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The minimum energy principle for the cutting process in theory

and experiment

Citation for published version (APA):

Dautzenberg, J. H., Veenstra, P. C., & van der Wolf, A. C. H. (1981). The minimum energy principle for the cutting process in theory and experiment. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0502). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1981 Document Version:

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(2)

RRK

01

IJPR

WT 0502

THE MINIMUM ENERGY PRINCIPLE FOR THE CUTTING PROCESS IN THEORY AND EXPERIMENT

Authors: J.H. Dautzenberg

P.C. Veenstra

A.C.H. van der Wolf

(3)

: .

j

THE

MINIMUM ENERGY PRINCIPLE

Fon

THE CUTTI~G PROCESS IN THEORY AND EXPERIMENT

,

J.H. Dautzenberg, P.C. Veenstra

,1)

and A.C.H. van der Wolf

(1).

SUmmary: The c tt" . .

i .

f h

d f . u Ing process IS modelled by assuming the presence of two deformation areas. The energy necessary or t e

A;

f~~:t~onin

these two areas

~a~

be written as a

fu~ction.of.

both. the shea: angle and the

fric~ional

force on the, tool.

be~

tlon of the theorem of minimum energy results In a differential equation. Numerical solution renders a relation

C

w~en

the shear angle and the normal ized frictional force. The material's properties are introduced through the constant

an

~

of LudWick's relation. The results of the analysis have been checked by carrying out cutting experiments under

severa cutting conditions and using different workpiece mater.ials as well as different tool materials.

!

Dr.Jr. J.H. Oautzenberg I

Division of Production Technology University of Technology Eindhoven

p .0. Box 513.

5600 MB EINDHOVEN

(4)

1. INTRODUCTION.

Quite a number of investigators tried to describe the cutting process as a process of plastic deformation. To this aim mainly the upper bound theorem and the slip line method have been applied. A good survey of the best known mathematical models is given in [1]. It must be concluded, however, that the agreement between theory and experiment in all the models proposed is not sat i sfactory.

Also in the present work the cutting process is considered to be a process of plastic deformation. while also the friction

phenomenon between tool and chip is described in terms of plasticity (Sec. 2).

The energy input to the process can be written as a function of the shear angle. Application of the principle of minimum energy yields a differential equation which can be numerically solved

if a boundary condition is introduced which can be derived from the upsetting test (Sec. 3). The solution is a relation between the normalized frictional force and the shear angle.

In order to account for the materials properties in the model the specific stress and the strain hardening exponent according to Ludwick are being used. The validity of the model has been tested by carrying out cutting tests on the workpiece materials C45 and X3BCrM05 whilst applying for different feeds as well as cutting

(5)

jspeeds. Moreover two different values of rake angle have been

j

;Used. The results in comparison with theory are shown in Sec. 4. The relevant data are taken from reports on earlier work [2,

13], where cutting force as well as the feed force have been

l

measured as a function of chip area ratio.

The value of the strain hardening exponent has been determined

I

through a tensile test whereas the specifice stress follows from

'performing a cutting test.

General discussion of the present work is given in Sec.

5.

2. A CUTTING MODEL.

Basicly the well known Merchant lamellae model for the cutting process has been adapted as shown in Fig. 1. It is assumed that two different processes of plastic deformation govern the cuttingl process:

- the process of shearing which forms the lamellae in the primary!

shear zone (shear plane).

I

- the process of friction which takes place in the area of contact

I

between tool and chip, to be called the secondary shear zone.

~ = shear angle Yo '" rake angle of v Vc f hc Ff Fv the tool

f

'" cutt i ng speed = chip speed

,V

=

feed - F f "" chip thickness

~

'" feed force = cutting force

fy

----.--.----.

Figure 1. Schematic representation of the cutting process in two dimensions.

(6)

3

. "--.- -....

The power (= Esp) in the deformation process is

(1) Esp

=

Ep + Es

with Ep

=

power in plastic deformation in the primary shear zone,

Es - power in plastic deformation in the secondary shear zone.

It hold for Ep [4J: (2) wi th Ep C1

de:

b f

-

Eend

=

effective stress in the shear zone

=

incremental effective strain in the shear zone

=

width of cut

= feed

v

=

cutting speed

€end = the maximum effective strain in the shear zone.

After introducing Ludwicks ' equation

a

=

cen and de - d (ti~ y),

being the incremental effective strain for pure shear

with shear angle y, integration of Eq. 2 renders:

(~-y ) n+1

(3)

E ___ C __ {cotan ~ + tan o} bfv

(7)

· 4

'.

with C = specific stress

~

=

shear angle in the primary shear zone

Yo

=

rake angle of the tool

n

=

strain hardening exponent.

With respect to Es It holds:

(4) Es

=

F v c

where Vc

=

speed of the chip.

From the geometry of the sin <1i (5) Vc

=

v cos (~-yo) Substitution of Eq.

(5)

(6)

E _ Fv sin <1i s - cos (~-y ) o

process it is obvious that:

in Eq. (4) del Ivers:

iwhere F

=

frictional force on the chip.

'With Eqs.

(3)

and

(6)

Eq. (1) can be written as:

(~_y ) n+1

(7) E

=

_C_ {cotan ~ + tan 0 } bfv + 'F" 5 in •

sp n+l

13

cos (t-yo)

The condition for minimum power follows through: dE

l

(8) -2E.

=

0

d<1i and hence:

1 dF F cos Yo

Cbf d<1i

= -

Cbf cos (<1i-y ) sin <1i

cotan

~

+ tan

(~-Yo)

n 0

{

13

}

x [ . 2 sin t cos (t-yo) +

13

sin t x 1 --:2:--- ] cos (~-yo)

Eq.

(9)

is a differential equation which can be solved

numerically once the boundary condition is known.

3.

THE BOUNDARY CONDITION.

In order to solve Eq.

(9)

the boundary condition must be known.

As to this the very beginning of the deformation process in the primary shear zone is considered. as shown in Fig. 2. Analoguous to the upsetting test the initiation of plastic deformation is

assumed to occur in a plane which is inclined over 450 with

respect to the upsetting force. If the flow stress in the shear plane is Tshi' the frictional force on the tool (Fw) in the contact zone can be written as:

(8)

-,_._--- , (10) F

I

"[' h" bf s I W '" 2 sin 450 sin y o

I

./ '''---'''~ . 5 : \ ' ... '"-."""'"

:For reasons of the definition of positive directions in the coordinate system applied, it is written:

. (11) F '" - F

w

which is the boundary condition for the case that the shear angle

amounts to 450

f

Figure 2. The initiation of the deformation process in cutting •

. 4.

RESULTS.

In order ro present results in dimensionless quantities the

frictional force is nor~lized relative to specific stress, width

of cut and feed, hence ~f' .

Numerical solution of Eq. t9}by means of the Runge-Kutta procedure is represented in Fig. 3, where the normalized frictional force has been plotted as a function of shear angle t. Parameters are

the rake angle (+ 60 and - 60 , resp.) and the strain hardening

exponent (n '" 0.2356).

The material's constants nand C, the strain hardening exponent and specific stress can be determined by means of the tensile

(9)

constants C and n, the original length 10 of the sample, the

instantaneous length 1 and the original area of cross section

Ao

according to:

{ I n _l } n 10

: (12) F

= CA

-I 0 10 I

:By means of appropriate numerical methods the values of both

C

and n can be determined. Results are shown in Table 1.

:Table 1.

'Values of C and n for

C45

and X38CrM05.

c

C45

X38CrMoS C-N-J mm2 n 1170 0.236 1013 0.200

iHowever,there is experimental evidence that the specific stress

:as measured by the tensile test is 10 - 30% less than in cutting,

:due to the combination of strain rate and temperature effects. .

For this reason the specific stress has been determined in a direct way from the cutting test. From the shear plane model

Fig. 1 can easily be analysed for the average shear stress in

.the shear plane to hold:

. , l 2 2'

.(13) Tshi med

=

VFy + Ff cos fb (t+a) sin t

Ff

where f3 = arc tan

r,

(14)

v cos y ~

=

a rc tan { o } (15) hc

f -

sin Yo hc

=

chip thickness Fv

=

cutting force F f

=

feed force

Since the shearing velocity is constant, the effective shearing strain is a linear function of time:

(16) E=kt

where k is a constant.

Hence the average

_ 1 t

effective stress after time tis:

( 17)

<Jmed

=

t

~

(10)

·" ... ,

7

- - - --- .

-From ( 18) plasticity tbeory O'med 'shi med

=

~ 'and thus it is known to hold: Tshi med (n+l) 13 C

=

-tn end

Substitution of experimental data for Tshi med as obtained from Eq. '(13) and (14) gives values for the specific stress under cutting ,conditions.

:The Figs. 4 to 11 included, shown experimental results. As mentioned before the relevant cutting data have been taken from earlier work [2, 3J.

·The frictional force on the tool has been calculated as:

! (20) Fw

=

Fv sin Yo + Ff cos Yo

lwhere Ff is the normal dynamometric force (feed force), whereas :Fv is the tangential component (cutting force).

I

!Figs. 4 and 5 refer to the behaviour of the workpiece materials C45 and X38CrM05 resp. for different feeds. The Figs. suggest a fair agreement between theory and experiment. It is observed

that there is a slight tendency for the shear angle to increase iwhen the feed decreases.

:Figs. 6 and 7 refer to the workpiece materials C45 and X38CrM05

; resp., for different cutting speeds. From these figures it 'appears that cutting speed has no significant influence on the ,relation investigated.

iFigs. 8 and

9

deal with the behaviour of the workpiece materials

iC45 and X38CrM05 resp. for five different tool materials. It cannot be concluded to any significant influence .

. Fig. 10 shows results for the two workpiece materials C45 and

X38CrM05 with different feeds, cutting speeds and tool materials •

. Fig. 11 gives results obtained with two workpiece materials c45

:and X38CrM05 and two different rake angles. As to be expected 'from theory two different groups of data points due to the effect.

of the rake angle can be distinguished.

5. DISCUSSION.

Application of the minimum energy principle in cutting and assuming a boundary condition from the upsetting test renders a theoretical relation between the normalized frictional force and the shear angle. The overall agreement between theory and experiment proves to be quite satisfactory. However, the

considerable scatter in experimental data hampers drawing of conclusions with respect to the influence of seperate influence factors. This scatter can originate from several sources.

(11)

---_._---, ,

8

- Measuring errors:

For the shear angle in the cutting model it holds Eq. (15). In this relation Yo and f are values taken at the beginning of the test. The thickness of the chip (h c ) is measured with a micro-meter after the cutting test. The upper side of the chip has a very irregularly shaped surface. Due to this the value of he

cannot exactly be measured and in many cases will be overestimated. This may be one of the reasons for the scatter in the Figs. 4-11. Another effect which causes scatter in the measurement of the shear angle is the formation of the built'"1Jpedge. This built-up edge roughens at low cutting speed the inner side of the chip and increases variability of chip thickness. Apart from this the widening of the chip is neglected. Probably it has no big

influence on the shear angle. Nevertheless it might be a cause for changing chip thickness.

- Systematical errors:

Moreover, it is a matter of discussion whether the value of the shear stress in the primary zone can also be applied in the ;secondary zone, which as a matter of fact has been done in the 'present work, thus neglecting the possible effects of different

temperatures. Also, the strain hardening of the workpiece in ,the second shear zone has been neglected. The effects of decrease!

jof specific stress by increase of temperature and increase by

!

;strain hardening may compensate more or less. However, it cannot ;be excluded that they have influence on the normal ized frictional

!

force.

:Also the strain hardening exponent depends on temperature. 'Fortunately, it follows from the analysis that its influence,

is mi nor.

ACKNOWLEDGMENTS.

The authors are indebted toMr.A. van Sorgen and Mr.M.Th.de Groot for their experimental assistance and Drs. N.A.L. Touwen for his assistance in processing the experimental results.

6.

REFERENCES.

[lJ. E.J.A. Armarego and R.H. Brown: The machining of metals, Prentice-Hall 1969, pg. 36-72.

[2J. A.G. Strous and H. Munnecom: Beitelkrachten bij draaien WPT 0138. Division of Production Technology, University of Technology Eindhoven (1965), The Netherlands.

[3J. H. Wagtelenberg: H.T.S. Afstudeerrapport. Division of

Production Technology (1974), University of Technology Eindhoven. The Netherlands.

[4J.

R. Hill: The mathematical theory of plasticity. Oxford at the Clarendom Press (1956) Oxford.

(12)

---.

' ,

rJ)

,t.., '",--, I , I

i

I

I

, IT: 0.2356 - Shear angle ~ 25 30 ,

Figure 3. The theoretical relation between the normal ized frictional force and the shear angle.

(1) U I-o 4-rc t: o

...

u I- 4-1.5 ~ .5 N rc E I-o 0'---_--1

o

~ Shear angle ~ 20

25

30 FEED o 0.10-0.24 [mm/rJ x 0.25-0.39 [mm/rJ ~ 0.40-0.54 [mm/rJ

+

0.55-0.69 [mm/rJ • 0.69-0.84 [mm/rJ - THEORY 35

Figure 4. Comparison of theory and experiment for the normalized frictional force as a function of the shear angle for different values of feed. \-Jorkpiece material C45.

Do not fole !

i 1~

(13)

,

.

OJ U I-a

....

1.5

~ 1 o I-U

I-....

"C OJ N

.5

t'D E I.. o Z F w Cbf

t

0 ' - - -....

o

10 - Shear angle <l) 20 25

30

- - - - , - - - , ---FEED (!) 0 . 1 5 [ mm/ r

J

'" 0.30 [mm/rJ t!l 0.45 [mm/r] -THEORY 35

Figure

5.

Comparison of theory and experiment for the normalized

frictional force as a function of the shear angle for different values of feed. Workpiece material X38CrMo5.

1 :5 OJ u I-a

....

t'D c: 0 ... U

I-....

"C OJ

.5

N t'D E I.. 0 Z 0 0 F w Cbf

t

_ Shear angle ~ 20 25

30

CUTTING SPEED (!) 1.00-1.99 [m/s]

x

2.00-2.99 [m/s] ~

3.00-3.99

[m/s] Q 4.00-5.01 [m/s] -,THEORY 35

45

Figure

6.

Comparison of theory and experiment for the normalized

frictional force as a function od the shear angle for different cutting speeds. \/orkpiece material c45.

(14)

1.5 Q) u I... 0 4-l'O c: 0

...

U I... 4-"U Q)

.5

N l'O E I... 0 Z

a

0 11 F w Cbf

t

)( ~ Shear angle ~ 20

25

)(

30

CUTTING SPEED o 1.0 [m/s] x 1.5 [m/s] CI 2.0 [m/s] +

2.5

[m/s] ~

3.0

[m/s] - THEORY 35

Figure

7.

Comparison of theory and experiment for the normalized

frictional force as a function of the shear angle for different cutting speeds. Workpiece material X38CrMo5.

(!.) U I... o 4-l'O c: o I...

4-1.5

15

.5 N l'O E I.. o Z F w Cbf

t

III CUTTING SPEED 2.0-4.0 [m/s] FEED 0.15 0.30 0.45 [~] r TOOLMATERIAL, 0 P 20 ')( P 40 CI M 40 + M 20 <t Kl0 - Shear angle ~ o~--~ ~~

_________

~

________

- L __________ ~ ___ ~

o

20

25

30 35

Figure

8.

Comparison of theory and experiment for the normalized

frictional force as a function of the shear angle for different tool materials. Workpiece material C45.

(15)

"

12 1.5 F w + TOOLMATERIAL Q) Cbf u

t

I-0 4-(!) P 20 III P 40 x + M 40 CI + M 20 ('(l c: 0 ~ Kl0 -+oJ U I- 4--0

.5

Q) N ('(l E I-0 :z 0 ~ Shear angle ~ 0 20

25

30

35

Figure

9.

Comparison of theory and experiment for the normalized

frictional force as a function of the shear angle for different tool materials. Workpiece material X38CrMo5.

1.5

Q) u I-0 4-('(l c: 0

...

U I- 4--0 Q) .5 N ('(l E I-0 :z 0 0 F w Cbf

t

~ Shear angle ~ 20

25

WORKPIECE MATERIAL

x

C45 (!) X38CrMo5 -THEORY 30 35

45

Fi~ure 10. Comparison of theory and experiment for the normalized frictional force as a function of the shear angle for different workpiece materials and various cutting conditions.

(16)

1.5

Q) u .... 0 4-III c:: 0 ... u .... 4-'"0 Q)

.5

N III E .... 0 :z

a

a

F w

Cbf

t

20 13

CUTTING SPEED RANGE 1-5 m/s FEED RANGE 0.1-0.7 mm/r ~:Yo

=

+6 C45. X38CrMo5 x:Yo' =

-6 ;

C45 - THEORY - Shear angle ~

25

30

35

Figure 11. The normalized frictional force for the workpiece materials C45 and X3BCrMo5 as a function of the shear angle for two different rake angles and various

conditions.

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