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The effect of grain size and strain rate on the lower yield stress of pure iron at 288°K
THE
EFFECT
OF GRAIN SIZE AND STRAIN RATE ON THE STRESS OF PURE IRON AT 288”K?
LOWER
YIELD
J. HARDING$ The results of previous investigations into the effect of grain size, strain rate and temperature on the lower yield stress of a-iron are briefly disoussed in terms of the theory of thermally activated flow. Lower yield stress measurements from room temperature tensile tests on high-purity iron specimens at strain rates throughout the range from 1O-4 s-l to 2360 s-l are presented. The results ere plotted to show the effect of grain size end are analysed in terms of the equations of thermal activation. Estimates of activation volume and activation energy from the present results agree well with those obtained by other workers at oonventional rates of strain. INFL~N~ DE LA TAILLE DU GRAIN ET DE LA VITESSE DE DEFORMATION SUR LA LIMITE ELASTIQ~~ I~ERIEURE DU FER PUR A 288°K Les r&sultatsde1 Etudes ant&ieures concernant l’intluence de la trsilledu grain, de le vitesse de d6formstion et de la tempbrature sup la limite Qlastique infbrieure du fer dosont disc&Q rapidement en fonction de la, th6orie de la. plasticit aotiv6e thermiqusment. L’auteur pr6sente des mesures de la contminte de limite Blastique infdrieure fournies par des essais de traction B la temp6rature ambiante effect&s sur des Qchantillons de fer de haute puret6, pour des vitesses de deiformation allant de 10-4 par seconde B 2350 par seconde. Las rcisultatssont port& sur des courbes pour montrer l’influenoe de la tail10 du grain, et sont analys6s au moyen des Equations d’aotivvation thermique. Las valeurs obtenues & partir de ces r6sultats pour le volume d’ectivation et 1’Qnergie d’activation sont en bon accord avec les valeurs obtenues par d’autres eheroheurs pour les vitesses de d$form&ion habituelles. DER
EINGLUB DER UNTERE
KORNGR~B~ UND ~GLEITGESC~I~DIGKEIT FLIEBGRENZE VON REINEM EISEN BE1 288°K
AUF
DIE
Die Ergebnisse friiherer Untersuchungen des Einflusses der KorngriiSe, der Abgleitgeschwindigk&t und der Temparatur auf die untere FlieDgrenze von a-Eisen werden im Hinblick auf die Theorie thermisch aktivierten FlieDens kurz diskutiert. Uber Messungen der unteren FlieDgrenze an hochreinen, bei Raumtemperatur zugverformten Eisenproben wird beriohtet; die Abgleitgesohwindigkeiten lagen zwischen 10e4 s-l und 2350 s-l. Die Ergebnisse sind aufgetragen, um den EinfiuR der Korngrijlje zu zeigen und werden an Hand der C&ichungen thermischer Aktivierung analysiert. Unsere Abschiitzungen des Aktivierungsvolumens und der Aktivierungsenergie stimmen gut mit Werten anderer, mit konventionellen Abgleitgesehwindigkeiten arbeitenden Aut,oren iiberein. INTRODUCTION
has been challenged on two counts. Firstly, while conflicting results regarding the temperature and strain rate dependence of 3, due in part, perhaps, to the dificulty of producing a wide range of grain sizes without at the same time altering other microstrucaV = bi + kvoY2 (1) tural features, have been obtained by different investigators,@-@ it is now generally concluded that where a* is the lower yield stress, a3 is the ‘friction’ such a dependence is either small or non-existent. stress opposing the motion of mobile dislocations and d Thus in tests at room temperature on high purity iron is the average grain diameter. Subsequently a similar specimens no change was found in 8$ over a range relationship was also found to hold for the flow stress.t3) of strain rates from 1O-3 s-l to 1O+3s-l. Further, The parameter kg in equation (I) was originally conalthough in tests on quenched and lightly aged sidered to be a measure of the stress concentration specimens a dependence of Ic, on temperature has required at the end of a slip band in one grain to been found,(lO) an increase in the extent of ageing unlock pinned dislocations in the adjacent grain. leads to an abrupt disappearance of this temperature Since the strong temperature and strain rate dependdependence at a limiting value of kg common to a wide ence of the yield stress in iron was associated with variety of pure irons and mild steels. While the dislocation unloc~ng, it was expected that the temperature dependence of ks found for the lightly parameter ks would show a similar strong temperature aged specimens is consistent with the process of and strain rate dependence. thermally activated unpinning of dislocations the More recently, however, this interpretation of Xr, abrupt change in behaviour after further ageing would indicate that a different mechanism operates t Received January 6, 1969. when dislocations are strongly pinned. Since metals $ Department of Engineering Science, University of are usually found in the annealed or normalised state, Bxford, Parks Road, Oxford. Early investigations into the effect of grain size on the yield stress of a-iron and mild steel(l*“) lead to the formulation of the empirical relationship
ACTA METALLURGICA,
VOL.
17, AUGUST
1969
949
ACTA
Thermally
Activated
METALLURGICA,
‘At hermal’ Region
Region
w* __----_-\I
0-I
L
I TEMPERATUFk
( OK )
FIG. 1. Temperature dependence of friction stress at constant strain rate. (schematic represent&ion).
strong pinning, and hence Ic, independent of temperature, is likely to be the most common condition. It has been suggested that under such conditions &, is a measure of the stress required to create new dislocations at or near the boundary of the adjacent grain,‘ll) a process relatively independent of thermal activation. If it is accepted that the parameter & shows little or no temperature dependence, the marked effect of temperature and strain rate on the yield and flow stress in iron and mild steel must appear in the friction stress term bi. Cracknell and Petch(12) first resolved (TV into two parts, an athermal component oil dependant on the structure, e.g. the presence of precipitates and such interstitials as carbon and nitrogen, and a thermal component (T* which varies strongly with temperature and strain rate. It is generally accepted that the athermal component of the friction stress arises from interactions between moving dislocations and barriers having long-range stress fields while the thermal component arises from interactions with barriers having localisecl stress fields. Thermal vibration of the lattice may assist in overcoming the latter but not the former. Experimentally determined friction stress-temperature curves having the form shown schematically in Fig. 1 are consistent with this explanation. In the athermal region above the critical temperature T, the thermal energy of the lattice is sufficient for all short-range barriers to be surmounted without delay. In the thermally activated region below T, the lattice energy is no longer sufficient and an additional, temperature clepenclant, component to the applied stress is required. At 0°K the thermal energy becomes zero and a limiting value of the friction stress, a,, say, is reached. The strain rate dependence of the friction stress follows a similar pattern except that at very high strain rates stresses in excess of a,,
VOL.
17,
1969
have been measured. Under these conditions a viscous clamping mechanism is thought to control the motion of dislocations.(l3) In an attempt to determine the physical origin of the thermal component of the friction stress many investigators have studied the temperature and strain rate sensitivities of the yield and flow stress of b.c.c. metals in the thermally activated region.(1*-20) Making the assumption that a single effective free energy of activation may be used to describe the ratecontrolling mechanism the mean dislocation velocity will be given by a relationship of the form,(lg) u = p * Yo{exp (-G/kT)
-
exp (-GJkT)}
(2)
where y. is an atomic vibration frequency, p is the distance moved after each successful activation and G and G, are the free energies of activation at an obstacle for motion in the direction of and in opposition to the local stress field. Except for temperatures close to T, the second term can be ignored and equation (2) may be rewritten in terms of strain rate to have the formc20) G= H -
TS = kT In (v/i)
(3)
where H and S are the free enthalpy and entropy of activation, 6 is the axial strain rate and v is a frequency factor given by y = Wpv,,
(4)
where 4 is an orientation factor, p is the density of mobile dislocations and b is the Burgers vector. Neglecting the effect of temperature on the elastio modulus the enthalpy of activation may be expressed in terms of the three external variables, temperature, strain rate and stress by the relationship H = --kT2
WVT);,.
In (+)lT
[aa*@
Similarly the activation dynamically by
volume,
(5)
defined thermo-
‘v = -2(aG/aa*)T
(6)
may be written as V = 2?6T{(a In @a*),
-
(a In @a*),}
(7)
In order that experimental measurements of temperature, strain rate and stress may be used in equations (5) and (7) to make estimates of H and V it has to be assumed that the frequency factor, v, remains the same from one experimental measurement to the next. This implies, as can be seen from equation (a), that the mobile dislocation density should not vary. In an attempt to meet this condition most
HARDING:
investigators technique
favour
the
involving
stress increments
use
thermal
being
LOWER
of
an
YIELD
STRESS
OF
PURE
951
IRON
experimental
or strain rate cycling,
measured
with respect
to a
stress-strain
curve corresponding to a basic temperature and strain rate. The disadvantage of this method is that the range of strain rates which may be used is To make measurements
strictly limited.
stress levels, therefore, rate
changes
temperature
are required.
at the highest
as well as strain
Consequently
it is not
possible to separate with certainty the apparent effects of stress from what may be intrinsic effects of temperature on calculated
values of
V.
In the present investigation the room-temperature lower yield stress of polycrystalline pure iron is measured
at strain rates from lop4 se1 to 2 x lo3 s-l.
By means of a Petch analysis the thermal component of the friction dependence
stress is isolated
determined.
may be evaluated from
and its strain rate
Thus the activation
volume
over a very wide range of stresses
experimental
results
obtained
at
one
test
temperature. EXPERIMENTAL
Tests
were performed
covering
DETAILS
in tension
at strain
rates
a range of more than seven orders of mag-
nitude,
from
10e4 to 2350 s-l,
using three different
designs of loading machine. At the lowest rates, in the range 10 4 s-l to lop2 s-l, a standard Instron screw-driven structed
machine,
fitted
with
tensile rig to improve
was employed. achieved
That
good
a specially
axiality
axiality
with this rig could
con-
of loading,
of loading
was
be seen from the large
yield drops often found in tests on the finer grained specimens. At
intermediate
strain
rates,
to about 100 s-l, a hydraulically machine
was used.
from
This machine
in detail elsewhere.(21)
about
0.2 s-l
operated rapid loading has been described
A typical
set of oscilloscope
records for a test at a nominal strain rate of 50 s-1 with this machine Fig. 2a. fixed
a load
cell in series with
the variation
the test. velocity
specimen
is shown in
The upper trace, derived from strain gauges
to
follows
on a fine grained
The lower transducers
(b) FIG. 2. Test oscillograms--fine grained specimens. (a) Hydraulic loading machine-upper trace, stress; lower trace, velocity. Time base, 1 ms/cm. (b) Impact loading machine-impact velocity, 9.7 m/s; timing pips at 2 ps intervals.
the
specimen,
specimen
of stress with time throughout
elastic
trace,
is required.
derived
in parallel
from
a pair
of
with the specimen,
is shown in Fig. 2b.
For the calculation
of
strain and strain rate during such a test a separate stress record at the relevant impact velocity The technique for calculating the dynamic
is used to calculate the levels of strain and strain rate
stress-strain curve from these records has been outlined elsewhere.‘**)
during the test. To attain the highest strain rates, approximately 500 s-l to 2500 s-l, tests were performed on a dropweight impact loading machine very similar in construction to that described previously.(aa) A typical
Specimens were machined from high-purity iron rods supplied by the National Physical Laboratory. The ingot from which they were obtained had been
oscilloscope obtained
record
of
specimen
stress
vs.
time
during an impact from 5 m on a fine grained
SPECIMEN
MATERIAL
hot rolled at 115O’C to a diameter of 16 mm. The rods were then swaged to a final diameter of 6.3 mm,
ACTA
962
METALLURGICA,
being cold worked by some 80 per cent in the process. The analysis of the specimen material is given in Table 1. 1. Analysis of high-purity iron (wt. %)
TABLE
C 0.004
Si 0.003
S 0.001
GRAIN
P 0.002 SIZE
0 0.0012
N 0.0005
H 0.00003
TREATMENTS
In order to permit an unambiguous determination of the effect of grain size on mechanical properties, specimens of as wide a range of grain sizes as possible are required. This, in turn, implies the application of a widely differing set of heat treatment processes. The criticism has been made that such widely differing heat treatments could also produce other microstructural differences affecting the mechanical properties of the various specimens, In the present work, therefore, an attempt was made to standardise as far as possible the various heat treatment procedures. All specimens were machined from the 6.3 mm diameter cold swaged rods. They were then annealed in vacua followed by slow cooling in the furnace. Originally it was hoped to obtain a sufficiently wide range of grain sizes by varying only the heating rate and the soaking time at a fixed maximum temperature. The former affects the number of sites at which recrystallisation commences, the latter the extent to which they may grow. In practice, however, it was also found necessary to vary the maximum (soaking) temperature and thus the rate at which the recrystallised regions grew from the nucleation sites. The four standard heat treatments that were used are listed in Table 2. It can be seen that for three of these treatments the soaking temperature differed by only 25“C. The four treatments listed, therefore, were considered to represent a reasonable compromise between the two opposing objectives, on the one hand the production of a wide range of grain sizes, on the other the application of not too dissimilar a set of heat treatmentxs. In practice, when on three occasions anomalous behaviour was found in subsequent mechanical tests it was due to grain boundary failure in the coarsest grained specimens. Since the soaking TABLE
2. Grain size heat treatment
Heating rate (“C/hr)
Soaking temperature (“C)
Soaking time (mins)
f: 1600 1600
720 880 900 905
60 :“o 600
Grain size range (mm-l/*) (g/mm*) 460-700 150-260 40-60 IO-16
4.6-5.2 3.6-4.0 2.5-2.8 1.7-2.0
VOL.
17,
1969
temperature for these specimens (905’C) was so close to the A, transition (910°C) this is perhaps not surprising. No consistent effect on mechanical behaviour of the biggest difference in soaking temperature, between specimens of the two finest grain sizes, was detected. As was found in earlier work,(g) when a strainanneal technique is applied to pure iron specimens a range of grain size values may result from any given heat treatment. In consequence, therefore, it was necessary, after testing, to measure the grain size of each specimen, the average grain diameter (d) being determined from a count of the number of grains (n) per square millimeter of a random cross-section of the specimen, using the relation d =
&/2
Furthermore, the geometry of the swaging process is such that a non-equiaxed structure is sometimes obtained, grains being produced during the course of recrystallisation which are elongated in the axial direction. This behaviour is most marked in the fine grained specimens and is almost completely absent from those of the coarsest grain size. To allow for this effect grain size determinations were made on longitudinal as well as transverse sections for each specimen, the mean of the two results being used. RESULTS
In the first part of the investigation three series of tests were performed, at nominal strain rates of 10W3s-l, in the Instron loading machine, 50 s-l, in the hydraulic loading machine, and 2000 s-l, equivalent to an impact velocity of 9.7 m/s, in the drop weight loading machine. Specimens covering the whole range of grain sizes were tested at each of these three rates. Typical stress-strain curves for coarse and fine grained specimens tested at the two lower rates and at the highest rate used in this investigation, equivalent to an impact velocity of 14.6 m/s, are shown in Figs. 3 and 4 respectively. The actual mean plastic strain rate applied during the test is quoted in each case, this being, in general, a little higher than that over the lower yield region. Lower yield stress levels were determined from the Instron test records and from oscilloscope traces similar to those of Fig. 2 for tests on specimens of differing grain size. The variation of the lower yield stress with the inverse square root of the grain diameter at the three strain rates is shown in Fig. 5. The results were sufficiently numerous for regression lines to be calculated at each rate, these being drawn as full lines in Fig. 5. The slopes of these lines, which are a
HARDING:
LOWER
YIELD
\ 10
2b
STRAIN
40
(PER CE$
FIG. 3. Stress-strain curves-coarse grained specimens. Approximate grain size, 10/mme. Mean strain rate; A, 10-s s-1; B, 43 s-‘; C, 2350 s-l.
measure of the parameter k, in equation (l), all fell within the range 0.76-0.87 MNm”12. In view of the limited range of grain sizes and the scatter in the lower yield stress levels, particularly at the higher strain rates, the accuracy with which kv can be measured by this technique is probably less than -&lo per cent. Nevertheless, the results confirm that there is no systematic change in k, over a very wide range of strain rate and show close agreement with the result of earlier work both in the magnitude and the strain rate independence of Ic,. Having determined the general trend of behaviour a further series of tests was performed at a number of intermediate strain rates, only a limited number of specimens being tested at each rate. The complete set of results is shown in Fig. 6, the dashed lines being taken from an earlier investigation(g) and the full lines devoid of experimental points from Fig. 5. At all strain rates except two, the lowest, 0.0001 s-l, and the highest, 2300 s-l, the slopes of the lines showing best fit with the experimental points fall within the same range, 0.76 to 0.87 MNm-a2, as found for the regression lines of Fig. 5. The slopes of the two lines where this was not the case have been arbitrarily chosen, therefore, to conform with the majority behaviour. Values of oi and kv, as defined in equation (l), derived from the straight lines of Fig. 6 are listed in Table 3. The two kg values determined from the lines of arbitrarily chosen slope are denoted by an asterisk. DISCUSSION
The results of Figs. 5 and 6 provide a clear indication that the Hall-Petch relationship of equation (1) is
STRESS
OF
PURE
953
IRON
obeyed when pure iron specimens are tested at room temperature over a very wide range of strain rates. This confirms, in greater detail, the results of a previous investigation.@) Also in agreement with the earlier work a significant effect of strain rate on kvT the slope of the lower yield stress-grain size relationship, could not be detected. This implies that ku should also be independent of temperature. Although an increase in le, has sometimes been found at low temperatures, such behaviour is usually thought to result from the onset of mechanical twinning. Some doubt is thrown on this explanation, however, when it is considered that the present results show no increase in ku at the impact loading rates even though mechanical twinning was frequently found under such conditions. Results from impact tests on high-purity iron single crystals’23) show that although twinning does occur in tests at room temperature, its extent is insufficient to modify significantly the yield behaviour. This may well be true also in the present polycrystalline tests, in which case no discrepancy arises between the results of this investigation and the interpretation placed on earlier results. A fuller 900
800
700
600
(v^ 500 E t z - 400
200
0
4b
i0
lb STRAIN
(PER CEN;
FIQ. 4. Stress-strain curves-fine grained specimens.Approximate grain size, 600/mms. Mean strain rate; A, lo-* s-1; B, 43 s-‘; C, 2360 s-l.
954
ACTA
METALLURGICA,
VOL.
17,
1969
FIG. 5. Effect of grain size and strain rate on the lower yield stress. Regression lines at three strain rates (6,=iimean strain rate during yield).
0.25
0
1
mm”@.)
5
4
6
FIG. 6. Effect of grain size and strain rate on the lower yield stress. Collected results.
TABLE 3. Effect of Strain rate on CT{ and &, for high-purity iron at 288%
(a) Present work Strain r&e at yield 1 errM$np k, MNm-W
10-4
IO-8
10-a
0.24
2.46
19.0
36.0
43.0
500
1400
2300
6.9 @.$I*
25.2 0.85
31.4 0.87
86.0 0.85
135 0.83
f95 0.85
213 0.85
231 0.81
318 0.85
366 0.76
363 0.83+
(b) Previous work’@) Mean strain rate s-1 CT~ MN/m* k; Mi%mz-a~ e
10-a 20.7 0.81
I 000 345 0.83
2600 394 0.81
HARDING:
LOWER
YIELD
clarification of this point must await the completion of pure iron impact tests at low temperatures. In the absence of such tests the present results may be considered to provide further evidence in support of an athermal mechanism for the propagation of yield from one grain to the next, as, for example, by the creation of new dislocations at or near the grain boundary. The strain rate dependence of the friction stress, bi, is shown in Fig. ‘7. This curve corresponds to that for the temperature dependence of the friction stress shown schematically in Fig. 1, the athermal component, oil, being given in both cases by the stress level at which the curve first becomes horizontal. In practice the curve of Fig. 7 continues to show a small strain rate dependence even at the lowest strain rate, 104 s-l, at which tests were performed. In addition, the point of zero slope is rather sensitive to the accuracy of the experimental results in the low strain rate region. However, an estimate of 7 MN/m2 for oi’ from Fig. 7 is unlikely to be greatly in error, the low value obtained reflecting the purity of the iron being studied. While it is difficult to conceive of a method by which the effect of strain rate on the mobile dislocation density could be determined, there is some evidence(24) that in molybdenum the total dislocation density for strains up to 10 per cent increases by no more than a factor of 2 when the applied strain rate is raised from 2 x 1O-5 s-l to 2 x lo3 s-l. Assuming that iron behaves in a similar way and that the percentage of mobile dislocations remains about the same at all strain rates, the (a In Y/%*), term in equation (7) can reasonably be neglected and the expression for
STRESS
OF PURE
IRON
“‘d 200 1
Fro. 8. Effect of Stress on Activation Volume. Conrad(=‘, -; Present work -(Of.
103 102 to’ STRAIN
too
16
RATE
d
8
(5-l f
Variation of friction stress with strain rate. Present work-(x ); Campbell and Harding@‘-(0)
FIU. 7.
After
activation volume reduces to V = 2kT(8 In _@u*)~
(7a)
Thus the activation volume at any given stress level is inversely proportional to the slope of the curve of Fig. 7. The variation of activation volume with the thermal component of the applied shear stress, r*, is given in Fig. 8, T* being defined by 7* = +i(f.r$ - 0;)
(7b)
and oi’ being taken as 7 MN/ms. The experimental results are in good agreement with the results of earlier analyses indicated by the full line which comes from the work of Conrad.‘“*) Since the earlier results were derived, in the main, from strain rate change experiments at relatively low rates and varying temperatures the agreement with the present results, obtained at a fixed temperature over a very wide range of strain rates, is of significance. At high stresses, r* > 150 MN/m2, the activation volume is almost independent of stress and has a value of about 12bs. In the absence of information from the present tests on the temperature sensitivity of the lower yield stress a direct calculation of the enthalpy of activation using equation (5) is not possible. Several investigators(16~1s~20~ have shown that H, determined in this way, varies in a linear manner with temperature at constant strain rate. Taken in conjunction with equation (3) this implies (lg) that H may be expressed in terms of the simple rate equation,
H = bT In (~‘/8) 104
955
(3)
where Y’ is a modified frequency factor, determined from the slope of the H-T plot, only equal to Y if the entropy of activation is small enough to be ignored. Equation (8) may be used to estimate the variation
ACTA
956
METALLURGICA,
H (eV)
1:1 ,‘-.,, 0
50
100
150
200
FIG. 9. Effect of Stress on Activation Enthalpy. Conrad”8J, -; Present work-( 0).
After
of H with strain rate and hence, from Fig. 7, with r*, provided a value can be placed on v’. A variety of experimental values are available. According to Conrad(l*) v’ may vary from between lo8 s-l, for impure iron (C + N + 0 > 0.02 wt.%) to 1011s-l for pure iron (C + N + 0 < 0.005 wt. %). Altshuler and Christian,(20) working with a very pure iron (C + N + 0 = 0.002 wt. %), found v’ to be about 107s-1 at low temperatures changing to about 1012s-i near room temperature. To obtain reasonable agreement at high stresses when the stress variation of H calculated from equation (8) for the present tests is compared with the results of earlier workos) it is necessary to use for v’ a value of lo* s-l. While this lies within the very wide range of experimentally determined values quoted above it is considerably smaller than the only measurement at room temperature which gave v’ = 1012s-l. This suggests that the frequency factor at room temperature and high strain rates may correspond to that at low temperatures and conventional strain rates. The variation of activation enthalpy with stress is shown in Fig. 9. At low stresses H approaches Ho = 0.7 eV (H = Ho when T* = 0). This is rather higher than the value of 0.53 eV quoted by Conrado@ and the values of 0.56 eV and 0.63 eV determined by Christian and Mastera. Certain features of the strain rate dependence curve of Fig. 7 conflict with the results of other investigators. Campbell and Ferguson,(25) who tested a 0.12 oA C mild steel in shear over a similar range of strain rates, found a linear relationship between the lower yield stress and the logarithm of the strain rate in the thermally activated region, implying that the activation volume is independent of stress. They also found a linear relationship for which a small
VOL.
17,
1969
strain rate dependence was still evident in the socalled ‘athermal’ region. Davidson et uZ.(as) tested a high-purity iron (C + 0 + N = 0.0019%) in compression at strain rates from 1O-4 s-l to lo3 s-l and found an almost linear strain rate dependence of the flow stress at 2 per cent strain throughout this range, there being no evidence of an athermal region. Although both Sakui and Morit2’) and Campbell and Cooper@) found a non-linear dependence of stress on the logarithm of the strain rate similar to that shown in Fig. 7 in each case the results could also be described with fair accuracy by two regions of linear dependence such as were found by Campbell and Ferguson.(25) While it is difficult to reconcile these conflicting results some comments may be made. It is usually assumed that the athermal resistance to dislocation motion is due to a constant internal stress. However, for reasons given by Li,(2g) a fluctuating internal stress field seems much more likely. Consequently at any given instant there will be some points within the crystal for which the internal stress field will be at such a level that the long-range barriers in that area are only just insurmountable. Under these conditions the thermal energy of the lattice could well be sufficient to effect the surmounting of the barrier, thereby introducing a limited thermal dependence in the otherwise ‘athermal’ region. The extent to which this effect will be apparent is likely to depend on the purity of the material being tested. Thus for very pure material, such as was used by Davidson et uZ.(~~)and in the present tests, where the ‘athermal’ component is, in any case, extremely small and where the spacing of the long-range barriers is expected to be of the order of the grain diameter, the rate dependence in the ‘athermal’ region is likely to be insignificant. In less pure materials such as that used by Campbell and Ferguson, where the long-range barriers are spaced more closely and play a bigger part in the overall deformation behaviour, a small rate dependence in the ‘athermal’ region, of the type detected experimentally, might reasonably be expected. The use of the lower yield stress in strain rate sensitivity measurements, as in the present tests and in the work of Campbell and Cooper,‘28) introduces complicating factors. Since the effective gauge length during lower yield is limited to the Liiders front region, the controlling strain rate may be as much as 2 orders of magnitude greater than the applied strain rate.(30) In consequence, to a first approximation, the whole of the curve of Fig. 7 should be moved parallel to the strain rate axis by this amount. The measured activation volumes of Fig. 8 will remain unchanged
HARDING:
LOWER
YIELD
since they depend only on the slope of the curve at specified stress levels. The modified frequency factor, v’, required in equation (8) to give the results of Fig. 9, will be increased, however, to a value somewhat nearer to that of 1012s-1 measured by Altshuler and Christian at room temperature. A second order effect, which could lead to a change in the activation volumes from those shown in Fig. 8, will arise if the ratio of the applied strain rate to that in the Liiders front varies with the rate at which the specimen is deformed. Such would be the case if either the width of the Liiders front or the number of Liiders bands involved in the deformation process were to change with strain rate. Campbell et aZ.t31) have shown that at high strain rates a large number of Liiders bands may be formed so that the ratio between the effective and the applied strain rate will be small. This corresponds to an increased curvature in the high strain rate region of Fig. 7 and thus to a reduction in the limiting value of the activation volume at high stresses below that shown in Fig. 8. At the lowest stresses, r* < 50 MN/m2 or g < 1 s--1, the activation volume, as given by equation (7a), increases very rapidly. Whether or not this is a genuine effect may be confirmed by using the full equation (2) to derive a more accurate expression for the activation volume at low stresses. The desired expression has the form (see appendix) 2kT(a In @a*), F = where +, is given by
2/(1 + &z/is)
.6, = 2v exp (-G,,/kT)
(9)
(10)
and G, is the total energy needed to overcome the obstacle. Thus when i > 6, equation (9) approximates to equation (7a) and no correction is needed. When 6 < d,, however, the activation volume is reduced significantly below that calculated from equation (7a). Taking for G, and v the values H,, = 0.7 eV and v’ = lo* s-l, as were used in equation (S), we obtain for 6, a value of 4 x lop4 s-i. A comparison of equation (10) with equation (3) suggests that .6,,may reasonably be interpreted in physical terms as twice the strain rate reached when thermal energy alone is sufficient to overcome the obstacle, i.e. &, approximately equals 10e4 s-l, twice the strain rate at which the curve of Fig. 7 first becomes horizontal. In view of the uncertainty involved in the choice of values for G,, and v the agreement between these two estimates of 6, is closer than might be expected. Since, by equation (9), the activation volume becomes infinitely large as the slope of Fig. 7, (au*/8 In &)*,, tends to 3
STRESS
OF
PURE
957
IRON
zero, whichever estimate of &, is used there will be
little difference between the activation volumes derived from equations (7a) and (9), even at the lowest stresses. It is clear from the foregoing discussion that neither a consideration of the Liiders strain effect nor the use of the full rate equation at low stresses will lead to activation volumes differing significantly from those calculated using equation (7a). Thus, with the one assumption that the frequency factor, v, is independent of strain rate, the results of the present investigation strongly support the conclusion that the activation volume follows a stress variation of the type shown in Fig. 8. CONCLUSIONS
The lower yield stress of pure iron tested at room temperature and strain rates from 10e4 5-l to 2350 s-l is shown to obey the Hall-Petch equation. The parameter Ic, is found to be independent of strain rate throughout this range, although mechanical twinning frequently occurred in tests at the highest loading rates. The strain rate independence of Ic, confirms previous suggestions that the propagation of yield from one grain to the next is governed by an athermal mechanism such as the creation of new dislocations at or near the grain boundary. The effect of strain rate on the Hall-Petch equation is confined to the thermal component of the friction stress which, in tests at room temperature, increases from zero at a strain rate of slightly less than 10-4 s-1 to about 200 MN/m2 at a strain rate of 2350 s--l, the athermal component being about 7 MN/m2. The variation of activation volume with stress at a fixed temperature closely follows the results obtained by other investigators in work at conventional strain rates and different temperatures. As has been argued elsewhere,~1s*20)the low activation volumes, about
12b3, found
support
MN/m2, trolling
at high stresses, r* > 150
the conclusion
mechanism
does
not
that the rate involve
con-
impurity
interactions. ACKNOWLEDGEMENTS
The author is grateful for many valuable discussions with Drs. J. D. Campbell and T. L. Briggs and for the assistance
of Mr.
experimental
R.
C. Stone
with
much
of the
work.
APPENDIX: DERIVATION OF THE ACTIVATION VOLUME AT LOW THERMAL STRESSES Let the work done during the activation process described by equation (2) be given by G=G,-
V*r*;
G, = Go +
V*T*
958
ACTA
METALLURGICA,
where G,, is the total energy needed to overcome the obstacle and V*T* is the work done by the thermal stress r*. Then equation (2) may be rewritten in the form 6 = 2~ sinh (V*r*/kT)
exp (-G,,/kT)
which gives V*r*/kT
= sinh-l (i/&J = (G, -
G)/kT
where &, = 2v exp (-G,,/kT). Differentiating with respect to (r* at constant temperature we obtain
From the definition of activation volume, equation (6), we have
and hence
which may be rearranged to give the required relationship (equation 9)
a In B v = d/(1 :k&)
( &r* ) T
REFERENCES E. N. H. M.
17, 1969
5. D. HULL
and I. L. MOOFORD, Phil. Meg. 3, 1213 (1968). 6. D. HULL, Acta Met. 9, 191 (1961). 7. T. J. FISCHHOF, D. Phil. dissertation. Oxford (1963). 8. G. R. BORWICK and MISS. E. S. FRASER, NEL Report No. 323 (1967). 9. J. D. CAMPBELL and J. HARDINO., Response of Metals to High VeZocity Defamation, p. 51. Interscience (1961). 10. R. M. FISHES, Ph.D dissertation, Cambridge (1962). 11. A. H. COTTRELL, The Relation between the Structure and Mechanical Properties of Metals, Vol. 2, p. 456. H.M.S.O. (1963). 12. A. CRACKNELL and N. J. PETCH, Acta Met. 3, 186 (1956). 13. W. G. FEROUSON, A. KUMAR and J. E. DORN, J. appl. Phys. 38, 1863 (1967). 14. Z. S. BASINSKI and J. W. CHRISTIAN, Au&. J. Phya. 13, 299 (1960). 15. H. CONRAD, J. Iron and Steel Inst. 198,364 (1961). 16. H. CONRAD and S. FREDERICK, Acta Met. 10,1013 (1962). 17. H. CONRAD and W. HAYES, Trans. Am. Sot. Metals 56, 125 (1963). 18. H. CONRAD, The Relation between the Structure a& Mechanical Proper&s of Metals, Vol. 2, p. 476. H.M.S.O. (1963). 19. J. W. CHRISTIAN and B. C. MASTERS, Proc. R. Sot. A281,
240 (1964). 20. T. L. ALTSHULER and J. W. CHRISTIAN, Phil. Trans. R. Sot. A261. 253 (1967). 21. R. H. COOPER and J: D. CAMPBELL, J. Mech. Engng Sci.
V -= 2kT
1. 2. 3. 4.
VOL.
0. Him, Proc. phys. Sot. B94, 747 (1951). J. PITCH, J. Iron Steel In&. 174,25 (1963). CONRAD and G. SCHOEK, Acta Met. 8, 791 (1960). M. HUTCHINSON, Phil. Mug. 8, 121 (1963).
9, 278 (1967). 22. J. HARDING, E. 0. WOOD and J. D. CAMPBELL, J. Mech. Engng Sci. 2, 88 (1960). 23. J. HARDING, PTOC. R. Sot. A299, 464 (1967). 24. A. GILBERT, B. A. WILCOX and G. T. HAHN, Phil. Mag. 12, 649 (1965). 25. J. D. CAMPBELL and W. G. FERCXJSON(to be published). 26. D. L. DAVIDSON, U. S. LINDHOLM and L. M. YEAKLEY, Acta Met. 14, 703 (1966). 27. S. SAKUI and T. MORI, Trans. Japan Inst. Metal 7, 71 (1966). 28. J. D. CAMPBELL and R. H. COOPER, Phyysical Basis of Yield and Fracture, p. 77, Institute of Physics and Physical Society (1966). 29. J. C. M. LI, Dislocation Dynamics, p. 87. McGraw-Hill (1968). 30. D. B. C. TAYLOR and L. E. MALVERN, Response of Metak to High l’e’elocity Deforoaation, p. 77. Interscienoe (1961). 31. J. D. CAMPBELL, R. H. COOPER and T. J. FISCHHOF, Dislocation Dynamics, p. 723. McGraw-Hill (1968).
EFFECT
OF GRAIN SIZE AND STRAIN RATE ON THE STRESS OF PURE IRON AT 288”K?
LOWER
YIELD
J. HARDING$ The results of previous investigations into the effect of grain size, strain rate and temperature on the lower yield stress of a-iron are briefly disoussed in terms of the theory of thermally activated flow. Lower yield stress measurements from room temperature tensile tests on high-purity iron specimens at strain rates throughout the range from 1O-4 s-l to 2360 s-l are presented. The results ere plotted to show the effect of grain size end are analysed in terms of the equations of thermal activation. Estimates of activation volume and activation energy from the present results agree well with those obtained by other workers at oonventional rates of strain. INFL~N~ DE LA TAILLE DU GRAIN ET DE LA VITESSE DE DEFORMATION SUR LA LIMITE ELASTIQ~~ I~ERIEURE DU FER PUR A 288°K Les r&sultatsde1 Etudes ant&ieures concernant l’intluence de la trsilledu grain, de le vitesse de d6formstion et de la tempbrature sup la limite Qlastique infbrieure du fer dosont disc&Q rapidement en fonction de la, th6orie de la. plasticit aotiv6e thermiqusment. L’auteur pr6sente des mesures de la contminte de limite Blastique infdrieure fournies par des essais de traction B la temp6rature ambiante effect&s sur des Qchantillons de fer de haute puret6, pour des vitesses de deiformation allant de 10-4 par seconde B 2350 par seconde. Las rcisultatssont port& sur des courbes pour montrer l’influenoe de la tail10 du grain, et sont analys6s au moyen des Equations d’aotivvation thermique. Las valeurs obtenues & partir de ces r6sultats pour le volume d’ectivation et 1’Qnergie d’activation sont en bon accord avec les valeurs obtenues par d’autres eheroheurs pour les vitesses de d$form&ion habituelles. DER
EINGLUB DER UNTERE
KORNGR~B~ UND ~GLEITGESC~I~DIGKEIT FLIEBGRENZE VON REINEM EISEN BE1 288°K
AUF
DIE
Die Ergebnisse friiherer Untersuchungen des Einflusses der KorngriiSe, der Abgleitgeschwindigk&t und der Temparatur auf die untere FlieDgrenze von a-Eisen werden im Hinblick auf die Theorie thermisch aktivierten FlieDens kurz diskutiert. Uber Messungen der unteren FlieDgrenze an hochreinen, bei Raumtemperatur zugverformten Eisenproben wird beriohtet; die Abgleitgesohwindigkeiten lagen zwischen 10e4 s-l und 2350 s-l. Die Ergebnisse sind aufgetragen, um den EinfiuR der Korngrijlje zu zeigen und werden an Hand der C&ichungen thermischer Aktivierung analysiert. Unsere Abschiitzungen des Aktivierungsvolumens und der Aktivierungsenergie stimmen gut mit Werten anderer, mit konventionellen Abgleitgesehwindigkeiten arbeitenden Aut,oren iiberein. INTRODUCTION
has been challenged on two counts. Firstly, while conflicting results regarding the temperature and strain rate dependence of 3, due in part, perhaps, to the dificulty of producing a wide range of grain sizes without at the same time altering other microstrucaV = bi + kvoY2 (1) tural features, have been obtained by different investigators,@-@ it is now generally concluded that where a* is the lower yield stress, a3 is the ‘friction’ such a dependence is either small or non-existent. stress opposing the motion of mobile dislocations and d Thus in tests at room temperature on high purity iron is the average grain diameter. Subsequently a similar specimens no change was found in 8$ over a range relationship was also found to hold for the flow stress.t3) of strain rates from 1O-3 s-l to 1O+3s-l. Further, The parameter kg in equation (I) was originally conalthough in tests on quenched and lightly aged sidered to be a measure of the stress concentration specimens a dependence of Ic, on temperature has required at the end of a slip band in one grain to been found,(lO) an increase in the extent of ageing unlock pinned dislocations in the adjacent grain. leads to an abrupt disappearance of this temperature Since the strong temperature and strain rate dependdependence at a limiting value of kg common to a wide ence of the yield stress in iron was associated with variety of pure irons and mild steels. While the dislocation unloc~ng, it was expected that the temperature dependence of ks found for the lightly parameter ks would show a similar strong temperature aged specimens is consistent with the process of and strain rate dependence. thermally activated unpinning of dislocations the More recently, however, this interpretation of Xr, abrupt change in behaviour after further ageing would indicate that a different mechanism operates t Received January 6, 1969. when dislocations are strongly pinned. Since metals $ Department of Engineering Science, University of are usually found in the annealed or normalised state, Bxford, Parks Road, Oxford. Early investigations into the effect of grain size on the yield stress of a-iron and mild steel(l*“) lead to the formulation of the empirical relationship
ACTA METALLURGICA,
VOL.
17, AUGUST
1969
949
ACTA
Thermally
Activated
METALLURGICA,
‘At hermal’ Region
Region
w* __----_-\I
0-I
L
I TEMPERATUFk
( OK )
FIG. 1. Temperature dependence of friction stress at constant strain rate. (schematic represent&ion).
strong pinning, and hence Ic, independent of temperature, is likely to be the most common condition. It has been suggested that under such conditions &, is a measure of the stress required to create new dislocations at or near the boundary of the adjacent grain,‘ll) a process relatively independent of thermal activation. If it is accepted that the parameter & shows little or no temperature dependence, the marked effect of temperature and strain rate on the yield and flow stress in iron and mild steel must appear in the friction stress term bi. Cracknell and Petch(12) first resolved (TV into two parts, an athermal component oil dependant on the structure, e.g. the presence of precipitates and such interstitials as carbon and nitrogen, and a thermal component (T* which varies strongly with temperature and strain rate. It is generally accepted that the athermal component of the friction stress arises from interactions between moving dislocations and barriers having long-range stress fields while the thermal component arises from interactions with barriers having localisecl stress fields. Thermal vibration of the lattice may assist in overcoming the latter but not the former. Experimentally determined friction stress-temperature curves having the form shown schematically in Fig. 1 are consistent with this explanation. In the athermal region above the critical temperature T, the thermal energy of the lattice is sufficient for all short-range barriers to be surmounted without delay. In the thermally activated region below T, the lattice energy is no longer sufficient and an additional, temperature clepenclant, component to the applied stress is required. At 0°K the thermal energy becomes zero and a limiting value of the friction stress, a,, say, is reached. The strain rate dependence of the friction stress follows a similar pattern except that at very high strain rates stresses in excess of a,,
VOL.
17,
1969
have been measured. Under these conditions a viscous clamping mechanism is thought to control the motion of dislocations.(l3) In an attempt to determine the physical origin of the thermal component of the friction stress many investigators have studied the temperature and strain rate sensitivities of the yield and flow stress of b.c.c. metals in the thermally activated region.(1*-20) Making the assumption that a single effective free energy of activation may be used to describe the ratecontrolling mechanism the mean dislocation velocity will be given by a relationship of the form,(lg) u = p * Yo{exp (-G/kT)
-
exp (-GJkT)}
(2)
where y. is an atomic vibration frequency, p is the distance moved after each successful activation and G and G, are the free energies of activation at an obstacle for motion in the direction of and in opposition to the local stress field. Except for temperatures close to T, the second term can be ignored and equation (2) may be rewritten in terms of strain rate to have the formc20) G= H -
TS = kT In (v/i)
(3)
where H and S are the free enthalpy and entropy of activation, 6 is the axial strain rate and v is a frequency factor given by y = Wpv,,
(4)
where 4 is an orientation factor, p is the density of mobile dislocations and b is the Burgers vector. Neglecting the effect of temperature on the elastio modulus the enthalpy of activation may be expressed in terms of the three external variables, temperature, strain rate and stress by the relationship H = --kT2
WVT);,.
In (+)lT
[aa*@
Similarly the activation dynamically by
volume,
(5)
defined thermo-
‘v = -2(aG/aa*)T
(6)
may be written as V = 2?6T{(a In @a*),
-
(a In @a*),}
(7)
In order that experimental measurements of temperature, strain rate and stress may be used in equations (5) and (7) to make estimates of H and V it has to be assumed that the frequency factor, v, remains the same from one experimental measurement to the next. This implies, as can be seen from equation (a), that the mobile dislocation density should not vary. In an attempt to meet this condition most
HARDING:
investigators technique
favour
the
involving
stress increments
use
thermal
being
LOWER
of
an
YIELD
STRESS
OF
PURE
951
IRON
experimental
or strain rate cycling,
measured
with respect
to a
stress-strain
curve corresponding to a basic temperature and strain rate. The disadvantage of this method is that the range of strain rates which may be used is To make measurements
strictly limited.
stress levels, therefore, rate
changes
temperature
are required.
at the highest
as well as strain
Consequently
it is not
possible to separate with certainty the apparent effects of stress from what may be intrinsic effects of temperature on calculated
values of
V.
In the present investigation the room-temperature lower yield stress of polycrystalline pure iron is measured
at strain rates from lop4 se1 to 2 x lo3 s-l.
By means of a Petch analysis the thermal component of the friction dependence
stress is isolated
determined.
may be evaluated from
and its strain rate
Thus the activation
volume
over a very wide range of stresses
experimental
results
obtained
at
one
test
temperature. EXPERIMENTAL
Tests
were performed
covering
DETAILS
in tension
at strain
rates
a range of more than seven orders of mag-
nitude,
from
10e4 to 2350 s-l,
using three different
designs of loading machine. At the lowest rates, in the range 10 4 s-l to lop2 s-l, a standard Instron screw-driven structed
machine,
fitted
with
tensile rig to improve
was employed. achieved
That
good
a specially
axiality
axiality
with this rig could
con-
of loading,
of loading
was
be seen from the large
yield drops often found in tests on the finer grained specimens. At
intermediate
strain
rates,
to about 100 s-l, a hydraulically machine
was used.
from
This machine
in detail elsewhere.(21)
about
0.2 s-l
operated rapid loading has been described
A typical
set of oscilloscope
records for a test at a nominal strain rate of 50 s-1 with this machine Fig. 2a. fixed
a load
cell in series with
the variation
the test. velocity
specimen
is shown in
The upper trace, derived from strain gauges
to
follows
on a fine grained
The lower transducers
(b) FIG. 2. Test oscillograms--fine grained specimens. (a) Hydraulic loading machine-upper trace, stress; lower trace, velocity. Time base, 1 ms/cm. (b) Impact loading machine-impact velocity, 9.7 m/s; timing pips at 2 ps intervals.
the
specimen,
specimen
of stress with time throughout
elastic
trace,
is required.
derived
in parallel
from
a pair
of
with the specimen,
is shown in Fig. 2b.
For the calculation
of
strain and strain rate during such a test a separate stress record at the relevant impact velocity The technique for calculating the dynamic
is used to calculate the levels of strain and strain rate
stress-strain curve from these records has been outlined elsewhere.‘**)
during the test. To attain the highest strain rates, approximately 500 s-l to 2500 s-l, tests were performed on a dropweight impact loading machine very similar in construction to that described previously.(aa) A typical
Specimens were machined from high-purity iron rods supplied by the National Physical Laboratory. The ingot from which they were obtained had been
oscilloscope obtained
record
of
specimen
stress
vs.
time
during an impact from 5 m on a fine grained
SPECIMEN
MATERIAL
hot rolled at 115O’C to a diameter of 16 mm. The rods were then swaged to a final diameter of 6.3 mm,
ACTA
962
METALLURGICA,
being cold worked by some 80 per cent in the process. The analysis of the specimen material is given in Table 1. 1. Analysis of high-purity iron (wt. %)
TABLE
C 0.004
Si 0.003
S 0.001
GRAIN
P 0.002 SIZE
0 0.0012
N 0.0005
H 0.00003
TREATMENTS
In order to permit an unambiguous determination of the effect of grain size on mechanical properties, specimens of as wide a range of grain sizes as possible are required. This, in turn, implies the application of a widely differing set of heat treatment processes. The criticism has been made that such widely differing heat treatments could also produce other microstructural differences affecting the mechanical properties of the various specimens, In the present work, therefore, an attempt was made to standardise as far as possible the various heat treatment procedures. All specimens were machined from the 6.3 mm diameter cold swaged rods. They were then annealed in vacua followed by slow cooling in the furnace. Originally it was hoped to obtain a sufficiently wide range of grain sizes by varying only the heating rate and the soaking time at a fixed maximum temperature. The former affects the number of sites at which recrystallisation commences, the latter the extent to which they may grow. In practice, however, it was also found necessary to vary the maximum (soaking) temperature and thus the rate at which the recrystallised regions grew from the nucleation sites. The four standard heat treatments that were used are listed in Table 2. It can be seen that for three of these treatments the soaking temperature differed by only 25“C. The four treatments listed, therefore, were considered to represent a reasonable compromise between the two opposing objectives, on the one hand the production of a wide range of grain sizes, on the other the application of not too dissimilar a set of heat treatmentxs. In practice, when on three occasions anomalous behaviour was found in subsequent mechanical tests it was due to grain boundary failure in the coarsest grained specimens. Since the soaking TABLE
2. Grain size heat treatment
Heating rate (“C/hr)
Soaking temperature (“C)
Soaking time (mins)
f: 1600 1600
720 880 900 905
60 :“o 600
Grain size range (mm-l/*) (g/mm*) 460-700 150-260 40-60 IO-16
4.6-5.2 3.6-4.0 2.5-2.8 1.7-2.0
VOL.
17,
1969
temperature for these specimens (905’C) was so close to the A, transition (910°C) this is perhaps not surprising. No consistent effect on mechanical behaviour of the biggest difference in soaking temperature, between specimens of the two finest grain sizes, was detected. As was found in earlier work,(g) when a strainanneal technique is applied to pure iron specimens a range of grain size values may result from any given heat treatment. In consequence, therefore, it was necessary, after testing, to measure the grain size of each specimen, the average grain diameter (d) being determined from a count of the number of grains (n) per square millimeter of a random cross-section of the specimen, using the relation d =
&/2
Furthermore, the geometry of the swaging process is such that a non-equiaxed structure is sometimes obtained, grains being produced during the course of recrystallisation which are elongated in the axial direction. This behaviour is most marked in the fine grained specimens and is almost completely absent from those of the coarsest grain size. To allow for this effect grain size determinations were made on longitudinal as well as transverse sections for each specimen, the mean of the two results being used. RESULTS
In the first part of the investigation three series of tests were performed, at nominal strain rates of 10W3s-l, in the Instron loading machine, 50 s-l, in the hydraulic loading machine, and 2000 s-l, equivalent to an impact velocity of 9.7 m/s, in the drop weight loading machine. Specimens covering the whole range of grain sizes were tested at each of these three rates. Typical stress-strain curves for coarse and fine grained specimens tested at the two lower rates and at the highest rate used in this investigation, equivalent to an impact velocity of 14.6 m/s, are shown in Figs. 3 and 4 respectively. The actual mean plastic strain rate applied during the test is quoted in each case, this being, in general, a little higher than that over the lower yield region. Lower yield stress levels were determined from the Instron test records and from oscilloscope traces similar to those of Fig. 2 for tests on specimens of differing grain size. The variation of the lower yield stress with the inverse square root of the grain diameter at the three strain rates is shown in Fig. 5. The results were sufficiently numerous for regression lines to be calculated at each rate, these being drawn as full lines in Fig. 5. The slopes of these lines, which are a
HARDING:
LOWER
YIELD
\ 10
2b
STRAIN
40
(PER CE$
FIG. 3. Stress-strain curves-coarse grained specimens. Approximate grain size, 10/mme. Mean strain rate; A, 10-s s-1; B, 43 s-‘; C, 2350 s-l.
measure of the parameter k, in equation (l), all fell within the range 0.76-0.87 MNm”12. In view of the limited range of grain sizes and the scatter in the lower yield stress levels, particularly at the higher strain rates, the accuracy with which kv can be measured by this technique is probably less than -&lo per cent. Nevertheless, the results confirm that there is no systematic change in k, over a very wide range of strain rate and show close agreement with the result of earlier work both in the magnitude and the strain rate independence of Ic,. Having determined the general trend of behaviour a further series of tests was performed at a number of intermediate strain rates, only a limited number of specimens being tested at each rate. The complete set of results is shown in Fig. 6, the dashed lines being taken from an earlier investigation(g) and the full lines devoid of experimental points from Fig. 5. At all strain rates except two, the lowest, 0.0001 s-l, and the highest, 2300 s-l, the slopes of the lines showing best fit with the experimental points fall within the same range, 0.76 to 0.87 MNm-a2, as found for the regression lines of Fig. 5. The slopes of the two lines where this was not the case have been arbitrarily chosen, therefore, to conform with the majority behaviour. Values of oi and kv, as defined in equation (l), derived from the straight lines of Fig. 6 are listed in Table 3. The two kg values determined from the lines of arbitrarily chosen slope are denoted by an asterisk. DISCUSSION
The results of Figs. 5 and 6 provide a clear indication that the Hall-Petch relationship of equation (1) is
STRESS
OF
PURE
953
IRON
obeyed when pure iron specimens are tested at room temperature over a very wide range of strain rates. This confirms, in greater detail, the results of a previous investigation.@) Also in agreement with the earlier work a significant effect of strain rate on kvT the slope of the lower yield stress-grain size relationship, could not be detected. This implies that ku should also be independent of temperature. Although an increase in le, has sometimes been found at low temperatures, such behaviour is usually thought to result from the onset of mechanical twinning. Some doubt is thrown on this explanation, however, when it is considered that the present results show no increase in ku at the impact loading rates even though mechanical twinning was frequently found under such conditions. Results from impact tests on high-purity iron single crystals’23) show that although twinning does occur in tests at room temperature, its extent is insufficient to modify significantly the yield behaviour. This may well be true also in the present polycrystalline tests, in which case no discrepancy arises between the results of this investigation and the interpretation placed on earlier results. A fuller 900
800
700
600
(v^ 500 E t z - 400
200
0
4b
i0
lb STRAIN
(PER CEN;
FIQ. 4. Stress-strain curves-fine grained specimens.Approximate grain size, 600/mms. Mean strain rate; A, lo-* s-1; B, 43 s-‘; C, 2360 s-l.
954
ACTA
METALLURGICA,
VOL.
17,
1969
FIG. 5. Effect of grain size and strain rate on the lower yield stress. Regression lines at three strain rates (6,=iimean strain rate during yield).
0.25
0
1
mm”@.)
5
4
6
FIG. 6. Effect of grain size and strain rate on the lower yield stress. Collected results.
TABLE 3. Effect of Strain rate on CT{ and &, for high-purity iron at 288%
(a) Present work Strain r&e at yield 1 errM$np k, MNm-W
10-4
IO-8
10-a
0.24
2.46
19.0
36.0
43.0
500
1400
2300
6.9 @.$I*
25.2 0.85
31.4 0.87
86.0 0.85
135 0.83
f95 0.85
213 0.85
231 0.81
318 0.85
366 0.76
363 0.83+
(b) Previous work’@) Mean strain rate s-1 CT~ MN/m* k; Mi%mz-a~ e
10-a 20.7 0.81
I 000 345 0.83
2600 394 0.81
HARDING:
LOWER
YIELD
clarification of this point must await the completion of pure iron impact tests at low temperatures. In the absence of such tests the present results may be considered to provide further evidence in support of an athermal mechanism for the propagation of yield from one grain to the next, as, for example, by the creation of new dislocations at or near the grain boundary. The strain rate dependence of the friction stress, bi, is shown in Fig. ‘7. This curve corresponds to that for the temperature dependence of the friction stress shown schematically in Fig. 1, the athermal component, oil, being given in both cases by the stress level at which the curve first becomes horizontal. In practice the curve of Fig. 7 continues to show a small strain rate dependence even at the lowest strain rate, 104 s-l, at which tests were performed. In addition, the point of zero slope is rather sensitive to the accuracy of the experimental results in the low strain rate region. However, an estimate of 7 MN/m2 for oi’ from Fig. 7 is unlikely to be greatly in error, the low value obtained reflecting the purity of the iron being studied. While it is difficult to conceive of a method by which the effect of strain rate on the mobile dislocation density could be determined, there is some evidence(24) that in molybdenum the total dislocation density for strains up to 10 per cent increases by no more than a factor of 2 when the applied strain rate is raised from 2 x 1O-5 s-l to 2 x lo3 s-l. Assuming that iron behaves in a similar way and that the percentage of mobile dislocations remains about the same at all strain rates, the (a In Y/%*), term in equation (7) can reasonably be neglected and the expression for
STRESS
OF PURE
IRON
“‘d 200 1
Fro. 8. Effect of Stress on Activation Volume. Conrad(=‘, -; Present work -(Of.
103 102 to’ STRAIN
too
16
RATE
d
8
(5-l f
Variation of friction stress with strain rate. Present work-(x ); Campbell and Harding@‘-(0)
FIU. 7.
After
activation volume reduces to V = 2kT(8 In _@u*)~
(7a)
Thus the activation volume at any given stress level is inversely proportional to the slope of the curve of Fig. 7. The variation of activation volume with the thermal component of the applied shear stress, r*, is given in Fig. 8, T* being defined by 7* = +i(f.r$ - 0;)
(7b)
and oi’ being taken as 7 MN/ms. The experimental results are in good agreement with the results of earlier analyses indicated by the full line which comes from the work of Conrad.‘“*) Since the earlier results were derived, in the main, from strain rate change experiments at relatively low rates and varying temperatures the agreement with the present results, obtained at a fixed temperature over a very wide range of strain rates, is of significance. At high stresses, r* > 150 MN/m2, the activation volume is almost independent of stress and has a value of about 12bs. In the absence of information from the present tests on the temperature sensitivity of the lower yield stress a direct calculation of the enthalpy of activation using equation (5) is not possible. Several investigators(16~1s~20~ have shown that H, determined in this way, varies in a linear manner with temperature at constant strain rate. Taken in conjunction with equation (3) this implies (lg) that H may be expressed in terms of the simple rate equation,
H = bT In (~‘/8) 104
955
(3)
where Y’ is a modified frequency factor, determined from the slope of the H-T plot, only equal to Y if the entropy of activation is small enough to be ignored. Equation (8) may be used to estimate the variation
ACTA
956
METALLURGICA,
H (eV)
1:1 ,‘-.,, 0
50
100
150
200
FIG. 9. Effect of Stress on Activation Enthalpy. Conrad”8J, -; Present work-( 0).
After
of H with strain rate and hence, from Fig. 7, with r*, provided a value can be placed on v’. A variety of experimental values are available. According to Conrad(l*) v’ may vary from between lo8 s-l, for impure iron (C + N + 0 > 0.02 wt.%) to 1011s-l for pure iron (C + N + 0 < 0.005 wt. %). Altshuler and Christian,(20) working with a very pure iron (C + N + 0 = 0.002 wt. %), found v’ to be about 107s-1 at low temperatures changing to about 1012s-i near room temperature. To obtain reasonable agreement at high stresses when the stress variation of H calculated from equation (8) for the present tests is compared with the results of earlier workos) it is necessary to use for v’ a value of lo* s-l. While this lies within the very wide range of experimentally determined values quoted above it is considerably smaller than the only measurement at room temperature which gave v’ = 1012s-l. This suggests that the frequency factor at room temperature and high strain rates may correspond to that at low temperatures and conventional strain rates. The variation of activation enthalpy with stress is shown in Fig. 9. At low stresses H approaches Ho = 0.7 eV (H = Ho when T* = 0). This is rather higher than the value of 0.53 eV quoted by Conrado@ and the values of 0.56 eV and 0.63 eV determined by Christian and Mastera. Certain features of the strain rate dependence curve of Fig. 7 conflict with the results of other investigators. Campbell and Ferguson,(25) who tested a 0.12 oA C mild steel in shear over a similar range of strain rates, found a linear relationship between the lower yield stress and the logarithm of the strain rate in the thermally activated region, implying that the activation volume is independent of stress. They also found a linear relationship for which a small
VOL.
17,
1969
strain rate dependence was still evident in the socalled ‘athermal’ region. Davidson et uZ.(as) tested a high-purity iron (C + 0 + N = 0.0019%) in compression at strain rates from 1O-4 s-l to lo3 s-l and found an almost linear strain rate dependence of the flow stress at 2 per cent strain throughout this range, there being no evidence of an athermal region. Although both Sakui and Morit2’) and Campbell and Cooper@) found a non-linear dependence of stress on the logarithm of the strain rate similar to that shown in Fig. 7 in each case the results could also be described with fair accuracy by two regions of linear dependence such as were found by Campbell and Ferguson.(25) While it is difficult to reconcile these conflicting results some comments may be made. It is usually assumed that the athermal resistance to dislocation motion is due to a constant internal stress. However, for reasons given by Li,(2g) a fluctuating internal stress field seems much more likely. Consequently at any given instant there will be some points within the crystal for which the internal stress field will be at such a level that the long-range barriers in that area are only just insurmountable. Under these conditions the thermal energy of the lattice could well be sufficient to effect the surmounting of the barrier, thereby introducing a limited thermal dependence in the otherwise ‘athermal’ region. The extent to which this effect will be apparent is likely to depend on the purity of the material being tested. Thus for very pure material, such as was used by Davidson et uZ.(~~)and in the present tests, where the ‘athermal’ component is, in any case, extremely small and where the spacing of the long-range barriers is expected to be of the order of the grain diameter, the rate dependence in the ‘athermal’ region is likely to be insignificant. In less pure materials such as that used by Campbell and Ferguson, where the long-range barriers are spaced more closely and play a bigger part in the overall deformation behaviour, a small rate dependence in the ‘athermal’ region, of the type detected experimentally, might reasonably be expected. The use of the lower yield stress in strain rate sensitivity measurements, as in the present tests and in the work of Campbell and Cooper,‘28) introduces complicating factors. Since the effective gauge length during lower yield is limited to the Liiders front region, the controlling strain rate may be as much as 2 orders of magnitude greater than the applied strain rate.(30) In consequence, to a first approximation, the whole of the curve of Fig. 7 should be moved parallel to the strain rate axis by this amount. The measured activation volumes of Fig. 8 will remain unchanged
HARDING:
LOWER
YIELD
since they depend only on the slope of the curve at specified stress levels. The modified frequency factor, v’, required in equation (8) to give the results of Fig. 9, will be increased, however, to a value somewhat nearer to that of 1012s-1 measured by Altshuler and Christian at room temperature. A second order effect, which could lead to a change in the activation volumes from those shown in Fig. 8, will arise if the ratio of the applied strain rate to that in the Liiders front varies with the rate at which the specimen is deformed. Such would be the case if either the width of the Liiders front or the number of Liiders bands involved in the deformation process were to change with strain rate. Campbell et aZ.t31) have shown that at high strain rates a large number of Liiders bands may be formed so that the ratio between the effective and the applied strain rate will be small. This corresponds to an increased curvature in the high strain rate region of Fig. 7 and thus to a reduction in the limiting value of the activation volume at high stresses below that shown in Fig. 8. At the lowest stresses, r* < 50 MN/m2 or g < 1 s--1, the activation volume, as given by equation (7a), increases very rapidly. Whether or not this is a genuine effect may be confirmed by using the full equation (2) to derive a more accurate expression for the activation volume at low stresses. The desired expression has the form (see appendix) 2kT(a In @a*), F = where +, is given by
2/(1 + &z/is)
.6, = 2v exp (-G,,/kT)
(9)
(10)
and G, is the total energy needed to overcome the obstacle. Thus when i > 6, equation (9) approximates to equation (7a) and no correction is needed. When 6 < d,, however, the activation volume is reduced significantly below that calculated from equation (7a). Taking for G, and v the values H,, = 0.7 eV and v’ = lo* s-l, as were used in equation (S), we obtain for 6, a value of 4 x lop4 s-i. A comparison of equation (10) with equation (3) suggests that .6,,may reasonably be interpreted in physical terms as twice the strain rate reached when thermal energy alone is sufficient to overcome the obstacle, i.e. &, approximately equals 10e4 s-l, twice the strain rate at which the curve of Fig. 7 first becomes horizontal. In view of the uncertainty involved in the choice of values for G,, and v the agreement between these two estimates of 6, is closer than might be expected. Since, by equation (9), the activation volume becomes infinitely large as the slope of Fig. 7, (au*/8 In &)*,, tends to 3
STRESS
OF
PURE
957
IRON
zero, whichever estimate of &, is used there will be
little difference between the activation volumes derived from equations (7a) and (9), even at the lowest stresses. It is clear from the foregoing discussion that neither a consideration of the Liiders strain effect nor the use of the full rate equation at low stresses will lead to activation volumes differing significantly from those calculated using equation (7a). Thus, with the one assumption that the frequency factor, v, is independent of strain rate, the results of the present investigation strongly support the conclusion that the activation volume follows a stress variation of the type shown in Fig. 8. CONCLUSIONS
The lower yield stress of pure iron tested at room temperature and strain rates from 10e4 5-l to 2350 s-l is shown to obey the Hall-Petch equation. The parameter Ic, is found to be independent of strain rate throughout this range, although mechanical twinning frequently occurred in tests at the highest loading rates. The strain rate independence of Ic, confirms previous suggestions that the propagation of yield from one grain to the next is governed by an athermal mechanism such as the creation of new dislocations at or near the grain boundary. The effect of strain rate on the Hall-Petch equation is confined to the thermal component of the friction stress which, in tests at room temperature, increases from zero at a strain rate of slightly less than 10-4 s-1 to about 200 MN/m2 at a strain rate of 2350 s--l, the athermal component being about 7 MN/m2. The variation of activation volume with stress at a fixed temperature closely follows the results obtained by other investigators in work at conventional strain rates and different temperatures. As has been argued elsewhere,~1s*20)the low activation volumes, about
12b3, found
support
MN/m2, trolling
at high stresses, r* > 150
the conclusion
mechanism
does
not
that the rate involve
con-
impurity
interactions. ACKNOWLEDGEMENTS
The author is grateful for many valuable discussions with Drs. J. D. Campbell and T. L. Briggs and for the assistance
of Mr.
experimental
R.
C. Stone
with
much
of the
work.
APPENDIX: DERIVATION OF THE ACTIVATION VOLUME AT LOW THERMAL STRESSES Let the work done during the activation process described by equation (2) be given by G=G,-
V*r*;
G, = Go +
V*T*
958
ACTA
METALLURGICA,
where G,, is the total energy needed to overcome the obstacle and V*T* is the work done by the thermal stress r*. Then equation (2) may be rewritten in the form 6 = 2~ sinh (V*r*/kT)
exp (-G,,/kT)
which gives V*r*/kT
= sinh-l (i/&J = (G, -
G)/kT
where &, = 2v exp (-G,,/kT). Differentiating with respect to (r* at constant temperature we obtain
From the definition of activation volume, equation (6), we have
and hence
which may be rearranged to give the required relationship (equation 9)
a In B v = d/(1 :k&)
( &r* ) T
REFERENCES E. N. H. M.
17, 1969
5. D. HULL
and I. L. MOOFORD, Phil. Meg. 3, 1213 (1968). 6. D. HULL, Acta Met. 9, 191 (1961). 7. T. J. FISCHHOF, D. Phil. dissertation. Oxford (1963). 8. G. R. BORWICK and MISS. E. S. FRASER, NEL Report No. 323 (1967). 9. J. D. CAMPBELL and J. HARDINO., Response of Metals to High VeZocity Defamation, p. 51. Interscience (1961). 10. R. M. FISHES, Ph.D dissertation, Cambridge (1962). 11. A. H. COTTRELL, The Relation between the Structure and Mechanical Properties of Metals, Vol. 2, p. 456. H.M.S.O. (1963). 12. A. CRACKNELL and N. J. PETCH, Acta Met. 3, 186 (1956). 13. W. G. FEROUSON, A. KUMAR and J. E. DORN, J. appl. Phys. 38, 1863 (1967). 14. Z. S. BASINSKI and J. W. CHRISTIAN, Au&. J. Phya. 13, 299 (1960). 15. H. CONRAD, J. Iron and Steel Inst. 198,364 (1961). 16. H. CONRAD and S. FREDERICK, Acta Met. 10,1013 (1962). 17. H. CONRAD and W. HAYES, Trans. Am. Sot. Metals 56, 125 (1963). 18. H. CONRAD, The Relation between the Structure a& Mechanical Proper&s of Metals, Vol. 2, p. 476. H.M.S.O. (1963). 19. J. W. CHRISTIAN and B. C. MASTERS, Proc. R. Sot. A281,
240 (1964). 20. T. L. ALTSHULER and J. W. CHRISTIAN, Phil. Trans. R. Sot. A261. 253 (1967). 21. R. H. COOPER and J: D. CAMPBELL, J. Mech. Engng Sci.
V -= 2kT
1. 2. 3. 4.
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0. Him, Proc. phys. Sot. B94, 747 (1951). J. PITCH, J. Iron Steel In&. 174,25 (1963). CONRAD and G. SCHOEK, Acta Met. 8, 791 (1960). M. HUTCHINSON, Phil. Mug. 8, 121 (1963).
9, 278 (1967). 22. J. HARDING, E. 0. WOOD and J. D. CAMPBELL, J. Mech. Engng Sci. 2, 88 (1960). 23. J. HARDING, PTOC. R. Sot. A299, 464 (1967). 24. A. GILBERT, B. A. WILCOX and G. T. HAHN, Phil. Mag. 12, 649 (1965). 25. J. D. CAMPBELL and W. G. FERCXJSON(to be published). 26. D. L. DAVIDSON, U. S. LINDHOLM and L. M. YEAKLEY, Acta Met. 14, 703 (1966). 27. S. SAKUI and T. MORI, Trans. Japan Inst. Metal 7, 71 (1966). 28. J. D. CAMPBELL and R. H. COOPER, Phyysical Basis of Yield and Fracture, p. 77, Institute of Physics and Physical Society (1966). 29. J. C. M. LI, Dislocation Dynamics, p. 87. McGraw-Hill (1968). 30. D. B. C. TAYLOR and L. E. MALVERN, Response of Metak to High l’e’elocity Deforoaation, p. 77. Interscienoe (1961). 31. J. D. CAMPBELL, R. H. COOPER and T. J. FISCHHOF, Dislocation Dynamics, p. 723. McGraw-Hill (1968).