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Predictions of the phase ratio at the lowest flow stress in superplastic titanium alloys
MATERIALS SCIENCE & ENGINEERING ELSEVIER
Materials Science and Engineering, A194 (1995) LI -L4
A
Letter
Predictions of the phase ratio at the lowest flow stress in superplastic titanium alloys A b h i j i t D u t t a a, D. Siva K u r n a r b aDefence Metallurgical Research Laboratory, Hyderabad-500258, India bMaturi Venkata Subbarao Engineering College, Hyderabad-500659, India Received 25 July 1994; in revised form 7 February 1994
Abstract During superplastic deformation of duplex titanium alloys a flow stress minimum is often observed at a temperature close to the optimum superplastic region which has been attributed to an optimum phase ratio present at that temperature. A mathematical analysis has been presented for predicting the phase ratio corresponding to the minimum flow stress at any temperature and strain rate.
Keywords: Titanium; Alloys; Plasticity; Stress
1. Introduction In the deformation of conventional metals or alloys under a conventional strain rate, the temperature dependence of flow stress is predictable in that the flow stress decreases monotonically with increase in temperature. However, under superplastic flow conditions, the flow stress in a two-phase titanium alloy attains a minimum at an intermediate temperature and increases on either side of it [1-4]. In a previous investigation [5], we have derived a constitutive equation relating the flow stress to the phase ratio and phase size. In the present investigation, we have used the abovementioned constitutive equation and by further manipulation of it have attempted to identify the critical phase ratio where the flow stress is minimum.
2. Experimental details Ti-6wt.%A1-4wt.%V alloy in two grain sizes (4.7 and 11.2 /~m) and VT-9 alloy (Ti-6.3wt.%Al2.7wt.%Mo-l.7wt.%Zr) in three different microstruc0921-5093/95/$9.50 © 1995 - Elsevier Science S.A. All rights reserved SSDI 0921-5093(94)02712-9
tural conditions (equiaxed, acicular, and mixed) were prepared by thermomechanical processing at the Defence Metallurgical Research Laboratory (DMRL), Hyderabad; the details have been given elsewhere [6]. The hydrogen-charging apparatus for altering the phase ratio at a particular temperature was designed and built at DMRL, Hyderabad; the detailed procedure has been given in [7].
3. Mathematical analysis The constitutive equation of flow stress as derived in Eq. (14) of [5] is as follows: (a c)T= ( VaK~'L~mb+ V~Ke'Lpb)(gc)rm
(1)
where (oc)r and (e¢)r are the flow stress and strain rate respectively at constant temperature. V~, and V~ are the volume fractions of a and fl phases respectively. L a and L~ are the mean free paths for a and /3 phases respectively. K a and K~ are the phase strength constants for a and/3 phases respectively, m is the strain rate sensitivity index and b the grain size exponent.
Letter /
L2
MaterialsScience and EngineeringA194(1995) L1-L 4 '~ TEMP* k
The assumption of no grain growth leads to the following relationship, as described in Eq. (16) of [5]: V~_ V~_ 0.5 L~
La
30 2.8
(2)
Lo
N
[5]:
i
~-L. 0
\ c C / T ,.
---o
•
-4
45 l
-1
3E
• - 4 -1 £ =1.3 x l O SEE
~
12 "8
i
750
(a)
To determine a minimum value of flow stress
5.5 53
/*,1 13.
0
(4)
1273
E:66x~3SE~ GS:~I.2 H~ 49
OS : 4 " 7 p m
4,
Eq. (3) can be rewritten as
x
1173
21¢
(3)
+G'(
1073
E E 2.0 -~
F 1.6
[Kj(1 -
= TEMP*k
1223
2-4
where L 0 is the mean free path at equal volume fractions (0.5) of both phases. This assumption leads to an equation of the following type, as derived in Eq. (17) of (Oc)T=(K~'V~, mb+l + Kj~Vj"b+I)ZLomb(gc)T m
1123
1023 3-2
i
eso
3.7 r 5
\~
33
st~
2.9 ;
90
----- T EMP*C
~
75o
i
(b)
i
8so
i
2,5 2.3
i
9s0
,- TEMP*C
Fig. 1. Temperature dependence of the flow stress for (a) finegrained superplastic and (b) coarse-grained Ti-6wt.%Al4wt.%V [3].
(5)
a(v )
Therefore ' -Ka(mb
+
J
1)[1-(Ve)c"b]+K/(mb+l)(Ve)c mb
(6)
36~-
where ( V#)c is the/3 volume fraction at minimum flow stress and
K#'- (-1--( V~)c]"b Ko'
(7)
(VAc /
(8) Z4
1 V~)c
(l+K,/Ka,)l/ml,
(9) 20
4. Results and discussion 16
Eq. (9) has to be carefully applied to determine the critical phase ratio corresponding to the lowest flow stress at any particular temperature and strain rate. Obviously, Eq. (9) does not provide a means for calculation of the critical temperature at which the minimum flow stress occurs. Moreover, the present analysis is only valid for superplastic conditions of fine equiaxed microstructures, high homologous temperatures and slow strain rates, because the flow stress minima are only observed under these conditions. For example, the coarse-grained microstructure in Fig. l(b) and the acicular microstructure in Fig. 2 do not demonstrate flow stress minima, while distinct flow stress minima are observed with superplastic microstructures as in Figs. l(a) and 2 ("equiaxed" and "mixed" microstruc-
1098
I 1123
i 1173
~ 1 3
1273
1323
TEMP K
Fig. 2. Temperature dependence of the flow stress of T i 6.3wt.%Al-2.7wt.%Mo-l.6wt.%Zr alloy with superplastic
(equiaxed and mixed) and non-superplastic (acicular) microstructures [4].
tures). The flow stress minima are observed for the fine-grained Ti-6wt.%AI-4wt.%V at 1 2 0 0 K ( V#~ 0.52) and for Ti-6.3wt.%Al-2.7wt.%Mo1.7wt.%Zr alloy at 1173-1198 K ( V# = 0.52-0.62). Table 1 on the contrary shows the variation in flow stresses with phase ratio (as governed by the weight percentage of H 2) at different constant temperatures at
Letter
/
Materials Science and Engineering A194(1995) L1-L4
L3
Table 1 Flow stress at different temperatures and hydrogen concentrations at g = 4.16 x 10-4 s- ~where the underlined values are the minimum flow stresses Temperature (n)
Flow stress (MPa)
1023 1073 1123 1173
Base alloy
Base alloy + 0.04 wt.% H
Base alloy + 0.11 wt.% H
Base alloy + 0.22 wt.% H
Base alloy + 0.32 wt.% H
121.51 52.45 32.78 6.04
54.60 27.11 13.46 9.97
33.11 13.40 14.88 10.38
19.69 17.86 14.30 8.67
23.57 17.11 ---
Table 2 Data for computing Kp'/K~' and (V~)c Temperature (K)
g (s - ~)
m
b
H (wt.%)
Vt~
(O~)x (MPa)
Ks//Kj
( V~)c
1073 1073
4.16 x 10 -4 4.16 x 10-4
0.56 0.56
5 5
0.0 0.04
0.32 0.46
52.45 27.11
0.517
0.56
tO
/~,a/ Z C) I---t..J "
0.8
06
LL
C~
o-1023, / ~ , . ~ 1
[] - 1073 K A - 1123 K
I m~ 02
Si- FREE Vt-9
• -
1173 K
•-
1198 K
• - 1223 K
I
J
1
0.1
0.2
03
HYDROGEN, W t % Fig. 3. Change in/3 volume fraction with hydrogen concentration at different temperatures for Ti-6.3wt.%AI-2.7wt.%Mo1.6wt.%Zr alloy.
exponent b. Padmanabhan and Davies [8] have reported the b values obtained by various researchers as ranging from 1 to 5, under conditions of optimal superplasticity. A value of 2 - 3 has been commonly quoted by different workers. However, in two-phase alloys the m e t h o d of determination itself might have given rise to different values of b. In our experiment the b value was determined from the elevated-temperature grain size when L~ = L a and V~ = V~, and not for the ambient temperature grain size. In [5] the b value was determined in various ways and a value close to 5 was consistently obtained. By using this value of b, the critical V~ was obtained as 0.56, which is quite close to the experimental value of 0.48. If the commonly quoted b value of 2 - 3 is substituted in Eq. (9), (V~)c values range from 0.6 to 0.64. 5. Conclusions
a particular strain rate. T h e change in phase ratio with H content is shown in Fig. 3 at different temperatures. W h e n Table 1 is read in conjunction with Fig. 3, it reveals that the minimum flow stress for Ti-6.3wt.%Al2 . 7 w t . % M o - l . 7 w t . % Z r alloy at 1173 K is observed with V~ = 0.52, at 1123 K with V~ = 0.48 (0.04 wt.% H), at 1073 K with V~=0.48 (0.11 wt.% H), and at 1023 K with V~ = 0.48 (0.22 wt.% H). In Table 2 the data necessary for calculating the critical V~ value for T i - 6 . 3 w t . % A I - 2 . 7 w t . % M o 1.7wt.%Zr at a temperature of 1073 K have been listed. Ka and K~ have been determined using Eq. (4). T h e m value in this region of temperatures and strains is 0.56 approximately. T h e other parameter which is required to be substituted in Eq. (9) is the grain size
(1) U n d e r the superplastic conditions of fine equiaxed grains, optimum strain rate (10- 5_ 10- 3 s- ~) and high homologous temperature (greater than 0.5 Tm) a flow stress minimum is observed at an optimum phase ratio. (2) T h e critical phase ratio as mentioned above can be calculated from the derived equation, at any particular temperature and strain rate.
References
[1] D. Lee and W.A. Backhofen, Trans. Metall. Soc. AIME, 239 (1967) 1034.
L4
Letter
/
Materials Science and Engineering A194(1995) L1-L4
[2] S.M.L. Sastry, R.J. Lederich, T.L. Machay and W.R. Kerr, J. Met., (1983) 48. [3] A. Dutta, N.C. Birla and A.K. Gupta, Trans. Indian. Inst. Met., 36 (3)(1983) 169. [41 A. Dutta, N.C. Birla and A.M. Rao, Trans. Indian Inst. Met., 40(3)(1987)195. [5] A. Dutta and A.K. Mukherjee, Mater. Sci. Eng., A138 (1991) 221.
[6] A. Dutta and N.C. Birla, Mater. Sci. TechnoL, 4 (1988) 341. [7] A. Dutta and N.C. Birla, in P. Lacome, R. Tricot and G. Beranger (eds.), Proc. 6th World Conf. on Titanium, Les Editions de Physique, Paris, 1988, p. 1185. [8] K.A. Padmanabhan and G.J. Davies, Superplasticity, Springer, Berlin, 1980, p. 65.
Materials Science and Engineering, A194 (1995) LI -L4
A
Letter
Predictions of the phase ratio at the lowest flow stress in superplastic titanium alloys A b h i j i t D u t t a a, D. Siva K u r n a r b aDefence Metallurgical Research Laboratory, Hyderabad-500258, India bMaturi Venkata Subbarao Engineering College, Hyderabad-500659, India Received 25 July 1994; in revised form 7 February 1994
Abstract During superplastic deformation of duplex titanium alloys a flow stress minimum is often observed at a temperature close to the optimum superplastic region which has been attributed to an optimum phase ratio present at that temperature. A mathematical analysis has been presented for predicting the phase ratio corresponding to the minimum flow stress at any temperature and strain rate.
Keywords: Titanium; Alloys; Plasticity; Stress
1. Introduction In the deformation of conventional metals or alloys under a conventional strain rate, the temperature dependence of flow stress is predictable in that the flow stress decreases monotonically with increase in temperature. However, under superplastic flow conditions, the flow stress in a two-phase titanium alloy attains a minimum at an intermediate temperature and increases on either side of it [1-4]. In a previous investigation [5], we have derived a constitutive equation relating the flow stress to the phase ratio and phase size. In the present investigation, we have used the abovementioned constitutive equation and by further manipulation of it have attempted to identify the critical phase ratio where the flow stress is minimum.
2. Experimental details Ti-6wt.%A1-4wt.%V alloy in two grain sizes (4.7 and 11.2 /~m) and VT-9 alloy (Ti-6.3wt.%Al2.7wt.%Mo-l.7wt.%Zr) in three different microstruc0921-5093/95/$9.50 © 1995 - Elsevier Science S.A. All rights reserved SSDI 0921-5093(94)02712-9
tural conditions (equiaxed, acicular, and mixed) were prepared by thermomechanical processing at the Defence Metallurgical Research Laboratory (DMRL), Hyderabad; the details have been given elsewhere [6]. The hydrogen-charging apparatus for altering the phase ratio at a particular temperature was designed and built at DMRL, Hyderabad; the detailed procedure has been given in [7].
3. Mathematical analysis The constitutive equation of flow stress as derived in Eq. (14) of [5] is as follows: (a c)T= ( VaK~'L~mb+ V~Ke'Lpb)(gc)rm
(1)
where (oc)r and (e¢)r are the flow stress and strain rate respectively at constant temperature. V~, and V~ are the volume fractions of a and fl phases respectively. L a and L~ are the mean free paths for a and /3 phases respectively. K a and K~ are the phase strength constants for a and/3 phases respectively, m is the strain rate sensitivity index and b the grain size exponent.
Letter /
L2
MaterialsScience and EngineeringA194(1995) L1-L 4 '~ TEMP* k
The assumption of no grain growth leads to the following relationship, as described in Eq. (16) of [5]: V~_ V~_ 0.5 L~
La
30 2.8
(2)
Lo
N
[5]:
i
~-L. 0
\ c C / T ,.
---o
•
-4
45 l
-1
3E
• - 4 -1 £ =1.3 x l O SEE
~
12 "8
i
750
(a)
To determine a minimum value of flow stress
5.5 53
/*,1 13.
0
(4)
1273
E:66x~3SE~ GS:~I.2 H~ 49
OS : 4 " 7 p m
4,
Eq. (3) can be rewritten as
x
1173
21¢
(3)
+G'(
1073
E E 2.0 -~
F 1.6
[Kj(1 -
= TEMP*k
1223
2-4
where L 0 is the mean free path at equal volume fractions (0.5) of both phases. This assumption leads to an equation of the following type, as derived in Eq. (17) of (Oc)T=(K~'V~, mb+l + Kj~Vj"b+I)ZLomb(gc)T m
1123
1023 3-2
i
eso
3.7 r 5
\~
33
st~
2.9 ;
90
----- T EMP*C
~
75o
i
(b)
i
8so
i
2,5 2.3
i
9s0
,- TEMP*C
Fig. 1. Temperature dependence of the flow stress for (a) finegrained superplastic and (b) coarse-grained Ti-6wt.%Al4wt.%V [3].
(5)
a(v )
Therefore ' -Ka(mb
+
J
1)[1-(Ve)c"b]+K/(mb+l)(Ve)c mb
(6)
36~-
where ( V#)c is the/3 volume fraction at minimum flow stress and
K#'- (-1--( V~)c]"b Ko'
(7)
(VAc /
(8) Z4
1 V~)c
(l+K,/Ka,)l/ml,
(9) 20
4. Results and discussion 16
Eq. (9) has to be carefully applied to determine the critical phase ratio corresponding to the lowest flow stress at any particular temperature and strain rate. Obviously, Eq. (9) does not provide a means for calculation of the critical temperature at which the minimum flow stress occurs. Moreover, the present analysis is only valid for superplastic conditions of fine equiaxed microstructures, high homologous temperatures and slow strain rates, because the flow stress minima are only observed under these conditions. For example, the coarse-grained microstructure in Fig. l(b) and the acicular microstructure in Fig. 2 do not demonstrate flow stress minima, while distinct flow stress minima are observed with superplastic microstructures as in Figs. l(a) and 2 ("equiaxed" and "mixed" microstruc-
1098
I 1123
i 1173
~ 1 3
1273
1323
TEMP K
Fig. 2. Temperature dependence of the flow stress of T i 6.3wt.%Al-2.7wt.%Mo-l.6wt.%Zr alloy with superplastic
(equiaxed and mixed) and non-superplastic (acicular) microstructures [4].
tures). The flow stress minima are observed for the fine-grained Ti-6wt.%AI-4wt.%V at 1 2 0 0 K ( V#~ 0.52) and for Ti-6.3wt.%Al-2.7wt.%Mo1.7wt.%Zr alloy at 1173-1198 K ( V# = 0.52-0.62). Table 1 on the contrary shows the variation in flow stresses with phase ratio (as governed by the weight percentage of H 2) at different constant temperatures at
Letter
/
Materials Science and Engineering A194(1995) L1-L4
L3
Table 1 Flow stress at different temperatures and hydrogen concentrations at g = 4.16 x 10-4 s- ~where the underlined values are the minimum flow stresses Temperature (n)
Flow stress (MPa)
1023 1073 1123 1173
Base alloy
Base alloy + 0.04 wt.% H
Base alloy + 0.11 wt.% H
Base alloy + 0.22 wt.% H
Base alloy + 0.32 wt.% H
121.51 52.45 32.78 6.04
54.60 27.11 13.46 9.97
33.11 13.40 14.88 10.38
19.69 17.86 14.30 8.67
23.57 17.11 ---
Table 2 Data for computing Kp'/K~' and (V~)c Temperature (K)
g (s - ~)
m
b
H (wt.%)
Vt~
(O~)x (MPa)
Ks//Kj
( V~)c
1073 1073
4.16 x 10 -4 4.16 x 10-4
0.56 0.56
5 5
0.0 0.04
0.32 0.46
52.45 27.11
0.517
0.56
tO
/~,a/ Z C) I---t..J "
0.8
06
LL
C~
o-1023, / ~ , . ~ 1
[] - 1073 K A - 1123 K
I m~ 02
Si- FREE Vt-9
• -
1173 K
•-
1198 K
• - 1223 K
I
J
1
0.1
0.2
03
HYDROGEN, W t % Fig. 3. Change in/3 volume fraction with hydrogen concentration at different temperatures for Ti-6.3wt.%AI-2.7wt.%Mo1.6wt.%Zr alloy.
exponent b. Padmanabhan and Davies [8] have reported the b values obtained by various researchers as ranging from 1 to 5, under conditions of optimal superplasticity. A value of 2 - 3 has been commonly quoted by different workers. However, in two-phase alloys the m e t h o d of determination itself might have given rise to different values of b. In our experiment the b value was determined from the elevated-temperature grain size when L~ = L a and V~ = V~, and not for the ambient temperature grain size. In [5] the b value was determined in various ways and a value close to 5 was consistently obtained. By using this value of b, the critical V~ was obtained as 0.56, which is quite close to the experimental value of 0.48. If the commonly quoted b value of 2 - 3 is substituted in Eq. (9), (V~)c values range from 0.6 to 0.64. 5. Conclusions
a particular strain rate. T h e change in phase ratio with H content is shown in Fig. 3 at different temperatures. W h e n Table 1 is read in conjunction with Fig. 3, it reveals that the minimum flow stress for Ti-6.3wt.%Al2 . 7 w t . % M o - l . 7 w t . % Z r alloy at 1173 K is observed with V~ = 0.52, at 1123 K with V~ = 0.48 (0.04 wt.% H), at 1073 K with V~=0.48 (0.11 wt.% H), and at 1023 K with V~ = 0.48 (0.22 wt.% H). In Table 2 the data necessary for calculating the critical V~ value for T i - 6 . 3 w t . % A I - 2 . 7 w t . % M o 1.7wt.%Zr at a temperature of 1073 K have been listed. Ka and K~ have been determined using Eq. (4). T h e m value in this region of temperatures and strains is 0.56 approximately. T h e other parameter which is required to be substituted in Eq. (9) is the grain size
(1) U n d e r the superplastic conditions of fine equiaxed grains, optimum strain rate (10- 5_ 10- 3 s- ~) and high homologous temperature (greater than 0.5 Tm) a flow stress minimum is observed at an optimum phase ratio. (2) T h e critical phase ratio as mentioned above can be calculated from the derived equation, at any particular temperature and strain rate.
References
[1] D. Lee and W.A. Backhofen, Trans. Metall. Soc. AIME, 239 (1967) 1034.
L4
Letter
/
Materials Science and Engineering A194(1995) L1-L4
[2] S.M.L. Sastry, R.J. Lederich, T.L. Machay and W.R. Kerr, J. Met., (1983) 48. [3] A. Dutta, N.C. Birla and A.K. Gupta, Trans. Indian. Inst. Met., 36 (3)(1983) 169. [41 A. Dutta, N.C. Birla and A.M. Rao, Trans. Indian Inst. Met., 40(3)(1987)195. [5] A. Dutta and A.K. Mukherjee, Mater. Sci. Eng., A138 (1991) 221.
[6] A. Dutta and N.C. Birla, Mater. Sci. TechnoL, 4 (1988) 341. [7] A. Dutta and N.C. Birla, in P. Lacome, R. Tricot and G. Beranger (eds.), Proc. 6th World Conf. on Titanium, Les Editions de Physique, Paris, 1988, p. 1185. [8] K.A. Padmanabhan and G.J. Davies, Superplasticity, Springer, Berlin, 1980, p. 65.