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An apparent strain analysis of orthogonal cutting
Int. J. Mach. Tool Des. Res.
Vol. 9, pp. 97-116.
Pergamon Press 1969.
Printed in Great Britain
AN APPARENT STRAIN ANALYSIS OF ORTHOGONAL CUTTING P. J. THOMPSON*, H. OGDEN~ and N. A. BUTTERWORTH~
(Received 24 November 1968) Abstract--The problems of obtaining a solution for orthogonal cutting based on experimental information obtained from materials which work-harden have been considered. A valid solution based entirely on experimental information and the apparent strain theory was obtained without the need to define a model of chip formation. The problems of relating results obtained from materials which have different work hardening characteristics have also been considered. Good correlation was obtained for a number of ductile metals on the basis of the simple postulate that the stress acting on the deformation zone in the chip is directly proportional to both the apparent coefficient of friction and the yield stress of the chip. This has shown that the work-hardening property of the material is an important parameter in defining the mechanics of orthogonal cutting. The appareat strain approach may be used to estimate the cutting forces from a simply determined cutting characteristic and the equivalent stress-strain diagram of the material. 1. I N T R O D U C T I O N ORTHOGONAL cutting is the process in which the tool forces and the direction o f cut lie in a plane, see Fig. 1. As a result o f relative m o t i o n between the cutting tool and the w o r k p i ece the material to be r e m o v e d passes t h r o u g h a d e f o r m a t i o n zone extending f r o m the tip o f the tool to the surface o f the workpiece to f o r m a chip. Several m o d e s o f cutting can result
I ~ U N D E F OTHICKNESS R M E D ~ICHIP
F¢ ~
DIP~ECTIONOFR O T A T ~ - ;
DEFORMATION ZONE
J
~
/
~
pLASTICSUBLAYER ~ CLEARANCEANGLE
/~ CUN TG I, R ~ o ~ G ~ ~ " -
1
" / / / / / / J J
j FiG. 1. Chip forces in orthogonal cutting. * Senior Lecturer in Production Engineering, Harris College, Preston. i" Senior Lecturer in Production Engineering, Harris College, Preston. :~Principal Lecturer in Production Engineering, Harris College, Preston. 97
RAKEANGLE
98
P.J. THOMPSON,H. OGDENand N. A. BUTTERWORTH
from this process which give rise to different chip characteristics. In the type of cutting investigated ductile materials are cut with the formation of a continuous chip. The material removed experiences large plastic strains in the deformation zone and is considerably work-hardened. These strains are induced in the chip by shearing of the material and they depend on the geometry of the deformation zone. This geometry is not entirely governed by the geometry of the cutting tool, for it also depends on the resistance offered by the rake face to the movement of the chip and the mechanical properties of the material. The resistance offered to the movement of the chip results in the application of a compressive force to the deformation zone, which regulates both the size of the deformation zone and the chip thickness. Both the total work done and the magnitude of the cutting force depend on the sum of the plastic work done during the formation of the chip and the work done in overcoming the resistance to the movement of the chip. I.zu) uJ~
uJ
I
FRICTIONAL WOP, K OF SLIDING DONE BY THE CHIP AS i'r MOVES JOVER THE P,AKE FACE. / PER UNIT VOLUME OF MATERIAL REMOVED
Ym
F1G. 2. The equivalent stress-strain diagram showing the terms used in the apparent strain analysis. The mechanics of orthogonal cutting has been the subject of much research work. This has included phenomenological investigations and upper bound solutions obtained from slip plane theory and models of cutting. The present work is based on an experimental approach which was first devised by Hill [1] and Johnson [2] and later extended by Pugh [3] and Thompson [4] for the analysis of extrusion. It is based on an effective or apparent equivalent strain, defined as the strain which bounds an area under the equivalent stressstrain diagram equal to the total specific energy, see Fig. 2. As considerable work is done in overcoming the resistance along the rake face the apparent equivalent strain is larger than the mean equivalent strain imparted to the chip. This latter strain is found by reducing the area under the equivalent stress-strain diagram by the amount of work done on the chip as it moves over the rake face. This approach enables the strain imparted to the chip and the apparent strain to be obtained from the equivalent stress-strain diagram of the material and measured components of the cutting force. It can, therefore, be found from materials which
An Apparent Strain Analysis of Orthogonal Cutting
99
work-harden and under cutting conditions which approximate to those employed commercially. By undertaking cutting tests at different rake angles distributions of both the mean and apparent equivalent strains for some materials have been obtained, these give equivalent strain characteristics for orthogonal cutting. It has been found that each material produces a different characteristic which depends, amongst other things, on the work-hardening characteristic of the material. Also, that the frictional equivalent strain, the difference between the apparent and the mean equivalent strains, is the ratio of the compressive stress acting on the deformation zone in the chip to the yield stress of the work-hardened chip. The experimental work has shown that this quantity is directly proportional to the apparent coefficient of' friction. It is postulated that the frictional equivalent strain is independent of the equivalent stress-strain diagram of the material. This empirical relationship enables the apparent equivalent strain characteristics for different materials to be related to each other. The information gained from the mean and apparent equivalent strain characteristics may be used in a similar way to that obtained from the models of cutting. Alone they are insufficient to determine the cutting tool force for a given undeformed chip thickness and material. The undeformed chip thickness, cutting ratio and resistance along the rake face, as measured by the apparent coefficient of friction, can be indicated on a single diagram; these relationships give the true cutting characteristics for the material. When this information is combined with the equivalent stress-strain diagram the apparent strain analysis enables the cutting forces to be predicted.
/ZA
F b T
/z t7 0~ o'1o-20" 3
~1~2E3
Y n W V
( g,Ae t
A g:A
~F
2. N O T A T I O N Apparent coefficient of friction ~-- F~/Fo Force Breadth of chip Shear stress Coulomb coefficient of friction ---- r/a Direct stress Rake angle Principal stresses Equivalent stress Equivalent strain Principal strains Yield or flow stress Indice Work per unit volume Work-hardening factor Ve][ocity Mean equivalent flow stress over the equivalent strain range 0 ~ A Mean equivalent flow stress over the equivalent strain range gm,~ga Chip thickness Cross sectional area of chip Apparent equivalent strain Mean plastic equivalent strain Frictional equivalent strain ---- ~a -- ~m = an/( Ym)F Friction factor
100
P.J. THOMPSON, H. OGDENand N. A. BUTTERWORTH r
k c, d , f
¢ R a
Sufficeg c t c~ 0 1 2 F P T m A R y
Cutting ratio = tl/t2 Constant of proportionality Constants Merchant's shear angle Friction angle = tan -1 (t~A) Force reading from dynamometer Dynamometer constant
Parallelto the direction of cut Normal to the direction of cut Parallelto the rake face Normal to the rake face Prior to cutting Aftercutting Frictional Plastic Total Mean Apparent Resultant At the yield point 3. T H E O R Y
3.1. Resistance to chip movement The conditions that exist at the chip-tool interface have been the subject of much research. Early work by Merchant [5] assumed that Coulomb sliding friction occurred over the entire area of contact between the chip and the tool. This assumption leads to a unique value of the coefficient of friction which is independent of the other cutting parameters. This does not agree with the experimental observations which have shown that the ratio of the component force along the rake face of the tool to the normal component varies with the rake angle and the undeformed chip thickness. Kattwinkel [6], Zorev [7], Thomsen [8] and Rubenstein [9] show that the contact stress between the chip and the tool is not constant and that it can be high enough to cause adhesion between the chip and the tool over part of the area of contact. As a result, the area of contact between the chip and the tool can be divided into two zones. Over the first zone, which is close to the tip of the tool, the contact stress is high, resulting in adhesion of the chip to the tool, a plastic sublayer in the chip, and no relative motion between the underface of the chip and the tool. It is suggested that this occurs even in the absence of built-up edge. Under these conditions the chip experiences further deformation as a result of the existence of the plastic sublayer. Over the second zone the contact stress is low and the surface shear stress in the chip is less than the yield shear stress, hence, the chip slides over the tool under conditions of Coulomb sliding friction. In spite of the inadequacy of the Coulomb coefficient of friction the ratio of the component force along the rake face to the normal component is a useful parameter of cutting and is much used in the literature. This ratio is described as the "apparent coefficient of friction" and is given by ,~,a =
F~ • Fo
(1)
An Apparent Strain Analysis of Orthogonal Cutting
101
Which may be written in terms of the measured components of the cutting force, thus /~A =
Ft cos ~ + Fe sin Fe cos ~ -- Ft sin a"
(2)
3.2. Materiul properties The material undergoing plastic flow in the deformation zone is subject to a system of complex stress. To relate the plastic work to the strains undergone by the material during the formation of the chip it is necessary to express in mathematical terms suitable mechanical properties of the material and for these properties to be independent of the stresses acting in the deformation zone. Such a mathematical relationship for stress is known as a plasticity condition or yield criterion. It has been shown by Taylor and Quinney [lO] that the Maxwell-von-Mises plasticity condition gives the best fit with experimental results obtained with ductile materials. If the plastic stress-strain or flow curve is plotted in terms of this function of stress and its corresponding strain function the same curve is obtained regardless of the stresses acting to induce plastic flow. These functions are known as equivalent stress and equivalent strain and are given by ~- = [1{(O-I __ 0-2)2 _t_ (0- 2 __ 0-3)2 _~_ (0-3 - - O"1)2}]1/2
(3)
and ~/2
A ~ =- ~
[(AE1 - - AE2) 2 ~- (AE2 - - AE3) 2 -~ (AE3 - - A~1)2]1/2.
(4)
It can be shown, see Johnson and Mellor [14], that the plastic work done can be expressed in terms of the equivalent stress and strain, thus
w~ ---- f 6 d ~.
(5)
For many materials both in the annealed and work-hardened state the equivalent stressstrain diagrams agree with the empirical relationship. 6 = (Y~=I) ~n.
(6)
Many workers have found from a large number of cutting tests carried out on commercial materials that the equivalent stress obtained from static property tests correlates with an equivalent stress calculated from the Merchant shear plane stresses at approximately the same equiw~lent strain, see Kobayashi and Thomsen [11], Thomsen, Schaller and Dohmen [12], and Zorev [7]. This represents an anomaly since correlation is achieved between static and dynamic conditions in spite of the extreme differences in strain rate and temperature. This anomaly is discussed fully by Thomsen [8] and Sata [13]. It is suggested that the effect of the induced temperature on the flow stress is offset by the strain rate. This work is taken to justify the use of statically determined equivalent stressstrain diagrams in the analysis of orthogonal cutting.
3.3. The apparent strain theory The apparent equivalent strain is defined as the equivalent strain which bounds an area under the equivalent stress-strain diagram eqflal to the total specific energy, thus ~A
W~-- [ 0
6d~---- Ymga.
(7)
102
P . J . THOMPSON, H. OGDEN and N. A. BLrrTERWORTH
Figure 2 shows the equivalent stress-strain diagram along with the apparent, mean and frictional equivalent strains. It is evident from these definitions that, ~A =
gm @ ~F.
(8)
To relate the apparent and mean equivalent strains it is necessary to determine the work done per unit volume of material removed in overcoming the resistance to movement of the chip, thus WE -- F~V~ tlbl Vc" From volume constancy it may be written,
(9)
tlbl Vc = t2b2 V~
giving, wF_F~ t2b2
--
(T~,
(10)
It should be noted that if adhesion between the chip and the tool takes place the "friction" work done per unit volume of material removed includes some plastic work. From the definition of frictional strain, the frictional work may be written, (11)
WE = ~e( Ym)F.
Combining equations (10) and (1 1) gives, ~V - -
F~
--
t262( Y) mF
~
(12)
( Ym )F"
Since the strains induced during orthogonal cutting are large the flow stress approaches a constant value thus (Ym)F-"- Y~-~. This shows that the frictional strain, the difference between the apparent and the mean equivalent strains, is also the ratio of the stress acting on the deformation zone in the chip to the yield stress of the material removed. The total specific energy may be written,
Fc
(13)
WT -- AI"
Using the principle of conservation of energy equations (7) and (13) may be combined to give, Fc - - #A Y m . (14) A1 Resolving forces in the direction of cutting gives, Fe = Fo cos c~+ F~ sin or Fc
=
(15)
F0[cos o~ -~ ~A sin ~]
since by definition
F~ Fo Combining equations (14) and (15) gives
Ym(:AA1 =
F0[cos ~ -[- P,A sin
a]
(16)
An Apparent Strain Analysis of Orthogonal Cutting
103
and equations (12) and (16) YmgAA1 : t2b2( Ym)FgF
-cos ~ + tz~i sin a],
/
P.A
hence EA =
1
~
EF,
(17)
9
where
Lcos O~-~
#3
l
(18)
/ZA sin (XJ
giving ~A-- 1 ~m --~b"
(19)
It is seen that the friction factor includes a work-hardened factor given by
3 - ( Ym Ym)F" As the flow stress approaches a constant value at the order of strains induced by orthogonal cutting this work-hardening factor may be written
Ym f l - Y~=EA"
(20)
Hence for tnaterials which obey the empirical law equation (6) the work-hardening factor becomes 1 l+n"
I0
0.9 0-6
RAKE ANGLE 0.7
4s
d~j.
35 d¢~j.
0-6
25 ¢k~. 0.5
is d ~ .
0.4
s
d¢9.
-s
0.3
d~.
- is tl~. 0-2 0-1 I
0 0
O.I
I
I
I
0.2
0.3
0,4
I
I
I
I
I
O'S 0"6 0.7 O.B 0"9 COEFFICIENT OF FPJC'IION
I
I
J__
f
I.O
H
1.2
1.3
I
1.4
FIo. 3. Cutting ratio-coefficient of friction relationships obtained from the single shear plane model.
104
P.J.
THOMPSON,H. OGDEN and N. A. BUTTERWORTH
3.4. A unique solution The experimental work, to be discussed later, shows that the apparent equivalent strains obtained with different materials do not agree. However, it is shown that the frictional equivalent strain is directly proportional to the apparent coefficient o f friction for the materials investigated. It is postulated that this empirical relationship is independent o f the material, giving
Y
c~=-eonst.:[~F]c~=c°nst" = k/LA
(21)
where k = c -- d.a.
(22)
Using this relationship and equations (17) and (18) the apparent equivalent strain may be written rfl[£4]~=eonst. = k (cos ~ + tZA sin ~).
(23)
The dimensionless group r/3gA should give a unique solution for a given rake angle and any material, the accuracy o f this is discussed later.
3.5. Merchant's analys& [5] In this analysis it is assumed that the material is rigid, perfectly plastic and that the deformation zone consists of a single shear plane inclined to the direction o f cut by an ,~=-20 de 9 .
3.0 r2.8
2"6
I
i 2.,4 ~" I
2.2 ~" i 20 1
i 1"6 i 1.4 I
Q VO[L 1"2
o~ ~-
I
0'4 II-0.2 t 0
I
0"2
, , , 0.,4 o6 o.,
, , ,. ,.o ,.2 ,,4
,. ,6
,',
,
2.o
COEFFICIENT OF FRICTION
FIG. 4. The influences of the cutting parameters on the mean equivalent strain according to the single shear plane model.
An Apparent Strain Analysis of Orthogonal Cutting
105
angle 4'. The mean equivalent strain induced in the chip according to the model of cutting is, gm = ½[cot ~ -]-
tan(~ -- a)].
(24)
Using this model an upper bound for orthogonal cutting may be obtained using the principle of minimum work, which gives the relationship 2~ +
a -
~ -
7"/" 2
(25)
The latter expression was subsequently modified by Merchant in an attempt to improve correlation with materials which work-harden. Figure 3 shows the relationship between the cutting ratio and the coefficient of friction obtained from equation (25). Figure 4 shows the mean equivalent strain obtained from equation (24) and (25) against the coefficient of friction. 4. E X P E R I M E N T A L P R O C E D U R E 4.1. All cutting tests were carried out on a 17 in. swing centre lathe machining tubular workpieces at constant cutting speed. Cutting forces were measured under orthogonal cutting conditions over a range of longitudinal feeds from 0.0012 to 0.020 in/rev. 4.2. Dynamometer
This was a three component strain gauge lathe-tool dynamometer manufactured to a design of Hawker-Siddeley, Manchester, and activated by an a.c. transducer meter. The demodulated signal from the meter was displayed on an u.v. recorder against a time base. The dynamometer was calibrated separately in two directions using dead weights and a lever system with a moment arm of 20 : 1. Records from the u.v. recorder were obtained from both measuring elements during calibration and the interaction was determined. The tool forces during cutting were obtained from: Rc = accFc q- aaFt I Rt = atcFt -]- attFc I" TABLE l. DYNAMOMETERCONSTANTSOBTAINED FROM CALIBRATION
Dynamometer constants
cm/lbf
ace act
0.0257 0.0015
arc a~t
0
0.0280
4.3. Work material
The workpieces were in the form of drawn tubing in the as-received condition. Brass tube 2½ in. o.d. 0.130 in. wall thickness Copper 70-73 ~ arsenic 0.02--0.06 ~ zinc remainder En 9 tube 27g in. o.d. 0.185 in. wall thickness.
(26)
106
P . J . THOMPSON, H. OGDEN and N. A. BUTTERWORTH
4.4. Cutting tools These were prepared from ~ in. square high-speed-steel tool bits of the following composition: C 0.8~ W 18.25~ Cr 4 . 8 ~ Va 1.25 ~
Co 5.0 ~o
Mo 0.75 ~o
The rake and clearance faces were commercially ground to a finish of approximately 24 c.l.a, and the tools were redressed between cutting operations. 4.5. Chip geometry The cutting ratio was obtained from measurement of the average thickness at the centre of five representative chips and from the calibrated longitudinal feed of the machine. The chip thickness was measured using a micrometer having modified ball anvils and giving least determination of 0.0001 in. 4.6. Stress-strain curves The static equivalent stress-strain curve data, Fig. 5, were obtained by plane strain compression tests carried out on samples of the material cut from the tube wall, according to the procedure outlined by Watts and Ford [15]. Plastic work-equivalent strain diagrams were obtained from these curves by graphical integration. The work-hardening factors were obtained using equation (20) for several points on each curve and found to be approxi. mately constant for each material, thus 70/30 brass = 0.9 En 9 Nc
120
--
I00
~
eo
~
60
= 0.85.
40,
~
20 0
I
I
I
I
I
I
I
I
I
I
I
I
I
0 N
120
---
IO0
x
80
~: 60 =,
4o
_~
20
=o
0
I
I
I
I
I
I
o.=
0-4
0.6
o.e
i,o
t'2
BQUIVALENT
I
I
I
I
I
I
f
1.4
f,6
i.a
2.o
2-2
2 "4.
2"6
STRAIN
FIG. 5. Equivalent stress-strain curves.
An Apparent Strain Analysis of Orthogonal Cutting 5. E X P E R I M E N T A L
107
RESULTS
In order to demonstrate the feasibility of the apparent strain method dry cutting tests were undert~.ken with 70/30 brass using rake angles between -- 15 deg and 45 deg at a cutting speed of 134 ft/min. This material was chosen because it was found to give a continuous chip without a built-up edge. The measured cutting forces were corrected for dynamometer interaction. The apparent coefficients of friction were determined from these cutting forces and are shown along with the measured cutting ratios in Fig. 6. The approximate trend lines corresponding to various undeformed chip thicknesses are also shown. SYMBOL • • • 0 r:l
BREADTH OF CUT ~ O'1301n. CUTTING SPEED ; 134ft/lilln.
Z~ 0
RAKEANGLE - 15 dcg. - 5 deg. 5 d¢ 9.
15 de9. 2Sdlg. 35 de 9. 45 d¢9.
0.8 t I - 0" 0 2 0 In.
0.7
./
t poolom.
~, ti'.0"00:39 in. ~ . . . = ~ " 0 " 0 0 " 1 t i . O. O0,9 i..
0"6
~
O'S
..,'=-zr~...~/
.~/D/~-)~.~ ~ / ~ O
~
.
/
r~/
0"4 0-3 0-2 0"I
0
I
I
I
I
0'1
0-2
0'3
0'4
I
1
I
I
I
I
I
0"5 0-6 0"7 0 ' 8 0 " 9 I'0 I.I APPARENT COEFFICIENT OF FRICTION
I
1'2
I
1,5
I
f
1"4
1"5
FIG. 6. The effects of the apparent coefficient of friction and the rake angle on the cutting ratio,
showing the undeformed chip thickness, for 70/30 brass.
The apparent equivalent strain was obtained for each test by converting the total specific energy, equation (13), to its corresponding equivalent strain using a plastic work-equivalent strain diagram, obtained from the equivalent stress-strain diagram. These strains are shown against the apparent coefficient of friction in Fig. 7. The mean equivalent strain for each test was derwed from equation (19) using the measured apparent equivalent strain and the calculated friction factor. These strains are shown against the apparent coefficient of friction in Fig. 8. Similar tests were carried out on En 9 steel using soluble-oil coolant at approximately the same cutting speed. The underface of the chips and the rake face of the tools showed evidence of built-up edge. The results were obtained as described above and are shown in Figs. 9 and 10. The frictional equivalent strains were obtained, using equation (17), at various rake angles, for the experimental data and for cutting data obtained from references [16] and [17]. These are shown against the apparent coefficient of friction in Fig. 11. It was shown that the friction~tl equivalent strain was directly proportional to the apparent coefficient of
P.J. THOMPSON,H. OGDEN and N. A. BUTTERWORTH
108
SYk~DL CUTTING SPEED: 134 f t / m l n .
,4.0
RAKE ANGLE
•
- IS dcg.
•
- 5 de 9.
3.8
•
S deg.
36
@ ~]
15dcg. 25 dcg. 35 dcg.
3"4
~)
4 S dcg.
3.2 3'O 2.8 2.6
• I
_z
/
./°
2,4 2-2 w _J 2.0 ~.8 ~-6 1.4 1.2 I.O 0.8
t
0"6
0
O'4
o
O'6 0.4 0"2t~
O.2 O
O-8
I
O
0.1
I
O1"2 0"3
I
I
O'4 O"S 0"6 O17 O!8 0"9 I~O APPARENT COEFFICIENT OF FRICTION
I
1!1
1.2
1!3
L ,.'s°
FIG. 7. The effects of the apparent coefficient of friction and the rake angle on the apparent
equivalent strain for 70/30 brass, showing the factor eArfl.
friction. The constant of proportionality was found to be independent of the material and to agree with the empirical law, equation (22), for which, c
---- 0 . 5 3 7
d =
0.0021.
As was shown in the theory this empirical relationship enables the apparent equivalent strains for different materials to be compared. This gave rise to the dimensionless group in equation (23) and shown in Fig. 7 for 70/30 brass and in Fig. 10 for En 9. It is also shown for a rake angle of 35 deg and different materials in Fig. 12, along with the apparent equivalent strains. The relationships between the cutting ratio and the apparent coefficient of friction for different materials at a rake angle of 35 deg are shown in Fig. 13. The work-hardening factors and the mean yield stresses of the chips are also shown in this figure.
An Apparent Strain Analysis of Orthogonal Cutting SYMBOL 4.0
109 RAKE ANGLE_
-- 15 de9.
--
CUTTING SPEED: 134 ft/min
--S dig. • O 13
3"8
/.
3-6 3.4 3.2 3.0
5 de 9. ISd¢ 9. 25 dig,
~.
3 5 dig. 4 5 dig.
2,8-2"6
2"4 2'2 2"0 1"8
1"6 1"4 I. 2 1.0 0"8 0.6 0"4 0'2 0 0
I
I
I
I
O'!
0'2
0-3
0-4
I 0-5
I
I
I
0-6
0.7
O.B
i 0.9
I
I
I
I
I
I
I'O
I'1
1"2
I-3
1"4
I-5
APPARENT COEFFICIENT OF FRICTION
FIG. 8. The effects of the apparent coefficient of friction and the rake angle on the mean equivalent strain for 70/30 brass.
These relationships were found to agree approximately with the following empirical law, r = f--
0.27/ZA
(27)
where f=
1.17/~ - - 0.26
and = 35 deg. This was used to estimate the relationship for a non-work-hardening material, curve (a) Fig. 13. The corresponding relationship obtained from Merchant's model is also shown, curve (b). 8
P. J. THOMPSON: H. OGDEN and N. A. BUTTERWORTH
110
5;YMBOL BREADTH OF CUT: CUTTING SPEED :
(~ f:'[ A
O'lBs;in, 135;ft/min.
RAKE ANGLE ISd¢ 9. 25;deg. 35 deg45 dcg-
0-8 0.7 t l = 0'015;5;in.
.....~_ ~. ....... ~tr/o.o,og., tro.oo631..
0.6
V
_(2
~
-
~
,
~
>
~
'
A
tl= OOOf91n,
0,5 0'4 0.3 0.2 0.I-0
0"I
0 "' 2
0 '3
0 4'
' 0"5
' 0"6
' 0"7
g
'8
' 0"9
,.'o
I! I
' 1"2
I' 3
' 1"4
I !fi;
1 1"6
, 1"7
APPARENT COEFFICIENT OF FRICTrON
FIG. 9. The effects of the apparent coefficient of friction and the rake angle on the cutting ratio, showing the undeformed chip thickness, for En 9. 6. D I S C U S S I O N
It was possible to obtain the mean and apparent equivalent strains, induced by orthogonal cutting, from the statically determined equivalent stress-strain diagram, the measured components of the cutting force and the chip geometry. An inaccuracy which could have occurred in the apparent equivalent strain determination was that due to the differences between both the temperature and strain rate induced by cutting and those induced during the plane strain compression test. The assumption that the statically determined equivalent stress-strain diagram was valid under the cutting conditions had not been investigated but was justified by the work of Thomsen [8] and Sata [13]. High cutting speeds were avoided and a coolant used when cutting En 9 steel in order to minimize the induced temperature. In addition it was found that increasing the speed to a maximum of 520 ft/min did not appreciably alter the apparent equivalent strain-apparent coefficient of friction relationship; but it may be significant that the duration of the tests were short. It was also found that this relationship was not affected by the surface roughness of the rake face within the range 2 to 24 c.l.a. Both the apparent and mean equivalent strains were shown against the apparent coefficient of friction. An alternative would have been to show these strains against the cutting ratio. However, it was shown that for a given rake angle and material there was a unique relationship between the cutting ratio and the apparent coefficient of friction. Also, it was considered that the apparent coefficient of friction was determined to a higher degree of accuracy than the cutting ratio and the apparent coefficient of friction was shown to undergo a large change over a wide range of rake angles which made it a more acceptable parameter. The apparent equivalent strain-apparent coefficient of friction characteristic showed a decrease in strain with decrease in the apparent coefficient of friction and an increase in rake angle. As small apparent coefficients of friction were obtained at large undeformed chip
An Apparent Strain Analysis of Orthogonal Cutting
111
CU'I-I'fNG SPEED: 13Sft/mln
4'0
SYMBOL
3-0
RAKE ANGLE
O
IS de9.
3-6
E] /',
2Sde 9. 3 S d e 9.
3"4
0
4Sdeg.
3"2 3"0 2'8 2"6 2-4 _Z
13
2.2 2"0
=a i-
f.B 1'6
n.. 1"4 KEY
1'2
PREDICTION BASED ON PROI~RTIONALITY BETWEEN FRICTIONAL EOUWALENT STRAIN AND APPARENT COI~:ICIENT OF FRICTION
I'0
I.O &
0"8
0-8
o
oo
0.6
oo'mG
0-6
0"4
0,4
O'2
O.2
0 C'I
I 0.2
I 0.3
I 0.4
I 0"5
I 0.6
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I I, I
I 1.2
i 1,3
I 1.4
I I.S
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O 1.7
APPARENT COEFFICENT OF FRICTION
FIG. 10. The effects of the apparent coefficient of friction and the rake angle on the apparent equivalent strain for En 9, showing the factor iar~. thicknesses, and as the apparent equivalent strain was a measure of the total specific energy, it follows that the energy required by orthogonal cutting was minimized by increasing the rake angle and the undeformed chip thickness. In order completely to define the machining properties of the material it is necessary to know the relationships between the cutting ratio, the apparent coefficient of friction and the undeformed chip thickness. It is shown in Figs. 6 and 9 that for a given material these relationships can be indicated on a single diagram. Such a diagram shows the cutting characteristics of the material which could not be deduced from the equivalent stress-strain diagram alone, since the relationships shown largely depend on the surface properties of the material and the tool. Figures 6 and 9 show that at all rake angles the cutting ratio decreases with an increase in the apparent coefficient of friction and a decrease in the undeformed chip thickness. The Merchant single shear plane model was used to derive theoretical relationships between the mean equivalent strain, the cutting ratio and the coefficient of friction, Figs.
P.J. THOMPSON,H. OGDENand N. A. BUTTERWORTH
112 I'O F
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FIG. 11. The effect of the apparent coefficient of friction on the frictional equivalent strain. 3 and 4. These relationships are very similar to those obtained experimentally using the apparent strain method. The main differences being that the changes in the predicted mean equivalent strains with variation in rake angle and coefficient of friction are smaller than those measured experimentally and that the cutting ratios are larger for a given rake angle and coefficient of friction. These differences indicated that the model gives a less severe mode of cutting than actually existed but this is modified by the fact that a rigid perfectly plastic material has been considered, a point which is discussed later. The frictional equivalent strain has been shown to be directly proportional to the apparent coefficient of friction for a given rake angle, see Fig. 11. Results obtained with different materials at a c o m m o n rake angle show a scatter about the mean trend but do not indicate a distribution that can be attributed to the mechanical properties of the materials. Therefore, it has been postulated that the direct proportionality between the frictional
An Apparent Strain Analysis of Orthogonal Cutting SYMBOL P~AKEANGLE O 0"9 70/30 BRASS • 0"85 En.9 r:l O'78 8S/IS BRASS (refencn(z16) / X O.9,5 $AE ,,12 AS RECEIVED(refcrena¢,~,) / • 0"96560~-T6 ALU~AINIUM(re(. . . . . i6) /
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equivalent strain and the apparent coefficient of friction is independent of material properties. Equation (12) shows that the frictional equivalent strain, the difference between the apparent and mean equivalent strains, is also the ratio of the stress acting in the chip on the deformation zone to the yield stress of the work-hardened material. The postulate, therefore, indicates that for a given apparent coefficient of friction and rake angle the stress acting on the deformation zone in the chip is proportional to the yield stress of the work-hardened material. It is interesting to consider the effect of this in relation to the work-hardening properties o f the materials. Consider the effect of cutting two materials of equal strength at large strains but having different work-hardening characteristics, such as a sample of a material which is work-hardened and a sample of the same material which is annealed. After cutting, the mechanical properties of the chips will be approximately similar and,
114
P.J. THOMPSON,H. OGDENand N. A. BUTTERWORTH SYMBO~ O (a). PREDICTION FROM EXPERIMENTAL TREND FOR A NON-WORK HARDENING MATERIAL
0
(b), PREDICTION BASED ON MERCHANT'S ANALYSIS FOR A NON- WORK HARDENING MATERIAL
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FIG. 13. The effects of the apparent coefficient of friction and the mechanical properties of the material on the cutting ratio, for a rake angle of 35 deg.
therefore, both the apparent coefficients of friction and the stresses acting on the deformation zone in the chips will be similar in magnitude. The effect of this for the material which has a low work-hardening factor is to produce spreading of the deformation zone to give a thick chip, since the undeformed material in the workpiece is soft and will deform more easily. This gives a low cutting ratio for the material with a low work-hardening factor. Experimental evidence to support this argument is shown in Fig. 13 for a rake angle of 35 deg. This figure clearly shows that low cutting ratios were associated with materials which have low working-hardening factors. It was found that these trends obeyed the empirical equation (27). The relationship between the cutting ratio and the apparent coefficient of friction for a non work-hardening material (/3 = 1) was predicted from this empirical law, curve (a) Fig. 13, and found to agree closely with that predicted by considering a non-work-hardening material and the single shear plane model, curve (b) Fig. 13. The effects described above indicate that the geometry of the deformation zone and hence the plastic strain induced in the chip are related to the resistance offered to the movement of the chip at the rake face and the work-hardening characteristic of the material. It follows that these factors also influence the induced apparent equivalent strain. The empirical law, equation (21), and equation (18) have been combined with equation (17) in order to predict the influence of these factors on the apparent equivalent strain and gave equation (23). This latter relationship indicates that the dimensionless group r/3~a should give a unique relationship for all materials. Figs. 7 and 10 show this group obtained with large rake angles for the 70/30 brass and the En 9 steel. It can be seen that this parameter lies within a narrow band over the range of apparent coefficient of friction induced.
An Apparent Strain Analysis of Orthogonal Cutting
115
Predictions obtained from equation (23) gave a family of closely interlocking lines which fall within the bands indicated in these figures. The influence of the materials on both this dimensionless group and the apparent equivalent strain are shown in Fig. 12 along with predicted lines obtained from equation (23) and the Merchant single shear plane model, for a rake a~gle of 35 deg. It can be seen that for a non-work-hardening material these two forms of prediction agree closely. On the basis of the empirical law equation (21) and using equation (17) it is possible to predict the distribution of the apparent equivalent strain for any work-hardening material. Such predictions have been made for En 9 steel, Fig. 10, and for other materials at a rake angle of 35 deg in Fig. 12. These indicate close agreement with the experimental results. Although this is a new approach which does not assume any particular mode of chip formation the similarities between the experimental results and the predictions based on the models of cutting are very clear. There are two parts to each solution obtained from the models of cutting. The first has been written in terms of the strain induced in the chip and the second in a relationship between the geometry of the deformation zone and the coefficient of friction, the latter often has been expressed in terms of a "shear angle relationship". The apparent strain method also leads to a solution with two parts. The first may be described as a process characteristic which shows either the mean or apparent equivalent strain against the apparent coefficient of friction, and the second shows the cutting characteristic for the material, the relationships between the cutting ratio, rake angle, apparent coefficient of friction and the undeformed chip thickness. The cutting characteristic is equivalent to the shear angle relationships obtained from the models of cutting. It is interesting to note that Mercha:at found it necessary to modify his shear angle relationship in order to make it agree with the results obtained from ductile metals by introducing a parameter which was a function of the mechanical properties of the material. Similarly it has been shown that the relationships which make up the cutting characteristic depend on the work-hardening factor, see Fig. 13. It has further been shown that the work-hardening factor influences the process characteristic, emphasizing that it is an important parameter in the mechanics of metal cutting.
7. S U M M A R Y It has been shown that the apparent strain approach leads to a valid solution for materials which work-harden and gives information unobtainable by other methods. The machining properties of a material can be represented on two diagrams, one showing the apparent equivalent strain and the other the cutting ratio both against the apparent coefficient of friction. These diagrams are similar for all ductile materials. Orthogonal cutting is more efficient when the rake angle, undeformed chip thickness and work-hardening factor for the material are large. The stress acting on the deformation zone in the chip has been shown to be directly proportional to both the apparent coefficient of friction and the yield stress of the workhardened material. This empirical law enables the apparent strain to be predicted for materials having different work-hardening properties. It is possible with the apparent strain theory to estimate the cutting force from the cutting characteristic of the material and the equivalent stress and strain diagram. It should be possible to analyse other cutting operations such as oblique and multi-point processes using the apparent strain theory.
116
P.J. THOMPSON,H. OGDEN and N. A. BUTTERWORTH
Acknowledgements--The authors wish to thank the Principal and Governors of Harris College for permission to undertake this work, and also Mr. R. Wood and Mr. J. Slater for their kind assistance during the machining tests. They are indebted to the following firms for their kind support of the investigation with materials, equipment and information: British Aircraft Corporation, Preston; Hawker-Siddeley Aviation, Manchester; Imperial Metal Industries (Kynoch), Birmingham; Titanium Metal and Alloys, London. They are grateful to Professor F. Koenigsberger, Professor W. Johnson, Dr. C. Rubenstein (University of Manchester) and Dr. D. H. Sansome (University of Aston in Birmingham) for their helpful comments during the preparation of this paper. REFERENCES R. HILL, Mathematical Theory of Plasticity O.U.P. (1950). W. JOHNSON,J. Inst. Metals 85, 403 (1957). H. LL D. PtrGH, Bulleid MemorialLectures, IIIB (3)27 (1965). P. J. THOMPSON,Hydrostatic extrusion of steel, M.Sc. Thesis, Aston Univ. (1968). M. E. MERCHANT,J. appl. Phys. 16, 267, 318 (1945). W. KATTWINKEL,lndustrie Anzeiger 525 (1957). [7] N. N. ZOREV,Proc. Int. Prod. Eng. Res. Conf., A.S.M.E., Pittsburgh, p. 42 (1963). [8] E. G. THOMSEN,C1RP Ann. XIV, 113 (1966). [9] C. RUBENSTEIN,Int. J. Mach. Tool Des. Res. 4, 123 (1965). [10] G. I. TAYLORand H. QLqNNEY,Phil. Trans. R. Soc. A. 232, 323 (1931). [11] S. KOBAYASHIand E. G. THOMSEN,J. Engng Ind. 81,251 (1959). [12] E. G. THOMSEN,E. SCHALLERand H. G. DOHMEN, Industrie-,4nzeiger 25 (1963). [13] T. SATA, Sc. paper 1.P.C.R. 53, 53, Sept (1959). [14] W. JOHNSONand P. B. MELLOR,Plasticity for Mechanical Engineers p. 62. Van Nostrand, London (1962). [15] A. B. WATTS and H. FORD, Proc. Inst. mech. Engrs B l B , 448 (1962). [16] D. M. EGGLESTONE,R. HERZOG and E. G. THOMSEN,,J'. Engng Ind. 81,263 (1959). [17] W. B. PALMERand P. L. B. OXLEY,Proc. Inst. mech. Engrs 173. 623 (1959). [1] [2] [3] [4] [5] [6]
Vol. 9, pp. 97-116.
Pergamon Press 1969.
Printed in Great Britain
AN APPARENT STRAIN ANALYSIS OF ORTHOGONAL CUTTING P. J. THOMPSON*, H. OGDEN~ and N. A. BUTTERWORTH~
(Received 24 November 1968) Abstract--The problems of obtaining a solution for orthogonal cutting based on experimental information obtained from materials which work-harden have been considered. A valid solution based entirely on experimental information and the apparent strain theory was obtained without the need to define a model of chip formation. The problems of relating results obtained from materials which have different work hardening characteristics have also been considered. Good correlation was obtained for a number of ductile metals on the basis of the simple postulate that the stress acting on the deformation zone in the chip is directly proportional to both the apparent coefficient of friction and the yield stress of the chip. This has shown that the work-hardening property of the material is an important parameter in defining the mechanics of orthogonal cutting. The appareat strain approach may be used to estimate the cutting forces from a simply determined cutting characteristic and the equivalent stress-strain diagram of the material. 1. I N T R O D U C T I O N ORTHOGONAL cutting is the process in which the tool forces and the direction o f cut lie in a plane, see Fig. 1. As a result o f relative m o t i o n between the cutting tool and the w o r k p i ece the material to be r e m o v e d passes t h r o u g h a d e f o r m a t i o n zone extending f r o m the tip o f the tool to the surface o f the workpiece to f o r m a chip. Several m o d e s o f cutting can result
I ~ U N D E F OTHICKNESS R M E D ~ICHIP
F¢ ~
DIP~ECTIONOFR O T A T ~ - ;
DEFORMATION ZONE
J
~
/
~
pLASTICSUBLAYER ~ CLEARANCEANGLE
/~ CUN TG I, R ~ o ~ G ~ ~ " -
1
" / / / / / / J J
j FiG. 1. Chip forces in orthogonal cutting. * Senior Lecturer in Production Engineering, Harris College, Preston. i" Senior Lecturer in Production Engineering, Harris College, Preston. :~Principal Lecturer in Production Engineering, Harris College, Preston. 97
RAKEANGLE
98
P.J. THOMPSON,H. OGDENand N. A. BUTTERWORTH
from this process which give rise to different chip characteristics. In the type of cutting investigated ductile materials are cut with the formation of a continuous chip. The material removed experiences large plastic strains in the deformation zone and is considerably work-hardened. These strains are induced in the chip by shearing of the material and they depend on the geometry of the deformation zone. This geometry is not entirely governed by the geometry of the cutting tool, for it also depends on the resistance offered by the rake face to the movement of the chip and the mechanical properties of the material. The resistance offered to the movement of the chip results in the application of a compressive force to the deformation zone, which regulates both the size of the deformation zone and the chip thickness. Both the total work done and the magnitude of the cutting force depend on the sum of the plastic work done during the formation of the chip and the work done in overcoming the resistance to the movement of the chip. I.zu) uJ~
uJ
I
FRICTIONAL WOP, K OF SLIDING DONE BY THE CHIP AS i'r MOVES JOVER THE P,AKE FACE. / PER UNIT VOLUME OF MATERIAL REMOVED
Ym
F1G. 2. The equivalent stress-strain diagram showing the terms used in the apparent strain analysis. The mechanics of orthogonal cutting has been the subject of much research work. This has included phenomenological investigations and upper bound solutions obtained from slip plane theory and models of cutting. The present work is based on an experimental approach which was first devised by Hill [1] and Johnson [2] and later extended by Pugh [3] and Thompson [4] for the analysis of extrusion. It is based on an effective or apparent equivalent strain, defined as the strain which bounds an area under the equivalent stressstrain diagram equal to the total specific energy, see Fig. 2. As considerable work is done in overcoming the resistance along the rake face the apparent equivalent strain is larger than the mean equivalent strain imparted to the chip. This latter strain is found by reducing the area under the equivalent stress-strain diagram by the amount of work done on the chip as it moves over the rake face. This approach enables the strain imparted to the chip and the apparent strain to be obtained from the equivalent stress-strain diagram of the material and measured components of the cutting force. It can, therefore, be found from materials which
An Apparent Strain Analysis of Orthogonal Cutting
99
work-harden and under cutting conditions which approximate to those employed commercially. By undertaking cutting tests at different rake angles distributions of both the mean and apparent equivalent strains for some materials have been obtained, these give equivalent strain characteristics for orthogonal cutting. It has been found that each material produces a different characteristic which depends, amongst other things, on the work-hardening characteristic of the material. Also, that the frictional equivalent strain, the difference between the apparent and the mean equivalent strains, is the ratio of the compressive stress acting on the deformation zone in the chip to the yield stress of the work-hardened chip. The experimental work has shown that this quantity is directly proportional to the apparent coefficient of' friction. It is postulated that the frictional equivalent strain is independent of the equivalent stress-strain diagram of the material. This empirical relationship enables the apparent equivalent strain characteristics for different materials to be related to each other. The information gained from the mean and apparent equivalent strain characteristics may be used in a similar way to that obtained from the models of cutting. Alone they are insufficient to determine the cutting tool force for a given undeformed chip thickness and material. The undeformed chip thickness, cutting ratio and resistance along the rake face, as measured by the apparent coefficient of friction, can be indicated on a single diagram; these relationships give the true cutting characteristics for the material. When this information is combined with the equivalent stress-strain diagram the apparent strain analysis enables the cutting forces to be predicted.
/ZA
F b T
/z t7 0~ o'1o-20" 3
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( g,Ae t
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~F
2. N O T A T I O N Apparent coefficient of friction ~-- F~/Fo Force Breadth of chip Shear stress Coulomb coefficient of friction ---- r/a Direct stress Rake angle Principal stresses Equivalent stress Equivalent strain Principal strains Yield or flow stress Indice Work per unit volume Work-hardening factor Ve][ocity Mean equivalent flow stress over the equivalent strain range 0 ~ A Mean equivalent flow stress over the equivalent strain range gm,~ga Chip thickness Cross sectional area of chip Apparent equivalent strain Mean plastic equivalent strain Frictional equivalent strain ---- ~a -- ~m = an/( Ym)F Friction factor
100
P.J. THOMPSON, H. OGDENand N. A. BUTTERWORTH r
k c, d , f
¢ R a
Sufficeg c t c~ 0 1 2 F P T m A R y
Cutting ratio = tl/t2 Constant of proportionality Constants Merchant's shear angle Friction angle = tan -1 (t~A) Force reading from dynamometer Dynamometer constant
Parallelto the direction of cut Normal to the direction of cut Parallelto the rake face Normal to the rake face Prior to cutting Aftercutting Frictional Plastic Total Mean Apparent Resultant At the yield point 3. T H E O R Y
3.1. Resistance to chip movement The conditions that exist at the chip-tool interface have been the subject of much research. Early work by Merchant [5] assumed that Coulomb sliding friction occurred over the entire area of contact between the chip and the tool. This assumption leads to a unique value of the coefficient of friction which is independent of the other cutting parameters. This does not agree with the experimental observations which have shown that the ratio of the component force along the rake face of the tool to the normal component varies with the rake angle and the undeformed chip thickness. Kattwinkel [6], Zorev [7], Thomsen [8] and Rubenstein [9] show that the contact stress between the chip and the tool is not constant and that it can be high enough to cause adhesion between the chip and the tool over part of the area of contact. As a result, the area of contact between the chip and the tool can be divided into two zones. Over the first zone, which is close to the tip of the tool, the contact stress is high, resulting in adhesion of the chip to the tool, a plastic sublayer in the chip, and no relative motion between the underface of the chip and the tool. It is suggested that this occurs even in the absence of built-up edge. Under these conditions the chip experiences further deformation as a result of the existence of the plastic sublayer. Over the second zone the contact stress is low and the surface shear stress in the chip is less than the yield shear stress, hence, the chip slides over the tool under conditions of Coulomb sliding friction. In spite of the inadequacy of the Coulomb coefficient of friction the ratio of the component force along the rake face to the normal component is a useful parameter of cutting and is much used in the literature. This ratio is described as the "apparent coefficient of friction" and is given by ,~,a =
F~ • Fo
(1)
An Apparent Strain Analysis of Orthogonal Cutting
101
Which may be written in terms of the measured components of the cutting force, thus /~A =
Ft cos ~ + Fe sin Fe cos ~ -- Ft sin a"
(2)
3.2. Materiul properties The material undergoing plastic flow in the deformation zone is subject to a system of complex stress. To relate the plastic work to the strains undergone by the material during the formation of the chip it is necessary to express in mathematical terms suitable mechanical properties of the material and for these properties to be independent of the stresses acting in the deformation zone. Such a mathematical relationship for stress is known as a plasticity condition or yield criterion. It has been shown by Taylor and Quinney [lO] that the Maxwell-von-Mises plasticity condition gives the best fit with experimental results obtained with ductile materials. If the plastic stress-strain or flow curve is plotted in terms of this function of stress and its corresponding strain function the same curve is obtained regardless of the stresses acting to induce plastic flow. These functions are known as equivalent stress and equivalent strain and are given by ~- = [1{(O-I __ 0-2)2 _t_ (0- 2 __ 0-3)2 _~_ (0-3 - - O"1)2}]1/2
(3)
and ~/2
A ~ =- ~
[(AE1 - - AE2) 2 ~- (AE2 - - AE3) 2 -~ (AE3 - - A~1)2]1/2.
(4)
It can be shown, see Johnson and Mellor [14], that the plastic work done can be expressed in terms of the equivalent stress and strain, thus
w~ ---- f 6 d ~.
(5)
For many materials both in the annealed and work-hardened state the equivalent stressstrain diagrams agree with the empirical relationship. 6 = (Y~=I) ~n.
(6)
Many workers have found from a large number of cutting tests carried out on commercial materials that the equivalent stress obtained from static property tests correlates with an equivalent stress calculated from the Merchant shear plane stresses at approximately the same equiw~lent strain, see Kobayashi and Thomsen [11], Thomsen, Schaller and Dohmen [12], and Zorev [7]. This represents an anomaly since correlation is achieved between static and dynamic conditions in spite of the extreme differences in strain rate and temperature. This anomaly is discussed fully by Thomsen [8] and Sata [13]. It is suggested that the effect of the induced temperature on the flow stress is offset by the strain rate. This work is taken to justify the use of statically determined equivalent stressstrain diagrams in the analysis of orthogonal cutting.
3.3. The apparent strain theory The apparent equivalent strain is defined as the equivalent strain which bounds an area under the equivalent stress-strain diagram eqflal to the total specific energy, thus ~A
W~-- [ 0
6d~---- Ymga.
(7)
102
P . J . THOMPSON, H. OGDEN and N. A. BLrrTERWORTH
Figure 2 shows the equivalent stress-strain diagram along with the apparent, mean and frictional equivalent strains. It is evident from these definitions that, ~A =
gm @ ~F.
(8)
To relate the apparent and mean equivalent strains it is necessary to determine the work done per unit volume of material removed in overcoming the resistance to movement of the chip, thus WE -- F~V~ tlbl Vc" From volume constancy it may be written,
(9)
tlbl Vc = t2b2 V~
giving, wF_F~ t2b2
--
(T~,
(10)
It should be noted that if adhesion between the chip and the tool takes place the "friction" work done per unit volume of material removed includes some plastic work. From the definition of frictional strain, the frictional work may be written, (11)
WE = ~e( Ym)F.
Combining equations (10) and (1 1) gives, ~V - -
F~
--
t262( Y) mF
~
(12)
( Ym )F"
Since the strains induced during orthogonal cutting are large the flow stress approaches a constant value thus (Ym)F-"- Y~-~. This shows that the frictional strain, the difference between the apparent and the mean equivalent strains, is also the ratio of the stress acting on the deformation zone in the chip to the yield stress of the material removed. The total specific energy may be written,
Fc
(13)
WT -- AI"
Using the principle of conservation of energy equations (7) and (13) may be combined to give, Fc - - #A Y m . (14) A1 Resolving forces in the direction of cutting gives, Fe = Fo cos c~+ F~ sin or Fc
=
(15)
F0[cos o~ -~ ~A sin ~]
since by definition
F~ Fo Combining equations (14) and (15) gives
Ym(:AA1 =
F0[cos ~ -[- P,A sin
a]
(16)
An Apparent Strain Analysis of Orthogonal Cutting
103
and equations (12) and (16) YmgAA1 : t2b2( Ym)FgF
-cos ~ + tz~i sin a],
/
P.A
hence EA =
1
~
EF,
(17)
9
where
Lcos O~-~
#3
l
(18)
/ZA sin (XJ
giving ~A-- 1 ~m --~b"
(19)
It is seen that the friction factor includes a work-hardened factor given by
3 - ( Ym Ym)F" As the flow stress approaches a constant value at the order of strains induced by orthogonal cutting this work-hardening factor may be written
Ym f l - Y~=EA"
(20)
Hence for tnaterials which obey the empirical law equation (6) the work-hardening factor becomes 1 l+n"
I0
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35 d¢~j.
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FIo. 3. Cutting ratio-coefficient of friction relationships obtained from the single shear plane model.
104
P.J.
THOMPSON,H. OGDEN and N. A. BUTTERWORTH
3.4. A unique solution The experimental work, to be discussed later, shows that the apparent equivalent strains obtained with different materials do not agree. However, it is shown that the frictional equivalent strain is directly proportional to the apparent coefficient o f friction for the materials investigated. It is postulated that this empirical relationship is independent o f the material, giving
Y
c~=-eonst.:[~F]c~=c°nst" = k/LA
(21)
where k = c -- d.a.
(22)
Using this relationship and equations (17) and (18) the apparent equivalent strain may be written rfl[£4]~=eonst. = k (cos ~ + tZA sin ~).
(23)
The dimensionless group r/3gA should give a unique solution for a given rake angle and any material, the accuracy o f this is discussed later.
3.5. Merchant's analys& [5] In this analysis it is assumed that the material is rigid, perfectly plastic and that the deformation zone consists of a single shear plane inclined to the direction o f cut by an ,~=-20 de 9 .
3.0 r2.8
2"6
I
i 2.,4 ~" I
2.2 ~" i 20 1
i 1"6 i 1.4 I
Q VO[L 1"2
o~ ~-
I
0'4 II-0.2 t 0
I
0"2
, , , 0.,4 o6 o.,
, , ,. ,.o ,.2 ,,4
,. ,6
,',
,
2.o
COEFFICIENT OF FRICTION
FIG. 4. The influences of the cutting parameters on the mean equivalent strain according to the single shear plane model.
An Apparent Strain Analysis of Orthogonal Cutting
105
angle 4'. The mean equivalent strain induced in the chip according to the model of cutting is, gm = ½[cot ~ -]-
tan(~ -- a)].
(24)
Using this model an upper bound for orthogonal cutting may be obtained using the principle of minimum work, which gives the relationship 2~ +
a -
~ -
7"/" 2
(25)
The latter expression was subsequently modified by Merchant in an attempt to improve correlation with materials which work-harden. Figure 3 shows the relationship between the cutting ratio and the coefficient of friction obtained from equation (25). Figure 4 shows the mean equivalent strain obtained from equation (24) and (25) against the coefficient of friction. 4. E X P E R I M E N T A L P R O C E D U R E 4.1. All cutting tests were carried out on a 17 in. swing centre lathe machining tubular workpieces at constant cutting speed. Cutting forces were measured under orthogonal cutting conditions over a range of longitudinal feeds from 0.0012 to 0.020 in/rev. 4.2. Dynamometer
This was a three component strain gauge lathe-tool dynamometer manufactured to a design of Hawker-Siddeley, Manchester, and activated by an a.c. transducer meter. The demodulated signal from the meter was displayed on an u.v. recorder against a time base. The dynamometer was calibrated separately in two directions using dead weights and a lever system with a moment arm of 20 : 1. Records from the u.v. recorder were obtained from both measuring elements during calibration and the interaction was determined. The tool forces during cutting were obtained from: Rc = accFc q- aaFt I Rt = atcFt -]- attFc I" TABLE l. DYNAMOMETERCONSTANTSOBTAINED FROM CALIBRATION
Dynamometer constants
cm/lbf
ace act
0.0257 0.0015
arc a~t
0
0.0280
4.3. Work material
The workpieces were in the form of drawn tubing in the as-received condition. Brass tube 2½ in. o.d. 0.130 in. wall thickness Copper 70-73 ~ arsenic 0.02--0.06 ~ zinc remainder En 9 tube 27g in. o.d. 0.185 in. wall thickness.
(26)
106
P . J . THOMPSON, H. OGDEN and N. A. BUTTERWORTH
4.4. Cutting tools These were prepared from ~ in. square high-speed-steel tool bits of the following composition: C 0.8~ W 18.25~ Cr 4 . 8 ~ Va 1.25 ~
Co 5.0 ~o
Mo 0.75 ~o
The rake and clearance faces were commercially ground to a finish of approximately 24 c.l.a, and the tools were redressed between cutting operations. 4.5. Chip geometry The cutting ratio was obtained from measurement of the average thickness at the centre of five representative chips and from the calibrated longitudinal feed of the machine. The chip thickness was measured using a micrometer having modified ball anvils and giving least determination of 0.0001 in. 4.6. Stress-strain curves The static equivalent stress-strain curve data, Fig. 5, were obtained by plane strain compression tests carried out on samples of the material cut from the tube wall, according to the procedure outlined by Watts and Ford [15]. Plastic work-equivalent strain diagrams were obtained from these curves by graphical integration. The work-hardening factors were obtained using equation (20) for several points on each curve and found to be approxi. mately constant for each material, thus 70/30 brass = 0.9 En 9 Nc
120
--
I00
~
eo
~
60
= 0.85.
40,
~
20 0
I
I
I
I
I
I
I
I
I
I
I
I
I
0 N
120
---
IO0
x
80
~: 60 =,
4o
_~
20
=o
0
I
I
I
I
I
I
o.=
0-4
0.6
o.e
i,o
t'2
BQUIVALENT
I
I
I
I
I
I
f
1.4
f,6
i.a
2.o
2-2
2 "4.
2"6
STRAIN
FIG. 5. Equivalent stress-strain curves.
An Apparent Strain Analysis of Orthogonal Cutting 5. E X P E R I M E N T A L
107
RESULTS
In order to demonstrate the feasibility of the apparent strain method dry cutting tests were undert~.ken with 70/30 brass using rake angles between -- 15 deg and 45 deg at a cutting speed of 134 ft/min. This material was chosen because it was found to give a continuous chip without a built-up edge. The measured cutting forces were corrected for dynamometer interaction. The apparent coefficients of friction were determined from these cutting forces and are shown along with the measured cutting ratios in Fig. 6. The approximate trend lines corresponding to various undeformed chip thicknesses are also shown. SYMBOL • • • 0 r:l
BREADTH OF CUT ~ O'1301n. CUTTING SPEED ; 134ft/lilln.
Z~ 0
RAKEANGLE - 15 dcg. - 5 deg. 5 d¢ 9.
15 de9. 2Sdlg. 35 de 9. 45 d¢9.
0.8 t I - 0" 0 2 0 In.
0.7
./
t poolom.
~, ti'.0"00:39 in. ~ . . . = ~ " 0 " 0 0 " 1 t i . O. O0,9 i..
0"6
~
O'S
..,'=-zr~...~/
.~/D/~-)~.~ ~ / ~ O
~
.
/
r~/
0"4 0-3 0-2 0"I
0
I
I
I
I
0'1
0-2
0'3
0'4
I
1
I
I
I
I
I
0"5 0-6 0"7 0 ' 8 0 " 9 I'0 I.I APPARENT COEFFICIENT OF FRICTION
I
1'2
I
1,5
I
f
1"4
1"5
FIG. 6. The effects of the apparent coefficient of friction and the rake angle on the cutting ratio,
showing the undeformed chip thickness, for 70/30 brass.
The apparent equivalent strain was obtained for each test by converting the total specific energy, equation (13), to its corresponding equivalent strain using a plastic work-equivalent strain diagram, obtained from the equivalent stress-strain diagram. These strains are shown against the apparent coefficient of friction in Fig. 7. The mean equivalent strain for each test was derwed from equation (19) using the measured apparent equivalent strain and the calculated friction factor. These strains are shown against the apparent coefficient of friction in Fig. 8. Similar tests were carried out on En 9 steel using soluble-oil coolant at approximately the same cutting speed. The underface of the chips and the rake face of the tools showed evidence of built-up edge. The results were obtained as described above and are shown in Figs. 9 and 10. The frictional equivalent strains were obtained, using equation (17), at various rake angles, for the experimental data and for cutting data obtained from references [16] and [17]. These are shown against the apparent coefficient of friction in Fig. 11. It was shown that the friction~tl equivalent strain was directly proportional to the apparent coefficient of
P.J. THOMPSON,H. OGDEN and N. A. BUTTERWORTH
108
SYk~DL CUTTING SPEED: 134 f t / m l n .
,4.0
RAKE ANGLE
•
- IS dcg.
•
- 5 de 9.
3.8
•
S deg.
36
@ ~]
15dcg. 25 dcg. 35 dcg.
3"4
~)
4 S dcg.
3.2 3'O 2.8 2.6
• I
_z
/
./°
2,4 2-2 w _J 2.0 ~.8 ~-6 1.4 1.2 I.O 0.8
t
0"6
0
O'4
o
O'6 0.4 0"2t~
O.2 O
O-8
I
O
0.1
I
O1"2 0"3
I
I
O'4 O"S 0"6 O17 O!8 0"9 I~O APPARENT COEFFICIENT OF FRICTION
I
1!1
1.2
1!3
L ,.'s°
FIG. 7. The effects of the apparent coefficient of friction and the rake angle on the apparent
equivalent strain for 70/30 brass, showing the factor eArfl.
friction. The constant of proportionality was found to be independent of the material and to agree with the empirical law, equation (22), for which, c
---- 0 . 5 3 7
d =
0.0021.
As was shown in the theory this empirical relationship enables the apparent equivalent strains for different materials to be compared. This gave rise to the dimensionless group in equation (23) and shown in Fig. 7 for 70/30 brass and in Fig. 10 for En 9. It is also shown for a rake angle of 35 deg and different materials in Fig. 12, along with the apparent equivalent strains. The relationships between the cutting ratio and the apparent coefficient of friction for different materials at a rake angle of 35 deg are shown in Fig. 13. The work-hardening factors and the mean yield stresses of the chips are also shown in this figure.
An Apparent Strain Analysis of Orthogonal Cutting SYMBOL 4.0
109 RAKE ANGLE_
-- 15 de9.
--
CUTTING SPEED: 134 ft/min
--S dig. • O 13
3"8
/.
3-6 3.4 3.2 3.0
5 de 9. ISd¢ 9. 25 dig,
~.
3 5 dig. 4 5 dig.
2,8-2"6
2"4 2'2 2"0 1"8
1"6 1"4 I. 2 1.0 0"8 0.6 0"4 0'2 0 0
I
I
I
I
O'!
0'2
0-3
0-4
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I
I
I
0-6
0.7
O.B
i 0.9
I
I
I
I
I
I
I'O
I'1
1"2
I-3
1"4
I-5
APPARENT COEFFICIENT OF FRICTION
FIG. 8. The effects of the apparent coefficient of friction and the rake angle on the mean equivalent strain for 70/30 brass.
These relationships were found to agree approximately with the following empirical law, r = f--
0.27/ZA
(27)
where f=
1.17/~ - - 0.26
and = 35 deg. This was used to estimate the relationship for a non-work-hardening material, curve (a) Fig. 13. The corresponding relationship obtained from Merchant's model is also shown, curve (b). 8
P. J. THOMPSON: H. OGDEN and N. A. BUTTERWORTH
110
5;YMBOL BREADTH OF CUT: CUTTING SPEED :
(~ f:'[ A
O'lBs;in, 135;ft/min.
RAKE ANGLE ISd¢ 9. 25;deg. 35 deg45 dcg-
0-8 0.7 t l = 0'015;5;in.
.....~_ ~. ....... ~tr/o.o,og., tro.oo631..
0.6
V
_(2
~
-
~
,
~
>
~
'
A
tl= OOOf91n,
0,5 0'4 0.3 0.2 0.I-0
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0 "' 2
0 '3
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' 0"5
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g
'8
' 0"9
,.'o
I! I
' 1"2
I' 3
' 1"4
I !fi;
1 1"6
, 1"7
APPARENT COEFFICIENT OF FRICTrON
FIG. 9. The effects of the apparent coefficient of friction and the rake angle on the cutting ratio, showing the undeformed chip thickness, for En 9. 6. D I S C U S S I O N
It was possible to obtain the mean and apparent equivalent strains, induced by orthogonal cutting, from the statically determined equivalent stress-strain diagram, the measured components of the cutting force and the chip geometry. An inaccuracy which could have occurred in the apparent equivalent strain determination was that due to the differences between both the temperature and strain rate induced by cutting and those induced during the plane strain compression test. The assumption that the statically determined equivalent stress-strain diagram was valid under the cutting conditions had not been investigated but was justified by the work of Thomsen [8] and Sata [13]. High cutting speeds were avoided and a coolant used when cutting En 9 steel in order to minimize the induced temperature. In addition it was found that increasing the speed to a maximum of 520 ft/min did not appreciably alter the apparent equivalent strain-apparent coefficient of friction relationship; but it may be significant that the duration of the tests were short. It was also found that this relationship was not affected by the surface roughness of the rake face within the range 2 to 24 c.l.a. Both the apparent and mean equivalent strains were shown against the apparent coefficient of friction. An alternative would have been to show these strains against the cutting ratio. However, it was shown that for a given rake angle and material there was a unique relationship between the cutting ratio and the apparent coefficient of friction. Also, it was considered that the apparent coefficient of friction was determined to a higher degree of accuracy than the cutting ratio and the apparent coefficient of friction was shown to undergo a large change over a wide range of rake angles which made it a more acceptable parameter. The apparent equivalent strain-apparent coefficient of friction characteristic showed a decrease in strain with decrease in the apparent coefficient of friction and an increase in rake angle. As small apparent coefficients of friction were obtained at large undeformed chip
An Apparent Strain Analysis of Orthogonal Cutting
111
CU'I-I'fNG SPEED: 13Sft/mln
4'0
SYMBOL
3-0
RAKE ANGLE
O
IS de9.
3-6
E] /',
2Sde 9. 3 S d e 9.
3"4
0
4Sdeg.
3"2 3"0 2'8 2"6 2-4 _Z
13
2.2 2"0
=a i-
f.B 1'6
n.. 1"4 KEY
1'2
PREDICTION BASED ON PROI~RTIONALITY BETWEEN FRICTIONAL EOUWALENT STRAIN AND APPARENT COI~:ICIENT OF FRICTION
I'0
I.O &
0"8
0-8
o
oo
0.6
oo'mG
0-6
0"4
0,4
O'2
O.2
0 C'I
I 0.2
I 0.3
I 0.4
I 0"5
I 0.6
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I 0'8
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I I, I
I 1.2
i 1,3
I 1.4
I I.S
I 1'6
,i
O 1.7
APPARENT COEFFICENT OF FRICTION
FIG. 10. The effects of the apparent coefficient of friction and the rake angle on the apparent equivalent strain for En 9, showing the factor iar~. thicknesses, and as the apparent equivalent strain was a measure of the total specific energy, it follows that the energy required by orthogonal cutting was minimized by increasing the rake angle and the undeformed chip thickness. In order completely to define the machining properties of the material it is necessary to know the relationships between the cutting ratio, the apparent coefficient of friction and the undeformed chip thickness. It is shown in Figs. 6 and 9 that for a given material these relationships can be indicated on a single diagram. Such a diagram shows the cutting characteristics of the material which could not be deduced from the equivalent stress-strain diagram alone, since the relationships shown largely depend on the surface properties of the material and the tool. Figures 6 and 9 show that at all rake angles the cutting ratio decreases with an increase in the apparent coefficient of friction and a decrease in the undeformed chip thickness. The Merchant single shear plane model was used to derive theoretical relationships between the mean equivalent strain, the cutting ratio and the coefficient of friction, Figs.
P.J. THOMPSON,H. OGDENand N. A. BUTTERWORTH
112 I'O F
SYMBOL
0"9 I
O
RAKE ANGLE 7 0 / 3 0 Bl:a.ASS En.9
A
O"8
0"7
J
(a). FOP, A RAKE ANGLE OF 45 deg.
/ O,6-O5 -O=l 0'3 O.2 O.I O
.-/
t
O O-I
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0"2
0.3
0"4
0"5
0.6
0.7
I
O.B
APPARENT
1"O
z
COEFFICIENT
SYMBOL MATF.PdAL 70/30 BRASS O
0,9
A rn
0"8
I
0.9
I
I
I.I
1.2
X
I
I
I
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1.7
(b~. FOP, A RAKE ANGLE OF 35dt~l.
•
MILD STEEL {reftrtnc¢ 17) @
I
1"3
OF FRICTION
En.9 8S/15 BP~SS (rGflrlnc¢ 16) SAE 1112 AS I~BCEWED (reflrlnce 16) 6061-T6 ALUMINIUM ('rcftclnc¢ 16)
0,7 0.6
I
1.0
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SAE 1112ANNEALED (referenc¢ 16~
O.S O'4 0.3
//
O-2
J O
0
J
O.I O
0
I
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0.2
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z
I
r
1
I
0,4
o,s
o,s
0.7
oa
APPARENT
I
I
I
I
I
0.9
~.0
H
~.2
=.3
COEFFICIENT
I
I
I
=.4
~.s
i-6
J 1.7
OF FRICTION
FIG. 11. The effect of the apparent coefficient of friction on the frictional equivalent strain. 3 and 4. These relationships are very similar to those obtained experimentally using the apparent strain method. The main differences being that the changes in the predicted mean equivalent strains with variation in rake angle and coefficient of friction are smaller than those measured experimentally and that the cutting ratios are larger for a given rake angle and coefficient of friction. These differences indicated that the model gives a less severe mode of cutting than actually existed but this is modified by the fact that a rigid perfectly plastic material has been considered, a point which is discussed later. The frictional equivalent strain has been shown to be directly proportional to the apparent coefficient of friction for a given rake angle, see Fig. 11. Results obtained with different materials at a c o m m o n rake angle show a scatter about the mean trend but do not indicate a distribution that can be attributed to the mechanical properties of the materials. Therefore, it has been postulated that the direct proportionality between the frictional
An Apparent Strain Analysis of Orthogonal Cutting SYMBOL P~AKEANGLE O 0"9 70/30 BRASS • 0"85 En.9 r:l O'78 8S/IS BRASS (refencn(z16) / X O.9,5 $AE ,,12 AS RECEIVED(refcrena¢,~,) / • 0"96560~-T6 ALU~AINIUM(re(. . . . . i6) /
3'O
2-6
•
0"8 MILD~'rEEL (ref. . . . . 17)
•
0"84
113
/ / /
] /
/
SAE '"2 ANNEALED~efefenot~ Q&I6)
&
/
2.4 2-2 2.0
/ •.
G
Z
•
~I,B
07 J
/
.
-J
O'S 0.6
pREDiCTION BASEDON PROPORTIONALITY BETWEEN FRICTIONALEQUIVALENTSTRAIN ANDAPPAF~ENTCOEFFICIENTOF FRICTION PREDICTION BASED ON k4ERCHANT% Ah~LYSIS FOP,A NON-WORK-HARDENING MATERIAL
0.4 0"2 0
,
0
O'[
,
0"2
,
0'3
,
0-4
,
O'S
t
0.6
,
'
.'
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0"7 O.B O 9
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,
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0.6
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v
0"4 0.2 I
O 0
0.1
I
0-2
I
I
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1
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0.3 0"4 0.5 0.6 0.7 0.8 0-9 I.O APPARENTCOEFFICIENTOF FPJCTION
1
I
I
I.I
1.2
I-3
1"4 I.S
FIG. 12. The effects of the work-hardening factor on the apparent equivalent strain and the factor ~ar~ for a rake angle of 35 deg.
equivalent strain and the apparent coefficient of friction is independent of material properties. Equation (12) shows that the frictional equivalent strain, the difference between the apparent and mean equivalent strains, is also the ratio of the stress acting in the chip on the deformation zone to the yield stress of the work-hardened material. The postulate, therefore, indicates that for a given apparent coefficient of friction and rake angle the stress acting on the deformation zone in the chip is proportional to the yield stress of the work-hardened material. It is interesting to consider the effect of this in relation to the work-hardening properties o f the materials. Consider the effect of cutting two materials of equal strength at large strains but having different work-hardening characteristics, such as a sample of a material which is work-hardened and a sample of the same material which is annealed. After cutting, the mechanical properties of the chips will be approximately similar and,
114
P.J. THOMPSON,H. OGDENand N. A. BUTTERWORTH SYMBO~ O (a). PREDICTION FROM EXPERIMENTAL TREND FOR A NON-WORK HARDENING MATERIAL
0
(b), PREDICTION BASED ON MERCHANT'S ANALYSIS FOR A NON- WORK HARDENING MATERIAL
I.O
MATERIAL 7 0 / 3 0 BRASS En 9 85/15 BRASS (reference 16)
X
SAE 1112 as received (reference 16)
•
6 0 6 1 - T 6 ALUMINIUM (reference 16)
0"9 0"8 ~='91Sj Ym = 115,0001bf/in a
/~= .965 ;
ym= 60,000 Ibf/m a
0-7 0-6 OS 014 /3 --78,; Ym" 9 2 , 0 0 0 Ibf/ln e
0"3
='90.;
Ym=lO0,O00 Ibf/in z
0"2 0"1 0 0
I O.I
I 0.2
I O-3
I t O.40.S
f 0.6
I 0.7
t O.8
I 0"9
I I.O
I I'I
I 1"2
I 1'3
l 1,4
I I.S
APPARENT COEFFICIENT OF FRICTION
FIG. 13. The effects of the apparent coefficient of friction and the mechanical properties of the material on the cutting ratio, for a rake angle of 35 deg.
therefore, both the apparent coefficients of friction and the stresses acting on the deformation zone in the chips will be similar in magnitude. The effect of this for the material which has a low work-hardening factor is to produce spreading of the deformation zone to give a thick chip, since the undeformed material in the workpiece is soft and will deform more easily. This gives a low cutting ratio for the material with a low work-hardening factor. Experimental evidence to support this argument is shown in Fig. 13 for a rake angle of 35 deg. This figure clearly shows that low cutting ratios were associated with materials which have low working-hardening factors. It was found that these trends obeyed the empirical equation (27). The relationship between the cutting ratio and the apparent coefficient of friction for a non work-hardening material (/3 = 1) was predicted from this empirical law, curve (a) Fig. 13, and found to agree closely with that predicted by considering a non-work-hardening material and the single shear plane model, curve (b) Fig. 13. The effects described above indicate that the geometry of the deformation zone and hence the plastic strain induced in the chip are related to the resistance offered to the movement of the chip at the rake face and the work-hardening characteristic of the material. It follows that these factors also influence the induced apparent equivalent strain. The empirical law, equation (21), and equation (18) have been combined with equation (17) in order to predict the influence of these factors on the apparent equivalent strain and gave equation (23). This latter relationship indicates that the dimensionless group r/3~a should give a unique relationship for all materials. Figs. 7 and 10 show this group obtained with large rake angles for the 70/30 brass and the En 9 steel. It can be seen that this parameter lies within a narrow band over the range of apparent coefficient of friction induced.
An Apparent Strain Analysis of Orthogonal Cutting
115
Predictions obtained from equation (23) gave a family of closely interlocking lines which fall within the bands indicated in these figures. The influence of the materials on both this dimensionless group and the apparent equivalent strain are shown in Fig. 12 along with predicted lines obtained from equation (23) and the Merchant single shear plane model, for a rake a~gle of 35 deg. It can be seen that for a non-work-hardening material these two forms of prediction agree closely. On the basis of the empirical law equation (21) and using equation (17) it is possible to predict the distribution of the apparent equivalent strain for any work-hardening material. Such predictions have been made for En 9 steel, Fig. 10, and for other materials at a rake angle of 35 deg in Fig. 12. These indicate close agreement with the experimental results. Although this is a new approach which does not assume any particular mode of chip formation the similarities between the experimental results and the predictions based on the models of cutting are very clear. There are two parts to each solution obtained from the models of cutting. The first has been written in terms of the strain induced in the chip and the second in a relationship between the geometry of the deformation zone and the coefficient of friction, the latter often has been expressed in terms of a "shear angle relationship". The apparent strain method also leads to a solution with two parts. The first may be described as a process characteristic which shows either the mean or apparent equivalent strain against the apparent coefficient of friction, and the second shows the cutting characteristic for the material, the relationships between the cutting ratio, rake angle, apparent coefficient of friction and the undeformed chip thickness. The cutting characteristic is equivalent to the shear angle relationships obtained from the models of cutting. It is interesting to note that Mercha:at found it necessary to modify his shear angle relationship in order to make it agree with the results obtained from ductile metals by introducing a parameter which was a function of the mechanical properties of the material. Similarly it has been shown that the relationships which make up the cutting characteristic depend on the work-hardening factor, see Fig. 13. It has further been shown that the work-hardening factor influences the process characteristic, emphasizing that it is an important parameter in the mechanics of metal cutting.
7. S U M M A R Y It has been shown that the apparent strain approach leads to a valid solution for materials which work-harden and gives information unobtainable by other methods. The machining properties of a material can be represented on two diagrams, one showing the apparent equivalent strain and the other the cutting ratio both against the apparent coefficient of friction. These diagrams are similar for all ductile materials. Orthogonal cutting is more efficient when the rake angle, undeformed chip thickness and work-hardening factor for the material are large. The stress acting on the deformation zone in the chip has been shown to be directly proportional to both the apparent coefficient of friction and the yield stress of the workhardened material. This empirical law enables the apparent strain to be predicted for materials having different work-hardening properties. It is possible with the apparent strain theory to estimate the cutting force from the cutting characteristic of the material and the equivalent stress and strain diagram. It should be possible to analyse other cutting operations such as oblique and multi-point processes using the apparent strain theory.
116
P.J. THOMPSON,H. OGDEN and N. A. BUTTERWORTH
Acknowledgements--The authors wish to thank the Principal and Governors of Harris College for permission to undertake this work, and also Mr. R. Wood and Mr. J. Slater for their kind assistance during the machining tests. They are indebted to the following firms for their kind support of the investigation with materials, equipment and information: British Aircraft Corporation, Preston; Hawker-Siddeley Aviation, Manchester; Imperial Metal Industries (Kynoch), Birmingham; Titanium Metal and Alloys, London. They are grateful to Professor F. Koenigsberger, Professor W. Johnson, Dr. C. Rubenstein (University of Manchester) and Dr. D. H. Sansome (University of Aston in Birmingham) for their helpful comments during the preparation of this paper. REFERENCES R. HILL, Mathematical Theory of Plasticity O.U.P. (1950). W. JOHNSON,J. Inst. Metals 85, 403 (1957). H. LL D. PtrGH, Bulleid MemorialLectures, IIIB (3)27 (1965). P. J. THOMPSON,Hydrostatic extrusion of steel, M.Sc. Thesis, Aston Univ. (1968). M. E. MERCHANT,J. appl. Phys. 16, 267, 318 (1945). W. KATTWINKEL,lndustrie Anzeiger 525 (1957). [7] N. N. ZOREV,Proc. Int. Prod. Eng. Res. Conf., A.S.M.E., Pittsburgh, p. 42 (1963). [8] E. G. THOMSEN,C1RP Ann. XIV, 113 (1966). [9] C. RUBENSTEIN,Int. J. Mach. Tool Des. Res. 4, 123 (1965). [10] G. I. TAYLORand H. QLqNNEY,Phil. Trans. R. Soc. A. 232, 323 (1931). [11] S. KOBAYASHIand E. G. THOMSEN,J. Engng Ind. 81,251 (1959). [12] E. G. THOMSEN,E. SCHALLERand H. G. DOHMEN, Industrie-,4nzeiger 25 (1963). [13] T. SATA, Sc. paper 1.P.C.R. 53, 53, Sept (1959). [14] W. JOHNSONand P. B. MELLOR,Plasticity for Mechanical Engineers p. 62. Van Nostrand, London (1962). [15] A. B. WATTS and H. FORD, Proc. Inst. mech. Engrs B l B , 448 (1962). [16] D. M. EGGLESTONE,R. HERZOG and E. G. THOMSEN,,J'. Engng Ind. 81,263 (1959). [17] W. B. PALMERand P. L. B. OXLEY,Proc. Inst. mech. Engrs 173. 623 (1959). [1] [2] [3] [4] [5] [6]