Experimental investigation of the strain rate versus stress dependence in Fe-6at.%Si single crystals

Materials Science and Engineering, 54 (1982) 151 - 161

151

Experimental Investigation of the Strain Rate Fe-6at.%Si Single Crystals

versus

Stress Dependence in

N. Z £ R U B O V . ~

Institute of Physics, Czechoslovak Academy of Sciences, Prague (Czechoslovakia) Y. Z. ESTRIN

Institute of Crystallography, Academy of Sciences of the U.S.S.R., Moscow (U.S.S.R.) (Received March 28, 1981; in revised form September 12, 1981)

(1)

SUMMARY

~ = PM be

The stress dependence o f the strain rate in Fe-6at.%Si single crystals was investigated b y means o f stress relaxation, strain rate cycling and creep experiments p e r f o r m e d at 295 and 475 K. The results obtained by the three techniques are in g o o d agreement with each other and s h o w that the spectrum o f localized obstacles to dislocation motion remains essentially unchanged during deformation in the investigated strain range (7 = O. 04 0.60). A n analysis o f the post-relaxation effects was made and the possible variations in the internal stress and mobile dislocation density during relaxation is discussed.

where p M is the density of mobile dislocations and b their Burgers vector. The derivation of the function v = v(r*,T) from mechanical tests encounters the following basic difficulties. Firstly, a separation of the applied stress into the thermal and the athermal components [2, 3] can be very complicated; in particular, for cases where the athermal c o m p o n e n t rG is the d o m i n a n t part of the shear stress r, the precision of the experimental determination of rG may be insufficient for an accurate determination of the thermal c o m p o n e n t [4]. Secondly, an extension of the range of the thermal stress r* desirable for a comprehensive analysis necessitates in many cases (cf. for example refs. 5 and 6) a variation in the strain. This can lead n o t only to a change in the athermal stress rG and in the density PM of mobile dislocations but also to a change in the spectrum of localized obstacles, which are overcome by a moving dislocation with the help of thermal activation and which determine the thermal stress r* [7]. Even small amounts of straining can distort the ~(r) curve which should be measured for a constant defect structure of the crystal. Finally, even if the strain variation along a ~(r) curve is small compared with the prestrain, some changes in rG and PM can occur along this curve [8], leading to additional complications. Three types of deformation conditions are conventionally used to obtain the ~(r) dependence. (i) The strain rate change after reaching a certain strain y under dynamic loading conditions can be used. This can be accompanied by changes in rG and PM as manifested in the yield drop phenomenon.

1. I N T R O D U C T I O N

The investigation of the dependence of the plastic strain rate ~ on the resolved shear stress r and the temperature T is an import a n t means of identifying the elementary processes of plastic deformation in crystals. Many attempts have been and are being made to derive, from the ~(r, T) dependence, the functional dependence of the average dislocation velocity v on the temperature T and the thermal c o m p o n e n t r* of the flow stress. This functional dependence is closely related to the force-distance profile for the dislocat i o n - o b s t a c l e ' i n t e r a c t i o n and thus provides information on the t y p e of the obstacles to dislocation m o t i o n (cf. for example ref. 1). Analysis of the macroscopic plasticity in terms of the mobility of individual dislocations is based on the well-known Orowan relation between ~ and v: 0025-5416/82/0000-0000/$02.75

© Elsevier Sequoia/Printed in The Netherlands

152 (ii) The stress relaxation after the dynamic prestraining up to some strain 7 can be employed. This process is accompanied by a plastic strain of the order of {r -- TG(3')}IG* where G* is the combined modulus of the sample and the testing machine (cf. for example ref. 9); here again a variation in both TG and PM along the relaxation curve can be expected. (iii) Finally the creep of specimens prestrained dynamically to a strain T can be employed. The load increments necessary to measure the stress dependence of the creep rate can also cause changes both in TG and in PM. In all three cases the prestraln ~ cannot be generally considered as a "parameter of state" uniquely characterizing the ~(T) curve obtained. Consequently, it cannot be expected a priori that the ~(r) curves for a given prestraln ~, and for a given deformation prehistory will coincide and give a unified " c o m p o u n d " curve if measured in the above three different deformation modes. The aim of the present paper is to demonstrate that for Fe-6at.%Si single crystals all three methods of measuring the ~(r) dependence are in good agreement with each other. Furthermore, it will be shown by comparing the ~(r) curves obtained for different prestrains that, in the strain range up to ~ = 0.6, new obstacles for dislocation motion produced during plastic deformation (e.g. dislocation intersections [7]) do n o t contribute essentially to the thermal stress r*, the work hardening manifesting itself only in the athermal part of the stress. Special attention will be paid to the possibilities of compensating for the changes in the dislocation ensemble during stress relaxation in order to obtain true ~ versus r* curves.

2. EXPERIMENTAL DETAILS Large oriented single crystals of Fe-6at.%Si alloy grown by floating zone melting were used for the preparation of cylindrical tensile specimens with a 3 mm gauge diameter and a 30 m m gauge length. Except for the final annealing, the specimens were prepared in the manner described previously [10]. The specimens utilized for the present experiments were annealed at 1073 K for 120 h in a closed ZrH2 purification system [11]. By this treat-

ment the carbon and nitrogen contents were reduced to less than 10 ppm. Two types of loading tests were carried out. (a) Tensile tests at a base strain rate {nominal) ~ of 5.5 × 10 -5 s-1 during which strain rate changes (d 2/~1 values of 5, 10 and 20) and interruptions to measure the stress relaxation were performed. These tests were carried out on a floor-model Instron machine at 295 and 475 K. (b) Tensile tests combined with creep experiments were made. Specimens dynamically prestrained at the base strain rate in the Instron machine were placed in a creep machine and deformed in creep at various loads. The load was increased stepwise during the test, starting from a level far below that of the prestrain. These tests were performed at room temperature only. The specimen temperature variation, even during the longest relaxation and creep tests, did not exceed +0.1 K. (The strain rate ~ ~ u c t associated with the sample deformation caused by these temperature fluctuations due to the difference between the thermal expansion coefficient of the sample and that of the material of the testing machine can be estimated to be negligible compared even with the lowest strain rates involved in experiment. Actually, for a typical duration texp of the prolonged tests of about 103 s and the difference As between the thermal expansion coefficients of about 10 -6 K -1, efluct <: Aa X 0.1 K/texp ~ 10 -l° s-1.)

3. RESULTS 3.1. Stress relaxation 3.1.1. Uncorrected results Shear stress-shear strain curves typical of middle oriented samples tested at 295 and 475 K are shown in Fig. l(a). The vertical marks indicate the strains at which stress relaxations were performed. The relaxations lasted for about 180 - 8000 s at 295 K and for about 1 - 1500 s at 475 K. By numerically differentiating the recorded load versus time dependence, the instantaneous plastic shear strain rate ~ was calculated from the relation

153

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where t is the time, l the actual specimen length, AP the load decrement, ~ an orientation factor and K the elastic compliance, defined from the slope of the elastic portion of the reloading curve. The results are presented as uncorrected data in Fig. 2 where lg ~; is plotted versus the applied stress r. For clarity, only the longest relaxations are presented. Figure 3 shows the data as lg ~ versus lg T. 3.1.2. Work hardening during relaxation An interpretation of the stress dependence of ~; in terms of the dislocation velocities requires the athermal stress T~ (usually termed the "back stress" or "internal stress") and the mobile dislocation density PM to remain constant along the ~(T) curve. For this reason possible changes in T¢ and PM (usually referred to as " w o r k hardening") during stress relaxation have been treated theoretically as well as experimentally [8, 12 - 18]. For the alloy under investigation, the observed post-relaxation effects indicate that the

work hardening during relaxation can be significant, especially at 475 K. As schematically shown in Fig. 4, the stress r a at which plastic deformation begins on reloading is higher than the stress r(0) at which the relaxation started. The work hardening, which is characterized by ~ r = TR -- T(0), strongly depends on temperature and relaxation depth while its dependence on prestrain is much less pronounced. At room temperature the stress increment 5 T is much smaller than the stress decrement AT during relaxation (typically, AT = 1 4 - 17 M P a a n d S T = 1 - 3 M P a f o r prolonged relaxations). At 475 K, in contrast, 5 T can be much greater than AT, as can be seen in Fig. 5. Except for very small prestrains at room temperature, a yield drop appears after reloading, indicating that the work hardening is, at least partly, reversible. 3.1.3. Correction o f relaxation data From the above, it is obvious t h a t rG and/ or PM change(s) during relaxation in the Fe-6at.%Si alloy. Therefore, a correction of

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Fig. 2. Plot of lg ~ vs. T for various shear strains at (a) T = 295 K and (b) T = 475 K: e, base strain rate; . . . . . , stress relaxation (uncorrected data); <% corrections according to eqn. (2); 1", corrections according to eqn. (3); ×, relaxation data corrected according to eqn. (2) using the ~T(t) dependence given in Fig. 3; m, strain rate cycling data; +, creep data. 10 .2

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Fig. 3. Plot of lg ~ vs. lg T for various shear strains at (a) T = 295 K and (b) T = 475 K. The symbols have the same meaning as in Fig. 2. t h e r e l a x a t i o n d a t a p l o t t e d in Figs. 2 a n d 3 is n e c e s s a r y t o c o m p e n s a t e f o r t h e v a r i a t i o n in t h e s e s t r u c t u r a l p a r a m e t e r s . F o l l o w i n g Dotsenko's [8] treatment of this problem, we divide the stress increment ~ r into an irrevers-

i b l e p a r t 5 r l a n d a r e v e r s i b l e p a r t ~ T2 ( s e e F i g . 4). I n F i g . 6, ~ Tt a n d GT2 a r e c o m p a r e d w i t h t h e t e r m 0 A A ~ w h e r e 0 A is t h e w o r k hardening coefficient for active (dynamic} d e f o r m a t i o n a n d Am is t h e s h e a r s t r a i n p r o -

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Fig. 5. Stress decrement A~- at t h e end of relaxation (relaxation depth) and stress increment 5T on reloading for a series of 11 stress relaxation tests of various durations carried out at 475 K in the strain interval ? = 0.39 - 0.50. duced during relaxation (for prolonged relaxations, 47 ~ 0.0015 at 295 K and 4 7 ~ 0 . 0 0 0 3 a t 4 7 5 K). As c a n b e seen, t h e r e is a g o o d c o r r e l a t i o n b e t w e e n 5 T1 a n d 0 A A~, : 5 rl is r e l a t i v e l y large f o r small strains w h e r e 0 A is large while a t high strains, w h e r e 0 A is small, 5 rl ~ 0. ( F o r t h e smallest ~/ t h e 5 r values d o n o t f o l l o w this p a t t e r n : t h e 8 rl values are m u c h g r e a t e r t h a n 0 A 4 7 , while 5 r2 = 0 (i.e. n o yield d r o p o c c u r s ) , at least at r o o m t e m p e r a t u r e . As t h e b e g i n n i n g o f d e f o r m a t i o n is c h a r a c t e r i z e d b y glide b a n d f o r m a t i o n a n d i n h o m o g e n e o u s strain [19, 20], a very detailed investigation would be

n e c e s s a r y f o r a discussion o f t h e early stage o f d e f o r m a t i o n . ) Thus, t h e c o m p o n e n t ~ r l u n d o u b t e d l y originates f r o m t h e t r u e w o r k h a r d e n i n g p r o d u c e d d u r i n g r e l a x a t i o n , i.e. f r o m an irreversible increase in t h e a t h e r m a l stress rG. As f o r 5 r2, t h e r e is o b v i o u s l y n o r e l a t i o n b e t w e e n this c o m p o n e n t a n d t h e t e r m 0 A A~/. As discussed in ref. 8, this reversible stress i n c r e m e n t c a n be c a u s e d e i t h e r b y a reversible increase in t h e average a t h e r m a l stress rG o w i n g to a redistribution of the mobile dislocation a s s e m b l y d u r i n g r e l a x a t i o n or b y a decrease in t h e m o b i l e d i s l o c a t i o n d e n s i t y PM. In b o t h cases, 5T2 can be m u c h larger t h a n 0 A A,/ a n d a yield p o i n t c a n b e e x p e c t e d a f t e r reloading. L e t us first c o n s i d e r an increase in rG d u r i n g r e l a x a t i o n , p M r e m a i n i n g c o n s t a n t . In this case t h e stress d e p e n d e n c e o f t h e plastic strain r a t e c a n yield s o m e i n f o r m a t i o n o n t h e f u n c t i o n v(r*) p r o v i d e d t h a t t h e i n s t a n t a n e o u s strain r a t e ~ (t) is p l o t t e d v e r s u s t h e corr e c t e f f e c t i v e stress, r * ( t ) = r ( 0 ) - - r G ( 0 ) - A t ( t ) - - ~ r ( t ) , or (if t h e a t h e r m a l stress r G ( 0 ) at t = 0 is n o t k n o w n ) v e r s u s Tco,,(t)

= T(O) -- aT(t)

-- 6 r(t)

(2)

Since t h e stress r¢o~ = T* + rG(0), t h e ~(rcorz) a n d ~ ( r * ) curves will b e o n l y s h i f t e d parallel w i t h r e s p e c t t o e a c h o t h e r o n t h e lg ~ v e r s u s r p l o t . S u c h c o r r e c t i o n s can b e easily carried o u t f o r ~E, rE c o r r e s p o n d i n g t o t h e e n d p o i n t

156

of a relaxation curve and are indicated by the arrows in Figs. 2 and 3. To be able to correct the whole ~(r) curve, and not only the ultimate point (TE, rE), the dependence of 6r on t must be known. For this purpose, 11 unequally long relaxations were performed at 475 K in the strain interval 7 = 0.39 0.50. The dynamic straining after each relaxation was as short as possible (47 ~ 0.01), just to allow the yield drop to occur and the stress-strain curve to "recover". In Fig. 5 the r and 5 r values for this series of relaxations are plotted versus time t. All these relaxations proved to be identical, so that the 5 r(t) curve can be used as a correction curve for any relaxation of this series. The result for the longest relaxation performed at 7 = 0.39 is shown in Fig. 3(b). Let us consider now the other possible reason for the observed post-relaxation effects, i.e. a change in PM during relaxation (re being considered as constant). If the mobile dislocation density decreases during relaxation, the strain rate ~ decreases faster than for constant PM. It is rather difficult to obtain the " c o m p e n s a t e d " strain rate versus stress dependence that would correspond to a constant PM equal to the initial mobile dislocation density p M ( 0 ) , a s we do n o t know h o w p M c h a n g e s during relaxation. Nevertheless, an estimate can be made. It is quite reasonable to suppose t h a t for r = ra the mobile dislocation density is still approximately the same as at the end of relaxation, i.e. PM = PME, while ~ has already reached the initial value, ~(0). As 6r~ < 6r2, we shall neglect the true work hardening during the relaxation 6 rl. The initial shape of the relaxation curve as well as the data on the strain rate changes seem to indicate that the empirical relation v ~ r m, with m = constant, holds at least for higher stresses (Fig. 3). From eqn. (1) the ~(r) curves for various PM (at 7 = constant} will be parallel in logarithmic or semilogarithmic coordinates; t h e y coincide when shifted with respect to each other along the ordinate axis [9]. Rewriting eqn. (1) in the form lg ~ = lg(bpM) + lg v for r = r(0), r = rE and r = rR, we obtain for ~corr (the corrected value of ~ that corresponds to r = rE at the unchanged mobile dislocation density PM ----PM(0)) an expression l g ~ o ~ , = l g ~ E + m l g I rr-(-~))f

(3)

where "YE is the strain rate measured at the end of the relaxation and m is estimated in the vicinity of r = r(0). After this correction has been carried out, the range of ~ covered during relaxation becomes smaller, especially at 475 K where m is very high. Within this range, however, the ~(r) dependence is almost the same as that obtained from eqn. (2). 3.2. Strain rate cycling In m a n y cases, strain rate cycling had been performed before the stress relaxation was carried out. Except for the initial stage of the stress-strain curve a yield point appeared after the strain rate had been changed, indicating a variation in PM and/or rG. For this reason the stress difference Ar between the upper yield stress and the stress level before the upward strain rate change was taken as a characteristic value of the strain rate sensitivity of the stress. The strain dependence of ~ r is shown in Fig. l(b), where some previous data are also plotted. To obtain the ~(r} dependence, the Ar values were added to the stress r(0) at which the next relaxation was started. It eliminated the difference in rG corresponding to the strain rate changes and to the relaxation and made a comparison between both sets of data possible. As seen from Figs. 2 and 3, the results of strain rate cycling agree very well with the relaxation data. 3.3. Creep To obtain strain rates even lower than those in the relaxation experiments, creep tests were performed on two dynamically prestrained samples. The prestrain 7 was 0.04 or 0.20. In the creep tests the strain rate increased immediately after each stress increment and then remained roughly constant, at least for a certain time interval. These strain rates are plotted in Figs. 2(a) and 3(a). As long as the total strain produced by creep is negligible, the creep data are in good agreem e n t with the relaxation data and fit a very similar ~(r) dependence. For the larger strain increments during the creep test, as observed at the three highest strain rates for the sample prestrained to 0.04, the ~(r) data appreciably deviate from those obtained by stress relaxation owing to the work hardening during creep. It appears very likely that during the creep t e s t s some changes in the dislocation structure also occur. It is expected t h a t the

157

changes will n o t be very large at r o o m temperature but may be substantial at elevated temperatures.

4. DISCUSSION

4.1. Changes in re and PM during relaxation It was clearly demonstrated in Section 3 t h a t changes in the dislocation ensemble which manifest themselves in work hardening do occur during stress relaxation. It was shown that a minor part of these changes is due to the irreversible work hardening during relaxation, while the major part m u s t be ascribed to reversible changes in re or in PM or in both. A decrease in the mobile dislocation density PM can occur by annihilation or by pinning of mobile dislocations. Above 113 K the plastic deformation of Fe-6at.%Si alloy is restricted to stage I of the stressstrain curve, the hardening being controlled by the elastic interaction of primary screw dislocations [ 21]. The m i n i m u m distance do at which two screw dislocations of the opposite sign can pass is proportional to 1/r [22]. For d < do the dislocations will be annihilated more frequently. However, by this mechanism, which is discussed in ref. 23, the mobile dislocation density can be reduced by some 20% at 295 K and by even less at 475 K while the correction according to eqn. (3) requires a decrease in p M of tWO orders of magnitude. A n o t h e r plausible explanation for a decrease in p M is the pinning of dislocations by solute atoms. Because interstitial solutes are only present in very small amounts, t h e y cannot play any significant role in this process. In fact, no serrations t h a t would indicate pinning by interstitial impurities were observed in flow curves in the range 375 - 475 K where interstitial solute atoms are mobile. However, for crystals of the same composition and purity as those used in this study, serrated flow was observed when strained at a strain rate of about 10 5 s-1 between 525 and 625 K, indicating pinning of dislocations by substitutional silicon atoms. Similarly, Cuddy and Leslie [24] reported serrated yielding on polycrystals of titanium-gettered Fe-Si in the temperature range 503 - 663 K. It was shown by Mulford and Kocks [25] that dynamic aging occurs over a wider temperature range than the region where macro-

scopic jerky flow is observed. Thus, despite the fact t h a t for our crystals the stressstrain curves were smooth at 475 K and the strain rate sensitivity was always positive, the observed post-relaxation yield point might result from static strain aging during stress relaxation. A simple estimate shows, however, that this effect can be excluded at room temperature. For this, we take the activation energy Q for the lattice diffusion of silicon in iron to be 2.3 eV [26], the pre-exponential factor D O in the diffusion coefficient being about 1 cm 2 s-1. Even though pipe diffusion might be involved in the strain-aging process, so that the activation energy is reduced by about a factor of 2 [24], the number of elementary jumps of silicon atoms during the time t, given roughly by N ~ - ~ - exp

~

(4)

is one or two orders of magnitude smaller than unity for the largest relaxation times of about 103 S when room temperature relaxation is considered. Thus, no strainaging effect can take place during relaxation at 295 K. A similar estimate for T = 475 K gives, on the contrary, N ~ 108, indicating that a prominent pinning by silicon atoms which migrate towards dislocations is possible at this temperature. Hence, it can be concluded that the mobile dislocation density can be reduced by pinning during stress relaxation at 475 K. However, the relative contribution of this process to the total effect observed cannot be estimated on the basis of the available data alone. Another possible reason for the postrelaxation peaks, namely an increase in re during relaxation, is consistent with the model of dislocation movement in a periodic internal stress field proposed by Li [27] and discussed in ref. 8. According to this m o d e l , the average internal stress (as measured in macroscopic experiments) depends n o t only on the internal stress field but also on the distribution of mobile dislocations in this field and increases when the applied stress r decreases. An estimate shows that, for the alloy under investigation, ~ T 2 should be much higher at 475 K than at room temperature and that, at the elevated temperature, ~ T 2 c a n be comparable with Ar (8r2 ~ 0.5 Ar).

158 From the present experiments we are n o t able to decide which is the real cause of the work hardening during relaxation and whether eqn. (2) or eqn. (3) should be used to correct the experimental data. Very probably it will be a combined effect. Since, however, both corrections discussed in Section 3.1.3 give very similar stress dependences of ~, either of them may be used to compensate for the work hardening in the process of stress relaxation.

4.2. Stress d e p e n d e n c e o f Let us first examine the stress dependence of ~ at a fixed strain. For the uncorrected relaxation data plotted as lg ~ versus lg r, the slope of the curves increased with decreasing ~ (Fig. 3). After corrections, using either eqn. (2) or eqn. (3), the dependence of lg on lg r becomes nearly linear. At 475 K the relaxation and the strain rate cycling data lie very accurately on a straight line (Fig. 3(b)). Thus, our results obtained by different deformation modes prove to be consistent with each other and to confirm the phenomenological relation v ~ r m (m = constant) which has been f o u n d in Fe-Si alloy single crystals of similar composition by measuring the velocity of leading dislocations in glide bands [19, 28, 29]. In our case the e x p o n e n t m at small strains is about 55 at 295 K, in good agreement with the m values given in ref. 19, and about 85 at 475 K. With increasing strain, m increases, which is consistent with the constancy of the activation area along the stress-strain curve (cf. Fig. l(b)), m being thus proportional to r. 4.3. Strain d e p e n d e n c e o f Z/(r ) curves Now let us discuss the strain dependence of the ~(r) curves. The parallelism of the curves presented in Fig. 2 proves that the spectrum of localized obstacles to dislocation motion remains unchanged within the strain range investigated (~/ = 0.04 - 0.60). This is consistent with the above-mentioned fact (Section 4.1) that the plastic deformation of Fe-6at.%Si alloy is restricted to stage I. Slip occurs exclusively on the primary slip system [10, 21] and, obviously, no straininduced localized obstacles (e.g. intersections of primary and secondary dislocations) are created during deformation. If we assume that

one solute mechanism is operative in the strain range under consideration, the constancy of the activation area (i.e. the same slope of the lg ~ versus r plots at ~ = constant (Fig. 2) as well as the constancy of the strain rate sensitivity (Fig. l(b))) means that the effective (thermal) stress r* also remains constant during straining, i.e. the observed work hardening (Fig. l(a)) is only due to an increase in the athermal stress re. This means, in turn, that the average velocity v of individual dislocations is strain independent and, because of eqn. (1), PM also remains constant along the stress-strain curve. (In contrast, the decrease in A r at very small strains, which is pronounced at 475 K and still apparent at 295 K, must u n d o u b t e d l y be attributed to an increase in PM, as discussed previously in ref. 30.) Consequently, all parallel curves of the "families" plotted in Fig. 2 must coincide after a reduction to the effective stress. This reduction necessitates, however, the separation of r into r* and re. A natural way of separating r* and re would be an extrapolation of the ~(r) curve to the zero value of ~. In fact, from theories of dislocation mobility [1, 31] it follows that, at low effective stresses, r* ~ k T / b A * (where A* is the activation area), v and hence depend linearly on r*, the intercept of the z/versus r plot with the r axis giving the value of re [ 32]. However, the ~ level under which the stress dependence of ~ becomes linear (this can be estimated as suggested in refs. 1 and 32) proves to be too low even compared with the lowest strain rates of the order of 10 -9 s-1 accessible in the present experiments. Instead, we can use another m e t h o d of separating the thermal and athermal stress components, based on our previous measurements of the critical resolved shear stress (CSSR), r0 and the strain rate sensitivity, which were carried out up to 1100 K [33, 34]. The strain rate sensitivity at the very beginning of deformation is nearly zero above 700 K [34], indicating that r* ~ 0, while re, which at small strains is mainly due to the solution hardening, is still very high (Fig. 7). Thus, we can take the r0 value at, say, 725 K as re and extrapolate it to lower temperatures (provided that the temperature dependence of the shear modulus is taken into account). Such an extrapolation implies t h a t the solute subsystem of the crystal

159 1

[

I

1

I

I

I

I

I

I

I

I-,

loo

o 0

200

400

600

800

1000 T K]

Fig. 7. T e m p e r a t u r e d e p e n d e n c e of the C R S S (e = 5.5 X 10 - 5 s - l ) . The data are taken f r o m ref. 34.

undergoes no change with increasing temperature up to 725 K (this seems to be justified at least for very small strains [34] ) and t h a t the contribution of the dislocation subsystem to rc is the same in the whole temperature range investigated. In this way we obtain the stress components r* and rG corresponding to the CRSS (r* = 65 MPa and r* = 26 MPa at 295 K and 475 K respectively). The decrease in the strain rate sensitivity at small strains (Fig. l(b)) has been interpreted as an increase in PM and, thus, a decrease in r* [30]. Following ref. 30, for 7 > 0.05 we obtain from the above extrapolation t h a t r* = 56 MPa at 295 K and r* = 11 MPa at 475 K. It is interesting to note t h a t at 475 K the effective stress obtained by this extrapolation m e t h o d is almost the same as the sum Ar + 8r for the longest relaxations. In fact, the relaxations performed at 7 = 0.04 and 7 = 0.39 (Fig. 2(b)) yield Ar + 5r = 13 MPa and Ar + 5r = 10 MPa respectively. This means, in turn, that the correction according to eqn. (2) is quite reasonable and that the stresses %o~r calculated from eqn. (2) for prolonged relaxations must already be very close to the athermal stress ra. At 295 K, on the contrary, the sum Ar + 8r (about 20 MPa for the longest relaxations) is much smaller than the value of r* obtained by extrapolation (56 MPa). From r, r* and m, the velocity-stress exp o n e n t m* which characterizes the individual dislocation mobility can be calculated from the relation m* = m r * I t (see for example ref. 8). The results are presented in Table 1. The values of m* at 295 K agree very well with m* = 25 + 5 f o u n d from stress relaxation

TABLE 1 Effective stresses and strain rate stress e x p o n e n t s T (K)

T

T (MPa)

T* (MPa)

m

rn*

295 295

0.04 0.20

143.5 173.5

65 56

55 73

24.9 23.6

475 475

0.07 0.39

123 160

11 11

85 130

7.6 8.9

experiments by Gupta and Li [35]. The values of m* and r* given in Table 1 are a little higher than those obtained on nearly identical single crystals from the strain rate change experiments [30]. In ref. 30, r*, rG and m* were estimated at 475 K using Michalak's technique [36]. Then TG was extrapolated to lower temperatures and the corresponding values of r* and m* were calculated. The inaccuracy of Michalak's method, as well as that of the extrapolation, accounts for the small discrepancy between the results of ref. 30 and the present results.

5. C O N C L U S I O N

In Section 1, some difficulties connected with the derivation of the function v(r*, T) from macroscopic mechanical tests were enumerated. From the present experiments the following conclusions concerning the possibility of such a derivation for Fe-6at.%Si alloy (and probably also applicable to other b.c.c, metal-based alloys deformed within stage I) can be drawn.

160

(i) As indicated by stress relaxation, strain rate cycling and creep tests performed at fixed prestrains and exhibiting good agreement at 295 and 475 K, the spectrum of localized obstacles to dislocation motion which can be overcome by thermal activation remains unchanged with increasing strain. The effective stress r*, the average velocity of individual dislocations and the mobile dislocation density pM are all strain independent, except at very small strains. (ii) Changes in the athermal stress rG and/or mobile dislocation density PM during stress relaxation, and most probably during creep tests as well, are important and can distort the measured ~(r) dependence. However, the data can be reasonably well corrected to compensate for these changes. (iii) Stress relaxation proves to yield reliable values of r* and rG at an elevated temperature (475 K). These values are consistent with the results obtained b y extrapolating the high temperature value of the CRSS (approaching rG) to 475 K and by taking into account the strain dependence of the strain rate sensitivity to determine r* for higher ~/. R o o m temperature data, on the contrary, do n o t exhibit such consistency: the relaxation depth achieved within the duration of experiment (even when corrected according to the above scheme) proves to be t o o small compared with the value of the effective stress at the beginning of relaxation, as estimated by the extrapolation of the high temperature rG value to r o o m temperature. For room temperature, the m e t h o d of determining the effective stress based on such an extrapolation appears to be suitable rather than that based on long-time relaxation tests. A knowledge of r* made it possible to characterize the individual dislocation mobility by determining the dislocation velocity-stress exponent m*.

ACKNOWLEDGMENTS

The authors are grateful to Professor V. L. Indenbom for his advice and attention during this work. Thanks are also due to Professor J. ~adek and his coworkers, Institute of Physical Metallurgy, Brno, for kindly providing the facilities for the creep tests and for their assistance in performing them. The

authors are indebted to Dr. S. Kadeekov~ for supplying the oriented single crystals and to Drs. B. ~est~k and V. Nov~k for many valuable discussions and for critically reading the manuscript.

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