Forces and plastic work in cutting

Dautzenberg, J.H.; Kals, J.A.G.; van der Wolf, A.C.H.

Published: 01/01/1983

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Citation for published version (APA):Dautzenberg, J. H., Kals, J. A. G., & van der Wolf, A. C. H. (1983). Forces and plastic work in cutting. (THEindhoven. Afd. Werktuigbouwkunde, Vakgroep Produktietechnologie : WPB; Vol. WPB0005). Eindhoven:Technische Hogeschool Eindhoven.

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Download date: 30. Apr. 2018

Forces and plastic work in cutting

J.H. DAUTZENBERG, J.A.G. KALS (1) and A.C.H. VAN DER WOLF (1)

University of Technology Eindhoven, Netherlands.

• _ ..... "::1 ,.-- ----

Prolific considerations in C.I.R.P. give rise to a further step in the development of a plastic cutting model. Basically twodifferent primary shear zone geometries are assumed: with one and with two shear planes respectively. For both geometriestwo energetically different approaches are developed. This leads to expressions for dimensionless friction force numbers.These are defined as the quotient of the friction force and the product of specific stress, feed and width of cut. Anadditional assumption that the supplied power is completely dissipated in the plastic deformation of the chip materialprovides expressions for dimensionless cutting force and feed force numbers. These three force numbers depend on the shearangle and the rake angle of the tool for a given value of the strain hardening exponent. All theoretical models are verifiedexperimentally for several workpiece materials, tool rake angles and cutting conditions. The plasticity values of thematerial necessary to quantify the force numbers are separately determined in tensile tests. Some of the numerous cuttingexperiments are represented graphically and they illustrate the general tendency of a satisfying agreement of theory andexperiment, especially for the model based on two shear planes.

,-1-J.::;

Dr.lr. J.H. DautzenbergDivision of Production TechnologyUniversity of Technology EindhovenP.O. Box 5135600 MB EindhovenNetherlands.

BCi

I../

.--_. --_.~- .. _- ,,-.--~_.

! 1. INTRODUCT ION

Recent discussions in C. I.R.P., partly represented in [1] and· [2], made it necessary to reflect at and to expand our cutting· model based on an energy criterion [3][4]. This model was based· on the following assumptions:

- the cutting process is a two dimensional plastic phenomenon.(plain strain condition).

- the process takes place in two regions: the primary and thesecondary shear zone.

I the deformation in the primary shear zone occurs in one plane.I - the friction force in the secondary shear zone is a functionII of the shear angle.I - the deformation process takes the least power consuming geo-I metry.

From this a differential equation had been derived for the nor­malized frictional force as a function of the shear angle. The

iworkpiece material properties were represented by the strainhardening exponent and the specific stress according to the well

: known Ludwik equation. The tool geometry is defined by the rakeangle of the tool. The differential equation could be solvednumerically with a boundary condition coming from the upsetting

, test.i Comparison of the theoretical and experimental results showedI that the agreement is good except for small shear angles.

'Moreover, it became evident that the deviation for small shearangles is the same for all investigated workpiece materials.

;50 the quantities used seem to be good but the relations can be: improved. Open to question was the assumption that the frictionforce in the secondary shear zone is a function of the shearangle. For the following new approaches this assumption is

!superfluous. Comparison of the new results with experimental data:shows that the theoretical values are too high (Model I). Next,i the primary shear zone has been split up into two shear zones

(Model I I). This model shows a better agreement with the ex­periments. Furthermore, a complete different approach has beenmade. Firstly a one plane primary shear zone is assumed, whilethe cutting process is split up in two deformation steps (ModelII I). The first step deals with shearing without friction. Thefriction effect is taken into account in a separate step.Accounting for the force equilibrium and the power balance for

; this deformation step a relation for the normalized frictionalforce is derived. It is called the friction force number in thefollowing. Comparison of the calculated and measured data showthat the calculated values are slightly to low. Next the primary·shear zone is split up again into two shear planes (Model IV). .

iFor this modification the agreement with the experiments appears. to be excellent. In order to intensify the testing possibilitiesthe cutting and feed force numbers are also calculated. For thispurpose a relation for the total cutting power consumption isused as well as a relation between the friction, cutting and feedforces. The large number of measurements could only partly berepresented. From the experiments it becomes clear that model IVprovides the best approach.The fOUir models are -in essence- presented in the followingchapters.

2. MODEL I

It is assumed [5J:- two dimensional plastic process (Fig. 1)- primary and secondary shear zone

one shear plane in the primary zone- minimum power geometryAs presented earlier [3J for this case we have the followingexpression for the power E1p in the primary zone:

C {cotan ~ + tan(~-Yo)}n+lE1p = "+1 l/3' bfv

C cos Y{ 0 }n+l bf (1)

= n+l 13' sin~ cos(~-Yo) v

with C = specific stressn = strain hardening exponent

Yo = rake angle of the the tool

~. = shear ~n91E!..

3/

b = width of cutf = feedv = cutting speed

; Figure 1: Schematic cutting process in two dimensions (planestrain).

This shear angle is determined geometrically by:

@) 4

(2)

(4)

E,s = cos(lp-Yo)

where F = friction force acting on the chip.w

Assuming a constant external friction force the power is minimumfor:

d (E, + E,) .s P = 0

I dlpI

i Hence, with Eqs(1), (3) and (4):

F, cos(2~-y)w 0

Cbf = 1::'3 • 2"~sln lp

. F1wwhere Cbf = friction force number (dimensionless).

----- -----cos y~-------

~ = arc tan{h }c

f - sin Yowhere h = chip thickness.c

:1 For the power in the secondary shear zone it holds:F v sin ~w

The total power consumption is represented by:/'

F1v v = E,p + E1s

where F1v = cutting force.

(6)

Hence,withEqs (6), (1) and (3):

F1v = _-1_ { I cos yo}n+1 F1w sin ep (])Cbf n+1 13 sin lp cos(<p-y ) + Cbf • cos(cp-y )

o 0

where 1v = cutting force number.tof

The condition of equilibrium of the tool forces leads to:

F,w = F,v sin Yo + F1f cos Yo

where F1f = feed force.

(8)

Hence,

I:~~- cos YoF

where 1f = feed force number.tof

Eqs (5), (]) and (9) prov ide the forces on the tool for a work-

«

l

~~./

piece material represented-by C and n and a geometry representedby y , band f.As sHown in Figs. 5 t 13 the force number curves calculated inaccordance with this model (I) deviate considerably from theexperimental cutting data.

3. MODEL II

This model [5J is based on the same assumptions as model I. Inthis model two shear planes are assumed to intersect each otheron the chip surface (Fig. 2) and shape a cavity at the tool tip.The angles between the shear planes and the connection line fromthe tool tip to the point of intersection of the two shear planesare taken equal (= a). The power in the primary zone is:

C - - n+1E2p = n+1 {£1 + E2} bfv (10)

<jJ.. cos (<jJ. - A. )I •

A:

i where generally:: cos;

E i = 73' sin

For shear plane

A.I ( 11)

(12 )

f

'e !

Figure 2: Geometrical representation for two shear planes.

(13)

For sh¢ar plane B:!

A 1=1t - Z; + Y2 1 2 0

1

I

<jJ 1= 1t+ <p - a -2 2

where Z; = arc tan

z;cos(lp-a)cos y + 2 sin <p cos a sin y{o o} (14)

sin(lp+a)cos yo

(15)

(16 )

/,,6

The power is minimum if:

a(E2 + E25 )P = 0

acp

a(E2 + E2 )P s = 0

auin which E

2s= E1s .

Numerical solution of Eqs (15) and (16) with respect to Eqs (10)F

f (14) gives the value of C~~ as a function of the shear angle cp.

The sh~ar angle cp in fact only a number representing the chipreductiion, is defined by Eq. (2). Conform model I, it holds forthe cutting force number:

Similarly it holds for the feed force number:

F2f F2v tan Yo F2w 1Cbf = - --C:":b"""':f::----'- + -Cb-f • -c-os-y­

o(18 )

The theoretical curves (II) from Eqs. (10) ;. (14), (1]) and (18)are compared with experimental results in Figs. (5) f (13).

4. MODEL II I

This model consists of two separate steps [6J:- frictionless cutting using one shear plane- addition of friction and its upsetting effectFor the rest it has the same geometry as model I. Fig. 3 givesa representation of the cutting process.E[1~~12Q!!!!_~~!~!Qg:The assumption of one shear plane in the

primary zone and frictionless cutting includes a chip thicknessequal to the feed.Because any other geometry requires additional deformation energy.This proposition leads geometrically to:

900+ y y

cp = _--=__0.;;. = 450 + --.£o 2 2

The same relation can be derived from Eqs. (1), (3) and (4) withthe addition that for frictionless cutting F

3w= o.

~9g!!lQr-_Qf_frl~!lQD: The friction force on the chip is indi-I

cated with F3w

. Because of equilibrium there is also a force F3wwith opposite sign in the primary zone parallel to the friction

./~'-'~'

t·, \~... ,)

( 7~>'

force on the tool. The power balance for this part reads:

F3w

v = F3w Vc + liE

where liE = upsetting power from f to h .c

f

Figure 3: Representation of cutting according to model II I.

(20)

With Eq. (1) it holds:C cos y 1

liE { 0 }n+ bfv-= n+l 13' sin ~ cos(~-y )o

C cos y 1{ 0 }n+ (21)

- n+l 13' sin cp cos(~ _y ) bfvo 0 0

Invariancy of volume leads to a relation between v and v .Eq. (20) gives wi th Eq. (21): c :..

(22)

·With Eq .. (b)-thecut-iTngforce number is:

F3v 1 cos Yo }n+1 sin ~ F3will = n+l {/3' sin ~ cos(~-y ) + cos(~-y} Cbf

o 0

(23 )

And-further wfth--Eq:---(9Y-for - the feed force number:

F3f F3v tan Yo F3wCbf = - Cbf + Cbf . cos Yo (24)

The theoretical curves (III) from Eqs. (22), (23) and (24) arecompared with experimental results in Figs. (5) f (13).

5. MODEL IV

The same basic assumptions are made as in model II I. Two shearplanes are assumed now (Fig. 4). For the power of frictionlesscutting it can be written [6J:

(25)

with CPOl

CP02

= shear angle of the first shear plane (I) forfrictionless cutting.

= shear angle of the second shear plane (II) forfrictionless cutting .

f

I~I II I

Y-

•,-_--. lYeI, -I-m--, ~---- --I' ----. J F, ---- r 4W

i J. II I• I/

Figure 4: Representation of model IV for a primary zone with twoshear planes.

Geometrically we have with Eq. (1):_ cos Yl£('01) = . y

IJ sin2

(45° + ~)

(26)

(27)

cos Y2e(CP02) = ----~-­

2 0 Y2IJ sin (45 + 2:)o

and Y2 = 90 + Y - Y1

In Eq. (26) the variable is Y1 taking Eq. (27) into account.If the process takes the geometry connected with minimum powerdissipation it holds:

,.,For the constructed shear angle CP2 in thefriction it is obtained with Eq. (1):

o Yo_ * cos(45 + 2:)e (CP2) = ----:;:----.------~* 'i~ 0 YI:f sin CP2 cos(cpi - 45 - 20

)

(28)

case of cutting with

(30)

where o Yosin(45 - -2)

tan{----.....::..---------}oos Y ~__~o + sin Y - cos(45° - -)tan cP 0 2

(32

Taking friction into account and in agreement with model III (Eq.22) the followin ex ression is obtained:

- n+ 1 - - n+F4w e(CP01)} - n+1 {e(CP02) + e(CP01)}Cbf = ---------{l-_-s-:-i~n-cp---}-------

cos(cp-y )o

Consequently for the cutting force number (with Eq. (7» is:

sin cpcos(cp-y )

o

and the feed force number with Eq. (9):

F4f F4v F4wCbf = - Cbf . tan Yo + Cbf cos Yo

The theoretical curves (IV) from Eqs. (32), (33) and (34)compared with experimental results in Figs. 5 f 13.

(34 )

are

I

~:. )/\ 1a'-....-..

I erE nt orce numbers.

DIN STANDARD %C %Si %Mn %P %SNO.

C15 1. 040 1 0.12-0.18 0.15-0.35 0.30-0.60 ::; 0.045 ::; 0.045

e22 1.0402 0.18-0.25 0.15-0.35 0.30-0.60 ::; 0.045 ::; 0.045

C45 1.0503 0.42-0.50 0.15-0.35 0.50-0.80 ::; 0.045 ::; 0.045

6. EXPERIMENTAL RESULTS

The experimental and theoretical values of the cutting, feed andfriction force numbers are represented in Figs. 5 + 13 as afunction of the chip reduction. The chip reduction is representedby the "shear angle". The experimental results are a representa­tive selection from more than 2000 different measurements. In

, order to obtain the values the following quantities were varied:- feed (between 0.1 - 0.5 mm/rev.)- cutting speed (from 1 to 3 m/s)- toolmaterial (at least three for each workpiece material)·- workpiece material (C15a C22, C45)- rake angle (_60

, 60, 18 ).

In these figures the variation of feed, cutting speed and toolmaterial is implicitely included. The experimental force numberscan be derived from the measured feed, width of cut and cuttingforces with Eq. (8). The specific stress C has been determinedas an average from three tensile tests (Table 1). For the strainhardening exponent only one value (= 0.24) has been taken,because the difference between the various workpiece materialsis very small and therefore neglectable. The values of thechipthlckness have been established by using the mass method [4J.Except for the experiments represented in Figs. 5, 8 and 11,which were carried out with a micrometer.Summarizing it can be concluded that model IV gives the bestagreement between theory and experiment for any of the threed·ff f

, Table 1: Plastic material properties and chemical composition ofthe workpiece materials.

7. DISCUSSION AND CONCLUSIONS

Numerous cutting experiments demonstrate a general tendency ofsatisfying agreement between theory and experiment, especiallyin the case of model IV. From these results it can be concludedthat, contrary to previously proposed models, cutting can beexplained by the plasticity theory. Apparently the necessaryplastic material properties can be derived from a tensile testor any other equivalent test. So, the knowledge of the plastic

11

quantities C and n of the workpiece material and the geometry ofthe process (f, b, Y ) enables to calculate the three forces onthe tool and the chi~ reduction if one of these values is alreadyknown. Although the agreement between theory and experiment israther well, especially in the case of model IV, still there issome scatter in the experimental results. This could be imputedto one or more of the following reasons:- For the determination of the C value results from tensile tests

at room temperature and at low strain rates were used. The Cvalue is necessary for the computation of the total power inthe primary zone. In order to get an impression of the tempera­ture in thi~ zone an approximate calculation can be made. 1f noenergy loss in this zone is assumed the maximum temperature riseis:

C cos Yo }n+1fiT =-- {13n+1 sin q> cos(qJ-Y ) poC0

where p = specific massc = specific heat

For the average temperature rise it holds:

(36)

Table 2 indicates the values which can be expected in thecutting experiments. The values of the shear angle extrema arederived from Figs. 5 + 13. Besides an enhanced temperature inthe primary zone, there is also a very high strain rate. Thisincreases the yield stress value contrary to the temperaturerise. In the first instance these effects will compensate eachothe~ [7J. Nevertheless there will be a deviation which

i

Matedial Shear angle variation Average temperature rise[oCJ

C15 12,5 - 27,5 317 - 134

C22 17,5 - 27 ,5 256 - 160

C45 20 - 35 302 - 181

Table 2: Estimated average temperature rise of the workpiecematerial in the primary zone.

increases with increasing temperature: for cutting with smallvalues of the shear angle.

- Up to now we assumed that the predeformation of the workpiecematerial is zero. Taking this effect in account in model IVresults in higher values of the friction force number, accordingto Eq. (32).

.. 12'-..

" "-----.. u. ". __

- For the sake of simplicity orthogonal cutting was assumed and achip flow in the direction of the feed force. Otherwise it wouldbe nedessary to account for a contribution of the thrust force.This correction might be very important for fine cutting.

- In the models I I and IV a cavity on the tool tip was assumed. Itis also possible to replace this cavity by workpiece material(built up edge). In that case the position of the theoreticalcurves in Figs. 5 f 13 will change a little.

- Another source of deviations could be the roundness of thecutting edge. Measurement gives values for the radius between20 - 90 ~m for unground tips. The relative magnitude of thisvalue compared with a common feed of 160 ~m in our experimentssuggests some effect.A similar effect can be expected from worn tips. Therefore thecutting experiments were always executed with fresh tips (VB <0.2 mm). Nevertheless such a tip has a friction zone on itsflank and gives a contribution to the feed and the cutting force.

- A measuring error can be introduced by the measurement of thechip length value of curled chips. This length is necessary todetermine the chip reduction with the mass method. For,thesechips the average of the inner and outer length has been taken.As there was found more accurate measurement of that lengthresults in a decrease of scatter.

Acknowledgements: The authors are indebted to Mr. A. van Sorgen,Mr. M.Th. de Groot and J.C.M. Manders for theirhelp in performing the experiments.

2 C[N/mmZ]C[N/mm J n ntens i Ie average tens i Ie averagetest test

...

0.23 0.220.23 830 810 8220.23 827

0.24 0.26 0.26 937 965 9800.29 10370.25 0.29 0.23 1377 1533 13390.15 1106

Referenc;es:

[lJ J.H. DautzenbergDiscussion on the paper: liThe minimum energy principle for thecutting process in theory and experiment ll

•

C.I.R.P. Annals Vol 30/2 pg. 603; 1981.[2] J.H. Dautzenberg

Discussion on the paper: liThe minimum energy princ.iple appl iedto the cutting process of various workpiece materials and toolrak~angIes ll

•

17

C. I.R.P. Annals Vol 31/2 pg. 581; 1982.[3J J.H. Dautzenberg, P.C. Veenstra and A.C.H. van der Wolf

The minimum ehergy principle for the cutting process in theoryand exper iment.C.I.R.P. Annals Vol 30/1, pg. 1-4; 1981.

[4J J.H'. Dautzenberg, J.A.W. Hijink and A.C.H. van der Wolf.The minimum energy principle applied to the cutting processof ~arious workpiece materials and tool rake angles.C.I.R.P. Annals Vol 31/1; pg. 91-96; 1982.

[5J P.N.G. Wijnands: Modelvorming voor het verspaningsproces m.b.v.de upper-bound theorie (Dutch) PT report Nr. 558, Universityof Technology, Eindhoven, Netherlands, 1983.

[6J J. Vosmer: private communication, will be published.[7J C.W. Macgregor and J.e. Fisher

A velocity-modified temperature for the plastic flow of metals.Journal of.applied mechanics, All-16, March 1966.

Fv

Cbf

t

\ rake angle _6°

" C45,\\\\ ',,-

'''-,~lll!JI"-..•••''''-,

-::---'~

rake angle 6°

* C15o cn+ C45

Fv

Cbf

,\ rake angle 18°

\ C45

\ \\ \ \\ \ \,\ \, \,

"- '\ "-,"- '~" "-,

"'- ~, ~,............ * "'-'...........

........... -- * ' .......

oo.\--_~ ~_-+- _

ro ~ ~ ~

_<P4020

o+---_~_--+-_~__-~-____<

104030

-<P

o-+--_~_-+-_~_-+-_~_--<

10 20

'. "

\ \ \ake angle 18°

\ .. \ C45

\ \ '.. ,\ \ '

.. \\

\ ... \

\\ ... \

\ \ .. '- \\.\ \.,,- '\ '

"-.. * * \"\'"'- * "

...................... .....

o.o+---_~__-~-_-~-____<

10 20 30 40

_<P

1.0

1.5

0.5

2.5

t

Fw

Cbf

2.0

4030

-<P20

o.o.\--_~__-~-_-_-____<

10

0.5

1.0

1.5

2.5

t

Fw

Cbf

2.0

4030_<P

20

\ \ rake angle _6°

\ \ C45

\ \'. \\. \\ '

0.0.\--_~_-+-_~ ~_--<

10

0.5

1.5

1.0

2.5

t

4030

-<P20

\ \ rake angle 18°

\ \. C45

.. \, \ ,.. \\ \ ,

.. \\ \ ... \\ \ ,

.. \\ \ ,.. \\ \.

\ * \\" ''\ \"-.. '--,,:,,-

"'- * * ','............ * "-

...........0.0 -+--_~__-~-_-~-=_

10

0.5

1.0

1.5

2.5

4030_ <P20

0.0 .\--_~__-~-_-~-----<

10

0.5

1.0

1.5

2.0

2.5

\ \ rake angle 6°

\ " *C15

\ \, 0 C22

\ \ \. + C45

.. \\00 \00 '01 .. '\\ .. ~ \ ,

\ l,* \ \* ' '

\01 . *' \

o .0*0 * \, \\ *0 +It + + \ \

\. \!ID!\ '- '\" 00 lit- t \ \

"- + + "'\ \

"" t+ + ' .. '"~ + +,,~,

........... :---.:...........

t

4030

-<P

\ , rake ~ngle _6°

\ \ C45

\ \\. \,

200.0 -+--_~_-+-_~_+--_~_----<

10

1.0

0.5

2.0

1.5

2.5

t

Figur~s 5-13:Theoretical and experimental results for different workpiecematerials and rake angles

Model I ---- Model II ---- Model III - - Model IV