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A model for steady state creep based on the motion of jogged screw dislocations
A MODEL
FOR STEADY STATE CREEP BASED ON THE MOTION OF JOGGED SCREW DISLOCATIONS* C. R.
BARRETT?
and
W.
D. NIXt
A theory of steady state creep is presented which is based on the diffusion controlled motion of jogged screw dislocations. Steady state creep is assumed to exist when the chemical force on the jogs is balanced by the force on the dislocation due to the applied stress. The resulting expression for the steady state creen rate is
where psis the mobile sorew disloc&tiondensity, 23 the self-diffusion c~ffi~ient, OL the number of atoms per unit cell, b the Burgers vector, op the lattice parameter, and /z is the average spacing between jogs. In order to compare this equation with experimental results, the stress dependence of ps is determined for an Fe + 3.0% Si alloy. The following expression is obtained:
Using this result, the above expression for B, can accurately predict the observed stress dependence and can give reasonable agreement with absolute creep rates. The predictions of this theory are compared with Garofelo’s empirical expression for steady state creep. UN MODELE
DE FLUAGE A VITESSE CONSTANTE BASE SUR LE DEPLACEMENT DES DISLOCATIONS-VIS AVEC CRANS
Les auteurs presentent une theorie du fluage a vitesse con&ante basee sur le d&placement, contrcilepar diffusion, des dislocations-vis avec trans. Ils admettent que le fluage a vitesse constante s’observe quand les forces de nature chimique qui s’exercent sur les dislocations du fait des contraintes appliquties. 11s en derivent une expression decrivant le fluage it vitesse eroissante: d, = 2PpaDor($)”
sinh (g),
ob ps est la densite en dislocations-vis mobiles, D le coefficient d’autodiffusion, a le nombre d’atomes parcellule unitaire, h le vecteur de Burgers, c(,,le parametre reticulaire et I la distance moyenne entre les trans. Dans le but de comparer oette relation aux r&&tats experimentaux on a determine la loi de variation de ps avec la tension appliquee pour un alliage Fe + 3.0 oASi. On a obtenu la relation suivante:
Partant de oe r&&tat, la relation donnant 2, permet de prevoir la loi de variation en fonction de la tension appliquee, en dormant un accord raisonnable avec les vitesses de flusge absolues. Les resultats de la prbente theorie sont compares a la relation empirique de Garofalo. EIN
MODELL FUR STATION~RES KRIECHEN, AUSGEHE~D VON DER BEWEGUNG VON SCH~AUBENVERSETZUNGEN MIT SPRttNGEN
Es wird eine Theorie entwickelt fur das station&e Kriechen, die die diffusionkontrollierte Bewegung von Schraubenversetzungen, die Spriinge enthalten, zugrunde legt. Es wird angenommen, da6 stationares Kriechen dann stattfindet, wenn die auf die Spriinge wirkende chemische Kraft der von der aul3eren Spannung auf die Versetzung ausgeiibten Kraft das Gleichgewicht halt. Fur die stationare Kriechgesohwindigkeit ergibt sich folgender Ausdruck: d, = 2rrp,Dsr(irsinh
(g),
wo ps die Dichte der bewegliehen Schraubenversetzungen, D der Selbstdiffusionskoeffizient, 0: die Zahl der Atome in der Elementarzelle, 6 der Burgersvektor, a, die Gitterkonstante und 3, die mittlere Entfernung der Spriinge ist. Urn dies%Gleichung mit den experimentellen Ergebnissen zu vergleichen, wird di.;s~~;;ungsabhangigkeit von ps fiir eine Fe + 3.0 % Si-Legierung bestimmt. Man erhalt folgenden
Unter Beniitzung dieses Resultates gestattet obiger Ausdruck fur 6,, die beobachtete Spannungsabhangigkeit richtig vorherzusagen und sine veriinftige Ubereinstimmung mit den absoluten Kriechgeschwindigkeiten zu ergeben. Die Vorhersagen dieser Theorie werden mit dem empirischen Ausdruck von Garofalo fur station&es Kriechen verglichen. * Received October 6, 1964; revised April 23, 1965. t Department of Materials Science, Stanford University, Stanford, California. ACTA META~LURGICA,
VOL. 13, DECEMBER
1966
1247
1248
ACTA
METALLURGICA,
1. INTRODUCTION
(1) where is is the steady state tensile strain rate, v the vibrational frequency at the jog, b the Burgers vector, d the distance between active slip planes, Qsd the activation energy for self diffusion, o the applied stress, k Boltzmann’s constant and T is the absolute temperature. Raymond and Dornt2) in a recent paper re-analyzed the thermally activated motion of jogged screw dislocations employing the same physical model used by Mott and incor~rat~g the concepts presented by Hirsch and Warringtonts) and Friedel.c4) They obtain an expression for the strain rate of the form b*(z h
1)v
pSexp (-Q,,/kT)
13,
1965 2. THEORY
Mott(l) was the first investigator to formulate a theory of high ~rn~rature creep based on the motion of jogged screw dislocations. Assuming jogs to be one atomic distance in height and the distance between jogs to be L, he proposed that the steady state creep rate be given by
PS =
VOL.
sinh (rb2A/hkT),
(2) where pS is the shear strain rate, z the coordination number, h the mean height of a moving jog, r the applied shear stress and ps is the mobile screw dislocation density. In order to make an explicit prediction of the stress and temperature dependence from equation (2) it is necessaryto evaluate ps, A and v. None of the parameters can be easily determined. While the concept of steady state creep controlled by the non-conservative motion of jogs on screw dislocations is attractive, insofar as it does not require the existence of the hypothetical dislocation arrangements present in some creep theories,(5*6) at present it is not possible to compare accurately the previous theoretical treatments with the experimental data. The purpose of the present paper is to present a calculation of the steady state creep rate based on the motion of jogged screw dislocations which can be more easily compared with experiment. In this treatment the quantity Y does not appear. This leaves ps and 1 as the parameters which must be evaluated. experimental evidence is presented for the variation of ps with temperature and applied stress. The final expression for the creep rate involves but one parameter which cannot be measured, namely A. Theoretical values of il are suggested.
When a jog in a screw dislocation does not lie in the slip plane, it is called a non-Conservative jog in that it can move with the gliding screw dislocation only by the emission or absorption of point defects (vacancies or interstitials). We may classify a non-conservative jog as one of two types: a vacancy-emitting jog or a vacancy-absorbing jog. In this treatment we consider both types of jogs. Consider a screw dislocation which contains a single vacancy-emitting jog. When the screw dislocation moves in response to a shear stress, the jog can maintain its position in the screw dislocation only by the emission of vacancies or by the absorption of interstitids. Since the concentration of interstitials existing in thermodynamic equilibrium is negligible, we can limit our attention to the case of vacancy emission. As vacancies are emitted by the jog, the local vacancy concentration (near the jog) is increased. If the non-Conservative jog motion is discontinued (by the removal of the applied shear stress, the vacancy concentration in the vicinity of the jog returns to the equilibrium value. However, continued motion of the jog enhances the local supersaturation of vacancies. The vacancy concentration near the jog is controlled by both the rate at which vacancies are being produced and the rate at which vacancies move away by diffusion. Since the jog in a screw dislocation can be considered as a very short segment of edge dislocation, the retarding force on the jog caused by the vacancy supersaturation can be computed as a chemical force. When the chemical force (dragging force) is equal to the driving force (applied stress), a steady state velocity is attained. This assumes that the lattice friction stress is negligible. We now consider the vacancy-absorbing jog. In general, there are two ways in which this jog can maintain its position in a gliding screw dislocation: by absorbing vacancies or by emitting interstitial& Friedel(7) has shown, however, that it is energetically more favorable for the jog to glide conservatively along the dislocation rather than move non-conservatively by the emission of interstitials. Therefore, we will limit our discussion to the ease of vacancy absorption. For this case, as in the previous, the diffusion controlled steady state velocity can be computed with reference to the chemical force. In this situation, however, there is a depression in the vacancy concentration near the jog which gives rise to a dragging force on the jog. In this treatment the creep rate is expressed aa 9s =
p@,
(3)
BARRETT
AND NIX:
CREEP
BY
MOTION
where 9, is the steady state shear strain rate, ps is the density of mobile screw dislocations, velocity
of the dislocations
and
b is the Burgers
Consider a segment of screw dislocation distance
in
a vacancy-producing
height.
The
exerted on the dislocation
JOGGED
express C,* as
chemical
dragging
force
exp
V
c 3)*-c,=~
-
.&*
hD,b
of length il
jog, one atomic
1249
DISLOCATIONS
G is the average
vector. which contains
OF
[E
+
2/F
d(P
+
+)I]
+ r2)
’
(6) and C,* as
by the jog is(*)
exp
c,* -co=
-A
f@=pln!$,
(
-
5 2D
&
T 2)
r2)1 1
[5 + 1/(E” +
’
+ 9)
(7)
0
where C, is the concentration
of vacancies in the vicin-
ity of the jog and Co is the equilibrium centration. velocity For
vacancy
C, and therefore f, are dependent
of the vacancy a similar
vacancy-absorbing
producing
dislocation
conon the
jog vV.
line which
contains
a
jog, the dragging force imposed by
the jog is In 2,
(5)
a where
C,
is the
absorbing
vacancy
the steady
concentration
respectively.
near
to compute
C,*
and
concentrations
near
and vacancy-absorbing
jogs
In general, the diffusion of vacancies
i.e., bulk diffusion core diffusion.
on the relative
which these two processes bilities are considered
in the matrix
to
occur.
by bulk diffusion
diffusion
in the matrix is rate controlling: in the matrix
when bulk (1) when
is so fast that the problem of a point
in an essentially
jog and diffusion
source
homogeneous
has predominantly
of vacancies
or sink of media and one type of
along the dislocation
core is much more rapid than lattice diffusion the problem reduces to the motion sink of vacancies.
so that
of a line source or
(1) Rapid bulk &f&ion. Formally, the problem of a moving point source of vacancies in an infinite homogeneous medium is identical with the moving point source heat flow problem originally treated by Rosenthal.cg) Using Rosenthal’s result it is possible to 3
distances
from
The boundary
concentration
condition
at infinitely
Since we are interested in a computation
chemical
large
the jog is equal to the equilibrium
force on the moving
concerned withC,*
andC,*
of the
jogs, we are primarily
at r = 0 and t = -b.
(just behind the moving
c * -c,=L
At
jogs) we have
&Deb2
and
c,* Strictly only
speaking,
for
-
-2% .
4rrD,b2
equations
(8) and (9) are valid
jogs moving in an infinite homogeneous A more realistic boundary condition, for
medium. example, vacancy
(Jo =
might
involve
concentration
the requirement
moving jog reach the equilibrium validity separate
of equation
value.
to
examine
condition
The solution
To estimate the
on the vacancy
tion near a source or sink. example,
the
from the the
(8) and (9) one can carry out a
computation
finite boundary
that
at a finite distance
for this case has not been found.
diffusion
(2) when a dislocation
rates with
The various possi-
There are two cases to be considered
vacancies
of
below.
2.1 Diffusion occurring~redominantly
to the motion
and dis-
Thus the exact calculation
and C,* will depend
reduces
is that the vacancy
the
or away from jogs will take place by two different
C,*
point source or sink for vacancies moving in a straight line in an infinite medium.
9
state vacancy
the vacancy-producing
processes,
These results
are obtained by treating the moving jog as if it were a
jog.
creep rate, it is first necessary
location
and D, is the
this position
In order to develop an equation for the steady state C,*;
where l and r are cylindrical
value. f, = y
vacancy
coordinates
lattice diffusion coefficient for vacancies.
that the vacancy
effect
of
a
concentra-
One can easily show, for concentration
C(r) near a
point vacancy source, fixed at the center of a sphere of changed when one radius ro, is not significantly changes
from
(r. > 20r). equations
infinite With
to finite boundary
this
reasoning
(8) and (9) to be valid
though the vacancy concentration equilibrium value at a finite distance. Substituting equations respectively we find
we
conditions may
expressions returns
expect even to
an
(8) and (9) into (4) and (5)
ACTA
1250
METALLURGICA,
VOL.
13,
1965
assumed that no conservative
and f, = -
y
In [I -
4 :bsc]
.
=. Steady state dislocation
(11)
0
The steady bining
velocities are attained when
the net force on the dislocation
motion of the jogs takes
place. t
line is zero.
This con-
state creep rate is obtained
equations
(3), (14),
expression
can be obtained
(12)
and 7bA = f,,
c, = and
r
is the
equations
applied
JL
(17)
XQ,
f-h3
D = xODv,
(13) stress.
shear
With
(18)
vacant sites at equilibrium, The following
= 4rD,b2CQ uup
c&=1, [exp
(g)
-
I] ,
,=4-DVb2C,[I-exp(-g)].
v, > v,
(15)
although
for
small
values
dislocation
producing tension
line will not
absorbing
remain
straight.
jogs will lag behind
when LX,= 0.5, cc9 = 0.5
dislocation
modifies
For purposes
is assumed that the average velocity containing
the force
of calculation
it
of a dislocation
both types of jogs is
Gyp
+
When a large fraction assumed
(equation
of vacancy
(16)
absorbing
ofvacancy
jogs respectively.
producing As a result
of the different jog velocities, the screw dislocation will not remain in the pure screw orientation but will assume the character
of a general dislocation
the regions between the jogs. For a screw dislocation with both
types
line in of jogs,
unless the jogs are equally spaced and alternate along the dislocation line, the bowing of the dislocation will, in general, give rise to a force on the jogs tending to move them conservatively along the dislocation line. If this force is sufficient to overcome the friction stress for conservative motion, then jogs may coalesce and either disappear or form jogs of several atomic distances
in
height.
For
the
(20)
present
case,
it is
jogs is
of the creep
rate with the applied stress becomes less pronounced as the stress increases, a result which is clearly in disaccord
with the experimental
absorbing
findings.(11-13)
dependence. attention
equal to the number of vacancy
In the following
to equations
(2) Rapid dislocation present.
For
the
When
jogs is either greater
jogs, the stress dependence
state creep rate is qualitatively and vacancy
absorbing
(19)), the variation
the fraction of vacancy producing
u,v,
where up and u, are the fractions
,
and when u, = 0, up = 1
than or approximately B=
(g)
The
balance on the different jogs and they move with the same overall velocity.
p,sinh
the vacancy
jogs until the added force due to the line
of the bowing
(19)
of
rb2AIkT, v, m v,. Thus if both types of jogs exist on a dislocation line, and the jogs are equally spaced, vacancy
when
a,=0
+, = 4rrDp (:r
the
coefficient.
general cases can be described;
~~-4~D/?(~~p,[I-exp(-~)],
general,
of
(14)
and
In
x,, the fraction
/l the number of atoms per
unit cell and D is the lattice self diffusion
as
are written
the jog velocities
these
in the final
if we note that
where a, is the lattice parameter, where
by com-
(16) for given
values of M, and CC,. Some simplification
dition is given by rbjZ = f,
(15) and
of the steady
similar to the observed sections
we limit our
(20) and (21). core diffusion,
calculation
of
one type of jog the
steady
state
vacancy sink of
concentration near a moving line source or vacancies we may again draw upon the Assuming an analogous heat flow calculation.“*) _FIn the present treatment the forces between jogs, which tend to move the jogs conservatively along the dislocation line, are ignored. This is in accord with Friedel’s postulate”” that the elastic distortions produced by jogs one atomic distance in height, are masked at distances greater than a few atomic dimensions by the stress field of the rest of the dislocation line. Thus, for the present study in which jogs of height b separated by a distance A(1 > b) are considered, the interjog forces should be negligible compared to the friction stress for conservative motion.
BARRETT
infinite homogeneous centrations
NIX:
AND
medium,
CREEP
the steady
BY
MOTION
state con-
right behind the jog are
c,* - c, =
OF
JOGGED
(20), however,
as the stress and temperature
ence predicted
by equations
identical
with that predicted stress
the
and
sinh (2) M &[exp (z) conclusions ture (23)
where K,(x)
is the modified
Bessel function
second kind of order zero. can be simplified
Equations
by noting
of the
(22) and (23)
that in nearly
all cases
1. Thus we can write
vb/2D, <
dependence
applicable
Dislocation
(20)
are
generally
(21) and (26) as well.
core diffusion may be important
for the
emitting.
absorbing
and vacancy
the dislocation
source
For this
can diffuse to or away from the jogs or through
the lattice.
of C,* and C,* thus entails calculating concentration
near a moving
or sink in a composite
system
of this type
is extremely
not attempted
-
(Z)
(24)
1 > 1
[I-exp(-g)],
(25)
w should of stress and temperature
mic nature.
As in the previous
be
only
a weak
due to its logarithcase v, > V, except
when rb2AlkT is small. Equation
(25)
leads
similar to equation
conditions
sidered
further.
combined
Equation
with equation
lis =
a creep
rate
2rrDfib2A a03w p
(24),
however,
be
assumed, and is
Even though C 2,* and C,* cannot be calculated it is dependence
of the jog
in this case will differ from that of the two
previous
models.
dislocation
That is, the activation
motion
energy for
may be closer to that for disloca-
(g)
and
bulk diffusion in self diffusion studies, dislocation
core
diffusion should become more important in controlling the jog velocity
as the temperature
of this observation
is lowered.
The
will be discussed in a
later section. 3. EXPERIMENTAL
RESULTS
In order to compare
-
I]
(26)
calculate known.
pre-exponential
Analo-
between grain boundary
AND
DISCUSSION
the stress dependence
the results of experiment, ps [exp
similar to that
(21) and (26) are identical
of bm/l in the
gous to the competition
of the
steady state creep rate predicted by equation (20) with
in practice.
Equations factor
can
(3) to give
which has a general stress dependence observed
expression
(19) and hence will not be con-
A
in the present study.
apparent that the temperature
implications to
(i.e., a
complicated,
tion core diffusion rather than bulk diffusion.
function
any
vacancy
velocity 2rD,bE, w,
equation
regardless of the boundary
Thus vl, and v, can be expressed as
vu,=
Thus
the stress or tempera-
line are alternately
calculation
w),
2 and
system with two regions of different diffusivities).
N 1.
and
than
all 2 > 2.
i.e.,
interest
case where jogs on the dislocation
point
[cap
for
the steady state vacancy
and
of
2.2 Dislocation core diffusion
either along
2n-D,bilC0
of
to equations
The calculation
21, =
l]
(20);
range
to or greater
made concerning
model, vacancies
where y is Euler’s constant,
by equation
temperature
equal
depend-
(21) and (26) are almost
for
rb21/kT is about
and
1251
DISLOCATIONS
except
for a
term.
The
physical basis upon which these expressions are derived is somewhat idealized, for in general there is no reason to expect only one type of jog to be present. For this reason, in the ensuing discussion we limit our attention only to the case where both types of jogs are present, i.e., equation (20). No generality in the treatment is lost by limiting our attention to equation
it is necessary to measure or
those parameters
which are not otherwise
For any given test condition
all the quantities
except ps and il are well known. Although the variation of dislocation
density
flow stress has been studied intensively
for low tem-
perature deformation,
with
there has been very little work
in the temperature range where creep deformation is important. McLean and Hale(r5) using transmission electron microscopy (TEM), measured the dislocation density within the subgrains of iron crept at 500 and 700°C under stresses ranging from 7.6 x 10’ to 8.5 x lo8 dynes/cm2. Although there is some scatter in the results, there is a definite indication that the
ACTA
1252
METALLURGICA,
VOL.
13,
1965
TABLE 1. Chemical composition of the Fe-3.0% Element
Si
Mn
MO
cu
Ni
CO
wt.70
3.0
0.12
0.01
0.15
0.04
0.01
Si alloy Cr 0.02
C 0.03
Fe Bal.
dislocation density increases with increasing stress. Also, Lytton et ~1.‘~~)using etch pit techniques showed
photographic
that the dislocation
location density, p, was then determined methods of Ham and Sharpe’18) in which
is a function
density in an Fe + 3.1%
of the applied
Their material,
however,
[OOl] polycrystalline defined subgrains urements portion
stress, varying
was strongly
sheet
Si alloy
and
during creep.
did
as 01’4.
oriented not
of the substructure dislocation
The reported
very localized
consisted
density
dislocation
meas-
of a relatively
formation.
from the peculiarities
of the dislocation
the data
et al., are probably
of Lytton
of
Aside
substructure, somewhat
erroneous as no special attempt was made to preserve the existing experiments
creep substructure
on cooling.
reported below, procedures
In the
were adopted
which were designed to enhance the probability the observed structure In
dislocation
that
is typical
of the
during the creep process.
an attempt
to
stress dependence conducted position
structure
determine
quantitatively
the
of pS a series of creep tests were
on an Fe-3.0% is listed in Table
Si alloy whose full com1.
Creep tests were con-
ducted
at 643°C over a stress range of 3 x lo* to 9 x lo8 dynes/cm 2. Following testing the dislocation density was measured using both TEM and etch pit techniques. Creep specimens
were in the form of 1.0 mm thick
sheet with a gage section 45 mm long and 5 mm wide. The average
grain size was approximately
All creep tests were conducted hydrogen
atmosphere
maintained arm.
utilizing
A typical
and an
0.3 mm.
in a dry de-oxidized
a constant
stress
Andrade-Chalmers
creep curve for a sample
p =
well
with some indication
network
The disusing the
(110)
form
were made on samples in which the major
uniform
plates were taken from 10 to 20 different
areas, well away from the edge of the foil.
was lever
tested at
(27)
2nli4
where n is the number of dislocation
intersections
both foil surfaces and A is the surface area.
with
Etch pit
counts were made both from carbon replicas and from optical micrographs,
using equation
the density
of etch pits.
necessitated
by
above light
the fact
(27) to relate p to
The use of replicas that
dislocation
was
densities
lo8 cm-2 are difficult to resolve with ordinary microscopy.
Whenever
both
electron
copy and etch pit densities are reported ticular set of test conditions,
micros-
for
a par-
the measurements
were
made on the same specimen. In Fig. 2 the steady state creep rates are shown as a function below
of the applied stress.
referred
to as the intermediate
characterized Above
shown as a function samples
into
observations figure.
within
of applied
the
the
steady
state
both
is of
subgrains
stress in Fig. 3.
can be made regarding
First, although
dislocation paoa,
of the form u‘“.
the stress dependence
in this series were strained
well
is commonly
stress region and is
by a stress dependence
5 x lo8 dynes/cm2
the form exp (&T) I The dislocation density
SO%,
The creep behavior
a stress of 5 x 10s dynes/cm2
is All
approximately region.
Several
the data in this
the etch pit and TEM
densities have the same stress dependence,
there
is a consistent
between the two sets of data.
factor
of 4 difference
The probabIe
reasons
4.8 x lo8 dynes/cm2 is shown in Fig. 1. Following
testing all specimens were cooled rapidly
under load from the test temperature to 200°C. cooling rate was approximately 150’C/min. specimen
was allowed
to cool slowly
The The
from 200°C to
.20 .18.16-
T:643’C E =a 6 x lo-65ec-’ *
room temperature to allow the carbon atoms to precipitate on the dislocations. Etch pitting was done using a chromium following
the
trioxide-glacial
method
of
acetic acid solution
Hibbard
and
Dunn.(l’)
Samples intended for TEM were lapped to a thickness ofO.13 mm following testing and then thinned in a 5% perchloric acid, 95% glacial acetic acid solution at a potential of 45-50 V. Dislocation density measurements following manner.
were made in the
For the electron microscopy
study,
1 2c
FIG. 1. A typical creep curve for Fe-3.0% Si tested at 643% in en atmosphere of dry hydrogen.
BARRETT
NIX:
AND
CREEP
BY
MOTION
OF
JOGGED
DISLOCATIONS
1253
Id”,
1 0
ELECTRON MICROSCOPY
a
ETCH PIT
i
1
1 3
4
5
6
STRESS, (dynes ICm’x
FIG.
2.
9
IO
are numerous,
bilities that not all dislocations Assuming
the more precise measurement above
a
stress
observed dislocation expression
of
including
the possi-
to a group of closely the TEM data to be
of dislocation
~4
density,
x lo8 dynes/ems
the
density can be represented by an
temperature
o is in dynes/cm2
and p is in cm-2.
the observed
deviates from equation
(23)
(28).
dislocation
Below density
This might be expected,
for at low stresses the observed
dislocation
density
would not be expected to be lower than that measured
density
TABLE
2.
of strain and
6
7
IO
16’)
within
the subgrains
of p at temperatures
The dislocation
was not
observed
to
Also, measurements
of 596, 643 and 743°C at a stress
of 4.8 x 10’ dynes/cm2
showed a slight increase in p
with increasing temperature. must be fulfilled before equation (28)
can be combined dislocation
with the theoretical
velocity
the creep rate.
These
the
equations
for
to give an accurate expression for
majority
of
subgrains
are primarily
conditions
dislocations
that all the dislocations
are (1) that the
observed
within
of screw orientation
the
and (2)
are mobile and contribute
to
the creep strain. resistance
will be attained
if there is no
to the glide of edge dislocations.
Dependence of the dislocation density, p, within subgrains on test variables p, cm-2 etch pit?
Temperature T, ‘C
Strain E
2.4 4.8 4.8 4.8 4.8 5.5 :::
643 643 643 743 596 643 643
0.20 0.20 0.15 0.20 0.20 0.21 0.21 0.22
1.9 6.8 6.6 5.8 7.6 8.2 2.0 1.6
10’ 10’ 10’ 10’ 10’ 10’ 108
8.6 x 10’ 2.9 x 108 -
7.6 8.7
643 643
0.22 0.24
2.6 x lo8 3.8 x lo*
8.4; lo8 1.4 x 100
Stress u, dynes/cm2
9
is given in Table 2, where the data in
The first condition
density as a function
tdynes/cm*x
vary during steady state creep.
in the annealed state. The dislocation
J
I,,,,
6
Fig. 3 are also listed in tabular form.
Two conditions
p = 2.1 x lo-rsoa where
I
5
FIQ. 3. Steady state dislocation density within the cells as a function of the creep stress.
of the form
4 x lo8 dynes/cm2
I
4 STRESS,
produce an etch pit or
that one etch pit may correspond spaced dislocations.
I
3
lb?
Steady state creep rate as a function of stress for Fe-3.0% Si tested at 643°C.
for this difference
then
6
7
x x x x x x x
p, cm-2 TEM
t The dislocation density in the as annealed material was approximately 7 x 10s cm-a (determined by etch pits).
That is,
1254
ACTA
METALLURGICA,
VOL.
13,
1965
where we have taken d, = PSI2 and CJ= 27. The average spacing between jogs, I, has never been measured directly.
For this reason it is only possible
to estimate a value for L. Ardell et uZ.(~~)showed that for most practical conditions in thermal equilibrium
the spacing between jogs
can be expressed
as
(ij exp (Gj,
A=
(30)
where E is the energy to form a jog of one atomic height and q is an entropy factor equal to 5-10.t impossible known. associated
to compute
il accurately
Seeger(20) suggested
height
can be
as pub3110where ,u is the shear modulus.
For iron this is approximately the average However,
that the misfit energy
with a jog of unit atomic
approximated
It is
since E is not well
jog
spacing
0.7 eV. With this value
is of the order
equation (30) accounts
dynamic equilibrium
of
lo3 A.
only for the thermo-
number of jogs on the dislocation
arises from the possible orientations of a jog around the d.sfozation 1 line. That is, the various distributions of jog orientations result in a configurational entropy term.
3
4
STRESS, PIG.
4.
(dynes
5
6
/ crn2
x Ida
78
9
IO
1
Comparison of equation (29) with experimental data for various values of 1.
when a dislocation ments glide quickly
loop
is generated,
the edge seg-
across the subgrain
leaving
the
screw segments to slowly work their way to the subgrain wall by the diffusion jogs.
The obvious
dislocations, dislocation
i.e., the long configurations
and pile-ups, structures
controlled
restraints
motion
to the motion range
stress fields from
such as dislocation
have not been observed
tangles
in creep sub-
during steady state creep, thereby
some support
to the above
of the of edge
adding
model.
There is no immediate way to establish what fraction of the dislocations within a subgrain are mobile.
As it seems likely that some fraction
observed
dislocations
equation
(28) will overestimate
of the
are sessile, it is apparent the density
that
of mobile
screw dislocations. However, in lieu of any more accurate measure of ps, this equation will be used throughout the remainder of the paper. Thus combining the theoretical expression given by equation (20) with the experimental
equation
(28) we obtain
I
o7
I
3
4 STRESS,
i, = 12.5 x 10-1sD/3 tzr
o3 sinh fig]
,
(29)
I
5 ( dynes/cm2
676
I
I
9
IO
x Idat
FIG. 5. Comparison of equation (29) with experimental data for 1, = 210 A.
-I
BARRETT
AND
NIX:
CREEP
BY
MOTION
OF
JOGGED
1255
DISLOCATIONS
FIG. 6. Comparison of the jogged screw dislocation theory (equation (31)) with Garofalo’s empirical creep equation for austenitic stainless steel tested et 704°C.
FIG. 7.
and jogs
formed
Comparison of the jogged screw dislocation theory (equation (31)) with Garofalo’s empirical creep equation for aluminum tested at 204°C.
by intersection
processes
are not
included. As the spacing between intersection jogs will depend in some complicated way on the dislocation arrangement, and hence on the applied stress, it is clear that only an estimate for ?, can be made. Equation (29) is illustrated in Fig. 4 for various values of 1 with D taken as 9.3 x lo-l5 cm2/sec.(21) As 4 is increased from 60 to 200 A the stress dependence of equation (29) approaches that of the experimental data, although the predicted creep rates are some ten times the observed
values.
Upon
in-
creasing il to 250 A the stress dependence
of equation
(29) begins to deviate from the experimental
results,
and further increases in ii only make the comparison worse. The optimum value of I for agreement between the experimentally determined and theoretically predicted stress dependence of i, is 210 A. This comparison is shown in Fig. 5 where the absolute magnitude of equation (29) has been reduced by a factor of 10. That the predicted rate is a factor of 10 high is probably a result of assuming that the measured dislocation
density is equal to pS; i.e., if at any
1256
ACTA
METALLURGICA,
VOL.
13,
196.5
FIGI. 8. Comparison of the jogged screw dislocation theory (equation (31)) with Garofalo’s equation for aluminum tested at 260°C.
empirical
r -. m
-
FIG. 9. Comparison of the jogged screw dislocation theory (equation (31)) with equation for aluminum tested at 647°C.
instant only about 10% of the observed dislocations are actually mobile screw dislocations then the comparison equation (29) and experiment is very good. The temperature dependence of the steady state creep rate predicted by the jogged screw dislocation theory can be examined with reference to equation (20). There are three factors which contribute to the overall tem~ratu~ dependence of d,; (1) the variat*ion of p,with temperature, (2) the temperature dependence of the hyperbolic sine term, and (3) the temperature dependence of D. Since the dislocation density, ps, is more than likely
Garofalo’s
empirical
related to the ratio of the applied stress to the shear modulus, it seems probable that the variation of ps with temperature is related to the temperature dependence of the elastic constants. In fact, Barrett et aZ.(22) have shown that the apparent temperature dependence of the activation energy for creep in aluminum and cadmium can be explained by taking account of the temperature dependence of the elastic modulus in this manner. The temperature dependence of sinh (-rb2L/kT) depends to a large extent on the temperature dependence of 1. If A is determined mainly by jogs in thermal
BARRETT
AND
NIX:
CREEP
BY
MOTION
OF
JOGGED
DISLOCATIONS
1257
Fro. 10 Comparison of the jogged screw dislocation theory (equation (31)) with Qarofalo’s empirical equation for aluminum -3.1% magnesium tested at 258°C. 3. Tabulation of empirical constants
TABLE
Ref.
Material
Garofalo’s empirical creep equation
Tamp
Jogged Screw Dislocation Theory
is = A” (sinh CL CT)% A”
(23) I;:; (30) (31)
Awtenitic Stainless steel Al Al Al Al-3.1 Mg
sBc-1 704°C 204°C 260°C 847OC 259°C
1.47 2.78 1.94 2.67 4.17
x x x x x
10-s 1OV IO-5 10-S 10-S
cm2/dyne 1.13 5.77 7.43 1.82 3.05
equilibrium, then il is proportional to exp (.e/kT) and the activation energy for creep will be less than that for self-diffusion. However, if 3, is determined by intersection jogs, as is likely judging by the magnitude of J.necessary for agreement between experiment and theory, then for normal creep stresses, the temperature dependence of sinh (d2@T) is negligible compared to the temperature dependence of the diffusion coefficient. As such, the overall activation energy for creep will be similar to that for self diffusion, a result which is in accord with experiment. For the case in which dislocation core diffusion is the rate controlling process for the diffusion of vacancies, the activation energy for creep should approach that for core diffusion. As mentioned earlier, it is likely that core diffusion will become more important as the tem~rature is lowered. In fact, activation energies lower than self diffusion values have been observed
x x x x x
8, = A’ (I~ sinh (B 0) 7s.
a
-
10-e
3.64
IO-9
5.00
10-S 10-s 10-S
4.55 1.24 2.23
B
A’
cm2/dyne
WC-'
8.88 1.41 7.76 3.10 7.55
x x x x x
m
10-34 10-28 10-Z’ 10-9 lo-=
2.31 2.82 2.75 1.67 3.82
x x x x x
10-a 10-s 10-s 10-G 1O-g
2.78 2.28 2.49 0.16 1.37
near 0.5-0.6 of the absolute melting temperature for a number of metals, including A1,(23) Agc2*) and CU.(~~*~~)Hence, it may be that these low activation energies correspond to situations where the velocity of the jogs is determined by dislocation core diffusion and not bulk lattice diffusion. Equation (29) may be rewritten in the general form e,?= A’om sinh (Ro),
(31)
where both A’ and R are temperature dependent. This equation bears some resemblance to Garofalo’s empirical equation(z7) which can be written as B, = A” (sinh tlo)“,
(32)
where R”, and u and n are constant at a given temperature. In Fig. 6-10, data for the steady state creep rates for stainless steel,t2*) aluminum(zs~~) and for an alloy of A1-3.1°/o Mg(a” are shown. From these data
ACTA
1258
Garofalo has determined his empirical equation
equation.
the constants A”, u and 12in The constants A’, m and B in
(31) have been determined
techniques)
METALLURGICA,
(with computer
and are also shown in Table 3.
Garofalo’s
empirical equation and the empirical equation which is derived from the jogged screw dislocation both shown in Figs. 6-10. indicates
Reference
that the empirical
from the theoretical
theory are
to these figures
equation
which results
treatment in this paper is equally
as general as Garofalo’s
result.
ACKNOWLEDGMENTS
The authors are grateful for the many helpful discussions with students and faculty in the Department of Materials assistance
Science
at Stanford
University.
of 0. L. Frost is also appreciated.
The Special
thanks are due Drs. A. J. Ardell, R. A. Huggins and 0. D. Sherby. This paper
Replicas
were made by D. Mattern. at the meeting
of the
Metallurgical
Society of the AIME in October
1964 in
Philadelphia,
Pa.
This search
work
was presented
was sponsored
Projects
Materials Research
Agency
by the Advanced
through
at Stanford
the
Center
Refor
University.
REFERENCES 1. N. F. MOTT, Conference on Creep and Fwxture of Metal8 at High Temperatures, p. 21. H.M. Stationery Office, London (1956). 2. L. RAYMOND and J. DORN, Trans. Met. Sot. AIME 230, 560 (1964). 3. P. HIRSCH and D. WARRINGTON, Phil. Mug. 6, 735 (1961).
VOL.
13,
1965
4. J. FRIEDEL, Les Di8lOCUtion8,p. 72. Gauthier-Villars, Paris (1956). J. AppZ. Phys. 26, 362 (1957). 5. J. WEERTMAN, 6. R. CHRISTY, J. AppZ. Phys. 60, 760 (1959). 7. J. FRIEDEL, Phil. Mug. 46, 1169 (1955). and C. HERRING, Imperfection8 in Nearly 8. J. BARDEEN Perfect CrystaZe, Chapter 10. Wiley, New York (1952). Trans. ASME66,849 (1946). 9. D.ROSENTHAL, p. 61. Pergemon Press,Oxford 10. J. FRIEDEL, DiSloCatiOn8, (1964). 11. J. DORN, J. Mech. Phys. Solida 3, 85 (1954). 12. 0. SHERBY. Acta Met. 10.135 (1962). 13. P. FELTHA&, Proc. Phys.‘Soc. iond.'B 66, 865 (1953). 14. H. CARSLAW and J. JAEGER, Conduction of Heat in Solids, p. 267. Oxford University Press, London (1959). and K. HALE, Structural Processes in Creep, 15. D. MCLEAN p. 19. Iron and Steel Inst., London (1961). 16. J. LYTTON,~. BARRETT and 0. SHERBY,~~ be published in Trans. Met. Sot. AIME. 17. W. HIBBARD, Jr. and C. DUNN, Acta Met. 4, 306 (1956). 18. HAM and N. SHARPE, Phil. Mug. 6, 1193 (1961). W.NIX, J.AppZ.Phy8.36,1727 19. A. ARDELL,H.REJSS~~~ (1965). 20. A. SEEQER, Report of the Conferenceon Defects in Crystalline Solida, p. 391. The Physical Society, London (1955). and M. COHEN, Acta Met. 9, 21. F. BUFFINGTON,K.HIRANO 434 (1961). 22. C. BARRETT, A. ARDELL and 0. SRERBY. Trans Met. Sot. AIME 230. 200 (1964). ActaMet.5,219(1957). 23. O.&IERBY,~~~~TTON'&~~~.DORN. 24. S. DUSHMAN, L. DUNBAR and H. HUTHSTEINER, J. Appl. Phys. 15, 108 (1944). P. FELTHAM and J.MEAKIN, Acta Met. 7, 614 (1959). f :: C. BARRETTand 0. SHERBY, Trans. Met. Sot. AIME 230, 1322 (1964). 27. F. GAROFALO, Trans. Met. Sot. AIME 227, 351 (1963). on Properties of Crystalline 28. F. GAROFALO, Symposium Solids. ASMT, Spec, Tech. Publication no. 283, p. 82 (1960). 191,909 29. I. S. SERVI and N. J. GRANT, Trans. AIME (1951). 30. J. H. HARPER and J. E. DORN, Acta Met, 5, 654 (1957). 31. J. E. DORN, NPL Conferenceon Creep and Fracture, p. 89. Philosophical Library, Inc., New York (1957).
FOR STEADY STATE CREEP BASED ON THE MOTION OF JOGGED SCREW DISLOCATIONS* C. R.
BARRETT?
and
W.
D. NIXt
A theory of steady state creep is presented which is based on the diffusion controlled motion of jogged screw dislocations. Steady state creep is assumed to exist when the chemical force on the jogs is balanced by the force on the dislocation due to the applied stress. The resulting expression for the steady state creen rate is
where psis the mobile sorew disloc&tiondensity, 23 the self-diffusion c~ffi~ient, OL the number of atoms per unit cell, b the Burgers vector, op the lattice parameter, and /z is the average spacing between jogs. In order to compare this equation with experimental results, the stress dependence of ps is determined for an Fe + 3.0% Si alloy. The following expression is obtained:
Using this result, the above expression for B, can accurately predict the observed stress dependence and can give reasonable agreement with absolute creep rates. The predictions of this theory are compared with Garofelo’s empirical expression for steady state creep. UN MODELE
DE FLUAGE A VITESSE CONSTANTE BASE SUR LE DEPLACEMENT DES DISLOCATIONS-VIS AVEC CRANS
Les auteurs presentent une theorie du fluage a vitesse con&ante basee sur le d&placement, contrcilepar diffusion, des dislocations-vis avec trans. Ils admettent que le fluage a vitesse constante s’observe quand les forces de nature chimique qui s’exercent sur les dislocations du fait des contraintes appliquties. 11s en derivent une expression decrivant le fluage it vitesse eroissante: d, = 2PpaDor($)”
sinh (g),
ob ps est la densite en dislocations-vis mobiles, D le coefficient d’autodiffusion, a le nombre d’atomes parcellule unitaire, h le vecteur de Burgers, c(,,le parametre reticulaire et I la distance moyenne entre les trans. Dans le but de comparer oette relation aux r&&tats experimentaux on a determine la loi de variation de ps avec la tension appliquee pour un alliage Fe + 3.0 oASi. On a obtenu la relation suivante:
Partant de oe r&&tat, la relation donnant 2, permet de prevoir la loi de variation en fonction de la tension appliquee, en dormant un accord raisonnable avec les vitesses de flusge absolues. Les resultats de la prbente theorie sont compares a la relation empirique de Garofalo. EIN
MODELL FUR STATION~RES KRIECHEN, AUSGEHE~D VON DER BEWEGUNG VON SCH~AUBENVERSETZUNGEN MIT SPRttNGEN
Es wird eine Theorie entwickelt fur das station&e Kriechen, die die diffusionkontrollierte Bewegung von Schraubenversetzungen, die Spriinge enthalten, zugrunde legt. Es wird angenommen, da6 stationares Kriechen dann stattfindet, wenn die auf die Spriinge wirkende chemische Kraft der von der aul3eren Spannung auf die Versetzung ausgeiibten Kraft das Gleichgewicht halt. Fur die stationare Kriechgesohwindigkeit ergibt sich folgender Ausdruck: d, = 2rrp,Dsr(irsinh
(g),
wo ps die Dichte der bewegliehen Schraubenversetzungen, D der Selbstdiffusionskoeffizient, 0: die Zahl der Atome in der Elementarzelle, 6 der Burgersvektor, a, die Gitterkonstante und 3, die mittlere Entfernung der Spriinge ist. Urn dies%Gleichung mit den experimentellen Ergebnissen zu vergleichen, wird di.;s~~;;ungsabhangigkeit von ps fiir eine Fe + 3.0 % Si-Legierung bestimmt. Man erhalt folgenden
Unter Beniitzung dieses Resultates gestattet obiger Ausdruck fur 6,, die beobachtete Spannungsabhangigkeit richtig vorherzusagen und sine veriinftige Ubereinstimmung mit den absoluten Kriechgeschwindigkeiten zu ergeben. Die Vorhersagen dieser Theorie werden mit dem empirischen Ausdruck von Garofalo fur station&es Kriechen verglichen. * Received October 6, 1964; revised April 23, 1965. t Department of Materials Science, Stanford University, Stanford, California. ACTA META~LURGICA,
VOL. 13, DECEMBER
1966
1247
1248
ACTA
METALLURGICA,
1. INTRODUCTION
(1) where is is the steady state tensile strain rate, v the vibrational frequency at the jog, b the Burgers vector, d the distance between active slip planes, Qsd the activation energy for self diffusion, o the applied stress, k Boltzmann’s constant and T is the absolute temperature. Raymond and Dornt2) in a recent paper re-analyzed the thermally activated motion of jogged screw dislocations employing the same physical model used by Mott and incor~rat~g the concepts presented by Hirsch and Warringtonts) and Friedel.c4) They obtain an expression for the strain rate of the form b*(z h
1)v
pSexp (-Q,,/kT)
13,
1965 2. THEORY
Mott(l) was the first investigator to formulate a theory of high ~rn~rature creep based on the motion of jogged screw dislocations. Assuming jogs to be one atomic distance in height and the distance between jogs to be L, he proposed that the steady state creep rate be given by
PS =
VOL.
sinh (rb2A/hkT),
(2) where pS is the shear strain rate, z the coordination number, h the mean height of a moving jog, r the applied shear stress and ps is the mobile screw dislocation density. In order to make an explicit prediction of the stress and temperature dependence from equation (2) it is necessaryto evaluate ps, A and v. None of the parameters can be easily determined. While the concept of steady state creep controlled by the non-conservative motion of jogs on screw dislocations is attractive, insofar as it does not require the existence of the hypothetical dislocation arrangements present in some creep theories,(5*6) at present it is not possible to compare accurately the previous theoretical treatments with the experimental data. The purpose of the present paper is to present a calculation of the steady state creep rate based on the motion of jogged screw dislocations which can be more easily compared with experiment. In this treatment the quantity Y does not appear. This leaves ps and 1 as the parameters which must be evaluated. experimental evidence is presented for the variation of ps with temperature and applied stress. The final expression for the creep rate involves but one parameter which cannot be measured, namely A. Theoretical values of il are suggested.
When a jog in a screw dislocation does not lie in the slip plane, it is called a non-Conservative jog in that it can move with the gliding screw dislocation only by the emission or absorption of point defects (vacancies or interstitials). We may classify a non-conservative jog as one of two types: a vacancy-emitting jog or a vacancy-absorbing jog. In this treatment we consider both types of jogs. Consider a screw dislocation which contains a single vacancy-emitting jog. When the screw dislocation moves in response to a shear stress, the jog can maintain its position in the screw dislocation only by the emission of vacancies or by the absorption of interstitids. Since the concentration of interstitials existing in thermodynamic equilibrium is negligible, we can limit our attention to the case of vacancy emission. As vacancies are emitted by the jog, the local vacancy concentration (near the jog) is increased. If the non-Conservative jog motion is discontinued (by the removal of the applied shear stress, the vacancy concentration in the vicinity of the jog returns to the equilibrium value. However, continued motion of the jog enhances the local supersaturation of vacancies. The vacancy concentration near the jog is controlled by both the rate at which vacancies are being produced and the rate at which vacancies move away by diffusion. Since the jog in a screw dislocation can be considered as a very short segment of edge dislocation, the retarding force on the jog caused by the vacancy supersaturation can be computed as a chemical force. When the chemical force (dragging force) is equal to the driving force (applied stress), a steady state velocity is attained. This assumes that the lattice friction stress is negligible. We now consider the vacancy-absorbing jog. In general, there are two ways in which this jog can maintain its position in a gliding screw dislocation: by absorbing vacancies or by emitting interstitial& Friedel(7) has shown, however, that it is energetically more favorable for the jog to glide conservatively along the dislocation rather than move non-conservatively by the emission of interstitials. Therefore, we will limit our discussion to the ease of vacancy absorption. For this case, as in the previous, the diffusion controlled steady state velocity can be computed with reference to the chemical force. In this situation, however, there is a depression in the vacancy concentration near the jog which gives rise to a dragging force on the jog. In this treatment the creep rate is expressed aa 9s =
p@,
(3)
BARRETT
AND NIX:
CREEP
BY
MOTION
where 9, is the steady state shear strain rate, ps is the density of mobile screw dislocations, velocity
of the dislocations
and
b is the Burgers
Consider a segment of screw dislocation distance
in
a vacancy-producing
height.
The
exerted on the dislocation
JOGGED
express C,* as
chemical
dragging
force
exp
V
c 3)*-c,=~
-
.&*
hD,b
of length il
jog, one atomic
1249
DISLOCATIONS
G is the average
vector. which contains
OF
[E
+
2/F
d(P
+
+)I]
+ r2)
’
(6) and C,* as
by the jog is(*)
exp
c,* -co=
-A
f@=pln!$,
(
-
5 2D
&
T 2)
r2)1 1
[5 + 1/(E” +
’
+ 9)
(7)
0
where C, is the concentration
of vacancies in the vicin-
ity of the jog and Co is the equilibrium centration. velocity For
vacancy
C, and therefore f, are dependent
of the vacancy a similar
vacancy-absorbing
producing
dislocation
conon the
jog vV.
line which
contains
a
jog, the dragging force imposed by
the jog is In 2,
(5)
a where
C,
is the
absorbing
vacancy
the steady
concentration
respectively.
near
to compute
C,*
and
concentrations
near
and vacancy-absorbing
jogs
In general, the diffusion of vacancies
i.e., bulk diffusion core diffusion.
on the relative
which these two processes bilities are considered
in the matrix
to
occur.
by bulk diffusion
diffusion
in the matrix is rate controlling: in the matrix
when bulk (1) when
is so fast that the problem of a point
in an essentially
jog and diffusion
source
homogeneous
has predominantly
of vacancies
or sink of media and one type of
along the dislocation
core is much more rapid than lattice diffusion the problem reduces to the motion sink of vacancies.
so that
of a line source or
(1) Rapid bulk &f&ion. Formally, the problem of a moving point source of vacancies in an infinite homogeneous medium is identical with the moving point source heat flow problem originally treated by Rosenthal.cg) Using Rosenthal’s result it is possible to 3
distances
from
The boundary
concentration
condition
at infinitely
Since we are interested in a computation
chemical
large
the jog is equal to the equilibrium
force on the moving
concerned withC,*
andC,*
of the
jogs, we are primarily
at r = 0 and t = -b.
(just behind the moving
c * -c,=L
At
jogs) we have
&Deb2
and
c,* Strictly only
speaking,
for
-
-2% .
4rrD,b2
equations
(8) and (9) are valid
jogs moving in an infinite homogeneous A more realistic boundary condition, for
medium. example, vacancy
(Jo =
might
involve
concentration
the requirement
moving jog reach the equilibrium validity separate
of equation
value.
to
examine
condition
The solution
To estimate the
on the vacancy
tion near a source or sink. example,
the
from the the
(8) and (9) one can carry out a
computation
finite boundary
that
at a finite distance
for this case has not been found.
diffusion
(2) when a dislocation
rates with
The various possi-
There are two cases to be considered
vacancies
of
below.
2.1 Diffusion occurring~redominantly
to the motion
and dis-
Thus the exact calculation
and C,* will depend
reduces
is that the vacancy
the
or away from jogs will take place by two different
C,*
point source or sink for vacancies moving in a straight line in an infinite medium.
9
state vacancy
the vacancy-producing
processes,
These results
are obtained by treating the moving jog as if it were a
jog.
creep rate, it is first necessary
location
and D, is the
this position
In order to develop an equation for the steady state C,*;
where l and r are cylindrical
value. f, = y
vacancy
coordinates
lattice diffusion coefficient for vacancies.
that the vacancy
effect
of
a
concentra-
One can easily show, for concentration
C(r) near a
point vacancy source, fixed at the center of a sphere of changed when one radius ro, is not significantly changes
from
(r. > 20r). equations
infinite With
to finite boundary
this
reasoning
(8) and (9) to be valid
though the vacancy concentration equilibrium value at a finite distance. Substituting equations respectively we find
we
conditions may
expressions returns
expect even to
an
(8) and (9) into (4) and (5)
ACTA
1250
METALLURGICA,
VOL.
13,
1965
assumed that no conservative
and f, = -
y
In [I -
4 :bsc]
.
=. Steady state dislocation
(11)
0
The steady bining
velocities are attained when
the net force on the dislocation
motion of the jogs takes
place. t
line is zero.
This con-
state creep rate is obtained
equations
(3), (14),
expression
can be obtained
(12)
and 7bA = f,,
c, = and
r
is the
equations
applied
JL
(17)
XQ,
f-h3
D = xODv,
(13) stress.
shear
With
(18)
vacant sites at equilibrium, The following
= 4rD,b2CQ uup
c&=1, [exp
(g)
-
I] ,
,=4-DVb2C,[I-exp(-g)].
v, > v,
(15)
although
for
small
values
dislocation
producing tension
line will not
absorbing
remain
straight.
jogs will lag behind
when LX,= 0.5, cc9 = 0.5
dislocation
modifies
For purposes
is assumed that the average velocity containing
the force
of calculation
it
of a dislocation
both types of jogs is
Gyp
+
When a large fraction assumed
(equation
of vacancy
(16)
absorbing
ofvacancy
jogs respectively.
producing As a result
of the different jog velocities, the screw dislocation will not remain in the pure screw orientation but will assume the character
of a general dislocation
the regions between the jogs. For a screw dislocation with both
types
line in of jogs,
unless the jogs are equally spaced and alternate along the dislocation line, the bowing of the dislocation will, in general, give rise to a force on the jogs tending to move them conservatively along the dislocation line. If this force is sufficient to overcome the friction stress for conservative motion, then jogs may coalesce and either disappear or form jogs of several atomic distances
in
height.
For
the
(20)
present
case,
it is
jogs is
of the creep
rate with the applied stress becomes less pronounced as the stress increases, a result which is clearly in disaccord
with the experimental
absorbing
findings.(11-13)
dependence. attention
equal to the number of vacancy
In the following
to equations
(2) Rapid dislocation present.
For
the
When
jogs is either greater
jogs, the stress dependence
state creep rate is qualitatively and vacancy
absorbing
(19)), the variation
the fraction of vacancy producing
u,v,
where up and u, are the fractions
,
and when u, = 0, up = 1
than or approximately B=
(g)
The
balance on the different jogs and they move with the same overall velocity.
p,sinh
the vacancy
jogs until the added force due to the line
of the bowing
(19)
of
rb2AIkT, v, m v,. Thus if both types of jogs exist on a dislocation line, and the jogs are equally spaced, vacancy
when
a,=0
+, = 4rrDp (:r
the
coefficient.
general cases can be described;
~~-4~D/?(~~p,[I-exp(-~)],
general,
of
(14)
and
In
x,, the fraction
/l the number of atoms per
unit cell and D is the lattice self diffusion
as
are written
the jog velocities
these
in the final
if we note that
where a, is the lattice parameter, where
by com-
(16) for given
values of M, and CC,. Some simplification
dition is given by rbjZ = f,
(15) and
of the steady
similar to the observed sections
we limit our
(20) and (21). core diffusion,
calculation
of
one type of jog the
steady
state
vacancy sink of
concentration near a moving line source or vacancies we may again draw upon the Assuming an analogous heat flow calculation.“*) _FIn the present treatment the forces between jogs, which tend to move the jogs conservatively along the dislocation line, are ignored. This is in accord with Friedel’s postulate”” that the elastic distortions produced by jogs one atomic distance in height, are masked at distances greater than a few atomic dimensions by the stress field of the rest of the dislocation line. Thus, for the present study in which jogs of height b separated by a distance A(1 > b) are considered, the interjog forces should be negligible compared to the friction stress for conservative motion.
BARRETT
infinite homogeneous centrations
NIX:
AND
medium,
CREEP
the steady
BY
MOTION
state con-
right behind the jog are
c,* - c, =
OF
JOGGED
(20), however,
as the stress and temperature
ence predicted
by equations
identical
with that predicted stress
the
and
sinh (2) M &[exp (z) conclusions ture (23)
where K,(x)
is the modified
Bessel function
second kind of order zero. can be simplified
Equations
by noting
of the
(22) and (23)
that in nearly
all cases
1. Thus we can write
vb/2D, <
dependence
applicable
Dislocation
(20)
are
generally
(21) and (26) as well.
core diffusion may be important
for the
emitting.
absorbing
and vacancy
the dislocation
source
For this
can diffuse to or away from the jogs or through
the lattice.
of C,* and C,* thus entails calculating concentration
near a moving
or sink in a composite
system
of this type
is extremely
not attempted
-
(Z)
(24)
1 > 1
[I-exp(-g)],
(25)
w should of stress and temperature
mic nature.
As in the previous
be
only
a weak
due to its logarithcase v, > V, except
when rb2AlkT is small. Equation
(25)
leads
similar to equation
conditions
sidered
further.
combined
Equation
with equation
lis =
a creep
rate
2rrDfib2A a03w p
(24),
however,
be
assumed, and is
Even though C 2,* and C,* cannot be calculated it is dependence
of the jog
in this case will differ from that of the two
previous
models.
dislocation
That is, the activation
motion
energy for
may be closer to that for disloca-
(g)
and
bulk diffusion in self diffusion studies, dislocation
core
diffusion should become more important in controlling the jog velocity
as the temperature
of this observation
is lowered.
The
will be discussed in a
later section. 3. EXPERIMENTAL
RESULTS
In order to compare
-
I]
(26)
calculate known.
pre-exponential
Analo-
between grain boundary
AND
DISCUSSION
the stress dependence
the results of experiment, ps [exp
similar to that
(21) and (26) are identical
of bm/l in the
gous to the competition
of the
steady state creep rate predicted by equation (20) with
in practice.
Equations factor
can
(3) to give
which has a general stress dependence observed
expression
(19) and hence will not be con-
A
in the present study.
apparent that the temperature
implications to
(i.e., a
complicated,
tion core diffusion rather than bulk diffusion.
function
any
vacancy
velocity 2rD,bE, w,
equation
regardless of the boundary
Thus vl, and v, can be expressed as
vu,=
Thus
the stress or tempera-
line are alternately
calculation
w),
2 and
system with two regions of different diffusivities).
N 1.
and
than
all 2 > 2.
i.e.,
interest
case where jogs on the dislocation
point
[cap
for
the steady state vacancy
and
of
2.2 Dislocation core diffusion
either along
2n-D,bilC0
of
to equations
The calculation
21, =
l]
(20);
range
to or greater
made concerning
model, vacancies
where y is Euler’s constant,
by equation
temperature
equal
depend-
(21) and (26) are almost
for
rb21/kT is about
and
1251
DISLOCATIONS
except
for a
term.
The
physical basis upon which these expressions are derived is somewhat idealized, for in general there is no reason to expect only one type of jog to be present. For this reason, in the ensuing discussion we limit our attention only to the case where both types of jogs are present, i.e., equation (20). No generality in the treatment is lost by limiting our attention to equation
it is necessary to measure or
those parameters
which are not otherwise
For any given test condition
all the quantities
except ps and il are well known. Although the variation of dislocation
density
flow stress has been studied intensively
for low tem-
perature deformation,
with
there has been very little work
in the temperature range where creep deformation is important. McLean and Hale(r5) using transmission electron microscopy (TEM), measured the dislocation density within the subgrains of iron crept at 500 and 700°C under stresses ranging from 7.6 x 10’ to 8.5 x lo8 dynes/cm2. Although there is some scatter in the results, there is a definite indication that the
ACTA
1252
METALLURGICA,
VOL.
13,
1965
TABLE 1. Chemical composition of the Fe-3.0% Element
Si
Mn
MO
cu
Ni
CO
wt.70
3.0
0.12
0.01
0.15
0.04
0.01
Si alloy Cr 0.02
C 0.03
Fe Bal.
dislocation density increases with increasing stress. Also, Lytton et ~1.‘~~)using etch pit techniques showed
photographic
that the dislocation
location density, p, was then determined methods of Ham and Sharpe’18) in which
is a function
density in an Fe + 3.1%
of the applied
Their material,
however,
[OOl] polycrystalline defined subgrains urements portion
stress, varying
was strongly
sheet
Si alloy
and
during creep.
did
as 01’4.
oriented not
of the substructure dislocation
The reported
very localized
consisted
density
dislocation
meas-
of a relatively
formation.
from the peculiarities
of the dislocation
the data
et al., are probably
of Lytton
of
Aside
substructure, somewhat
erroneous as no special attempt was made to preserve the existing experiments
creep substructure
on cooling.
reported below, procedures
In the
were adopted
which were designed to enhance the probability the observed structure In
dislocation
that
is typical
of the
during the creep process.
an attempt
to
stress dependence conducted position
structure
determine
quantitatively
the
of pS a series of creep tests were
on an Fe-3.0% is listed in Table
Si alloy whose full com1.
Creep tests were con-
ducted
at 643°C over a stress range of 3 x lo* to 9 x lo8 dynes/cm 2. Following testing the dislocation density was measured using both TEM and etch pit techniques. Creep specimens
were in the form of 1.0 mm thick
sheet with a gage section 45 mm long and 5 mm wide. The average
grain size was approximately
All creep tests were conducted hydrogen
atmosphere
maintained arm.
utilizing
A typical
and an
0.3 mm.
in a dry de-oxidized
a constant
stress
Andrade-Chalmers
creep curve for a sample
p =
well
with some indication
network
The disusing the
(110)
form
were made on samples in which the major
uniform
plates were taken from 10 to 20 different
areas, well away from the edge of the foil.
was lever
tested at
(27)
2nli4
where n is the number of dislocation
intersections
both foil surfaces and A is the surface area.
with
Etch pit
counts were made both from carbon replicas and from optical micrographs,
using equation
the density
of etch pits.
necessitated
by
above light
the fact
(27) to relate p to
The use of replicas that
dislocation
was
densities
lo8 cm-2 are difficult to resolve with ordinary microscopy.
Whenever
both
electron
copy and etch pit densities are reported ticular set of test conditions,
micros-
for
a par-
the measurements
were
made on the same specimen. In Fig. 2 the steady state creep rates are shown as a function below
of the applied stress.
referred
to as the intermediate
characterized Above
shown as a function samples
into
observations figure.
within
of applied
the
the
steady
state
both
is of
subgrains
stress in Fig. 3.
can be made regarding
First, although
dislocation paoa,
of the form u‘“.
the stress dependence
in this series were strained
well
is commonly
stress region and is
by a stress dependence
5 x lo8 dynes/cm2
the form exp (&T) I The dislocation density
SO%,
The creep behavior
a stress of 5 x 10s dynes/cm2
is All
approximately region.
Several
the data in this
the etch pit and TEM
densities have the same stress dependence,
there
is a consistent
between the two sets of data.
factor
of 4 difference
The probabIe
reasons
4.8 x lo8 dynes/cm2 is shown in Fig. 1. Following
testing all specimens were cooled rapidly
under load from the test temperature to 200°C. cooling rate was approximately 150’C/min. specimen
was allowed
to cool slowly
The The
from 200°C to
.20 .18.16-
T:643’C E =a 6 x lo-65ec-’ *
room temperature to allow the carbon atoms to precipitate on the dislocations. Etch pitting was done using a chromium following
the
trioxide-glacial
method
of
acetic acid solution
Hibbard
and
Dunn.(l’)
Samples intended for TEM were lapped to a thickness ofO.13 mm following testing and then thinned in a 5% perchloric acid, 95% glacial acetic acid solution at a potential of 45-50 V. Dislocation density measurements following manner.
were made in the
For the electron microscopy
study,
1 2c
FIG. 1. A typical creep curve for Fe-3.0% Si tested at 643% in en atmosphere of dry hydrogen.
BARRETT
NIX:
AND
CREEP
BY
MOTION
OF
JOGGED
DISLOCATIONS
1253
Id”,
1 0
ELECTRON MICROSCOPY
a
ETCH PIT
i
1
1 3
4
5
6
STRESS, (dynes ICm’x
FIG.
2.
9
IO
are numerous,
bilities that not all dislocations Assuming
the more precise measurement above
a
stress
observed dislocation expression
of
including
the possi-
to a group of closely the TEM data to be
of dislocation
~4
density,
x lo8 dynes/ems
the
density can be represented by an
temperature
o is in dynes/cm2
and p is in cm-2.
the observed
deviates from equation
(23)
(28).
dislocation
Below density
This might be expected,
for at low stresses the observed
dislocation
density
would not be expected to be lower than that measured
density
TABLE
2.
of strain and
6
7
IO
16’)
within
the subgrains
of p at temperatures
The dislocation
was not
observed
to
Also, measurements
of 596, 643 and 743°C at a stress
of 4.8 x 10’ dynes/cm2
showed a slight increase in p
with increasing temperature. must be fulfilled before equation (28)
can be combined dislocation
with the theoretical
velocity
the creep rate.
These
the
equations
for
to give an accurate expression for
majority
of
subgrains
are primarily
conditions
dislocations
that all the dislocations
are (1) that the
observed
within
of screw orientation
the
and (2)
are mobile and contribute
to
the creep strain. resistance
will be attained
if there is no
to the glide of edge dislocations.
Dependence of the dislocation density, p, within subgrains on test variables p, cm-2 etch pit?
Temperature T, ‘C
Strain E
2.4 4.8 4.8 4.8 4.8 5.5 :::
643 643 643 743 596 643 643
0.20 0.20 0.15 0.20 0.20 0.21 0.21 0.22
1.9 6.8 6.6 5.8 7.6 8.2 2.0 1.6
10’ 10’ 10’ 10’ 10’ 10’ 108
8.6 x 10’ 2.9 x 108 -
7.6 8.7
643 643
0.22 0.24
2.6 x lo8 3.8 x lo*
8.4; lo8 1.4 x 100
Stress u, dynes/cm2
9
is given in Table 2, where the data in
The first condition
density as a function
tdynes/cm*x
vary during steady state creep.
in the annealed state. The dislocation
J
I,,,,
6
Fig. 3 are also listed in tabular form.
Two conditions
p = 2.1 x lo-rsoa where
I
5
FIQ. 3. Steady state dislocation density within the cells as a function of the creep stress.
of the form
4 x lo8 dynes/cm2
I
4 STRESS,
produce an etch pit or
that one etch pit may correspond spaced dislocations.
I
3
lb?
Steady state creep rate as a function of stress for Fe-3.0% Si tested at 643°C.
for this difference
then
6
7
x x x x x x x
p, cm-2 TEM
t The dislocation density in the as annealed material was approximately 7 x 10s cm-a (determined by etch pits).
That is,
1254
ACTA
METALLURGICA,
VOL.
13,
1965
where we have taken d, = PSI2 and CJ= 27. The average spacing between jogs, I, has never been measured directly.
For this reason it is only possible
to estimate a value for L. Ardell et uZ.(~~)showed that for most practical conditions in thermal equilibrium
the spacing between jogs
can be expressed
as
(ij exp (Gj,
A=
(30)
where E is the energy to form a jog of one atomic height and q is an entropy factor equal to 5-10.t impossible known. associated
to compute
il accurately
Seeger(20) suggested
height
can be
as pub3110where ,u is the shear modulus.
For iron this is approximately the average However,
that the misfit energy
with a jog of unit atomic
approximated
It is
since E is not well
jog
spacing
0.7 eV. With this value
is of the order
equation (30) accounts
dynamic equilibrium
of
lo3 A.
only for the thermo-
number of jogs on the dislocation
arises from the possible orientations of a jog around the d.sfozation 1 line. That is, the various distributions of jog orientations result in a configurational entropy term.
3
4
STRESS, PIG.
4.
(dynes
5
6
/ crn2
x Ida
78
9
IO
1
Comparison of equation (29) with experimental data for various values of 1.
when a dislocation ments glide quickly
loop
is generated,
the edge seg-
across the subgrain
leaving
the
screw segments to slowly work their way to the subgrain wall by the diffusion jogs.
The obvious
dislocations, dislocation
i.e., the long configurations
and pile-ups, structures
controlled
restraints
motion
to the motion range
stress fields from
such as dislocation
have not been observed
tangles
in creep sub-
during steady state creep, thereby
some support
to the above
of the of edge
adding
model.
There is no immediate way to establish what fraction of the dislocations within a subgrain are mobile.
As it seems likely that some fraction
observed
dislocations
equation
(28) will overestimate
of the
are sessile, it is apparent the density
that
of mobile
screw dislocations. However, in lieu of any more accurate measure of ps, this equation will be used throughout the remainder of the paper. Thus combining the theoretical expression given by equation (20) with the experimental
equation
(28) we obtain
I
o7
I
3
4 STRESS,
i, = 12.5 x 10-1sD/3 tzr
o3 sinh fig]
,
(29)
I
5 ( dynes/cm2
676
I
I
9
IO
x Idat
FIG. 5. Comparison of equation (29) with experimental data for 1, = 210 A.
-I
BARRETT
AND
NIX:
CREEP
BY
MOTION
OF
JOGGED
1255
DISLOCATIONS
FIG. 6. Comparison of the jogged screw dislocation theory (equation (31)) with Garofalo’s empirical creep equation for austenitic stainless steel tested et 704°C.
FIG. 7.
and jogs
formed
Comparison of the jogged screw dislocation theory (equation (31)) with Garofalo’s empirical creep equation for aluminum tested at 204°C.
by intersection
processes
are not
included. As the spacing between intersection jogs will depend in some complicated way on the dislocation arrangement, and hence on the applied stress, it is clear that only an estimate for ?, can be made. Equation (29) is illustrated in Fig. 4 for various values of 1 with D taken as 9.3 x lo-l5 cm2/sec.(21) As 4 is increased from 60 to 200 A the stress dependence of equation (29) approaches that of the experimental data, although the predicted creep rates are some ten times the observed
values.
Upon
in-
creasing il to 250 A the stress dependence
of equation
(29) begins to deviate from the experimental
results,
and further increases in ii only make the comparison worse. The optimum value of I for agreement between the experimentally determined and theoretically predicted stress dependence of i, is 210 A. This comparison is shown in Fig. 5 where the absolute magnitude of equation (29) has been reduced by a factor of 10. That the predicted rate is a factor of 10 high is probably a result of assuming that the measured dislocation
density is equal to pS; i.e., if at any
1256
ACTA
METALLURGICA,
VOL.
13,
196.5
FIGI. 8. Comparison of the jogged screw dislocation theory (equation (31)) with Garofalo’s equation for aluminum tested at 260°C.
empirical
r -. m
-
FIG. 9. Comparison of the jogged screw dislocation theory (equation (31)) with equation for aluminum tested at 647°C.
instant only about 10% of the observed dislocations are actually mobile screw dislocations then the comparison equation (29) and experiment is very good. The temperature dependence of the steady state creep rate predicted by the jogged screw dislocation theory can be examined with reference to equation (20). There are three factors which contribute to the overall tem~ratu~ dependence of d,; (1) the variat*ion of p,with temperature, (2) the temperature dependence of the hyperbolic sine term, and (3) the temperature dependence of D. Since the dislocation density, ps, is more than likely
Garofalo’s
empirical
related to the ratio of the applied stress to the shear modulus, it seems probable that the variation of ps with temperature is related to the temperature dependence of the elastic constants. In fact, Barrett et aZ.(22) have shown that the apparent temperature dependence of the activation energy for creep in aluminum and cadmium can be explained by taking account of the temperature dependence of the elastic modulus in this manner. The temperature dependence of sinh (-rb2L/kT) depends to a large extent on the temperature dependence of 1. If A is determined mainly by jogs in thermal
BARRETT
AND
NIX:
CREEP
BY
MOTION
OF
JOGGED
DISLOCATIONS
1257
Fro. 10 Comparison of the jogged screw dislocation theory (equation (31)) with Qarofalo’s empirical equation for aluminum -3.1% magnesium tested at 258°C. 3. Tabulation of empirical constants
TABLE
Ref.
Material
Garofalo’s empirical creep equation
Tamp
Jogged Screw Dislocation Theory
is = A” (sinh CL CT)% A”
(23) I;:; (30) (31)
Awtenitic Stainless steel Al Al Al Al-3.1 Mg
sBc-1 704°C 204°C 260°C 847OC 259°C
1.47 2.78 1.94 2.67 4.17
x x x x x
10-s 1OV IO-5 10-S 10-S
cm2/dyne 1.13 5.77 7.43 1.82 3.05
equilibrium, then il is proportional to exp (.e/kT) and the activation energy for creep will be less than that for self-diffusion. However, if 3, is determined by intersection jogs, as is likely judging by the magnitude of J.necessary for agreement between experiment and theory, then for normal creep stresses, the temperature dependence of sinh (d2@T) is negligible compared to the temperature dependence of the diffusion coefficient. As such, the overall activation energy for creep will be similar to that for self diffusion, a result which is in accord with experiment. For the case in which dislocation core diffusion is the rate controlling process for the diffusion of vacancies, the activation energy for creep should approach that for core diffusion. As mentioned earlier, it is likely that core diffusion will become more important as the tem~rature is lowered. In fact, activation energies lower than self diffusion values have been observed
x x x x x
8, = A’ (I~ sinh (B 0) 7s.
a
-
10-e
3.64
IO-9
5.00
10-S 10-s 10-S
4.55 1.24 2.23
B
A’
cm2/dyne
WC-'
8.88 1.41 7.76 3.10 7.55
x x x x x
m
10-34 10-28 10-Z’ 10-9 lo-=
2.31 2.82 2.75 1.67 3.82
x x x x x
10-a 10-s 10-s 10-G 1O-g
2.78 2.28 2.49 0.16 1.37
near 0.5-0.6 of the absolute melting temperature for a number of metals, including A1,(23) Agc2*) and CU.(~~*~~)Hence, it may be that these low activation energies correspond to situations where the velocity of the jogs is determined by dislocation core diffusion and not bulk lattice diffusion. Equation (29) may be rewritten in the general form e,?= A’om sinh (Ro),
(31)
where both A’ and R are temperature dependent. This equation bears some resemblance to Garofalo’s empirical equation(z7) which can be written as B, = A” (sinh tlo)“,
(32)
where R”, and u and n are constant at a given temperature. In Fig. 6-10, data for the steady state creep rates for stainless steel,t2*) aluminum(zs~~) and for an alloy of A1-3.1°/o Mg(a” are shown. From these data
ACTA
1258
Garofalo has determined his empirical equation
equation.
the constants A”, u and 12in The constants A’, m and B in
(31) have been determined
techniques)
METALLURGICA,
(with computer
and are also shown in Table 3.
Garofalo’s
empirical equation and the empirical equation which is derived from the jogged screw dislocation both shown in Figs. 6-10. indicates
Reference
that the empirical
from the theoretical
theory are
to these figures
equation
which results
treatment in this paper is equally
as general as Garofalo’s
result.
ACKNOWLEDGMENTS
The authors are grateful for the many helpful discussions with students and faculty in the Department of Materials assistance
Science
at Stanford
University.
of 0. L. Frost is also appreciated.
The Special
thanks are due Drs. A. J. Ardell, R. A. Huggins and 0. D. Sherby. This paper
Replicas
were made by D. Mattern. at the meeting
of the
Metallurgical
Society of the AIME in October
1964 in
Philadelphia,
Pa.
This search
work
was presented
was sponsored
Projects
Materials Research
Agency
by the Advanced
through
at Stanford
the
Center
Refor
University.
REFERENCES 1. N. F. MOTT, Conference on Creep and Fwxture of Metal8 at High Temperatures, p. 21. H.M. Stationery Office, London (1956). 2. L. RAYMOND and J. DORN, Trans. Met. Sot. AIME 230, 560 (1964). 3. P. HIRSCH and D. WARRINGTON, Phil. Mug. 6, 735 (1961).
VOL.
13,
1965
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