The relation between effective deformation and micro-hardness in a state of large plastic deformation

The relation between effective deformation and

micro-hardness in a state of large plastic deformation

Citation for published version (APA):

Ramaekers, J. A. H., & Veenstra, P. C. (1970). The relation between effective deformation and micro-hardness in a state of large plastic deformation. C.I.R.P., 18, 541-545.

Document status and date: Published: 01/01/1970

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The Relation between

Effective Deformation and Micro-Hardness

in a State of Large Plastic Deformation

J. A. H. RAMAEKERS and P. C. VEENSTRA

Laboratory of Production Engineering, Technological University, Eindhoven

SUMMARY. It is shown that the equations of Voce and Palm and of Ludwik describe the relation between hard-ness and effective deformation in a defective way in the case oflarge deformation such as prevails in punching.

Generalisation of Nadai's equation however leads to a reliable non-linear function between hardness and effective stress. The result is applied to the punching process, giving an expression for the punching force and an indication towards the physical background of the shearing factor in punching.

RESUME. II est demontre que les deux equations de Voce et Palm et de Ludwik deerivent une relation entre la durete et la deformation plastique dans Ie cas de large deformation cororoe on Ie trouve dans 1a technique du poin<;onnage.

D'autre part, la generalisation de l'6quation de Nadai fournit une fonction non-lineaire precise entre la durete et Ja charge effective. Le resultat est applique au poin .. onnage, donnant une expression pour la force necessaire et une indication concernant la nature physique du phenomene de cisaillement au poim;onnage.

ZUSAMMENFASSUNG. In der vorliegenden Arbeit wird gezeigt, daB die Gleichungen nach Voce, Palm und Ludwik den Zusaromenhang zwischen Harte und logarithmischer Formiinderung fOr groBe Formanderungen, wie sie zum Beispiel beim Stanzen auftreten, nieht richtig beschreiben konnen. Eine zuverlassige nichtlineare Beziehung zwischen Harte und Vergleichsspannung erhiilt manjedoch durch Verrallgemeinerung der Nadai'schen FlieBkurve. Die Anwendung dieses Ergebnisses beim StanzprozeB ergibt einen Ausdruck fUr die Stanzkraft und ermoglicht eine beschriinkte physikalische Deutung des Scherfaktors.

1. INTRODUCTION

IT is a well known fact that due to work hardening the hardness of metals is increased by deformation.

Palm[4] gives a complementary function in terms of hardness

Therefore the measurement of micro-hardness is applied to determine the distribution of deforma-tion in forming processes [1-3].

As however the analytical relations available to describe the dependence of hardness on deforma-tion[4,5] are only valid in a limited range of

deformation, investigations have been carried out with the aim of obtaining more general validity.

By performing tensile tests combined with hard-ness measurements on the tensile specimen the strain (effective deformation) can be related to hardness. The validity of the expressions obtained has been checked in a case oflarge plastic deforma-tion such as prevails in the punching process.

2. VOCE'S EQUATION '{oce[5]fornlulates

if

=

0'00 - (0' 00 -:- 0'0) e-l/Oc (1) 541

and shows that

Hence it follows that

8 I

C

HV = C1 +Czii (3)

From tensile tests for which the results are shown in Fig. 1 and then subjecting the experimental data to non-linear regression analysis the constants in equation (1) have been determined as listed in Table 1.

Regression analysis has also been applied to experimental data obtained by hardness measure-ments on the tensile test specimen according to equation (3) from which Fig. 2 results. The values of the constants C1 and Cz are also listed in

542 J. A. H. RAMAEKERS and P. C. VEENSTRA

Table 1

Goo Go Be Load HVoo Ho C1 C2

Material (N/mm2) (N/mm2 ) (gf) (N/mm2) (N/mm2) (N/mm2) low carbon steel 610 210 0-200 100 2400 1200 610 2-93 KMS63H 890 250 0-751 50 2590 1140 590 2-24 600

zr~

..

<II III ~

-

III 400 200 O~---~0~-2'---'0~'4'---~O~"6---0~-~8---~1-0 straj n

Fig. 1. Comparison between the deformation equations (work-hardening functions) according to Voce

[equation (1)), Ludwik (equation (4)] and Nadal [equation (6)], for low carbon steel.

However, when applying equation (3) to a state of large plastic deformation such as prevails in the region of shearing in the punching process (as indicated in Fig. 5) it is found that for the carbon steel investigated

HVmax

>

3000N/mm2

>

HVoo

from which it follows

This is not compatible with equation (1) and Table 1 from which is taken the condition

Umax

<

0'00 610 N/mm2

For this reason equations (1) and (2) are considered not to be applicable to a state of large deforma-tion[6].

3. LUDWIK'S EQUATION

According to literature, Ludwik proposed [6]

if

=

ao+K[e]1 (4)

Combining equations (3) and (4) as shown in Fig. 2, it follows that

HV = Ho+Hl[e]1 (5)

Application of regression analysis to the experi-mental data from tensile tests leads to the constants of equation (4) as listed in Table 2.

The fact that the yield stress 0'0 proves to be

negative creates some doubt as to the physical correctness ofthis.

Combining however the results of tensile tests on the one hand with hardness measurements on

the other, as shown in Fig. 2, it is concluded that

Table 2 0"0 K Material Equation (N/mm2) (N/mm2) low (4) -820 1480 0·09 carbon steel (3)-(5) 178 530 0·465

By introducing the latter value into the regression analysis the constants of equations (4) and (5) have been re-determined as shown in the bottom line of Table 2.

Figure 6, however, shows that hardness measure-ments in punching nevertheless indicate that Ludwik's equation does not hold in a state oflarge deformation. 2S00r---~---~

'"

..

..

_..

c:

.,.

..

.:: 100°0

l

0 100

I

200

I I

300 st rat n

I

1

I

400 500 600 al re . . , N mm' Fig. 2. Micro-hardness as a function both of effective deformation [equation (5)] and effective stress [equation

(3)] (Table 2).

4. NADAl'S EQUATION Starting from Nadai's equation

u

C[e]" (6)

it is assumed that, analogous to Palm's equation, the dependence of hardness on effective deformation is governed by

(7)

Here ii stands for the effective deformation

genera-Load (gO 100 100 Ho H1 C1 Cz (N/mmZ) (N/mmZ) (N/mm2) 1135 1550 610 2-93 alr---~~

if

-t $ t ra j r.

Fig. 3. Definition of the dependence of

/),.(jH on EH [equation (8)].

ted by the forming process while SH represents

the average effective deformation locally added by the indentation connected with the hardness measurement.

It is remarked that the latter value depends on the width of the indentation and on the local distribution of deformation as controlled by the work-hardening properties of the material. From Fig. 3 it follows that

AUH

=

Ul-U

= C(e+BH)n-C[S]n

or

(8)

Substitution in equation (7) yields

(9) From the experimental data as shown in Fig. 4,

Au H is derived as

t?-Au =

-H

u+a

The quantity (a) being a constant it follows that

HV=-

H(

u+--

a

2 )

544 J. A. H. RAMAEKERS and P. C. VEENSTRA 2 5 0 0 . - - - -____ ~ ! 20001--

15001--./1.

_..

/

I-"7'

..

., .,

..

"1000t--..,

..

.c: SOO~

I

I

I

O~---~20~O.---'4~OO---~6*OO~~ stress,

.Ji"

mm'

Fig. 4. Hardness as a function of effective stress for low

carbon steel and brass-KMS 63 H [equation (10)] (Table 3).

It is observed that the presence of initial deforma-tion 80 of the material changes equation (6) into

(11)

not affecting the result formulated in equation (10). Table 3 shows the results of regression analysis applied to equations (6), (10) and (11).

It is noticed that the quantity HICin equation (10) has the same meaning as

Hv

oo/u

00 in equations (1)

and (2)[4]. The result obtained, proving that the relation between hardness and effective stress is

non-linear as expressed by equation (10), is sup-ported by experimental data shown in literature[7] .

5, VERIFICATION IN PUNCHING

By carrying out hardness measurements on a specimen obtained by punching according to the principle shown in Fig. 5 it can be checked whether equations (3)-(5) and equations (6) and (10) will hold in the case of large plastic flow in a state of combined stress.

Fig. 5. Principle of hardness measurement in punching. Definition of the geometrical quantity holh.

The maximum value of hardness in the shearing zone has been measured in a process of incre-mental punching.

Making use of equations (3) and (10) the effective stress cT_max is calculated and is plotted against the

L ho € n h '

Fig. 6. Result of applying Ludwik's equation [equation

(13)] to punching. Table 3

e

n

eo

Load HIe a Material (N/mm2) (gO (N/mm2₎ low carbon steel 680 0'264 O· 100 3'47 270 KMS63H 710 0'495 0'11 50 2·75 310

geometrical quantity

holh

as a measure of deforma-tion. The results are shown in Fig. 6 using Ludwik's equation and in Fig. 7 when applying Nadai's equation. The latter curve can be represented analytically by = [kin

hoJ"

C h (12) 2 · 0 , - - - . , 1'5, h -In

T '

E

Fig. 7. Result of applying Nadai's equation [equation

(12)] to punching.

covering the entire range of deformation investi-gated while the former

~= if (J [ k/ln~ h

Jl

K h (13)

deviates considerably from the measuring points in the region oflarge strains.

When assuming that the mechanics of punching is based on pure shearing, the instantaneous punching force is given by

2n [ h

In

F= .y3RhC

kIn ; (14)

and hence the maximum force by

Fmax =

~~RhoC[;J~

(15)

when neglecting the influence of friction.

In Table 4 comparative experimental data[S] have been listed.

From the theory of plastic instability it is easily shown that the tensile strength of a material can be expressed in terms of

(16)

or

(17)

in the case where initial deformation 80 is present. Substitution in equation (15) gives

~

Fmax = 2nRho(JB0.y3 (1S)

or Fmax 2nRho S f(J 8₀ (19) Where Sf

kH

(20)

.y3

represents the shearing factor in punching.

Obviously k is approximately a constant

charac-teristic for the process describing the relation between the geometrical measure of deformation

ho/h and some average of effective deformation

physically being present in the entire shearing zone.

Table 4 C n k F eq.15 Fmeasured AF Material (NJmm2) (N.104) (N.104) (%) Al 99·3 148 0·264 2-92 3'85 4·06 5·4 CIO 695 0'106 2·92 20'0 21'6 7-4 Ma-8 683 0·218 2-92 18'1 18·8 3·7 Low carbon steel 680 0·264 2-92 3·54 3·94 H)'O REFERENCES

1. YOKOYAMA, K. and SUZUKI, T., Bull. Tokyo Inst. Tech. No. 65, 15 (1965).

2. DANNENMANN, E., STECK, E. and WILHELM, H., Bander, Bleche, Rohre 9, 388 (1968).

3. THOMSEN, T., Glatte Lochwande beim Lochen von Grobblechen. Dissertation TH Darmstadt (1966).

4. PALM, J. H., Trans. Am. Inst. Min. metall. Engrs 185, 904 (1949).

5. VOCE, E., Metallurgia 51. 219 (1955).

6. HILL, R., The Mathematical Theory of Plasticity, Chapter 1. Oxford University Press (1950).

7. SANDIN, A., VDI Ber. No. 41, 77 (1961).