Deformation in Zr–1Nb–1Sn–0.1Fe using stress relaxation technique

Materials Science and Engineering A328 (2002) 324– 333 www.elsevier.com/locate/msea

Deformation in Zr–1Nb –1Sn–0.1Fe using stress relaxation technique Rajeev Kapoor *, Shashikant L. Wadekar, Jayanta K. Chakravartty Materials Science Di6ision, Bhabha Atomic Research Centre (B.A.R.C.), Mumbai 400085, India Received 18 May 2001; received in revised form 31 August 2001

Abstract Deformation behavior of Zr–1Nb–1Sn–0.1Fe was studied using the stress relaxation technique. Stress relaxation experiments were carried out over a range of temperatures (296–765 K) and for strains up to 0.12. The stress– time data were analyzed to obtain the activation volume and enthalpy. It was found that in the strain rate range of 10 − 4 –10 − 6 s − 1 and in the temperature range of 296–570 K, the activation volume and enthalpy do not vary with strain. From this and the magnitude of the activation volume and its variation with thermal stress, either the Peierls stress or the dislocation– interstitial interaction is the rate controlling short range barrier to dislocation motion. The time independent stress component obtained using decremental unloading technique, called here as the remnant stress, was observed to have a large temperature dependence. By using a relation in which the activation free energy is a function of thermal stress, it was found that, in general, the remnant stress cannot be used to represent the athermal stress. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Zirconium alloys; Stress relaxation; Thermal activation; Deformation

1. Introduction In the last decade a new family of zirconium alloys, containing Nb, Sn and Fe additions to Zr have been studied as potential fuel cladding materials for the use in nuclear reactors [1 – 6]. The addition of Nb and reduction in the content of Sn from Zircaloy leads to an improved corrosion resistant alloy [5,6]. Addition of Fe helps in precipitate formation thereby increasing creep strength. In addition to improved corrosion resistance, the irradiation stability and creep resistance are superior [5,7] for alloys of usual form Zr – 1wt.%Nb – 1wt.%Sn –(0.1 –0.4)wt.% Fe. Nikulina et al. [7] studied Zr – 1Nb –1Sn –0.4Fe after hotworking in the b region and annealing at 590 °C. They observed partial recrystallization with grain size of 5 mm and second phase precipitates having about 1 mm globular shape. These precipitates were indicated to be either orthorhombic (Zr, Nb)3Fe, or hexagonal Zr(Nb, Fe)2. Fine precipitates of tetragonal Zr4Sn were also identified. Nikulin et * Corresponding author. Tel.: +91-22-559-318; fax: + 91-22-5505151. E-mail address: [email protected] (R. Kapoor).

al. [3] observed that quenching and subsequent annealing of Zr –1Nb –1.3Sn –0.4Fe produced precipitate free a-Zr with fine grain eutectoid interlayers, which had high ductility. On the other hand, heat treatment which produced coarse secondary particles resulted in a loss of ductility. Kutty et al. [1] and Murty et al. [4] carried out creep studies on Zr –Nb –Sn–Fe system, identifying different mechanisms based on the activation energy in various temperature regimes. A considerable amount of work has been done to understand the deformation behavior of common zirconium alloys, e.g. Zr –Nb, Zr –Sn, Zircaloy-2 and Zircaloy-4, whereas limited work has been carried out on ternary and quaternary zirconium alloy systems, such as Zr –Nb –Sn and Zr – Nb –Sn–Fe alloys. The present work was undertaken to study the activation parameters as a function strain and thermal stress for Zr –1Nb –1Sn –0.1Fe using the stress –strain rate and stress –temperature relation in the temperature range of 295 –600 K. The stress –strain rate relation was obtained using the stress relaxation technique. Stress relaxation experiments involve deforming the sample to a predetermined stress or strain value, stopping the cross-head movement and recording the load as a func-

0921-5093/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 8 4 9 - 4

R. Kapoor et al. / Materials Science and Engineering A328 (2002) 324–333

tion of time [8– 12]. It is important to correctly separate the flow stress into a thermal and an athermal part, because identifying deformation mechanisms become easier if the activation parameters are represented as a function of thermal stress. In general, during stress relaxation, the stress as well as the absolute value of the stress rate, both decrease with time. At long times the stress rate approaches zero, and the stress becomes nearly independent of time. Some earlier researchers have referred to this time independent stress as the athermal or internal stress. Instead we refer to this ‘observed’ time independent stress (during relaxation) as a remnant stress, |r. Correspondingly the stress which varies with time is referred to as the transient stress, |t. The remnant stress |r, can be obtained in a quicker manner by the decremental unloading technique [11]. Here the sample is unloaded in small steps and allowed to relax for a given time. This decremental unloading is carried out until no relaxation is observed to occur (in some cases the stress increases with time). The corresponding transient stress at the start of the relaxation, |t, is |t =|o −|r, where |o is the stress at t = 0. Fig. 1 schematically shows the stress– time curve during a relaxation experiment with subsequent decremental unloading. The athermal stress on the other hand, is by definition the stress at which d(|/v)/dT = 0. Here v is the shear modulus dependence on temperature. Previously the stress relaxation technique has been used to study the deformation behavior of zirconium based alloys [13–18]. In addition previous studies have used the decremental unloading technique to determine the athermal stress [14,17,19– 22]. The work presented here was carried out with the following objectives. 1. To use the stress– strain rate relation (from stress relaxation) and stress– temperature relation to obtain the activation volume and activation enthalpy,

and thus determine the rate controlling deformation mechanism. 2. To obtain the remnant and transient stresses, as defined above, during stress relaxation experiments and relate them to the athermal and thermal stresses as obtained from stress–temperature data.

2. Experimental procedure Zr –1wt.%Nb –1wt.%Sn –0.1wt.%Fe alloy obtained from Nuclear Fuel Complex (NFC), Hyderabad, India, was used in the present study. The processing carried out at NFC was as follows: double vacuum arc melted ingot (300 mm diameter), hot extrusion at 800 °C to 150 mm diameter billet, b quenching, hot extrusion to 19 mm diameter bars, cold drawing to 16 mm diameter rods, and annealing at 732 °C. The microstructure consists primarily of 10 mm equiaxed a-grains, with 200–500 nm sized precipitates dispersed within the grain at a spacing of about 3 mm. Samples with 30 mm gauge length and 6 mm diameter, were loaded in tension using a screw driven Instron machine (model 1185). Samples were deformed in tension at a nominal strain rate of 10 − 4 s − 1 at 297, 377, 471, 568, 666, 764 K. A three zone resistance heating furnace was used. It took about 35–40 min to heat the sample from room temperature and to stabilize it at the desired temperature. At each temperature stress relaxations were carried out at five different strain levels (0.002, 0.02, 0.06, 0.08, 0.12). The exact value of the strains varied at each temperature. It is required that from the stress–time data, the stress rate, |; , be obtained from which the plastic strain rate, m; p, and activation volume can be calculated (see Appendix A). |; can be obtained by fitting the stress– time data to an equation and then taking the derivative of that equation. Earlier researchers [9,10,23] have used a power-law relation relating stress during relaxation to time, namely, | =|a + K(t+ c) − n, where K, c, and n are constants and |a is the athermal stress component. In the present study an equation of a similar form has been used except that the constants are expressed differently, that is: |= |o + B[(at + 1) − n − 1],

Fig. 1. Schematic stress –time plot during relaxation, depicting the remnant stress |r, the transient stress |t, and their relation with the stress at start of relaxation |o.

325

(1)

where |o is the stress at start of relaxation (t= 0), B, a, and n are the fitting parameters. The fit was done using a non-linear regression analysis with initial values of B, a, and n being specified. The program iteratively finds the best B, a, and n values which satisfy the least square fit criteria. The Appendix A briefly describes the procedure to calculate the plastic strain rate and the experimental activation volume once the stress rate is calculated using the fitted parameters B, a, and n. The elastic effects of the machine and sample are taken into

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326

strain data at varying temperatures and stress relaxation experiments; and (2), dealing with remnant stress behavior, as obtained by the decremental unloading technique.

3.1. Stress–temperature relation The shear modulus in general decreases with temperature. In the 200–800 K range, the shear modulus (v) dependence on temperature (T) can be represented as [24]: v(T) = 1−5.45× 10 − 4 T= v%. vo

Fig. 2. (a) Stress –strain curves at 10 − 4 s − 1. (b) |/v% vs. T plot at different strains.

account through the compliance C (mm/KN), while calculating m; p (Eq. (A.2)). The activation volume is obtained from (Eq. (A.6)). The activation enthalpy is: DH = −

)

)

)

(ln m; p (|* (ln m; p = (1/kT |* (|* T (1/kT m;

(2)

where |* is the thermal stress component. In the present study all deformations were carried out at a nominal strain rate of 10 − 4 s − 1, thus Eq. (2) can be written as: DH = − V(10 − 4 s − 1, T)T

(|* − 4 − 1 (10 s , T). (T

(3)

After relaxing for about 500 s, the samples were decrementally unloaded to obtain the remnant stress. The stress was observed for a maximum of 60 s during the unloading steps.

3. Results The results are presented in two parts, (1), dealing with activation parameters as obtained from stress–

Thus the stress corrected for the temperature dependence of the shear modulus is henceforth written as |/v%. Fig. 2(a) shows the true stress–true strain curves up to UTS, of Zr–1Nb –1Sn–0.1Fe deformed at a nominal strain rate of 10 − 4 s − 1 at temperatures ranging from 298 to 765 K. Fig. 2(b) is the corresponding |/v% versus temperature plot at different strains. For each strain the shear modulus corrected stress decreases with increasing temperature up to about 570 K after which it shows a slight increase with increasing temperature. Strain rate change tests were carried out at 673 K the strain rate sensitivity obtained was very near zero becoming negative as strain rate decreased, thus suggesting that dynamic strain aging occurs at about 670 K. Thorpe and Smith [25] also observed a sharp fall in the strain rate sensitivity in Zr–1Nb at around 650–700 K, and attributed this to dynamic strain aging. From Fig. 2(b) it is seen that 575 K is close to the temperature at which the thermal stress becomes zero and the remnant stress obtained at this temperature is taken as the athermal stress. Using this the temperature corrected thermal stress, |*/v%, is obtained.

3.2. Acti6ation 6olume and enthalpy 6ariation During a stress relaxation test, the strain rate is continuously decreasing with time, thus the activation volume as calculated from Eq. (A.6) varies with both stress and strain rates1. In order to plot activation volume (V) at one strain rate and given strains, V at the start of each relaxation were interpolated to three strains 0.3, 0.6, 0.9. The V/b 3 versus |* plot at 10 − 4 s − 1 is shown in Fig. 3, where b is the Burgers vector= 3.2× 10 − 10 m. Here increasing value of |* implies decreasing value of T. It is seen that the activation volume increases with decreasing stress, and is unaffected by strain. The activation volume as shown in 1 Strain rate change tests were conducted at 296 K and the observed stress – strain rate relation was compared with that obtained during stress relaxation. The results established by both these methods were similar.



R. Kapoor et al. / Materials Science and Engineering A328 (2002) 324–333

Fig. 3 is used in Eq. (3) for calculating the activation enthalpy, DH. #|*/#T is obtained from Fig. 2(b) by fitting a spline curve to stress– temperature data and then taking the derivative. DH versus |* as calculated from Eq. (3) is also shown in Fig. 3 on the second y axis. DH increases with decreasing thermal stress. DH at |*=0, called DHo, extrapolates to about 1.8 eV.

3.3. Estimation of |ˆ *, Go, m; o From classical dislocation theory, the plastic strain rate is: m; p = hbzmw(|*, T) (4) where h is constant, b is burgers vector, zm is the mobile dislocation density, w(|*, T) is the velocity of dislocations as a function of the thermal stress |*, and temperature T. It is assumed that during thermal activation the dislocation velocity is a function of the thermal stress |*, and is independent of the athermal stress |a. w is usually related to |* through an Arrhenius relation [26–29]:

6 =uw exp

− DG(|*) kT

327

n

(5)

Kocks [30] suggested an empirical relation between DG and |*:

  n |* |ˆ *

DG =Go 1−

p q

(6)

Here | and Go are the thermal stress and activation free energy at 0 K, respectively. From Eqs. (4)–(6):

 

|* = |ˆ * 1−

n

kT m; o ln Go m; p

1/q 1/p

,

(7)

where m; o = hbzmuw. If the |* and T are related as Eq. (7), a plot of (|*)p versus T 1/q at constant strain rate, should yield a straight line. Ono [31], suggested that p=1/2, q= 3/2 results in a barrier shape profile that fits many predicted barrier shapes. Note that in Eq. (7) there is no shear modulus temperature dependence, but experimental data inherently contain shear modulus temperature dependence. Therefore, the experimentally obtained stress is corrected for this temperature dependence. A plot of (|*/v%)1/2 versus T 2/3 should produce a straight line, with: intercept=

'

|ˆ * v%

and

slope= −



k m; ln o Go m;

' 2/3

|ˆ * v% (8)

Fig. 4 is a plot of (|*/v%)1/2 versus T 2/3 at different strains, from which is obtained an intercept of 32.9 and a slope of 0.393. Thus: |ˆ * = 1060 MPa v%

and

k m; o ln − = 1.33× 10 − 3 Go 10 4

  n

(9)

Also from Eq. (7) for p= 1/2 and q= 3/2: ln m; p = ln m; o − Fig. 3. Experimental activation volume and activation enthalpy vs. thermal stress, at a strain rate of 10 − 4 s − 1.

|* Go 1− kT |ˆ *

1/2 3/2

(10)

Thus a least-square linear fit of ln m; p versus 1/T[1− (|*/|ˆ *)1/2]3/2 (where |ˆ * value is taken from Eq. (9) should result in ln m; o as intercept and − Go/k as slope. From such a fitting of the data at about 0.08 strain for 298, 377, 471, 568 K, the strain rate pre-exponential factor is estimated to lie in the range, 107 s − 1 B m; o B109 s − 1. Taking m; o = 108 s − 1 and using Eqs. (9) and (10), Go = 1.8 eV. Thus from the above exercise, the parameters of short range barrier during thermal activation in the 296–570 K range are estimated as, |ˆ */v% =1060 MPa, m; o = 108 s − 1, Go = 1.8 eV.

3.4. Remnant stress beha6ior

Fig. 4. Plot of (|*/v%)1/2 vs. T 2/3 at different strains.

The remnant stress |r, was obtained by decremental unloading technique. The corresponding transient stress at the start of the relaxation, |t, was obtained as, |t = |o − |r (Fig. 1). In Fig. 5 is shown the total stress |, the remnant stress |r, and the transient stress |t

328

R. Kapoor et al. / Materials Science and Engineering A328 (2002) 324–333

Fig. 5. Total stress |, remnant stress |r, and transient stress |t, vs. strain, at four temperatures.

versus strain for different temperatures. |t is independent of strain (except at small strain) whereas |r increases with strain. The strain dependence of the remnant stress is similar to that of the total stress. A plot of |r/v% and |t/v% versus T for fixed strains is shown in Fig. 6. Guiu [32] suggested that if the remnant stress represents the athermal stress, then for a given microstructure it should vary as the shear modulus varies with temperature. As is seen in the present experiments, the normalized remnant stress (|r/v%) has a strong temperature dependence, suggesting that the remnant stress does not represent the athermal stress.

independent of strain. At this point it would be important to consider the stress dependence on temperature history. Temperature history dependence implies that the microstructure evolves differently at different tem-

4. Discussion

4.1. Representation of |* and |a Fig. 2(b) shows that the normalized stress, |/v%, versus T for different strains is only shifted parallel to the stress axis. Plots of (|*/v%)1/2 versus T 2/3 (Fig. 4) fall on top of one another, suggesting that the temperature sensitivity of normalized stress at a given temperature is

Fig. 6. Normalized remnant and transient stress vs. temperature for fixed strains.

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329

4.2. Acti6ation 6olume and enthalpy 6ariation The activation volumes obtained are small at 297 and 377 K, and larger at 471 and 568 K. This is so because the thermal stress represents different regions at different temperatures. If the activation volume for 297, 377, 471 and 568 K are plotted together, as shown in Fig. 8 they appear as different parts of the same curve. Eq. (7) can be rewritten as:

 !  "n

m; p = m; o exp −

Go |* 1− kT |ˆ *

p q

.

(12)

Taking natural log of Eq. (12) and differentiating w.r.t. |* the experimental activation volume is obtained as: Fig. 7. Temperature change tests from 296 to 570 K.

V= kT

Fig. 8. Activation volume vs. thermal stress as obtained by stress relaxation at different temperatures.

peratures and since |a depends on microstructure, |a is affected by temperature history. The following experiment was performed to check the stress dependence on temperature history: sample deformed at 298 K to 0.035 strain, unloaded, deformed at 377 K to 0.066 strain, unloaded, deformed at 471 K to 0.095 strain, unloaded, deformed at 570 K to 0.13 strain, unloaded. All samples were deformed at 10 − 4 s − 1. Fig. 7 shows the results of this experiment. For each reloading, the flow stress nearly matches with the stress of the directly loaded samples, suggesting a temperature history independence of the flow stress. From this it can be concluded that in Zr– 1Nb – 1Sn – 0.1Fe, for 296 K BTB575 K deformed at 10 − 4 s − 1 to 0.15 strain, the thermal stress component depends only on strain rate and temperature, and the athermal stress component depends only on the strain, being independent of loading path, i.e.: |(m, m; , T)=|*(m; , T)+|a(m)

(11)

!  "  

( ln m; Gopq |* = 1− (|* |ˆ * |ˆ *

p q−1

|* |ˆ *

p−1

.

(13)

Using Go = 1.8 eV, |ˆ *=1060 MPa, p=1/2 and q= 3/2, a V− |* relation is obtained as shown by the solid line in Fig. 8. The calculated values and the experimental values match reasonably well, suggesting one activation mechanism in the 296–570 K temperature range. Further, as observed from Fig. 3, neither the activation volume nor the activation enthalpy vary significantly with strain, suggesting that the short range barriers do not evolve with deformation in the 297–570 K temperature range, and 0.02–0.12 strain range. Consider some of the common short range barriers to dislocation motion [33,34]. The dislocation–dislocation interaction mechanism has an activation volume ranging from about 102 –104 b 3, with the activation volume and enthalpy varying with strain. Conservative jog motion has an activation volume ranging from 10 to 1000 b 3, with the activation volume varying with strain. Climb has an activation volume of about 1–10 b 3, and an activation enthalpy corresponding to self diffusion energy (The self diffusion energy for Zr is about 3 eV). Cross slip has an activation volume ranging from about 10 to 1000 b 3, with the activation volume and enthalpy being independent of temperature. Peierls stress has an activation volume ranging from about 10 to 100 b 3, with the activation volume and enthalpy being independent of strain. Dislocation– interstitial interaction has an activation volume ranging from about 10 to 1000 b 3 with the activation parameters being dependent on the point defect concentration. The dislocation–interstitial interaction may depend on strain if the point defect concentration or distribution is altered during deformation. For the case of a-Zr, either Peierls stress [35,36] or dislocation–interstitial interaction [37–39] have been suggested as the short range barriers to dislocation motion. In addition to this it is also possible that both the Peierls stress and interstitial indirectly influence the double kink formation required to overcome the Peierls stress [40–42].

330

R. Kapoor et al. / Materials Science and Engineering A328 (2002) 324–333

In the present study, the activation volumes obtained are small and are independent of strain. This suggests that the short range barrier to dislocation motion is either the Peierls stress or the dislocation– interstitial interaction stress. From the experimental results obtained it is difficult to firmly establish one mechanism over the other. That would require a study of the variation of the activation parameters with interstitial content.

4.3. Dependence of |r on temperature As seen from Figs. 5 and 6, |r/v% varies with strain and also with temperature, whereas |t/v% varies with neither strain nor significantly with temperature. Some earlier workers have also observed a similar temperature dependence of the remnant stress [14,17,19–22]. Tiwari et al. [14] observed the remnant stress (which they call internal stress) in a-Zr to be temperature dependent and suggested the cause to be a temperature history dependence effect, but did not explicitly verify it. Mehrotra and Tangri [17] observed the remnant stress (which they called the athermal stress) in aZircaloy to be temperature dependent and attributed it to recovery effects, but did not explicitly verify it. Tung and Sommer [19] also observed the remnant stress (called the internal stress) in a-Ti to be temperature dependent, attributing it to history effects with no explicit experiment to show it. The reason for the |r dependence on T could be either that the stress has a temperature history dependence or that recovery takes place as temperature increases. In the present study, the athermal stress was independent of temperature history as shown in Fig. 9. In addition, the recovery effect was checked by reloading a sample at 296 K after relaxation at 296, 377, 471, 571, 675 K, as shown in Fig. 9. The recovery effect is about 50 MPa as marked on the figure. Also shown on the figure is the reload stress of a sample initially deformed at 666 K (including five relaxations) and

subsequently deformed at 296 K. This shows a combined temperature history dependence and recovery effect. As marked on the figure the reload flow stress is lower by about 70 MPa. From Fig. 6 the difference between |r/v% at 296 K and at 675 K, at 0.08 strain is about 140 MPa. Thus it is clear that the observed |r/v% dependence on temperature is not completely due to either temperature history or recovery effects.

4.4. Calculated |* 6ersus time In this section the thermal stress during relaxation is back calculated as a function of time. It is assumed that plastic deformation occurs by thermal activation as given by Eq. (7) and the recovery affects are negligible. The activation parameters are taken as estimated previously, |ˆ *= 1060 MPa, m; o = 108 s − 1, Go = 1.8 eV. During decremental unloading a criteria has to be set to identify a ‘zero’ stress rate. It is expected that this criteria depends primarily on the sensitivity of the load cell and the observation time during relaxation following decremental unloading. The usual method is to observe the load change in a fixed time interval. If the load change is negligible in this time interval, the load is supposed to have reached its remnant value. The time interval chosen should be such as to prevent recovery at higher temperatures. In the present analysis the limiting load rate was taken to be 3× 10 − 3 N s − 1. Correspondingly the smallest measurable strain rate from Eq. (A.3) is about 10 − 8 s − 1. Implications of the changes in the minimum load rate are discussed later. The thermal stress versus time relation was numerically calculated as follows. 1. Using a range of strain rates from 10 − 4 to 10 − 8 s − 1 the corresponding thermal stress (|*) was calculated from Eq. (7). 2. The corresponding stress rate (|; =|; *), was calculated using Eq. (A.3). 3. The time corresponding to m; p, |; and |* was calculated using: t=

&

|*

|* o

Fig. 9. Sample reloaded at 296 K after relaxing at 296, 377, 471, 571, 675 K and sample reloaded at 296 K after deforming at 666 K.

d|* |; *

(14)

where | *o is the value of |* at m; p = 10 − 4 s − 1, i.e. the start of relaxation. Fig. 10(a) shows the calculated |* versus time. |* is plotted up to the minimum measurable stress rate. It is clearly seen that the stress obtained at the smallest measurable stress rate of 3×10 − 3 N s − 1 varies with temperature. This value of stress corresponds to the experimentally obtained remnant stress. If instead of integrating Eq. (14) to a finite |*, the integral is carried out to |*=0, the complete stress relaxation curve is obtained. Such a plot is shown in Fig. 10(b), with time on a log scale. Superimposed on this is the loci of the maximum and minimum strain rates. The maximum

R. Kapoor et al. / Materials Science and Engineering A328 (2002) 324–333

thus:

 

m; c = m; o exp −

Go kT

331

(16)

At a given temperature if m; p \ m; c then |*\ 0, and if m; p 5 m; c then |*=0. m; c decreases with decreasing T, and m; c = 0 at T= 0 K. During relaxation, the plastic strain rate decreases with time till it reaches the critical strain rate (Eq. (16)), at which the thermal stress becomes zero. If the critical strain rate m; c, at a temperature T, lies in the window [m; max, m; min], i.e. m; max \ m; c \ m; min, then the remnant stress will correspond to the athermal stress. The critical strain rate in turn depends on the activation parameters m; o and Go. In the present experiments on Zr –1Nb –1Sn–0.1Fe, the critical strain rates and the corresponding times for complete relaxation are given in Table 1. It may be noted that the critical strain rate is very sensitive to temperature. A doubling of temperature changes the critical strain rate by 13 orders. Also seen from Fig. 10(b) is that improving the minimum measurable load rate by an order or two would only change the remnant stress by a small amount. It can be stated that, in general, the remnant stress obtained by the decremental unloading technique, does not represent the athermal stress.2 It should be noted here that the nature of the argument and calculations will not change for different materials. What would change are the activation parameters m; o and Go which would in turn modify the variation of critical strain rate with temperature. Fig. 10. Calculated (a) |* vs. time till minimum measurable strain rate, (b) |* vs. log time for complete relaxation. Table 1 Calculated critical strain rates and complete relaxation times at different temperatures Temperature (K)

Critical strain rate (s−1)

Relaxation time (s)

296 377 471 571

2×10−23 9×10−17 6×10−12 10−8

1017 1011 107 104

strain rate m; max, is the strain rate just before relaxation, and the minimum strain rate m; min, corresponds to the minimum measurable stress rate. Loci of three sets of minimum measurable strain rates are shown. The stress corresponding to m; min is the remnant stress. Thus during relaxation, the thermal stress variation with time can only be seen through the strain rate window of [m; max, m; min]. From Eq. (7) it is seen that, at a given temperature there exists a critical strain rate m; c, such that |* = 0, which gives: kT m; o 1/q 1/p 1− ln =0 (15) Go m; c

 

n

4.5. Dependence of |r on strain and temperature in terms of |* and |a The remnant stress can be represented in the form of Eq. (11). The remnant stress is the stress at the minimum measurable strain rate (m; min): |r(m, T)= |*(m; min, T)+ |a(m).

(17)

At the start of the relaxation |=|o, and m; =m; max |o(m, T)= |*(m; max, T)+ |a(m).

(18)

From Eqs. (17) and (18), and |t = |o − |r (from Fig. 1): |t = |*(m; max, T)− |*(m; min, T).

(19)

From these equations it is seen that |r has a strain dependence similar to |a, and a temperature dependence similar to |*. Also |t is independent of strain. Figs. 5 and 6 support the above interpretations. 2 The change of mobile dislocation density, zm, has not been considered here. If zm decreases with |* [43 – 45], the strain rate during relaxation would decrease further resulting in a higher remnant stress obtained.

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332

5. Conclusions

where:

1. The flow stress of Zr – 1Nb – 1Sn – 0.1Fe at 296– 570 K and 10 − 4 –10 − 6 s − 1 can be separated in two components, a thermal stress depending only on instantaneous strain rate and temperature, and an athermal stress depending only on deformation. 2. In the 296–570 K regime the activation volume and activation enthalpy were found to be independent of strain up to 0.12 strain. The magnitude of the activation volume, its variation with thermal stress, and it being independent of strain, suggests that either the Peierls stress or the dislocation– interstitial interaction stress is the rate controlling short range barrier to dislocation motion. 3. It is found that in Zr– 1Nb – 1Sn – 0.1Fe the normalized remnant stress was temperature dependent, and was shown that, in general, for the case of a single thermally activated process, the remnant stress does not represent the athermal stress. The remnant stress represents the athermal stress when the critical strain rate lies in the strain rate observation window.

Y=



n

(1+ e) Lo + X , C yr 2o

ro is the initial radius of sample and e is the strain at relaxation. Typical values of the parameters are, Lo = 30, ro = 6 mm, C= 0.07 mm K − 1 N − 1, and X ranges from 2 to 5 mm depending on strain. From Eqs. (1) and (A.3) the strain rate is analytically obtained as: m; p =

− |; Bna = (at+ 1) − (n + 1). Y Y

(A.4)

Taking natural log of Eq. (A.4) and differentiating w.r.t. |*(|; *= |; ):

 

(ln m; p (ln m; p (t n+ 1 (at+ 1)n = = . n (|* (t (| B

(A.5)

From which the experimental activation volume V is obtained as: V=kT

)

 

(ln m; p n+ 1 (at+ 1)n = kT . (|* T n B

(A.6)

Acknowledgements The authors gratefully acknowledge insightful discussions with Dr S. Banerjee, Head Materials Science Division, and Director Materials Group, Bhabha Atomic Research Centre. Thanks are also due to Dr T.K. Sinha and Dr S.K. Ray for useful discussions and to Dr G.K. Dey in TEM studies.

Appendix A In general the plastic strain rate, m; p, is calculated from [46]: m; p =

X: − P: C , Lo +X− PC

(A.1)

where P is the load, P: is the load rate, Lo is the original length, X is the cross-head displacement, X: is the cross-head velocity, and C is the combined compliance (units of length/force) of the machine and the sample. The value of C is obtained from the load– displacement curve. In the case of stress relaxation X: = 0, Eq. (A.1) becomes: m; p =

P: . P −(Lo +X)/C)

(A.2)

If (Lo +X)/C P, which is valid for the present study, then: m; p :

− |; , Y

(A.3)

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