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New constitutive equation for two-phase superplastic titanium alloys
Materials Science and Engineering, A 138 ( 1991 ) 221-226
221
New constitutive equation for two-phase superplastic titanium alloys Abhijit Dutta Defence Metallurgical Research Laboratory, Hyderabad -- 500 258 (India)
Amiya K. Mukherjee Department of Mechanical Engineering, University of California, Davis, CA 95616 (U.S.A.)
(Received 23 July 199(I)
Abstract In two-phase a + fl titanium alloys, the phase ratio and consequently the phase sizes vary with temperature. The flow stress of this type of alloy is not only affected by the thermal contribution of temperature but also by changes in the phase ratio and sizes. This prompted us to develop a constitutive equation in a superplastic titanium alloy (Ti-6.3A1-2.VMo-I.VZr), incorporating volume fraction and phase sizes. The phase ratio was altered at constant temperature by means of charging with hydrogen. The constants of the constitutive equation were evaluated. Flow stresses were then predicted at the optimum superplastic temperature of 1073 K and strain rates of 1.9x 10 -3 s -1 and 7.1 x 10 -4 s -~ for various fl volume fractions. The predicted flow stresses were found to correlate fairly well with the experimentally determined values.
1. Introduction The various constitutive equations describing superplastic flow usually relate strain rate with flow stress, temperature and grain size and the most widely used equation [1] is as follows: O|/m goc L ~ e x p
Q) -
(1)
where L is the grain size or mean free path, b is the grain size exponent, Q is the activation energy and the other terms have the usual meaning. It is, however, difficult in a two-phase alloy to specify the grain size as it is different in different phases. At the same time the phase ratio should have an effect in the constitutive equation, especially in the a +/3 titanium alloy where the diffusivity of the /3 phase is two orders of magnitude higher than that of the a phase [2]. The change in phase ratio with increasing temperature might lead to error in determining the activation energy from the usual Arrhenius-type plot. Suery and Baudelet have developed a constitution equation incorporating a volume fraction term for an a-/3 brass [3]. In their analysis, the contribution of the 0921-5093/91/$3.50
grain size of the softer phase was not considered; moreover, the stress exponent was assumed to be 2 instead of considering the actual strain rate sensitivity or stress exponent. In the present investigation we have tried to develop a constitutive equation incorporating a volume fraction term and phase sizes in the equation and also retaining m and grain size exponent terms. Since charging with hydrogen in an a + fl titanium alloy provides a means of altering the volume fraction of a / f l phases without changing the temperature, an attempt has been made to validate the derived equation by charging with hydrogen in a superplastic Ti-6.3A1-2.VMo1.VZr alloy.
2. Experimental details A Ti-6.3A1-2.VMo-l.VZr alloy ingot was prepared in the Defence Metallurgical Research Laboratory, Hyderabad, India. Suitable thermomechanical processing, described elsewhere [4], converted the microstructure to extremely fine grain size (about 4.1 /zm). The fl transus was determined as 1263 + 2 K. © Elsevier Sequoia/Printedin The Netherlands
222 The hydrogen charging unit was designed and fabricated in the Defence Metallurgical Research Laboratory. The unit consisted of a stainless steel tube over which a sliding tubular furnace with control was mounted. Inlets were provided for passing hydrogen and argon gas. The volume proportions were controlled through two needle valves and flow rate meters. After hydrogen charging, the concentrations were determined by the weight gain method (to the nearest 0.1 mg). Tensile testing involving incremental changes in the cross-head velocity was carried out in a 10 ton Instron machine under argon atmosphere.
where e is the flow stress; v is the volume fraction; and a,/3 and c are subscripts denoting pure a phase, pure fl phase and combined matrix respectively. It will also be assumed that gc=ga= ~fl
(7)
This isostrain rate assumption was found by Hamilton et al. [6] to be well suited for describing the flow stress behaviour of two-phase alloys of titanium. Following eqn. (2), the flow stress for the two phases can be written separately as follows: o
3. Analysis and results
(o~),,r= K j L f
The rule of mixtures and an isostrain rate condition has been assumed in deriving the constitutive equation of flow in two-phase alloys, as described in the following section.
3.1. Volume fraction and grain s&e dependence of fiow stress Flow stress has often been related to grain size by the following expression [5]: (o)e, rOCL ~
(2)
where (o)~,r is the flow stress at constant strain rate and temperature, L is the grain size (mean free path) and a is a constant. Similarly, strain rate is also related to grain size through the following relationship [5 ]: 1
(g)o,r~--~
(3)
where (g)o,r is the strain rate at constant flow stress and temperature and b is a constant. On the other hand, flow stress is connected to strain rate according to
(O)r,L = Kg m
(4)
where (o)r, L is the flow stress at constant temperature and grain size. By manipulating the terms of eqns. (2)-(4) the constants a and b may be related through
a =mb
(5)
Assuming that the two-phase alloy behaves as a composite of two constituent phases (a and/3) and therefore obeys the rule of mixture, the combined flow stress Oc can be written as
Oc=VaOa +V~O ~
(6)
(8)
(9)
where L~ and L~ are the a phase and t3 phase grain size respectively and K~', K j and a are constants. Incorporating the strain rate dependence of flow stress from eqn. (4), eqns. (8) and (9) can be written as (Oa)T=gattaaEa m
(10)
Since ~ = ec (o~)r = K,~'La"(gc)r ~
(11)
Similarly
(off) r = K~'L~"( gc)rm
(12)
Combining eqns. (6), ( 11 ) and (12)
(oc)T=(Volq'L°+
(13)
where (a¢)r is the flow stress of the alloy at constant temperature and (gc)r is the strain rate of the alloy at constant temperature. Putting a = mb from eqn. (5), eqn. (13) can be written as
(oc)r=(VaK 'L "b+ VeKjLemb)(gc)r "
(14)
In order to simplify eqn. (14), if the grain size is assumed to be the same as the phase size, i.e., a - a or fl-fl boundaries do not exist (in the present investigation the a - a or fl-fl boundaries were not visible under an optical microscope), the volume fraction V can be related to the grain size or mean free path L in the following way:
VdVe= LJL e
(15)
By adjusting the terms we get
V~/L~ = V,ff L a = 0.5/L 0
(16)
where L 0 is the mean free path of both a and fl phases when the volume fractions are equal, i.e.
L 0 = L~ = L~ when V~= V~= 0.5. Combining eqns. (14) and (16) the following expression for flow stress can be written: (o~)r=(K ' V ( f '+] + Kfl'Vyb+])2L,,mb(gc)vm
s1F
223
55 I,.3
(17) 3S
The above equation is applicable only when the phase sizes and grain sizes are assumed to be the same. In the event of the existence of large numbers of a - a or t3-t 3 boundaries in the microstructure, the grain sizes of each phase can be measured separately at a given temperature and eqn. ( 1 ) can be used for greater accuracy. The validity of this theoretical analysis can be tested by determining the flow stress (at constant temperature) with different phase proportions. This was feasible in the present investigation by the hydrogenation of test samples at different levels. Knowledge of the constants b,/Ca' and K~' is also required for this study. Methods for determining these constants are discussed in the following sections. 3.2. Determination of parameter b The parameter b is usually determined by measuring the slope of an g vs. L plot on a logarithmic scale. In a two-phase alloy such as the present sample, it seems reasonable to assume that the grain size dependence of the constituent phases is the same, i.e. to say that the exponent b may be assumed to be the same for both a and fl phases. Even with this assumption it is difficult to determine the exponent b when the phase sizes are not the same. Therefore two combinations of alloy compositions and temperatures were chosen where nearly equal phase proportions existed so that the grain sizes were nearly equal. The proportions are nearly equal for the base alloy at 1173 K and for the 0.04 wt.% hydrogenated alloy at 1073 K. The initial average grain size L for the base alloy with 0.04 wt.% H was 3.77 /~m at a temperature of 1073 K. The grain size was deliberately coarsened by soaking at a temperature of 1123 K for 30 min and 60 min respectively. The resulting grain sizes were 4.57 ktm and 5.4 ktm respectively. The double logarithmic stress vs. strain rate plots for these three grain sizes at 1073 K are shown in Fig. 1. The stress level ( o = 2.5 MPa) was carefully chosen so that the m values for all the three grain sizes remained nearly the same (m = 0.6). The value of the exponent b was determined from the slope of a In g vs. In L plot and was 5.0 as shown in Fig. 2. The exponent b
I
TEMPERATURE~073K
~'%P/
03
-12
-11
-10
- 9
-8 In
-7
~,
-6
-S
-z,
-3
s e c -1
Fig. 1. In o vs. In g for the base alloy plus 0.04 wt.% H at 1073 K for three grain sizes: zx, 5.4/~m; • 4.57/~m; o 3.77 ,um. 10.3
~" 9.7 m 10.1e.5 ._= 9.3
' 1~" _=
0.9
/
t. ~ = -72s m
: OM~-O.t~7
8.1
-10
b -" 5
.S
TEMPERALLOY AT01~S. AS*001~1073 E KWt%H, -m" " ;i;"" -11
I
1.3
1.4,
, I
1.5
I
l.a
I
1.'7
In L , pm Fig. 2. Determination of the grain size exponent b for the 0.04 wt.% hydrogenated base alloy at 1073 K: e, 1ng=-7.25, m=0.44-0.47, b~-5.3; o, o=2.5 MPa, m=0.6, b=5.
was also determined by plotting ( 1/m)ln o against lnL at constant g (= 7.25 s-~). The b value was calculated to be 5.3 as illustrated in Fig. 2. Similarly, the flow stress vs. strain rate plots for the base alloy were obtained at 1173 K for three grain sizes (3.58, 5 and 6.1/~m) as shown in Fig. 3. The latter two grain sizes were obtained by soaking the test samples at 1223 K for 1 h and 2 h respectively, before testing. The b values were
224 /,.3
°5
8.1
3.5 -6
2.7
-7 r, "s- 1.9
Io e'-"
c 1.1
/
BASE ALLOY TEMPERATURE 1173K
0.3
-W
- 0.5,~ -12
-11
-10
-9
-8
-7
-6
-5
-/.,
-3
In ~., sec -1 Fig. 3. In o v s . In ~ plots for the base alloy at 1173 K for three grain sizes: zx, 8.1/~m; o, 5 pm; o, 3.58 pm.
determined at three different stress levels as illustrated in Fig. 4 and also at one constant strain rate as shown in Fig. 5. In each case the value of b was 5.0. Hence it can be concluded that the grain size exponent b has a value close to 5.0 at different temperatures and strain rates in the superplastic deformation region. However, the exponent a changes according to the m value through the relationship a = mb.
3.3. Determination of constants K a' and K 8' To determine K j and K 8' it is necessary to know the flow stresses o~ and o 8 (the flow stresses for the 'all-a' and 'all-fl' phase respectively). By increasing the concentration of hydrogen in the base alloy, a complete/3 microstructure can be attained but in that case the grain size may increase quite substantially. Hence the flow stress at the required superplastic grain size is not obtainable. It was therefore decided to extrapolate the logarithmic flow stress vs./3 volume fraction plot to obtain o~ and o 8 as illustrated in Fig. 6 for the base alloy at 1073 K. It is possible to show by series expansion, logarithmic expansion and approximation of eqn. (17) that In a is a polynomial function of V8 (see the Appendix). Three strain rates were chosen for plotting In a against V~: two were in the superplastic region
1.2
1.3
1.4
1.5
1.6
1.7
1.11
2.0
InL, tjm
Fig. 4. Determination of the grain size exponent b for the base alloy at 1173 K (constant stress): A , a = 1 . 1 5 MPa, m = 0 . 5 ; o, ~ = 2 . 1 MPa, m = 0 . 6 6 - 0 . 7 2 ; m, o = 1 . 7 MPa, m = 0.72. T h e slopes of the lines give b ~ 5.
3.8
3.4
3.0
e~
2.6
.=_ 2.2
b c
1.a
I"-'IE
/ 1.4
BASE ALLOY
/
TEHIP 11731(
/
,o
V ' l : ; , '?5
0.6
1.2
I 1.3
| 1./,.
I 1.5
I 1.6
I 1.7
I 1.8
1.9
InL, pm Fig. 5. Determination of the grain size exponent b for the base alloy at 1173 K (constant strain rate ~ = - 9 . 7 5 s - l ; m = 0.7): b ~ 5.
225 (os0:50)l:,~= V~K~'L~¢'°+ ~ K / L # "°
65
= 0.5Ka'L0"" + 0.5K~'Lo "°
i.e. (os,,:50)r,e = [(Kj + K s' )/2]L,, ~''
m
n
where (050:50)r,e is the flow stress at a 50:50 volume ratio of the phases at constant temperature and strain rate; a 0 = c o n s t a n t = m, ob where m0 is the strain rate sensitivity index at a 50:50 volume ratio of the phases. The constants K~', K s' and a can be determined from eqns. (18)-(20). Equations (18) and (19) can also be written in the following form by incorporating a strain rate term:
3.7
2.9
1.1 BASE ALLOY * 0.0/* v t % TEMPERATURE
1.3
0.5
0.3
O -
In $. = -6.25 seC 1 In I~ = - 7 . 2 5 sec "1
-
fl z
1073 K
•
A -
In I~ = 4 . 1 5
X -
CALCULATEO In I F
sec "1
1
,
I
t
2
.t.
.6
.8
1.0
Fig. 6. In o vs. fl v o l u m e fraction plots at 1073 K for the base alloy plus 0.04 wt.% H: o, g = - 6 . 2 5 s-~; o, g = - 7 . 2 5 s - J ; A, g = - 8.15 s ~; × , calculated values of In a.
( l n g = - 7.25 and - 8.15 respectively) and the other was marginally beyond these values. It is apparent from eqn. (6) that any deviation from a 50:50 volume ratio of the phases on either side leads to an apparent increase in grain size (not due to conventional grain growth) of the major phase, at the cost of a decrease in grain size of the minor phase. Obviously, the grain sizes of the two phases are equal (L~ = L~ = L0) when V~ = V~. (It can be reasonably assumed that the phase size is the same as the grain size since there is no contiguity of phases at the 50:50 volume ratio.) When the microstructure turns either completely a or completely fl, the grain size should be double L 0 (the grain size at the 50:50 phase ratio). The flow stresses of the two phases can then be written as follows: (18)
and (o~)r,~ = K~'L~~= K~'(ZL0)"
( a~)r,¢ = K~( 2L,,)" g .....
(21)
( o~)r,c = K~(2Lo)"g ....
(22)
From the above equations K~ and K s can be obtained at any strain rate.
,B VOL. FRACTION
(o~)r.,: = K~'L,fl = K,'(2L0)"
(20)
(19)
where (aa)r, ~ and (a~)r,~ are the flow stresses for 100% a and 100% fl phases respectively at constant temperatures and strain rate and K j , K~' and a are constants. The flow stress at a 50 : 50 volume ratio of the phases may be written as follows:
3.4. Estimation of flow stress from constitutive equation From the knowledge of K~ and K s at any strain rate and temperature it is now possible to calculate the flow stress at a particular strain rate and temperature by using me constitutive equation (13) or (17), whichever is applicable. The calculated flow stresses are shown in Fig. 6 by crosses which deviated marginally from the experimentally obtained values. Flow stress values were not calculated at higher strain rates (shown with a dotted line) which is in the non-superplastic region and it is very likely that both the b value and the a value might change in this region. 4. Conclusions (1) A constitutive equation for a two-phase titanium alloy has been developed by assuming an isostrain rate condition for the two phases and applying the rule of mixtures. (2) At any particular temperature the phase ratio can be altered by varying the hydrogen content. Thus the effect of phase ratio on flow stress can be determined at the given temperature. (3) The predicted flow stress at 1073 K at two strain rates correlates well with the experimentally measured values.
a+fl
Acknowledgements The authors would like to thank Dr. P. Rama Rao, Director, Defence Metallurgical Research
226
Laboratory, Dr. N. C. Birla and Dr. M. C. Pandey for their encouragement and advice. One of the authors (1. K. Mukherjee) would like to thank the United Nations Development Program for a travel grant.
By expanding (1-Nil) rob+l, eqn. (A1) can be written as (Oc)r,~= K j { 1 - ( m b +
1)Vt~+[(mb+1)mb/2!lV~ 2
-1-.., -- g~ mb+l "l-(gflt/gat)Vfl rob+l}
(°c)r'~=Ka'{1-(mb+ l)V~+
References 1 K. A. Padmanabhan and G. J. Davies, Superplasticity, Springer, Berlin, 1980, p. 83. 2 A.K. Ghosh and R. Raj, Acta Metall., 29 ( 1981 ) 607. 3 M. Suery and B. Baudelet, Res. Mech., 2 (1981) 163. 4 A. Dutta and N . C. Birla, Mater. Sci. Technol., 4 (1988) 341. 5 K. A. Padmanabhan and G. J. Davies, Superplasticity, Springer, Berlin, 1980, pp. 65-66. 6 C.H. Hamilton, A. K. Ghosh and M. M. Mahoney, in D. E Hasson and C. H. Hamilton (eds.), Advanced Processing Methods for Titanium, Metallurgical Society of AIME, Warrendale, PA, 1981, pp. 120-144.
(13)
(mb+ 1)mb V~ 2!
+ . . . + V F +~[(Ke'/KJ)-
1]}
(14)
By taking the logarithm on both sides, eqn. (A4) can be written as ln(oc)~e--lnK~ •
+(I In 1 -
(rob+ 1)V~
t
mb(mb 2! + 1) V~2-'"- V,,b+L x[(K/j/K,')-I]I )
(A5)
Appendix: Derivation of o as a function of V~
Defining the terms in braces as x, eqn. (A5) reduces to
At a constant temperature T and strain rate g eqn. (17) reduces to
In( a c)T,~ = In K J + In( 1 - x)
(Oc)T,~ -----Ka'Va mb+l + K#'Vfl mb+l
(A1)
X2 X3
(A6)
)
(A7)
(since L0, m and b are constants) (Oc)T,t=Ka'[(1-- Nil)mb+l +(Kj/Ka')V~rob+'] (A2)
ln( Oc)r,~= constant + polynomial function of Vs.
221
New constitutive equation for two-phase superplastic titanium alloys Abhijit Dutta Defence Metallurgical Research Laboratory, Hyderabad -- 500 258 (India)
Amiya K. Mukherjee Department of Mechanical Engineering, University of California, Davis, CA 95616 (U.S.A.)
(Received 23 July 199(I)
Abstract In two-phase a + fl titanium alloys, the phase ratio and consequently the phase sizes vary with temperature. The flow stress of this type of alloy is not only affected by the thermal contribution of temperature but also by changes in the phase ratio and sizes. This prompted us to develop a constitutive equation in a superplastic titanium alloy (Ti-6.3A1-2.VMo-I.VZr), incorporating volume fraction and phase sizes. The phase ratio was altered at constant temperature by means of charging with hydrogen. The constants of the constitutive equation were evaluated. Flow stresses were then predicted at the optimum superplastic temperature of 1073 K and strain rates of 1.9x 10 -3 s -1 and 7.1 x 10 -4 s -~ for various fl volume fractions. The predicted flow stresses were found to correlate fairly well with the experimentally determined values.
1. Introduction The various constitutive equations describing superplastic flow usually relate strain rate with flow stress, temperature and grain size and the most widely used equation [1] is as follows: O|/m goc L ~ e x p
Q) -
(1)
where L is the grain size or mean free path, b is the grain size exponent, Q is the activation energy and the other terms have the usual meaning. It is, however, difficult in a two-phase alloy to specify the grain size as it is different in different phases. At the same time the phase ratio should have an effect in the constitutive equation, especially in the a +/3 titanium alloy where the diffusivity of the /3 phase is two orders of magnitude higher than that of the a phase [2]. The change in phase ratio with increasing temperature might lead to error in determining the activation energy from the usual Arrhenius-type plot. Suery and Baudelet have developed a constitution equation incorporating a volume fraction term for an a-/3 brass [3]. In their analysis, the contribution of the 0921-5093/91/$3.50
grain size of the softer phase was not considered; moreover, the stress exponent was assumed to be 2 instead of considering the actual strain rate sensitivity or stress exponent. In the present investigation we have tried to develop a constitutive equation incorporating a volume fraction term and phase sizes in the equation and also retaining m and grain size exponent terms. Since charging with hydrogen in an a + fl titanium alloy provides a means of altering the volume fraction of a / f l phases without changing the temperature, an attempt has been made to validate the derived equation by charging with hydrogen in a superplastic Ti-6.3A1-2.VMo1.VZr alloy.
2. Experimental details A Ti-6.3A1-2.VMo-l.VZr alloy ingot was prepared in the Defence Metallurgical Research Laboratory, Hyderabad, India. Suitable thermomechanical processing, described elsewhere [4], converted the microstructure to extremely fine grain size (about 4.1 /zm). The fl transus was determined as 1263 + 2 K. © Elsevier Sequoia/Printedin The Netherlands
222 The hydrogen charging unit was designed and fabricated in the Defence Metallurgical Research Laboratory. The unit consisted of a stainless steel tube over which a sliding tubular furnace with control was mounted. Inlets were provided for passing hydrogen and argon gas. The volume proportions were controlled through two needle valves and flow rate meters. After hydrogen charging, the concentrations were determined by the weight gain method (to the nearest 0.1 mg). Tensile testing involving incremental changes in the cross-head velocity was carried out in a 10 ton Instron machine under argon atmosphere.
where e is the flow stress; v is the volume fraction; and a,/3 and c are subscripts denoting pure a phase, pure fl phase and combined matrix respectively. It will also be assumed that gc=ga= ~fl
(7)
This isostrain rate assumption was found by Hamilton et al. [6] to be well suited for describing the flow stress behaviour of two-phase alloys of titanium. Following eqn. (2), the flow stress for the two phases can be written separately as follows: o
3. Analysis and results
(o~),,r= K j L f
The rule of mixtures and an isostrain rate condition has been assumed in deriving the constitutive equation of flow in two-phase alloys, as described in the following section.
3.1. Volume fraction and grain s&e dependence of fiow stress Flow stress has often been related to grain size by the following expression [5]: (o)e, rOCL ~
(2)
where (o)~,r is the flow stress at constant strain rate and temperature, L is the grain size (mean free path) and a is a constant. Similarly, strain rate is also related to grain size through the following relationship [5 ]: 1
(g)o,r~--~
(3)
where (g)o,r is the strain rate at constant flow stress and temperature and b is a constant. On the other hand, flow stress is connected to strain rate according to
(O)r,L = Kg m
(4)
where (o)r, L is the flow stress at constant temperature and grain size. By manipulating the terms of eqns. (2)-(4) the constants a and b may be related through
a =mb
(5)
Assuming that the two-phase alloy behaves as a composite of two constituent phases (a and/3) and therefore obeys the rule of mixture, the combined flow stress Oc can be written as
Oc=VaOa +V~O ~
(6)
(8)
(9)
where L~ and L~ are the a phase and t3 phase grain size respectively and K~', K j and a are constants. Incorporating the strain rate dependence of flow stress from eqn. (4), eqns. (8) and (9) can be written as (Oa)T=gattaaEa m
(10)
Since ~ = ec (o~)r = K,~'La"(gc)r ~
(11)
Similarly
(off) r = K~'L~"( gc)rm
(12)
Combining eqns. (6), ( 11 ) and (12)
(oc)T=(Volq'L°+
(13)
where (a¢)r is the flow stress of the alloy at constant temperature and (gc)r is the strain rate of the alloy at constant temperature. Putting a = mb from eqn. (5), eqn. (13) can be written as
(oc)r=(VaK 'L "b+ VeKjLemb)(gc)r "
(14)
In order to simplify eqn. (14), if the grain size is assumed to be the same as the phase size, i.e., a - a or fl-fl boundaries do not exist (in the present investigation the a - a or fl-fl boundaries were not visible under an optical microscope), the volume fraction V can be related to the grain size or mean free path L in the following way:
VdVe= LJL e
(15)
By adjusting the terms we get
V~/L~ = V,ff L a = 0.5/L 0
(16)
where L 0 is the mean free path of both a and fl phases when the volume fractions are equal, i.e.
L 0 = L~ = L~ when V~= V~= 0.5. Combining eqns. (14) and (16) the following expression for flow stress can be written: (o~)r=(K ' V ( f '+] + Kfl'Vyb+])2L,,mb(gc)vm
s1F
223
55 I,.3
(17) 3S
The above equation is applicable only when the phase sizes and grain sizes are assumed to be the same. In the event of the existence of large numbers of a - a or t3-t 3 boundaries in the microstructure, the grain sizes of each phase can be measured separately at a given temperature and eqn. ( 1 ) can be used for greater accuracy. The validity of this theoretical analysis can be tested by determining the flow stress (at constant temperature) with different phase proportions. This was feasible in the present investigation by the hydrogenation of test samples at different levels. Knowledge of the constants b,/Ca' and K~' is also required for this study. Methods for determining these constants are discussed in the following sections. 3.2. Determination of parameter b The parameter b is usually determined by measuring the slope of an g vs. L plot on a logarithmic scale. In a two-phase alloy such as the present sample, it seems reasonable to assume that the grain size dependence of the constituent phases is the same, i.e. to say that the exponent b may be assumed to be the same for both a and fl phases. Even with this assumption it is difficult to determine the exponent b when the phase sizes are not the same. Therefore two combinations of alloy compositions and temperatures were chosen where nearly equal phase proportions existed so that the grain sizes were nearly equal. The proportions are nearly equal for the base alloy at 1173 K and for the 0.04 wt.% hydrogenated alloy at 1073 K. The initial average grain size L for the base alloy with 0.04 wt.% H was 3.77 /~m at a temperature of 1073 K. The grain size was deliberately coarsened by soaking at a temperature of 1123 K for 30 min and 60 min respectively. The resulting grain sizes were 4.57 ktm and 5.4 ktm respectively. The double logarithmic stress vs. strain rate plots for these three grain sizes at 1073 K are shown in Fig. 1. The stress level ( o = 2.5 MPa) was carefully chosen so that the m values for all the three grain sizes remained nearly the same (m = 0.6). The value of the exponent b was determined from the slope of a In g vs. In L plot and was 5.0 as shown in Fig. 2. The exponent b
I
TEMPERATURE~073K
~'%P/
03
-12
-11
-10
- 9
-8 In
-7
~,
-6
-S
-z,
-3
s e c -1
Fig. 1. In o vs. In g for the base alloy plus 0.04 wt.% H at 1073 K for three grain sizes: zx, 5.4/~m; • 4.57/~m; o 3.77 ,um. 10.3
~" 9.7 m 10.1e.5 ._= 9.3
' 1~" _=
0.9
/
t. ~ = -72s m
: OM~-O.t~7
8.1
-10
b -" 5
.S
TEMPERALLOY AT01~S. AS*001~1073 E KWt%H, -m" " ;i;"" -11
I
1.3
1.4,
, I
1.5
I
l.a
I
1.'7
In L , pm Fig. 2. Determination of the grain size exponent b for the 0.04 wt.% hydrogenated base alloy at 1073 K: e, 1ng=-7.25, m=0.44-0.47, b~-5.3; o, o=2.5 MPa, m=0.6, b=5.
was also determined by plotting ( 1/m)ln o against lnL at constant g (= 7.25 s-~). The b value was calculated to be 5.3 as illustrated in Fig. 2. Similarly, the flow stress vs. strain rate plots for the base alloy were obtained at 1173 K for three grain sizes (3.58, 5 and 6.1/~m) as shown in Fig. 3. The latter two grain sizes were obtained by soaking the test samples at 1223 K for 1 h and 2 h respectively, before testing. The b values were
224 /,.3
°5
8.1
3.5 -6
2.7
-7 r, "s- 1.9
Io e'-"
c 1.1
/
BASE ALLOY TEMPERATURE 1173K
0.3
-W
- 0.5,~ -12
-11
-10
-9
-8
-7
-6
-5
-/.,
-3
In ~., sec -1 Fig. 3. In o v s . In ~ plots for the base alloy at 1173 K for three grain sizes: zx, 8.1/~m; o, 5 pm; o, 3.58 pm.
determined at three different stress levels as illustrated in Fig. 4 and also at one constant strain rate as shown in Fig. 5. In each case the value of b was 5.0. Hence it can be concluded that the grain size exponent b has a value close to 5.0 at different temperatures and strain rates in the superplastic deformation region. However, the exponent a changes according to the m value through the relationship a = mb.
3.3. Determination of constants K a' and K 8' To determine K j and K 8' it is necessary to know the flow stresses o~ and o 8 (the flow stresses for the 'all-a' and 'all-fl' phase respectively). By increasing the concentration of hydrogen in the base alloy, a complete/3 microstructure can be attained but in that case the grain size may increase quite substantially. Hence the flow stress at the required superplastic grain size is not obtainable. It was therefore decided to extrapolate the logarithmic flow stress vs./3 volume fraction plot to obtain o~ and o 8 as illustrated in Fig. 6 for the base alloy at 1073 K. It is possible to show by series expansion, logarithmic expansion and approximation of eqn. (17) that In a is a polynomial function of V8 (see the Appendix). Three strain rates were chosen for plotting In a against V~: two were in the superplastic region
1.2
1.3
1.4
1.5
1.6
1.7
1.11
2.0
InL, tjm
Fig. 4. Determination of the grain size exponent b for the base alloy at 1173 K (constant stress): A , a = 1 . 1 5 MPa, m = 0 . 5 ; o, ~ = 2 . 1 MPa, m = 0 . 6 6 - 0 . 7 2 ; m, o = 1 . 7 MPa, m = 0.72. T h e slopes of the lines give b ~ 5.
3.8
3.4
3.0
e~
2.6
.=_ 2.2
b c
1.a
I"-'IE
/ 1.4
BASE ALLOY
/
TEHIP 11731(
/
,o
V ' l : ; , '?5
0.6
1.2
I 1.3
| 1./,.
I 1.5
I 1.6
I 1.7
I 1.8
1.9
InL, pm Fig. 5. Determination of the grain size exponent b for the base alloy at 1173 K (constant strain rate ~ = - 9 . 7 5 s - l ; m = 0.7): b ~ 5.
225 (os0:50)l:,~= V~K~'L~¢'°+ ~ K / L # "°
65
= 0.5Ka'L0"" + 0.5K~'Lo "°
i.e. (os,,:50)r,e = [(Kj + K s' )/2]L,, ~''
m
n
where (050:50)r,e is the flow stress at a 50:50 volume ratio of the phases at constant temperature and strain rate; a 0 = c o n s t a n t = m, ob where m0 is the strain rate sensitivity index at a 50:50 volume ratio of the phases. The constants K~', K s' and a can be determined from eqns. (18)-(20). Equations (18) and (19) can also be written in the following form by incorporating a strain rate term:
3.7
2.9
1.1 BASE ALLOY * 0.0/* v t % TEMPERATURE
1.3
0.5
0.3
O -
In $. = -6.25 seC 1 In I~ = - 7 . 2 5 sec "1
-
fl z
1073 K
•
A -
In I~ = 4 . 1 5
X -
CALCULATEO In I F
sec "1
1
,
I
t
2
.t.
.6
.8
1.0
Fig. 6. In o vs. fl v o l u m e fraction plots at 1073 K for the base alloy plus 0.04 wt.% H: o, g = - 6 . 2 5 s-~; o, g = - 7 . 2 5 s - J ; A, g = - 8.15 s ~; × , calculated values of In a.
( l n g = - 7.25 and - 8.15 respectively) and the other was marginally beyond these values. It is apparent from eqn. (6) that any deviation from a 50:50 volume ratio of the phases on either side leads to an apparent increase in grain size (not due to conventional grain growth) of the major phase, at the cost of a decrease in grain size of the minor phase. Obviously, the grain sizes of the two phases are equal (L~ = L~ = L0) when V~ = V~. (It can be reasonably assumed that the phase size is the same as the grain size since there is no contiguity of phases at the 50:50 volume ratio.) When the microstructure turns either completely a or completely fl, the grain size should be double L 0 (the grain size at the 50:50 phase ratio). The flow stresses of the two phases can then be written as follows: (18)
and (o~)r,~ = K~'L~~= K~'(ZL0)"
( a~)r,¢ = K~( 2L,,)" g .....
(21)
( o~)r,c = K~(2Lo)"g ....
(22)
From the above equations K~ and K s can be obtained at any strain rate.
,B VOL. FRACTION
(o~)r.,: = K~'L,fl = K,'(2L0)"
(20)
(19)
where (aa)r, ~ and (a~)r,~ are the flow stresses for 100% a and 100% fl phases respectively at constant temperatures and strain rate and K j , K~' and a are constants. The flow stress at a 50 : 50 volume ratio of the phases may be written as follows:
3.4. Estimation of flow stress from constitutive equation From the knowledge of K~ and K s at any strain rate and temperature it is now possible to calculate the flow stress at a particular strain rate and temperature by using me constitutive equation (13) or (17), whichever is applicable. The calculated flow stresses are shown in Fig. 6 by crosses which deviated marginally from the experimentally obtained values. Flow stress values were not calculated at higher strain rates (shown with a dotted line) which is in the non-superplastic region and it is very likely that both the b value and the a value might change in this region. 4. Conclusions (1) A constitutive equation for a two-phase titanium alloy has been developed by assuming an isostrain rate condition for the two phases and applying the rule of mixtures. (2) At any particular temperature the phase ratio can be altered by varying the hydrogen content. Thus the effect of phase ratio on flow stress can be determined at the given temperature. (3) The predicted flow stress at 1073 K at two strain rates correlates well with the experimentally measured values.
a+fl
Acknowledgements The authors would like to thank Dr. P. Rama Rao, Director, Defence Metallurgical Research
226
Laboratory, Dr. N. C. Birla and Dr. M. C. Pandey for their encouragement and advice. One of the authors (1. K. Mukherjee) would like to thank the United Nations Development Program for a travel grant.
By expanding (1-Nil) rob+l, eqn. (A1) can be written as (Oc)r,~= K j { 1 - ( m b +
1)Vt~+[(mb+1)mb/2!lV~ 2
-1-.., -- g~ mb+l "l-(gflt/gat)Vfl rob+l}
(°c)r'~=Ka'{1-(mb+ l)V~+
References 1 K. A. Padmanabhan and G. J. Davies, Superplasticity, Springer, Berlin, 1980, p. 83. 2 A.K. Ghosh and R. Raj, Acta Metall., 29 ( 1981 ) 607. 3 M. Suery and B. Baudelet, Res. Mech., 2 (1981) 163. 4 A. Dutta and N . C. Birla, Mater. Sci. Technol., 4 (1988) 341. 5 K. A. Padmanabhan and G. J. Davies, Superplasticity, Springer, Berlin, 1980, pp. 65-66. 6 C.H. Hamilton, A. K. Ghosh and M. M. Mahoney, in D. E Hasson and C. H. Hamilton (eds.), Advanced Processing Methods for Titanium, Metallurgical Society of AIME, Warrendale, PA, 1981, pp. 120-144.
(13)
(mb+ 1)mb V~ 2!
+ . . . + V F +~[(Ke'/KJ)-
1]}
(14)
By taking the logarithm on both sides, eqn. (A4) can be written as ln(oc)~e--lnK~ •
+(I In 1 -
(rob+ 1)V~
t
mb(mb 2! + 1) V~2-'"- V,,b+L x[(K/j/K,')-I]I )
(A5)
Appendix: Derivation of o as a function of V~
Defining the terms in braces as x, eqn. (A5) reduces to
At a constant temperature T and strain rate g eqn. (17) reduces to
In( a c)T,~ = In K J + In( 1 - x)
(Oc)T,~ -----Ka'Va mb+l + K#'Vfl mb+l
(A1)
X2 X3
(A6)
)
(A7)
(since L0, m and b are constants) (Oc)T,t=Ka'[(1-- Nil)mb+l +(Kj/Ka')V~rob+'] (A2)
ln( Oc)r,~= constant + polynomial function of Vs.