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The influence of alloying and temperature on the elastic constants of tantalum
,Joumal
of the Less-Common
Elsevier
Sequoia
Metals
S.A., Lausanne
- Printed
265
in The Netherlands
THE INFLUENCE OF ALLOYING AND TEMPERATURE ELASTIC CONSTANTS OF TANTALUM
Ll. A.
;ZKMSTRONG
Ckpartment of Metalluvgy (Received
AND
B.
ON THE
I,. MORDIKE
and Materials
S&me.
Ulziversity
of Liverpool,
Liverpool
(Gt. Britain)
April 18th, 1970)
The elastic compliance constants of TaRe, TaW, TaMo and TaNb single crystals have been determined over the temperature range So0-~4000K. These constants were then used to derive the bulk modulus, shear moduli C44, and C’= ;(CllClz) and the anisotropy factor. The results were also used to interpret solid solution hardening in tantalum-base alloys.
INTRODUCTION
It is often useful to know the concentration and temperature dependence of the elastic constants. Unfortunately these are not available for b.c.c. metals, particularly transition metals. Hitherto, estimates of the concentration dependence of the elastic constants have been obtained by linear extrapolation from the values for the pure metals or, in some cases, estimated from measurements on polycrystalline samples. In the latter case not all the elastic constants can be obtained and, in any case, polycrystalline moduli are easily influenced by a residual texture in the rod. The present work describes the results of an investigation into the concentration and temperature dependence of the elastic moduli of the tantalum-base alloys TaRe, TaW, TaMo and TaNb. EXPERIMENTAL
(a) Preparation
PROCEDURE
,4ND SPECIMES
PREPARATIOh-
of the alloy>
Tantalum and the alloying element, both in the form of powder of about 1000 ,um, were mixed, compressed into pellets and melted into an ingot in a R.F. heating furnace (silver boat technique)l. The melting operation was carried out three or four times until a vacuum of IO -5 torr could be maintained. Vacuum melting largely removed the interstitial impurities, after which the ingots could be cold swaged to 4 mm diam. rod without intermediate annealing. The rod was cut into lengths 12 cm long and an X-ray fluorescent analysis carried out on solutions prepared from samples taken from each end of the rod. If the concentration varied by more than 3% the rods were remelted. J. Less-Common
Metals.
22
(1970)
265- ~74
266
D. A. ARMSTRONG,
B. 1;. MORDIKE
(b) Specimelz pre$aratiolz
Single crystals were grown from the swaged rods by electron-beam zone melting. After an outgassing treatment, one zone was passed at a speed of I cm/min. The pressure was usually - 1.10-5 torr. Seed crystals and a tiltable top chuck were used to ensure that the correct crystal orientation was obtained. (OOI), (011) and (III) oriented crystals were prepared. Close control of the melting conditions was necessary to ensure straight, non-elliptical crystals. Samples of the completed specimens were analysed to obtain the final solute concentration. Representative samples were also analysed for interstitial content. The alloys used and the interstitial content are shown in Tables I and II. The interstitial content given is typical for all crystals, irrespective of substitutional element concentration. The precise orientation of the specimen was determined, and hence its orientation factor, H=c@/P+~~y2y2+j392, where M, ,8 and ^J are the direction cosines of the rod axis. TABLE SOLUTE
I CONCENTR.ATION
Solute
At.%
Molybdenum
1.35 3.4 4.7 2.2
Tungsten
4.25 2.3 3.8 5.3 3.95 8.25
Rhenium
Niobium
TABLE
II
INTERSTITIAL
CONCENTRATION
$x&m. by wt.
Element
40 15 2
Oxygen Nitrogen Hydrogen Carbon
2.5
(c) Merssztrement of the resolzalzt frequencies The elastic constants were determined using the thin rod resonance technique2. The equipment used for the room temperature, low temperature, and high temperature measurements was similar in principle but differed in detail due to the difficulties associated with control of temperature, oxidation, spurious resonances, damping etc. The apparatus will be described elsewherea. The elastic constants were first determined at i-oom temperature. Theresonant frequencies are related to the elastic moduli. In the case of ~oly~ystal~ine specimens, the longitudinal resonances yield values of Young’s modulus, E, and the torsional resonant frequencies yield the shear modulus, G. For single crystals, the resonant J. Less-Common
Metals,
2.z (Igyro)
265-274
ELASTIC
COKSTANTS
frequencies
207
OF TANTALUM
are dependent
on the orientation.
The apparent
shear moduli are given by the general expressions
or effective
Young’s
and
(I) and (2).
Longitudinal E
__!!Y__
=
I 1)
n”Kn ( I +&AT)
Torsional (‘
-cW?
_
T
(2)
___
~~(1 +cxAT)
where K, = I - (n’n~~zd2812),listhelength of thespecimen, measuredatroomtemperature To,.f the resonant frequency, Q the density, 01 the average coefficient of thermal expansion, AT the temperature of measurement T, -To, 12the harmonic, ,L[Poisson’s ratio, and d the diameter. The effective moduli expressions I -;-=L
bII
-
I
(k
are related
to the elastic
compliance
constants
So 1~~
(3) and (4)
=
s11- iz(S,,-S,~)-S44]H
(3)
.s44+2[2(s11-.51~)
(4)
-S44]H.
It can be seen that
is a constant independent of orientation. This provides a consistency check between different rods before eqns. (3) and (4) are solved. The (001) crystals yield Sii and 54 directly since H=o. The (011) and (III) crystals can be used to obtain independent values of Sii and S44 by using the two E values or G values, respectively. Alternatively, Sii can be obtained from 2G and an E value, and S44 from 2E and a G value Average values for Sii and S44 are obtained which are used with EH, GH from non (001) crystals to calculateSi2. The concentration dependence of the Sij was plotted at room temperature for all the alloy systems. The best straight line was drawn through the points originating from the values for pure tantalum 4. The concentration dependence at room temperature was confirmed by using additional alloys to those quoted in Table I. The values of Sii, SE and .S44 lying on the straight line were read off at the concentrations of the alloys studied at high and low temperatures. At high- and low temperatures it is not practical to undertake a large number of measurements and average as was done at room temperature. The temperature dependence of E and G will, however, be more accurately reproduced by a particular crystal than the absolute value. The high- and low temperature results were obtained by determining the temperature dependence of Sii, SE and S44 and displacing the curve where necessary to pass through the well substantiated room-temperature value. A particular resonance was identified at room temperature and tracked from room temperature to high temperature and back to room temperature. Otherwise, the problems of distinguishing between spurious and specimen resonances could have
268
D.A. ARMSTRONG,
B. L. MORDIKE
been unsurmountable. Identification of low-temperature resonances was not particularly difficult. The frequency of the resonance was measured at regular tem~rature intervals, from which a plot of IIEH and I/GH against T can be plotted. For (001) crystals this was the desired temperature dependence of Sri and S44. A plot of c=
i
;;+If Pcrw
against temperature was prepared for each orientation (Fig. I). These were compared and if they exhibited the same temperature dependence, i.e., the crystals were consistent, then the Sii (T) and!%4 (T) obtained from the (001) crystal were pinned to the room-temperature value. The non (001) crystals were used to evaluate Siz(l’). &2(T) was again pinned at the room temperature value. All the alloys were investigated below room temperature whereas only the highest concentration ahoys were studied at high temperatures. In the case of Ta-Re only low temperature results are available.
0
500
000
IS00
(“Kl Fig. 1.Typical consistency plot, C= I/EH+ I/ZGN, for a Ta-q.7Mo alloy. _
RESULTS
The values obtained at room temperature (zo’C) are given in Table III. (b) Low temperatures
The concentration dependence of Sal was plotted at several temperatures below room temperature when it became apparent that the concentration dependence was unaffected by change in temperature within the accuracy of measurement, Consequently, it was relatively easy to calculate the influence of alloying and temperature knowing the .&i(T) for pure tantalum6 and (d.Sij/dc)RTfor each alloy system.
Only one concentration for each system was studied. Sg, ‘OS.T for these systems and pure tantalum are plotted in Figs. 2, 3, 4. It cannot be assumed that the same (dSljldc)KT will obtain over such a wide temperature range, although a linear J. Less-Common Metals, 2s (1970) 265-274
EL.-\STIC CONST.-\STS OF TASTALUM TABLE ROOM For
II1
TEMPERATURE
high
and
low
VALUES
temperature
USED
AS PINNING
POINTS
measurements .S,j measured in 10-13 cm”/dyn( 10-12 m?/N)
I’ure tantalum
0.00
c).gor
1r.ooq
2.561
Srolybtlcnum
I.35
0.790 6.020 6.5’8
12.130 12.358 12.498
2.jlO 2.433 ~~2.381
4.2j
6.770 6.470
1r.o3o 12.060
~ 2.470 2.380
2.3 3.8
6.507 0.355
5.3
O.ILp
LI.912 II.858 II.803
2.4’5 ~ 2 323 r.rrg
3.95 8.25
6.790 6.658
12.480 11.995
3.4 4.7 Tungsten Rhenium
Niobium
2.2
-2.5ro -2.48j
1
Fig. 2.
SII as a function of temperature
relationship will still probably hold. Consequently the actual measured were used to evaluate the effect of alloying on the various constants.
values of S
DISCITSSIONS OF RESULTS
It is intended to present here primarily the influence of temperature on the elastic constants of the alloys rather than to discuss the influence of alloying at particular temperatures. For general convenience in discussion and use of the results
D. A. ARMSTRONG, B. L. MORDIKE
)
I5 )O -T[OK]
Fig. 3. $4 as a function of temperature.
0
Fig. 4. SU as a function of temperature. J. Less-Commo~z
Metals,
22
(1970)
265-274
ELASTIC CONSTANTS
OF TANTALUM
271
customary to use combinations of the stiffness coefficients which have a physical significance. The stiffness coefficients CQ were calculated from the compliance constants using the following relationships.
it is
c11=(s11+s12)/(s11-&)(s11+2.512)
(5)
c12=
(0)
-sl2/(sll-slz)(s11+zsl2)
(7)
c44=1/s44
the
The coefficient C44 represents the resistance to shear between direction. The other constants calculated were:
(100)
planes in
(010)
B =
c11+c12
3 C, =
Cll--Cl2
2
2c44 A
=
-the
bulk modulus indicating
resistance
to hydrostatic
compression -coefficient
(8)
indicating the resistance to shear between
planes in (rio) -anisotropy
directrons.
(110)
(9)
factor.
c11-Cl2
Figures 5, 6, 7, and 8, show the temperature dependence of B, C’, A and C~J for the high concentration alloys and pure tantalum. Other concentrations show a similar dependence. The shear moduli, C’ and C44, decrease continuously with temperature, the alloy curves running approximately parallel. The bulk modulus is reasonably constant up to 400°K and then decreases rapidly. The anisotropy first decreases with increasing temperature but increases again above -600°K.
Fig. 5. Cd4as a function of temperature.
Fig. 6. Variation of anisotropy with temperature. J. LeSS-&WW%3n
Metals.
22
(1970) 265-274
D. A. ARMSTRONG, B. L. MORDLKE
272
4.0
0
500
Fig. 7. Influence of temperature on C’.
Fig. 8. Influence of temperature on the bulk modulus, R.
Fig. 9. Correlation of elastic misfit parameter with the increase in flow stress for Ta alloys.
A search of the literature revealed no data on the influence of alloying on the elastic constants of tantalum or on the temperature dependence of the constants for pure tantalum, and hence no comparisons can be drawn. There is, as yet, insufficient experimental data to attempt to explain the eIastic constants in terms of the electronic structure of the transition metals. The anisotropy of b.c.c. transitionmetals is J.
Less-Common
Metals, 22 (‘970)
265-274
ELASTIC
CONSTANTS
OF
273
TANTALUM
very small compared with other cubic metals and is slightly dependent on temperature, which has been interpreted in terms of the antiferromagnetic model of the b.c.c. transition
metals5,6.
It is not, however,
possible to estimate
the absolute
magnitude
of the elastic constants. APPLICATION
TO
SOLID
SOLUTIOK
HARDENIh‘G
In solid solution hardening7 dislocations interact with the stress fields of solute atoms. The stress fields may result from the difference in size of the solute and solvent atoms or from the difference in elastic properties (elastic misfit) of the two atoms. Alloying changes the macroscopic elastic moduli and with it the anisotropy of the material, and it increases the flow stress compared with the pure metal by an amount, Ato. Azo depends upon the testing parameters, e.g., temperature and strain rate, as well as on the solute atom parameters, size difference, elastic misfit, its distribution and concentration. At high temperatures the increase in flow stress is due solely to elastic interactions between dislocations and solute atoms8 thus: Azo=A. I
G.F C’ db
e”=1)dc; E is a combination of the size misfit parameter, &b, and the modulus misfit parameter &G.C is the concentration, G the elastic modulus, b the lattice constant and A a constant N ro-3. For the alloy systems investigated the increase in flow stress on alloying was measured for a fixed concentration, C=3.5 at.“//o.A correlation of A70 with the various misfit parameters permits an estimate of the relative importance of the two elastic interactions. The size misfit parameter can be calculated from lattice parameter data in the literature. To calculate the elastic misfit parameter, .Sij from this investigation were used. However,
it is important
is theoretically
compatible
to make a suitable
with the physical
choice of modulus,
picture
i.e., one which
of the dislocation-solute
atom
interaction. It has been shown that the modulus K33 as defined by HEADS must be used for the interaction of a screw dislocation and a solute atom because this modulus determines the stress field of a screw dislocation in the b.c.c. lattice. For an isotropic crystal K33=C44. Other moduli have been used by other workerslO,ll and these can result in very different misfit parameters, even though the absolute values arc similar. A comparison of At0 and eG(K33) and misfit parameters based on other definitions of the modulus leads to the conclusion that only FG(K~~) can explain both the absolute and relative magnitude of solid solution hardening in tantalum by rhenium, tungsten, molybdenum and niobium. l’igure 9 shows the correlation between AZ, and the misfit parameter &G determined from the room temperature values of the elastic constants. A range of Azo is indicated to take into account any possible changes in the interstitial hardening due to alloying. The points for all four alloys lie on a straight line. &~(K33) is thus agood parameter to describe solid solution hardening. Since the line passes through the origin, the interaction between a solute atom is, in fact, purely elastic in the temperature range in which At” was measured. The correlation between Ato and the misfit parameter based on the elastic J. L~SS-CO~WON Metals,
22 (1970) 2G,j-27-1
D. A. ARMSTRONG,
274
B. L. MORDIKE
constants measured at 1200’K is also given in Fig. g. To obtain these values it was necessary to assume a linear concentration dependence of St, for the values of EG(&) at C2: 3.5at.% to be calculated. The results show that the use of room-temperature values in the past has been justified. REFERENCES I G. RUDOLPHANDB.L.MORDIKE,Z. Metallk.,58(1967)708. 2 F. FGRSTER,Z.Metallk., 29 (1937) log. 3 D.A. ARMSTRONGANDB.L.MORDIKE,~II~~~~~. 4 D.A.ARMSTRONGANDB.L.MORDIKE,Z. Metallk. ,61(1g7o)inpress. =jD. I. BOLEF,J.A#. Phys., 32 (1961) IOO. 6 I. ISENBERG,Phys. Rev., 83 (1951) 637. 7 P. HAASEN,in R. W. CAHU (ed.), Physical Metallurgy, North HollandPub. Co.,Amsterdam.
Ig65,Chapter 17. 8 B.L.MORDIKE AND K.D. RoGAuscH,J.M~~~v..SC~.,~(I~~~)~~. g A K HEAD, Phys StatusSolidi, 6 (1964) 461 IO G. KOSTORZ,Z. Metallk., 59 (1968) 914. II T. E. MITCHELLANDP.L.RAFFO,C~~.J. Phys.45(‘967) 1047. J. Less-Common
Metals, 2.2 (1970) 265-274
of the Less-Common
Elsevier
Sequoia
Metals
S.A., Lausanne
- Printed
265
in The Netherlands
THE INFLUENCE OF ALLOYING AND TEMPERATURE ELASTIC CONSTANTS OF TANTALUM
Ll. A.
;ZKMSTRONG
Ckpartment of Metalluvgy (Received
AND
B.
ON THE
I,. MORDIKE
and Materials
S&me.
Ulziversity
of Liverpool,
Liverpool
(Gt. Britain)
April 18th, 1970)
The elastic compliance constants of TaRe, TaW, TaMo and TaNb single crystals have been determined over the temperature range So0-~4000K. These constants were then used to derive the bulk modulus, shear moduli C44, and C’= ;(CllClz) and the anisotropy factor. The results were also used to interpret solid solution hardening in tantalum-base alloys.
INTRODUCTION
It is often useful to know the concentration and temperature dependence of the elastic constants. Unfortunately these are not available for b.c.c. metals, particularly transition metals. Hitherto, estimates of the concentration dependence of the elastic constants have been obtained by linear extrapolation from the values for the pure metals or, in some cases, estimated from measurements on polycrystalline samples. In the latter case not all the elastic constants can be obtained and, in any case, polycrystalline moduli are easily influenced by a residual texture in the rod. The present work describes the results of an investigation into the concentration and temperature dependence of the elastic moduli of the tantalum-base alloys TaRe, TaW, TaMo and TaNb. EXPERIMENTAL
(a) Preparation
PROCEDURE
,4ND SPECIMES
PREPARATIOh-
of the alloy>
Tantalum and the alloying element, both in the form of powder of about 1000 ,um, were mixed, compressed into pellets and melted into an ingot in a R.F. heating furnace (silver boat technique)l. The melting operation was carried out three or four times until a vacuum of IO -5 torr could be maintained. Vacuum melting largely removed the interstitial impurities, after which the ingots could be cold swaged to 4 mm diam. rod without intermediate annealing. The rod was cut into lengths 12 cm long and an X-ray fluorescent analysis carried out on solutions prepared from samples taken from each end of the rod. If the concentration varied by more than 3% the rods were remelted. J. Less-Common
Metals.
22
(1970)
265- ~74
266
D. A. ARMSTRONG,
B. 1;. MORDIKE
(b) Specimelz pre$aratiolz
Single crystals were grown from the swaged rods by electron-beam zone melting. After an outgassing treatment, one zone was passed at a speed of I cm/min. The pressure was usually - 1.10-5 torr. Seed crystals and a tiltable top chuck were used to ensure that the correct crystal orientation was obtained. (OOI), (011) and (III) oriented crystals were prepared. Close control of the melting conditions was necessary to ensure straight, non-elliptical crystals. Samples of the completed specimens were analysed to obtain the final solute concentration. Representative samples were also analysed for interstitial content. The alloys used and the interstitial content are shown in Tables I and II. The interstitial content given is typical for all crystals, irrespective of substitutional element concentration. The precise orientation of the specimen was determined, and hence its orientation factor, H=c@/P+~~y2y2+j392, where M, ,8 and ^J are the direction cosines of the rod axis. TABLE SOLUTE
I CONCENTR.ATION
Solute
At.%
Molybdenum
1.35 3.4 4.7 2.2
Tungsten
4.25 2.3 3.8 5.3 3.95 8.25
Rhenium
Niobium
TABLE
II
INTERSTITIAL
CONCENTRATION
$x&m. by wt.
Element
40 15 2
Oxygen Nitrogen Hydrogen Carbon
2.5
(c) Merssztrement of the resolzalzt frequencies The elastic constants were determined using the thin rod resonance technique2. The equipment used for the room temperature, low temperature, and high temperature measurements was similar in principle but differed in detail due to the difficulties associated with control of temperature, oxidation, spurious resonances, damping etc. The apparatus will be described elsewherea. The elastic constants were first determined at i-oom temperature. Theresonant frequencies are related to the elastic moduli. In the case of ~oly~ystal~ine specimens, the longitudinal resonances yield values of Young’s modulus, E, and the torsional resonant frequencies yield the shear modulus, G. For single crystals, the resonant J. Less-Common
Metals,
2.z (Igyro)
265-274
ELASTIC
COKSTANTS
frequencies
207
OF TANTALUM
are dependent
on the orientation.
The apparent
shear moduli are given by the general expressions
or effective
Young’s
and
(I) and (2).
Longitudinal E
__!!Y__
=
I 1)
n”Kn ( I +&AT)
Torsional (‘
-cW?
_
T
(2)
___
~~(1 +cxAT)
where K, = I - (n’n~~zd2812),listhelength of thespecimen, measuredatroomtemperature To,.f the resonant frequency, Q the density, 01 the average coefficient of thermal expansion, AT the temperature of measurement T, -To, 12the harmonic, ,L[Poisson’s ratio, and d the diameter. The effective moduli expressions I -;-=L
bII
-
I
(k
are related
to the elastic
compliance
constants
So 1~~
(3) and (4)
=
s11- iz(S,,-S,~)-S44]H
(3)
.s44+2[2(s11-.51~)
(4)
-S44]H.
It can be seen that
is a constant independent of orientation. This provides a consistency check between different rods before eqns. (3) and (4) are solved. The (001) crystals yield Sii and 54 directly since H=o. The (011) and (III) crystals can be used to obtain independent values of Sii and S44 by using the two E values or G values, respectively. Alternatively, Sii can be obtained from 2G and an E value, and S44 from 2E and a G value Average values for Sii and S44 are obtained which are used with EH, GH from non (001) crystals to calculateSi2. The concentration dependence of the Sij was plotted at room temperature for all the alloy systems. The best straight line was drawn through the points originating from the values for pure tantalum 4. The concentration dependence at room temperature was confirmed by using additional alloys to those quoted in Table I. The values of Sii, SE and .S44 lying on the straight line were read off at the concentrations of the alloys studied at high and low temperatures. At high- and low temperatures it is not practical to undertake a large number of measurements and average as was done at room temperature. The temperature dependence of E and G will, however, be more accurately reproduced by a particular crystal than the absolute value. The high- and low temperature results were obtained by determining the temperature dependence of Sii, SE and S44 and displacing the curve where necessary to pass through the well substantiated room-temperature value. A particular resonance was identified at room temperature and tracked from room temperature to high temperature and back to room temperature. Otherwise, the problems of distinguishing between spurious and specimen resonances could have
268
D.A. ARMSTRONG,
B. L. MORDIKE
been unsurmountable. Identification of low-temperature resonances was not particularly difficult. The frequency of the resonance was measured at regular tem~rature intervals, from which a plot of IIEH and I/GH against T can be plotted. For (001) crystals this was the desired temperature dependence of Sri and S44. A plot of c=
i
;;+If Pcrw
against temperature was prepared for each orientation (Fig. I). These were compared and if they exhibited the same temperature dependence, i.e., the crystals were consistent, then the Sii (T) and!%4 (T) obtained from the (001) crystal were pinned to the room-temperature value. The non (001) crystals were used to evaluate Siz(l’). &2(T) was again pinned at the room temperature value. All the alloys were investigated below room temperature whereas only the highest concentration ahoys were studied at high temperatures. In the case of Ta-Re only low temperature results are available.
0
500
000
IS00
(“Kl Fig. 1.Typical consistency plot, C= I/EH+ I/ZGN, for a Ta-q.7Mo alloy. _
RESULTS
The values obtained at room temperature (zo’C) are given in Table III. (b) Low temperatures
The concentration dependence of Sal was plotted at several temperatures below room temperature when it became apparent that the concentration dependence was unaffected by change in temperature within the accuracy of measurement, Consequently, it was relatively easy to calculate the influence of alloying and temperature knowing the .&i(T) for pure tantalum6 and (d.Sij/dc)RTfor each alloy system.
Only one concentration for each system was studied. Sg, ‘OS.T for these systems and pure tantalum are plotted in Figs. 2, 3, 4. It cannot be assumed that the same (dSljldc)KT will obtain over such a wide temperature range, although a linear J. Less-Common Metals, 2s (1970) 265-274
EL.-\STIC CONST.-\STS OF TASTALUM TABLE ROOM For
II1
TEMPERATURE
high
and
low
VALUES
temperature
USED
AS PINNING
POINTS
measurements .S,j measured in 10-13 cm”/dyn( 10-12 m?/N)
I’ure tantalum
0.00
c).gor
1r.ooq
2.561
Srolybtlcnum
I.35
0.790 6.020 6.5’8
12.130 12.358 12.498
2.jlO 2.433 ~~2.381
4.2j
6.770 6.470
1r.o3o 12.060
~ 2.470 2.380
2.3 3.8
6.507 0.355
5.3
O.ILp
LI.912 II.858 II.803
2.4’5 ~ 2 323 r.rrg
3.95 8.25
6.790 6.658
12.480 11.995
3.4 4.7 Tungsten Rhenium
Niobium
2.2
-2.5ro -2.48j
1
Fig. 2.
SII as a function of temperature
relationship will still probably hold. Consequently the actual measured were used to evaluate the effect of alloying on the various constants.
values of S
DISCITSSIONS OF RESULTS
It is intended to present here primarily the influence of temperature on the elastic constants of the alloys rather than to discuss the influence of alloying at particular temperatures. For general convenience in discussion and use of the results
D. A. ARMSTRONG, B. L. MORDIKE
)
I5 )O -T[OK]
Fig. 3. $4 as a function of temperature.
0
Fig. 4. SU as a function of temperature. J. Less-Commo~z
Metals,
22
(1970)
265-274
ELASTIC CONSTANTS
OF TANTALUM
271
customary to use combinations of the stiffness coefficients which have a physical significance. The stiffness coefficients CQ were calculated from the compliance constants using the following relationships.
it is
c11=(s11+s12)/(s11-&)(s11+2.512)
(5)
c12=
(0)
-sl2/(sll-slz)(s11+zsl2)
(7)
c44=1/s44
the
The coefficient C44 represents the resistance to shear between direction. The other constants calculated were:
(100)
planes in
(010)
B =
c11+c12
3 C, =
Cll--Cl2
2
2c44 A
=
-the
bulk modulus indicating
resistance
to hydrostatic
compression -coefficient
(8)
indicating the resistance to shear between
planes in (rio) -anisotropy
directrons.
(110)
(9)
factor.
c11-Cl2
Figures 5, 6, 7, and 8, show the temperature dependence of B, C’, A and C~J for the high concentration alloys and pure tantalum. Other concentrations show a similar dependence. The shear moduli, C’ and C44, decrease continuously with temperature, the alloy curves running approximately parallel. The bulk modulus is reasonably constant up to 400°K and then decreases rapidly. The anisotropy first decreases with increasing temperature but increases again above -600°K.
Fig. 5. Cd4as a function of temperature.
Fig. 6. Variation of anisotropy with temperature. J. LeSS-&WW%3n
Metals.
22
(1970) 265-274
D. A. ARMSTRONG, B. L. MORDLKE
272
4.0
0
500
Fig. 7. Influence of temperature on C’.
Fig. 8. Influence of temperature on the bulk modulus, R.
Fig. 9. Correlation of elastic misfit parameter with the increase in flow stress for Ta alloys.
A search of the literature revealed no data on the influence of alloying on the elastic constants of tantalum or on the temperature dependence of the constants for pure tantalum, and hence no comparisons can be drawn. There is, as yet, insufficient experimental data to attempt to explain the eIastic constants in terms of the electronic structure of the transition metals. The anisotropy of b.c.c. transitionmetals is J.
Less-Common
Metals, 22 (‘970)
265-274
ELASTIC
CONSTANTS
OF
273
TANTALUM
very small compared with other cubic metals and is slightly dependent on temperature, which has been interpreted in terms of the antiferromagnetic model of the b.c.c. transition
metals5,6.
It is not, however,
possible to estimate
the absolute
magnitude
of the elastic constants. APPLICATION
TO
SOLID
SOLUTIOK
HARDENIh‘G
In solid solution hardening7 dislocations interact with the stress fields of solute atoms. The stress fields may result from the difference in size of the solute and solvent atoms or from the difference in elastic properties (elastic misfit) of the two atoms. Alloying changes the macroscopic elastic moduli and with it the anisotropy of the material, and it increases the flow stress compared with the pure metal by an amount, Ato. Azo depends upon the testing parameters, e.g., temperature and strain rate, as well as on the solute atom parameters, size difference, elastic misfit, its distribution and concentration. At high temperatures the increase in flow stress is due solely to elastic interactions between dislocations and solute atoms8 thus: Azo=A. I
G.F C’ db
e”=1)dc; E is a combination of the size misfit parameter, &b, and the modulus misfit parameter &G.C is the concentration, G the elastic modulus, b the lattice constant and A a constant N ro-3. For the alloy systems investigated the increase in flow stress on alloying was measured for a fixed concentration, C=3.5 at.“//o.A correlation of A70 with the various misfit parameters permits an estimate of the relative importance of the two elastic interactions. The size misfit parameter can be calculated from lattice parameter data in the literature. To calculate the elastic misfit parameter, .Sij from this investigation were used. However,
it is important
is theoretically
compatible
to make a suitable
with the physical
choice of modulus,
picture
i.e., one which
of the dislocation-solute
atom
interaction. It has been shown that the modulus K33 as defined by HEADS must be used for the interaction of a screw dislocation and a solute atom because this modulus determines the stress field of a screw dislocation in the b.c.c. lattice. For an isotropic crystal K33=C44. Other moduli have been used by other workerslO,ll and these can result in very different misfit parameters, even though the absolute values arc similar. A comparison of At0 and eG(K33) and misfit parameters based on other definitions of the modulus leads to the conclusion that only FG(K~~) can explain both the absolute and relative magnitude of solid solution hardening in tantalum by rhenium, tungsten, molybdenum and niobium. l’igure 9 shows the correlation between AZ, and the misfit parameter &G determined from the room temperature values of the elastic constants. A range of Azo is indicated to take into account any possible changes in the interstitial hardening due to alloying. The points for all four alloys lie on a straight line. &~(K33) is thus agood parameter to describe solid solution hardening. Since the line passes through the origin, the interaction between a solute atom is, in fact, purely elastic in the temperature range in which At” was measured. The correlation between Ato and the misfit parameter based on the elastic J. L~SS-CO~WON Metals,
22 (1970) 2G,j-27-1
D. A. ARMSTRONG,
274
B. L. MORDIKE
constants measured at 1200’K is also given in Fig. g. To obtain these values it was necessary to assume a linear concentration dependence of St, for the values of EG(&) at C2: 3.5at.% to be calculated. The results show that the use of room-temperature values in the past has been justified. REFERENCES I G. RUDOLPHANDB.L.MORDIKE,Z. Metallk.,58(1967)708. 2 F. FGRSTER,Z.Metallk., 29 (1937) log. 3 D.A. ARMSTRONGANDB.L.MORDIKE,~II~~~~~. 4 D.A.ARMSTRONGANDB.L.MORDIKE,Z. Metallk. ,61(1g7o)inpress. =jD. I. BOLEF,J.A#. Phys., 32 (1961) IOO. 6 I. ISENBERG,Phys. Rev., 83 (1951) 637. 7 P. HAASEN,in R. W. CAHU (ed.), Physical Metallurgy, North HollandPub. Co.,Amsterdam.
Ig65,Chapter 17. 8 B.L.MORDIKE AND K.D. RoGAuscH,J.M~~~v..SC~.,~(I~~~)~~. g A K HEAD, Phys StatusSolidi, 6 (1964) 461 IO G. KOSTORZ,Z. Metallk., 59 (1968) 914. II T. E. MITCHELLANDP.L.RAFFO,C~~.J. Phys.45(‘967) 1047. J. Less-Common
Metals, 2.2 (1970) 265-274