The strain rate sensitivity increases with temperature but decreases with hydrogen ... when the flow stress or strain rate is increased. At a fixed strain rate, ...
Effect of Strain Rate and Temperature on the Flow Stress of/3-Phase Titanium-Hydrogen Alloys O.N. SENKOV and J.J. JONAS Compression tests were carried out on seven titanium-hydrogen alloys containing hydrogen concentrations up to 31 at. pct. All the experiments were performed within the [3-phase field at strain rates of 0.001 to 1.0 s-L The dependences of the steady-state flow stress on strain rate, temperature, and hydrogen concentration were determined. The strain rate sensitivity increases with temperature but decreases with hydrogen concentration. The experimental activation energy of deformation decreases when the flow stress or strain rate is increased. At a fixed strain rate, it decreases when the hydrogen concentration is increased. However, when measured at a fixed steady-state stress, the activation energies are nearly the same for all the alloys. The steady-state flow stress increases with hydrogen concentration as can be expressed by both linear and quadratic dependences. The flow behavior of the alloys can also be described in terms of thermally activated glide and the relation
where the constants v and 2if-/o are independent of hydrogen concentration, while the parameter K decreases exponentially when the hydrogen concentration is increased.
I.
INTRODUCTION
TITANIUM alloys have a large affinity for hydrogen, the addition of which stabilizes the more ductile bcc [3 phase. As a result, the temperature of the (a + [3) --) [3 transformation decreases from 1155 K for nearly pure titanium to 573 K for alloys of eutectoid composition (39 at. pct H).m The stabilization of the [3 phase by hydrogen can be used to facilitate the processing of titanium alloys as well as to modify the microstructure and final properties.t2,3, a]
Despite the commercial importance of these materials, the mechanical properties of the hydrogen-stabilized [3 phase have not been studied extensively, and only a few publications devoted to their properties are currently available. t5,6,7] In the present article, the effect of dissolved hydrogen on the flow stress of [3 titanium is characterized over a wide range of temperatures and strain rates. II.
EXPERIMENTAL
The base material used in this study was a titanium of technical purity with the chemical composition given in Table I. Cylindrical specimens 8 mm in diameter and 12 mm in height were prepared for compression testing. The specimens were alloyed by holding them at 1073 K in pure hydrogen atmospheres of increasing pressure for 2.5 hours. Six series of specimens with hydrogen contents of 5, 9, 12, 15, 23.5, and 31.5 at. pct. were prepared in this way. The hydrogen levels were determined by weighing the specimens before and after hydrogenation to the nearest 0.0002
O.N. SENKOV, Visiting Scientist, and J.J. JONAS, Professor, are with the Department of Metallurgical Engineering, McGill University, Montreal, PQ, Canada H3A 2A7. Manuscript submitted August 18, 1995. METALLURGICAL AND MATERIALS TRANSACTIONS A
g. The mechanical tests were carried out on an MTS hydraulic compression machine at strain rates of 0.001 to 1.0 s -1 and over the temperature range 750 to 1300 K. The specimens were heated to temperature at 1 K/s and then given a 10-minute soak. When held at temperatures above 950 K in a pure argon atmosphere, the specimens lost some of their hydrogen contents. By contrast, the alloying levels were maintained up to 1240 K when the sampies were heated in air, because an oxidized film formed on the specimen surfaces, which prevented the escape of hydrogen.iS] Oxidized layer thicknesses of about 30 /xm were developed in samples held in air at 1300 K for 30 minutes. As a result, the specimens displayed different flow behaviors depending on whether they had been deformed in air or in argon. However, specimens with the samefinal hydrogen concentrations (measured after straining) displayed essentially the same behavior independently of the prior annealing environment. The concentrations of oxygen and nitrogen in the samples that had been heated in air did not change. All subsequent tests were therefore performed in air, and the hydrogen content of each specimen was remeasured after deformation. The grain size in these specimens was around 100 /zm. The experimental method is described in more detail in a companion article.t81
IlL
RESULTS
A. Determination of the (t~ + [3)/[3 Transformation Temperatures The temperature o f the (a + [3)//3 transformation in specimens of different hydrogen concentrations was determined from the dependence of the flow stress on inverse temperature (Figure 1). For this purpose, the flow stress at e = 0.3 was used, and the temperature at which the slope of the cr vs 1/T curve changed abruptly was established. VOLUME 27A, MAY 1996~1303
Table I.
A1 0.426
Chemical Composition of Base Specimens (Atomic Percent)
V
Fe
Si
0.255 0.058
0.070
H
C
O
N
0.047 0.067 0.011 0.001
The transformation temperatures did not depend on strain rate, although they could be evaluated more readily when the lowest strain rate was used, especially in the alloys containing 15 and 23 pct hydrogen. The 31 pct hydrogen material did not undergo a phase transformation within the experimental temperature range. The (or + /3)//3 transformation temperatures measured in this way are illustrated in Figure 2 as a function of hydrogen concentration. The location of the cd(a + /3) transus, determined on specimens containing 0.05 and 5 pct hydrogen, is included in Figure 2. These results are in good agreement with the Ti-H phase diagram obtained by other methods v~ and identified in Figure 2 by means of dashed lines. B. Flow Curves Typical flow curves measured in the/3-phase field on two of the present titanium-hydrogen alloys are presented in Figure 3. After an initial period of strain hardening, steadystate flow sets in at a strain that decreases with increasing temperature and decreasing strain rate. A slight amount of flow softening is evident at 973 K. Marked serrations, indicating the occurrence of dynamic strain aging, can be seen on the flow curves obtained at low strain rates. In a separate series of tests, strain rate changes were performed during testing. Some typical results are presented in Figure 4. It can be seen that the new steady state is reached after strains of no more than 0.05 after the strain rate change. Furthermore, the new value of the steady-state stress is almost identical to that determined in continuous testing. Such behavior indicates that strain rate change tests can be employed to study the rate dependence of the steadystate flow stress. Because of the relatively short work hardening transients, each such test can provide as many data points as six or seven continuous tests. C. Rate and Temperature Dependence of the Steady-State Flow Stress The rate dependences of the steady-state flow stress measured in this way at a series of temperatures are shown in Figure 5. For all the hydrogen levels shown, the rate dependence can be described by the relation
~ ~m
[11
The rate sensitivity m -- (8 log ~r/8 log e)r that corresponds to the slope of the log o- vs log ~ curves is plotted in Figure 6(a) as a function of temperature at a series of hydrogen concentrations. It increases with temperature in the usual way but decreases when the hydrogen concentration is increased. As indicated in the diagram, the rate sensitivity for each of the alloys is linearly related to the inverse temperature and can be described by the expression m = m o - B1/RT 1304~VOLUME 27A, MAY 1996
[2]
The dependences of the parameters mo and B 1 on hydrogen concentration are given in Figures 6(b) and (c). The effect of inverse temperature on the steady-state stress is depicted in Figure 7 for two different testing strain rates. Data for two further strain rates were presented in Figure 1. Within the/3 phase, the behavior is approximately linear, so that the data can be represented by the relation tr ~ exp (mQ/RT)
[3]
where Q = -2.3 R (0 log PJ0(1/T))~ is the activation energy for steady-state flow. The values of mQ = 2.3 R[0 log tr/O(1/T)]~ that correspond to the slopes of these log tr vs 1/T curves are plotted in Figure 8(a) as a function of strain rate. For each of the alloys, these dependences can be described by the expression mQ = mQl - 2.3B2 loge
[4]
where mQ~ is the value o f m Q at & = 1 s-L The influence of hydrogen concentration on mQl is presented in Figure 8(b), from which it can be seen that this term decreases as the alloying level is increased. The dependence of the parameter B2 on hydrogen concentration is given in Figure 6(c). It can be seen that the values of B2 are approximately equal to those of B~, which is not fortuitous. As explained in more detail in the Appendix, the strain-rate-dependent term in Eq. [4] is the equivalent of the temperature-dependent term in Eq. [2]; i.e., Eqs. [1] and [3] contain the common term ~-Blrar ~ exp ( - 2 . 3 B l log PJRT) - exp (-2.3B2 log PJRT). By combining Eqs. [ 1] through [4], the constitutive equation describing the steady-state flow stress o f / 3 titanium alloys can be expressed in the form o- = A~ e m exp (mQffRT) - AI e,,o exp (mQ/RT)
[5]
The dependence of the parameter A~ on hydrogen concentration is depicted in Figure 8(c), from which it can be seen that it increases with alloying level. The values of the parameters At, mo, B~, and mQ~ are summarized in Table II for each of the alloys. The activation energy Q was calculated from the experimentally determined values of m and mQ for each of the alloys. For this purpose, the following formula was used: Q = at - 2.3 (B~/m) log e
[6]
Relations [2] and [4] are taken into account in Eq. [6]. The values of Q determined in this way at two different strain rates and two different temperatures are plotted in Figure 8(d) as a function of hydrogen concentration. It can be seen that Q decreases when the strain rate and temperature are increased. At the same temperature and strain rate, the activation energy decreases with hydrogen concentration. D. Dependence of Steady-State Flow Stress on Hydrogen Content Within the temperature range 1000 to 1270 K, the steady-state stress increased linearly with hydrogen concentration when the latter exceeded about 10 to 12 pct, as depicted in Figure 9. This dependence can also be considered a quadratic one, as also indicated in the diagram and described by the relation METALLURGICAL AND MATERIALS TRANSACTIONS A
I000
1000 1
I
i
100
I
r~
f
7
I 10 I
.--'~
10
0.00
tJ
~0~
I
100
I o 1.0_~
i
1
1
0.8
0.7
0.9
1.1
1
1.2
1.3
0.8
0.7
0.9
1
1.1
1.2
1.3
lO00ff (K~)
1000/T ( K "l)
(b)
(a)
I000
1000
__Lm o0-001s-1 23~ ~ ol.0s-1 -i0.001s-1 .l.0s-1 31~6H~
G"
100 "-" 100
........--.o-
o.--.'-
o
10
IS
!
,x
ta 0.001 s-I
1
~ 1 7 6s T ~
0.8
0.85
0.9
0.95
1
1.05
1o 1.1
0.8
j
~i
i 0.9
1000/T (K ~)
1
1.I
1.2
1.3
1000/T (K j )
(c)
(~
Fig. l--Dependence of the flow stress at e = 0.3 on inverse temperature determined at strain rates of 0.001 s -I (t3) and 1.0 s -~ (o) in alloys containing the following concentrations of hydrogen, in atomic percent: (a) 0.05, (b) 5, (c) 12, and (d) 23 and 31.
1200
tr = tr (0) (1 + xCH z)
1100
Here, X is a parameter that depends on temperature and strain rate and (7(0) represents the steady-state flow stress of/3 titanium in the absence of hydrogen addition. At temperatures below the allotropic transformation, the values of o(0) were obtained by extrapolating the experimental tr vs CH dependence to (7. = 0. This procedure led to the following equations for o(0) and X:
1000 \ 900
i
I
800
I
E
a 7O0 60O
tl+l~
I
[7]
" -x
or (0) = 0.43 • ~ (o.353-158/Dexp (4780//') (in MPa)
! ! ! !
X = 387
• ~-o.o7
exp (-3950/T)
[8a] [8b]
L ,
500 0
I
i
I
10
20
30
40
Atomic Percent Hydrogen Fig. 2--The Ti-H phase diagram. Dashed lines are reproduced from Ref. 1; the solid and open circles were determined in the present work. METALLURGICAL AND MATERIALS TRANSACTIONS A
IV.
DISCUSSION
The experimental data indicate that the steady-state flow stress increases while the rate sensitivity m and experimental activation energy Q measured at a constant strain rate decrease with hydrogen concentration. This result is someVOLUME 27A, MAY 1996~1305
30 0.05 % H
1193 K, 1.0 s"l
25 ,--, 20 t~ " r 15
?=
f
r~ 10
/
"
1303 K, 0.1 s q
f
1303 K, 0.01 sq
1303 K, 0.001 s~ 0 0
I
I
I
I
I
0.1
0.2
0.3
0.4
0.5
0.6
Strain (a) 120 31%H 973 K, 1.0 s l
100 ,-.
80
""
60
1073 K, 1.0 s "t
40 20
1233 K, 0_~001 s q . . . . . . . . . .
0
0
I
I
0.1
0.2
mally activated glide is employed and the glide activation energies for the alloys are considered to be stress dependent, decreasing with the stressY ~ Because an increase in hydrogen concentration leads concurrently to a decrease in the pre-exponential factor (discussed subsequently), the effect of the decrease in the activation energy can be overcome in this way. The dependence of the rate sensitivity on temperature and hydrogen concentration can also be given a physical interpretation in this way. It should be noted that the activation energy Q calculated from Eq. [6] on the basis of the experimentally determined values of mQ and m (Section C) is a complex function of strain rate and temperature and cannot be expressed as a function of the flow stress in a simple manner7 ~ Nevertheless, the stress dependence of Q can be determined from plots of strain rate vs 1/T at constant stress, where the strain rates are obtained by interpolation from Figure 5. Examples of the curves obtained in this way are given in Figure 10 for the alloys containing 12 and 23 pct hydrogen. The stress dependence of Q at each of the seven hydrogen levels investigated here is illustrated in Figure 11. It can be seen that Q does decrease with stress in the normal way. When the flow stress is held constant, Q depends weakly on the hydrogen level. The zero stress values of the activation energies Q0 ranged only from 190 to 200 kJ/mol over the full range of hydrogen concentrations. Such behavior, which includes a dependence of the strain rate sensitivity on temperature, can be described by the following relation:tl0~
-_ _
= Kl o"~~exp - --~--] exp
I
I
I
0.3
0.4
0.5
0.6
[91
Strain
(b)
=K2o'n~
Fig. 3--Typical/3-phase flow curves for (a) titanium and (b) Ti-31 pct H alloy.
,0s 0.3 '30
0.1
2o
k-~)exp(V~-~)
where K l is a material constant; K2 = K~ exp (ASo/k); k is the Boltzmann constant; AGo, AHo, and ASo are the Gibbs free energy, enthalpy, and entropy of activation at zero stress, respectively; v is the activation volume; and no is the stress sensitivity of the pre-exponential term. The zero stress Gibbs free energy is commonly expressed as a function of the shear modulus /z as AGo = a/~b 3, where b is the Burgers vector and a is the parameter that characterizes the activation mechanism. Then, AHo = a/xob3 and ASo = -o&3(dtx/dT), where/Xo is the shear modulus at 0 K. A fit of Eq. [9] to the experimental data was made for each of the alloys separately using the MATLAB* program. *MATLAB is a trademark of MathWorks, Inc., Sherborn, MA.
0
0.1
0.2
0.3
0.4
0.5
0.6
Strain Fig. 4--Flow curves for the strain-rate change tests compared with the tests at constant strain rate. Ti-23 pet H alloy.
what unusual in that decreases in activation energy are usually associated with flow stress reductions. This contradiction can be overcome, however, if the formalism of ther1306---VOLUME 27A, MAY 1996
The stress sensitivity n o of the pre-exponential term was estimated for each of the alloys using the method described in References 11 and 12. This method permits the separation of the stress dependences of the preexponential and exponential terms when glide is controlled by linear elastic obstacles. The values of no obtained in this way ranged from 3.8 to 4.1, and an average value of n o = 4.0 was therefore employed for all the alloys. The activation volume v was found to depend very weakly on stress above 20 MPa but increased rapidly when the stress was reduced to zero. However, because the contribution of the exponential term, METALLURGICAL AND MATERIALS TRANSACTIONS A
30
40
o.o5
21
Hl 24
15
20
...S
12
9
12
r~
o
6
o
0.001
0.01
1233 K 1273 K 1303 K Eq. (5) Eq. (9)
0.1
Strain Rate
r~ 8
o
1
1273K Eq. (5) -- Eq. (9)
4
0.001
(s "1)
0.01
0.1
1
Strain Rate (s "t)
(a)
(b) 50
50 ~ 1 at.% 9 H -
g
25
,-, 25 t~
20
20
15
15 n o t~ o
10 l
1073 K 1123 K 1173 K 1203 K Eq. (5)
"TJ > J f
f
o o a o
10
- Eq. (9)
1043 K , 1073 K 1123 K 1203 K Eq. (5) i
. . . . . . Eq. (9) i
5
0.001
~176
0.01
0.1
1
5
0.001
Strain Rate (sq) (c)
0.01
0.1
1
Strain Rate (s 4)
(a) Fig. 5--Strain-rate dependence of the steady-state flow stress displayed by the Ti-H alloys containing (a) 0.05, (b) 5, (c) 9, (d) 12, (e) 15, (f) 23, and (g) 31 at. pet H.
exp (v~r/kT), is negligible at low stresses in comparison with that of the pre-exponential term, v was considered to be stress independent. The parameters ~t/o = 1.97 _ 0.06 eV (190 _ 6 kJ/mole) and v = 8.0 _ 1.0-10 -2s m 3 (4.8 __+ 0.6.10 -4 m3/mole) were found to be nearly identical for all the alloys. Because ~/-/o = oqz0 b3, the experimental values of A/-/0 are equivalent to a = 0.6. The pre-exponential term K2 was also calculated with the aid of the MATLAB program and found to decrease sharply with the hydrogen concentration (Figure 12), according to the expression K2 = K3 exp (-30CH 1.5)
[10]
where K3 = 260 s -~ MPa -4. This sharp decrease in K2 can be interpreted in terms of the effect of hydrogen on the density of mobile dislocations. Indeed, the pre-exponential term K2o 4 in Eq. [9] is generally considered to contain the Burgers vector, attempt frequency, density of mobile dislocations or of thermally activatable sites, area swept by a dislocation on successful activation, and the entropy term.tin2,15] Given that the activation volume does not appear to change much over the experimental range of hyMETALLURGICAL AND MATERIALS TRANSACTIONS A
drogen concentrations, the two order of magnitude reduction of the pre-exponential can be most readily associated with a decrease in the density of mobile dislocations or of thermally activatable sites. Alternatively, it can be attributed to the decrease in the temperature dependence of the shear modulus, dtz/dT, associated with the addition of hydrogen, as discussed in more detail elsewhere, tl6,~7~ The dependences of the flow stress on strain rate at different temperatures calculated from Eq. [9] are plotted in Figure 5 as dashed lines. Good agreement is observed with the experimental data for each of the alloys. These results indicate that dissolved hydrogen does not noticeably change AHo and v but has a strong influence on the magnitude of the pre-exponential term. Similar conclusions regarding the effects of dissolved hydrogen have been drawn in References 13 and 14. Conversion of the activation volume v = 4.8 9 10 _4 m3/mol into Burgers vector units (using a Taylor factor M = 2) leads to v ~ 68b 3. Such a value of v, together with a = 0.6, suggests that steady-state flow may be controlled by the motion of jogged screw dislocations.tm8~ In this case, the activation volume is given as v = b2L, where L is the VOLUME 27A, MAY 199~-1307
100
60
.~176
50
, ~ 30 ~
24
~
18
~ 40
f
12
~
I
o I I
o §
1003 K1 1073 K[_ 1173K / 1193 K (
~"
Eq. !5!1
0.01
o
973 K
o +
1163K 1183 K Eq. (5) 9Eq. (9)
10
t
0.001
j -~"J~
20
. . . . . . t~q. t~) i 6
30
0.1
0.001
1
0.01
0.1
1
Strain Rate (s q)
Strain Rate (sl )
(t)
(e) 100
50 40 ~
r~
30 20
-~# ...... tx o
t
[ ......
'
10 0.001
0.01
9"/3K 1073 K 1193 K 1233 K
Eq. (5) Eq, (9)
0.1
Strain Rate
(s l )
(g) Fig. 5--Continued Strain-rate dependence of the steady-state flow stress displayed by the Ti-H alloys containing (a) 0.05, (b) 5, (c) 9, (d) 12, (e) 15, (f) 23, and (g) 31 at. pet H.
average spacing between jogs. The thermal equilibrium value of L can in turn be estimated using the equationt~81
Zeq
=
(b/q) exp (~/RT)
[11]
where Uj =__ 0 . 2 / ~ b 3 is the energy required to form a unit jog and q is the number of alternative jog orientations. In the absence of applied stress, q = 5 to 10. When a stress is applied, q is assumed to decrease to one. Setting T = 1273 K for/3 titanium,/z = 15.3 GN/m 2, b = 2.86 10 -t~ m , [19] and q = 1, a thermal equilibrium jog spacing Leq of about 60b is obtained. This value is close to that determined experimentally (L = 68h). It should be noted that similar values of L were estimated for/3 titanium in Reference 6. In the case of the jog-dragging mechanism, the activation energy for the process is generally considered to be that for self-diffusion.rio1 The self-diffusion activation energies measured by various authors in/3 titanium lie within the interval 130 to 260 kJ/mol, increasing with temperature up to the melting pointY ~ All these data can be approximated by a single activation energy of 152 kJ/mol. 09] Limited data for the high-temperature deformation of /3 titanium 151analyzed according to two different methods in
1308--VOLUME 27A, MAY 1996
References 6 and 7 led to activation energies of 172 and 209 kJ/mol, respectively. It can therefore be seen that the value of AHo = 190 kJ/moi obtained in the present study is in good agreement with the literature data on both the self-diffusion and high-temperature deformation of/3 titanium. V.
CONCLUSIONS
1. The a/(a + fl) and (a + fl)lfl transformation temperatures determined from the dependence of the flow stress on inverse temperature in specimens of different hydrogen concentrations are in good agreement with the known Ti-H diagram. 2. Specimens deformed in the /3-phase field display a steady state of flow following a period of strain hardening. Work hardening disappears at a strain that decreases with temperature and decreasing strain rate. 3. The rate and temperature dependence of the steady-state flow stress for each of the alloys can be described by the relation cr = A ~ exp (mQ~/RT). The rate sensitivity m decreases when the hydrogen concentration and in-
METALLURGICAL AND MATERIALSTRANSACTIONS A
100 T (K) 1273
0.24
I
1173
1073
973
I
I
I
873
u .05 % H o5%H a9%H x12%H x15%H
0.22 0.2
50 40 ca 30
0.18
20
0.16 0.14
10
/
a15%H o23%H +31%H
o
0.85
0.75 0.12 0.75
I
I
I
0.85
0.95
1.05
1.15
1000/T (K 1)
0.95 IO00/T ( K "l) (a)
J zJ j.J
0.4 30 0.35
J
24
1
~a18
0.3
o .05 % H x5%H x9%H o12%H a15%H o23%H § i
12 0.25
0
1.15
60
(a)
0.2
1.05
I
I
I
I
I
I
5
10
15
20
25
30
35
Atomic Percent Hydrogen
(b)
6f 0.75
0.85
0.95 1000/T (K "1)
1.05
1.15
O) Fig. 7 Inverse temperature dependence of the steady-state flow stress at strain rates o f ( a ) 0.I s ~ and (b) 0.01 s-L
1.5
o
o
0.5 !-o BJi[ o B2 0
0
I
I
I
I
I
I
5
10
15
20
25
30
35
Atomic Percent Hydrogen (c) Fig. 6 - - ( a ) Temperature dependence of the rate sensitivity m. Effect of hydrogen on the parameters (b) m o and (c) B, (Eq. [2]) and B 2 (Eq. [4]).
METALLURGICAL AND MATERIALS TRANSACTIONS A
verse temperature are increased. The parameter mQl decreases and A~ increases with hydrogen concentration; they do not depend on temperature and strain rate. 4. The steady-state flow stress increases with hydrogen concentration, as can be expressed by both linear and quadratic dependences. 5. The activation energies for steady-state flow Q = - R [ 0 In P_/O(1/T)],,were determined for the series of alloys. At a fixed strain rate, they decrease when the hydrogen concentration is increased. However, when calculated at the same steady-state stress, the activation energies are nearly the same for all the alloys and decrease when the flow stress is increased. The zero flow stress activation energy Qo does not depend on hydrogen concentration; its value is close to that for self-diffusion. 6. When the behavior of titanium-hydrogen alloys is analyzed in the framework of thermally activated glide, it is found that hydrogen does not alter markedly the ac-
VOLUME 27A, MAY 1996--1309
45
45
4O
40 35 O
3O
o
o
mQ~l =40~- 6)C~
30
c~
25
20
25 20
15 0.001
0.Ol
~
15
0.1
0
[
I
I
!
I
I
5
10
15
20
25
30
Strain Rate (s "1)
35
Atomic Percent Hydrogen
(a)
(b) 20O [
180
O
I
160
O
140 120
o,,93K1
I 100 J
0
0
I
I
I
5
10
15
20
I
25
30
35
Atomic Percent Hydrogen (c) Fig. 8--(a ) Strain-rate dependence of [5]), and (d) Q (Eq. [6]).
Table II.
AI
(At. Pct)
(MPa 9 Sin0)
0.05 5 9 12 15 23.5 31.5
0.38 0.71 0.85 1.27 1.39 5.16 7.61
I
0
5
"~] I
I
I
I
10 15 20 25 Atomic Percent Hydrogen
Ql
I
30
35
(a9
mQ determined at seven different hydrogen levels. Dependences on hydrogen concentration of (b) rnQ, (c) A~ (Eq.
Dependence of the Parameters in Equation [5] on Hydrogen Concentration
H
o 1293 K I
80
I
,os,
BI
mQl
mo
(kJ/mol)
(kJ/tool)
0.271 0.287 0.300 0.312 0.323 0.363 0.370
0.46 0.79 1.0O 1.16 1.21 1.62 1.85
41 35.5 34 31 31 21 20
ACKNOWLEDGMENTS The authors are indebted to Professors F. Montheillet and E.G. Ponyatovsky and to Dr. T.M. Maccagno for numerous stimulating discussions. They are thankful to the Natural Sciences and Engineering Research Council of Canada for financial support. ONS acknowledges with gratitude the period of sabbatical leave accorded by the Institute of Solid State Physics, Russian Academy of Sciences (Chemogolovka, Russia). APPENDIX
tivation parameters for steady-state flow; it does, however, decrease significantly the pre-exponential term. Such a behavior can be interpreted in terms of the decrease in density of mobile dislocations or of thermally activatable sites that is brought about by hydrogen addition. 1310~VOLUME 27A, MAY 1996
It is first assumed that the steady-state flow stress of titanium-hydrogen alloys obeys the following viscoplastic power law: o - = ~7(T)" ~,,~r/
[A1]
where m(T) is a temperature-dependent rate sensitivity and rl(T ) is a temperature-dependent viscosity. Then, the actiMETALLURGICAL AND MATERIALS TRANSACTIONS A
70
80
1173 K
o
60
o 0.01 s-1 a0.1 s-1 ol.Os-1
50 r~ o~
g
0.001 s-1
70
40 30
1073 K
z ~J= , .
I
,,I~/" - i~,
10
30
15.9
I
I
I
10
20
30
20
40
~
16.1
18.7
. . . .
22.2
0
o 0.001 s-1
Z=
1233 K
17.6
--, t~
.. 196
'-"
30
10 I
I
10
20
30
o 1.0
s-1
11
/7-
.o
21.5
(D
10 I
z Jr 17
20
0
40
X = 15.5
o 0.001 s-I o 0.01 s-1 A0.1 s-1
4O
20
0
I
30
60
14.2
50
. f
30
I
20
(b)
r
: 00101ls~ll
I
10
Atomic Percent Hydrogen
,,9 /, I
1193 K
r/3
j
a
10
60
40
,
~
Atomic Percent Hydrogen (a)
50
/
91. s -0~
40
. 13.5
0 0
0.001 s-1 o 0.01 s-1 A0.1 s-1
X = 13.4
o
50
. 11 5 ,'~ '
. . . . .
20
60
./z j
~
= 9.8
40
0
Atomic Percent Hydrogen
10
20
30
40
Atomic Percent Hydrogen
(a)
(c)
Fig. 9 - - ( a ) through (d) Dependence of the steady-state flow stress on hydrogen concentration at few selected temperatures. The value of the parameter
g in Eq. [7] correspondingto each curve is displayed.
m = m o - B1/RT ~7 = Al exp (mQ]RT)
vation energy Q of steady-state flow can be calculated from the relation
mQ=_mR(Oln~l = R ( d l n ~ 7 + dm \0 (1/T)], \d (l/T) ~
In
~)
[A2]
Experimentally, the value of mQ was not found to depend on temperature (Section III-C). One can therefore conclude that the derivative terms on the right side of Eq. [A2] are constants, so that d In ~ - - a
d (l/T)
and
dm
d(1/r)
-
/3
[A3]
[A5] [A61
where mo and A1 are integration constants. From the combination of Eqs. [All, [A5], and [A6], one obtains O" =
A 1
~m exp (mQ1/RT) =- Al ~mo exp (mQ/RT)
[A7]
The effect of the temperature dependence of the rate sensitivity (Eq. [A1]) on the activation energy of plastic flow has been treated recently in Reference 9, on which the preceding derivation is based.
The combination of Eqs. [A2] and [A3] leads to REFERENCES
mQ = R a -
R / 3 1 n ~ = mQ~ - Bl ln g
[A4]
where Q1 = Rcdm is the value of Q at a strain rate of 1 s -l and B1 = R/3 is a constant. Integrating Eq. [A3] and substituting for the parameters a and/3 in terms of Q~ and Bl give METALLURGICAL AND MATERIALS TRANSACTIONS A
1. A. San-Martin and F.D. Manchester: Butt. Alloy Phase Diagrams, 1987, vol. 8, pp. 30-42. 2. F.H. Froes, D. Eylon, and C. Suryanarayana: JOM, 1990, vol. 42, pp.
26-29. 3. H. Yoshimura, K. Kimura, M. Hayashi, M. Ishii, T. Hanamura, and J. Takamura:Mater. Trans. JIM, 1994, vol. 35, pp. 266-72. VOLUME 27A, MAY 199~-1311
200[
10 a, MPa
180 0.1
170
29
160
.=.
0.01
(Y
150 140
0.001
"6
130 0.0001
,
- .....
,
~
7
''~
9 '
120
0.82
0.87
0.92
0.97
0
20
40
1000/T (K n)
60
80
100
Stress ( M P a )
(a) Fig. 1 l--Stress dependences of the activation energies of the present series of Ti-H alloys.
10
1000 ~, MPa
"--..70
0.1
"--.060 "---050
0.01
0.001
~,
100
ff
lO
0.0001 0.82
0.92
1.02
1.12
1.22
1000/T ( K " )
(b) Fig. 10~Arrhenius plots of the interpolated strain rate corresponding to selected values of the flow stress for Ti-H alloys containing (a) 12 pct and (b) 23 pct hydrogen. 4. O.N. Senkov and 1.O. Bashkin: in Metallurgical Processes for the Year 2000 andBeyond, H.Y. Sohn, ed., TMS, Warrendale, PA, 1994, vol. 1. pp. 271-80. 5. H. Biihler and H.W. Wagener: Bander, Bleche, Rohre, 1965, vol. 6, pp. 625-30, 667-68, and 677-84. 6. J.J. Jonas and J.-P. Immarigeon: Z. Metallkd., 1969, vol. 60, pp. 22731. 7. H. Conrad, M. Doner, and B. deMeester: in Titanium Science and Technology, R.I. Jaffee and H.M. Burte, eds, Plenum Press, NY, 1973, vol. 2, pp. 969-1005. 8. O.N. Senkov and J.J. Jonas: Metall. Mater. Trans. A, 1996, in press. 9. F. Montheillet: unpublished research, 1995. Ecole des Mines de St. Etienne, St. Etienne, France. 10. U.F. Kocks, A.S. Argon, and M.F. Ashby: in Thermodynamics and Kinetics of Slip, vol. 19, Progress in Materials Science, B. Chalmers, J.W. Christian, and T.B. Massalski, eds., Pergamon Press, Oxford, United Kingdom, 1975. 11. T. Surek, L.G. Kuon, M.J. Luton, and J.J. Jonas: in Rate Processes in Plastic Deformation of Materials, J.C.M. Li and A.K. Mukherjee, eds., ASM, Cleveland, OH, 1975, pp. 629-55.
1312--VOLUME 27A, MAY 1996
l
I
0
5
I
I
I
I
I
10
15
20
25
30
35
Atomic Percent Hydrogen Fig. 12--Dependence of the parameter K~ of Eq. [9] on hydrogen concentration. 12. T. Surek, M.J. Luton, and J.J. Jonas: Phys. Status Solidi B, 1973, vol. 57, pp. 647-59. 13. E.G. Ponyatovsky, O.N. Senkov, and I.O. Bashkin: Phys. Met. Metallogr., 1991, vol. 72 (2), pp. 194-200. 14. H. Dong and A.W. Thompson: Mater. Sci. Eng. A, 1994, vol. 188 (1-2), pp. 43-49. 15. MJ. Luton and J.J. Jonas: Acta Metall., 1970, vol. 18, pp. 511-17. 16. O.N. Senkov and J.J. Jonas: Symp. in Advances in the Science and Technology of Titanium Alloy Processing, TMS, Anaheim, CA, 1996. 17. O.N. Senkov and J.J. Jonas: in Hot Workability and TMP of Ferrous
and Non-Ferrous Alloys, 35th Annual Conference of Metallurgists, The Metallurgical Society of CIM, Montreal, Quebec, 1996, in press. 18. A. Ardell, H. Reiss, and W. Nix: J. Appl. Phys., 1965, vol. 36, pp. 1727-32. 19. HA. Frost and M.F. Ashby: Deformation-Mechanism Maps, Pergamon Press, Oxford, United Kingdom, 1982, pp. 49-52. 20. H. Nakamura and M. Koiwa: Iron Steel Inst. Jpn. Int., 1991, vol. 31, pp. 757-66. 21. D. Lazarus: Diffusion in Body-Centered Cubic Metals, ASM, Cleveland, OH, 1965, p. 155.
METALLURGICALAND MATERIALSTRANSACTIONS A
