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The mechanism of the work-hardening in F.C.C. metals
THE
MECHANISM
OF
THE
WORK-HARDENING I.
IN
F.C.C.
METALS*
KOVACSf
It has been shown by Sseda (Thesis, Paris 1960) that the number of point defects produced during plastic deformation is proportional to the plastic work. In this paper a simple model is presented which leads to the proportionality not only between the plastic work and the point defect concentration but also between dislocation density as well. Experimental proof is presented that the work hardening process in Stages II and III is governed by the equation (derived theoretically)
dN T,, = a,,ub2 3 dY valid only for Stage II, while for Stage III it leads to
which leads to the well known connection r1 CCN”’ the expression r, cc N113. MECANISME
DE
LA
CONSOLIDATION
DANS
LES
METAUX
C.F.C.
Saads a montre (These, Paris 1960) que le nombre de defauts ponctuels produits pendant la deformaDans ce papier, on presente un modele simple qui tion plastique est proportionnel au travail plestique. conduit B la proportionnalite au travail plastique non seulement de la oonoentration en defauts ponctuels meis aussi de la densite de dislocations. On donne une preuve experimentale du fait que le processus de consolidation dans les stedes II et III est gouverne par l’equation (Btablie de facon theorique):
qui conduit 8. la relation bien connue rr cc N’12, valable seulement stade III, elle conduit rl l’expression 7, a N1’3.
DER
VERFESTIGUNGSMECHANISMUS
IN
pour le stade II, tandis que pour le
K.F.Z.
METALLEN
Saade hat gezeigt (Dissertation Paris 1960), dass die Zahl der bei plastischer Verformung erzeugten Punktfehler proportional zur plastischen Verformung ist. In dieser Arbeit wird ein einfaches Model1 vorgeschlagen, welches zu dieser Proportionalit& nicht nur zwischen plastisoher Verformung und Punktfehlerkonzentration, sondern euch zwischen letzterer und der Versetzungsdichte ftihrt. Es wird experimentell nachgewiesen, dass der Verfestigungsprozess in den Bereichen II und III durch die theoretisch hergeleitete Beziehung
dN T,, = ao,ub2,
dy
beschrieben wird, die zu der bekennten III den Ausdruck rr cc N1’B ergibt.
1.
Beziehung
rr cc N i’2 im Bereich II fiihrt, wiihrend sie fiir Bereich
INTRODUCTION
and the square of the Burgers vector, b2. The simplest
The purpose of this paper is to present a theoretical and experimental
choice off
r, = upbN”2.
description of the work hardening process both in Stages II and III. In this section, for the sake of comparison,
a brief summary
of the previous theories
is given. It can be shown by a dimensionality
argument that
the critical resolved shear stress, 7,. of a crystal can be expressed
(b2N)lj2 and in this case
isfm
treatment which leads to the uniform
All theories for the linear stage of the work hardening lead to this formula. It is assumed in the literature that it is valid for the subsequent, parabolic stage of the work hardening too. 3 and g.(i)
in the form(i)
The value of cc lies between
The only formula which gives a parabolic connection between
;
(2)
stress and strain is due to Taylor.(2)
consequences
= f(b2N),
from
this
formula
will
The
be discussed
briefly in the next paragraph. where p is the shear modulus, function
of the product
of the dislocation
* Received February 23, 1967; t Institute for Experimental University, Budapest. ACTA
METBLLURGICA,
and f is an arbitrary
VOL.
density,
N
revised April 6, 1967. Physics, LorBnd Eijtviis 15, NOVEMBER
1967
2.
DETERMINATION TION
DURING
OF
THE
PLASTIC
DEFECT
PRODUC-
DEFORMATION
Let a crystal be in a certain state of work hardening characterised by the shear stress 7,. The increase of 1731
ACTA
1732
METALLURGICA,
VOL.
15, 1967
this stress by drr gives rise to an increase in the macro-
of the jogs (the number of the intersections),
scopic
the number
strain, dy.
flow dislocation
During
sources are activated
and a great number These loops-because spread
the process of the plastic
out along
in the crystal
a mean
free path,
I, causing
an By means of equations we obtain
strainf3)
dy K blS2dn,
(3) and (5) and assumption
(3)
where oh is the increase of the number of the dislocation loops in unit volume. The increase of the disdensity is clearly
dn, = where A,
of a dislocation
in a slip plane
requires the passing through a “forest” crossing
its slip plane.
A, dy, iUb3 T,
is a constant.
The results obtained
movement
This expression
of dislocations
In the following
it will be
defects
experimental From
of view
of the work
that it is in agreement with equation
only
(i) The mean free path, distance,
this effect.
Two other
with equation
1, is proportional
to the
the dislocation
of the
1 between
(ii) The stress necessary to move the dislocations through the forest is inversely proportional to 1 (or I,)
equation
These assumptions defect concentration Using the expressions
lead to connections
between the
and the macroscopic
parameters.
(4) and (5) one can easily obtain
dN = ‘“T&J, Pb2
(6)
where A, is a proportionality factor. This equation shows that the increase of the dislocation density is proportional
to the increment
of the plastic
work.
For estimating
the value of A, one can take ,ub2 as the energy of a dislocation of unit length. A, gives therefore the rat’io of the stored energy to the plastic work, which, according to Ref. 4, is about 0.05, SO pb2 dN A, = ~ E 0.05. 77 dY A similar relationship
can be obtained
(7) for the point
defects generated during plastic deformation as well. In this case we have to take into account, that when dislocations intersect each other a jog is formed on them. The moving jogs generally produce a point defect in each successive step. The increase of their number must be proport’ional, therefore, to the number
(1) and coincides
(2) in the linear stage of hardening.
For the Stage III there is, however, some contradicCombining
(5)
hardening
It can be seen,
tion between the present model and Taylor’s
forest ;
Pb TrcC-. 1,
during
findings.@-@
the point
tion has to counteract
are made :
produced
plastic deformation are proportional to each other. This statement is in good agreement with previous
equation (6) seems to be fundamental.
average
was first
show that the number of the
and the point
supposed that the force required to move the dislocabasic assumptions
(1)
derived by Saada.c5) dislocations
The
to a
of dislocation loops are formed. of the barriers in the slip planes
increase in the macroscopic
location
and to
of the atomic steps corresponding distance 1,. So we can write
equations
theory.
(2) and (4) we find the Taylor’s
between stress and strain 7,. = a’,u(by/l,)1/2
This equation gives the experiment’ally observed parabolic law if, and only if, 1, is a constant. It is contrary to our assumption
given in equation
be quite improbable of the dislocations high dislocation
(5).
It seems to
indeed, that the mean free path should remain constant
densities
as found
up to so
in the parabolic
stage. In order to investigate
experimentally
the validity
of equation (6) in Stage III, it is necessary to measure, besides the stress-strain
curves, some other parameters
also, which could connect the variation of the dislocation density with the plastic work. This parameter may be, for instance, the change of the electrical resistivity during plastic deformation at low temperature.
If one accepts the resistivity of the point defects
and the dislocations to be additive, then the total resistivity change must be proportional to the plastic work also in Stage III if equation In the next described.
section
3. EXPERIMENTAL
The
stress-strain
(6) is valid there.
the experimental
and
method
is
METHOD
resistivity
change-strain
curves of polycrystalline f.c.c. metals (Au, Ag, Cu, Al) deformed by torsion were measured at liquid nitrogen temperature.
KOVACS:
The
torsional
deformation
sample is always concurrent
WORK
of
HARDENING
a polycrystalline In
IN
F.C.C.
the sample, which means that
with an elongation.@)
order to measure this elongation reliably a small tensile stress (less than half of the yield stress) was also applied. The total mean strain in such a sample can be given in t’he following formclO)
JW,
0,)
(9)
= ‘T
the effect
Differentiating
we have
= 7,(a, O,),
(12)
ad@,) =-_2&f
s
1
a3
This result shows that the torque measured statically
where (9 is the angle of the plastic torsion. quantit’y can be determined graphically.
that equation
T(a,@,) where T, = 7,1RstiC= p ~
d6’ ,, (1 + [AZ(6')/Zo]}3'2
The method of the resistivity measurement described previously. (lo) A large number
7(r, @,)r2 dr, (11)
of M(a, 0,).
0, (which can be done exactly),
and 0 is given by
0
proved
0
both sides of this equation in respect of a at a constant
where a,, and IOare the initial radius and length of the
@=
a4 = 2~ 1’
where ,u is the shear modulus and ~(0,) is the angle of the elastic torsion at which the elastic stresses compensate
wire, Al is the elongation
1733
METALS
This
has been of data
after the cessation of the plastic the flow stress
7
belonging
flow, directly
gives
to the strain yn = a . @,,
so we have
2WYa) T(Ya)= ~ 7ra3 .
(9) is valid in all the available
strain ranges independently from the ratio of the two terms on the right hand side.(lc~ll)
This conclusion
3.1 Determination
mination of the shape of the stress-strain
is not valid,
(13)
of course, for the flow
stresses inside the wire. Using this formula it can be proved that the deter-
of the stress-strain
curve by means of
torsional deformation It is necessary to deal somewhat the problem
of the stress-strain
more in detail with curve for torsional
deformation. In this case the measurable parameter is not the flow stress itself, but the torque at which the plastic torsion takes place. in t,he following
the elastic modulus does not depend on the amount of Then one can obtain for the stress incredeformation. ment from equation (12)
This torque can be written
Cl7
form
M(a, 0,)
-=--3 dT~
= 27~ a 7(r, @Jr2 dr, f0
(10)
a1A @ aA1 VA
where the index A denotes the data corresponding
where a is the external radius of the wire, 7 is the flow
elastic
assumption
is the angle per unit
modulus
length belonging to the plastic torsion previously applied. A theory for the calculation of the stress-
is a constant. Applying the usual a21 = aA 21A, we have (for p = const.) 7 -
To
TA
curve was worked
pp
pA
.
P
(14)
13’2’
out by N&dai.(12) He assumed that the strain varies
which is a linear connection
linearly along the radius of the wire, further that the flow stress at a given radius depends on the local strain
and the quantity p;!13f2. The angle v can be measured with good accuracy
only.
the following
On the basis of these assumptions
between
a connection
the flow stress and the torque-twist
to
a strain of ya lying in the range of strain in which the
stress at the radius r and 0,
strain curve from the torque-twist
curve doesn’t
require the measurement of the torque itself. Let us suppose that-at least in a certain range of strain-
curve
way.
Applying
between
the flow stress in
first a certain torque,
after the cessation of the plastic flow the removal
of
can be derived.
the elastic stresses in the specimen leads to a backlash
These assumptions, however, are not trivial. To avoid them, the following can be done. Let us suppose that by tjhe application of a given torque on a specimen the torsional plastic tlow takes place. After a certain amount, of deformation the flow will be stopped even
of the twisted wire. This backlash directly gives the angle 9 of the elastic torsion. The angle due to the backlash lies between 20” and 150”, which provides a
if the torque remains on the sample.
made in this way led to good agreement with the direct measurements.c11*13)
t,orque (10) is compensated
In this case the
by the elastic stresses of
good accuracy for the measurements (the error of the angle determination is less than 1’). Measurements
1734
ACTA 4. EXPERIMENTAL
4.1 Stress-strain
METALLURGICA,
VOL.
15, 1967
RESULTS
curves
The Stage I of the work hardening which is due to the easy glide cannot be expected to be observed in polycrystals,
since operation
of one glide system only
could not lead to the simultaneous deformation of all the grains. Figures 14
show the stress-strain
and
uniform
curves (choosing
yA = 0.2) for Cu, Au, Ag and Al (a curves).
The initial
parts of the curvesare also plotted to showtheexistence of the linear stage. parabolic
To prove
stage a r/r,, 2 -
the existence
specimens were annealed at temperatures figures for 3 hr (except time was 30min).
of the
yllz plot was used.
The
given on the FIG. 2. Stress-strain
Al, for which the annealing
The purity of the samples is 99.999 %
for Cu (supplied from Johnson and Matthey, and Al, while in 99.99 % for Au and Ag.
London)
curve of Au.
the connection between stress and strain can be written as r =
X2(72
+
for Stage II,
Y)
and in the parabolic
(15)
stage for Stage III,
?- = X&3 + Y1’2),
where x2, ys and x3, y3 are constants.
(16)
If equation (6) is
valid in these two stages, the resistivity-strain
connec-
tion can be written in the form AP
=
AP,(Y,
+
Sr)r
for Stage II,
(17)
AP
=
APAY,
+
$Y~‘~)Y
for Stage III
(18)
and
Figures 7(a, b) and 8(a, b) show the quantity 0
LO
a.5
20
15
FIG. 1. Stress-strain
a*
1
curve of Cu.
It can be seen from the figures that the parabolic connection
between stress and strain is valid only up to
a certain
limit of strain (of about 70-80x), which depends on the material. After this part of the stress strain curves, at very high strains a new, Stage IV does appear. As we shall see, the resistivity change shows also very clearly the existence of this stage. It is also
as a function of y112and y respectively. that
the
equations
resistivity-strain in definite
shows the connection
curves
strain
fulfil
intervals.
the
between resistivity
plastic work in the strain range 0 2 y 2 1 directly. This plot was obtained and stress-strain change-plastic existence
by using the resistivity-strain
curves. The linearity of the resistivity work
of equations
connection
together
with
curves
Simultaneously with the stress-strain determinations resistivity-strain curves were also measured. The results obtained for the same samples of the four metals
in the following
way.
In the linear stage
I 0
0,5
the
(17) and (18) means that the
of Ag.
constructed
7(c)
change and
in Stage IV than in Stage III, with the only exception
given in the previous section are shown on Figs. 5 and 6. To prove the validity of equation (6) and the existence of the different stages further plots were
above
Figure
a parabolic one with parameters different from those in Stage III. The “rate” of hardening, &/dy1f2 is less
4.2 Resistivity-strain
Aply
It can be seen
1,o
FIG. 3. Stress-strain
a
20 curve of Ag.
7’
KOVACS:
10
a.5
t5
FIG. 4. Stress-strain
defect production the equations the quantity
HARDENING
IN
F.C.C.
1735
METALS
a
20 curve of Al.
in Stages II and III is governed
by
(6) and (8) and proves the constancy
of
A, in this range.
In agreement these curves
WORK
with the stress-strain
measurements
also show the appearance
Stage IV approximately the resistivity-strain
above 0.8 strain.
of the new In this stage
curve becomes linear (Figs. 5 and
(cl FIG. 7. The connection between resistivity change and strain of Au, Ag, Cu in the different stages of the deformation. 5.
0
0,5
FIG. 5. Resistivity
6) and equation
(5
CO change-strain
J-
2.0
2s
curves of Au, Ag and Cu.
(6) loses its validity.
Similar obser-
vation has been made for the resistivity of point defects
CONCLUSIONS
The experimental facts presented in the previous sections prove that the work hardening process in face centred cubic metals in Stages II and III (which latter takes place in polycrystalline
in Al by Ceresara et uZ.(ll) Further measurements
have shown that the param-
eter Ap3 does not depend on the previous heat treatment and on impurity
content.
Figure
9 shows an
example for two Cu samples with strongly impurity contents.
0
0.5
FIG. 6. Resistivity
to change-strain
1.5
different
7,
=
dN q,,ub2 -
where cc,, is a constant linear stage
7
(a)
by
(19)
,
dY
equal to about
dy =
curve of Al.
metals at liquid nitrogen
temperature about up to 80% strain) is governed the equation
20.
$ dT+_’ q2
(b)
FIQ. 8. The same as in Fig. 7 for Al.
For the
ACT_% METALLURGICA,
1536
VOL.
rate.
15,
1967
With this expression
we have
where O,(O) = [Q02~,(0)] l12. Using the values of 0, = 13.5 kp/mm2 and ~~(0) = 16 kp/mm2 for copper(14), one gets O,(O) = 17 kp/mmz. The above results show that equation only in the Stage II. stress is approximately of the dislocation
(20) is valid
In Stage III the resolved shear proportional
density.
This
to the cube root conclusion
clearly
shows up the cause of the difficulties in Taylor’s t’heory. 0
LO
a5
FIB.
9.
where 0,
b5
P 0
65
;f
The same a8 in Fig. 7 for Cu samples with different impurity contents. is the rate of hardening.
With
this ex-
pression equation (19) leads to the following connection between resolved shear stress and dislocation
density
7,. = upU6N1j2,
(20)
is inversely proportional dislocations dislocations).
0,
112
( 1
tc==
2X,-
to the mean free path of the
In Stage II the activated
are the best orientated
of the applied stress. The distribution of the newly generated dislocations therefore is restricted to certain slip systems.
Shortly,
the distribution
in quite a good agreement III
results
between
stress and
equation
Eliminating
a
(5), equation
to Nw1j3. Inserting (21) is obtained.
(&,F)““. density from equations 0,
1. F.
4. 5. 6.
where 73 is the stress at which the Stage III begins to
9. 10.
appear. 0, is independent from the temperature, 73 depends on it in the following way’14)
12.
but
11.
13.
73 = T3(0)eeBT, on the strain
R. N.
NABARRO,
Z. S. BASINSKI
and
D.
B.
HOLT,
Adv. Phys. 13, 193 (1964). 2. G. I. TAYLOR, Proc. R. Sot. A145, 388 (1934). 3. A. H. COTTRELL, Dislocations and Plastic Flow in Crystals,
7. 8.
0,s = $02T3,
which depends
of this work. REFERENCES
(21)
(20) and (21) we obtain for the parameter
where B is a constant,
discussion
(19) yields
/3$F3Nli3,
the dislocation
the average intersecting
ACKNOWLEDGMENTS valuable
where
p =
and, therefore,
The author is grateful to Prof. Dr. E. Nagy for his
032 dy.
By means of this expression
The
generated in t’his stage
given slip plane will be proportional
be written as
2TrdTr ”
71. =
of the dislocations
this into equation
the connection
strain can approximately
slip systems will be activated.(l.14)
distribution
distance between the forest dislocations
with experimental
to
(20).
the possible
will be three dimensional
300
on single crystals.
In Stage
is proportional
In Stage III, however, cross slips take place and all
Taking as a mean value of u = 0.35 one obtains 02-1 p =
can be con-
In this case the average
N-l12, which leads directly to equation
P
slip systems
ones in respect to the direction
distance between forest dislocations
.
(20) and (21) (5) the stress
(or the average distance between t,he forest
sidered as two dimensional.
where
obtained
A simple interpretation for equations is the following. According to equation
14.
p. 17. Clarendon Press (1953). A. SEEKER and H. KRONM~~LLER, Phil. Mag. 7,897 (1962). G. SAADA, Thesis, Paris (1960). T. H. BLEWITT, R. R. COLTMAN and J. K. REDUN, Rep. Conf. on Defects in Crystalline Solids. The Physical Society (1957). H. R. PEIFFER, Acta Met. 11, 435 (1963). I. Koviics, E. NACY and P. FELTHAM, Phil. Mag. 9, 797 (1964). J. H. POYNTING, Proc. R. Sot. 80, 534 (1912). I. KovLcs and E. NACY, Phys. Status Solidi 8,795 (1965). S. CERESARA, H. ELKHOLY and T. FEDERIGHI, Phys. Status Solidi 2, 509 (1965). A. N~DAI, Theoy of Flow and Fracture of Solids, 2nd edition, Vol. I, p. 347. McGraw-Hill (1950). I. KovAcs and P. FELTRAM. Phvs. Status Solidi 3. 2379 (1963). A. SEEGER, Encyclopedia of Physics, Vol. VII/B, p. 156 Springer-Verlag (1958). ”
MECHANISM
OF
THE
WORK-HARDENING I.
IN
F.C.C.
METALS*
KOVACSf
It has been shown by Sseda (Thesis, Paris 1960) that the number of point defects produced during plastic deformation is proportional to the plastic work. In this paper a simple model is presented which leads to the proportionality not only between the plastic work and the point defect concentration but also between dislocation density as well. Experimental proof is presented that the work hardening process in Stages II and III is governed by the equation (derived theoretically)
dN T,, = a,,ub2 3 dY valid only for Stage II, while for Stage III it leads to
which leads to the well known connection r1 CCN”’ the expression r, cc N113. MECANISME
DE
LA
CONSOLIDATION
DANS
LES
METAUX
C.F.C.
Saads a montre (These, Paris 1960) que le nombre de defauts ponctuels produits pendant la deformaDans ce papier, on presente un modele simple qui tion plastique est proportionnel au travail plestique. conduit B la proportionnalite au travail plastique non seulement de la oonoentration en defauts ponctuels meis aussi de la densite de dislocations. On donne une preuve experimentale du fait que le processus de consolidation dans les stedes II et III est gouverne par l’equation (Btablie de facon theorique):
qui conduit 8. la relation bien connue rr cc N’12, valable seulement stade III, elle conduit rl l’expression 7, a N1’3.
DER
VERFESTIGUNGSMECHANISMUS
IN
pour le stade II, tandis que pour le
K.F.Z.
METALLEN
Saade hat gezeigt (Dissertation Paris 1960), dass die Zahl der bei plastischer Verformung erzeugten Punktfehler proportional zur plastischen Verformung ist. In dieser Arbeit wird ein einfaches Model1 vorgeschlagen, welches zu dieser Proportionalit& nicht nur zwischen plastisoher Verformung und Punktfehlerkonzentration, sondern euch zwischen letzterer und der Versetzungsdichte ftihrt. Es wird experimentell nachgewiesen, dass der Verfestigungsprozess in den Bereichen II und III durch die theoretisch hergeleitete Beziehung
dN T,, = ao,ub2,
dy
beschrieben wird, die zu der bekennten III den Ausdruck rr cc N1’B ergibt.
1.
Beziehung
rr cc N i’2 im Bereich II fiihrt, wiihrend sie fiir Bereich
INTRODUCTION
and the square of the Burgers vector, b2. The simplest
The purpose of this paper is to present a theoretical and experimental
choice off
r, = upbN”2.
description of the work hardening process both in Stages II and III. In this section, for the sake of comparison,
a brief summary
of the previous theories
is given. It can be shown by a dimensionality
argument that
the critical resolved shear stress, 7,. of a crystal can be expressed
(b2N)lj2 and in this case
isfm
treatment which leads to the uniform
All theories for the linear stage of the work hardening lead to this formula. It is assumed in the literature that it is valid for the subsequent, parabolic stage of the work hardening too. 3 and g.(i)
in the form(i)
The value of cc lies between
The only formula which gives a parabolic connection between
;
(2)
stress and strain is due to Taylor.(2)
consequences
= f(b2N),
from
this
formula
will
The
be discussed
briefly in the next paragraph. where p is the shear modulus, function
of the product
of the dislocation
* Received February 23, 1967; t Institute for Experimental University, Budapest. ACTA
METBLLURGICA,
and f is an arbitrary
VOL.
density,
N
revised April 6, 1967. Physics, LorBnd Eijtviis 15, NOVEMBER
1967
2.
DETERMINATION TION
DURING
OF
THE
PLASTIC
DEFECT
PRODUC-
DEFORMATION
Let a crystal be in a certain state of work hardening characterised by the shear stress 7,. The increase of 1731
ACTA
1732
METALLURGICA,
VOL.
15, 1967
this stress by drr gives rise to an increase in the macro-
of the jogs (the number of the intersections),
scopic
the number
strain, dy.
flow dislocation
During
sources are activated
and a great number These loops-because spread
the process of the plastic
out along
in the crystal
a mean
free path,
I, causing
an By means of equations we obtain
strainf3)
dy K blS2dn,
(3) and (5) and assumption
(3)
where oh is the increase of the number of the dislocation loops in unit volume. The increase of the disdensity is clearly
dn, = where A,
of a dislocation
in a slip plane
requires the passing through a “forest” crossing
its slip plane.
A, dy, iUb3 T,
is a constant.
The results obtained
movement
This expression
of dislocations
In the following
it will be
defects
experimental From
of view
of the work
that it is in agreement with equation
only
(i) The mean free path, distance,
this effect.
Two other
with equation
1, is proportional
to the
the dislocation
of the
1 between
(ii) The stress necessary to move the dislocations through the forest is inversely proportional to 1 (or I,)
equation
These assumptions defect concentration Using the expressions
lead to connections
between the
and the macroscopic
parameters.
(4) and (5) one can easily obtain
dN = ‘“T&J, Pb2
(6)
where A, is a proportionality factor. This equation shows that the increase of the dislocation density is proportional
to the increment
of the plastic
work.
For estimating
the value of A, one can take ,ub2 as the energy of a dislocation of unit length. A, gives therefore the rat’io of the stored energy to the plastic work, which, according to Ref. 4, is about 0.05, SO pb2 dN A, = ~ E 0.05. 77 dY A similar relationship
can be obtained
(7) for the point
defects generated during plastic deformation as well. In this case we have to take into account, that when dislocations intersect each other a jog is formed on them. The moving jogs generally produce a point defect in each successive step. The increase of their number must be proport’ional, therefore, to the number
(1) and coincides
(2) in the linear stage of hardening.
For the Stage III there is, however, some contradicCombining
(5)
hardening
It can be seen,
tion between the present model and Taylor’s
forest ;
Pb TrcC-. 1,
during
findings.@-@
the point
tion has to counteract
are made :
produced
plastic deformation are proportional to each other. This statement is in good agreement with previous
equation (6) seems to be fundamental.
average
was first
show that the number of the
and the point
supposed that the force required to move the dislocabasic assumptions
(1)
derived by Saada.c5) dislocations
The
to a
of dislocation loops are formed. of the barriers in the slip planes
increase in the macroscopic
location
and to
of the atomic steps corresponding distance 1,. So we can write
equations
theory.
(2) and (4) we find the Taylor’s
between stress and strain 7,. = a’,u(by/l,)1/2
This equation gives the experiment’ally observed parabolic law if, and only if, 1, is a constant. It is contrary to our assumption
given in equation
be quite improbable of the dislocations high dislocation
(5).
It seems to
indeed, that the mean free path should remain constant
densities
as found
up to so
in the parabolic
stage. In order to investigate
experimentally
the validity
of equation (6) in Stage III, it is necessary to measure, besides the stress-strain
curves, some other parameters
also, which could connect the variation of the dislocation density with the plastic work. This parameter may be, for instance, the change of the electrical resistivity during plastic deformation at low temperature.
If one accepts the resistivity of the point defects
and the dislocations to be additive, then the total resistivity change must be proportional to the plastic work also in Stage III if equation In the next described.
section
3. EXPERIMENTAL
The
stress-strain
(6) is valid there.
the experimental
and
method
is
METHOD
resistivity
change-strain
curves of polycrystalline f.c.c. metals (Au, Ag, Cu, Al) deformed by torsion were measured at liquid nitrogen temperature.
KOVACS:
The
torsional
deformation
sample is always concurrent
WORK
of
HARDENING
a polycrystalline In
IN
F.C.C.
the sample, which means that
with an elongation.@)
order to measure this elongation reliably a small tensile stress (less than half of the yield stress) was also applied. The total mean strain in such a sample can be given in t’he following formclO)
JW,
0,)
(9)
= ‘T
the effect
Differentiating
we have
= 7,(a, O,),
(12)
ad@,) =-_2&f
s
1
a3
This result shows that the torque measured statically
where (9 is the angle of the plastic torsion. quantit’y can be determined graphically.
that equation
T(a,@,) where T, = 7,1RstiC= p ~
d6’ ,, (1 + [AZ(6')/Zo]}3'2
The method of the resistivity measurement described previously. (lo) A large number
7(r, @,)r2 dr, (11)
of M(a, 0,).
0, (which can be done exactly),
and 0 is given by
0
proved
0
both sides of this equation in respect of a at a constant
where a,, and IOare the initial radius and length of the
@=
a4 = 2~ 1’
where ,u is the shear modulus and ~(0,) is the angle of the elastic torsion at which the elastic stresses compensate
wire, Al is the elongation
1733
METALS
This
has been of data
after the cessation of the plastic the flow stress
7
belonging
flow, directly
gives
to the strain yn = a . @,,
so we have
2WYa) T(Ya)= ~ 7ra3 .
(9) is valid in all the available
strain ranges independently from the ratio of the two terms on the right hand side.(lc~ll)
This conclusion
3.1 Determination
mination of the shape of the stress-strain
is not valid,
(13)
of course, for the flow
stresses inside the wire. Using this formula it can be proved that the deter-
of the stress-strain
curve by means of
torsional deformation It is necessary to deal somewhat the problem
of the stress-strain
more in detail with curve for torsional
deformation. In this case the measurable parameter is not the flow stress itself, but the torque at which the plastic torsion takes place. in t,he following
the elastic modulus does not depend on the amount of Then one can obtain for the stress incredeformation. ment from equation (12)
This torque can be written
Cl7
form
M(a, 0,)
-=--3 dT~
= 27~ a 7(r, @Jr2 dr, f0
(10)
a1A @ aA1 VA
where the index A denotes the data corresponding
where a is the external radius of the wire, 7 is the flow
elastic
assumption
is the angle per unit
modulus
length belonging to the plastic torsion previously applied. A theory for the calculation of the stress-
is a constant. Applying the usual a21 = aA 21A, we have (for p = const.) 7 -
To
TA
curve was worked
pp
pA
.
P
(14)
13’2’
out by N&dai.(12) He assumed that the strain varies
which is a linear connection
linearly along the radius of the wire, further that the flow stress at a given radius depends on the local strain
and the quantity p;!13f2. The angle v can be measured with good accuracy
only.
the following
On the basis of these assumptions
between
a connection
the flow stress and the torque-twist
to
a strain of ya lying in the range of strain in which the
stress at the radius r and 0,
strain curve from the torque-twist
curve doesn’t
require the measurement of the torque itself. Let us suppose that-at least in a certain range of strain-
curve
way.
Applying
between
the flow stress in
first a certain torque,
after the cessation of the plastic flow the removal
of
can be derived.
the elastic stresses in the specimen leads to a backlash
These assumptions, however, are not trivial. To avoid them, the following can be done. Let us suppose that by tjhe application of a given torque on a specimen the torsional plastic tlow takes place. After a certain amount, of deformation the flow will be stopped even
of the twisted wire. This backlash directly gives the angle 9 of the elastic torsion. The angle due to the backlash lies between 20” and 150”, which provides a
if the torque remains on the sample.
made in this way led to good agreement with the direct measurements.c11*13)
t,orque (10) is compensated
In this case the
by the elastic stresses of
good accuracy for the measurements (the error of the angle determination is less than 1’). Measurements
1734
ACTA 4. EXPERIMENTAL
4.1 Stress-strain
METALLURGICA,
VOL.
15, 1967
RESULTS
curves
The Stage I of the work hardening which is due to the easy glide cannot be expected to be observed in polycrystals,
since operation
of one glide system only
could not lead to the simultaneous deformation of all the grains. Figures 14
show the stress-strain
and
uniform
curves (choosing
yA = 0.2) for Cu, Au, Ag and Al (a curves).
The initial
parts of the curvesare also plotted to showtheexistence of the linear stage. parabolic
To prove
stage a r/r,, 2 -
the existence
specimens were annealed at temperatures figures for 3 hr (except time was 30min).
of the
yllz plot was used.
The
given on the FIG. 2. Stress-strain
Al, for which the annealing
The purity of the samples is 99.999 %
for Cu (supplied from Johnson and Matthey, and Al, while in 99.99 % for Au and Ag.
London)
curve of Au.
the connection between stress and strain can be written as r =
X2(72
+
for Stage II,
Y)
and in the parabolic
(15)
stage for Stage III,
?- = X&3 + Y1’2),
where x2, ys and x3, y3 are constants.
(16)
If equation (6) is
valid in these two stages, the resistivity-strain
connec-
tion can be written in the form AP
=
AP,(Y,
+
Sr)r
for Stage II,
(17)
AP
=
APAY,
+
$Y~‘~)Y
for Stage III
(18)
and
Figures 7(a, b) and 8(a, b) show the quantity 0
LO
a.5
20
15
FIG. 1. Stress-strain
a*
1
curve of Cu.
It can be seen from the figures that the parabolic connection
between stress and strain is valid only up to
a certain
limit of strain (of about 70-80x), which depends on the material. After this part of the stress strain curves, at very high strains a new, Stage IV does appear. As we shall see, the resistivity change shows also very clearly the existence of this stage. It is also
as a function of y112and y respectively. that
the
equations
resistivity-strain in definite
shows the connection
curves
strain
fulfil
intervals.
the
between resistivity
plastic work in the strain range 0 2 y 2 1 directly. This plot was obtained and stress-strain change-plastic existence
by using the resistivity-strain
curves. The linearity of the resistivity work
of equations
connection
together
with
curves
Simultaneously with the stress-strain determinations resistivity-strain curves were also measured. The results obtained for the same samples of the four metals
in the following
way.
In the linear stage
I 0
0,5
the
(17) and (18) means that the
of Ag.
constructed
7(c)
change and
in Stage IV than in Stage III, with the only exception
given in the previous section are shown on Figs. 5 and 6. To prove the validity of equation (6) and the existence of the different stages further plots were
above
Figure
a parabolic one with parameters different from those in Stage III. The “rate” of hardening, &/dy1f2 is less
4.2 Resistivity-strain
Aply
It can be seen
1,o
FIG. 3. Stress-strain
a
20 curve of Ag.
7’
KOVACS:
10
a.5
t5
FIG. 4. Stress-strain
defect production the equations the quantity
HARDENING
IN
F.C.C.
1735
METALS
a
20 curve of Al.
in Stages II and III is governed
by
(6) and (8) and proves the constancy
of
A, in this range.
In agreement these curves
WORK
with the stress-strain
measurements
also show the appearance
Stage IV approximately the resistivity-strain
above 0.8 strain.
of the new In this stage
curve becomes linear (Figs. 5 and
(cl FIG. 7. The connection between resistivity change and strain of Au, Ag, Cu in the different stages of the deformation. 5.
0
0,5
FIG. 5. Resistivity
6) and equation
(5
CO change-strain
J-
2.0
2s
curves of Au, Ag and Cu.
(6) loses its validity.
Similar obser-
vation has been made for the resistivity of point defects
CONCLUSIONS
The experimental facts presented in the previous sections prove that the work hardening process in face centred cubic metals in Stages II and III (which latter takes place in polycrystalline
in Al by Ceresara et uZ.(ll) Further measurements
have shown that the param-
eter Ap3 does not depend on the previous heat treatment and on impurity
content.
Figure
9 shows an
example for two Cu samples with strongly impurity contents.
0
0.5
FIG. 6. Resistivity
to change-strain
1.5
different
7,
=
dN q,,ub2 -
where cc,, is a constant linear stage
7
(a)
by
(19)
,
dY
equal to about
dy =
curve of Al.
metals at liquid nitrogen
temperature about up to 80% strain) is governed the equation
20.
$ dT+_’ q2
(b)
FIQ. 8. The same as in Fig. 7 for Al.
For the
ACT_% METALLURGICA,
1536
VOL.
rate.
15,
1967
With this expression
we have
where O,(O) = [Q02~,(0)] l12. Using the values of 0, = 13.5 kp/mm2 and ~~(0) = 16 kp/mm2 for copper(14), one gets O,(O) = 17 kp/mmz. The above results show that equation only in the Stage II. stress is approximately of the dislocation
(20) is valid
In Stage III the resolved shear proportional
density.
This
to the cube root conclusion
clearly
shows up the cause of the difficulties in Taylor’s t’heory. 0
LO
a5
FIB.
9.
where 0,
b5
P 0
65
;f
The same a8 in Fig. 7 for Cu samples with different impurity contents. is the rate of hardening.
With
this ex-
pression equation (19) leads to the following connection between resolved shear stress and dislocation
density
7,. = upU6N1j2,
(20)
is inversely proportional dislocations dislocations).
0,
112
( 1
tc==
2X,-
to the mean free path of the
In Stage II the activated
are the best orientated
of the applied stress. The distribution of the newly generated dislocations therefore is restricted to certain slip systems.
Shortly,
the distribution
in quite a good agreement III
results
between
stress and
equation
Eliminating
a
(5), equation
to Nw1j3. Inserting (21) is obtained.
(&,F)““. density from equations 0,
1. F.
4. 5. 6.
where 73 is the stress at which the Stage III begins to
9. 10.
appear. 0, is independent from the temperature, 73 depends on it in the following way’14)
12.
but
11.
13.
73 = T3(0)eeBT, on the strain
R. N.
NABARRO,
Z. S. BASINSKI
and
D.
B.
HOLT,
Adv. Phys. 13, 193 (1964). 2. G. I. TAYLOR, Proc. R. Sot. A145, 388 (1934). 3. A. H. COTTRELL, Dislocations and Plastic Flow in Crystals,
7. 8.
0,s = $02T3,
which depends
of this work. REFERENCES
(21)
(20) and (21) we obtain for the parameter
where B is a constant,
discussion
(19) yields
/3$F3Nli3,
the dislocation
the average intersecting
ACKNOWLEDGMENTS valuable
where
p =
and, therefore,
The author is grateful to Prof. Dr. E. Nagy for his
032 dy.
By means of this expression
The
generated in t’his stage
given slip plane will be proportional
be written as
2TrdTr ”
71. =
of the dislocations
this into equation
the connection
strain can approximately
slip systems will be activated.(l.14)
distribution
distance between the forest dislocations
with experimental
to
(20).
the possible
will be three dimensional
300
on single crystals.
In Stage
is proportional
In Stage III, however, cross slips take place and all
Taking as a mean value of u = 0.35 one obtains 02-1 p =
can be con-
In this case the average
N-l12, which leads directly to equation
P
slip systems
ones in respect to the direction
distance between forest dislocations
.
(20) and (21) (5) the stress
(or the average distance between t,he forest
sidered as two dimensional.
where
obtained
A simple interpretation for equations is the following. According to equation
14.
p. 17. Clarendon Press (1953). A. SEEKER and H. KRONM~~LLER, Phil. Mag. 7,897 (1962). G. SAADA, Thesis, Paris (1960). T. H. BLEWITT, R. R. COLTMAN and J. K. REDUN, Rep. Conf. on Defects in Crystalline Solids. The Physical Society (1957). H. R. PEIFFER, Acta Met. 11, 435 (1963). I. Koviics, E. NACY and P. FELTHAM, Phil. Mag. 9, 797 (1964). J. H. POYNTING, Proc. R. Sot. 80, 534 (1912). I. KovLcs and E. NACY, Phys. Status Solidi 8,795 (1965). S. CERESARA, H. ELKHOLY and T. FEDERIGHI, Phys. Status Solidi 2, 509 (1965). A. N~DAI, Theoy of Flow and Fracture of Solids, 2nd edition, Vol. I, p. 347. McGraw-Hill (1950). I. KovAcs and P. FELTRAM. Phvs. Status Solidi 3. 2379 (1963). A. SEEGER, Encyclopedia of Physics, Vol. VII/B, p. 156 Springer-Verlag (1958). ”