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High temperature tensile deformation behavior of β-Ti alloys
Materials Science and Engineering, A129 (1990) 229-237
229
High Temperature Tensile Deformation Behavior of l -Ti Alloys M. N. VUAYSHANKAR and S. ANKEM
Engineering Materials Group, Department of Chemical and Nuclear Engineering, University of Maryland, College Park, MD 20742 (U.S.A.) (Received February 20, 1990; in revised form April 19, 1990)
Abstract The high temperature tensile deformation behavior of fl-Ti alloys has been investigated as function of strain rate, temperature and type of alloying elements. It was found that the magnitude of the flow stress drop increases with strain rate and solute concentration and decreases with temperature. For a given temperature and strain rate the extent of the drop decreased with a decrease in the relative atomic size between the solvent (titanium) and solute (manganese and vanadium) and a decrease in the diffusivity of the alloying element in the fl phase. In addition, it was found that prestraining to a stress below the peak stress significantly reduces the peak stress and the magnitude of the flow stress drop. On the basis of these results it is suggested that the initial flow stress drop is due to the rapid multiplication of dislocations. The steady state behavior after the initial flow stress drop was attributed to dynamic recovery leading to the formation of subgrains. 1. Introduction fl-Ti alloys are heat treatable and deep hardenable. They have good formability, high strength, good corrosion resistance and low elastic modulus. Owing to these attractive properties, they find use in various industrial applications. Furthermore, most of the a and a - f l alloys are processed at high temperatures in the fl phase field. To design new fl alloys with enhanced properties or to optimize the processing conditions for the near a and a - f l alloys, it is necessary to understand the deformation behavior of titanium alloys at high temperatures in the fl phase field. A number of investigators [1-4] have shown that abrupt flow softening followed by steady state behavior occurs in fl-Ti alloys at high temperatures. These earlier studies attribute the flow 0921-5093/90/$3.50
softening as well as the steady state behavior to dynamic recovery. Similar flow stress drops observed by Jonas et al. [5] in fl-Zr-Nb alloys were attributed by them to the declustering of solute atoms. Recently, Ankem et al. [6] suggested a possibility that the initial flow stress drop is related to the multiplication of mobile dislocations and the subsequent steady state is related to dynamic recovery. Apart from these [1-6] studies, no other significant information is available as to the mechanisms of high temperature deformation of ~-Ti alloys. Therefore a systematic study was undertaken to determine the effect of prestrain, heat treatments, diffusivity of alloying elements, temperature and strain rate on the high temperature deformation behavior of fl-Ti alloys. For these studies, T i - M n and T i - V alloys were used as the model systems. 2. Experimental procedure The chemical compositions of the titanium alloys used in this investigation are shown in Table 1. These alloys were annealed in vacuum at different temperatures for 200 h followed by water quenching. As indicated in Table 1, the annealing temperatures are above the respective fl transii. It is known that quenching these alloys
TABLE 1 Composition, heat treatment and test temperatures of the alloys used in this investigation
System
Solute, Mn or V (wt.% (at.%))
Ti-Mn Ti-Mn
9.4 (8.3) 13.0(11.5)
Ti-V
14.8 (14.0)
Annealing temperature
Test temperature
(K)a
(K)
1023 973 1023 973
1023 973 1023 973
aThe alloys were annealed at the indicated temperatures for 200 h followed by water quenching. © Elsevier Sequoia/Printed in The Netherlands
230
from these temperatures results in a metastable fl phase. High temperature tensile tests were conducted in vacuum. Cylindrical-pin-type tensile specimens with a gauge length of 6.35 cm were used for these tests [6]. Temperature variation was controlled to within _+5 K. Tensile tests were conducted at 973 and 1023 K at strain rates ranging from 0.000 11 to 0.026 s- 1. To check reproducibility, tests were repeated under identical conditions. The variation was found to be within _+5%. Therefore data corresponding to a single specimen for each testing condition were used in the stress-strain plots. From the load-displacement curves, true stress-true strain curves were calculated after subtracting machine displacements [6, 7]. Standard metallographic techniques were used for optical and scanning electron microscopy (SEM) studies [8, 9]. For transmission electron microscopy (TEM) studies, discs 3 mm in diameter and 500 /tm thick were spark cut and mechanically ground to 300 /~m. These discs were electropolished with a solution of 10 ml perchloric acid, 300 ml, methanol and 100 ml nbutanol at 243 K [9]. The final step consisted of electrothinning with a Fischione twin-jet electropolishing unit using a solution of zinc chloride, aluminum chloride, methanol and n-butanol at 298-303 K [9]. 3. Results The true stress-true strain plots corresponding to 1023 K for different strain rates for the Ti-13wt.%Mn alloy are shown in Fig. 1. As can be seen, the flow stresses and the flow stress drop increase with an increase in strain rate. The true stress-true strain curves corresponding to 973 K for the same alloy, which were obtained in an earlier investigation [5], are shown in Fig. 2. Comparison of Figs. 1 and 2 indicates that for a given strain and strain rate the flow stresses and the magnitude of the flow stress drop decrease with an increase in temperature. The true stress-true strain curves corresponding to 1023 K for the Ti-9.4wt.%Mn alloy are shown in Fig. 3. For a given strain and strain rate the flow stresses and the extent of the flow stress drop increase with an increase in solute content-compare Figs. 1 and 3. The true stress-true strain curves corresponding to 973 K for the T i 14.8wt.%V alloy are shown in Fig. 4. As is evi-
45
310
279
HEATTREATIAENT:1023 K. 200 HRS,,W.O.
4O
TESTTEMPERATURE:t 023 K 248
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217 3O 186
~
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10
.oouons
5
31 ,ooo11/8
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0
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O2
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Fig. 1. Effect of strain rate on the stress-strain behavior of a fl-Ti- 13wt.%Mn alloy.
45
310
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HEATTREA'I~IENT:073 K, 200 HRO.,W.O. ~TURE:
27g
9"/3K
35
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0 0
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.0~
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I I 0.1 0.15 TRUE 8TRAIN
I 0.2
0 0,25
Fig. 2. Effect of strain rate on the stress-strain behavior of a fl-Ti- 13wt.%Mn alloy.
dent from Figs. 2 and 4, at similar strain and strain rates the flow stresses as well as the flow stress drop for the T i - M n alloy are higher. To determine the mechanisms of the abrupt flow stress drops and to study the difference in
231 45
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310 HEAT TREATMENT : 1(]~1K. 200 HRS..W.Q. 279
HEAT TREA~JIENT: 10~!3K. 200 HRS..W.O.
40
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TEST TBMPERATURE : 1023 K STRAIN RATE : 0.026tS
TEST TEMPERATUR~E: 1023K 248 35
248. 35 A
217 30
TESTED WITHOUTANY PRESTRAIN 217
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186 25
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Fig. 3. Effect of strain rate on the stress-strain behavior of a fl-Ti-9.4wt.%Mn alloy.
45
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HEAT TREATMENT : 973 K. 200 HRS,. W,Q.
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.
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Fig. 5. Comparison of the stress-strain behavior of a flTi-13wt.%Mn alloy with and without prestrain.
tests are shown in Figs. 5-10. The test conditions are also included in the figures. The significance of these tests will be discussed in the next section. Optical micrographs of Ti-13wt.% Mn alloy before and after deformation are shown in Fig. 11. The deformed microstructure shows the formation of subgrains. Similar micrographs of Ti-14.Swt.%V alloy are shown in Fig. 12. The deformed microstructure of this alloy also shows the formation of subgrains. Transmission electron micrographs of the Ti-13wt.%Mn and Ti14.Swt.%V alloys after deformation are shown in Fig. 13..They clearly indicate subgraln formation and dislocation activity inside the subgrains.
10 62
4. Discussion
.00011~ 5
31
0 0
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I
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0.05
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0.2
0 0.25
TRUE STRAIN
Fig. 4. Effect of strain rate on the stress-strain behavior of a fl-Ti- 14.8wt.%V alloy.
behavior between the Ti-Mn alloy and the T i - V alloys, several special tests were conducted. These included strain interrupt tests using the same specimens or testing the specimens with different heat treatments. The results of these
The crystal structure of the fl phase is b.c.c. The systems Ti-Mn and T i - V are fl eutectoid and fl isomorphous respectively [10]. The atomic sizes of titanium, manganese and vanadium are 1.47, 1.12 and 1.32 A respectively as indicated in Table 2. Given that the relative size difference between titanium and manganese is more than that between titanium and vanadium, solid solution strengthening from the solute manganese is expected to be more. As expected, the flow stresses for a given strain, strain rate and temperature of the Ti-11.5at.%Mn are higher than
232 45
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Fig. 6. Comparison of the stress-strain behavior of a 8Ti-13wt.%M_n alloy with and without prestrain.
45
I 0.15
Fig. 8. Comparison of the effect of heat treatment on the stress-strain behavior of a fl-Ti- 13wt.%Mn alloy.
310
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TEST1TddPERATURJ::1)73K STIU~ flATE : 0.~V8
279
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Fig. 7. Effect of prestrain and heat treatment on the stress-strain behavior of a fl-Ti-13wt.%Mn alloy.
Fig. 9. Effect of prestrain and heat treatment on the stress-strain behavior of a ~8-Ti-14.8wt.%V alloy.
those of the Ti-14at.%V--see Figs. 2 and 4. Furthermore, the flow stresses also increased with an increase in the manganese content--compare Figs. 1 and 3--as expected.
Abrupt flow stress drops in t h e / ; alloys indicate some type of dislocation-solute atom interaction since flow stress drops were not observed in pure fl-Ti [11]. From Table 2 it is clear that the
233 45
310 TEST TEMPERATURE : 973 K 8TRAIN RATE : 0~2(1~
40
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HEAT TREA1t/F.NT 35
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973 K, 200 HRS., W.O., 1123K. 2 HRS.. FC TO 973 K
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Fig. 10. Comparison of the effect of heat treatment on the stress-strain behavior of a fl-Ti-14.8wt.%V alloy.
atomic sizes of both manganese and vanadium are less than that of titanium. Therefore elastic interactions exist between dislocations and the solute atoms, i.e. Cottrell locking. Such interactions can result in flow stress drops in substitutional alloys [12-15]. The magnitude of these interactions depends on the relative atomic size difference between the solvent and the solute [15]. The interaction energy increases as the relative size difference increases. In addition, for a given solute the magnitude of the interaction energy increases with increase in solute content [15]. Therefore these factors must be considered in rationalizing the flow stress drop behavior. The occurrence of an abrupt flow stress drop in the tensile stress-strain curves of solid solution alloys has been explained [16] with two different theories, i.e. the static and dynamic theories. In the static theory the dislocations are assumed to be pinned by solute atoms and the flow stress drop is attributed to the breakaway of dislocations from their atmospheres [16]. This theory has been applied to explain the flow stress drop in iron containing interstitial impurities such as carbon [15]. An: alternative explanation proposed by Johnston and Gilman [17], the dynamic theory, attributes the flow stress drop to the low initial mobile dislocation density and the subsequent rapid multiplication of dislocations. Abrupt
Fig. 11. Comparison of microstructures of a Ti-13wt.%Mn alloy (a) before and (b) after deformation. Test conditions: 973K, 0.026 s -1, 25% strain. Note the formation of subgrains.
flow stress drops have been observed in several materials [17-19] containing low initial mobile dislocation densities. Johnson and Gilman [17] showed that in the case of LiF the sharp flow stress drop was a consequence of the low dislocation density. An increase in the initial dislocation density reduced the extent of the flow stress drop. Hahn [20] modelled the flow stress drop behavior of b.c.c, materials. This model accounts for the dependence of the drop on the initial mobile dislocation density and strain rate. While some of the results of the present investigation can be explained on the basis of either the static or the dynamic theory, the latter appears to be the most probable mechanism for the following reasons. First of all, the initial dislocation density in the fl phase was found to be low. A
234
Fig. 12. Comparison of microstructures of a Ti-14.8wt.%V alloy (a) before and (b) after deformation. Test conditions: 973K, 0.026 s -j, 25% strain. Note the formation of subgrains.
similar observation was made earlier by Ankem and Margolin [21]. T E M observation of the deformed specimen immediately after the initial flow stress drop showed a significant increase in dislocation density, suggesting an increase in dislocation density during the drop. The other reason is related to the fact that when the specimen was deformed to a strain of 0.03 and annealed at the same temperature for 12 h and reloaded, the flow stress was lower than the initial peak stress. If the drop was due to just unlocking then the flow stress upon reloading should have reached the initial peak stress. These observations suggest that the flow stress drop is related to another mechanism, i.e. multiplication of dislocations. Therefore in the following analyses it is assumed that the flow stress drop is related to the multiplication of mobile dislocations. The velocity of dislocations in a material is a function of stress [17] and can be represented by an empirical equation of the form
Fig. 13. Transmission electron micrographs of titanium alloys after deformation: (a) Ti-13wt.%Mn alloy showing dislocation structure inside subgrains; (b) Ti-14.8wt.%V alloy showing subgrain formation. Test conditions: 973 K, 0.026 s- ~, 25% strain.
v=
(;0) n
(1)
where a is the applied stress, a0 is a reference stress and, n is the stress exponent. The strain rate imposed on a material can be expressed by the relationship
=pbv
(2)
where p is the mobile dislocation density, b is the Burgers vector and, v is the velocity of dislocations. In materials containing a low initial mobile dislocation density, the velocity of dislocations has
235 TABLE 2 Atomic size and diffusivity of solvent (titanium) and solute (manganese or vanadium)
Atomic size (A) Diffusivity in fl-Ti at 973 K
( )< 1011 cm z s-I) a
Ti
Mn
V
1.47 3.3
1.12 16
1.32 1.7
TABLE 3 Comparison of theoretically predicted and experimentally determined peak stress ratios corresponding to two different strain rates Strain rates corresponding to the peak stress ratios
Peak stress ratio for two different strain rates
~ (s ~) 42 (s 1)
Experimental
Theoretical
1.97
1.99
4.06
3.98
5.36
5.15
2.8
2.6
aFrom ref. 22.
to be high to match the applied strain rate, as is evident from eqn. (2). From eqn. (1), higher velocity results in higher flow stress. This leads to rapid multiplication and an increase in the mobile dislocation density. Once the dislocation density increases, the velocity drops, as can be seen from eqn. (2). From eqn. (1), this implies that the flow stress should drop to a lower value, resulting in flow stress drops. It is to be noted that the total dislocation density itself need not be low; what is needed is a low initial mobile dislocation density for the flow stress drops to occur. The multiplication mechanism outlined above readily accounts for the observed strain rate dependence of the flow stress drop--Figs. 1-4. As mentioned earlier, the velocity of dislocations is dependent on the imposed strain rate. An increase in strain rate results in an increase in the dislocation velocity. Thus the flow stress increases with an increase in strain rate. Once multiplication occurs, the flow stress decreases, giving rise to the flow stress drop. An increase in strain rate should result in an increase in the magnitude of the flow stress drop. This can be seen in Figs. 1-4. Theoretical calculations were made to predict the maximum (peak) stress as a function of strain rate for the Ti-13wt.%Mn alloy, corresponding to the test temperature of 973 K, using the following relationship [20]: o m = A g 1/"
(3)
where am is the stress corresponding to a strain of 0.005, g is the strain rate, and A is a constant. From eqn. (3), the maximum stress can be determined if the values of A and n are known. Since the value of A was not known, given that the initial microstructures of the specimens tested at different strain rates were similar, ratios of the maximum stresses at two different strain rates were computed with the following relationship: Orm~'
(~1) l]n
O.m~"~
(~2)1/n
(4)
0.026 0.0026 0.026 0.00026 0.026 0.00011 0.0026 0.000l 1
where, am~' is the peak stress at strain rate gl and am g2is the peak stress at strain rate g2" For these calculations a value of 0.3 was used for the exponent 1/n. This value was obtained from a previous study [6]. In that study the strain rate sensitivity m for fl-Ti-13wt.%Mn alloy was determined as 0.3 and m is equal to 1 / n [20]. Theoretical ratios determined by this method are compared with the experimentally obtained ratios in Table 3. As can be seen, the theoretical ratios are close to the experimental ratios. The difference in the magnitude of the flow stress drop--Figs. 1 and 3--can be explained on the basis of solute concentration. The equilibrium solute concentration near a dislocation is given by the expression [15]
exp() where C o is the average concentration and U is the interaction energy. Therefore for a given temperature an increase in C o results in higher C. This in turn results in higher flow stress drops. Comparison of the flow stress drop of Ti-9.4wt.%Mn and Ti-13wt.%Mn alloys--Figs. 3 and 1--clearly shows that the extent of the drop increases with solute content. The effect of the nature of the solute on the flow stress drop can be seen by comparing Figs. 2 and 4, the stress-strain curves for the Ti-ll.5at.% Mn and Ti-14at.%V alloys respectively. They show that the magnitude of the drop is higher for the T i - M n alloy as compared to the Ti-V alloy. This is consistent with the fact that
236 the relative size difference between titanium and manganese is higher than that between titanium and vanadium and the larger size difference is expected to result in stronger dislocation-solute atom interactions. Stronger interactions result in higher peak stresses and larger flow stress drops. It is known that the friction stress in b.c.c. materials is dependent on temperature [14]. A n increase in temperature results in a decrease in the friction stress. Since the velocity of dislocations is a function of stress, it is expected that an increase in temperature should result in a decrease in the extent of flow stress drop. Comparison of Figs. 1 and 2 shows an increase in magnitude of the flow stress drop with a decrease in test temperature. Studies on the effect of prestrain on the flow stress drops--Figs. 5 and 6--are also consistent with the suggestion that the flow stress drop is related to the multiplication of dislocations. Prestraining to a stress level below the peak stress results in a decrease in the peak stress--see Figs. 5 and 6. Moreover, the magnitude of the flow stress drop after prestraining is lower for higher prestrains. Similar behavior observed in the past [18, 19] has been attributed to prior multiplication. The behavior observed in the Ti-Mn system can also be attributed to prior multiplication. It is interesting to note that the decrease in magnitude of the peak stress increases with increasing prestrain--compare Figures 5 and 6. Evidently, the extent of increase in the mobile dislocation density depends on the initial prestrain, being higher for larger prestrains below the peak stress. Special tests involving either strain interrupts followed by annealing or annealing at temperatures higher than the test temperature also indicate that the abrupt flow stress drop is related to the low initial mobile dislocation density. As can be seen from Figs. 7 and 9, the flow stress drop is recovered after annealing. This can be attributed to a decrease in the mobile dislocation density during annealing. As expected, the magnitude of the drop as well as the peak stress are higher for the Ti-ll.5at.%Mn alloy owing to the higher diffusivities of manganese in the fl phase [22] and larger relative size difference between titanium and manganese as compared to titanium and vanadium--see Table 2. Similar reasoning can also explain the higher flow stress drop and peak stress for the Ti-11.5at.%Mn alloy as compared to the Ti-14at.%V alloy--see Figs. 8 and 1 0 after annealing.
As mentioned before, Jonas et aL [5] observed similar flow stress drops in fl-Zr-Nb alloys and the magnitude of the drop increased with solute content. Increasing the annealing time prior to their tests resulted in an increase in magnitude of the flow stress drop. On the basis of these observations they [5] have suggested that the flow stress drop is due to the declustering of solute atoms. One of their reasons for this suggestion was that clustering is expected in the Z r - N b system because it exhibits a miscibility gap in the fl phase field. As far as the mechanism for the flow stress drops in Ti-Mn and Ti-V alloys is concerned, it does not appear to be due to the declustering phenomenon. First of all, the flow stress drops were observed whether there was a miscibility gap in the fl phase field (Ti-V system) or not (Ti-Mn system). If the declustering phenomenon was to be the mechanism in the fl-Ti alloys, then prestraining below the maximum stress should not have resulted in lower flow stress drops. Furthermore, the increase in peak stress after annealing at higher temperatures in the fl phase field cannot be explained on the basis of the declustering phenomenon because clustering should have been less prevalent at higher temperatures and this should have resulted in lower peak stresses. Rather, annealing at higher temperatures resulted in an increase in the peak stress and the magnitude of the flow stress drop. Therefore the authors strongly suggest that the flow stress drop observed in fl-Ti alloys is related to the multiplication of mobile dislocations as discussed above. So far, the discussion has been confined to the flow stress drops occurring in the initial stages of deformation. The steady state behavior observed in these alloys subsequent to the flow stress drop indicates that a dynamic restoration mechanism is operative in this region. In the past, several investigators [2, 23-26] have suggested that the steady state behavior is due to the process of dynamic recovery leading to the formation of subgrains. In the present investigation, subgrain formation was also observed--see Figs. 11-13. The authors agree with the earlier investigators [2, 23-26] that the steady state behavior subsequentto the flow stress drop is related to the formation of subgrains. 5. Conclusions
(1) Abrupt flow stress drops followed by steady state behavior were observed in fl-Ti-Mn
237
and fl-Ti-V alloys. The magnitude of the flow stress drop increased with strain rate and solute concentration and decreased with temperature. (2) Prestraining to a stress level below the peak stress resulted in a decrease in the peak stress as well as a decrease in the extent of flow stress drop. The extent of decrease in peak stress increased with amount of prestrain. (3) Post-deformation annealing treatment resulted in a recovery of the peak stress as well as the flow stress drop. Higher annealing temperatures also resulted in an increase in the peak stress and the extent of flow stress drop. (4) The abrupt flow stress drops were attributed to the multiplication of mobile dislocations. The extent of the flow stress drop was found to be higher for the Ti-Mn alloy as compared to the T i - V alloy. This difference was attributed to the higher relative atomic size difference between the solvent titanium and manganese and the higher diffusivities of manganese in fl-Ti. (5) The steady state behavior subsequent to the flow stress drop was attributed to the process of dynamic recovery leading to the formation of subgrains, as suggested by several investigators in the past. Acknowledgments The authors would like to express their gratitude to Messrs. D. J. McNeish, D. E. Thomas and S. R. Seagle and their associates of the RMI company for making the alloys used in this investigation with great care at a nominal cost. The authors are grateful to Dr. R. J. Arsenault for helpful discussions. The authors are indebted to Mr. I. L. Caplan of DTRC and Dr. A. H. Rosenstein of A F O S R for their keen interest and constant encouragement throughout the course of this investigation. This work was supported by the Air Force Office of Scientific Research on Grant AFOSR-85-0367. References 1 S. M. L. Sastry, P. S. Pao and K. K. Sankaran, in H. Kimura and O. Izumi (eds.), Titanium '80, Science and
Technology, AIME, New York, 1980, p. 874. 2 C. H. Hamilton, in S. P. Agrawal (ed.), Proc. Conf. on Superplastic Forming, ASM, Metals Park, OH, 1984, p. 122. 3 D.L. Bourell and H. J. McQueen, J. Appl. Metalworking, (1987) 15. 4 N. Furushiro and S. Hori, in H. Kimura and O. Izumi (eds.), Titanium '80, Science and Technology, AIME, New York, 1980, p. 1067. 5 J. J. Jonas, B. Heritier and M. J. Luton, Metall. Trans. A, 10(1979) 611. 6 S. Ankem, J. G. Shyue, M. N. Vijayshankar and R. J. Arsenault, Mater. Sci. Eng., A l l l (1989) 51. 7 J. E. Hockett and E P. Gilfis, Mater. Res. Stand., 13 (1970) 251. 8 I. Weiss, E H. Froes and D. Eylon, Metall. Trans. A, 15 (1984) 1493. 9 G. E Van Der Voort, Metallography Principles and Practice, McGraw-Hill, New York, 1984. 10 J. L. Murray, in T. H. Massalki (ed.), Binary Alloy Phase Diagrams, Vol. 2, ASM, Metals Park, OHI 1986, pp. 1598, 2145. 11 H. Oikawa, K. Nishimura and M. X. Cui, Scr. Metall., 19 (1985)826. 12 L. I. Van Tome and G. Thomas, Acta MetaU., 14 (1966) 621. 13 G.R. Wilms, J. Less Common Met., 6 (1964) 169. 14 R. E. Smallman, Modern Physical Metallurgy, Butterworths, London, 1985. 15 E.O. Hall, Yield Point Phenomenon in Metals and Alloys, Plenum, New York, 1970. 16 N. E Fiore and C. L. Bauer, Prog. Mater. Sci., 13 (1967) 85. 17 W, G. Johnston and J. J. Gilman, J. Appl. Phys., 30 (1959) 129. 18 R. J. Arsenault, Trans. Metall. Soc. A1ME, 230 (1964) 1570. 19 J. R. Patel and A. R. Chaudhuri, J. Appl. Phys., 34 (1963) 2788. 20 G.T. Hahn, Acta Metall., 10 (1962) 727. 21 S. Ankem and H. Margolin, Metall. Trans. A, 17 (1986) 2209. 22 R. C. West (ed.), Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 64th edn., 1983-1984, p. F51. 23 E Griffith and C. Hammond, Acta Metall., 20 (1972) 935. 24 G. C. Morgan and C. Hammond, in G. Lutjering, U. Zwicker and W. Bunk (eds.), Titanium '84, Science and Technology, AIME, New York, 1984, p. 717. 25 T. Sheppard and J. Norley, Mater. Sci. Technol. (October 1988) 903. 26 J. J. Jones and M. J. Luton, in J. J. Burke and V. Weirs (eds.), Advances in Deformation Processing, Vol. 4, Plenum, New York, 1978.
229
High Temperature Tensile Deformation Behavior of l -Ti Alloys M. N. VUAYSHANKAR and S. ANKEM
Engineering Materials Group, Department of Chemical and Nuclear Engineering, University of Maryland, College Park, MD 20742 (U.S.A.) (Received February 20, 1990; in revised form April 19, 1990)
Abstract The high temperature tensile deformation behavior of fl-Ti alloys has been investigated as function of strain rate, temperature and type of alloying elements. It was found that the magnitude of the flow stress drop increases with strain rate and solute concentration and decreases with temperature. For a given temperature and strain rate the extent of the drop decreased with a decrease in the relative atomic size between the solvent (titanium) and solute (manganese and vanadium) and a decrease in the diffusivity of the alloying element in the fl phase. In addition, it was found that prestraining to a stress below the peak stress significantly reduces the peak stress and the magnitude of the flow stress drop. On the basis of these results it is suggested that the initial flow stress drop is due to the rapid multiplication of dislocations. The steady state behavior after the initial flow stress drop was attributed to dynamic recovery leading to the formation of subgrains. 1. Introduction fl-Ti alloys are heat treatable and deep hardenable. They have good formability, high strength, good corrosion resistance and low elastic modulus. Owing to these attractive properties, they find use in various industrial applications. Furthermore, most of the a and a - f l alloys are processed at high temperatures in the fl phase field. To design new fl alloys with enhanced properties or to optimize the processing conditions for the near a and a - f l alloys, it is necessary to understand the deformation behavior of titanium alloys at high temperatures in the fl phase field. A number of investigators [1-4] have shown that abrupt flow softening followed by steady state behavior occurs in fl-Ti alloys at high temperatures. These earlier studies attribute the flow 0921-5093/90/$3.50
softening as well as the steady state behavior to dynamic recovery. Similar flow stress drops observed by Jonas et al. [5] in fl-Zr-Nb alloys were attributed by them to the declustering of solute atoms. Recently, Ankem et al. [6] suggested a possibility that the initial flow stress drop is related to the multiplication of mobile dislocations and the subsequent steady state is related to dynamic recovery. Apart from these [1-6] studies, no other significant information is available as to the mechanisms of high temperature deformation of ~-Ti alloys. Therefore a systematic study was undertaken to determine the effect of prestrain, heat treatments, diffusivity of alloying elements, temperature and strain rate on the high temperature deformation behavior of fl-Ti alloys. For these studies, T i - M n and T i - V alloys were used as the model systems. 2. Experimental procedure The chemical compositions of the titanium alloys used in this investigation are shown in Table 1. These alloys were annealed in vacuum at different temperatures for 200 h followed by water quenching. As indicated in Table 1, the annealing temperatures are above the respective fl transii. It is known that quenching these alloys
TABLE 1 Composition, heat treatment and test temperatures of the alloys used in this investigation
System
Solute, Mn or V (wt.% (at.%))
Ti-Mn Ti-Mn
9.4 (8.3) 13.0(11.5)
Ti-V
14.8 (14.0)
Annealing temperature
Test temperature
(K)a
(K)
1023 973 1023 973
1023 973 1023 973
aThe alloys were annealed at the indicated temperatures for 200 h followed by water quenching. © Elsevier Sequoia/Printed in The Netherlands
230
from these temperatures results in a metastable fl phase. High temperature tensile tests were conducted in vacuum. Cylindrical-pin-type tensile specimens with a gauge length of 6.35 cm were used for these tests [6]. Temperature variation was controlled to within _+5 K. Tensile tests were conducted at 973 and 1023 K at strain rates ranging from 0.000 11 to 0.026 s- 1. To check reproducibility, tests were repeated under identical conditions. The variation was found to be within _+5%. Therefore data corresponding to a single specimen for each testing condition were used in the stress-strain plots. From the load-displacement curves, true stress-true strain curves were calculated after subtracting machine displacements [6, 7]. Standard metallographic techniques were used for optical and scanning electron microscopy (SEM) studies [8, 9]. For transmission electron microscopy (TEM) studies, discs 3 mm in diameter and 500 /tm thick were spark cut and mechanically ground to 300 /~m. These discs were electropolished with a solution of 10 ml perchloric acid, 300 ml, methanol and 100 ml nbutanol at 243 K [9]. The final step consisted of electrothinning with a Fischione twin-jet electropolishing unit using a solution of zinc chloride, aluminum chloride, methanol and n-butanol at 298-303 K [9]. 3. Results The true stress-true strain plots corresponding to 1023 K for different strain rates for the Ti-13wt.%Mn alloy are shown in Fig. 1. As can be seen, the flow stresses and the flow stress drop increase with an increase in strain rate. The true stress-true strain curves corresponding to 973 K for the same alloy, which were obtained in an earlier investigation [5], are shown in Fig. 2. Comparison of Figs. 1 and 2 indicates that for a given strain and strain rate the flow stresses and the magnitude of the flow stress drop decrease with an increase in temperature. The true stress-true strain curves corresponding to 1023 K for the Ti-9.4wt.%Mn alloy are shown in Fig. 3. For a given strain and strain rate the flow stresses and the extent of the flow stress drop increase with an increase in solute content-compare Figs. 1 and 3. The true stress-true strain curves corresponding to 973 K for the T i 14.8wt.%V alloy are shown in Fig. 4. As is evi-
45
310
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HEATTREATIAENT:1023 K. 200 HRS,,W.O.
4O
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Fig. 1. Effect of strain rate on the stress-strain behavior of a fl-Ti- 13wt.%Mn alloy.
45
310
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27g
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Fig. 2. Effect of strain rate on the stress-strain behavior of a fl-Ti- 13wt.%Mn alloy.
dent from Figs. 2 and 4, at similar strain and strain rates the flow stresses as well as the flow stress drop for the T i - M n alloy are higher. To determine the mechanisms of the abrupt flow stress drops and to study the difference in
231 45
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310 HEAT TREATMENT : 1(]~1K. 200 HRS..W.Q. 279
HEAT TREA~JIENT: 10~!3K. 200 HRS..W.O.
40
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TEST TBMPERATURE : 1023 K STRAIN RATE : 0.026tS
TEST TEMPERATUR~E: 1023K 248 35
248. 35 A
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Fig. 3. Effect of strain rate on the stress-strain behavior of a fl-Ti-9.4wt.%Mn alloy.
45
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HEAT TREATMENT : 973 K. 200 HRS,. W,Q.
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.
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Fig. 5. Comparison of the stress-strain behavior of a flTi-13wt.%Mn alloy with and without prestrain.
tests are shown in Figs. 5-10. The test conditions are also included in the figures. The significance of these tests will be discussed in the next section. Optical micrographs of Ti-13wt.% Mn alloy before and after deformation are shown in Fig. 11. The deformed microstructure shows the formation of subgrains. Similar micrographs of Ti-14.Swt.%V alloy are shown in Fig. 12. The deformed microstructure of this alloy also shows the formation of subgrains. Transmission electron micrographs of the Ti-13wt.%Mn and Ti14.Swt.%V alloys after deformation are shown in Fig. 13..They clearly indicate subgraln formation and dislocation activity inside the subgrains.
10 62
4. Discussion
.00011~ 5
31
0 0
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0.2
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TRUE STRAIN
Fig. 4. Effect of strain rate on the stress-strain behavior of a fl-Ti- 14.8wt.%V alloy.
behavior between the Ti-Mn alloy and the T i - V alloys, several special tests were conducted. These included strain interrupt tests using the same specimens or testing the specimens with different heat treatments. The results of these
The crystal structure of the fl phase is b.c.c. The systems Ti-Mn and T i - V are fl eutectoid and fl isomorphous respectively [10]. The atomic sizes of titanium, manganese and vanadium are 1.47, 1.12 and 1.32 A respectively as indicated in Table 2. Given that the relative size difference between titanium and manganese is more than that between titanium and vanadium, solid solution strengthening from the solute manganese is expected to be more. As expected, the flow stresses for a given strain, strain rate and temperature of the Ti-11.5at.%Mn are higher than
232 45
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Fig. 6. Comparison of the stress-strain behavior of a 8Ti-13wt.%M_n alloy with and without prestrain.
45
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Fig. 8. Comparison of the effect of heat treatment on the stress-strain behavior of a fl-Ti- 13wt.%Mn alloy.
310
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Fig. 7. Effect of prestrain and heat treatment on the stress-strain behavior of a fl-Ti-13wt.%Mn alloy.
Fig. 9. Effect of prestrain and heat treatment on the stress-strain behavior of a ~8-Ti-14.8wt.%V alloy.
those of the Ti-14at.%V--see Figs. 2 and 4. Furthermore, the flow stresses also increased with an increase in the manganese content--compare Figs. 1 and 3--as expected.
Abrupt flow stress drops in t h e / ; alloys indicate some type of dislocation-solute atom interaction since flow stress drops were not observed in pure fl-Ti [11]. From Table 2 it is clear that the
233 45
310 TEST TEMPERATURE : 973 K 8TRAIN RATE : 0~2(1~
40
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973 K, 200 HRS., W.O., 1123K. 2 HRS.. FC TO 973 K
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atomic sizes of both manganese and vanadium are less than that of titanium. Therefore elastic interactions exist between dislocations and the solute atoms, i.e. Cottrell locking. Such interactions can result in flow stress drops in substitutional alloys [12-15]. The magnitude of these interactions depends on the relative atomic size difference between the solvent and the solute [15]. The interaction energy increases as the relative size difference increases. In addition, for a given solute the magnitude of the interaction energy increases with increase in solute content [15]. Therefore these factors must be considered in rationalizing the flow stress drop behavior. The occurrence of an abrupt flow stress drop in the tensile stress-strain curves of solid solution alloys has been explained [16] with two different theories, i.e. the static and dynamic theories. In the static theory the dislocations are assumed to be pinned by solute atoms and the flow stress drop is attributed to the breakaway of dislocations from their atmospheres [16]. This theory has been applied to explain the flow stress drop in iron containing interstitial impurities such as carbon [15]. An: alternative explanation proposed by Johnston and Gilman [17], the dynamic theory, attributes the flow stress drop to the low initial mobile dislocation density and the subsequent rapid multiplication of dislocations. Abrupt
Fig. 11. Comparison of microstructures of a Ti-13wt.%Mn alloy (a) before and (b) after deformation. Test conditions: 973K, 0.026 s -1, 25% strain. Note the formation of subgrains.
flow stress drops have been observed in several materials [17-19] containing low initial mobile dislocation densities. Johnson and Gilman [17] showed that in the case of LiF the sharp flow stress drop was a consequence of the low dislocation density. An increase in the initial dislocation density reduced the extent of the flow stress drop. Hahn [20] modelled the flow stress drop behavior of b.c.c, materials. This model accounts for the dependence of the drop on the initial mobile dislocation density and strain rate. While some of the results of the present investigation can be explained on the basis of either the static or the dynamic theory, the latter appears to be the most probable mechanism for the following reasons. First of all, the initial dislocation density in the fl phase was found to be low. A
234
Fig. 12. Comparison of microstructures of a Ti-14.8wt.%V alloy (a) before and (b) after deformation. Test conditions: 973K, 0.026 s -j, 25% strain. Note the formation of subgrains.
similar observation was made earlier by Ankem and Margolin [21]. T E M observation of the deformed specimen immediately after the initial flow stress drop showed a significant increase in dislocation density, suggesting an increase in dislocation density during the drop. The other reason is related to the fact that when the specimen was deformed to a strain of 0.03 and annealed at the same temperature for 12 h and reloaded, the flow stress was lower than the initial peak stress. If the drop was due to just unlocking then the flow stress upon reloading should have reached the initial peak stress. These observations suggest that the flow stress drop is related to another mechanism, i.e. multiplication of dislocations. Therefore in the following analyses it is assumed that the flow stress drop is related to the multiplication of mobile dislocations. The velocity of dislocations in a material is a function of stress [17] and can be represented by an empirical equation of the form
Fig. 13. Transmission electron micrographs of titanium alloys after deformation: (a) Ti-13wt.%Mn alloy showing dislocation structure inside subgrains; (b) Ti-14.8wt.%V alloy showing subgrain formation. Test conditions: 973 K, 0.026 s- ~, 25% strain.
v=
(;0) n
(1)
where a is the applied stress, a0 is a reference stress and, n is the stress exponent. The strain rate imposed on a material can be expressed by the relationship
=pbv
(2)
where p is the mobile dislocation density, b is the Burgers vector and, v is the velocity of dislocations. In materials containing a low initial mobile dislocation density, the velocity of dislocations has
235 TABLE 2 Atomic size and diffusivity of solvent (titanium) and solute (manganese or vanadium)
Atomic size (A) Diffusivity in fl-Ti at 973 K
( )< 1011 cm z s-I) a
Ti
Mn
V
1.47 3.3
1.12 16
1.32 1.7
TABLE 3 Comparison of theoretically predicted and experimentally determined peak stress ratios corresponding to two different strain rates Strain rates corresponding to the peak stress ratios
Peak stress ratio for two different strain rates
~ (s ~) 42 (s 1)
Experimental
Theoretical
1.97
1.99
4.06
3.98
5.36
5.15
2.8
2.6
aFrom ref. 22.
to be high to match the applied strain rate, as is evident from eqn. (2). From eqn. (1), higher velocity results in higher flow stress. This leads to rapid multiplication and an increase in the mobile dislocation density. Once the dislocation density increases, the velocity drops, as can be seen from eqn. (2). From eqn. (1), this implies that the flow stress should drop to a lower value, resulting in flow stress drops. It is to be noted that the total dislocation density itself need not be low; what is needed is a low initial mobile dislocation density for the flow stress drops to occur. The multiplication mechanism outlined above readily accounts for the observed strain rate dependence of the flow stress drop--Figs. 1-4. As mentioned earlier, the velocity of dislocations is dependent on the imposed strain rate. An increase in strain rate results in an increase in the dislocation velocity. Thus the flow stress increases with an increase in strain rate. Once multiplication occurs, the flow stress decreases, giving rise to the flow stress drop. An increase in strain rate should result in an increase in the magnitude of the flow stress drop. This can be seen in Figs. 1-4. Theoretical calculations were made to predict the maximum (peak) stress as a function of strain rate for the Ti-13wt.%Mn alloy, corresponding to the test temperature of 973 K, using the following relationship [20]: o m = A g 1/"
(3)
where am is the stress corresponding to a strain of 0.005, g is the strain rate, and A is a constant. From eqn. (3), the maximum stress can be determined if the values of A and n are known. Since the value of A was not known, given that the initial microstructures of the specimens tested at different strain rates were similar, ratios of the maximum stresses at two different strain rates were computed with the following relationship: Orm~'
(~1) l]n
O.m~"~
(~2)1/n
(4)
0.026 0.0026 0.026 0.00026 0.026 0.00011 0.0026 0.000l 1
where, am~' is the peak stress at strain rate gl and am g2is the peak stress at strain rate g2" For these calculations a value of 0.3 was used for the exponent 1/n. This value was obtained from a previous study [6]. In that study the strain rate sensitivity m for fl-Ti-13wt.%Mn alloy was determined as 0.3 and m is equal to 1 / n [20]. Theoretical ratios determined by this method are compared with the experimentally obtained ratios in Table 3. As can be seen, the theoretical ratios are close to the experimental ratios. The difference in the magnitude of the flow stress drop--Figs. 1 and 3--can be explained on the basis of solute concentration. The equilibrium solute concentration near a dislocation is given by the expression [15]
exp() where C o is the average concentration and U is the interaction energy. Therefore for a given temperature an increase in C o results in higher C. This in turn results in higher flow stress drops. Comparison of the flow stress drop of Ti-9.4wt.%Mn and Ti-13wt.%Mn alloys--Figs. 3 and 1--clearly shows that the extent of the drop increases with solute content. The effect of the nature of the solute on the flow stress drop can be seen by comparing Figs. 2 and 4, the stress-strain curves for the Ti-ll.5at.% Mn and Ti-14at.%V alloys respectively. They show that the magnitude of the drop is higher for the T i - M n alloy as compared to the Ti-V alloy. This is consistent with the fact that
236 the relative size difference between titanium and manganese is higher than that between titanium and vanadium and the larger size difference is expected to result in stronger dislocation-solute atom interactions. Stronger interactions result in higher peak stresses and larger flow stress drops. It is known that the friction stress in b.c.c. materials is dependent on temperature [14]. A n increase in temperature results in a decrease in the friction stress. Since the velocity of dislocations is a function of stress, it is expected that an increase in temperature should result in a decrease in the extent of flow stress drop. Comparison of Figs. 1 and 2 shows an increase in magnitude of the flow stress drop with a decrease in test temperature. Studies on the effect of prestrain on the flow stress drops--Figs. 5 and 6--are also consistent with the suggestion that the flow stress drop is related to the multiplication of dislocations. Prestraining to a stress level below the peak stress results in a decrease in the peak stress--see Figs. 5 and 6. Moreover, the magnitude of the flow stress drop after prestraining is lower for higher prestrains. Similar behavior observed in the past [18, 19] has been attributed to prior multiplication. The behavior observed in the Ti-Mn system can also be attributed to prior multiplication. It is interesting to note that the decrease in magnitude of the peak stress increases with increasing prestrain--compare Figures 5 and 6. Evidently, the extent of increase in the mobile dislocation density depends on the initial prestrain, being higher for larger prestrains below the peak stress. Special tests involving either strain interrupts followed by annealing or annealing at temperatures higher than the test temperature also indicate that the abrupt flow stress drop is related to the low initial mobile dislocation density. As can be seen from Figs. 7 and 9, the flow stress drop is recovered after annealing. This can be attributed to a decrease in the mobile dislocation density during annealing. As expected, the magnitude of the drop as well as the peak stress are higher for the Ti-ll.5at.%Mn alloy owing to the higher diffusivities of manganese in the fl phase [22] and larger relative size difference between titanium and manganese as compared to titanium and vanadium--see Table 2. Similar reasoning can also explain the higher flow stress drop and peak stress for the Ti-11.5at.%Mn alloy as compared to the Ti-14at.%V alloy--see Figs. 8 and 1 0 after annealing.
As mentioned before, Jonas et aL [5] observed similar flow stress drops in fl-Zr-Nb alloys and the magnitude of the drop increased with solute content. Increasing the annealing time prior to their tests resulted in an increase in magnitude of the flow stress drop. On the basis of these observations they [5] have suggested that the flow stress drop is due to the declustering of solute atoms. One of their reasons for this suggestion was that clustering is expected in the Z r - N b system because it exhibits a miscibility gap in the fl phase field. As far as the mechanism for the flow stress drops in Ti-Mn and Ti-V alloys is concerned, it does not appear to be due to the declustering phenomenon. First of all, the flow stress drops were observed whether there was a miscibility gap in the fl phase field (Ti-V system) or not (Ti-Mn system). If the declustering phenomenon was to be the mechanism in the fl-Ti alloys, then prestraining below the maximum stress should not have resulted in lower flow stress drops. Furthermore, the increase in peak stress after annealing at higher temperatures in the fl phase field cannot be explained on the basis of the declustering phenomenon because clustering should have been less prevalent at higher temperatures and this should have resulted in lower peak stresses. Rather, annealing at higher temperatures resulted in an increase in the peak stress and the magnitude of the flow stress drop. Therefore the authors strongly suggest that the flow stress drop observed in fl-Ti alloys is related to the multiplication of mobile dislocations as discussed above. So far, the discussion has been confined to the flow stress drops occurring in the initial stages of deformation. The steady state behavior observed in these alloys subsequent to the flow stress drop indicates that a dynamic restoration mechanism is operative in this region. In the past, several investigators [2, 23-26] have suggested that the steady state behavior is due to the process of dynamic recovery leading to the formation of subgrains. In the present investigation, subgrain formation was also observed--see Figs. 11-13. The authors agree with the earlier investigators [2, 23-26] that the steady state behavior subsequentto the flow stress drop is related to the formation of subgrains. 5. Conclusions
(1) Abrupt flow stress drops followed by steady state behavior were observed in fl-Ti-Mn
237
and fl-Ti-V alloys. The magnitude of the flow stress drop increased with strain rate and solute concentration and decreased with temperature. (2) Prestraining to a stress level below the peak stress resulted in a decrease in the peak stress as well as a decrease in the extent of flow stress drop. The extent of decrease in peak stress increased with amount of prestrain. (3) Post-deformation annealing treatment resulted in a recovery of the peak stress as well as the flow stress drop. Higher annealing temperatures also resulted in an increase in the peak stress and the extent of flow stress drop. (4) The abrupt flow stress drops were attributed to the multiplication of mobile dislocations. The extent of the flow stress drop was found to be higher for the Ti-Mn alloy as compared to the T i - V alloy. This difference was attributed to the higher relative atomic size difference between the solvent titanium and manganese and the higher diffusivities of manganese in fl-Ti. (5) The steady state behavior subsequent to the flow stress drop was attributed to the process of dynamic recovery leading to the formation of subgrains, as suggested by several investigators in the past. Acknowledgments The authors would like to express their gratitude to Messrs. D. J. McNeish, D. E. Thomas and S. R. Seagle and their associates of the RMI company for making the alloys used in this investigation with great care at a nominal cost. The authors are grateful to Dr. R. J. Arsenault for helpful discussions. The authors are indebted to Mr. I. L. Caplan of DTRC and Dr. A. H. Rosenstein of A F O S R for their keen interest and constant encouragement throughout the course of this investigation. This work was supported by the Air Force Office of Scientific Research on Grant AFOSR-85-0367. References 1 S. M. L. Sastry, P. S. Pao and K. K. Sankaran, in H. Kimura and O. Izumi (eds.), Titanium '80, Science and
Technology, AIME, New York, 1980, p. 874. 2 C. H. Hamilton, in S. P. Agrawal (ed.), Proc. Conf. on Superplastic Forming, ASM, Metals Park, OH, 1984, p. 122. 3 D.L. Bourell and H. J. McQueen, J. Appl. Metalworking, (1987) 15. 4 N. Furushiro and S. Hori, in H. Kimura and O. Izumi (eds.), Titanium '80, Science and Technology, AIME, New York, 1980, p. 1067. 5 J. J. Jonas, B. Heritier and M. J. Luton, Metall. Trans. A, 10(1979) 611. 6 S. Ankem, J. G. Shyue, M. N. Vijayshankar and R. J. Arsenault, Mater. Sci. Eng., A l l l (1989) 51. 7 J. E. Hockett and E P. Gilfis, Mater. Res. Stand., 13 (1970) 251. 8 I. Weiss, E H. Froes and D. Eylon, Metall. Trans. A, 15 (1984) 1493. 9 G. E Van Der Voort, Metallography Principles and Practice, McGraw-Hill, New York, 1984. 10 J. L. Murray, in T. H. Massalki (ed.), Binary Alloy Phase Diagrams, Vol. 2, ASM, Metals Park, OHI 1986, pp. 1598, 2145. 11 H. Oikawa, K. Nishimura and M. X. Cui, Scr. Metall., 19 (1985)826. 12 L. I. Van Tome and G. Thomas, Acta MetaU., 14 (1966) 621. 13 G.R. Wilms, J. Less Common Met., 6 (1964) 169. 14 R. E. Smallman, Modern Physical Metallurgy, Butterworths, London, 1985. 15 E.O. Hall, Yield Point Phenomenon in Metals and Alloys, Plenum, New York, 1970. 16 N. E Fiore and C. L. Bauer, Prog. Mater. Sci., 13 (1967) 85. 17 W, G. Johnston and J. J. Gilman, J. Appl. Phys., 30 (1959) 129. 18 R. J. Arsenault, Trans. Metall. Soc. A1ME, 230 (1964) 1570. 19 J. R. Patel and A. R. Chaudhuri, J. Appl. Phys., 34 (1963) 2788. 20 G.T. Hahn, Acta Metall., 10 (1962) 727. 21 S. Ankem and H. Margolin, Metall. Trans. A, 17 (1986) 2209. 22 R. C. West (ed.), Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 64th edn., 1983-1984, p. F51. 23 E Griffith and C. Hammond, Acta Metall., 20 (1972) 935. 24 G. C. Morgan and C. Hammond, in G. Lutjering, U. Zwicker and W. Bunk (eds.), Titanium '84, Science and Technology, AIME, New York, 1984, p. 717. 25 T. Sheppard and J. Norley, Mater. Sci. Technol. (October 1988) 903. 26 J. J. Jones and M. J. Luton, in J. J. Burke and V. Weirs (eds.), Advances in Deformation Processing, Vol. 4, Plenum, New York, 1978.