ra, and the average effective strain of surface

8/6/2019 Ra, And the Average Effective Strain of Surface

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Wear, 34 (1975) 77 - 840 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

77

R, AND THE AVERAGE EF FE CTIVE STRAIN OF SURF ACEASPE RITIES DE FORMED IN METAL-WORKING PROCESSE S*

N. BAY, T. WANHEIM and A. S. PETERSEN

Department of Mechanical Processing of Materials, A.M.T., Technical University ofDenmark (Denmark)

(Received February 20, 1975)

Summary

Based upon a slip-line an alysis of t he plas tic deform at ion of su rfaceas per ities, a th eory is developed deter min ing t he R,-value (c.1.a.) an d th eavera ge effective str ain in th e sur face layer wh en deform ing as per ities inmetal-working processes. The ratio between R, and R,,,, th e R,-value afteran d before deform at ion, is a fun ction of the nominal n orm al pressur e an dth e initial slope y. of t he sur face asperities. The last par am eter does n otinfluence R, significan tly. The a verage effective str ain E in th e deformed sur -face layer is a fun ction of t he n omin al norm al pr essu re an d yo. E is highly

dependent on yo, F increasing with increasing yo. It is shown th at th e R,-value an d the str ain ar e ha rdly affected by th e norm al pr essur e un til intera ct-ing deform at ion of t he as per ities begins, th at is un til the limit of Amont onslaw is reached. After th at R, decrea ses and t he st ra in increas es ra pidly withthe normal pressure, R, appr oaching zero whilst F appr oaches a limitingvalue dependin g on t he initia l slope of th e as perit ies.

Nomenclature

Q-n9

I:R,R 00t2

AdA

m

nominal friction stressnominal normal pressureequivalent yield stressyield str ess in pure shearc.l.a.-valueinitia l R,-valuedistance between ridgesvert ical displacemen t of toolar ea of the deform ed zonear ea of the inst an ta neous deform at ion zonefriction factor

*Paper presented at the Tribology Session of the Eighth Israel Conference onMechanical Engineering, Technion, Haifa, 23 - 24 September, 1974.


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79

00 1.0 2.0 3.0 40 6Fig. 1. The real area of contact as a functionfriction factor.

Fig. 2. Ini tial geometry of surface asperities.

(a)

0

8t

0

bt

of the nominal normal pressure and the

(b)

0

bLJt

0bt

0

Ut

Ii30btFig. 3. Development of slip-line fields and corresponding hodographs with increasing Q!for M = 0.

For (Y ncreasing from a1 to 0~2,~ decreases from y 1 to 7 2. From thehodograph, Fig. 3(c), it is seen that the bottom of the valley moves upwardsat the same velocity ut as the surface moves downwards. The volumeconstancy then gives:


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80

tanyz =8 tanyi (1 + tanyi)

(3 + tanyi)2

1 (3)

ffp =- 2

Similar ly th e hodograp h corr esponding to F ig. 3(e) sh ows tha t the bott ommoves upwa rds a t the sam e velocity ut a s th e surface moves downwards andgives:

tanys =5tany2-12+4J5tany2+9

8--any2

3 + tanys(Y -

3 -5+tany3

(4)

The hodograph in Fig. 3(g) shows tha t the bott om moves upwar ds at avelocit y tw ice th e downwa rd s velocity of th e su rfa ce ut . The law of volum econs ta ncy th en yields:

tarry, =12 tan-r3 (2 + tany3)

2(5+ tw312 (5)

a 4=-3

For LY e4,y

remains constant and equal to y4.

3. Determination of R,

A th eoret ical d eter min at ion of the resu lting sur face geomet ry of ameta l-work ed specimen ha s so far not been possible, but based upon th eres ult s foun d by t he slip-line th eory [ 11, for th e cas e of zero friction it ispossible to det er min e th e R,-valu e (c.1.a.) as a fun ction of y. an d q/2k. Asshown in section 2 the very simple init ial geomet ry chosen will res ult in arelat ive simp le geomet ry of th e deform ed ridges, an d th ereby:

R, =R- =

aO Eo(l -e

where RaO is th e R,-value of t he u nd eform ed ridges an d is given by:

(6)

Rt t=wo

00=-

8

Corr esponding values between OL nd y ar e deter min ed by eqns . (2) - (5).To deter min e p as a fun ction of q/2k, the relationship between (Y and

q/2k shown in Fig. 1 is used . F igure 4 shows th e p-value plott ed as a fun c-

tion of t he nominal norma l pressu re with +yo as a para meter. I t is seen t hat pis near ly independent of y. which seems reasonable. F ur th ermore p onlydecrea ses slightly with increasing q/2k as long as q/2k < 0.6, wher eas pdecrea ses ra pidly an d approximat ely linea r with q/2k in the case of q/2k >


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81

Fig. 4. p = RJR,, as a function of q/2k and 70.

1.009

OS070605QL0.3

0.2

0.100- -0.0 1.0 2.0 30

440-z

Fig. 5. p = RJR,0 as a function of q/2k and m; yo = 5.

0.6. The r eason for th is cha nge is th at t he slip-line field r eaches th e bott omof th e ridge, Fig. 3, which mea ns th at th e bott om begins to rise. At q/2k =2.4 th e &-value ha s decreased to 10% of t he initia l value and at q/2le-valueshigher than 3.0, R, is van ishing.

The an alysis is based upon an as su mp tion of zero friction between tooland specimen, but it is reasonable t o assu me th at th e relationship between pan d (Y s ap pr oximat ely un affected by th e friction factor. Un der th is ass um p-tion the relationship between p, q/2k an d m has been determ ined (Fig. 5)

for th e case of y. = 5, using Fig. 1 and eqn. (6).As seen, p is nea rly independ ent of q/2k for small q/2k valu es, formedium q/2k-values, p decreases ap proximat ely linea rly with q/212 and forlarge q/2k-values, p appr oaches zero. Fu rt herm ore p decrea ses with increas-ing friction factor corr esponding to th e fact th at increa sing friction factorcau ses th e real ar ea of cont act to increase.

4. Determination of the average effective strain

A th eoretical determ inat ion of t he average effective str ain in th e sur-

face layer of a met al-work ed specimen , th at is in th e deform ed zone of th eridges, is import an t. It is necessar y to k now th e stra in in order to u se th efriction th eory [l] in pr actice and kn owledge of th e str ain also pr ovidesinform at ion concern ing ha rd ness an d wear ability of th e specimen sur face.


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82

L t -iFig . 6 . Nota t ion .

Bay an d Wanh eim [2] ha ve for th e cas e of cone inden ta tion described ameth od to deter mine th e avera ge effective str ain in th e deform at ion zone.The theory was based upon th e law of conservat ion of energy an d the samemet hod of an alysis is ap plied here.

In the following an alysis th e notat ion shown in Fig. 6 is us ed and againconfined t o th e case of zero friction, th a t is m = 0. Compressing the tool adistan ce z into th e su rface a sperities resu lts in an extern ally supplied energyW, given by:

The int erna l energy dissipation Wi is given by:

integrat ing over ar ea since plan e str ain deform at ion was assu med. Conserva-tion of ener gy now leads to We = Wi tha t i s:

wher e E is th e avera ge effective st ra in in the deform ed zone Ad of thespecimen . Replacing int egrat ion by sum ma tion a nd looking only a t a st ep-wise deform at ion , gives:

where

(7)

AZi Zi-2i-l-=

t t

The corr espondence between th e i-value an d th e sta ge of deform at ion is sochosen that i = 1, 2, 3, 4, 5, . corr espon ds to sta ge b, d, f, h, k, . (seeFig. 3). Geomet rical con sider a tions now give AZi/t (see ap pen dix).

To d eterm ine th e real str ain incremen t th e Ad,-values which should beinserted in eqn. (7) are given by:

Adi = Ai for i = 1,2

Adi = Az for i > 2


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83

Fig. 7. F as a function of q/2& and 70.

where Ai is the area of the instantaneous deformation zone. The reason forthis is that the deformation zone for i > 2 is reduced and everywhere enclos-ed in the deformation zone present at stage 2, which is not quite obviousfrom Fig. 3 but proves true when making a closer investigation. Al and A2

are found by geometrical considerations (see appendix). The values of(q/2k), to be inserted in eqn. (7) are found from Fig. 1. Since the deformedzone of the ridges in stage 1, Fig. 3, is enclosed too in the deformation atstage 2, the average effective strain in the deformed zone is now given by:

- -Ei =tZl -?2+iA& (i>2)

2

It should be noted that c, in the case of i > 2 differs from the average effec-

tive strain E, in the instantaneous deformation zone, E, being greater thanEi since the deformation zone is reduced for i > 2. An exact determination ofE, is rather complicated, since it involves considerations concerning thestrain distribution in the deformation zone, and will therefore not beattempted here. Slip-line fields for i > 2 show that the deformation zone issteadily reduced with increasing cu; assuming a uniform strain distributionin the instantaneous deformation zone, the following expression for e,could be used:

E m i= f i (i = 1,2)

Emi = Em,i-1 + AG * A.2fi._ (i > 2)A3 and A4 are determined by gekmetrical considerations (see appendix)whereas Ai for i > 4 is measured on the slip-line field using a planimeter.


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yo= 5

yQ=lO-

x0=5*

00 00 10 20 30

Fig. 8. E, as a function of q/2k and 70.

Figure 7 shows the r elationsh ip between th e average effective str ain f in th edeform ed zone of th e specimen , q/2k an d yo. It is seen th at E is appr oxima -tely constant for q/2k less th an about 1.3, th at is as long as th e deform at ionzones ar e isolat ed an d Amont ons law is obeyed. For q/2k > 1.3, E sud denlyincreases with q/212 app roaching a limiting value depend ing on 70. Thereason for th e sudden increase in E is th at th e deform ed zone does n otincrease fur th er. It is also seen th at < increases with increasing y. as expected.

In Fig. 8 the average effective str ain E, in the inst an ta neous deform a-tion zone is plott ed as a fun ction of q/2k an d 70. The t endencies ar e th esa me as for th e E-cur ves, th e only differen ce being th at E, increas es to agreater extent tha n E.

5. Appendix

AZ, t a n 7 0 (1 + t w d-=

t 4(2+tanyl)

A 2 3 t w 3 (1 + t a v 3 )-=

t 2 (5 + tanQ

A 2 2 t i n y 1 (1 -t a w, )-=

t 2 (3 + tanyi)2

A 2 4 t a w, (1 - t=v31-=

t 3 (5 + tanys)s

Azi = (I- hi-A2 - (1 - cd2 tan74 ( i > 4 )t 4

Ai l+tany,-=t2 4 (3 + tany,)2

[(7T-l)tanyi +3+n1,

A2 Z+n-tany2 A 3 11 + 477 h73 (2 + t m73)-=16

-=

t 2 t2 4 (5 + tany,)2

A 4 3+n-tany4-=t2 36

1 T. Wanheim, N. Bay and A. S. Petersen, Wear, 28 (1974) 251.2 N. Bay and T. Wanheim, Int. J. Prod. Res., 12 (1974) 195.

ra, and the average effective strain of surface

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