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On the evaluation of the deformation kinetics in titanium using stress relaxation
201
Journal of the Less-Common Metals, 34 (1974) 201-207 c; Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
ON THE EVALUATION OF THE DEFORMATION TITANIUM USING STRESS RELAXATION
S. P. AGRAWAL,
G. A. SARGENT
Department of Metallurgical Ky 40506 (U.S.A.) (Received
KINETICS
IN
and H. CONRAD
Engineering
and Materials
Science,
CJniaersit,v CT)’Kenlucky.
Lexington.
July 24, 1973)
SUMMARY
The theoretical considerations in the conventional analysis using stress relaxation technique to determine deformation kinetics of metals are critically examined. It is shown that the underlying assumptions are not well obeyed in cc-titanium above room temperature. It is suggested, therefore, that the conventional approach may not be used in titanium in the entire temperature range up to three-tenths its melting point. Alternatively, a more direct approach to obtain the activation parameters is briefly outlined.
INTRODUCTION
The mechanical properties of materials are generally studied by two main experimental techniques. The most commonly employed is the tensile test in which the uniaxial load on the specimen is measured as a function of elongation at a constant crosshead speed or nominal strain rate. An alternative technique is the creep test in which the elongation of the specimen is measured as a function of time at constant load or stress. Occasionally, however, an alternative mechanical test has been utilized, the “stress relaxation test”, in which the drop in load is measured as a function of time at constant total strain. In this test, the load, strain rate and plastic strain are all varying. However, to a first approximation the plastic strain may be considered to be constant. Felthamlm3 was the first to use the stress relaxation method to evaluate the activation parameters, such as the activation volume and the activation energy, for the plastic flow in several metals in order to study the kinetics of deformation. He was able to do so by establishing an empirical logarithmic relationship between the relaxation in tensile stress and time, which is given as1 00
-
CT =
Sre, log1dl +vt)
where o0 is the initial value of the tensile stress, CJ the value of tensile stress at time t, v a constant independent of time and S,,, a characteristic time-independent parameter of the relaxation, given as Sre, = -do/d
log, ,, t .
S. P.
202
AGRAWAL,
G. A. SARGENT,
H. CONRAD
His analysis, however, did not take the elastic deformation of the tensile machine, which can be considerable, into account. This and a few other modi~cations, such as including the effect of work hardening on the activation parameters (which was shown to be less than 1% of the total value, hence negligible), were introduced into Feltham’s analysis in the research to follow4-8. However, the empirical logarithmic relationship between the relaxation in stress and time was still assumed in all of the above treatments. Also assumed therein was a linear dependence of the activation energy upon the effective shear stress acting on the dislocation, as originally proposed by Seeger’ and Thornton and HirschlO and given by AGMAG~-v~*
(3)
where AG is the Gibbs free-energy of activation ( = AH - TAS where AH is the enthalpy of activation, T the temperature and AS the entropy of activation), AC, is the total work required to bring the system into the activated state and is equal to. that contributed during the activation event by the externally applied force pius the excess mechanical work required to bring the system isothermally and reversibly into the activated state. u is the activation volume, defined as
z* is the effective shear stress (= r- tcl, where r is the applied shear stress and zP the athermal component of the flow stress). It is the purpose of this note to show that although these assumptions, namely eqn. 3, seem reasonable for several metals’-*, usually at low temperatures ( T< 300 K) where z* is large so that v is relatively independent of r*, they are not valid in ~-titanium at temperatures above 300 K. Rather, a more direct approach to obtain the activation parameters which has been described previously”, is more appropriate. 1.1. THEORETICAL CONSIDERATIONS STRESS RELAXATION
IN
THE
CONVENTIONAL
ANALYSIS
USING
It is now generally accepted that the shear strain rate for a thermally activated deformation process in metals can be given as
j=jeexp
-g C > #here POis the pre-exponential factor and is related to the number of dislocation elements per unit volume contributing to the deformation, the area swept out by one such element after activation, the Burgers vector of these dislocations and an atomic frequency. AG is the Gibbs free energy of activation, k the Boltzmann’s constant and T the temperature. Feltham’ assumed that the work done by the effective stress on the dislocation segment involved in thermal activation decreases the activation energy linearly, as was first proposed by Seeger’ and Thornton and Hirsch”, and is expressed by eqn. 3. Upon differentiating both sides of eqn. 3 with respect to the effective shear stress, z*, at constant temperature one obtains (5)
DEFORMATION
KINETICS
203
IN Ti
The definition of the activation workers ” - l5 is
volume U* employed by Conrad and co-
JAG r ‘*=
-
az* =’
Further, the two activation volumes v and v* represent the same quantity for: (i) constant dislocation segment length, I, whereupon U*=u= tbx, where x is the activation distance and (ii) the case when the obstacles are randomly dispersed, whereupon u* = v =$lbx. One therefore obtains from eqns. 5 and 6 gf?.
1,-T;; /,‘O.
(7)
It can now be shown that both the quantities on the left hand side of eqn, 7 individually approach zero only under certain conditions: (a) the first quantity, aAG,/dr*l., when approximated to zero assumes that the total work required to bring the system into the activated state is independent of the effective stress at any given temperature. In other words, the total area under the force-activation distance curve at a given temperature is independent of the effective stress, or the force-activation distance curve is independent of the effective stress. This has been shown to be a thermodynamically consistent assumption for titaniumi2. 14.i5. (b) the second quantity, z,*(av/&*)l,, may be approximated to zero when either of the following two conditions is met.
au 1
is linite. 6) TT*-OandIv aT*T However, as z+O,
(3
Az*+O and u-+co. Thus, an indeterminate
solution is obtained.
-&/T~0 and Z$ is finite.
This condition is satisfactorily met at low temperatures where r? is of appreciable magnitude and AZ* >O but Av+O. In other words, at low temperatures, where T* is large, u tends to be independent of z* and one obtains a constant value of u over a range of T* (usually corresponding to 7’~ 300 K). It can now clearly be seen that eqn. 3 was found to be valid in the conventional analysis’ on account of the fact that au/&*/, 40, for a finite value of T; for T< 300 K. It is also obvious that for dv/Sz*l,+O, other conditions remaining the same, eqn. 3 cannot be justified. The latter is usually the case in most metals at higher temperatures (T > 300 K) since a small change in z* results in an appreciable change in v. 1.2. EXPERIMENTAL
QUANTITIES
From eqns. (4) and (6) one can derive directly
204
S. P. AGRAWAL,
aAG lJ*= -az*
,&9 Or, upon
=kT
a ln V/i0
az*
T
considering
the total applied
H. CONRAD
T
(f. = con&.).
az* T
G. A. SARGENT,
(84 stress, r, rather
than the effective stress r*,
(& = const)
where r~ is the total tensile stress and M the Taylor’s factor, such that o=Mr. In agreement with the condition (b) (ii) in the previous section, 0 was indeed found to vary linearly with In 8 in conventional strain rate cycling tests16 at low temperatures (Fig. l), thus resulting in a value of the slope
which is constant with cr, i.e., av/&r=O. Also, shown in Fig. 1 is the non-linear behavior of G with In i: at higher temperatures using the same testing technique as abover6. In this case, (a In 6)/&l, is varying with c and one obtains a significant dependence of the activation volume, v, on 0, which can be described by the following equation.
In stress relaxation the stress decreases with increasing time, i.e., A(T is negative, and the strain may be considered constant to a first approximation. The quantity which is conventionally obtained experimentally in this case is (a In t)/&l, and is related to the activation volume as follows’. V
MkT
=-
a In i: ao
T
=--
a In t a0 T’
(11)
Similar to the relationship observed between c and In i at low temperatures in Fig . 12it was observed’ ’ that during stress relaxation tests CJdecreased linearly with In t at low temperatures (Fig. 2). Also shown in Fig. 2 is the a-ln t relationship at high temperatures, which is clearly non-linear: an observation analogous to that at similar temperatures in strain ratechange tests. Thus, one obtains a value of v independent of g (au/&r = 0) at low temperatures and varying with CJ(au/% # 0) at high temperatures. This further reinforces the earlier conclusion that the constancy of volume condition (condition (b) (ii) in the previous section) may be met in a-titanium only at low temperatures. It was therefore concluded that the conventional approach of determining v* was not appropriate in the entire range of temperature (below 0.3 T,) in or-titanium.
DEFORMATION
205
KINETICS IN Ti
Data from Monteiro,Sonthonam Reed- H%II o-T~fCommercial Pwtyl
and
Fig. 1. Flow stress W. log strain rate for a-titanium (commercial purity)plate. Stress (in psi units) - log strain rate data for these temperatures obtained from Monteiro et al.“.
Fig. 2. Stress vs. log time during stress relaxation tests on cc-titanium (0.2 at.% O,,) wire.
2.1. A MORE DIRECT APPROACH FOR THE DETERMINATION
OF v AND AG
The relationship between stress and strain which produces the relaxation curve is determined by the stiffness of the tensile testing machine and the elastic modulus~ of the specimen. It has been shown 4.5. l1 that the rate of plastic deformation during relaxation can be related to the rate of relaxation itself as follows. $=-KC%.
(121
Here K is a constant which includes the specimen geometry and the specimen and the machine modulii; thus, it is equivalent to the combined modulii of the
S. P.
206
AGRAWAL,
G. A. SARGENT,
H. CONRAD
machine and the specimen. A measure of K can be obtained directly from the slope of the elastic part of the stress-strain curve and, thus, the instantaneous strain rate during stress relaxation can be determined from the rate of decrease in stress. Further, on substituting for i: from eqn. 12 into eqn. (9(a)) one obtains (13)
where M is the Taylor factor relating shear stress and strain to the tensile counterparts and is considered to be constant in the entire range of stress. Thus, simply by measuring the slopes and stresses at two times along the stress relaxation curve, the activation volume can be derived directly from the stress relaxation curve. This approach was employed in evaluating the thermal activation parameters for the plastic flow of cc-titanium at low temperatures (below 0.3 T ) in a separate paper l1 . Figure 3 shows that the Gibbs free energy of acti:ation for plastic flow derived from the stress relaxation tests using this more direct approach (Fig. 3(a)) is in good accord with that obtained from the more usual method of strain rate cycling tests (Fig. 3(b)); data for the strain rate cycling tests were obtained from Conrad and Jones”. The equations for AG” (corresponding to the case where the force-activation distance curve is independent of temperature, in addition to being independent of the applied stress) and AQ (corresponding to the case where the force activation distance barrier is propor2.0
I Stress
I
I
I
I
a-Ti(0.2
2.
I
1 St&
a
Relaxation
a-Ti
at.%Oeql
G.S.=5-42pm.
Cyclikg at.%
b
Oeq)
= 2.5/m
i=3.3xl0-4sec-’ I.
1.5
a
(0.2
G.S.
i=3.3x10-4sec-I
7 0 ”
Rate
> 3 ”
1.0
I.1
a
0.
0.5
0
I
400
200
600
I
I
200
400
Fig. 3. (b) AG” and obtained from Conrad
I 600
T(“K)
T(“K)
Fig. 3. (a) AG” and At?’ us. temperature
I
from stress relaxation
At? vs. temperature and Jones”.
from
strain
rate
tests. cycling
tests.
Stress-temperature
data
DEFORMATION
KINETICS
IN Ti
207
tional to the shear modulus p) used here were those derived by deMeester et ~1.‘~ and Schoeck’*. C,, (ref. 19) was chosen as the appropriate modulus to compute AG’ since it had been previously established I5 that the plastic deformation of polycrystalline titanium was controlled by slip on the {lOTO} (1120) system. ACKNOWLEDGEMENTS
The authors acknowledge financial support for this investigation by the Qffice of Aerospace Research, United States Air Force, under Contract F 33615-69C1027, A. Adair, technical monitor.
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
P. Feltham, Phil. Msg., 6 (1961) 259. P. Feltham, Phil. Msg., 6 (1961) 847. P. Feltham, Phil. Msg., 8 (1963) 989. F. Guiu and P. L. Pratt, Phys. Status Solidi, 6 (1964) 111. G. A. Sargent, Acta Met., 13 (1965) 663. R. B. Clough, Phys. Status Solidi, 17 (1966) K175. G. A. Sargent and H. Conrad, Scripta Met., 3 (1969) 43. R. B. Clough and L. J. Demer, Phys. Status Solidi, 36 (1969) 221. A. Seeger, Bristol Conf. Defects Crystalline Solids, The Physical Society, London, 1955, p. 328. P. R. Thornton and P. B. Hirsch, Phil. Mag., 3 (1958) 738. S. P. Agrawal, G. A. Sargent and H. Conrad, Met. Trans., in press. B. deMeester, M. Doner and H. Conrad, Dept. Met. Eng. Mater. Sci., Univ. Kentucky, Lexington, Ky., unpublished research, 1972. H. Conrad, Can. .I. Phys., 45 (1967) 581. B. delvleester, C. Yin, M. Doner and H. Conrad, John E. Dorn Mem. Symp., Rate processes in plastic deformation, Cleveland, Oct. 16 and 17, 1972, in press. K. Okazaki and H. Conrad, Acta Met., in press. S. N. Monteiro, A. T. Santhanam and R. E. Reed-Hill, The Science, Technology and Application of Titanium, Pergamon, Oxford, 1970, p. 503. H. Conrad and R. Jones, The Science, Technology and Application of Titanium, Pergamon, Oxford, 1970, p. 489. G. Schoeck, Phys. Status Solidi, 8 (1965) 499. E. S. Fisher and C. J. Renken, Phys. Rev., 135 No. 2A (1964) A482.
Journal of the Less-Common Metals, 34 (1974) 201-207 c; Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
ON THE EVALUATION OF THE DEFORMATION TITANIUM USING STRESS RELAXATION
S. P. AGRAWAL,
G. A. SARGENT
Department of Metallurgical Ky 40506 (U.S.A.) (Received
KINETICS
IN
and H. CONRAD
Engineering
and Materials
Science,
CJniaersit,v CT)’Kenlucky.
Lexington.
July 24, 1973)
SUMMARY
The theoretical considerations in the conventional analysis using stress relaxation technique to determine deformation kinetics of metals are critically examined. It is shown that the underlying assumptions are not well obeyed in cc-titanium above room temperature. It is suggested, therefore, that the conventional approach may not be used in titanium in the entire temperature range up to three-tenths its melting point. Alternatively, a more direct approach to obtain the activation parameters is briefly outlined.
INTRODUCTION
The mechanical properties of materials are generally studied by two main experimental techniques. The most commonly employed is the tensile test in which the uniaxial load on the specimen is measured as a function of elongation at a constant crosshead speed or nominal strain rate. An alternative technique is the creep test in which the elongation of the specimen is measured as a function of time at constant load or stress. Occasionally, however, an alternative mechanical test has been utilized, the “stress relaxation test”, in which the drop in load is measured as a function of time at constant total strain. In this test, the load, strain rate and plastic strain are all varying. However, to a first approximation the plastic strain may be considered to be constant. Felthamlm3 was the first to use the stress relaxation method to evaluate the activation parameters, such as the activation volume and the activation energy, for the plastic flow in several metals in order to study the kinetics of deformation. He was able to do so by establishing an empirical logarithmic relationship between the relaxation in tensile stress and time, which is given as1 00
-
CT =
Sre, log1dl +vt)
where o0 is the initial value of the tensile stress, CJ the value of tensile stress at time t, v a constant independent of time and S,,, a characteristic time-independent parameter of the relaxation, given as Sre, = -do/d
log, ,, t .
S. P.
202
AGRAWAL,
G. A. SARGENT,
H. CONRAD
His analysis, however, did not take the elastic deformation of the tensile machine, which can be considerable, into account. This and a few other modi~cations, such as including the effect of work hardening on the activation parameters (which was shown to be less than 1% of the total value, hence negligible), were introduced into Feltham’s analysis in the research to follow4-8. However, the empirical logarithmic relationship between the relaxation in stress and time was still assumed in all of the above treatments. Also assumed therein was a linear dependence of the activation energy upon the effective shear stress acting on the dislocation, as originally proposed by Seeger’ and Thornton and HirschlO and given by AGMAG~-v~*
(3)
where AG is the Gibbs free-energy of activation ( = AH - TAS where AH is the enthalpy of activation, T the temperature and AS the entropy of activation), AC, is the total work required to bring the system into the activated state and is equal to. that contributed during the activation event by the externally applied force pius the excess mechanical work required to bring the system isothermally and reversibly into the activated state. u is the activation volume, defined as
z* is the effective shear stress (= r- tcl, where r is the applied shear stress and zP the athermal component of the flow stress). It is the purpose of this note to show that although these assumptions, namely eqn. 3, seem reasonable for several metals’-*, usually at low temperatures ( T< 300 K) where z* is large so that v is relatively independent of r*, they are not valid in ~-titanium at temperatures above 300 K. Rather, a more direct approach to obtain the activation parameters which has been described previously”, is more appropriate. 1.1. THEORETICAL CONSIDERATIONS STRESS RELAXATION
IN
THE
CONVENTIONAL
ANALYSIS
USING
It is now generally accepted that the shear strain rate for a thermally activated deformation process in metals can be given as
j=jeexp
-g C > #here POis the pre-exponential factor and is related to the number of dislocation elements per unit volume contributing to the deformation, the area swept out by one such element after activation, the Burgers vector of these dislocations and an atomic frequency. AG is the Gibbs free energy of activation, k the Boltzmann’s constant and T the temperature. Feltham’ assumed that the work done by the effective stress on the dislocation segment involved in thermal activation decreases the activation energy linearly, as was first proposed by Seeger’ and Thornton and Hirsch”, and is expressed by eqn. 3. Upon differentiating both sides of eqn. 3 with respect to the effective shear stress, z*, at constant temperature one obtains (5)
DEFORMATION
KINETICS
203
IN Ti
The definition of the activation workers ” - l5 is
volume U* employed by Conrad and co-
JAG r ‘*=
-
az* =’
Further, the two activation volumes v and v* represent the same quantity for: (i) constant dislocation segment length, I, whereupon U*=u= tbx, where x is the activation distance and (ii) the case when the obstacles are randomly dispersed, whereupon u* = v =$lbx. One therefore obtains from eqns. 5 and 6 gf?.
1,-T;; /,‘O.
(7)
It can now be shown that both the quantities on the left hand side of eqn, 7 individually approach zero only under certain conditions: (a) the first quantity, aAG,/dr*l., when approximated to zero assumes that the total work required to bring the system into the activated state is independent of the effective stress at any given temperature. In other words, the total area under the force-activation distance curve at a given temperature is independent of the effective stress, or the force-activation distance curve is independent of the effective stress. This has been shown to be a thermodynamically consistent assumption for titaniumi2. 14.i5. (b) the second quantity, z,*(av/&*)l,, may be approximated to zero when either of the following two conditions is met.
au 1
is linite. 6) TT*-OandIv aT*T However, as z+O,
(3
Az*+O and u-+co. Thus, an indeterminate
solution is obtained.
-&/T~0 and Z$ is finite.
This condition is satisfactorily met at low temperatures where r? is of appreciable magnitude and AZ* >O but Av+O. In other words, at low temperatures, where T* is large, u tends to be independent of z* and one obtains a constant value of u over a range of T* (usually corresponding to 7’~ 300 K). It can now clearly be seen that eqn. 3 was found to be valid in the conventional analysis’ on account of the fact that au/&*/, 40, for a finite value of T; for T< 300 K. It is also obvious that for dv/Sz*l,+O, other conditions remaining the same, eqn. 3 cannot be justified. The latter is usually the case in most metals at higher temperatures (T > 300 K) since a small change in z* results in an appreciable change in v. 1.2. EXPERIMENTAL
QUANTITIES
From eqns. (4) and (6) one can derive directly
204
S. P. AGRAWAL,
aAG lJ*= -az*
,&9 Or, upon
=kT
a ln V/i0
az*
T
considering
the total applied
H. CONRAD
T
(f. = con&.).
az* T
G. A. SARGENT,
(84 stress, r, rather
than the effective stress r*,
(& = const)
where r~ is the total tensile stress and M the Taylor’s factor, such that o=Mr. In agreement with the condition (b) (ii) in the previous section, 0 was indeed found to vary linearly with In 8 in conventional strain rate cycling tests16 at low temperatures (Fig. l), thus resulting in a value of the slope
which is constant with cr, i.e., av/&r=O. Also, shown in Fig. 1 is the non-linear behavior of G with In i: at higher temperatures using the same testing technique as abover6. In this case, (a In 6)/&l, is varying with c and one obtains a significant dependence of the activation volume, v, on 0, which can be described by the following equation.
In stress relaxation the stress decreases with increasing time, i.e., A(T is negative, and the strain may be considered constant to a first approximation. The quantity which is conventionally obtained experimentally in this case is (a In t)/&l, and is related to the activation volume as follows’. V
MkT
=-
a In i: ao
T
=--
a In t a0 T’
(11)
Similar to the relationship observed between c and In i at low temperatures in Fig . 12it was observed’ ’ that during stress relaxation tests CJdecreased linearly with In t at low temperatures (Fig. 2). Also shown in Fig. 2 is the a-ln t relationship at high temperatures, which is clearly non-linear: an observation analogous to that at similar temperatures in strain ratechange tests. Thus, one obtains a value of v independent of g (au/&r = 0) at low temperatures and varying with CJ(au/% # 0) at high temperatures. This further reinforces the earlier conclusion that the constancy of volume condition (condition (b) (ii) in the previous section) may be met in a-titanium only at low temperatures. It was therefore concluded that the conventional approach of determining v* was not appropriate in the entire range of temperature (below 0.3 T,) in or-titanium.
DEFORMATION
205
KINETICS IN Ti
Data from Monteiro,Sonthonam Reed- H%II o-T~fCommercial Pwtyl
and
Fig. 1. Flow stress W. log strain rate for a-titanium (commercial purity)plate. Stress (in psi units) - log strain rate data for these temperatures obtained from Monteiro et al.“.
Fig. 2. Stress vs. log time during stress relaxation tests on cc-titanium (0.2 at.% O,,) wire.
2.1. A MORE DIRECT APPROACH FOR THE DETERMINATION
OF v AND AG
The relationship between stress and strain which produces the relaxation curve is determined by the stiffness of the tensile testing machine and the elastic modulus~ of the specimen. It has been shown 4.5. l1 that the rate of plastic deformation during relaxation can be related to the rate of relaxation itself as follows. $=-KC%.
(121
Here K is a constant which includes the specimen geometry and the specimen and the machine modulii; thus, it is equivalent to the combined modulii of the
S. P.
206
AGRAWAL,
G. A. SARGENT,
H. CONRAD
machine and the specimen. A measure of K can be obtained directly from the slope of the elastic part of the stress-strain curve and, thus, the instantaneous strain rate during stress relaxation can be determined from the rate of decrease in stress. Further, on substituting for i: from eqn. 12 into eqn. (9(a)) one obtains (13)
where M is the Taylor factor relating shear stress and strain to the tensile counterparts and is considered to be constant in the entire range of stress. Thus, simply by measuring the slopes and stresses at two times along the stress relaxation curve, the activation volume can be derived directly from the stress relaxation curve. This approach was employed in evaluating the thermal activation parameters for the plastic flow of cc-titanium at low temperatures (below 0.3 T ) in a separate paper l1 . Figure 3 shows that the Gibbs free energy of acti:ation for plastic flow derived from the stress relaxation tests using this more direct approach (Fig. 3(a)) is in good accord with that obtained from the more usual method of strain rate cycling tests (Fig. 3(b)); data for the strain rate cycling tests were obtained from Conrad and Jones”. The equations for AG” (corresponding to the case where the force-activation distance curve is independent of temperature, in addition to being independent of the applied stress) and AQ (corresponding to the case where the force activation distance barrier is propor2.0
I Stress
I
I
I
I
a-Ti(0.2
2.
I
1 St&
a
Relaxation
a-Ti
at.%Oeql
G.S.=5-42pm.
Cyclikg at.%
b
Oeq)
= 2.5/m
i=3.3xl0-4sec-’ I.
1.5
a
(0.2
G.S.
i=3.3x10-4sec-I
7 0 ”
Rate
> 3 ”
1.0
I.1
a
0.
0.5
0
I
400
200
600
I
I
200
400
Fig. 3. (b) AG” and obtained from Conrad
I 600
T(“K)
T(“K)
Fig. 3. (a) AG” and At?’ us. temperature
I
from stress relaxation
At? vs. temperature and Jones”.
from
strain
rate
tests. cycling
tests.
Stress-temperature
data
DEFORMATION
KINETICS
IN Ti
207
tional to the shear modulus p) used here were those derived by deMeester et ~1.‘~ and Schoeck’*. C,, (ref. 19) was chosen as the appropriate modulus to compute AG’ since it had been previously established I5 that the plastic deformation of polycrystalline titanium was controlled by slip on the {lOTO} (1120) system. ACKNOWLEDGEMENTS
The authors acknowledge financial support for this investigation by the Qffice of Aerospace Research, United States Air Force, under Contract F 33615-69C1027, A. Adair, technical monitor.
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
P. Feltham, Phil. Msg., 6 (1961) 259. P. Feltham, Phil. Msg., 6 (1961) 847. P. Feltham, Phil. Msg., 8 (1963) 989. F. Guiu and P. L. Pratt, Phys. Status Solidi, 6 (1964) 111. G. A. Sargent, Acta Met., 13 (1965) 663. R. B. Clough, Phys. Status Solidi, 17 (1966) K175. G. A. Sargent and H. Conrad, Scripta Met., 3 (1969) 43. R. B. Clough and L. J. Demer, Phys. Status Solidi, 36 (1969) 221. A. Seeger, Bristol Conf. Defects Crystalline Solids, The Physical Society, London, 1955, p. 328. P. R. Thornton and P. B. Hirsch, Phil. Mag., 3 (1958) 738. S. P. Agrawal, G. A. Sargent and H. Conrad, Met. Trans., in press. B. deMeester, M. Doner and H. Conrad, Dept. Met. Eng. Mater. Sci., Univ. Kentucky, Lexington, Ky., unpublished research, 1972. H. Conrad, Can. .I. Phys., 45 (1967) 581. B. delvleester, C. Yin, M. Doner and H. Conrad, John E. Dorn Mem. Symp., Rate processes in plastic deformation, Cleveland, Oct. 16 and 17, 1972, in press. K. Okazaki and H. Conrad, Acta Met., in press. S. N. Monteiro, A. T. Santhanam and R. E. Reed-Hill, The Science, Technology and Application of Titanium, Pergamon, Oxford, 1970, p. 503. H. Conrad and R. Jones, The Science, Technology and Application of Titanium, Pergamon, Oxford, 1970, p. 489. G. Schoeck, Phys. Status Solidi, 8 (1965) 499. E. S. Fisher and C. J. Renken, Phys. Rev., 135 No. 2A (1964) A482.