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Non-elastic unloading of copper and aluminum
NON-ELASTIC
UNLOADING
A. R. ROSENFIELD
OF COPPER
AND ALUMINUM*
and B. L. AVERBACH?
The mechanical hysteresis curves of copper and aluminum single crystals and polycrystalline samples were measured at 25 and - 196°C in the plastic strain range of lo-“. A departure from elastic unloading is ascribed to the motion of piled-up glide dislocations toward their sources. The stress at which reverse plastic flow starts on unloading is associated with the force required for the glide dislocations to cut through a forest of intersecting dislocations with the formation of a jog. The jog formation energy is calculated to be 1.0 eV at 25°C and 1.4 eV at - 196°C for copper, and 0.2 eV at 25°C and 0.4 eV at - 196°C for aluminum. The temperature dependence of the jog formation energy is ascribed to the constriction of stacking faults on the intersection of the partial dislocations. DECHARGEMENT
NON-ELASTIQUE
DU
CUIVRE
ET
DE
L’ALUMINIUM
Les auteurs ont releve les courbes d’hysteresis mecanique de monocristaux et de polycristaux de 11s cuivre et d’aluminium, a 25 et -196”C, dans un domaine de deformations plastiques de 10d4. attribuent le comportement non-elastique qu’ils observent au mouvement vers leurs sources de dislocations empilees. 11s mettent en relation la contrainte Q laquelle debute la deformation plastique inverse, et la force a mettre en jeu pour qu’une dislocation traverse une for& de dislocations avec formation d’un cran. Les auteurs calculent une Bnergie de for nation de cran de 1,O eV a 25°C et 1,4 eV a - 196°C pour le cuivre, et de 0,2 eV a 25°C et 0,4 eV it - 196°C pour l’aluminium. Le fait que l’energie de formation de cran varie avec la temperature est attribue au pincement de fautes d’empilement it l’intersection des dislocations partielles. NICHTELASTISCHE
ENTLASTUNG
VON
KUPFER
UND
ALUMINIUM
Die mechanischen Hysteresekurven van Einkristallen und polykristallinen Proben van Kupfer und Aluminium wurden bei 25 und - 196°C im Bereich plastischer Verformung van 10e4 gemessen. Eine Abweichung von elastischer Entlastung wird daruf zuriickgeftihrt, da13 sich aufgestaute Gleitversetzungen zu ihren Quellen zuriickbewegen. Die Spannung, bei der beim Entlasten das entgegengesetzte plastische sum Durohschneiclen eines Fliel3en beginnt, hiingt mit der Kraft zusammen, die Gleitversetzungen Die Sprungbildungsenergie fiir Waldes schneidender Versetzungen unter Sprungbildung benotigen. Kupfer wurde zu 1,0 eV bei 25°C und I,4 eV bei - 196°C berechnet, die fur Aluminium zu 0,2 eV bei 25°C und 0,4 eV bei - 196°C. Die Temperaturabhiingigkeit der Sprungbildungsenergie wird auf das Einmhniiren der Stapelfehler am Schnittpunkt van Halbversetzungen zuriickgeftihrt.
Measurements
1. INTRODUCTION
strain at the bottom
of mechanical
observed in single crystals of copper and aluminum,(2) iron,(s) zinc(*) and alpha brass.(G)
been made in connection the Bauschinger
effect.
hysteresis curves have
with studies of fatigue and These measurements
involve detailed
substantial plastic deformation, measurements have been made
crystals
after small plastic
Roberts and Brown(*) have interpreted
and few on single
this reverse
plastic flow to the unbowing of dislocation segments and the reverse motion of glide dislocations.
flow is
However, in order to accomplish sufficient motion to account for the reverse plastic strain, it appears necessary to cut through the dislocation forest, and it
formed
Plastic
curve has been
associated with the motion of dislocations, but a central problem remains in understanding the reasons that dislocations
strains.
usually
of the unloading
under load do not retrace
seemed that a study of the plastic flow during unload-
their paths on the removal of the load and flow back
ing could lead to measurements
into the respective sources.
to form a jog in dislocations. In order to simplify the interpretation, experiments were carried out in single
stresses
near
the
This is particularly
initiation
of
plastic
true at
flow
since
of the work required
relatively small changes in the dislocation arrangement occur in this region. It is evident that some kind
was thus anticipated
of dislocation
would not be altered greatly by the small amount of
rearrangement
crystals with plastic strains of the order of lo-*.
occurs during unloading
and this results in a non-elastic
unloading
curve.
The
tation. 2. PROCEDURE
VOL.
10, JANUARY
1962
AND
RESULTS
Single crystals were grown by the Bridgman technique in the form of t-in. diameter unthreaded tensile
* This work was sponsored by the Office of Naval Research under contract Nonr-1841(35). Received June 14, 1961. t Department of Metallurgy, Massachusetts Institute of Technology, Cambridge, Massachusetts. METALLURGICA,
It
arrangement
plastic flow and the action of a single slip system in this microstrain region would simplify the interpre-
non-linear behavior at the beginning of tensile unloading has been observed by Thompson et uZ.(~) in aluminum single crystals and an enhanced reverse
ACTA
that the dislocation
bars with a 2-m. gage section. 71
The copper was 99.999
72
ACTA METALLURGICA,
Sheor Strain (10e6) FIG. 1. Load-unload curve for CU crystal at - 196°C.
per cent purity and the crystals were annealed at 3OO’C for 24 hr in vacuum. The aluminum was 99.995 per cent purity and the crystals were annealed at 26O’C for 1 hr in air. Electrical resistance strain gages were mounted on opposite sides of each specimen and the strain was measured on loading and unloa~ng to a sensitivity of 2 x 10-s. Care was taken to achieve axiality and readings from each gage on a specimen were within 5 per cent of the average strain. The loading was carried out in a Hounsfield tensometer. A typical loading and unloading curve for a copper crystal at -196°C is shown in Fig. 1. A residual plastic strain of about 200 x 1O-6 was introduecd in this sample. Deviations from elastic behavior on continuous loading appear to occur in the vicinity of 0.2 kg/mm2 but measurements of residual strain on the repeated unloading of similar crystals(“) has shown that plastic flow of the order of 1O-6 may be detected at 0.02 kgjmm2. On unloading, a deviation from elastic behavior was observed in the vicinity of 0.1 kg/mm2 and a net reverse plastic flow of about 10 x 10-B was measured on complete unloading. The stress at which unloading becomes plastic was designated as 7,. Similar behavior was observed for the same copper crystal tested at room temperature and these data are
Shear Strain (10e6) Fra. 2. Load-unload curve for Cu crystal at 25°C.
VOL. 10, 1962
shown in Fig. 2. The crystal was annealed between successive tests and experience indicated that the effects of the previous test were wiped out by these anneals as long as the total flow was in the easy glide microstrain region, i.e. less than 1O-4. The data for the aluminum crystals were all similar in form. Non-elastic behavior on unloading was also observed for polycrystalline aluminum and copper and a typical hysteresis curve is shown in Fig. 3. The tensile stress at which reverse plastic flow set in was designated as cU, and the corresponding shear stress T, was taken as half this value. A small region of non-elastic unloading was also observed at the beginning of unloading (see Fig. 3) and this was ascribed to the presence of microcreep at the high stresses,(l) this type of non-elastic unloading was more common for polycrystals than for single crystals. Values of T, are listed in Table 1. The elastic moduli were also measured and these values are also tabulated.
tomJJ;flu, lGOX 200 300 400 Strain (IOea)
i 500
FIG. 3. Stress-strain curve of copper at room temperature showing mechanical hysteresis. 3. DISCUSSION
assume that the initial u~oading is elastic and that reverse dislocation motion is negligible until the stress r, is reached. We consider that the glide dislocations do not flow back immediately on unloading because of the presence of intersecting dislocations. A pile-up of reversing glide dislocations against the forest occurs until the applied stress is lowered to the critical stress on unloading, 7%. At this stress the glide dislocation moves through the forest with the formation of jogs and reverse plastic flow is observed. Not all of the glide dislocations which have been generated are available for this reverse flow since some have passed out of the crystal or been absorbed at subboundaries and other interfaces. The extent of the reverse plastic flow is thus limited. The jog formation energy may be computed in the following manner. We assume that the force required to move a glide dislocation through the first tree in the Let
us
ROSENFIELD
AVERBACH:
AND
TABLE
Material
13 13 12 12 polycr. polycr. 25 25
cu
E Al Al Al Al Al
cc-Brass (Ref. 5) Zinc (Ref. 4)
_. ~__. ’ (w~2) T” 1
E(25”C) 17,000 17,500 16,000 16,900 13,000 14,700 7,000 7,300 7,100 7,300 7,300 7,800
~
1.03 1.06 1.05 1.04 1.03 1.09
~ (kggm2) 0.29 0.45 0.42 0.49 1.40 1.29 0.12 0.77 0.12 0.28 2.10 2.80
0.27 0.22 0.21 0.26 0.31 0.38 0.46 0.18 0.50 0.39 0.08 0.17
1
0.60
2.40
0.25
25
1
0.80
1.20
0.67
1Average
over the distance w, and zero else-
thus reduced by a factor 3.2.
vector b is given by
energy equations Ej = rjblw
energy and 7-j is the
to cut through
the tree.
Using
the
stacking
fault
energies
of
1.3 1.4 1.1 2.8
~ I 1 ~
1.8 2.6
0.31
density is also 1, is
The corresponding
jog
are thus:
Ej = 4.07, (copper polycrystals)
(lc)
and Ej = 1.0~~ (aluminum
The action
distance w is taken as the width of the extended glide dislocations.
1 ‘“~~~~~~
higher by a factor of 10 and the forest spacing,
(1)
the jog formation
is
stress required
=I
assumed that the resultant dislocation
where. For a spacing between trees, 1, the work done by a unit length of glide dislocation with a Burgers
where Ej
~ z
0.08 0.10 0.09 0.13 0.44 0.49 0.05 0.14 0.06 0.11 0.18 0.47
I
forest is constant
mpppp
25
I
_
_. _~~
E(-196°C)
- 196 25 -196 25 -196 25 -196
;: polycr. polycr.
73
UNLOADING
results
W)
!
CU CU CU CU
Experimental
Test temp.
Crystal No.
I
1.
NON-ELASTIC
polycrystals).
(ld)
We now consider the relationship between the stress to make a jog, 7j and TV, the measured critical
25 ergs/cm2 for copper and 100 ergs/cm2 for aluminum
unloading
previously
cation forest (T,) is equal to the sum of the applied
determined
by
the
crystals,(6) the values of w are 2.8 and 5 x 1O-s cm for aluminum. vector,
authors
on
similar
1O-8 cm for copper The partial Burgers
x
b, = a/l/S, where a is the lattice parameter.
stress.
stress (T) and the combined dislocations
linear
crystal
strained
determined similar
between
to about
by an X-ray
crystals
about 5 formation
spacing
10-4.
rocking
dislocations
in
lop5 cm. Using energy becomes
This was recently curve methodc2) for
these
values
the
Ej = 13~~ (copper single crystals)
Tt =
a
and was shown to have a value of
x
single crystals)
where 7i is expressed in kg/mm2 and
7 +
TD =
flow will
-Ti.
(2)
corresponds
curve such that
clear that the back stress
(a)
TD
to a point
7 =
Tu.
under
.
load
--F-+
-r,
(lb)
Ed in eV. (b)
at
-
r=r.
r = ry
FIG. 4. Schematic
It
is
must be negative in order
maximum
(la)
density was not measured for these polycrystals but it may be estimated as follows: The stress required to initiate forward plastic flow is about 10 times higher for polycrystals than for single crystals;(6) it will be
pp.)
fields of the
Reverse
jog
In polycrystals the stresses are high enough so that slip probably occurs on more than one set of planes and the forest density is thus higher. The forest
BA-(4
back-stress (T,).
We assume that this condition on the stress-strain
and E, = 3.17, (aluminum
in the pile-up
occur when (see Fig. 4)
The linear spacing of the forest, 1, was taken as the average
The stress on any tree in the dislo-
representation
of unloading.
74
ACTA
METALLURGICA,
for reverse Aow to occur by means of glide dislocations cutting through trees and returning to the source. The largest negative values of r0 occur close to the end of the pile-up. On unloading, as 7 decreases toward 7,, 7p becomes more negative due to elastic relaxation against the tree. However, it is difficult to choose a proper value for To, because the positions of the dislocations in the pile-up eannot he calculated precisely. A reasonable choice is that the value of r, is such that 7, = ?-j. An alternative analysis yields a similar result for the relationship between 7, and ri. During loading, dislocations have intersected trees in the forward direction and built up a stress in equilibrium with the external stress. Thus, 7, - 7P = 7j, where 7, is the m.aximum forward stress reached. On unloading, 7, - TD= --7j,andr,n - r, = 2~~. An inspection of Figs. l-3 shows that 7, m +rrn and 7,‘ m TV. The assumption that =rtL w -rj thus appears reasonable. Although values of T,/T, show considerable scatter, the average is quite close to r,/~, m 8. Carrington et LzI.(~) have derived an equation relating 7j and dislocation spacing. Using their values of dislocation spacing for iron, (about 1000 A) measured by means of electron microscope on thin films, i-f becomes approximately 700 g/mm”. This agrees fairly well with the value of 7, measured by Sullivan et al. (420 g/mm2).t3) In the case of polycrystals the number of dislocations in a pile-up N, may be calculated on the assumption that there is on the average only one source per grain operating. This is reasonable in view of the observations of Gay et aZ.(8) which indicate that there are about four sources per grain at 1 per cent deformation in many pure materials. The total plastic strain is then given by E = gANb (3) where 9 is the number of grains per unit volume, and A is the average cross sectional area of a grain. For strains of lo-* and a grain size of 0.25 mm, N = 20. The values of 7j shown for polycrystals in Table 2 were calculated on this assumption. The values of the jog formation energy are listed in Table 2. The values for copper are considerably higher than for aluminum, but the aluminum shows a greater effect of temperature. The values derived from the single crystal data are considered more reliable since fewer assumptions are involved. Considering the temperature dependence of ri (and thus E,:) shown in Table 1, it is evident that the temperature dependence is greater than that shown by the elastic constants. An almost identical temperature dependence was observed
VOL.
~~
10, 1962 TABLE 2. Jog formation energies .--____ __ - ._ .._.
-._
Material
j -.~
Cu
i
E Al
;
T-p
WI
1 ~&l-2) ..___~
Single crystals -__--.
.___
25 - 196 25 -196
I -
CU
Al Al
1
~_...
;:; 0:4 _.. . ..--
.____.
Folycrystals
cu
-.
~~
0.08 0.11 0.05 0.12
..^._.-
EAeV)
1
.____~~
25
0.41 -196 1 0.44 25 0.16 -196 0.45 / ~_______.__..._.. - ....-._ ..--
j
:i
~_
0:2 0.5
^ -._
for the stress required to produce an initial yield of 10V6, and the activation energy was associated with the energy required to change the stacking fault area.c6) Similarly, the energy of jog formation may be divided into two parts: (1) the energy required to constrict the partial dislocation and (2) the energy required to create a new segment of dislocation line. The energy required to create a new segment would be expected to have the same temperature dependence as the elastic constants which is small compared to the temperature dependence of the constriction of stacking faults.c6) It is thus concluded that the temperature dependence of the jog formation energy arises mainly from the constriction of partial dislocations. 4. SUMMARY The energy of jog formation has been inferred from the onset of plastic strain on the tensile unloading of single crystals after small plastic strains. The temperature dependence of the jog formation energy was associated with the constriction of partial dislocations during the jog forming process. ACKNOWLEDGMENTS
The authors would like to acknowledge the sponsorship of the Office of Naval Research. REFERENCES 1. N. TKOMPSON, C. K. COOGANand J. 0. RIDER, J. Inst. Met 84, 73 (1955). 2. M. J, HORDON and B. L. AVERBACH, Acta Met. 9, 237 (1961). 3. C. P. SULLIVAN, B. L. AVERBACH and M. COHEN, to be published.
4. J. M. ROBERTS and N. BROWN, Trans. Amer. Inst. &fin. (Metatl.) Engrs. 218, 454 (I 960). 5. J. D. ME~KIN and H. G. F. WILSDORF, Trans. Amer. Inst. Min. (Metall.) Engrs. 218, 745 (1960). 6. A. R. ROSENFIELD and B. L. AVERBACEI, Acta Met. 8, 624 (1960). 7. W. CAREINGTON, K. F. HALE and D. MCLEAN, PTOC. Roy. Sac. A259, 203 (1960). 8. P. GAY, P. B. HIRSCH and A. KELLY, Acta Cry&, Camb. 7, 41 (1954).
UNLOADING
A. R. ROSENFIELD
OF COPPER
AND ALUMINUM*
and B. L. AVERBACH?
The mechanical hysteresis curves of copper and aluminum single crystals and polycrystalline samples were measured at 25 and - 196°C in the plastic strain range of lo-“. A departure from elastic unloading is ascribed to the motion of piled-up glide dislocations toward their sources. The stress at which reverse plastic flow starts on unloading is associated with the force required for the glide dislocations to cut through a forest of intersecting dislocations with the formation of a jog. The jog formation energy is calculated to be 1.0 eV at 25°C and 1.4 eV at - 196°C for copper, and 0.2 eV at 25°C and 0.4 eV at - 196°C for aluminum. The temperature dependence of the jog formation energy is ascribed to the constriction of stacking faults on the intersection of the partial dislocations. DECHARGEMENT
NON-ELASTIQUE
DU
CUIVRE
ET
DE
L’ALUMINIUM
Les auteurs ont releve les courbes d’hysteresis mecanique de monocristaux et de polycristaux de 11s cuivre et d’aluminium, a 25 et -196”C, dans un domaine de deformations plastiques de 10d4. attribuent le comportement non-elastique qu’ils observent au mouvement vers leurs sources de dislocations empilees. 11s mettent en relation la contrainte Q laquelle debute la deformation plastique inverse, et la force a mettre en jeu pour qu’une dislocation traverse une for& de dislocations avec formation d’un cran. Les auteurs calculent une Bnergie de for nation de cran de 1,O eV a 25°C et 1,4 eV a - 196°C pour le cuivre, et de 0,2 eV a 25°C et 0,4 eV it - 196°C pour l’aluminium. Le fait que l’energie de formation de cran varie avec la temperature est attribue au pincement de fautes d’empilement it l’intersection des dislocations partielles. NICHTELASTISCHE
ENTLASTUNG
VON
KUPFER
UND
ALUMINIUM
Die mechanischen Hysteresekurven van Einkristallen und polykristallinen Proben van Kupfer und Aluminium wurden bei 25 und - 196°C im Bereich plastischer Verformung van 10e4 gemessen. Eine Abweichung von elastischer Entlastung wird daruf zuriickgeftihrt, da13 sich aufgestaute Gleitversetzungen zu ihren Quellen zuriickbewegen. Die Spannung, bei der beim Entlasten das entgegengesetzte plastische sum Durohschneiclen eines Fliel3en beginnt, hiingt mit der Kraft zusammen, die Gleitversetzungen Die Sprungbildungsenergie fiir Waldes schneidender Versetzungen unter Sprungbildung benotigen. Kupfer wurde zu 1,0 eV bei 25°C und I,4 eV bei - 196°C berechnet, die fur Aluminium zu 0,2 eV bei 25°C und 0,4 eV bei - 196°C. Die Temperaturabhiingigkeit der Sprungbildungsenergie wird auf das Einmhniiren der Stapelfehler am Schnittpunkt van Halbversetzungen zuriickgeftihrt.
Measurements
1. INTRODUCTION
strain at the bottom
of mechanical
observed in single crystals of copper and aluminum,(2) iron,(s) zinc(*) and alpha brass.(G)
been made in connection the Bauschinger
effect.
hysteresis curves have
with studies of fatigue and These measurements
involve detailed
substantial plastic deformation, measurements have been made
crystals
after small plastic
Roberts and Brown(*) have interpreted
and few on single
this reverse
plastic flow to the unbowing of dislocation segments and the reverse motion of glide dislocations.
flow is
However, in order to accomplish sufficient motion to account for the reverse plastic strain, it appears necessary to cut through the dislocation forest, and it
formed
Plastic
curve has been
associated with the motion of dislocations, but a central problem remains in understanding the reasons that dislocations
strains.
usually
of the unloading
under load do not retrace
seemed that a study of the plastic flow during unload-
their paths on the removal of the load and flow back
ing could lead to measurements
into the respective sources.
to form a jog in dislocations. In order to simplify the interpretation, experiments were carried out in single
stresses
near
the
This is particularly
initiation
of
plastic
true at
flow
since
of the work required
relatively small changes in the dislocation arrangement occur in this region. It is evident that some kind
was thus anticipated
of dislocation
would not be altered greatly by the small amount of
rearrangement
crystals with plastic strains of the order of lo-*.
occurs during unloading
and this results in a non-elastic
unloading
curve.
The
tation. 2. PROCEDURE
VOL.
10, JANUARY
1962
AND
RESULTS
Single crystals were grown by the Bridgman technique in the form of t-in. diameter unthreaded tensile
* This work was sponsored by the Office of Naval Research under contract Nonr-1841(35). Received June 14, 1961. t Department of Metallurgy, Massachusetts Institute of Technology, Cambridge, Massachusetts. METALLURGICA,
It
arrangement
plastic flow and the action of a single slip system in this microstrain region would simplify the interpre-
non-linear behavior at the beginning of tensile unloading has been observed by Thompson et uZ.(~) in aluminum single crystals and an enhanced reverse
ACTA
that the dislocation
bars with a 2-m. gage section. 71
The copper was 99.999
72
ACTA METALLURGICA,
Sheor Strain (10e6) FIG. 1. Load-unload curve for CU crystal at - 196°C.
per cent purity and the crystals were annealed at 3OO’C for 24 hr in vacuum. The aluminum was 99.995 per cent purity and the crystals were annealed at 26O’C for 1 hr in air. Electrical resistance strain gages were mounted on opposite sides of each specimen and the strain was measured on loading and unloa~ng to a sensitivity of 2 x 10-s. Care was taken to achieve axiality and readings from each gage on a specimen were within 5 per cent of the average strain. The loading was carried out in a Hounsfield tensometer. A typical loading and unloading curve for a copper crystal at -196°C is shown in Fig. 1. A residual plastic strain of about 200 x 1O-6 was introduecd in this sample. Deviations from elastic behavior on continuous loading appear to occur in the vicinity of 0.2 kg/mm2 but measurements of residual strain on the repeated unloading of similar crystals(“) has shown that plastic flow of the order of 1O-6 may be detected at 0.02 kgjmm2. On unloading, a deviation from elastic behavior was observed in the vicinity of 0.1 kg/mm2 and a net reverse plastic flow of about 10 x 10-B was measured on complete unloading. The stress at which unloading becomes plastic was designated as 7,. Similar behavior was observed for the same copper crystal tested at room temperature and these data are
Shear Strain (10e6) Fra. 2. Load-unload curve for Cu crystal at 25°C.
VOL. 10, 1962
shown in Fig. 2. The crystal was annealed between successive tests and experience indicated that the effects of the previous test were wiped out by these anneals as long as the total flow was in the easy glide microstrain region, i.e. less than 1O-4. The data for the aluminum crystals were all similar in form. Non-elastic behavior on unloading was also observed for polycrystalline aluminum and copper and a typical hysteresis curve is shown in Fig. 3. The tensile stress at which reverse plastic flow set in was designated as cU, and the corresponding shear stress T, was taken as half this value. A small region of non-elastic unloading was also observed at the beginning of unloading (see Fig. 3) and this was ascribed to the presence of microcreep at the high stresses,(l) this type of non-elastic unloading was more common for polycrystals than for single crystals. Values of T, are listed in Table 1. The elastic moduli were also measured and these values are also tabulated.
tomJJ;flu, lGOX 200 300 400 Strain (IOea)
i 500
FIG. 3. Stress-strain curve of copper at room temperature showing mechanical hysteresis. 3. DISCUSSION
assume that the initial u~oading is elastic and that reverse dislocation motion is negligible until the stress r, is reached. We consider that the glide dislocations do not flow back immediately on unloading because of the presence of intersecting dislocations. A pile-up of reversing glide dislocations against the forest occurs until the applied stress is lowered to the critical stress on unloading, 7%. At this stress the glide dislocation moves through the forest with the formation of jogs and reverse plastic flow is observed. Not all of the glide dislocations which have been generated are available for this reverse flow since some have passed out of the crystal or been absorbed at subboundaries and other interfaces. The extent of the reverse plastic flow is thus limited. The jog formation energy may be computed in the following manner. We assume that the force required to move a glide dislocation through the first tree in the Let
us
ROSENFIELD
AVERBACH:
AND
TABLE
Material
13 13 12 12 polycr. polycr. 25 25
cu
E Al Al Al Al Al
cc-Brass (Ref. 5) Zinc (Ref. 4)
_. ~__. ’ (w~2) T” 1
E(25”C) 17,000 17,500 16,000 16,900 13,000 14,700 7,000 7,300 7,100 7,300 7,300 7,800
~
1.03 1.06 1.05 1.04 1.03 1.09
~ (kggm2) 0.29 0.45 0.42 0.49 1.40 1.29 0.12 0.77 0.12 0.28 2.10 2.80
0.27 0.22 0.21 0.26 0.31 0.38 0.46 0.18 0.50 0.39 0.08 0.17
1
0.60
2.40
0.25
25
1
0.80
1.20
0.67
1Average
over the distance w, and zero else-
thus reduced by a factor 3.2.
vector b is given by
energy equations Ej = rjblw
energy and 7-j is the
to cut through
the tree.
Using
the
stacking
fault
energies
of
1.3 1.4 1.1 2.8
~ I 1 ~
1.8 2.6
0.31
density is also 1, is
The corresponding
jog
are thus:
Ej = 4.07, (copper polycrystals)
(lc)
and Ej = 1.0~~ (aluminum
The action
distance w is taken as the width of the extended glide dislocations.
1 ‘“~~~~~~
higher by a factor of 10 and the forest spacing,
(1)
the jog formation
is
stress required
=I
assumed that the resultant dislocation
where. For a spacing between trees, 1, the work done by a unit length of glide dislocation with a Burgers
where Ej
~ z
0.08 0.10 0.09 0.13 0.44 0.49 0.05 0.14 0.06 0.11 0.18 0.47
I
forest is constant
mpppp
25
I
_
_. _~~
E(-196°C)
- 196 25 -196 25 -196 25 -196
;: polycr. polycr.
73
UNLOADING
results
W)
!
CU CU CU CU
Experimental
Test temp.
Crystal No.
I
1.
NON-ELASTIC
polycrystals).
(ld)
We now consider the relationship between the stress to make a jog, 7j and TV, the measured critical
25 ergs/cm2 for copper and 100 ergs/cm2 for aluminum
unloading
previously
cation forest (T,) is equal to the sum of the applied
determined
by
the
crystals,(6) the values of w are 2.8 and 5 x 1O-s cm for aluminum. vector,
authors
on
similar
1O-8 cm for copper The partial Burgers
x
b, = a/l/S, where a is the lattice parameter.
stress.
stress (T) and the combined dislocations
linear
crystal
strained
determined similar
between
to about
by an X-ray
crystals
about 5 formation
spacing
10-4.
rocking
dislocations
in
lop5 cm. Using energy becomes
This was recently curve methodc2) for
these
values
the
Ej = 13~~ (copper single crystals)
Tt =
a
and was shown to have a value of
x
single crystals)
where 7i is expressed in kg/mm2 and
7 +
TD =
flow will
-Ti.
(2)
corresponds
curve such that
clear that the back stress
(a)
TD
to a point
7 =
Tu.
under
.
load
--F-+
-r,
(lb)
Ed in eV. (b)
at
-
r=r.
r = ry
FIG. 4. Schematic
It
is
must be negative in order
maximum
(la)
density was not measured for these polycrystals but it may be estimated as follows: The stress required to initiate forward plastic flow is about 10 times higher for polycrystals than for single crystals;(6) it will be
pp.)
fields of the
Reverse
jog
In polycrystals the stresses are high enough so that slip probably occurs on more than one set of planes and the forest density is thus higher. The forest
BA-(4
back-stress (T,).
We assume that this condition on the stress-strain
and E, = 3.17, (aluminum
in the pile-up
occur when (see Fig. 4)
The linear spacing of the forest, 1, was taken as the average
The stress on any tree in the dislo-
representation
of unloading.
74
ACTA
METALLURGICA,
for reverse Aow to occur by means of glide dislocations cutting through trees and returning to the source. The largest negative values of r0 occur close to the end of the pile-up. On unloading, as 7 decreases toward 7,, 7p becomes more negative due to elastic relaxation against the tree. However, it is difficult to choose a proper value for To, because the positions of the dislocations in the pile-up eannot he calculated precisely. A reasonable choice is that the value of r, is such that 7, = ?-j. An alternative analysis yields a similar result for the relationship between 7, and ri. During loading, dislocations have intersected trees in the forward direction and built up a stress in equilibrium with the external stress. Thus, 7, - 7P = 7j, where 7, is the m.aximum forward stress reached. On unloading, 7, - TD= --7j,andr,n - r, = 2~~. An inspection of Figs. l-3 shows that 7, m +rrn and 7,‘ m TV. The assumption that =rtL w -rj thus appears reasonable. Although values of T,/T, show considerable scatter, the average is quite close to r,/~, m 8. Carrington et LzI.(~) have derived an equation relating 7j and dislocation spacing. Using their values of dislocation spacing for iron, (about 1000 A) measured by means of electron microscope on thin films, i-f becomes approximately 700 g/mm”. This agrees fairly well with the value of 7, measured by Sullivan et al. (420 g/mm2).t3) In the case of polycrystals the number of dislocations in a pile-up N, may be calculated on the assumption that there is on the average only one source per grain operating. This is reasonable in view of the observations of Gay et aZ.(8) which indicate that there are about four sources per grain at 1 per cent deformation in many pure materials. The total plastic strain is then given by E = gANb (3) where 9 is the number of grains per unit volume, and A is the average cross sectional area of a grain. For strains of lo-* and a grain size of 0.25 mm, N = 20. The values of 7j shown for polycrystals in Table 2 were calculated on this assumption. The values of the jog formation energy are listed in Table 2. The values for copper are considerably higher than for aluminum, but the aluminum shows a greater effect of temperature. The values derived from the single crystal data are considered more reliable since fewer assumptions are involved. Considering the temperature dependence of ri (and thus E,:) shown in Table 1, it is evident that the temperature dependence is greater than that shown by the elastic constants. An almost identical temperature dependence was observed
VOL.
~~
10, 1962 TABLE 2. Jog formation energies .--____ __ - ._ .._.
-._
Material
j -.~
Cu
i
E Al
;
T-p
WI
1 ~&l-2) ..___~
Single crystals -__--.
.___
25 - 196 25 -196
I -
CU
Al Al
1
~_...
;:; 0:4 _.. . ..--
.____.
Folycrystals
cu
-.
~~
0.08 0.11 0.05 0.12
..^._.-
EAeV)
1
.____~~
25
0.41 -196 1 0.44 25 0.16 -196 0.45 / ~_______.__..._.. - ....-._ ..--
j
:i
~_
0:2 0.5
^ -._
for the stress required to produce an initial yield of 10V6, and the activation energy was associated with the energy required to change the stacking fault area.c6) Similarly, the energy of jog formation may be divided into two parts: (1) the energy required to constrict the partial dislocation and (2) the energy required to create a new segment of dislocation line. The energy required to create a new segment would be expected to have the same temperature dependence as the elastic constants which is small compared to the temperature dependence of the constriction of stacking faults.c6) It is thus concluded that the temperature dependence of the jog formation energy arises mainly from the constriction of partial dislocations. 4. SUMMARY The energy of jog formation has been inferred from the onset of plastic strain on the tensile unloading of single crystals after small plastic strains. The temperature dependence of the jog formation energy was associated with the constriction of partial dislocations during the jog forming process. ACKNOWLEDGMENTS
The authors would like to acknowledge the sponsorship of the Office of Naval Research. REFERENCES 1. N. TKOMPSON, C. K. COOGANand J. 0. RIDER, J. Inst. Met 84, 73 (1955). 2. M. J, HORDON and B. L. AVERBACH, Acta Met. 9, 237 (1961). 3. C. P. SULLIVAN, B. L. AVERBACH and M. COHEN, to be published.
4. J. M. ROBERTS and N. BROWN, Trans. Amer. Inst. &fin. (Metatl.) Engrs. 218, 454 (I 960). 5. J. D. ME~KIN and H. G. F. WILSDORF, Trans. Amer. Inst. Min. (Metall.) Engrs. 218, 745 (1960). 6. A. R. ROSENFIELD and B. L. AVERBACEI, Acta Met. 8, 624 (1960). 7. W. CAREINGTON, K. F. HALE and D. MCLEAN, PTOC. Roy. Sac. A259, 203 (1960). 8. P. GAY, P. B. HIRSCH and A. KELLY, Acta Cry&, Camb. 7, 41 (1954).