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Effect of stress reversals on the work hardening behaviour of polycrystalline copper
Acta metal/. Vol. 34, No. 8, pp. 1553-1562, Printed in Great Britain
1986
OOOI-6160/86 $3.00 + 0.00
Pergamon Journals Ltd
EFFECT OF STRESS REVERSALS ON THE WORK HARDENING BEHAVIOUR OF POLYCRYSTALLINE COPPER N. CHRISTODOULOU, Materials
Science Branch,
0.
T. WOO
and
S. R. MacEWEN
Atomic Energy of Canada Limited, Chalk River Nuclear Chalk River, Ontario, Canada KOJ 1JO
Laboratories,
[Received 23 May 1985; in revised form 12 December 1985) Abstract-The flow behaviour of polycrystalline copper is determined under conditions where the loading direction is reversed after increasing amounts of prestress (or prestrain). It is shown that the Bauschinger effect (defined as the response during the early stages of reverse flow) first increases and then saturates as the prestress is raised. In the same stress range, the reverse work hardening rate, 0,) is higher than that determined from a continuous test 0,. Beyond this stage, Br becomes less than Or. The difference Ati = Or - 0, attains non-negligible values over an extended stress (strain) interval of reverse loading before it becomes equal to zero. Concurrently, the reverse flow stress a, tends to saturate before it begins to rise again. It is suggested that the region of “almost” constant flow stress and the lower 0, coincide with the partial dissolution of dislocation tangles and cell-walls observed by TEM. The rearrangement of dislocation substructure that appears to take place in this stress (strain) interval is treated by employing a two-component composite model. R&sum&-Now avons Ctudie l’ecoulement plastique dans le cuivre polycristallin dam des conditions od l’on inverse la direction de la charge apres des taux croissants de precontrainte (ou de predeformation). Nous montrons que l’effet Bauschinger (defini comme la rtponse au tours des premiers stades de l’ecoulement inverse) augmente tout d’abord, puis se sature lorsque la precontrainte augmente. Dans le mime domaine de contraintes, le taux d’tcrouissage inverse 0, est plus grand que celui que l’on determine a partir d’un essai continu Or. Au deli de ce stade, f$ devient inferieur a tIr. La difference 0 = Br - tIr atteint des valeurs non ntgligeables dans un grand intervalle en contrainte (deformation) de charge inverse avant qu’elle ne devienne &gale a zero. En m&me temps, la contrainte d’ecoulement inverse Q, tend vers une saturation avant de commencer a croitre de nouveau. Nous pensons que la region de contrainte d’ecoulement “presque” constante et de faible valeur de 0, coincide avec la dissolution partielle des tcheveaux de dislocations et des parois de cellules observtes par MET. Nous traitons le rearrangement de la sous-structure de dislocations qui a lieu dans cet intervalle de contrainte (deformation) en utilisant un modele composite a deux composantes. Zusammenfassung-Das FlieBverhalten von polykristallinem Kupfer wird fur Bedingungen bestimmt, bei denen die Lastrichtung nach zunehmenden BetrHgen der Vorverformung umgekehrt wird. Der Bauschinger-Effekt (definiert als das Verhalten im ersten Stadium des umgekehrten FlieDens) nimmt zuerst zu und geht dann in die Sittigung mit zunehmender Vorspannung. Im selben Spannungsbereich ist die die Verfestigungsrate or in umgekehrter Richtung gr6Ber als diejenige, die aus einem kontinuierlichen Versuch bestimmt wird (0,). Oberhalb dieses Bereiches wird 0, kleiner als 8,. Die Differenz Af3 = 0,- 8, nimmt nicht-vemachlassigbare Werte in einem grol3eren Spannungs- (Dehnungs-) Interval1 der umgekehrten Belastung an, bevor sie zu Null wird. Gleichzeitig neight die Spannung fur umgekehrtes FlieBen o, zur Slttigung, bevor sie wieder ansteigt. Es wird nahegelegt, dag der Bereich einer nahezu konstanten FlieOspannung und eines niedrigeren 8, zusammenfallen mit der teilweisen Aufliisung von Versetzungsknaueln und Zellwlnden, wie sie im Elektronenmikroskop beobachtet worden ist. Die Umordnung in der Versetzungsstruktur, die in diesem Spannungs- (Dehnungs-) Interval1 aufzutreten scheint, wird mit einem Model1 einer zweikomponentigen Legierung beschrieben.
1. INTRODUCTION
The flow stress of work hardening materials during monotonic loading can be adequately described, with some restrictions, by a single state variable [l-6]. In these theories, the state variable can be closely related to the “average dislocation density”, p [7], although various authors have used other definitions. It is well tHere the subscripts r and f imply reverse and forward straining,
respectively.
Also, t is the true plastic
strain,
i.e. t = L,- u/E, where (I is the total strain and E is the Young’s AM
34/8--F
modulus. 1553
known, however, that the work hardening behaviour under reverse loading cannot be uniquely characterized by changes in p [8,9] alone, and additional state variables are needed for its description. Hasegawa et al. [lo] showed that in Al the decrease in p during a stress reversal at room temperature was less than the uncertainty of the measurement. Turner [8,9], and Hasegawa and co-workers [l&-14] that _ _ demonstrated the arrangement of dislocations, rather than the average density, changed during the reversal. After the initial transient both the flow stress cr and the work hardening rate 0, = da/de,? in the reverse
1554
CHRISTODOULOU
et al.:
EFFECT OF STRESS REVERSALS
direction fell below the values that they would have attained during forward straining. Hasegawa et al. [l&13] recently studied the Bauschinger effect as a function of temperature in order to establish the effect of static recovery on the behaviour of Al under reverse loading. The aims of the present study are to determine the effects of stress reversals on the Bauschinger effect (as defined by the parameter /I,, the ratio of the proportional limit in reverse flow to the flow stress at the point of unloading), and on the stress dependence of or at higher reverse strains. The function &(]a 1) will be employed for characterizing the behaviour in the reverse direction beyond the initial stage, because parameters related to strain or energy [12, 15, 161 cannot readily be used in a manner consistent with a state variable approach [ 141.The present work is also an attempt to correlate transients in the work hardening behaviour in the reverse direction with changes in the distribution and density of the dislocations when they form a cell-structure.
2. EXPERIMENTAL PROCEDURES AND MATERIALS Samples of oxygen-free high-conductivity copper, as described in [6], with a gauge length of 25.4 mm, a uniform gauge diameter of 9.5 and 12.7 mm heads, were machined from 12.7 mm hot extruded bars and were annealed under vacuum [6] at 723 K for 60 min prior to testing. The annealed samples had a close to random crystallographic texture and an average grain size of 0.02 mm. The experiments were conducted in a MTS Alpha System. The alignment of this machine was sufficiently precise so that an annealed sample could be compressed up to a true strain E N 0.1 before buckling. The constant true strain rate of 5 x 10u5 s-l was measured and controlled by an extensometer attached to the sample. After establishing that the flow curves in tension and compression, up to E N 0.15, were identical, the following experiments were carried out: (i) Samples were first compressed to or = 70, 118 and 156 MPa (i.e. t N 0.02, 0.04 and 0.08, respectively) and then pulled in tension. (ii) Samples were first pulled to q N 227, 265 and 277 MPa (i.e. t -0.18, 0.28 and 0.33, respectively) and then cylinders were cut from the uniform section of the gauge length and compressed. Molybdenum disulphide was used as a lubricant and the compression platens were covered with thin teflon sheets. The cylinders initially had an aspect ratio of 1.5 and grooves were introduced on their end surfaces [17] for the retention of the lubricant. (iii) Samples were first pulled to er = 227,265 and 277 MPa and then compressed until they buckled. This set of experiments was conducted in order to verify to what extent the machining
ON WORK HARDENING
and the introduction of grooves would alter the work hardening behaviour after the load reversal. It was found that 8, was essentially unaffected by the remachining.
3. EXPERIMENTAL RESULTS 3.1. Flow stress after the stress reversal The flow behaviour upon a stress reversal involves more than one phenomenon [13]. In the early stages of reverse flow, the material can be characterized by parameters [15, 181 describing the Bauschinger effect. In the present study, the Bauschinger effect is measured by means of the parameter 8, = a,/ar. Here eY is the yield stress in the reverse direction, defined as the proportional limit, and has the same sign as or if the material yields during unloading. An example of a stress reversal going from compression to tension is shown in Fig. 1, which defines also AC,, the reverse strain during unloading. The variation of the parameters fi, and 86, with err can be seen in Table 1. Although the determination of /I, may not be the only or the best way to describe the Bauschinger effect, it is evident from Table 1 that /I, and A4 initially increase and then level off to a constant value as the material approaches the UTS (ours = 267 MPa in tension). Abel and Muir [15, 181 reported that in mild steel and various Cu alloys the parameter (1 + &,) continuously increased with prestrain. In their experiments, however, the prestrains were always lower than 0.1, and in this range, the results from Table 1 are indicating a similar trend. The true stress-true accumulated plastic strain curves for three different prestress levels (i.e. or N 156, 227 and 265 MPa) are shown in Fig. 2(a) to (c). The dotted line was determined experimentally using a second specimen. It can be seen that the reverse flow curve always stays below the monotonic one, and that after high prestresses tends to remain almost constant before beginning to rise again [Fig. 2(b) MO0261
E'l06GPa
Qt t____-160. -.0075
I -.0671
-.0667
-0663
-.0659
-.o55
TRUE PLASTIC STRAIN
Fig. 1. Stress reversal after -0.09 prestrain in compression; er (Z 160 MPa) is the prestress, eY (- 85 MPa) the proportional limit during reverse reverse loading, and At, the reverse strain when the load is zero.
CHRISTODOULOU Table Direction of prestrain
1. Dependence
et al.:
EFFECT
OF STRESS
REVERSALS
ON WORK
1555
HARDENING
of 8, and AC, on or
i * 5 I to-’ a,(MPa)
Compression Compression Compression Tension Tension Tension
uy (MPa)
70 118 156 227 265 277
B,
13 30 85 130 146 152
0.18 0.25 0.54 0.57 0.55 0.55
5.0 6.0 1.2 1.8 I.6 1.6
x x x x x x
s-1
T=293K
A% 10-s lo-’ 10-d lO-4 lO-4 10m4
H I aa
and (c)l. The strain interval over which the flow stress remains approximately constant increases with increasing prestress. 3.2. Work hardening behaviour after the load reversal The flow behaviour following a stress reversal can be more clearly visualized if the work hardening rate is plotted against the flow stress [13, 191, as shown in
--
COHPRESSION I I 0.1 0.2
0 TRUE
P p
PLASTIC
0.3
STRAIN
Q
260
i; $ L !?I
180 Et = 0.18 120
if? I-
b) 60
COHPRESSION 0.2 TRUE PLASTIC
I
0.1 E
0.3 STRAIN
3001 /R
-
ABSOLUTE TRUE STRESS u(MPa)
Fig. 3. Dependence of the plastic work hardening rate on absolute stress during forward (curve 1) and reverse flow, after (2) 70 MPa (camp.), (3) 118MPa (camp.), (4) 156 MPa (camp.), (5) 227 MPa (tension), (6) 265 MPa (tension), and (7) 277 MPa (tension).
Fig. 3. Curve 1 (obtained from a continuous experiment) is 0 in tension, whereas curves 2-7 correspond to 6 after the stress was reversed at the indicated prestress values (dotted lines). After the load reversal, 6, is initially higher than f&, but decreases rapidly as the specimen goes through the elastic-plastic transition. Beyond this initial stage, ~9~drops below Br for an extended period of stress (and strain) before becoming equal again to or at much higher flow stresses. For low prestresses (a, < 160 MPa) 8, measured at a reverse flow stress equal to or is higher than of measured at cr,; for high prestresses the opposite is true. The transient in the work hardening behaviour beyond the initial stages of reverse loading is analyzed by plotting the difference A6 = or- 8, against either 101or accumulated strain; A@is identified as the “height” of the transient, and its dependence on 1~1 and 6 is shown in Fig. 4(a) and (b), respectively. The stress and the strain increments, Ao and AC, required for A0 to become zero, as given in Table 2, are a measure of the “length” of the 8, transient. Both the “height” A&_ and the “length” of the transient as measured by Aa, first increase and then decrease with or. It is not clear whether AC exhibits the same trend. Even though the last data point in Table 2 was reproducible, further experimental work is needed to establish the decrease in AC for uf > 280 MPa.
6‘ i 0.21
(cl
CWPRESSIOK
3
TRUE PLASTIC STRAIN
I
I
0.4
05
c
Fig. 2. True stress/true accumulated plastic strain flow curves after (a) (~ru 156 MPa (in compression), (b) or = 227 MPa (in tension), and (c) cr IT 265 MPa (in tension).
Table 2. Stress and strain increments &r) +(MPa) 70 II8 156 227 265 277
Au and Ar as a function
of
(et)
Au (MPa)
A&
A%,, (MPa)
(0.02) (0.04) (0.08) (0.18) (0.28) (0.33)
98 125 96 57 36 33
0.09 0.19 0.23 0.25 0.30 0.27
259 314 543 364 239 202
1556
et al.: EFFECT OF STRESS REVERSALS ON WORK HARDENING
CHRISTODOULOU
PRESTRMN
(2)
.,
+ 0.02
131 0 004 I41 a 0.06 151 A O.IB
400
0
90
160
ABSOLUTE
TRUE
0
240
STRESS
v(
.
0.28
I71
.
0.33
0.4
0.2 ACCUM.
MPo)
(6)
TRUE
STRAIN
0.6 4
Fig. 4. (a) Dependence of A6 = l?r- Oron absolute stress after a load reversal at the indicated prestresses. (b) Dependence of A0 on accumulated plastic true strain after a load reversal at the indicated prestrains.
The strain rate dependence of the reverse work hardening transient was investigated by the following experiment. A sample was first compressed to
OFHC Cu 1. 293K i I a-‘1 -_--
2
5 x 10-1 3 x ,0-s
a, N 156 MPa (t N 0.08) at i, = 5 x lo-’ SC’ and subsequently pulled at i2 = 5 x 10e3 s-‘. The result compared to that obtained by pulling at 5 x 10-5s-1 is shown in Fig. 5. It is evident that with the exception of the rate sensitivity of the flow stress, the t$ transient is only slightly affected by a 100 times increase in strain rate.
2wO-
I 8
IOOO-
3.3. TEM observations 0.
’
160
60 ABSOLUTE
TRUE
The flow curve and the work hardening rate of a tensile sample that was first compressed to 0.04 strain and then pulled are shown in Fig. 6(a) and (b). TEM was used for the examination of the dislocation microstructures characteristic of points A-D in Fig.
240
STRESS
o(MPaI
Fig. 5. Dependence of the reverse work hardening rate on strain rate.
200 ANNEALED
OF “C
Cu
. ..*..
130
. . .. .* :
.
.
,....’ . . . . .
. .
.
...”
10
o
.
C
(a)
B
\
g
:
.:
. f
A
,
TRUE
PLASTIC
I
STRAIN
q
.-’
SE-S
T = 263
I I - 0.01
-0.03
i
. . *
7,)*’
1
m
:.
up.
--COMPRESSION TENSION
I(
I
I 0.01
003
c
(1,
0.’
60 ABSOLUTE
160
TRUE STRESS
240
u(HPa)
Fig. 6. (a) Stress reversal after -0.04 prestrain in compression. (b) Dependence of 0 on absolute stress for (a). Note that -0.01 reverse plastic strain is needed for 0, z Or (region between points B and C).
CHRISTODOULOU
et al.:
EFFECT OF STRESS REVERSALS
Fig. 7. (a) Dislocation substructure at point A (in Fig. represents 1 pm. (b) Dislocation arrangement at point B cell-straddling dislocations. (c. d) Dislocation configuration in some regions
6(a) and (b) [20]. Foils were prepared from sections cut normal to the axes of the deformed specimens. The substructure at A (er= 118 MPa, cr N 0.04) consisted largely of dislocation cells N l-3 ym in size with relatively clear interiors [Fig. 7(a)]. The cell walls contained a high density of tightly-knit dislocations, while dislocations in the cell interiors generally terminated at the cell-walls. Larger scale rough cell structures were also observed in other areas of the foil. Both distributions resemble those observed by Steeds [21] in (111) Sections of Cu single crystals deformed to late stage II and early stage III. Reversing the strain by 0.007 (point B, i.e. er = 117 MPa) produced some structures such as that shown in Fig. 7(b). There is evidence here of cell walls being dissolved or partially reduced to tangles, with individual dislocations straddling the cells. The breaking-up of an
ON WORK HARDENING
1557
6). In all micrographs, the length indicator showing dissolution of cell-walls (at X) and at point D. Note the absence of cell-structure (d).
otherwise continuous tight dislocation wall is shown at X. Although the structure shown in Fig. 7(b) was quite common, the simultaneous presence of tight cells similar to those seen in Fig. 7(a) (i.e. before the reversal) imply that in Cu the dissolution of cell walls is only partial during the reversal and not as dramatic as was observed in Al [lo, 131. After 0.015 reverse strain (point C, er z 128 MPa) dissolution of the cell-walls was clearly evident; by point D (a, 1: 150 MPa, c, N 0.04) the substructure consisted of: (i) regions where the walls were tightly-knit (as at point A), and (ii) areas where there was no distinct cell-structure and the dislocations were arranged almost randomly [Fig. 7(c) and (d)]. When the stress reversal took place after a higher prestrain (i.e. er N 0.20) there was a region of almost constant flow stress [Fig. 2(b)] before or began to rise.
I558
CHRISTODOULOU et al.: ‘EFFECT OF STRESS REVERSALS ON WORK HARDENING
Fig. 8. (a) Microstructure of a sample deformed 0.20 in tension. (b) Dislocation substructure of a sample that was pulled 0.20 and compressed 0.02. (c) Dislocation substructure after 0.2 strain in tension and 0.07 in compression.
The micrographs of a sample predeformed in tension and then compressed to a strain of 0.02 and 0.07 are shown in Fig. 8. From Fig. 8(a) (ct N 0.20) it can be seen that the cell-interiors are relatively free of dislocations, whereas after 0.02 compressive strain [Fig. S(b)] many contain a high density of dislocations. Concurrently, partial dissolution of cell-walls is taking place, as at lower prestrains (Fig. 7). However, it now requires much larger reverse strains to reestablish the tightly-knit cell structure. Even after a compressive strain of 0.07 [Fig. S(c)], evidence of loosely-knit, partially-dissolved walls remains. For samples prestrained 0.35 in tension, and then compressed various amounts, the same general processes of the breaking-up of cell-walls and rearrangement of dislocations took place, even though the initial cell structure was smaller and had more tightly-knit walls. 4. DISCUSSION
The reverse flow behaviour, as characterized by the work hardening rate, has two distinct stages: (a) The very early response during which the deforming piece yields prematurely (i.e. the Bauschinger effect) and the work hardening rate or > 6,, and (b) The transient region in which 6, stays below Br for a stress (strain) interval that depends on prestress (prestrain).
If it is assumed that back stresses, of the type considered by Pedersen et al. [16] and Nix et al. [22], build up during monotonic loading and evolve to a different state during the stress reversal, both stages of reverse flow can be explained. 4.1. The Bauschinger efict : 8, > 6, Hasegawa and co-workers [l&14] and Kocks ec al. [23] attributed the premature reverse yielding to the polarization of the substructure, i.e. to the accumulation of excess dislocations of opposite sign in either side of dislocation cell-walls. Mughrabi [24-261 has recently proposed a composite model in which ‘the polarization of dislocation tangles was necessary for accommodating the difference between the elastic and plastic shear strains of the walls and the cell-interiors. If z^, and f2 are the flow (shear} stresses of the “hard” (dislocation dense) and “soft” (dislocation poor) regions, respectively, the flow stress of the composite 2^is given by [2428] z^= (1 -f)z^, +p*
(1)
where f is the volume fraction of cell-interiors. For the present analysis, the following assumptions will be made. (i) The thermally-activated components of Q, and fZ are negligible. (ii) Grain-to-grain interaction stresses can be ignored. The latter is justified because
CHRISTODOULOU
et al.:
EFFECT
OF STRESS
neutron diffraction analysis? showed line broadening, whereas no detectable shift of the peak of the diffraction lines was observed with increasing deformation. The stress Q is related to the macroscopic flow stress oc by the expression CT~ = hE, where M is the average Taylor factor. Given that f N 0.8 for Cu [24-281, it follows that most of the material is flowing at Q, < t^. Therefore, if the flow stress in the cell-interior is represented as Q, = z^+ tb @a) Zb= -(1 -.f)
(?, -?-J
(2b)
and since zb is negative, it opposes flow in the forward direction and aids flow in the reverse direction. The stress state when the material has formed a cell-structure and is under load can be depicted schematically as done by Nix ef al. [22], Fig. 9(a). When the applied stress z(=cr/M) reaches 2^(=ar/M), i.e. when zi and r2 reach z^, and fZ, respectively, the material flows plastically. Mughrabi [24-261 and Ungar et al. [27,28] have used an X-ray line broadening analysis to determine the internal stresses that develop when the dislocations form a cell-structure, and reported that the ratio f,/z^ decreases, and
REVERSALS
ON WORK
HARDENING
1559
crystals near 0.5 T,. These “delayed unloading strains” were attributed to the backward glide and rearrangement of dislocations in the cell-interiors under the influence of the internal back stress, and thus are another manifestation of the processes that determine AC,. 4.2. Work hardening during reverse loading The work hardening during reverse loading is characterized by two features: the value of B, when the reverse flow stress, ]erlr becomes equal to 1~~~1, and the extended period of almost constant flow stress during which 8, passes through a minims. It can be seen from Fig. 3 that for low prestrains (CT,< 160 MPa) 0, is greater than of when )crrlN letl. For higher prestrains, the opposite is true. Hasegawa and Kocks [19] have recently investigated the work hardening behaviour in Al crystals, oriented for multiple slip, that had been deformed, statically annealed, and then deformed again in the same direction as the prestrain. They defined Type-I recovery when the work hardening rate at the prestress level was higher than that before the anneal, and (HARD)
-T
IHARD
FLOW STRESS
Fig. 9. (a) State of internal stresses in a two-component composite model (from [22]). (b) Evolution (schematic) of z^ and i, with shear strain y. At point A, i = ir and t2 = $r.
1560
CHRISTODOULOU et al.: EFFECT OF STRESS REVERSALS ON WORK HARDENING
Type-II recovery when the opposite was true. Type-II recovery was promoted by increasing the annealing time or temperature, and thus increasing the amount of rearrangement of dislocations in the cell-walls during the anneal. It was also suggested that Type-II recovery was enhanced by deforming to higher prestresses, possibly as a result of the dislocation processes occurring during Type-I recovery taking place by dynamic recovery during the prestrain. The work hardening rate during stress reversals shows a remarkable similarity to the Type-I, Type-II behaviour observed following static annealing. Thus, it would appear that the rearrangement of dislocation substructure during static annealing may be very similar to that occurring during reverse plastic flow. Yakou et al. [14] have demonstrated that Al and Fe exhibit a work hardening transient during which 0, passes through a minimum. The extent of the transient? initially increased and then decreased with increasing temperature, that is, as dynamic recovery became more important. In polycrystalline Cu, a minimum in or was observed only for test temperatures greater than 723 K [lo], possibly because the prestrain was too low. The transient in Br was attributed to the dissolution and reformation of the cell-walls, as observed by TEM [lo]. It was also reported that the total dislocation density in Al [lo] decreased by about 15% during the first 0.01 reverse strain. However, since the 8, transient lasted for larger reverse strains than did the changes in dislocation density, and given the uncertainty of the measurement (+40%), it is reasonable to conclude that changes in p alone are insufficient to fully describe the effects of a stress reversal on work hardening. Therefore, it seems most reasonable to seek the identity of structure parameters that describe reverse flow in variables that take into account both the density and arrangement of dislocations, and the manner in which these vary with strain, as suggested by Turner and Hasegawa [9] and Swearengen and Holbrook [3 11. 4.3. Structure
variables related to the 0, transient
Reverse flow behaviour beyond the elastic-plastic transition is characterized by the macroscopic flow stress and the work hardening rate that are both lower than those measured in the forward direction. The origin of the softening can be attributed to changes in f, z^,or z^*[equation (1)] that take place during the reversal. A detailed analysis would require the determination of the evolution off, r^,and fz with accumulated strain in both loading directions and would require the simultaneous solution of a set of coupled equations [32]. This option is beyond the
scope of the present work. Instead, it is worth considering the following limiting case. It was shown (Figs 7 and 8) that some of the cell-walls break up and disintegrate during the reversal, and that the dislocation density in the cellinteriors initially increases and then decreases as cell-walls reform during flow in the reverse direction. The rearrangement of the dislocation microstructure suggests that bothfand z^2initially increase during the transient, and then decrease as the cell-walls reform and the dislocation density within the cells decreases. Walls that do not break up appear to be largely unaltered, suggesting that z^, is unaffected by the reversal. Rewriting equation (1), the flow stress of the composite in either direction can be given by Q = ?, -f.A? where A? = Q, - fz. Assuming that fir= Z^,r: Ar = z^r- ?, = -frAz^, +f,A;,
(34
The evolution offf with y during monotonic loading can be determined from the data of Ungar et al. [27] and Giittler [33], as shown in Fig. 10. The evolution off, with y can be established as follows. Due to the dislocation rearrangement, fz in the reverse direction can be expressed as Q,, = ?*t + sz^z
(3b)
A?, = A?, - X2.
(3c)
and therefore
Here, 62, represents the change in ?* as a result of the dislocation rearrangement during the reversal. From equations (3a-3c)
(44 and
From equation (2), (z^,,- z^t)/Qr= -u, where and experimentally it has been tl = (1 -fr)Afr/?r, shown [24,29] that CIsaturates at a value of -0.23 after prestrains of -0.08. Therefore, fzr = (1 - ~~)z^r
I
0 tThe similarity in the evolution of B,, and AtJ (Au or Ac) with or is likely to be coincidental; p,, describes the flow during the very ear/y stages of reverse flow, whereas A0 (Au or AC) refers to changes occurring much lafer.
(la)
0.2
I
I
I
I
0.4
06
0.6
IO
ACCUM.
SHEAR
STRAIN
y
Fig. 10. Evolution offduring forward (data from [25] and [33]) and reverse loading [equation (6)].
CHRISTODOULOU ef al.: EFFECT OF STRESS REVERSALS ON WORK HARDENING Table 3. Ar, i, and fr as a function of y for the transient observed after or z 156 MPa
and
Y,,
r^,(MPa)
AT (MPa)
ft
f, [equation (6)]
0.30
58
4.4
0.73
0.82
0.34
60
5.0
0.72
0.83
0.40
65
7.2
0.72
0.85
0.46
69
8.7
0.71
0.87
0.52
72
8.6
0.70
0.86
0.61
78
9.6
0.70
0.86
0.76
84
9.0
0.69
0.83
4,, =
(d) A one-component model characterized by a single structure parameter, such as r^, cannot satisfactorily explain the transient that occurs during reverse flow. A two-component model, such as the one proposed by Mughrabi [24,25] or Nix et aI. [22,32], and characterized by three microstructural parameters f, z^, and z^,, can at least qualitatively describe the transient. Acknowledgements-The authors would like to thank Dr U. F. Kocks of the Los Alamos National Laboratories and Professor J. J. Jonas of McGill University for many useful discussions. Sincere thanks are due to J. F. Mecke and J. F. Watters for their technical assistance, to Dr B. Cox for his comments on the manuscript, and to Professor W. D. Nix for allowing us to modify and use one of his figures.
[1 + olf,/(l -ft)]t^r, and A?^,= I&/( 1 -fr).
Equations
1561
(4a) and (4b) then become
and
REFERENCES
If S?r is negligible, the lower reverse flow stress is attributed only to variations infr with y; then f, 2: (1 -ff) Ar/(%)
+ff.
(6)
As an example, the values of A7 and r^,from Fig. 2(a), and ff from Fig. 10 were measured and are listed in Table 3. The evolution off, with y was then derived from equation (6) and plotted in Fig. 10. It is evident that f, evolves in the same manner as A6, Fig. 4; however, it attains a maximum at y N 0.46 (t 1: 0.1 S), a strain much larger than that at which A0 is maximum [curve 4 in Fig. 4(b)]. It thus seems that the change inf, alone cannot account for the 13,transient, and suggests that the effect of 6z^,is not negligible, at least initially, during the reversal. If 62, achieves a maximum during the early stages of reverse flow, the maximum in f, would be larger and would occur at a lower value of y. The difference Ad = & - 0, = (Or- 0,)/M* = Al?/M* can be derived by differentiating relations (4a) or (5a) with respect to y, as was suggested in [29]. 5. CONCLUSIONS (a) The Bauschinger effect was determined by measuring the proportional limit in the reverse direction. Its dependence on prestress or (and prestrain cr) was assessed by the parameters p, and Ac, that saturate as er (or tr) increases. It follows that the ratio ob/ar (= 02/ar - 1) also approaches saturation in a similar manner. (b) The work hardening rate 0, beyond the early stages of reverse flow drops below Orfrom a continuous test. The “height” and “length” of the 13~ transient first increase and then decrease with increasing or (tr). (c) The transient during reverse flow can be correlated with TEM observations which show the partial dissolution of cell-walls and changes of dislocation density in the cell-interiors.
1. U. F. Kocks, Strength of Metals and Alloys (edited by P. Haasen et al.), Vol. 3, p. 1661. Pergamon Press, Oxford (1979). 2. U. F. Kocks, Constitutive Equations in Plasticity (edited by A. S. Argon), p. 81, MIT Press, Cambridge, Mass. (1975). 3. U. F. Kocks, J. Engng Mater. Tech., A.S.M.E. 98, 76 (1976). 4. E. W. Hart, J. Engng Mater. Tech., A.S.M.E. 98, 193 (1976). 5. U. F. Kocks, J. J. Jonas and H. Mecking, Acta metall. 27, 419 (1979). and J. Jonas, Acta metall. 32, 1655 6. N. Christodoulou (1984). 7. H. Mecking and U. F. Kocks, Acta metall. 29, 1865 (1981). 8. A. P. L. Turner, Metall. Trans. lOA, 225 (1979). 9. A. P. L. Turner and T. Hasegawa, Mechanical Testing for Deformation Model Development (edited by R. W. Rohde and J. C. Swearengen) p. 322. ASTM-STP-765 (1982). Muter. Sci. 10. T. Hasegawa, T. Yakou and S. Karashima, Engng 20, 267 (1975). 11. T. Hasegawa and T. Yakou, Scripta metall. 8, 951 (1974). 12. T. Hasegawa, Y. Yakou, M. Shimizu and S. Karashima, Trans. Japan Inst. Metals 17, 414 (1976). 13. T. Hasegawa and T. Yakou, Scripta metall. 14, 1083 (1980). 14. T. Yakou, T. Hasegawa and S. Karashima, Trans. Japan Inst. Metals. In press. 15. A. Abel and H. Muir, Phil. Mug. 26, 459 (1982). 16. 0. B. Pedersen, L. M. Brown and W. M. Stobbs, Acta metall. 29, 1843 (1981). 17. M. J. Luton and J. J: Jonas, Proc. Second Int. Conf. Strength Metals and Alloys, p. 1100. Am. Sot. Metals, 18. 19.
Metals Park, Ohio (1970). A. Abel and H. Muir, Phil. Mng. 27, 585 (1973). T. Hasegawa and U. F. Kocks, Acta metall. 27, 1705
(1979). 20. 0. T. Woo, N. Christodoulou and J. F. Mecke, Proc. 42nd Annual Meeting of the Electron Microscopy Society of America (edited by G. W. Bailey), p. 474 (1984). 21. J. W. Steeds, Proc. R. Sot. A 292, 343 (1966). 22. W. D. Nix, J. C. Gibeling and K. P. Fuchs. Ref. 191. p. 301 (1982). 23. U. F. Kocks, T. Hasegawa and R. 0. Scattergood, Scripta metall. 14, 449 (1980). 24. H. Mughrabi, Acta metall. 31, 1367 (1983). L
.I
1562
CHRISTODOULOU
et al.:
EFFECT OF STRESS REVERSALS
25. H. Mughrabi, Proc. Fourth Int. Conf on Continuum Models of Discrete Systems (edited by 0. Brulin and R. K. Hsieh), p. 241. North-Holland, Amsterdam (1981). 26. H. Mughrabi, Proc. 5th Int. Conf. on Strength of Metals and Alloys (edited by P. Haasen et al.), Vol. 3, p. 1615. Pergamon Press, Oxford (1979). 27. T. Ungar, H. Mughrabi, D. Rijnnpagel and M. Wilkens, Acta metall. 32, 333 (1984). 28. T. Ungar, H. Mughrabi and M. Wilkens, Microstructural Characterization of Materials by NonMicroscopical Techniques, Proc. 5th RIS0 Int. Symp.
ON WORK HARDENING
Metall. Mater. Sci. (edited by N. Hessel et al.), p. 539 (1984). 29. N. Christodoulou, S. R. MacEwen and 0. T. Woo, Proc. 7th Int. Conf. Strength Metal Alloys (edited by H. J. McQueen et ai), Vol.-l, p. 111 (1985). 30. T. Haseeawa. T. Yakou and U. F. Kocks, Acta metall. 30, 235 71982). 31. J. C. Swearengen and J. H. Holbrook, Res. mechanica 13, 93 (1985). 32. W. D. Nix and J. C. Gibeling, ASM-Mater. Sci. Semin. on Flow and Fracture at Elevated Temperatures (edited by R. Raj) (1985). 33. E. Giittler, Phil. Mag. 28, 1057 (1973).
1986
OOOI-6160/86 $3.00 + 0.00
Pergamon Journals Ltd
EFFECT OF STRESS REVERSALS ON THE WORK HARDENING BEHAVIOUR OF POLYCRYSTALLINE COPPER N. CHRISTODOULOU, Materials
Science Branch,
0.
T. WOO
and
S. R. MacEWEN
Atomic Energy of Canada Limited, Chalk River Nuclear Chalk River, Ontario, Canada KOJ 1JO
Laboratories,
[Received 23 May 1985; in revised form 12 December 1985) Abstract-The flow behaviour of polycrystalline copper is determined under conditions where the loading direction is reversed after increasing amounts of prestress (or prestrain). It is shown that the Bauschinger effect (defined as the response during the early stages of reverse flow) first increases and then saturates as the prestress is raised. In the same stress range, the reverse work hardening rate, 0,) is higher than that determined from a continuous test 0,. Beyond this stage, Br becomes less than Or. The difference Ati = Or - 0, attains non-negligible values over an extended stress (strain) interval of reverse loading before it becomes equal to zero. Concurrently, the reverse flow stress a, tends to saturate before it begins to rise again. It is suggested that the region of “almost” constant flow stress and the lower 0, coincide with the partial dissolution of dislocation tangles and cell-walls observed by TEM. The rearrangement of dislocation substructure that appears to take place in this stress (strain) interval is treated by employing a two-component composite model. R&sum&-Now avons Ctudie l’ecoulement plastique dans le cuivre polycristallin dam des conditions od l’on inverse la direction de la charge apres des taux croissants de precontrainte (ou de predeformation). Nous montrons que l’effet Bauschinger (defini comme la rtponse au tours des premiers stades de l’ecoulement inverse) augmente tout d’abord, puis se sature lorsque la precontrainte augmente. Dans le mime domaine de contraintes, le taux d’tcrouissage inverse 0, est plus grand que celui que l’on determine a partir d’un essai continu Or. Au deli de ce stade, f$ devient inferieur a tIr. La difference 0 = Br - tIr atteint des valeurs non ntgligeables dans un grand intervalle en contrainte (deformation) de charge inverse avant qu’elle ne devienne &gale a zero. En m&me temps, la contrainte d’ecoulement inverse Q, tend vers une saturation avant de commencer a croitre de nouveau. Nous pensons que la region de contrainte d’ecoulement “presque” constante et de faible valeur de 0, coincide avec la dissolution partielle des tcheveaux de dislocations et des parois de cellules observtes par MET. Nous traitons le rearrangement de la sous-structure de dislocations qui a lieu dans cet intervalle de contrainte (deformation) en utilisant un modele composite a deux composantes. Zusammenfassung-Das FlieBverhalten von polykristallinem Kupfer wird fur Bedingungen bestimmt, bei denen die Lastrichtung nach zunehmenden BetrHgen der Vorverformung umgekehrt wird. Der Bauschinger-Effekt (definiert als das Verhalten im ersten Stadium des umgekehrten FlieDens) nimmt zuerst zu und geht dann in die Sittigung mit zunehmender Vorspannung. Im selben Spannungsbereich ist die die Verfestigungsrate or in umgekehrter Richtung gr6Ber als diejenige, die aus einem kontinuierlichen Versuch bestimmt wird (0,). Oberhalb dieses Bereiches wird 0, kleiner als 8,. Die Differenz Af3 = 0,- 8, nimmt nicht-vemachlassigbare Werte in einem grol3eren Spannungs- (Dehnungs-) Interval1 der umgekehrten Belastung an, bevor sie zu Null wird. Gleichzeitig neight die Spannung fur umgekehrtes FlieBen o, zur Slttigung, bevor sie wieder ansteigt. Es wird nahegelegt, dag der Bereich einer nahezu konstanten FlieOspannung und eines niedrigeren 8, zusammenfallen mit der teilweisen Aufliisung von Versetzungsknaueln und Zellwlnden, wie sie im Elektronenmikroskop beobachtet worden ist. Die Umordnung in der Versetzungsstruktur, die in diesem Spannungs- (Dehnungs-) Interval1 aufzutreten scheint, wird mit einem Model1 einer zweikomponentigen Legierung beschrieben.
1. INTRODUCTION
The flow stress of work hardening materials during monotonic loading can be adequately described, with some restrictions, by a single state variable [l-6]. In these theories, the state variable can be closely related to the “average dislocation density”, p [7], although various authors have used other definitions. It is well tHere the subscripts r and f imply reverse and forward straining,
respectively.
Also, t is the true plastic
strain,
i.e. t = L,- u/E, where (I is the total strain and E is the Young’s AM
34/8--F
modulus. 1553
known, however, that the work hardening behaviour under reverse loading cannot be uniquely characterized by changes in p [8,9] alone, and additional state variables are needed for its description. Hasegawa et al. [lo] showed that in Al the decrease in p during a stress reversal at room temperature was less than the uncertainty of the measurement. Turner [8,9], and Hasegawa and co-workers [l&-14] that _ _ demonstrated the arrangement of dislocations, rather than the average density, changed during the reversal. After the initial transient both the flow stress cr and the work hardening rate 0, = da/de,? in the reverse
1554
CHRISTODOULOU
et al.:
EFFECT OF STRESS REVERSALS
direction fell below the values that they would have attained during forward straining. Hasegawa et al. [l&13] recently studied the Bauschinger effect as a function of temperature in order to establish the effect of static recovery on the behaviour of Al under reverse loading. The aims of the present study are to determine the effects of stress reversals on the Bauschinger effect (as defined by the parameter /I,, the ratio of the proportional limit in reverse flow to the flow stress at the point of unloading), and on the stress dependence of or at higher reverse strains. The function &(]a 1) will be employed for characterizing the behaviour in the reverse direction beyond the initial stage, because parameters related to strain or energy [12, 15, 161 cannot readily be used in a manner consistent with a state variable approach [ 141.The present work is also an attempt to correlate transients in the work hardening behaviour in the reverse direction with changes in the distribution and density of the dislocations when they form a cell-structure.
2. EXPERIMENTAL PROCEDURES AND MATERIALS Samples of oxygen-free high-conductivity copper, as described in [6], with a gauge length of 25.4 mm, a uniform gauge diameter of 9.5 and 12.7 mm heads, were machined from 12.7 mm hot extruded bars and were annealed under vacuum [6] at 723 K for 60 min prior to testing. The annealed samples had a close to random crystallographic texture and an average grain size of 0.02 mm. The experiments were conducted in a MTS Alpha System. The alignment of this machine was sufficiently precise so that an annealed sample could be compressed up to a true strain E N 0.1 before buckling. The constant true strain rate of 5 x 10u5 s-l was measured and controlled by an extensometer attached to the sample. After establishing that the flow curves in tension and compression, up to E N 0.15, were identical, the following experiments were carried out: (i) Samples were first compressed to or = 70, 118 and 156 MPa (i.e. t N 0.02, 0.04 and 0.08, respectively) and then pulled in tension. (ii) Samples were first pulled to q N 227, 265 and 277 MPa (i.e. t -0.18, 0.28 and 0.33, respectively) and then cylinders were cut from the uniform section of the gauge length and compressed. Molybdenum disulphide was used as a lubricant and the compression platens were covered with thin teflon sheets. The cylinders initially had an aspect ratio of 1.5 and grooves were introduced on their end surfaces [17] for the retention of the lubricant. (iii) Samples were first pulled to er = 227,265 and 277 MPa and then compressed until they buckled. This set of experiments was conducted in order to verify to what extent the machining
ON WORK HARDENING
and the introduction of grooves would alter the work hardening behaviour after the load reversal. It was found that 8, was essentially unaffected by the remachining.
3. EXPERIMENTAL RESULTS 3.1. Flow stress after the stress reversal The flow behaviour upon a stress reversal involves more than one phenomenon [13]. In the early stages of reverse flow, the material can be characterized by parameters [15, 181 describing the Bauschinger effect. In the present study, the Bauschinger effect is measured by means of the parameter 8, = a,/ar. Here eY is the yield stress in the reverse direction, defined as the proportional limit, and has the same sign as or if the material yields during unloading. An example of a stress reversal going from compression to tension is shown in Fig. 1, which defines also AC,, the reverse strain during unloading. The variation of the parameters fi, and 86, with err can be seen in Table 1. Although the determination of /I, may not be the only or the best way to describe the Bauschinger effect, it is evident from Table 1 that /I, and A4 initially increase and then level off to a constant value as the material approaches the UTS (ours = 267 MPa in tension). Abel and Muir [15, 181 reported that in mild steel and various Cu alloys the parameter (1 + &,) continuously increased with prestrain. In their experiments, however, the prestrains were always lower than 0.1, and in this range, the results from Table 1 are indicating a similar trend. The true stress-true accumulated plastic strain curves for three different prestress levels (i.e. or N 156, 227 and 265 MPa) are shown in Fig. 2(a) to (c). The dotted line was determined experimentally using a second specimen. It can be seen that the reverse flow curve always stays below the monotonic one, and that after high prestresses tends to remain almost constant before beginning to rise again [Fig. 2(b) MO0261
E'l06GPa
Qt t____-160. -.0075
I -.0671
-.0667
-0663
-.0659
-.o55
TRUE PLASTIC STRAIN
Fig. 1. Stress reversal after -0.09 prestrain in compression; er (Z 160 MPa) is the prestress, eY (- 85 MPa) the proportional limit during reverse reverse loading, and At, the reverse strain when the load is zero.
CHRISTODOULOU Table Direction of prestrain
1. Dependence
et al.:
EFFECT
OF STRESS
REVERSALS
ON WORK
1555
HARDENING
of 8, and AC, on or
i * 5 I to-’ a,(MPa)
Compression Compression Compression Tension Tension Tension
uy (MPa)
70 118 156 227 265 277
B,
13 30 85 130 146 152
0.18 0.25 0.54 0.57 0.55 0.55
5.0 6.0 1.2 1.8 I.6 1.6
x x x x x x
s-1
T=293K
A% 10-s lo-’ 10-d lO-4 lO-4 10m4
H I aa
and (c)l. The strain interval over which the flow stress remains approximately constant increases with increasing prestress. 3.2. Work hardening behaviour after the load reversal The flow behaviour following a stress reversal can be more clearly visualized if the work hardening rate is plotted against the flow stress [13, 191, as shown in
--
COHPRESSION I I 0.1 0.2
0 TRUE
P p
PLASTIC
0.3
STRAIN
Q
260
i; $ L !?I
180 Et = 0.18 120
if? I-
b) 60
COHPRESSION 0.2 TRUE PLASTIC
I
0.1 E
0.3 STRAIN
3001 /R
-
ABSOLUTE TRUE STRESS u(MPa)
Fig. 3. Dependence of the plastic work hardening rate on absolute stress during forward (curve 1) and reverse flow, after (2) 70 MPa (camp.), (3) 118MPa (camp.), (4) 156 MPa (camp.), (5) 227 MPa (tension), (6) 265 MPa (tension), and (7) 277 MPa (tension).
Fig. 3. Curve 1 (obtained from a continuous experiment) is 0 in tension, whereas curves 2-7 correspond to 6 after the stress was reversed at the indicated prestress values (dotted lines). After the load reversal, 6, is initially higher than f&, but decreases rapidly as the specimen goes through the elastic-plastic transition. Beyond this initial stage, ~9~drops below Br for an extended period of stress (and strain) before becoming equal again to or at much higher flow stresses. For low prestresses (a, < 160 MPa) 8, measured at a reverse flow stress equal to or is higher than of measured at cr,; for high prestresses the opposite is true. The transient in the work hardening behaviour beyond the initial stages of reverse loading is analyzed by plotting the difference A6 = or- 8, against either 101or accumulated strain; A@is identified as the “height” of the transient, and its dependence on 1~1 and 6 is shown in Fig. 4(a) and (b), respectively. The stress and the strain increments, Ao and AC, required for A0 to become zero, as given in Table 2, are a measure of the “length” of the 8, transient. Both the “height” A&_ and the “length” of the transient as measured by Aa, first increase and then decrease with or. It is not clear whether AC exhibits the same trend. Even though the last data point in Table 2 was reproducible, further experimental work is needed to establish the decrease in AC for uf > 280 MPa.
6‘ i 0.21
(cl
CWPRESSIOK
3
TRUE PLASTIC STRAIN
I
I
0.4
05
c
Fig. 2. True stress/true accumulated plastic strain flow curves after (a) (~ru 156 MPa (in compression), (b) or = 227 MPa (in tension), and (c) cr IT 265 MPa (in tension).
Table 2. Stress and strain increments &r) +(MPa) 70 II8 156 227 265 277
Au and Ar as a function
of
(et)
Au (MPa)
A&
A%,, (MPa)
(0.02) (0.04) (0.08) (0.18) (0.28) (0.33)
98 125 96 57 36 33
0.09 0.19 0.23 0.25 0.30 0.27
259 314 543 364 239 202
1556
et al.: EFFECT OF STRESS REVERSALS ON WORK HARDENING
CHRISTODOULOU
PRESTRMN
(2)
.,
+ 0.02
131 0 004 I41 a 0.06 151 A O.IB
400
0
90
160
ABSOLUTE
TRUE
0
240
STRESS
v(
.
0.28
I71
.
0.33
0.4
0.2 ACCUM.
MPo)
(6)
TRUE
STRAIN
0.6 4
Fig. 4. (a) Dependence of A6 = l?r- Oron absolute stress after a load reversal at the indicated prestresses. (b) Dependence of A0 on accumulated plastic true strain after a load reversal at the indicated prestrains.
The strain rate dependence of the reverse work hardening transient was investigated by the following experiment. A sample was first compressed to
OFHC Cu 1. 293K i I a-‘1 -_--
2
5 x 10-1 3 x ,0-s
a, N 156 MPa (t N 0.08) at i, = 5 x lo-’ SC’ and subsequently pulled at i2 = 5 x 10e3 s-‘. The result compared to that obtained by pulling at 5 x 10-5s-1 is shown in Fig. 5. It is evident that with the exception of the rate sensitivity of the flow stress, the t$ transient is only slightly affected by a 100 times increase in strain rate.
2wO-
I 8
IOOO-
3.3. TEM observations 0.
’
160
60 ABSOLUTE
TRUE
The flow curve and the work hardening rate of a tensile sample that was first compressed to 0.04 strain and then pulled are shown in Fig. 6(a) and (b). TEM was used for the examination of the dislocation microstructures characteristic of points A-D in Fig.
240
STRESS
o(MPaI
Fig. 5. Dependence of the reverse work hardening rate on strain rate.
200 ANNEALED
OF “C
Cu
. ..*..
130
. . .. .* :
.
.
,....’ . . . . .
. .
.
...”
10
o
.
C
(a)
B
\
g
:
.:
. f
A
,
TRUE
PLASTIC
I
STRAIN
q
.-’
SE-S
T = 263
I I - 0.01
-0.03
i
. . *
7,)*’
1
m
:.
up.
--COMPRESSION TENSION
I(
I
I 0.01
003
c
(1,
0.’
60 ABSOLUTE
160
TRUE STRESS
240
u(HPa)
Fig. 6. (a) Stress reversal after -0.04 prestrain in compression. (b) Dependence of 0 on absolute stress for (a). Note that -0.01 reverse plastic strain is needed for 0, z Or (region between points B and C).
CHRISTODOULOU
et al.:
EFFECT OF STRESS REVERSALS
Fig. 7. (a) Dislocation substructure at point A (in Fig. represents 1 pm. (b) Dislocation arrangement at point B cell-straddling dislocations. (c. d) Dislocation configuration in some regions
6(a) and (b) [20]. Foils were prepared from sections cut normal to the axes of the deformed specimens. The substructure at A (er= 118 MPa, cr N 0.04) consisted largely of dislocation cells N l-3 ym in size with relatively clear interiors [Fig. 7(a)]. The cell walls contained a high density of tightly-knit dislocations, while dislocations in the cell interiors generally terminated at the cell-walls. Larger scale rough cell structures were also observed in other areas of the foil. Both distributions resemble those observed by Steeds [21] in (111) Sections of Cu single crystals deformed to late stage II and early stage III. Reversing the strain by 0.007 (point B, i.e. er = 117 MPa) produced some structures such as that shown in Fig. 7(b). There is evidence here of cell walls being dissolved or partially reduced to tangles, with individual dislocations straddling the cells. The breaking-up of an
ON WORK HARDENING
1557
6). In all micrographs, the length indicator showing dissolution of cell-walls (at X) and at point D. Note the absence of cell-structure (d).
otherwise continuous tight dislocation wall is shown at X. Although the structure shown in Fig. 7(b) was quite common, the simultaneous presence of tight cells similar to those seen in Fig. 7(a) (i.e. before the reversal) imply that in Cu the dissolution of cell walls is only partial during the reversal and not as dramatic as was observed in Al [lo, 131. After 0.015 reverse strain (point C, er z 128 MPa) dissolution of the cell-walls was clearly evident; by point D (a, 1: 150 MPa, c, N 0.04) the substructure consisted of: (i) regions where the walls were tightly-knit (as at point A), and (ii) areas where there was no distinct cell-structure and the dislocations were arranged almost randomly [Fig. 7(c) and (d)]. When the stress reversal took place after a higher prestrain (i.e. er N 0.20) there was a region of almost constant flow stress [Fig. 2(b)] before or began to rise.
I558
CHRISTODOULOU et al.: ‘EFFECT OF STRESS REVERSALS ON WORK HARDENING
Fig. 8. (a) Microstructure of a sample deformed 0.20 in tension. (b) Dislocation substructure of a sample that was pulled 0.20 and compressed 0.02. (c) Dislocation substructure after 0.2 strain in tension and 0.07 in compression.
The micrographs of a sample predeformed in tension and then compressed to a strain of 0.02 and 0.07 are shown in Fig. 8. From Fig. 8(a) (ct N 0.20) it can be seen that the cell-interiors are relatively free of dislocations, whereas after 0.02 compressive strain [Fig. S(b)] many contain a high density of dislocations. Concurrently, partial dissolution of cell-walls is taking place, as at lower prestrains (Fig. 7). However, it now requires much larger reverse strains to reestablish the tightly-knit cell structure. Even after a compressive strain of 0.07 [Fig. S(c)], evidence of loosely-knit, partially-dissolved walls remains. For samples prestrained 0.35 in tension, and then compressed various amounts, the same general processes of the breaking-up of cell-walls and rearrangement of dislocations took place, even though the initial cell structure was smaller and had more tightly-knit walls. 4. DISCUSSION
The reverse flow behaviour, as characterized by the work hardening rate, has two distinct stages: (a) The very early response during which the deforming piece yields prematurely (i.e. the Bauschinger effect) and the work hardening rate or > 6,, and (b) The transient region in which 6, stays below Br for a stress (strain) interval that depends on prestress (prestrain).
If it is assumed that back stresses, of the type considered by Pedersen et al. [16] and Nix et al. [22], build up during monotonic loading and evolve to a different state during the stress reversal, both stages of reverse flow can be explained. 4.1. The Bauschinger efict : 8, > 6, Hasegawa and co-workers [l&14] and Kocks ec al. [23] attributed the premature reverse yielding to the polarization of the substructure, i.e. to the accumulation of excess dislocations of opposite sign in either side of dislocation cell-walls. Mughrabi [24-261 has recently proposed a composite model in which ‘the polarization of dislocation tangles was necessary for accommodating the difference between the elastic and plastic shear strains of the walls and the cell-interiors. If z^, and f2 are the flow (shear} stresses of the “hard” (dislocation dense) and “soft” (dislocation poor) regions, respectively, the flow stress of the composite 2^is given by [2428] z^= (1 -f)z^, +p*
(1)
where f is the volume fraction of cell-interiors. For the present analysis, the following assumptions will be made. (i) The thermally-activated components of Q, and fZ are negligible. (ii) Grain-to-grain interaction stresses can be ignored. The latter is justified because
CHRISTODOULOU
et al.:
EFFECT
OF STRESS
neutron diffraction analysis? showed line broadening, whereas no detectable shift of the peak of the diffraction lines was observed with increasing deformation. The stress Q is related to the macroscopic flow stress oc by the expression CT~ = hE, where M is the average Taylor factor. Given that f N 0.8 for Cu [24-281, it follows that most of the material is flowing at Q, < t^. Therefore, if the flow stress in the cell-interior is represented as Q, = z^+ tb @a) Zb= -(1 -.f)
(?, -?-J
(2b)
and since zb is negative, it opposes flow in the forward direction and aids flow in the reverse direction. The stress state when the material has formed a cell-structure and is under load can be depicted schematically as done by Nix ef al. [22], Fig. 9(a). When the applied stress z(=cr/M) reaches 2^(=ar/M), i.e. when zi and r2 reach z^, and fZ, respectively, the material flows plastically. Mughrabi [24-261 and Ungar et al. [27,28] have used an X-ray line broadening analysis to determine the internal stresses that develop when the dislocations form a cell-structure, and reported that the ratio f,/z^ decreases, and
REVERSALS
ON WORK
HARDENING
1559
crystals near 0.5 T,. These “delayed unloading strains” were attributed to the backward glide and rearrangement of dislocations in the cell-interiors under the influence of the internal back stress, and thus are another manifestation of the processes that determine AC,. 4.2. Work hardening during reverse loading The work hardening during reverse loading is characterized by two features: the value of B, when the reverse flow stress, ]erlr becomes equal to 1~~~1, and the extended period of almost constant flow stress during which 8, passes through a minims. It can be seen from Fig. 3 that for low prestrains (CT,< 160 MPa) 0, is greater than of when )crrlN letl. For higher prestrains, the opposite is true. Hasegawa and Kocks [19] have recently investigated the work hardening behaviour in Al crystals, oriented for multiple slip, that had been deformed, statically annealed, and then deformed again in the same direction as the prestrain. They defined Type-I recovery when the work hardening rate at the prestress level was higher than that before the anneal, and (HARD)
-T
IHARD
FLOW STRESS
Fig. 9. (a) State of internal stresses in a two-component composite model (from [22]). (b) Evolution (schematic) of z^ and i, with shear strain y. At point A, i = ir and t2 = $r.
1560
CHRISTODOULOU et al.: EFFECT OF STRESS REVERSALS ON WORK HARDENING
Type-II recovery when the opposite was true. Type-II recovery was promoted by increasing the annealing time or temperature, and thus increasing the amount of rearrangement of dislocations in the cell-walls during the anneal. It was also suggested that Type-II recovery was enhanced by deforming to higher prestresses, possibly as a result of the dislocation processes occurring during Type-I recovery taking place by dynamic recovery during the prestrain. The work hardening rate during stress reversals shows a remarkable similarity to the Type-I, Type-II behaviour observed following static annealing. Thus, it would appear that the rearrangement of dislocation substructure during static annealing may be very similar to that occurring during reverse plastic flow. Yakou et al. [14] have demonstrated that Al and Fe exhibit a work hardening transient during which 0, passes through a minimum. The extent of the transient? initially increased and then decreased with increasing temperature, that is, as dynamic recovery became more important. In polycrystalline Cu, a minimum in or was observed only for test temperatures greater than 723 K [lo], possibly because the prestrain was too low. The transient in Br was attributed to the dissolution and reformation of the cell-walls, as observed by TEM [lo]. It was also reported that the total dislocation density in Al [lo] decreased by about 15% during the first 0.01 reverse strain. However, since the 8, transient lasted for larger reverse strains than did the changes in dislocation density, and given the uncertainty of the measurement (+40%), it is reasonable to conclude that changes in p alone are insufficient to fully describe the effects of a stress reversal on work hardening. Therefore, it seems most reasonable to seek the identity of structure parameters that describe reverse flow in variables that take into account both the density and arrangement of dislocations, and the manner in which these vary with strain, as suggested by Turner and Hasegawa [9] and Swearengen and Holbrook [3 11. 4.3. Structure
variables related to the 0, transient
Reverse flow behaviour beyond the elastic-plastic transition is characterized by the macroscopic flow stress and the work hardening rate that are both lower than those measured in the forward direction. The origin of the softening can be attributed to changes in f, z^,or z^*[equation (1)] that take place during the reversal. A detailed analysis would require the determination of the evolution off, r^,and fz with accumulated strain in both loading directions and would require the simultaneous solution of a set of coupled equations [32]. This option is beyond the
scope of the present work. Instead, it is worth considering the following limiting case. It was shown (Figs 7 and 8) that some of the cell-walls break up and disintegrate during the reversal, and that the dislocation density in the cellinteriors initially increases and then decreases as cell-walls reform during flow in the reverse direction. The rearrangement of the dislocation microstructure suggests that bothfand z^2initially increase during the transient, and then decrease as the cell-walls reform and the dislocation density within the cells decreases. Walls that do not break up appear to be largely unaltered, suggesting that z^, is unaffected by the reversal. Rewriting equation (1), the flow stress of the composite in either direction can be given by Q = ?, -f.A? where A? = Q, - fz. Assuming that fir= Z^,r: Ar = z^r- ?, = -frAz^, +f,A;,
(34
The evolution offf with y during monotonic loading can be determined from the data of Ungar et al. [27] and Giittler [33], as shown in Fig. 10. The evolution off, with y can be established as follows. Due to the dislocation rearrangement, fz in the reverse direction can be expressed as Q,, = ?*t + sz^z
(3b)
A?, = A?, - X2.
(3c)
and therefore
Here, 62, represents the change in ?* as a result of the dislocation rearrangement during the reversal. From equations (3a-3c)
(44 and
From equation (2), (z^,,- z^t)/Qr= -u, where and experimentally it has been tl = (1 -fr)Afr/?r, shown [24,29] that CIsaturates at a value of -0.23 after prestrains of -0.08. Therefore, fzr = (1 - ~~)z^r
I
0 tThe similarity in the evolution of B,, and AtJ (Au or Ac) with or is likely to be coincidental; p,, describes the flow during the very ear/y stages of reverse flow, whereas A0 (Au or AC) refers to changes occurring much lafer.
(la)
0.2
I
I
I
I
0.4
06
0.6
IO
ACCUM.
SHEAR
STRAIN
y
Fig. 10. Evolution offduring forward (data from [25] and [33]) and reverse loading [equation (6)].
CHRISTODOULOU ef al.: EFFECT OF STRESS REVERSALS ON WORK HARDENING Table 3. Ar, i, and fr as a function of y for the transient observed after or z 156 MPa
and
Y,,
r^,(MPa)
AT (MPa)
ft
f, [equation (6)]
0.30
58
4.4
0.73
0.82
0.34
60
5.0
0.72
0.83
0.40
65
7.2
0.72
0.85
0.46
69
8.7
0.71
0.87
0.52
72
8.6
0.70
0.86
0.61
78
9.6
0.70
0.86
0.76
84
9.0
0.69
0.83
4,, =
(d) A one-component model characterized by a single structure parameter, such as r^, cannot satisfactorily explain the transient that occurs during reverse flow. A two-component model, such as the one proposed by Mughrabi [24,25] or Nix et aI. [22,32], and characterized by three microstructural parameters f, z^, and z^,, can at least qualitatively describe the transient. Acknowledgements-The authors would like to thank Dr U. F. Kocks of the Los Alamos National Laboratories and Professor J. J. Jonas of McGill University for many useful discussions. Sincere thanks are due to J. F. Mecke and J. F. Watters for their technical assistance, to Dr B. Cox for his comments on the manuscript, and to Professor W. D. Nix for allowing us to modify and use one of his figures.
[1 + olf,/(l -ft)]t^r, and A?^,= I&/( 1 -fr).
Equations
1561
(4a) and (4b) then become
and
REFERENCES
If S?r is negligible, the lower reverse flow stress is attributed only to variations infr with y; then f, 2: (1 -ff) Ar/(%)
+ff.
(6)
As an example, the values of A7 and r^,from Fig. 2(a), and ff from Fig. 10 were measured and are listed in Table 3. The evolution off, with y was then derived from equation (6) and plotted in Fig. 10. It is evident that f, evolves in the same manner as A6, Fig. 4; however, it attains a maximum at y N 0.46 (t 1: 0.1 S), a strain much larger than that at which A0 is maximum [curve 4 in Fig. 4(b)]. It thus seems that the change inf, alone cannot account for the 13,transient, and suggests that the effect of 6z^,is not negligible, at least initially, during the reversal. If 62, achieves a maximum during the early stages of reverse flow, the maximum in f, would be larger and would occur at a lower value of y. The difference Ad = & - 0, = (Or- 0,)/M* = Al?/M* can be derived by differentiating relations (4a) or (5a) with respect to y, as was suggested in [29]. 5. CONCLUSIONS (a) The Bauschinger effect was determined by measuring the proportional limit in the reverse direction. Its dependence on prestress or (and prestrain cr) was assessed by the parameters p, and Ac, that saturate as er (or tr) increases. It follows that the ratio ob/ar (= 02/ar - 1) also approaches saturation in a similar manner. (b) The work hardening rate 0, beyond the early stages of reverse flow drops below Orfrom a continuous test. The “height” and “length” of the 13~ transient first increase and then decrease with increasing or (tr). (c) The transient during reverse flow can be correlated with TEM observations which show the partial dissolution of cell-walls and changes of dislocation density in the cell-interiors.
1. U. F. Kocks, Strength of Metals and Alloys (edited by P. Haasen et al.), Vol. 3, p. 1661. Pergamon Press, Oxford (1979). 2. U. F. Kocks, Constitutive Equations in Plasticity (edited by A. S. Argon), p. 81, MIT Press, Cambridge, Mass. (1975). 3. U. F. Kocks, J. Engng Mater. Tech., A.S.M.E. 98, 76 (1976). 4. E. W. Hart, J. Engng Mater. Tech., A.S.M.E. 98, 193 (1976). 5. U. F. Kocks, J. J. Jonas and H. Mecking, Acta metall. 27, 419 (1979). and J. Jonas, Acta metall. 32, 1655 6. N. Christodoulou (1984). 7. H. Mecking and U. F. Kocks, Acta metall. 29, 1865 (1981). 8. A. P. L. Turner, Metall. Trans. lOA, 225 (1979). 9. A. P. L. Turner and T. Hasegawa, Mechanical Testing for Deformation Model Development (edited by R. W. Rohde and J. C. Swearengen) p. 322. ASTM-STP-765 (1982). Muter. Sci. 10. T. Hasegawa, T. Yakou and S. Karashima, Engng 20, 267 (1975). 11. T. Hasegawa and T. Yakou, Scripta metall. 8, 951 (1974). 12. T. Hasegawa, Y. Yakou, M. Shimizu and S. Karashima, Trans. Japan Inst. Metals 17, 414 (1976). 13. T. Hasegawa and T. Yakou, Scripta metall. 14, 1083 (1980). 14. T. Yakou, T. Hasegawa and S. Karashima, Trans. Japan Inst. Metals. In press. 15. A. Abel and H. Muir, Phil. Mug. 26, 459 (1982). 16. 0. B. Pedersen, L. M. Brown and W. M. Stobbs, Acta metall. 29, 1843 (1981). 17. M. J. Luton and J. J: Jonas, Proc. Second Int. Conf. Strength Metals and Alloys, p. 1100. Am. Sot. Metals, 18. 19.
Metals Park, Ohio (1970). A. Abel and H. Muir, Phil. Mng. 27, 585 (1973). T. Hasegawa and U. F. Kocks, Acta metall. 27, 1705
(1979). 20. 0. T. Woo, N. Christodoulou and J. F. Mecke, Proc. 42nd Annual Meeting of the Electron Microscopy Society of America (edited by G. W. Bailey), p. 474 (1984). 21. J. W. Steeds, Proc. R. Sot. A 292, 343 (1966). 22. W. D. Nix, J. C. Gibeling and K. P. Fuchs. Ref. 191. p. 301 (1982). 23. U. F. Kocks, T. Hasegawa and R. 0. Scattergood, Scripta metall. 14, 449 (1980). 24. H. Mughrabi, Acta metall. 31, 1367 (1983). L
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EFFECT OF STRESS REVERSALS
25. H. Mughrabi, Proc. Fourth Int. Conf on Continuum Models of Discrete Systems (edited by 0. Brulin and R. K. Hsieh), p. 241. North-Holland, Amsterdam (1981). 26. H. Mughrabi, Proc. 5th Int. Conf. on Strength of Metals and Alloys (edited by P. Haasen et al.), Vol. 3, p. 1615. Pergamon Press, Oxford (1979). 27. T. Ungar, H. Mughrabi, D. Rijnnpagel and M. Wilkens, Acta metall. 32, 333 (1984). 28. T. Ungar, H. Mughrabi and M. Wilkens, Microstructural Characterization of Materials by NonMicroscopical Techniques, Proc. 5th RIS0 Int. Symp.
ON WORK HARDENING
Metall. Mater. Sci. (edited by N. Hessel et al.), p. 539 (1984). 29. N. Christodoulou, S. R. MacEwen and 0. T. Woo, Proc. 7th Int. Conf. Strength Metal Alloys (edited by H. J. McQueen et ai), Vol.-l, p. 111 (1985). 30. T. Haseeawa. T. Yakou and U. F. Kocks, Acta metall. 30, 235 71982). 31. J. C. Swearengen and J. H. Holbrook, Res. mechanica 13, 93 (1985). 32. W. D. Nix and J. C. Gibeling, ASM-Mater. Sci. Semin. on Flow and Fracture at Elevated Temperatures (edited by R. Raj) (1985). 33. E. Giittler, Phil. Mag. 28, 1057 (1973).