- Email: [email protected]
Note on the heating effect of moving dislocations
ACTA
560
METALLURGICA,
VOL.
4,
1956
verschiedener Spew. Suszeptibilitiit Ti-Al-Legierungen bei 20°C
30
20
10
LO
Gew. ‘lo Al FIG. 3.
Man kann sich unschwer
Phasendiagramme
struieren, welche den mitgeteilten nung
tragen.
Heranziehen
Dieselben weiterer
wiirden
es deshalb
suchungen
Daten
Charakter
vor, diese Frage
menhang mit den Ergebnissen metallogrnphischen
jedoch
experimenteller
mehr oder weniger spekulativen Wir ziehen
Ergebnissen
kon-
temperature
besitzen.
im ZusamUnter-
A. MUENSTER K. SAGEL
der Metallgesellschaft
U. ZWICKER
1. H. R. OGDEN, 0. J. MAYICUTH, W. L. FINLAY, und R. J. JaFFEE Trans. Amer. Inst. Min. Met. &%g. _ 191. 1150 (1951). 2. E. S. BUMPS, H. 0. KESSLER, und M. HANSEN Trans. Amer. Inst.Nin. Met. Eng. 194, 609 (1952). 3. A. A. MC&UII,LAN J. Inst.Met. 83, 181 (1954). 4. W. ROSTOKER J. Metals 4, 209 (1952). 5. H. W. WORNER J. Inst. Met. 81, 521 (1952/53). 6. W. L. FINLAY, R. J. JAFFEE, R. W. PARCEL, und R. D. DURSTEIN J. Metnls 6, 25 (1954). 7. H. R. OGDEN, D. J. MAYKUTH, W. L. FINLAY, und R. J. JAFFEE J. Met& 5, 267 (1953). 8. A. KNAPPWOST 2. Elektrochemie 59, 561 (1951).
plastic deformation.
Note on the Heating Effect of Moving Dislocations* and Weiner have
proposed that thermal stresses produced during fatigue are severe enough to create microcracks;
slip
plane
by
calculated
some
moving in many
200°C.
along slip aspects
of
In this note the heating effect is
in a different
manner,
of Freudenthal
We take the dislocation
and some of the
and Weiner are discussed. to be a moving line-source
of heat of strength q = b-rV erg cm-l set-1 where 6 is its Burgers 7 is the applied
vector,
shear stress.
V its velocity, The temperature
and at
(x,y) ist2) T=
‘-~ e’/”
2nK x,y
are
rectangular
dislocation, behind
K&r/R), r2 = x2 + ~2. co-ordinates
centered
y = 0 is the slip-plane,
the dislocation.
and K is the thermal
* Received March 9, 1956.
Freudenthal
the
planes is in fact of importance
Frankfurt/Main References
near
The heating effect of dislocations
assumptions
Aus dem Metall- Laboratorium
In a recent paper,(l)
“thermal
ohne
zu diskutieren.
A.G.
localized
flashes,”
einen
der zur Zeit laufenden
und riintgenographischen
these stresses are due to highly
associated with repeated and reversed slip processes, which in the extreme case may raise the
Rech-
the
K is the heat conductivity diffusivity (K divided by the
specific heat per unit volume). K,(z) has the asymptotic forms K,(z) -
on
and x is positive
--In z,
The Bessel function
z>
1;
LETTERS
TO
the scale of the temperature be many
atomic
pattern.
distances,
It will
so that near the
dislocation
(compare
alloy, of
24&T,
Seitzt3)).
in the two
T = .:Lg ln A r 77
procession
Even for fairly extreme values of
the slip-plane
T is of the order of only a fraction of a
degree a few at,omic distances
from the dislocation.
The question now is, do the temperature succession of dislocations
fields of a
add up to give an appreciable
We can make an estimate in two extreme cases.
Consider a procession
of n equally-spaced dislocations 2 the maximum spread over a distance jz. Then if A> temperature will be about
Similarly,
we would
for the hard light
suggest
a temperature-rise
about two degrees rather than fifty degrees. This discrepancy, of course, arises from the differing
values,
the constants,
rise?
561
of about half a degree.
A = 2K/V fixes
EDITOR
effective length of 10-r cm, giving a temperature-rise
The lengt’h
usually
THE
the
of
models,
of the length
dislocations.
case in which
We
dislocations
with a spacing
have move
and
dislocations
uniformly
interatomic
Weiner
distances.
Bunching
slip-plane
forces,
but it is unlikely
could
reach
may
the
and Weiner.
consider
occur
value
source, whereas by
and the temperature at a distance
will be n times what it would be (inter-dislocation
l/n
single dislocation. If A< 1, the maximum
which
.un
is identical
equation
nA
~mq
2nK
temperature
for a
will be about
s(-~-I - -a
T=
spacing)
2r
dr
A/n
suggested
by
was so severe that the distribution pile-up
and
Weiner’s
(5):
and the
If the bunching
was similar to that
of n dislocations,
the distance
between first and last, il, would be about n2 times the distance between the leading pair,(d) as against equally
2nh’
together,
then exceed
strength of the material.
of a static
spacing
Freudenthal
would require very
the array would
n times this distance
4
wit)h Freudenthal
theoretical
five
of retarding
that the average
high stresses to keep the dislocations the stress around
of
about
of the dislocations
because
Their distribution
along
by their
a procession
separated
in the
freely
determined
rate of emission from the Frank-Read Freudenthal
3, of the considered
spaced
the value
for Freudenthal
procession.
We
and Weiner’s
believe
of il for 500 dislocations,
from the simple picture
then
that
about
1O-4 cm
of the Frank-Read
source,
might be reduced to 1O-2 cm if bunching should occur, but that the figure of 10e4 cm is too small. The effect of stacking individual
where m is the spacing between successive dislocations
nearby
in unit,s of b. As before, there will be a temperature-
the possible
rise about’ n times that due to a single dislocation,
Weiner’s
provided the factor (A/J.)* is not too unfavorable. To determine the actual temperature rise, we have
by different arguments,
to estimate if one of appropriate.
plausible
the approximations A> 1, A< V has a maximum value of
fraction of the velocity It’ is commonly on one slip-plane source;
values for V, A, n, and decide
of sound, say i V,,-
assumed
that
il is some
lo5 cm/set.
all the dislocations
come from the same Frank-Read
if the sweeping velocity
of the source is about
parallel
slip-planes
rise in temperature.
equation
by Orowan.c6) temperature
by Cottrellc5) and before that
varies linearly with the amount
the slip-planes, packed
temperature. necessary
so that a large shear on many closely-
planes
could
that
simultaneously,
give
an
appreciable
For this magnified the
nearby
annealed material, 1 N 10e4 cm, and, with n = 500,R becomes about 10-r cm. Thus, A< ;1, and the second Freuform of our equation is the most appropriate. denthal and Weiner achieve a temperature rise of about 8°C on a slip-plane in aluminum, with 500 dislocations moving at Q8, in a procession 1O-4 cm in length; whereas our model suggests that this procession has an
evidence
presented
by
rise
in
heating effect’ it is
slip
and Freudenthal
suggests that’, during fatigue,
In
of glide
on each plane and inversely with the spacing between
about
source.
and
obtained
For a given shear stress, the rise in
that this is not highly probable.
of the Frank-Read
considerably
Freudenthal
7 is the same as that,
the same as the velocity of the moving dislocations, the spacing between successive dislocations will be 1, t’he length
slip processes into
increases
processes
occur
and Weiner suggest
Brawn(7)
The microscopical and
Forsyth’s)
slip only occurs simul-
taneously on planes a micron or more apart. the “striations” widening and intensifying during the test by the addition of individual slip processes one after the other. Furthermore, if displacements of 1500 A were to occur on many planes 200 A apart, as Freudenthal and Weiner suggest, the material would suffer the equivalent of a homogeneous shear of seven atomic
562
ACTA
~~T_~LLU~GICA,
diameters on each atom plane; and this is clearly not achieved over large volumes of the material. The nearest approximation to this case is fine flip, in which the displacement is about 50 k on many slip-planes 200 pi apart. Here the maximum rise in temperat~e might be a few degrees if many of t-he slip processes occurred sim~taneously. It seems inlprobable then that there can be appreciable local heating near the glide-plane, except perhaps at very high rates of loading. The creation of cracks during normal reversed stressing may be explained by the pile-up of dislocations,‘lO) and the softening processes by the local generation of vacancies.(ll) J. D. ESHELBY
VOL.
4,
1956
.ti mole 600 IPrecipitation
FIG. 1. Rate of evolution of energr- ve~‘sus t,irne-euwe for a meesurelnent at !%?.$“C.
thermostat with a very large heat capacity,t3) The specimen was located in a cent‘ral hole in the thermostat and a series of thermocouples, connected to a References sensitive galvanometer, was arranged between the 1. A. M. FREUDENTHAL and J. H. WEINER J. Appl. Phys. specimen and the wall of the thermostat in such a 27, 44 (1956). way that the deflection of the galvanometer was 2. 3-f. S. CARSLAW and J. C. JAEGERConcluction of Heat in Solids p. 234, Oxford (1947). pro~rtioi~a~ to the rise in temperature of the specimen 3. F. SE~TZA&awes in Physics 1, 43 (1952). 4. J. D. ESHELBY, F. C. FRARK, and F. R. N. N~BAREO caused by evolution of heat due to the tra~lsformation. Phil. Lwag,42, 351 (1951). As the heat liberated during the grain enlargement 5. A. H. COTTRELLDislocations and P.?astic Flow in Cry&a& of a metal is very small-at il. grain size of 0.01 mm p. 6 Oxford (1963). 6. E. OROWAN in Principles of Theological Meaewement the total boundary energy is of the magnit’ude of p. 180 Nelson (1949). 0.1 cal/mole-extreme precautions must be taken 7. A. F. BROWN Advances in Physics 1, 427 (1952). 8. P. J. E. FORSYTHJ. Inst. Metals 80, 181 (1952/53); 82, ttioreduce disturbing effects, which will be discussed 449 (1953,!54): P. J. E. FORSYTHand C. A. STUBBINGTON in a later article. The zinc specimens used ha.d the R.A.E. Report Met. 76 (1953), .iWel.78 (1954). 9. D. KuHL~A~~-W~LSDORF and H. WILSDORST Acta iWet. form of granules of a size of about 3 mm with the com4, 394 (1953). position: As < 0.000015~~“, Fe < 006~~. and Pb < 10. N. F. MOTT J. Pkys. 5’0~. Japan 10 (S), 650 (1955). 11. T. BROOM and J. H. MOLIXEUX J. Inst. .&f&a& 88, 528 O.O5O/ / 0‘ The granules were put, into a thin-walled (1954j55). aluminum container fitting into the measuring cell. * Received March 14, 1956. The junction of a Pt-PtRh peltiercouple was located at the center of the cont’ainer for calibration purposes. In Fig. I the rate of evolution of energy- versus Calorimetric Measurements of the time-curve for a measurelnent at 282.8% is shown. Grain-boundary Energy in Zinc* Although the mea,sureme~ts were carried out in an Although relative lain-boundary energies can be at,mosphere of argon, it was almost impossible to easily obtained, absolute values of these energies are avoid a small oxidation effect of the specimens, due more di&uit to estimate. The values reported in to rests of oxygen absorbed at the surface of the the literature are all indirect in the sense that they granules and the container. This oxidation effect have been obtained by comparing the energy of a gives an approximately exponential initial fall of grain boundary with that of an external surface, the measuring curves. The effect of the grain growth the surface energy of which is determined, e.g., from in the specimen, from a size of the grains of somewhat the dihedral angles formed between the soIid and a less than 0.1 mm to abont t mm, appears as a small liquid of known surface tension. This procedure, evolution of heat with a maximum after about though correct in principle, seems not to be very 3 hours, as shown Fig. 1. The form of the curve is accurate in practice. The present investigation is a bell-shaped, which may be expected for a coalescencedirect measurement of the grain-boundary energy in process. As the oxidation effect a’nd the growt,h of zinc with an isothermal calorimetric method, practiced the grains have different energies of activation, earlier.(ls 2, t.he temperatures of the measurements have been The calorimetric apparatus used for the measurechosen so that the maximum of the latter effect does ments was an electrically heated and regulated not occur until the rate of oxidation ha,s diminished.
560
METALLURGICA,
VOL.
4,
1956
verschiedener Spew. Suszeptibilitiit Ti-Al-Legierungen bei 20°C
30
20
10
LO
Gew. ‘lo Al FIG. 3.
Man kann sich unschwer
Phasendiagramme
struieren, welche den mitgeteilten nung
tragen.
Heranziehen
Dieselben weiterer
wiirden
es deshalb
suchungen
Daten
Charakter
vor, diese Frage
menhang mit den Ergebnissen metallogrnphischen
jedoch
experimenteller
mehr oder weniger spekulativen Wir ziehen
Ergebnissen
kon-
temperature
besitzen.
im ZusamUnter-
A. MUENSTER K. SAGEL
der Metallgesellschaft
U. ZWICKER
1. H. R. OGDEN, 0. J. MAYICUTH, W. L. FINLAY, und R. J. JaFFEE Trans. Amer. Inst. Min. Met. &%g. _ 191. 1150 (1951). 2. E. S. BUMPS, H. 0. KESSLER, und M. HANSEN Trans. Amer. Inst.Nin. Met. Eng. 194, 609 (1952). 3. A. A. MC&UII,LAN J. Inst.Met. 83, 181 (1954). 4. W. ROSTOKER J. Metals 4, 209 (1952). 5. H. W. WORNER J. Inst. Met. 81, 521 (1952/53). 6. W. L. FINLAY, R. J. JAFFEE, R. W. PARCEL, und R. D. DURSTEIN J. Metnls 6, 25 (1954). 7. H. R. OGDEN, D. J. MAYKUTH, W. L. FINLAY, und R. J. JAFFEE J. Met& 5, 267 (1953). 8. A. KNAPPWOST 2. Elektrochemie 59, 561 (1951).
plastic deformation.
Note on the Heating Effect of Moving Dislocations* and Weiner have
proposed that thermal stresses produced during fatigue are severe enough to create microcracks;
slip
plane
by
calculated
some
moving in many
200°C.
along slip aspects
of
In this note the heating effect is
in a different
manner,
of Freudenthal
We take the dislocation
and some of the
and Weiner are discussed. to be a moving line-source
of heat of strength q = b-rV erg cm-l set-1 where 6 is its Burgers 7 is the applied
vector,
shear stress.
V its velocity, The temperature
and at
(x,y) ist2) T=
‘-~ e’/”
2nK x,y
are
rectangular
dislocation, behind
K&r/R), r2 = x2 + ~2. co-ordinates
centered
y = 0 is the slip-plane,
the dislocation.
and K is the thermal
* Received March 9, 1956.
Freudenthal
the
planes is in fact of importance
Frankfurt/Main References
near
The heating effect of dislocations
assumptions
Aus dem Metall- Laboratorium
In a recent paper,(l)
“thermal
ohne
zu diskutieren.
A.G.
localized
flashes,”
einen
der zur Zeit laufenden
und riintgenographischen
these stresses are due to highly
associated with repeated and reversed slip processes, which in the extreme case may raise the
Rech-
the
K is the heat conductivity diffusivity (K divided by the
specific heat per unit volume). K,(z) has the asymptotic forms K,(z) -
on
and x is positive
--In z,
The Bessel function
z>
1;
LETTERS
TO
the scale of the temperature be many
atomic
pattern.
distances,
It will
so that near the
dislocation
(compare
alloy, of
24&T,
Seitzt3)).
in the two
T = .:Lg ln A r 77
procession
Even for fairly extreme values of
the slip-plane
T is of the order of only a fraction of a
degree a few at,omic distances
from the dislocation.
The question now is, do the temperature succession of dislocations
fields of a
add up to give an appreciable
We can make an estimate in two extreme cases.
Consider a procession
of n equally-spaced dislocations 2 the maximum spread over a distance jz. Then if A> temperature will be about
Similarly,
we would
for the hard light
suggest
a temperature-rise
about two degrees rather than fifty degrees. This discrepancy, of course, arises from the differing
values,
the constants,
rise?
561
of about half a degree.
A = 2K/V fixes
EDITOR
effective length of 10-r cm, giving a temperature-rise
The lengt’h
usually
THE
the
of
models,
of the length
dislocations.
case in which
We
dislocations
with a spacing
have move
and
dislocations
uniformly
interatomic
Weiner
distances.
Bunching
slip-plane
forces,
but it is unlikely
could
reach
may
the
and Weiner.
consider
occur
value
source, whereas by
and the temperature at a distance
will be n times what it would be (inter-dislocation
l/n
single dislocation. If A< 1, the maximum
which
.un
is identical
equation
nA
~mq
2nK
temperature
for a
will be about
s(-~-I - -a
T=
spacing)
2r
dr
A/n
suggested
by
was so severe that the distribution pile-up
and
Weiner’s
(5):
and the
If the bunching
was similar to that
of n dislocations,
the distance
between first and last, il, would be about n2 times the distance between the leading pair,(d) as against equally
2nh’
together,
then exceed
strength of the material.
of a static
spacing
Freudenthal
would require very
the array would
n times this distance
4
wit)h Freudenthal
theoretical
five
of retarding
that the average
high stresses to keep the dislocations the stress around
of
about
of the dislocations
because
Their distribution
along
by their
a procession
separated
in the
freely
determined
rate of emission from the Frank-Read Freudenthal
3, of the considered
spaced
the value
for Freudenthal
procession.
We
and Weiner’s
believe
of il for 500 dislocations,
from the simple picture
then
that
about
1O-4 cm
of the Frank-Read
source,
might be reduced to 1O-2 cm if bunching should occur, but that the figure of 10e4 cm is too small. The effect of stacking individual
where m is the spacing between successive dislocations
nearby
in unit,s of b. As before, there will be a temperature-
the possible
rise about’ n times that due to a single dislocation,
Weiner’s
provided the factor (A/J.)* is not too unfavorable. To determine the actual temperature rise, we have
by different arguments,
to estimate if one of appropriate.
plausible
the approximations A> 1, A< V has a maximum value of
fraction of the velocity It’ is commonly on one slip-plane source;
values for V, A, n, and decide
of sound, say i V,,-
assumed
that
il is some
lo5 cm/set.
all the dislocations
come from the same Frank-Read
if the sweeping velocity
of the source is about
parallel
slip-planes
rise in temperature.
equation
by Orowan.c6) temperature
by Cottrellc5) and before that
varies linearly with the amount
the slip-planes, packed
temperature. necessary
so that a large shear on many closely-
planes
could
that
simultaneously,
give
an
appreciable
For this magnified the
nearby
annealed material, 1 N 10e4 cm, and, with n = 500,R becomes about 10-r cm. Thus, A< ;1, and the second Freuform of our equation is the most appropriate. denthal and Weiner achieve a temperature rise of about 8°C on a slip-plane in aluminum, with 500 dislocations moving at Q8, in a procession 1O-4 cm in length; whereas our model suggests that this procession has an
evidence
presented
by
rise
in
heating effect’ it is
slip
and Freudenthal
suggests that’, during fatigue,
In
of glide
on each plane and inversely with the spacing between
about
source.
and
obtained
For a given shear stress, the rise in
that this is not highly probable.
of the Frank-Read
considerably
Freudenthal
7 is the same as that,
the same as the velocity of the moving dislocations, the spacing between successive dislocations will be 1, t’he length
slip processes into
increases
processes
occur
and Weiner suggest
Brawn(7)
The microscopical and
Forsyth’s)
slip only occurs simul-
taneously on planes a micron or more apart. the “striations” widening and intensifying during the test by the addition of individual slip processes one after the other. Furthermore, if displacements of 1500 A were to occur on many planes 200 A apart, as Freudenthal and Weiner suggest, the material would suffer the equivalent of a homogeneous shear of seven atomic
562
ACTA
~~T_~LLU~GICA,
diameters on each atom plane; and this is clearly not achieved over large volumes of the material. The nearest approximation to this case is fine flip, in which the displacement is about 50 k on many slip-planes 200 pi apart. Here the maximum rise in temperat~e might be a few degrees if many of t-he slip processes occurred sim~taneously. It seems inlprobable then that there can be appreciable local heating near the glide-plane, except perhaps at very high rates of loading. The creation of cracks during normal reversed stressing may be explained by the pile-up of dislocations,‘lO) and the softening processes by the local generation of vacancies.(ll) J. D. ESHELBY
VOL.
4,
1956
.ti mole 600 IPrecipitation
FIG. 1. Rate of evolution of energr- ve~‘sus t,irne-euwe for a meesurelnent at !%?.$“C.
thermostat with a very large heat capacity,t3) The specimen was located in a cent‘ral hole in the thermostat and a series of thermocouples, connected to a References sensitive galvanometer, was arranged between the 1. A. M. FREUDENTHAL and J. H. WEINER J. Appl. Phys. specimen and the wall of the thermostat in such a 27, 44 (1956). way that the deflection of the galvanometer was 2. 3-f. S. CARSLAW and J. C. JAEGERConcluction of Heat in Solids p. 234, Oxford (1947). pro~rtioi~a~ to the rise in temperature of the specimen 3. F. SE~TZA&awes in Physics 1, 43 (1952). 4. J. D. ESHELBY, F. C. FRARK, and F. R. N. N~BAREO caused by evolution of heat due to the tra~lsformation. Phil. Lwag,42, 351 (1951). As the heat liberated during the grain enlargement 5. A. H. COTTRELLDislocations and P.?astic Flow in Cry&a& of a metal is very small-at il. grain size of 0.01 mm p. 6 Oxford (1963). 6. E. OROWAN in Principles of Theological Meaewement the total boundary energy is of the magnit’ude of p. 180 Nelson (1949). 0.1 cal/mole-extreme precautions must be taken 7. A. F. BROWN Advances in Physics 1, 427 (1952). 8. P. J. E. FORSYTHJ. Inst. Metals 80, 181 (1952/53); 82, ttioreduce disturbing effects, which will be discussed 449 (1953,!54): P. J. E. FORSYTHand C. A. STUBBINGTON in a later article. The zinc specimens used ha.d the R.A.E. Report Met. 76 (1953), .iWel.78 (1954). 9. D. KuHL~A~~-W~LSDORF and H. WILSDORST Acta iWet. form of granules of a size of about 3 mm with the com4, 394 (1953). position: As < 0.000015~~“, Fe < 006~~. and Pb < 10. N. F. MOTT J. Pkys. 5’0~. Japan 10 (S), 650 (1955). 11. T. BROOM and J. H. MOLIXEUX J. Inst. .&f&a& 88, 528 O.O5O/ / 0‘ The granules were put, into a thin-walled (1954j55). aluminum container fitting into the measuring cell. * Received March 14, 1956. The junction of a Pt-PtRh peltiercouple was located at the center of the cont’ainer for calibration purposes. In Fig. I the rate of evolution of energy- versus Calorimetric Measurements of the time-curve for a measurelnent at 282.8% is shown. Grain-boundary Energy in Zinc* Although the mea,sureme~ts were carried out in an Although relative lain-boundary energies can be at,mosphere of argon, it was almost impossible to easily obtained, absolute values of these energies are avoid a small oxidation effect of the specimens, due more di&uit to estimate. The values reported in to rests of oxygen absorbed at the surface of the the literature are all indirect in the sense that they granules and the container. This oxidation effect have been obtained by comparing the energy of a gives an approximately exponential initial fall of grain boundary with that of an external surface, the measuring curves. The effect of the grain growth the surface energy of which is determined, e.g., from in the specimen, from a size of the grains of somewhat the dihedral angles formed between the soIid and a less than 0.1 mm to abont t mm, appears as a small liquid of known surface tension. This procedure, evolution of heat with a maximum after about though correct in principle, seems not to be very 3 hours, as shown Fig. 1. The form of the curve is accurate in practice. The present investigation is a bell-shaped, which may be expected for a coalescencedirect measurement of the grain-boundary energy in process. As the oxidation effect a’nd the growt,h of zinc with an isothermal calorimetric method, practiced the grains have different energies of activation, earlier.(ls 2, t.he temperatures of the measurements have been The calorimetric apparatus used for the measurechosen so that the maximum of the latter effect does ments was an electrically heated and regulated not occur until the rate of oxidation ha,s diminished.