(PDF) Work hardening and rate sensitivity material coefficients for OFHC Cu and 99.99% A1

Acta metall. Vol. 32, No. 10, pp. 1655-1668, 1984 0001-6160/84 $3.00+0.00 Printed in Great Britain Pergamon Press Ltd

WORK HARDENING AND RATE SENSITIVITY MATERIAL COEFFICIENTS FOR OFHC Cu AND 99.99% A1

N. CHRISTODOULOUt and J. J. JONAS Department of Metallurgical Engineering, McGill University, 3450 University Street, Montreal,

P.Q., Canada H3A 2A7

(Received 21 November 1983; in revised form 29 March 1984)

Abstract--Large strain tensile tests were carried out on OFHC Cu and 99.99% A1 with the aim of determining the first and second order work hardening and rate sensitivity coefficients. The tests were performed at room temperature and 473 K and at constant true strain rates in the range 5 x 10 -4 to 10 J s -]. The use of a diameter transducer to measure and control the rate of reduction of the neck diameter of the tensile specimens is described. In this way, the strain rate at the minimum cross-section was held constant well beyond the point of maximum load. A second diameter sensor for use at 473 K is also described. In order to determine the coefficients for Cu at 698 K, constant strain rate compression tests were performed. The detailed dependence of the material coefficients on stress, strain rate and temperature is characterized. It is shown that the values of the rate sensitivity of the work hardening rate B, beyond the maximum load are less than one, but are not negligible. As a result, the rate sensitivity at constant work hardening rate N is not the material coefficient that controls the growth of strain rate gradients at large strains. It is also shown that the actual value of the coefficient B, is substantially higher than the one obtained from the single state parameter approach.

Rrsumg--Nous avons effectu6 des essais de traction ~ forte drformation sur du Cu OFHC et de rA1 99,99% afin de d&erminer les coefficients du premier et du second ordre de l'rcrouissage et de la sensibilit6 fi la vitesse. Les essais 6talent effecturs h la temprrature ambiante et fi 473 K pour des vitesses de drformation vraie constantes comprises entre 5 x 10 -4 et 10 ~ s ~. Nous drcrivons rutilisation d'un transducteur de diamrtre pour mesurer et contrrler la vitesse de rrduction de la striction des 6chantillons de traction. De cette fa9on, la vitesse de drformation ~i la section minimale &air maintenue constante bien au-del~ du point de charge maximale. Nous drcrivons 6galement un deuxibme d&ecteur de diamrtre utilis6

473 K. Pour drterminer les coefficients du cuivre h 698 K, nous avons effectu6 des essais de compression fi vitesse de drformation constante. Nous avons prrcis6 en d&ail la variation des coefficients du matrriau en fonction de la contrainte, de la vitesse de drformation et de la temprrature. On montre que les valeurs de la sensibilit6 fi la vitesse du taux d'rcrouissage B, au-delfi de la charge maximale sont infrrieures ~ un, mais qu'elles ne sont pas nrgligeables. I1 en rrsulte que la sensibilit6 fi la vitesse ~ taux d'rcrouissage N constant n'est pas le coefficient du matrriau qui contrrle la croissance des gradients de vitesse de d~formation aux fortes d+forrnations. Nous montrons 6galement que la valeur rrelle du coefficient B, est notablement plus 61evre que celle que donne l'approche du paramrtre d'rtat unique.

Zusammenfassung--OFHC-Kupfer und 99,99% A1 wurden im Zugversuch mit groBen Dehnungen verformt, um die Koeffizienten von Verfestigung und Geschwindigkeitsempfindlichkeit erster und zweiter Ordnung zu bestimmen. Die Versuche wurden bei 293 und 473 K mit wahren Dehngeschwindigkeiten zwischen 5 x 10 -4 und 10 -~ s -~ durchgeffihrt. Das Verfahren wird beschrieben, mit einem Durch- messersensor die Schrumpfungsrate des Einschnfirdurchmessers der Zugprobe zu messen und zu regeln. Dieses Verfahren kann den minimalen Querschnitt bis welt fiber den Punkt maximaler Last hinaus konstant halten. Ein zweiter Durchmessersensor wird ffir Anwendungen bei 473 K beschrieben. Um die Koeffizienten ffir Kupfer bei 698 K zu bestimmen, wurde bei konstanter Dehngeschwindigkeit im Druckversuch verformt. Die Abh/ingigkeit der verschiedenen Koeffizienten von Spannung, Temperatur und Dehngeschwindigkeit werden ermittelt. Es wird gezeigt, dab die Dehngeschwindigkeitsempfindlichkeit des Verfestigungskoeffizienten B, oberhalb des Lastmaximums kleiner als eins, aber nicht vernachl/issigbar ist. Demnach ist die Dehngeschwindigkeitsempfindlichkeit bei konstantem Verfestigungskoeffizienten N nicht der Materalkoeffizient, der das Wachsen von Gradienten in der Dehngeschwindigkeit bei groBen Dehnungen kontrolliert. AuBerdem ist der tats~ichliche Koeffizient B, betriichtlich h6her als der, welcher sich aus der N/iherung mit einem einzigen Zustandsparameter ergibt.

1. INTRODUCTION

The integrated form of the plastic strain is not a well-defined deformat ion variable, because it does not reflect in a unique way the current microstructure of a deformed piece (and particularly the density and

tPresent address: Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada K0J 1J0.

arrangement of dislocations). Har t [1] was one of the first to suggest that, instead of the plastic strain, an evolutionary variable (called " the hardness state") has to be employed to describe the memory of the material for its past history. Kocks in 1975 suggested that there exists a somewhat different state parameter (called the "mechanical threshold") [2] that can also describe work-hardening in a satisfactory way. It should be pointed out, however, that a single state

1655

1656 CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND Al

variable is not adequate [3] for describing phenomena in which the load is cyclic or during the first few percent or tenths of a percent of strain when the strain rate is imposed or changed abruptly during monotonic loading.

For the present study, the one state variable theory developed by Kocks et al. [4] was adopted. In that work, Kocks and co-workers assumed that the flow stress a is a unique function of the true strain rate and a structure parameter ~. They proposed the following evolutionary law

(Qlna/~t)x = m (aln~/Ot)~ + H.~ (1)

to describe the evolution of stress, a, with time, t, at a particular location, x, in the material. Here, the dimensionless material coefficients, M = (9 lna / aln~)~ and H=(alna/O~)~ depend only on the current values of true stress a and true strain rate 4. According to equation (1), these two sets of material coefficients are sufficient for the complete description of the mechanical response of a plastically deforming specimen.

Although equation (1) is of fairly general utility, there are applications that require the determination of differences between material elements. Kocks et al. (4) obtained the appropriate second order differential equation by differentiating equation (1) with respect to position x.% Subsequently, in treating the specific case of a tensile bar, they derived the following expression

MY" + [H + B, -- 1 -- (g/~2).p] y, - [ C - ( g / ~ z ) . a ] Y = 0 . (2)

Here Y=(OlnA/ax) , is the area gradient, Y ' = (OY/Oe)x = - ( 0 ln~/Ox), the strain rate gradient, and Y"=(ozY/aez)x=-[a(g/~z)/ox], the acceleration gradient. The definitions of the other dimensionless material coefficients are

B~(a, ~) = (OH/O ln~),; C (a, ~) = -- (aH/O lna)~

P(a, 4) = --(0m/01n~),; Q(a, ~) = (aM/Olna)~. (3)

The coefficient B, is the rate sensitivity of the work hardening rate H, defined at constant stress. It can also be defined at constant state as B, = (OH/O ln~)~. The relation between B, and B, is given by

B~ = B~ + M C . (4)

These coefficients are particularly useful when the development of strain rate gradients during testing or forming is of interest. Under such conditions, the variation of H and M with position must be known, and these can in turn be specified if the dependences of a and ~ on position are also known [6]. In their

?It should be noted that the coordinate x here refers to a scale which is fixed to material elements in the specimen. It is thus a Lagrangian and not a laboratory or Eulerian coordinate. The position x of a material element does not therefore have to be updated as a sample elongates (or contracts) during the course of an experiment (5).

work, Kocks et al. introduced a new kind of strain rate sensitivity, defined by the following equation

N = (6 lna/a ln~)n. (5)

They also claimed in the same study that the coefficient N is very important, because it controls "catastrophic" neck growth. The aim of the present investigation was to measure these parameters for OFHC Cu at room temperature (0.22 Tin) and at 698 K (0.51 Tin), and for 99.99% A1 at room temperature (0.32 Tin) and at 473 K (0.51 Tin). These quantities were determined by performing tensile and compression tests, as described below.

2. EXPERIMENTAL MATERIALS AND PROCEDURES

Two basic materials were used in this study; oxygen-free high conductivity copper (99.95o/0 Cu) and 99.99% aluminum. The Cu was obtained in the form of 12.7 mm diameter hot extruded rods. The main impurities present were silver (55 ppm), mag- nesium (64 ppm) and sulfur (80 ppm). After machining, the samples were annealed for 45 min at 723 K in a dynamic vacuum of 2 x 10 -6 torr. The resulting average grain size was approx. 0.02 mm.

The 99.99~o A1 samples were received in the form of 19.05 mm thick A1 plate. The plate was produced by cold rolling Direct Chill cast A1 ingots. The tensile axis of the specimens coincided with the rolling direction. After machining, the samples were annealed for 60 min in a salt bath at 648 K. The average grain size produced was approx. 0.08 mm.

The grain size was intentionally kept small for two reasons. The first is that the measurement of true strain was achieved by attaching a modified Linear Variable Differential Transformer (LVDT) to the specimen, which monitored the minimum specimen diameter continuously. Had the grains been coarse, the diameter which the LVDT was monitoring would have varied significantly around the sample per- imeter. The second is that the flow stress evaluated at the cross-section where the measurement of strain was made had to be the average over a significant number of individual crystals in order to be represen- tative of the sample as a whole. That is, the material was designed to be "polycrystalline" and not "multi- crystalline".

The tensile tests were performed by using tensile samples that had an hourglass shape. A constant true strain rate was imposed through the LVDT, which was connected in turn to the strain control mode of an MTS machine. The latter was interfaced with a PDP 11/04 mini-computer that provided the desired input signal to the servocontroller by means of real time programming. The detailed procedure is described in Ref. [7].

The LVDT was modified [8] for performing tensile tests at 473 K using an Instron environmental cham-

CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND AI 1657

ber for heating the A1 specimens and controlling the temperature to within ___ 0.5 K. For this purpose, the diameter transducer was placed as a unit inside the chamber.

In order to test the Cu at 0.51 Tm (698 K), a series of compression experiments were carried out. Ten- sion tests could not be performed at this temperature because the available LVDT could not be operated at temperatures higher than 493K. The height-to- diameter ratio of the compression samples was kept constant at 1.5 and their end surfaces were grooved [9] to retain the colloidal graphite lubricant applied to minimize the friction between the sample ends and the test equipment. The specimens remained approximately cylindrical during testing with no apparent barrelling.

A CENTORR model M60 front loading vacuum furnace was employed for the hot compression tests. A detailed description of the test assembly can be found in Ref. [10]. With this system, a vacuum of 2 × 10 -6 torr was consistently achieved in high temperature operation. In this way, oxidation of the Cu was avoided. The accuracy of temperature measurement was about _ 1 K.

The true stress/true strain curves obtained from the raw data were smoothed and corrected by the method

200[ - - [ A

i AL

~ 160~-

09 09 I~l120

~ 8o

I00

T = 2 9 3 K / 4

/ / / / ' ~ / o) ~ , i o- 2 / / ' ~ .I ,o -2

V c) 5xl (y5 ol ,0_-34 e) 5xlO

i i 0 4 018 ll2 I 6 0

TRUE STRAIN ( (a)

~ 8 0

~o40

of Bridgman [11] to allow for the apparent increase in flow stress because of the development of tri- axiality (for more details, see Ref. [8]). Also, in order to evaluate the degree of "circularity" of the minimum cross-section (where the strain was measured), the contours of specimens deformed to different strains were analysed by means of a profile projector [7, 8]. In the case of the Cu (grain size 0.02 mm) it was not possible to detect any nonuniformity in the roundness of the minimum cross-section. In the case of the AI (grain size 0.08 mm) and at a true strain of e ~- 0.5, the minimum diameter varied by as much as + 1.3 x 10 -2 mm; at a strain of e --~ 1.0 this variation increased to +2.5 x 10-2mm. The corresponding uncertainties in the calculation of the true strains are + 0.004 and + 0.01. Nevertheless, the flow curves for both materials were very reproducible at strains well beyond the Considrre value.

3. RESULTS AND DISCUSSION

Two types of experiments were performed on the two materials and at each testing temperature. These were: (i) the continuous and (ii) the strain rate change tests, both of which were performed at constant true strain rate. The range of strain rates used for the

Cu a b

, ~ " a) I0 -t / b) I0 -z

/ c) I0 -s f d ) 5 X I0 -4

500

400 g_ g

500

~ 2oo

~ lOG

200

T = 293 K

I I I 0 3 0.6 0.9

TRUE STRAIN E (b)

-- 10"2 s_ 1

$ -1

y c~ T = 698 K

160

r~ , s -

'G 12o

1.2

I I I I I 0 J I t I t 0.2 0.4 0.6 0.8 i.O t.2 0.2 0./,, 0.6

TRUE STRAIN ( TRUE STRAIN ( (c) (d)

Fig. 1. True stress-true strain curves determined at the indicated strain rates and temperatures. The flow curves in (a)-(c) were obtained from tensile tests using the diametral transducer, whereas those of (d) were

obtained from compression tests.

I 0.8

1658 CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND A1

continuous experiments was 5 x 10 -4 to 10 -~ s -t. The upper strain rate limit was imposed in order to avoid the temperature increase associated with deformation at strain rates higher than 10 -~ s-L For the strain rate change tests, the base strain rate was increased by factors of 5, 10, 20 or 100 as indicated in more detail on each figure.

3.1. Continuous flow curves Typical flow curves for the four combinations of

material plus hom*ologous temperature are presented in Fig. 1. The important feature of the tensile curves [Fig. l(a)-(c)] is the extended range of strains obtained with the aid of the diametral transducer. True strains of around 0.8 were reached in the hot compression tests, as shown in Fig. l(d). Beyond that, the samples exhibited some barrelling.

The flow curves obtained in this fashion were very smooth, a characteristic which was important for the determination of the work hardening slopes. All the tensile tests were stopped prior to fracture, which occurred at strains well beyond those corresponding to the maximum load, a feature which constitutes an essential ingredient in the present study. This is

because the values of the coefficients beyond the UTS are of particular interest with regard to flow localization during certain metal forming processes.

3.2. The work hardening coefficient (WHC) H

The WHC (H) was derived from the continuous flow curves after the latter were smoothed slightly [7]. In Fig. 2, the dependence of the WHC on stress and strain rate is shown for the four sets of experimental conditions. It is evident that H is a decreasing function of stress, and that the rate dependence of H, as given by B, = (aH/a ln~), is greater for A1 than for Cu. Although the solution to equation (2) depends on the value of B, and not on B+, the latter is also important, as it is more closely linked to the funda- mental work hardening behaviour of the material. Accordingly, in Fig. 3, the dependence of H on the state parameter ~ is presented for the four experimental conditions. This was obtained by noting that the stress change Alna resulting from a small rate increase Aln~ is equal to the instantaneous rate sensitivity M, i.e.

Aln o/Aln~ = M. (6)

4 0 =

t T = 293 K

~* 20

2 " ~ ~ = 5 X 10 "L" S-1

=: 10 ~ 5 X 10 -2 s -1

"::I lOO .200 300 ~,oo 500

TRUE STRESS ct(MFa) (a)

T=475 K

~ .~-=5 x l d 2 i I

k \ ..,:5,16';'

2 0 60 I 0 0 TRUE STRESS cr(Mpo)

(c)

Av~

=2C

T = 2 9 3 K

- ~ ~ - H - ~ - - - , - - ' ~ ' 7 - . ~ ~=~ '(~ . . . . . . . . r . . . . 4 0 8 0 120

TRUE STRESS o ' (MPo) (b)

8 - T = 698 K

\\ 6 E = 10-2 s -1

Z 4

2 H=I

40 80 120 1/,0 200

TRUE STRESS o-(MPa) (d)

Fig. 2. Dependence of the work hardening coefficient H on stress and strain rate for (a) Cu at room temperature, (b) A1 at room temperature, (c) A1 at 473 K and (d) Cu at 698 K. Note that H decreases

with a and increases with ~.

CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND A1 1659

.,~ 6

b c t~ "i7 SE

-w A ~ 6

T=C~93 K

=Sxl0~s" ~ =5x10% 4

_ " ~ ~ _ , ,°o ,oo ,oo -7o;-

1" MPa (a)

T=293K

x ~ =SxlO-~s4

=5x104s " ~ ~ \

20 "t" MPa (b)

-- ~ ~ ~ - - 4 =Sxl0"ZS" T:A~3K

~ =2x10-~s4

- 1 20 6 0 60 80

r MPa (c)

T=C~98K

~ - ~ x ~ _ , : lO-,~-,

t ,- ~ = 2xlO'3s-' - - ' ~ ' ~

~0 80 120 160

r MPa (d)

Fig. 3. Dependence of the work hardening coefficient H on the structure parameter r and strain rate for (a) Cu at room temperature, (b) A1 at room temperature, (c) A1 at 473 K and (d) Cu at 698 K. Note that

H is essentially rate insensitive in Cu, while the contrary is true for the A1

If Alntr is measured when the strain rate is changed from the basic strain rate i0 at which z is defined to a higher one ~, then equation (6) becomes

In(a/z) = Mln(i/go) (7a)

o r

z = a ( i / i0) -M. (7b)

Note that the "hardness state" quantity z is rate insensitive and that the behaviour of H shown in Fig. 3 illustrates that the latter is nearly rate insensitive in Cu, whereas it is clearly rate sensitive in the A1. Thus

Fig. 3 indicates that, at 0.22 and even at 0.51 Tin, the behaviour of the present Cu depends mainly on the state of work-hardening, and less on i (i.e. it falls in the "cold-working" regime), whereas, at 0.32 and 0.51 Tin, the behaviour of the A1 distinctly depends on z and ~ (indicating that it is a "dynamic recovery" material). The max imum load condition is indicated in Figs 2 and 3 by the line H = t.

From Figs 1 and 2, it is evident that H drops down to 1 at around 0.2 to 0.3 strain for all materials and temperatures, i.e. when the stress is increased to sufficiently high levels. In the A1 at 0.32 and 0.51 Tin,

1660 CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND A1

this occurs at stresses above 50 MPa, which correspond to t~/G levels of about 2 × 10 -3. For Cu, the equivalent a/G levels lie between 5 and 6 × 10 -a, depending on the strain rate; i.e. they are about three times higher. This difference can be attributed to two factors: (i) the difference in purity, whereby there is a component of solution hardening in the Cu and (ii) the difference in stacking fault energy.

3.3. The instantaneous rate sensitivity M

For the measurement of M, strain rate change tests were imposed at increments of strain of 0.05 and on a different sample for each value of strain. The experiment was continued for at least a strain of 0.3 at the higher strain rate ~2, before it was stopped. The technique is illustrated in Fig. 4. In this case, the copper sample was deformed continuously at ~ = 5 × 10 ~ s -~ to a strain ofe ~ 0.91 (point A). The strain rate was suddenly changed to ~= = 10-~s -~, and the flow curve followed the full line (representing the actual transient) until it rejoined the "normal" work hardening relation at point C. The one state parameter theory assumes that the flow curve goes from A directly to B and then to C by following the broken line corresponding to continuous work hardening at the same stress levels. Such a description, while inaccurate (because it represents the curve EC by two segments EB and BC), is nevertheless closer to the true behaviour than a transient composed of ED and DF, i.e. the "transient" associated with the direct passage from one continuous flow curve to the next. The coefficient M, was therefore determined from the quantity ln(aJt~J/ln(~/~J (see Fig. 4).

It is of interest that in the early stages of straining, the Cu exhibited small yield drops (i.e. short transients), an example of which is illustrated in Fig. 5(a). These transients typically extended over strains of about 0.001 (0.1~). At higher strains, both types of transient were observed, as illustrated in Fig. 5(b). The length of the short transient was around 0.003 strain in this case, whereas that of the long one was

45O

~4:55

I1: ~ 4 2 0

~ 4 0 5

C U(295 K ) continuous flow ~ . .

• ¢ i M= (Ncz/o~l/In( ~2/~111

/ . - / ~ / , ~ H = ( e / o - I) ,,

39c / el=sx'io-4 s-' ',

~,o~ I I [ I

:57 ~0-78 0 84 0.90 0.96 I-~)2 1.08 TRUE STRAIN ,~

Fig. 4. Determination of the instantaneous stress change At7 that follows a strain rate change. The "long" transient (full line) is neglected and, instead, Aa is measured by back- extrapolating the discontinuous high strain rate flow curve

as shown.

! ] CU (293 K)

~ 182

w1711 1"42 MPa ertod short

~- . . . . . . . . T - ~ tronsient

17 '0 l ~ ~ ~110"4 ' l I I

0'10 0"11 0'12 TRUE STRAIN

(a)

-[ ~. CU(293 K)

4351-

. . . . . . . r . . . . ? I inverted short

r v , 1 ~ - / - ~ tronsient " i: ',

.9o]

~ec

~-70

TRUE STRAIN (b)

AL(293K)

_J . . . . . . _ r / ~ i

" - - ' J r . . . . . . . i i

" ~ ~/-~ nor m al Ion(J 1-4 _, i / " I tronslent

e=SxlO ~' ~ I

60 0[5 0 6 0 7 TRUE STRAIN

(c) Fig. 5. Examples of the three types of transient observed experimentally. (a) Cu at room temperature and in the early stages of straining (inverted short transient only), (b) Cu at 293 K and at higher strains (combination of inverted short and normal long transients), (c) AI at 293 K (normal long transient only). Normal transients (without yield drops) were also observed in the Cu at 698 K and the AI at 473 K.

of the order of 0.05. In the case of the Cu at 698 K, on the other hand, or of the AI at any temperature, only the long transients were observed, as shown in Fig. 5(c). There are several possible reasons why the Cu exhibited inverted transients at room temperature which disappeared at the higher temperature. One is the solute pinning effect attributable to the presence of 200 ppm of impurities (Ag + Mg + S). By contrast, the A1 contained a considerably lower level of solute. A second possible reason for the absence of the yield drop in the AI and in the Cu at 0.51 Tm is that the hom*ologous temperatures pertaining to these three

CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND A1 1661'

sets of experimental conditions (0.32, 0.51 and 0.51, respectively) are higher than for the Cu at room temperature (0.22) [7].

The detailed dependence of M on a and ~ is shown in Fig. 6. In Fig. 6(a), the behaviour of the Cu and AI at room temperature is summarized. For refer- ence, the mean strain rate associated with each curve is given in the diagram. The Cu values go through a minimum, whereas the A1 ones do not [7]. It is worth mentioning here that Melander and Thuvander [12] found that M decreased with flow stress (i.e. with increasing strain) for the austenitic stainless steel they tested. This corresponds to the present Cu "low strain" behaviour. By contrast, in the OFHC Cu, they reported that M increased with stress (strain). The latter observation can be readily explained in that their data, determined over the stress range

¢1 ,-i

o OO5

. 0 ' 0 3

0.01

200-330MPa, and at a strain rate of 5 × 10-4s -1, correspond to "high strain" data in the present context [see curve (3) of Fig. 6(a)]. It should be added that Melander and Thuvander determined their coefficients by performing constant crosshead speed tests, i.e. the strain rate was a decreasing function of the strain (and stress), and was not held constant during testing, as in the present investigation.

In Fig. 6(b), the experimental M's determined in AI at 473 K are shown. The main difference between the room and the high temperature data is the magnitude of M and the slope of the individual curves. In particular, M at 473 K is around 3 to 4 times larger than at room temperature. The shapes of the curves, on the other hand, remain the same.

The dependence of M on tr and ~ in Cu at 698 K was determined by means of compression tests, as

TRUE S T R E S S c r ( M P a ) I00 2 0 0 3 0 0 4 0 0 5 0 0 (CU) I I I I I I I ~ovS.I ~lS_ I ~2S" I

IO0(AL) (e (I) 3"16x10 "2 I 0"1 10"2 1"4xtO -2 . CU~ (213.16.ro "3 ro "3 to "2

A L ~/ ( (3)224xld 5 SxlO "4 10 -2 / J'~ (4) 1"58x10 "3 5xl (~4 51(10":3

fl (4)~9/ A L t o ( 5 ) Sx,O -5 5.10-4 r~lO t

4/,<-

gg X

~ 0 - 8 T : 2 9 3 K

0"6

0 4 t ~ i n i i 0 2 4 6 8 I0 12 14xlO -3

10 -3 5xlO -3 2,Z:3x I0 -3

i0 -2 5xlO -2 2 23x10 -2

AL / c

/ - -

o-/G (a)

t T : 2 9 3

0 .04

I.-,

b 0-03 c

0.02

0.01

f l l ~'~4"4 7 x 10"5 s-I

i u(698 K) 2

. 2 L ~ 24x10- 6 I

~i s-i 42 s-~

• 2xlO -3 io -2

.f o I 0 - 2 5 x l O - 2

.2/~ A1(293 K)

l I 1 I I I I I l 0 2 4 6 8 I O x l O -~'

0 2 O"/G 4 x 10-3 O' /G

(b) (c) Fig. 6. Dependence of the instantaneous rate sensitivity M on flow stress, strain rate and temperature. At room temperature, the Cu curves go through a minimum and the values increase with strain rate. The A1 M values and the Cu ones at 698 K always increase with a and decrease with 4. (a) A1 and Cu at 293 K,

(b) A1 at 473 K, (c) Cu at 698 K.

1662 CHRISTODOULOU and JONAS: WORK H A R D E N I N G AND RATE SENSITIVITY OF Cu AND A1

Table 1. Comparison of the continuous rate sensitivity m, the instantaneous sensitivity M and the "structure" rate sensitivity N at different temperatures

Temperature Material K(T,v ) a/G M m M / m Nx= I M/N#= I

Cu 293 (0.22) 8 × 10 - 3 0.006 0.010 0.60 0.023 0.26 CU 698 (0.51) 5.3 x 10 -3 0.038 0.061 0.62 0.057 0.67 AI 293 (0.32) 3 x 10 -3 0.013 0.032 0.40 0.076 0.17 AI 473 (0.51) 2.5 x 10 -3 0.045 0.090 0.50 0.145 0.31

described above. This dependence is demonstrated in Fig. 6(c). The important difference between the room and high temperature Cu data is that the shape of the curves itself changes. The minimum observed at room temperature is no longer present at 698 K. An expla- nation of this difference is presented below, when the Haasen plot (Fig. 7) is discussed.

The apparent strain rate sensitivity m = dlna/dln~ that can be read from the continuous flow curves for Cu and A1 is compared with the instantaneous M at the same stress level in Table 1. As can be seen, the ratio M / m is between 0.4 and 0.6. This means that when m is employed in simulations of flow localization, the material rate sensitivity is overestimated by a factor of about two. Normally, when the rate sensitivity m is used, the rate sensitivity of the work hardening rate B, is neglected. Thus, the employment of m instead of M (which "overstabilizes" plastic flow [4]), compensates to some extent for the neglect of B~ (which is "destabilizing"). Nevertheless, as will be shown in a later publication [13], the description of the material behaviour in terms of m alone does not lead to as good agreement between simulations of flow localization and experimental results as when both M and B, [i.e. equation (2)] are utilized.

Before proceeding further, it is worth considering the errors involved in the measurement of M. The accuracy of this type of measurement depends almost entirely on the determination of tr 2 (Fig. 4). In general, it was found that the maximum variation in the back-extrapolated value of a (i.e. tr2) did not exceed _1 MPa. Such an absolute error is re- sponsible for a varying relative error in M, the variation depending on the level of the flow stress. In the early stages of straining, + 1 MPa led to maximum errors of + 30%. At large strains, the latter was reduced to around _ 10%. As a result, the values of M for Cu are considerably more accurate than those for A1. However, given that M plays its greatest role beyond the UTS, the determination of M in this range is sufficiently accurate for the present purposes.

t i t should be noted that the results depicted in Fig. 7(a) differ slightly from those presented earlier by Chris- todoulou et al. [7], because the latter workers determined a 2 (Fig. 4) at low strains by back-extrapolating the long transient, which was difficult to distinguish from the basic flow curve. This erroneous procedure was avoided at large strains because the back-extrapolation was performed from the proper part of the flow curve.

3.4. The Haasen plot

In order to assess the influence of impurities on the two materials, a Haasen plot [14] was constructed. This is depicted in Fig. 7(a), where both the abscissa and the ordinate have been normalized by the shear modulus.t The stress change Atr is the "instantaneous" one measured as was shown in Fig. 4 (i.e.

6 x I() ' T=293 K

7 , [ , / 4 / oAL ~o~Sxl() 3 S -I

, 2

^ V :i I I I I ' I 4 o ' /G 8 12xlO -3

°~,L °'ycu (a)

24 x 10 -5

20 ~ =4.47x~/

"~ 16

Cu (698K1 J J " 4 / / / v

/ /

I [ I I 0 2 4 6 8 lOxlO -3

G/G (b)

Fig. 7. Haasen plot (Mtr vs tr) for (a) Cu and AI at room temperature and (b) Cu at 698 K. The small positive intercept in the case of the Cu (a) is attributable to detectable solute hardening associating with the presence of = 200 ppm

Ag 4-Mg + S. This effect seems to disappear at 698 K.

CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND A1 1663

AB). As mentioned earlier [equation (6)] the term Alncr/Alni is equal to M. The Atr measurements for the Cu at low stresses were carried out on an ct-MTS machine at the Chalk River Nuclear Laboratories of Atomic Energy of Canada Limited. It appears from the CRNL data that there is a vertical intercept, while the A1 curve goes through the origin. However, it must be pointed out that the magnitude of the intercept is small (Aa/Aln~ -~ 0.3) compared to that of Kocks [14] for INCONEL 600 (Aa/Aln~ "-, 2.0). It therefore appears that the component of the Cu flow stress due to solute hardening is small and that the main contribution to tr is from dislocation inter- action.

It is of interest that the minima exhibited in the M curves of Cu at room temperature [Fig. 60)] are related to the presence of intercepts on curves such as that of Fig. 7(a). On the latter diagram, the value of M is given by the slope of the straight line connecting an arbitrary point of the Haasen plot to the origin. When there is no intercept and the plot curves upwards, M increases monotonically with stress (and strain). When the curvature is of the same sign but there is an intercept, the slopes (M values) first decrease then increase with stress. In this way, the occurrence of the minimum is linked to the presence of a detectable solute effect. The decrease and sub- sequent increase is also consistent with a passage from "low temperature" to "high temperature" behaviour as the strain is increased, as discussed in more detail by Christodoulou et al. [7].

In order to verify whether the solute effect persists when the temperature is raised to 698 K, a Haasen plot was also constructed for the Cu at 0.51 T,,, Fig. 7(b). From these data, it cannot be stated conclusively that the curves extrapolate through zero. However, from the trend of the data and from the observed M - a dependence [Fig. 6(c)], it appears likely that the solute atoms do not contribute significantly to the flow stress at 698 K.

4. SECOND ORDER COEFFICIENTS

The material coefficients discussed above were based on the first derivatives of the flow stress. The ones involving the second derivatives of a will now be treated. These are: the rate and stress sensitivity of the WHC, i.e. B~ = (OH/~ln~)o and C = --(OH/Olna)~, respectively; and the rate and stress sensitivity of the instantaneous rate sensitivity M, i.e. P = - ( O M / c~ln~), and Q = (OM/Olna)~, respectively. In addi- tion, the coefficient N = (Olna/Oln~)n will be con- sidered in this section because N = Bo/C is linked to both B, and C.

4.1. The rate sensitivity B o of the WHC

The coefficient B was defined in equation (3) as B~ = (~H/Oln~)~, i.e. as the change in the WHC H accompanying a strain rate change evaluated at con- stantflow stress. This change in H can be measured

in three ways. First, directly; this method is shown schematically in Fig. 8(a) and involves measurements from continuous curves only. Second, indirectly, by a method like that of Fig. 8(a), but carried out on an H-T plot (Fig. 3), rather than an H - a plot (Fig. 2). This leads to values of B~ = (0H/aln~)~, from which Bo can be obtained from Bo = Be + MC. Finally, B~ can also be determined from strain rate change data, see Fig. 8(b), from which Bo can be derived as before, using the appropriate values of MC.

In the present work, B o was evaluated directly by the method of Fig. 8(a), using the continuous curves of Fig. 2, and B, was measured independently, from strain rate change tests, employing the technique of Fig. 8(b). In general, the indirect values of B, obtained from B~ and the strain rate change tests, were higher by about 9% than the direct ones obtained from the continuous curves. It should, however, be noted that the accuracy of measurement of B~ is not as good as that of B,.

From Fig. 2, it can be readily seen that, below a certain stress, H is independent of strain rate, for a given material and temperature. This is because the initial work hardening rate H0 is rate insensitive (15) and only depends on temperature. For Cu, H0 was found to fall between 25 and 35, H0 being approximately equal to G/(Fa,.), where G is the shear modulus, 25 < F < 35 and try is the yield stress. For A1, H0 < 15 with F > 65. As a result, for both materials, Bo is close to zero at low stresses and then increases when the work hardening curves begin to deviate at larger strains, see Fig. 9. This is also likely to be the

• B A H _ H2- HI

i i o"

(a)

Ino"

" ~ - -Hi _ H3- H l In~ ~ l ~ -SHa~ "f-'-H ' H'r= "I ni~a/~, )

tE (b)

Fig. 8. (a) Measurement of the coefficient B from the H-a relations derived from continuous flow curves. (b) Calcu- lation of B, by measuring B, from a strain rate change test

and using the expression B~ = B~ + MC.

1664 C H R I S T O D O U L O U and JONAS: W O R K H A R D E N I N G A N D R A T E SENSITIVITY OF Cu A N D AI

general behaviour Of simple materials other than AI and Cu.

An interesting feature of Fig. 9 is that, at maximum load, B~ for Cu is 0.10 and 0.19 at T n = 0.22 and 0.51, respectively, whereas for A1, B~ is 0.35 and 0.44 at Tn = 0.32 and 0.51, respectively. Thus, it is apparent that B~ increases with temperature and material purity. There may also be a contribution from the increase in stacking fault energy. These B, differences have a practical significance which can be seen more clearly by plotting the B~ values against the WHC H, as illustrated in Fig. 10. In Fig. 10(a), an overall picture of the evolution of B, is shown for most of the work hardening range. Here, increasing flow stress (i.e. the normal course of an experiment) corresponds to decreasing values of H (i.e. to moving from right to left on the abscissa). Initially, Cu at room temperature has the highest B~, with Cu at 698 K following, A1 at room temperature next, and AI at 473 K being last. However, the range of practical importance is in the vicinity of the maximum load, H = 1. It is when H is lower than, say, 2 (i.e. when the test approaches the UTS and exceeds it), that the rate of flow localization becomes noticeable. Because of the importance of this range, B, is plotted against H in the expanded diagram of Fig. 10(b). It can readily be seen that Cu at room temperature has the lowest B,, followed by Cu at 698 K, AI at room temperature, and finally A1 at 473 with the highest values. Thus, the vertical order near H = 0.2 appears to be the same as the horizontal order (in the direction of diminishing H) at maximum H.

From Fig. 10(b), it is evident that, at low hom*ologous temperatures (Cu at room temperature), flow localization cannot be substantially inhibited by B~ at strains beyond the UTS. This is because B, < 0.1 and so cannot contribute significantly to the stability

o CU_ _ - a . . . . 293K

0 8 '/~i~'-- CU . 698K [ ~

~% AL . . . . . . .

~/ '~'a~ / \" o 035 " , '~;- ~'~ \ • 0 4 4

I \°'e2~i----a'X- . . . . I 6 '.+I "o..;'-, ' , \ . . . . . . . . . . . . ~ i m "* I I 1

2 6 IOx'lO 3 0" /0

]Fig. 9. Dependence of the rate sensitivity of the work hardening rate B. on flow stress. B. is init ial ly zero, increases rapidly to a max im um in the early stages o f straining, and then decreases to low values beyond the UTS. At H = 1 (points identified by arrows), B~ = 0.10 and 0.19 for Cu at 293 and 698 K and 0.35 and 0.44 for Al at 293 and 473 K,

respectively.

~ 0 6

~ o 4

; 1: o)~t I

: .2 m ~. , l ' a ,~ . ,%

,1.. I : ~ . . . . . . . . . . . . . . . . - . - : - ' " / P,J

O 2 4 6 -f l I0" 15 20 25- 30 H = (~InO'l~ ( )(

(a)

f I . . J " / ', A L(4?'3KL - e" " / "

F "b e 4 ~

t I . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . : I , . -

o ~l~':" . . . . . 014 018 ll2 I '6 210

H :(~lno-/3()(

(b) Fig. 10. Dependence of B~ on the work hardening coefficient H. Note that, as deformation proceeds, H decreases. In (a), the overall B, vs H behaviour is shown. In (b), the behaviour of B, in the vicinity of maximum load (H = 1) is illustrated.

requirement [4] H + B, >__ 1. As the temperature is increased, B, increases, and the retardation of the rate of strain concentration becomes more and more effective. This will be substantiated in detail in a later publication [13]. However, it is worth mentioning that the present values of B, for A1 at 0.51 T,, and at H = 1 are considerably lower than those predicted by Kocks et al. [4]. They suggested that at temperatures around 0.7 Tin, B, is likely to be in the range 3-200. Such high values of Bo should retard the onset of flow localization to a very considerable degree. It is clear from the present results, however, that B, is well below the range they expected, and so has less effect than originally supposed. Nevertheless, when H drops below 1, B, falls in the range 0.3-0.4 in the A1. Thus, the stability condition H + B, > 1 can be main- tained until H drops to 0.6-0.7 because of the magnitude of B,. In this way, the non-negligible values of Bo at high stresses make a significant contribution to inhibiting flow localization.

Before proceeding further, the effect of under- estimating B, will be discussed when the latter is defined in terms of the single state variable approach, as described in Figs 4 and 8, for example. The reason why such an examination is required is because of the importance of B, in reducing the rate of localization

CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND A1 1665

420 g ~E b 410

I.- to 4 0 0

i -

I I I I 3 9 0 0"86 0 ' 9 4 I 02

TRUE STRAIN e"

Fig. 11. Expanded plot of Fig. 4. The values of the work hardening coefficient H at different stress levels are indicated

as HI = Ojal, HE = 02/a2 and H3 = 03/aa.

cFy a A0_~_

~ ' e ' l = 5x lO S

beyond the UTS, which is the region where accurate values are "needed". This point can be clarified by considering Fig. 11, which is an expanded version of Fig. 4. Note that the short transient is not shown for simplicity. At point A, /-/1 = 0.35; similarly at B, //2 = 0.82 and at C, //3 = 0.43. The one parameter rate sensitivity of H at constant structure B e is B~ = (H 3 --HO/ln(~2/~ 0 ~_ 0.03. Using the relationship B, = B~ + MC, and substituting the values of B,, M =8.2 x 10 -3 and C = 1.3, the one parameter value of B~ -~0.04, in agreement with the one evaluated from the continuous curves. When B~ is calculated by back extrapolation in this way, the long transient is of course neglected. However, it could be argued that the actual rate sensitivity of H at constant structure B* = ( H 2-H~)/Aln~ should be used instead. It is greater than B~, which is an underestimate of the "true" rate sensitivity of the WHC. An estimate of B* (see Fig. 1 l) suggests that it is equal to 0.16, which is more than 5 times larger than B~. The corresponding value of B* = B* + M C -~ 0.17, is again considerably larger than B,. However, the employment of B* (or B*) instead of B, requires an appropriate analysis, for example, one based on a two structure parameter model that takes into account the long transient [16]. The view taken here is that B* cannot be used in equation (2) because the derivation of this relation requires the material behaviour to be describable by a single state variable. When B, < B* is employed instead, which is consistent with the requirements, the inappropriately low value of B (with respect to the actual behaviour) is in part compensated by the inappropriately high value of M (again with respect to the actual behaviour). In an analogous fashion, the single state variable analysis requires M to be based on a Air in Fig. 11 of AC, rather than the actual value A B < AC. Thus, even though M (single state variable)> M (actual), the former must be used, because the latter is incompat- ible with the assumptions of the analysis.

4.1. The stress sensitivity C o f the W H C

The coefficient C was defined above as C = - (all~ 81ntr)~; in physical terms, it is the rate of change of WHC with flow stress during a constant strain rate test. From a practical point of view, it is the negative

of the slope of the H vs lncr curve, and is related to the curvature term C ' = -(O21ntr/~e:)~ = --(3H/8e)~ by the expression C = C'/H. Its calculation, however, can present difficulties because a small amount of scatter in the H-lntr data results in a large variation in the C coefficient. To reduce this to a minimum, the flow curves were smoothed repeatedly until the resulting H - l n a plot was essentially without noise. For this purpose, C was calculated by assuming that the dependence of the WHC on a can be approximated by

H = Hiexp(aa + btr 2 -J¢- C0 "3) (8)

where Hi, a, b and c are strain rate dependent and can be determined at a given strain rate by a suitable least-squares analysis. As an example, the constants in relation (8) for Cu at room temperature and for

~--" 5 X 10-4S -I are H,=46.69, a = -0.01251, b = c = 0, for which the correlation coefficient is 0.996. The degree to which the above curve fits the experimental data is shown in Fig. 12. It can be seen that the agreement is very good for H < 20, i.e. for e > 0.01. This covers practically the entire a-e range. Expressions such as relation [8] are particularly useful for computer simulations because they can be readily differentiated with respect to Incr.

The coefficient C was described in turn by

C = - ( a + 2btr + 3ca2)aH (9)

where a, b and c are the coefficients defined in equation (8), and tr and H are the current values of the flow stress and the WHC, respectively. The values of C calculated from equation (9) agreed quite well with the C's evaluated from the raw data after the latter were smoothed, indicating that the smoothing process did not deform the C vs tr relationship. The dependence of C on tr is summarized in Fig. 13 and compared with the dependence of the WHC H on ~r. It is apparent that C is a decreasing function of tr and temperature. It was also observed that C increases, but not strongly, with strain rate [7].

2O; t t-

• It IO~

~o H = Hoexp(o o-) ~ = 5 x i 6 4 $~ \

\ 9b T=295 K

I I 0 0

I I ~ i a 200 :500 4 0 0 5 0 0

TRUE S T R E S S o" ( M P o)

Fig. 12. Fitting of the H--a data (O) by the expression H = Hj exp(aa) (full line). Here, H i = 46.69, a = -0.01251,

and the correlation coefficient equals 0.996.

1666 CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND A1

I0 \ CU(295K) I0

~ _ _ C =__ (~ H/~lno.)( #

IO0 300 5-00 0 TRUE STRESS o'(MPo) (a)

1 H = (bln~/~//

I 2O

AL(4? 3 K) E=Zx0d 3 ,,-~

40 60 80 TRUE STRESS o" (MPo) (c)

H=(~I ~ / /~

~ AL(Z95K )

\ ,HIII I

I I 1 I I

H : ( ? m c r / ~ ~

I I I ,w . I00 0 40 120 200

TRUE STRESS o-(MPa) (d)

Fig. 13. Dependence of the curvature term C on tr. Note the decrease of C with stress. C always remains considerably higher than H.

4.2. Strain rate sensitivity N at constant W H C H

In the work of Kocks and co-workers [4], a new kind of strain rate sensitivity was defined; i.e. N = (d lna /d ln~)~, which is also equal to Bo/C [equation (5)]. In their paper, the above authors suggested that N is of considerable importance beyond the UTS because of the way it was predicted to influence the rate of flow localization. In order to assess this possibility, the magnitude of N during straining will now be discussed.

In order to compare N with ease for Cu and AI at the two experimental temperatures, it was plotted against H, as shown in Fig. 14. It is evident that N is an increasing function of (r,t the temperature, and indirectly, of the stacking fault energy and material purity. At maximum load (H = 1), N is 0.023, 0.057, 0.076 and 0.145 for Cu at 0.22 and 0.51 Tin, and AI at 0.32 and 0.51 Tin, respectively (see Table 1). These values are higher in both materials than the corresponding continuous and instantaneous rate sensitivities rn and M, which are also tabulated in Table 1. If the argument of Kocks et al. [4] about the coefficient N controlling the rate of flow localization

tNote that H decreases when a increases.

024

o AL(473K)

0 2C o AL(295K)

• CU (698 K)

01( ~ 6 CU(293 K)

z , Bo-

O'f2 . = (~) InO'/()ln() ='~,

0'08

0-0~

I Z H =Cdlno../~(),e Fig. 14. Dependence of N, the rate sensitivity at constant work hardening rate, on H. Note the increase of N with hom*ologous temperature, stacking fault energy and

material purity.

CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND AI 1667

at large strains were correct, the above observation would be important. However, as will be shown analytically in a forthcoming paper [13], N is not likely to be an important parameter. Kocks and co-workers [4] arrived at the opposite conclusion because of the simplifying assumptions they made regarding the flow localization equation. In order to get a tractable solution, they assumed that B, ~> 1 for 0 < H < 1, leading to the replacement of the term (H + Bo - 1) by B,. However, as shown above, B~ is less than 1 in this range, at least in the present materials and under the current experimental conditions, so that somewhat different approximate solu- tions apply [17].

Before leaving this topic, it should be mentioned that the value of N for A1 at maximum load and 0.51 Tm is actually higher than the one estimated by Kocks et al. [4]. This is because their estimate was based on data for 304 stainless steel, a material which is not as pure as the AI used in the present instance.

4.3. The rate and stress sensitivities P and Q o f the instantaneous rate sensitivity M

The coefficients P and Q are multipliers of the term ~'/~2=(t~lne/Oe)x contained in the second order differential equation (2). However, the term ~./~2 can frequently be neglected, depending on the values of P and Q; their magnitudes are given here so that the importance of this term can be assessed.

The coefficient P =--(t3M/t31ni), expresses the change in M when the strain rate is varied and the stress is held constant. Conversely, Q = (dM/3 lntr)~ is the change in M when the stress is varied and the strain rate is held constant. It can be readily seen from Fig. 6 that, for Cu at room temperature, P is negative at first, becoming positive later. Similarly, Q is negative before the minimum and positive after- wards. For all the other cases, P and Q are always positive. With respect to their magnitudes, it should be added that P is of the order of 10 -4 at room temperature and between 10 -3 and 10 -2 at 0.51 Tm. In a similar way, Q is around 10 -3 at room temperature and increases to approximately 2 x 10 -2 at 0.51 Tin. Thus, P and Q are unlikely to be of practical importance, as long as ~./~2 remains relatively small, i.e. < 102. This term is expected, however, to attain large values when M is very low and the flow is becoming highly localized, as may happen when Cu is deformed at room temperature and the strain is well beyond the Considrre value.

5. CONCLUSIONS

(1) Useful tensile tests can be carried out at constant true strain rate to strains well beyond the maximum load with the aid of a diametral transducer and a servo-controlled testing machine. The present transducer was employed up to 473 K (0.51 Tm for A1); higher temperatures can be reached by incor- porating suitable LVDT's.

(2) An instantaneous rate sensitivity M = (tglntr/ aln~)~ consistent with a single state parameter description of metal flow can be accurately determined by back-extrapolating through the "short" and "long" transients in the flow stress that follow a strain rate change. M decreases then increases with strain in Cu at room temperature, an effect ascribed to the solute hardening resulting from the presence of approximately 200 ppm of Ag, Mg and S. This effect disappears at 0.51 T~. It is entirely absent in the A1 tested, which was of higher purity, and in which M increases monotonically with strain. The instantaneous sensitivity M was approximately half the corresponding continuous rate sensitivity at all hom*ologous temperatures.

(3) The curvature coefficient C = (dH/alntr)~ decreases with stress (and strain) and depends weakly on strain rate. It is apparent that C > H, and that C < 10 over most of the experimental range.

(4) The rate sensitivity of the WHC B~ first increases (at small strains) and then decreases. At H = 1, B ,=0 .1 , 0.19, 0.35 and 0.44 in Cu at room temperature and 698 K, and in A1 at room temperature and 473 K, respectively. It is an increasing function of temperature, material purity and stacking fault energy. Although B, does not attain values as high as those predicted by Kocks et al. [4], it nevertheless makes a significant contribution to delaying flow localization, as it falls in the range 0.05 < B~ < 0.5 at strains beyond maximum load.

(5) The assumption made by Kocks et al. [4] in their work, i.e. that Bo >> 1 in the vicinity of H = l, does not hold under the present conditions. Thus, the rate of localization does not depend on the rate sensitivity N = B,/C. The coefficient N is an increasing function of stress (strain), stacking fault energy and material purity and is approximately twice the continuous rate sensitivity m.

(6) The actual value of Bo at large strains determined from strain rate changes without back- extrapolation through the long transient (B*) is substantially higher than the single state parameter value of Bo, or the value of B, obtained from continuous H curves. Nevertheless, B* cannot be used in analyses based on single state parameter work hardening behaviour.

(7) The coefficients P and Q fall in the approximate range 10 -4 to 10 -2 and increase with hom*ologous temperature. They can be neglected as long as the term k'/~ 2 is roughly less than 102.

Acknowledgements--The authors are indebted to the Natu- ral Sciences and Engineering Research Council of Canada, the Ministry of Education of Quebec (FCAC program) and McGill University (Graduate Faculty Research Fellowship) for financial assistance, as well as to Dr U. F. Kocks of the Los Alamos National Laboratories for his many useful comments and suggestions. Special thanks are due to Dr S. R. MacEwen of the Chalk River Nuclear Laboratories for his advice and interest and to Joe Mecke, also of CRNL, for carrying out some of the low strain experiments.

[668 CHRISTODOULOU and JONAS: WORK HARDENING AND RATE SENSITIVITY OF Cu AND A1

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Constitutive Equations in Plasticity (edited by A. S. Argon), p. 81. MIT Press, Cambridge, MA (1975).

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11. P. W. Bridgman, McGraw-Hill, New York (1952). 12. A. Melander and A. Thuvander, Res. Mech. 5, 129

(1982). 13. N. Christodoulou and J. J. Jonas. To be published. 14. U. F. Kocks, Fifth Int. Conf. on Strength of Metals and

Alloys (edited by P. Haasen et al.), Vol. 3, p. 1661 (1979).

15. U. F. Kooks, J. Engng Mater. Teehnol. (ASME) 98, 76, 1976.

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17. S. L. Semiatin and J. J. Jonas, Monogr. Formability and Workability o f Metals: Plastic Instability and Flow Localization. Am. Soc. Metals, Metals Park, OH. In press.

(PDF) Work hardening and rate sensitivity material coefficients for OFHC Cu and 99.99% A1 - DOKUMEN.TIPS (2024)

References