermal conductivity of the work material. The limits, 0 ~< fl ~< 1, are also imposed. The strain at AB is given by

1 cos c~ ?ab = ~ sin ¢ cos (¢ -- ~) (16)

B.Y. Lee and Y.S. Tarng/ Prediction of specific cutting force 75

The average temperature at the tool/chip interface from which the average shear flow stress at the interface is determined is taken as

1 - f l Fs cos ~ Tint = Tw -~ + ~ TM (17)

p St1 b cos (4 - ~)

where TM is the maximum temperature rise in the chip and the factor (0 < ~ ~< 1) allows for Tin t being an average value. Using numerical methods

Boothroyd [11] has calculated TM by assuming a rectangular plastic zone (heat source) at the tool/chip interface, his results agreeing well with experimentally measured temperatures. If the thickness of the plastic zone is taken as 5t2, where ~ is the ratio of this thickness to the chip thickness t2, then Boothroyd's results can be represented by the equation:

, , ,c., , ,,, h ]

where Tc which is the average temperature rise in the chip, is given by

F sin ~b Tc - (19)

pStlb cos (~b-~)

and h is the tool/chip contact length, which can be calculated from the equation:

t ls in0 {1_~ Ca } h = cos,~ sin ¢ 3 [1+2(~/4-4~)-Cn] (2O)

which is derived by taking moments about B of the normal stresses on AB to find the position of R and then assuming that the normal stress distribution at the tool face is uniform so that R intercepts the tool at distance hi2 from B. The maximum shear strain rate at the tool/chip interface, which is also needed in determining the shear flow stress, is found from the equation:

V ~int =~22 (21)

where V (Fig. 1) is the rigid chip velocity. The above equations are now sufficient for the calculation of the cutting

forces, temperatures etc., for given cutting conditions as long as the appropriate work-material properties and the values of C in Eqns. (8), (9) and (20) and

in Eqns. (18) and (21) are known. Briefly, the method used is to calculate, for a range of values of 4, the resolved shear stress at the tool/chip interface from the resultant cutting force obtained from the stresses on AB, that is

F (22) Tint - - h w

and then for the same range of values to calculate the temperature Tint and strain rates ?int at the tool/chip interface and hence the corresponding values of

76 B.Y. Lee and Y.S. Tarng/Prediction of specific cutting force

shear flow stress kchip (the influence of strain above a strain of Eint ----- 1 is assumed to have negligible effect on the flow stress). The solution is taken as the value of ¢ which gives tint = kchip, as the assumed model of chip formation is then in equilibrium.

To determine C, Oxley and Hastings [12] considered the stress boundary condition at the cutting edge (B in Fig. 1). For a uniform normal stress at the interface the average normal stress is given by:

N (23) a N ~ h w

This stress can be found also from the stress boundary condition at B by working from A along AB. If AB turns through angle (¢ - ~) (over a negligible distance) to meet the interface at right angles, as it must do if the interface is assumed to be a direction of maximum shear stress, then it can be shown that:

a N = ~-- 2a -- 2 Cn (24)

and C can now be determined from the condition that 6 g and a~ must be equal. In considering the plastic zone at the tool/chip interface, Oxley and Hastings

[13] have proposed that its thickness 5t2 can be determined from minimum- work considerations. From eqns. (17), (18) and (21) it can be seen that as 5 is reduced, the temperature and strain rate both increase, with Ti,t tending to some finite value and ~int tending to infinity as 5 approaches zero. Usually the flow stress of metals increases with increase in strain rate and decreases with increase in temperature. When this applies it is found that for given cutting conditions a value of 5 exists which gives a combination of strain rate and temperature that minimizes the shear flow stress kchip. This in turn is found to minimize the rate of both frictional work F V and total work Fc U and it is assumed that in practice 5 will take up values satisfying this minimum-work condition. It has been shown by Oxley and Hastings [12] that the values of C and 5 predicted in this way are in good agreement with experimental results.

4. Ver i f i ca t ion

In applying the orthogonal machining theory to make predictions of the specific cutting force ks and the cutting force ratio kr, the work material flow stress properties, expressed in terms of al and n in eqn. (10), were determined from high-speed compression tests results obtained by Oyane et al. [14] for a range of plain carbon steels. The thermal properties used were taken from the experimental measurements of Woolman and Mottram [15]. The work materials in the present machining experiments were steels of chemical composition 0.20% C, 0.72% Mn, 0.15% Si, 0.015% S, 0.015% A1; and 0.38% C, 0.77% Mn, 0.1% Si, 0.15% P. Curves of al and n for these steels obtained from the results of Oyane et al. (which are the most suitable results available), using the


appropriate carbon content are given in Fig. 2 plotted against velocity-modified temperature Tmod, this latter being a frequently-used parameter that com- bines the effects of strain rate and temperature and can be written as:

1 where T(K) is the temperature, ~ is the uniaxial strain rate and v and ~0 are constants which are taken in the present study to be 0.09 and 1 s- 1, respectively [16]. The curves of al and n (Fig. 2) can be seen to indicate a clear dynamic strain-ageing range where al increases with increase in temperature. To obtain the flow stress for given values of strain, strain rate and temperature from these curves the method used is to determine Tmoa from the strain rate and temperature and hence the corresponding values of al and n which can then be substituted together with the strain in eqn. (9) to give the stress. Uniaxial flow-stress results are related to plane-strain machining conditions in the usual way, with, for example:

n O ' l ~ a b , f i (26)

Tab ~ab = x//~ (27)

ab=fi (28)

For the thermal properties, the influence of the carbon content on the specific heat S is found to be small and equation

S(Jkg -1K 1)=420+0.504 T(°C) (29)

1400.0

1000.0

~ 600.0

Q

200.0 200

0.8

t ° " ,t~

\ 0 .4 "

O" 1 \ \

\ x 0.2 n x .~

4oo 800 8oo 1ooo

v e l o c i t y m o d i f i e d t e m p e r a t u r e , Tmod

Fig. 2. Flow stress and strain-hardening index versus velocity-modified temperature: ( ) 0.2% carbon steel; ( - - ) 0.38% carbon steel.


can be used for bo th of the s teels considered. However , the re is a m a r k e d inf luence on the t he rm a l conduc t iv i ty K and a l lowance has been made for va r i a t i ons in K wi th ca rbon con ten t and o the r a l loy ing e lements on the basis of the resu l t s of W o o l m a n and M o t t r a m [15]. The equa t ion ob ta ined in this way is

K ( W m 1 K - , ) = 5 4 . 1 7 _ 0 . 0 2 9 8 T ( ° C ) (30)

for the 0.2% ca rbon steel and

K ( W m - 1 K 1)=52.61-0 .0281 T(°C) (31)

for the 0.38% ca rbon steel. For bo th ma te r i a l s the densi ty, p, is t a k e n as 7862 kg m - 3.

4000.0 (a)

3000.0

Z 2000,0

IO00.O

0.0

1.5

,~o 36o ,so c u t t i n g s p e e d ( m / r a i n )

(b)

1.2-

0.9-

0.8

0,3

o.o o 1~o 360 450 c u t t i n g s p e e d ( m / r a i n )

Fig. 3. Predicted and experimental specific cutting forces and cutting force ratios: solid symbols indicate experimental results with a built-up edge (chip width=4.0 mm; rake angle = 5°; 0.2% carbon steel).


Based on the above knowledge of the material properties, a series of com- puter simulations has been carried out to resemble the testing conditions of an experimental investigation [16]. The cutting conditions used in the experiments were: ~=5 ° and -5° ; t1=0.125, 0.25 and 0.5 mm; U=25, 50, 100, 200 and 400 m min -1 (for t~=0.5 mm the maximum speed was limited to 200 m min 1 because of power restrictions) and b = 4.0 mm.

Figures 3 and 4 show the predicted (solid lines) and experimental specific cutting forces and cutting force ratios with 0.2% carbon steel. It is shown that both the specific cutting force k~ and the cutting force ratio kr decrease with increase in undeformed chip thickness tl, cutting speed U, and rake angle ~. The agreement between the predicted and experimental values of k~ and kr is good for those cutting conditions in which there was no built-up edge. The results also show that the built-up edge disappears with increase in cutting speed, U, and undeformed chip thickness, tl, similar results for the prediction

4000.0

3000.0

~ 2000.0

lO00.O

0.0

1.5

(a)

t~ ~ °'2e~a~

o. , , i , , i • ,

150 300

c u t t i n g s p e e d ( m / v a i n )

( b )

1 . 2 -

0 . 9 -

0 . 6

0.3 ¸

0.0 ~o s6o

cutting speed(m/rain) 4-50

Fig. 4. As for Fig. 3 for r ake angle = - 5 °.

80 B.Y. Lee and Y.S. Tarng / Prediction of specific cutting force

4000.0

3000.0

2000.0

1000.0

0.0

1.5

1.2 ]

0,9-

0.6-

0.3-

0.0

(a)

Ito ado ~o cutting speed(refrain)

(b)

~do sdo cutting speed(m/rain)

Fig. 5. As for Fig. 3 for 0.38% carbon steel.

of the specific cutting force, ks, and the cutting force ratio, kr, being shown in Figs. 5 and 6 for the 0.38% carbon steel.

5. C o n c l u s i o n s

An analytical model for the prediction of the specific cutting force and the cutting force ratio for a wide range of cutting conditions has been developed in this paper, good agreement being observed between the theoretical predictions and experimental verifications of the specific cutting force and the cutting force ratio in turning. Since the specific cutting force and the cutting force ratio can be estimated reasonably, an investigation of cutting dynamics in turning based on this model will be exploited in subsequent work. In addition to this, an extension of the predictive machining theory to predict the specific cutting force and the cutting force ratio in milling will be reported also in the near future.

4000.0

3000.0

2000.0

1000.0 -

0.0

1.5

B.Y. Lee and Y.S. Tarng/ Prediction of specific cutting force

(a)

x.~o sdo ~o cutting speed(m/raln)

(b)

81

1.2

0.9

o . 8

0.3

o . o , i , , i . , 0 150 300 4,50

c u t t i n ~ s p e e d ( m / r a i n )

Fig. 6. As for Fig. 4 for 0.38% carbon steel.

R e f e r e n c e s

[1] F. Koenigsberger and J. Tlusty, Structural of Machine Tools, Pergamon, Oxford, 1971. [2] J. Tlusty and F. Ismail, Basic non-linearity in machining chatter, Ann. CIRP, 30(1)

(1980) 299-304. [3] T. Kaneko, H. Sato, Y Tani and M. O-hori, Self-excited chatter and its marks in turning,

J. Eng. Ind., ASME, 106 (1984) 222-228. [4] K. Jemielniak and A. Widota, Numerical simulation of non-linear chatter vibration in

turning, Int. J. Mach. Tools Manuf., 29(2) (1989) 239-247. [5] J. Tlusty, Dynamics of cutting forces in end milling, Ann. CIRP, 24(1) (1975) 21-25. [6] P. Gygax, Dynamics of single-tooth milling, Ann. CIRP, 28(1) (1979) 65-70. [7] H. Fu, DeVor, R. and S. Kapoor, A mechanistic model for the prediction of the force

system in face milling operations, J. Eng. Ind., ASME, 106 (1984) 81-88. [8] Y. Altintas and P. Chan, In-process detection and suppression of chatter in milling, Int.

J. Mach. Tools Manuf., 32(3) (1992) 329-347. [9] P.L.B. Oxley, The Mechanics of Machining: An Analytical Approach to Assessing

Machinability, Ellis Horwood, Chichester, 1989.


[10] M. Stevenson and P.L.B. Oxley, An experimental investigation of the influence of speed and scale on the strain-rate in a zone of intense plastic deformation, Proc. Inst. Mech. Eng., 184 (1970) 561 576.

[11] G. Boothroyd, Temperature in orthogonal metal cutting, Proc. Inst. Mech. Eng., 177 (1963) 789-802.

[12] P.L.B. Oxley and W. Hastings, Predicting the strain-rate in the zone of intense shear in which the chip is formed in machining from the dynamic flow stress properties of the workpiece material and the cutting conditions, Proc. R. Soc. London, A356 (1977) 395-410.

[13] P.L.B. Oxley and W. Hastings, Minimum work as a possible criterion for determining the frictional conditions at the tool/chip interface in machining, Phil. Trans R. Soc., A282 (1976) 565 584.

[14] M. Oyane, F. Takashima, K. Osakada and H. Tanaka, The behavior of some steels under dynamic compression, Proc. lOth Japan Congress on Testing Materials, 1967, pp. 72 76.

[15] J. Woolman and R. Mottram, The Mechanical and Physical Properties of the British En Steel, British Iron and Steel Research Association, Pergamon Oxford, 1964.

[16] W. Hastings, P. Mathew and P.L.W. Oxley, A machining theory for predicting chip geometry, cutting forces etc. from work material properties and cutting conditions, Proc. R. Soc. London., A371 (1980) 569-587.

Documents

Prediction of specific cutting force and cutting force ratio in turning