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The peak of flow stress in the L12 structure and the elimination of Kear-Wilsdorf locks
Materials Science and Engineering, A164 (1993) 281-285
281
The peak of flow stress in the L 12 structure and the elimination of Kear-Wilsdorf locks Georges Saada and Patrick Veyssi~re Laboratoire d'Etude des Microstructures*, BP 72, 92322 Chftillon Cedex (France)
Abstract The conditions that determine the peak of flow stress in L12 alloys are explored in relation to the elimination of the Kear-Wilsdorf (KW) configurations. Two alternative hypotheses are explored according to whether the Kear-Wiisdorf locks are destroyed by transformation and glide in the primary octahedral plane or disappear upon activation of extensive cross-slip on the cube plane. It is shown that the consistency between present predictions and the experimental behaviour of a variety of single crystals with L 12 structure is qualitatively good.
1. Introduction
The atypical mechanical properties of L12 alloys have been intriguing materials scientists for nearly 35 years and a number of basic questions have remained unsatisfactorily answered. The configuration that consists in a screw superdislocation dissociated in two superpartials bordering an antiphase boundary (APB) extended in a cube plane, known as the Kear-Wilsdorf (KW) lock, has played a central role in the analysis of the mechanical properties of L12 alloys. Below the temperature at which the flow stress peaks, slip occurs on octahedral planes and there is general agreement on the fact that the cross-slip transformation of a glissile screw superdislocation into a sessile KW configuration is instrumental in explaining the atypical increase in flow stress with increasing test temperature (see for instance ref. 1). Altogether, mechanical tests, transmission electron microscope observations, atomistic simulations and theoretical models have made clear that the format i o n - b e it partial or complete--the stability as well as the destruction of KW locks play a key role in an explanation of the atypical mechanical properties of L 12 alloys. It should be recalled that, although it is established that in most orientations deformation proceeds on the primary cube plane, experiments conducted with the applied load oriented near [001] have shown that the peak is not necessarily associated with a transition in the nature of primary slip from octahedral to cubic. * Unit6 Mixte CNRS-ONERA, UMR- 104. 0921-5093/93/$6.00
The present paper is aimed at exploring the idea that the flow stress peak originates from the massive destruction of KW locks upon application of an external stress.
2. The properties of Kear-Wiisdorf barriers
We summarize several properties of the cross-slip of a screw superdislocation between an octahedral and a cube plane which are of interest in the following development [2]. 2.1. The dissociated screw superdislocation under no applied stress Let the subscripts o and c define quantities relevant to the octahedral and the cube planes respectively. The ratio of the antiphase boundary (APB) energies is chosen as Z=7o/7c
(la)
which equivalently can be defined as z = ~,c/Ao
(lb)
where the distances ;t o and ~,c correspond to )~ox= E/7o,c
(2)
2E is the prelogarithmic term in the expression of the self-energy of an undissociated screw superdislocation. We also define the following elasticity parameter A+2 a -A31/2
(3)
where A is the usual Zener anisotropy coefficient. © 1993 - Elsevier Sequoia. All rights reserved
282
G. Saada, P. Veyssi&e
/
We consider the transition path in which superpartial SO is ascribed to slip in the octahedral plane while S~ is confined to the cube plane (Fig. 1) and we limit our analysis to that of the forces, F o and Fc, that are applied to superpartials So and Sc respectively:
Flow stress in L12 structure
and the companion superpartials experience the following force in the octahedral plane Fo = y~(3 -~/2 - z)
(8)
Then, when z > 3 - ]/2, the superdislocation is stable in the cube plane. It follows from comparison of eqns. (6) and (8), that the nucleation of a KW configuration (transition C--" O) is not submitted to the same driving forces as its destruction (transition C --, O). In the following, we take a = 0.93
(9a)
and we assume with
z> a
fJ = alo z + 3]/21¢z + 21o1¢
(4c)
When allowed to dissociate in the octahedral plane only, a screw superdislocation is at equilibrium (Fo = 0) when the separation between its superpartials satisfies
lo=,~o
(5)
However, each superpartial is submitted to a net force in the cube plane that writes Fc = y¢(1 - z/a)
(6)
when z < a, the superdislocation is at equilibrium although F c is not cancelled. Conversely, when z > a, Fc tends to promote the extension of the superdislocation in the cube plane. In the latter plane both superpartials are then submitted to a force couplet which is the torque term introduced by Yoo [3]. Similarly, when equilibrium is ascribed to the cube plane (Fc = 0), then
xa troll
tf~l/ ~o (7 > 0
0
X3 [101]
Xl
[0101
(C |
q)ca> Ol~) $c / Fig. 1. The geometrical conditions under which an infinitely long screw superdislocation, dissociated in the (1 l i ) primary octahedral plane, may transform into a KW configuration. Superpartials SO and S¢ border an APB on the octahedral (width lo) and on the cube (width lc) planes respectively.
(9b)
which is the case for almost every documented L12 alloy [2]. However, since the difference ( z - a ) varies between almost zero (0.07) to 0.87 [2], it should be realized that the uncertainty on z is such that F¢ may be mistaken by one order of magnitude. 2.2. The role o f an appliedstress The properties of the dihedral configuration of Fig. 1 are formally the same as under no applied stress (Section 2.1 ), provided now that Z where Z = Yo - cPoab
(10)
7c+~cob
is substituted for z in the discussion, q% and q0c are the Schmid factors on the octahedral and the cube planes respectively. Upon application of an external load, the relative stability of a screw superdislocation, depending on whether it is dissociated in a cube or in an octahedral plane, can be modified by appropriate adjustment of Z with respect to the above critical values (3 ]/z and a). Finally, it is shown in ref. 2 that, irrespective of the applied load, a dihedral configuration is never under stable equilibrium. When it is assumed for the sake of simplicity that lattice friction can block occasionally the leading superpartial as it moves on the cube plane, then one finds that incomplete dihedral locks can be stabilized. 2.3. Stress-induced destruction of Kear- Wilsdorf locks From eqns. (8)-(10), the critical applied stress above which a KW lock is transformed into an O configuration is given by the condition (11)
Z<3-U2
which, in the absence of friction stresses, is equivalent tO
z31/2- 1 s { z ' N } = N + 3 ~/~
s{z, 0} l + N 3 1/2
(12)
G. Saada, P. Veyssi~re
/
where N =cpc
(13a)
~o
ab
s=q~o 7c
(13b)
N, the usual stress orientation parameter [4], is positive; s can be regarded as a dimensionless quantity, expressed in 7c/b units, which is proportional to the level of shear stress resolved in the octahedral plane.
3. Determination of saturation stresses
3.1. Hypotheses In this discussion, we make the three following assumptions. (a) The rate of creation of KW locks increases both with temperature and with the force acting in the cube plane. This force is the sum of F c (eqn. (6)) and of the shear stress in the cube plane. Accordingly, the larger N, the easier the formation of KW segments. (b) Once nucleated, the process of transformation of screw segments yields fully formed sessile KW locks. (c) KW locks control dislocation multiplication in the primary octahedral plane (see the dynamical simulations of Mills and Chrzan [5]). It is the density of KW locks relative to non-screw segments that limits the rate of dislocation expansion. The larger this relative density, the shorter the average mobile mixed segments and the larger the resistance of the crystal to deformation [2]. No measurement of appropriate dislocation densities has been made available so far in order to check point (a). This point is however consistent with the generally accepted property that KW formation is a thermally aided process. An important consequence of point (a) is that the rate at which the flow stress increases with temperature is an increasing function of N. By contrast, assumptions (b) and (c) differ from those made classically to explain the positive temperature dependence of the flow stress [6, 7], but they are in good agreement with the following experimental properties: (i) the existence of KW locks at low temperature or in the absence of any applied stress on the cube plane [8, 9]; (ii) the fact that the flow stress anomaly disappears at very low levels of permanent strain, that is in the microdeformation domain [10]; (iii) very large strain hardening rates [9-11]; (iv) a low strain rate sensitivity [10, 12]. The relevance of point (i) to the present hypotheses is obvious. Points (ii) and (iii) are a consequence of the fact that the more the sample is deformed, the more
Flow stress in L I~ structure
283
KW locks are created. Moreover, the larger the restriction to slip, the larger the strain hardening. However, since the dislocations which produce deformation in the upper temperature range of flow stress anomaly are non-screw in character, their intrinsic slip motion is that of regular glissile and the strain rate sensitivity should not be large. In other words, the activation area, which is the average area swept by dislocations which contribute to plastic strain, relates to those dislocations that are not sitting in a locked configuration. These views are in addition supported by creep experiments which show that primary creep is anomalous and yields a large number of KW segments and that cube glide operates on the cross-slip plane at temperatures below the peak Tp [13]. Consistently, KW configurations are observed to undergo a local bending in the cube plane when tests are conducted under constant strain rate and the amplitude of bending increases with temperature [9]. The bending in the cube plane may hinder further motion of mixed segments when regarded as regular kinks [14]. Under these conditions, the yield stress should increase with temperature until it reaches the fracture stress. However, before this upper limit is attained, two alternative mechanisms may operate which can oppose stress augmentation. (1) Dislocations may glide in the cube plane resulting in the elimination of KW segments by expansion in this plane. We assume that the rate-limiting factor in this process is the existence of a friction force fc( T ) in the cube plane that decreases with temperature. For consistency with eqns. (12) and (13b), re(T) is expressed in 7c/b units. (2) KW locks may be destroyed under an appropriately applied stress (Section 2.2) by transformation into a screw dislocation glissile in the octahedral plane. We recall that, consistent with experimental measurements [4], point (a) implies that when N2 > N1, then ds( T, N 2)/d T > ds( T, N 2)/d T and
s( T, N2)> s( T,N,) The variations of the critical resolved shear stress s as a function of temperature, for two such values of N, are plotted in Fig. 2 in order to study the implications of the two saturation mechanisms above.
3.2. Saturation by destruction of Kear-Wilsdorf locks In this case the flow stress ceases to increase when it reaches the level defined by eqn. (12). It is clear from Fig. 2(a), that the intersection of the horizontal lines, which represent eqn. (12) for each value of N, and of
284
G. Saada, P. Veyssi~re
/
Flow stress in L12 structure 2
s
tjr)m
S
s(z,NI)
*
s(z,N)
I I (a)
TP2
IT Tp1
I
(b)
I
r
Tp2 Tpl
Fig. 2. Graphical determinations of the flow stress peak as the intersection of the flow stress curves (plotted for two values of N
such that N2>N1) with (a) the resolved shear stress that destroys KW locks by a cross-slip process back to the primary octahedral plane s(z,N), and (b) the thermally activated friction on the cube plane that opposes dislocationexpansionon this plane.
the flow stresses s( T, N2) and s( T, N 1), define two saturation stresses s* such that s2*(N2) < sl*(N 1)
(14)
Furthermore, it is obvious that Tp2 is smaller than Tp1.
3.3. Saturation by cube slip In contrast, the elimination of KW locks by slip on the cube plane occurs when the stress resolved in the cube plane is larger or equal to lattice friction in this plane. Hence, the peak corresponds in this case to
Ns*( N)= fc( T )
(15)
This is determined graphically in Fig. 2(b), where the flow stresses s(T, Nz) and s(T, N1) are now superimposed with the curves f~/N 1 and fc/Nz, again defining two saturation points s2* and Sl*. Under the hypothesis of the peak being controlled by slip on the cube plane, one again obtains the relation sz*(Nz)< sl*(N1), with
(a)
(b)
Fig. 3. Prediction of the variation of flow stress with temperature depending on whether the stress level at which KW segments may further expand on the cube plane is located (a) above s(z, N) or (b) below s(z,N). In the former case, a plateau is predicted whose extension cannot be quantified. Experimentally, the plateau should correspond to a wide rounded peak and there are such experimental peaks in the literature.
that yields KW destruction by cross-slip is lower than the critical cube friction stress. Figure 3(b) corresponds to the opposite case. In the first case the saturation process is the destruction of the KW locks, in the second case the saturation process is cubic glide. It should be emphasized that in the real deformation experiments the temperature at which elimination occurs, both by back cross-slip or by cube slip, is not defined accurately, essentially because of stress concentration and by virtue of thermally activated processes. Consequently, the real macroscopic behaviour of L12 crystals should be significantly rounded with respect to the present theoretical curves. This effect may contribute to the progressive decrease in the large activation volumes that are measured in the upper temperature range of the anomalous domain [12], when the flow stress exhibits its inflexion towards its eventual saturation.
Tpl> Tp2. 4. Comparison with experiments and conclusion
3.4. Comments In fact, the above two mechanisms for elimination of KW locks should occur in domains of temperatures and stresses which are not predictable from simple arguments. They both predict an increase in do/dT with increasing N and this is observed experimentally [4]. However, there are no available measurements of the friction stress on the cube plane that would enable us to conduct a more accurate comparison between the present predictions and experiments. We can nevertheless consider schematically the nature of the peak under two distinct situations which are plotted in Fig. 3. Figure 3(a) corresponds to the case where the stress
We have not considered in detail the mechanism of KW creation, and it is thus clear that the particular properties of tension compression asymmetry are beyond the scope of the present analysis. Regarding a comparison with experimental determinations of peak flow stresses and peak temperatures as a function of load orientation, the present hypotheses yield a good qualitative agreement. However, the role of friction, both on the cube and the octahedral planes, together with its response to several dislocation velocities remains one of the major unknowns in this problem [14]. This is important because Ni3A1 has been pre-
G. Saada,P. Veyssi~re
/
Flow stress in L1, structure
285
pared in non-stoichiometric compositions and most generally with the help of ternary addition, sometimes in considerable amounts, in order to facilitate the growth of single crystals. Consequently, solution strengthening or other composition related phenomena are liable to hinder analysis of the peak stress significantly. Equation (12) and in particular the relation
Acknowledgments
s{z,N} = slz,O}/(1 + N f 3 )
1 P.B. Hirsch, Prog. Mater. Sci., 36 (1992) 63. P. B. Hirsch, Philos. Mag. A, 65(1992) 569. 2 G. Saada and P. Veyssi6re, Philos. Mag. A, 66 (1992) 6, 1081. 3 M.H. Yoo, Scripta Metall., 20(1986) 915. 4 D.P. Pope and S. S. Ezz, Int. Met. Rev., 29 (1984) 136. 5 M. J. Mills and D. C. Chrzan, Acta Metall. Mater., (1992) in press. 6 S. Takeuchi and E. Kuramoto, Acta Metall., 21 (1973) 415. 7 V. Paidar, D. E Pope and V. Vitek, Acta Metall., 32 (1984) 435. 8 A. Korner, Philos. Mag. A, 58(1988) 507. 9 C. Bontemps-Neveu, Th~se, l'Universit6 de Paris-Sud, 1991. 10 P.H. Thornton, R. Davies and T. L. Johnston, Metall. Trans., 1 (1970)207. 11 J. Oliver, Th~se, Universit6 de Paris-Sud, 1992. 12 J. Bonneville and J. L. Martin, in L. A. Johnson, D. P. Pope and J. O. Stiegler (eds.), High Temperature Ordered Intermetallics Alloys IV, Vol. 213, The Materials Research Society, Pittsburgh, PA, 1991, p. 629. 13 K. Hemker, M. J. Mills, K. R. Forbes, D. D. Stenbergh and W. D. Nix, in T. C. Lowe, A. D. Rollet, P. S. Follanabee and G. E. Daehn, Modelling the Deformation of Crystalline Solids, The Minerals, Metals & Materials Society, 1991, p. 411. 14 G. Saada and P. Veyssi~re,Philos. Mag. Lett., 64( 1991 ) 365. 15 Y.Q. Sun, PhD Thesis, University of Oxford, 1990.
has been checked for consistency with experimental results in rather pure L12 alloys, such as Ni3AI(Hf ) [9] and Ni3Ga [15]. In both cases good quantitative agreement is obtained on the orientation dependence of the peak stress, even when the load is oriented in the vicinity of the [001] axis. The present analysis does not account for the fact that, under certain load orientations, deformation above the peak temperature is dominated by slip that differs from the cube cross-slip plane. It is clear that this should modify the conclusions of the present analysis, since the flow stress is no longer associated with the value of q~c but with the lowest schmid factor on the cube plane. However, whereas it is true that above the peak temperature there is considerable evidence of slip on the cube plane with lowest Schmid factor, the situation is much less clear in the immediate vicinity of the peak where evidence of slip traces on the cube cross-slip plane have been reported [9]. These properties will be addressed more in more detail in a forthcoming paper.
The authors are happy to thank Professor P. B. Hirsch and Dr. D. Dimiduk for stimulating discussions.
References
281
The peak of flow stress in the L 12 structure and the elimination of Kear-Wilsdorf locks Georges Saada and Patrick Veyssi~re Laboratoire d'Etude des Microstructures*, BP 72, 92322 Chftillon Cedex (France)
Abstract The conditions that determine the peak of flow stress in L12 alloys are explored in relation to the elimination of the Kear-Wilsdorf (KW) configurations. Two alternative hypotheses are explored according to whether the Kear-Wiisdorf locks are destroyed by transformation and glide in the primary octahedral plane or disappear upon activation of extensive cross-slip on the cube plane. It is shown that the consistency between present predictions and the experimental behaviour of a variety of single crystals with L 12 structure is qualitatively good.
1. Introduction
The atypical mechanical properties of L12 alloys have been intriguing materials scientists for nearly 35 years and a number of basic questions have remained unsatisfactorily answered. The configuration that consists in a screw superdislocation dissociated in two superpartials bordering an antiphase boundary (APB) extended in a cube plane, known as the Kear-Wilsdorf (KW) lock, has played a central role in the analysis of the mechanical properties of L12 alloys. Below the temperature at which the flow stress peaks, slip occurs on octahedral planes and there is general agreement on the fact that the cross-slip transformation of a glissile screw superdislocation into a sessile KW configuration is instrumental in explaining the atypical increase in flow stress with increasing test temperature (see for instance ref. 1). Altogether, mechanical tests, transmission electron microscope observations, atomistic simulations and theoretical models have made clear that the format i o n - b e it partial or complete--the stability as well as the destruction of KW locks play a key role in an explanation of the atypical mechanical properties of L 12 alloys. It should be recalled that, although it is established that in most orientations deformation proceeds on the primary cube plane, experiments conducted with the applied load oriented near [001] have shown that the peak is not necessarily associated with a transition in the nature of primary slip from octahedral to cubic. * Unit6 Mixte CNRS-ONERA, UMR- 104. 0921-5093/93/$6.00
The present paper is aimed at exploring the idea that the flow stress peak originates from the massive destruction of KW locks upon application of an external stress.
2. The properties of Kear-Wiisdorf barriers
We summarize several properties of the cross-slip of a screw superdislocation between an octahedral and a cube plane which are of interest in the following development [2]. 2.1. The dissociated screw superdislocation under no applied stress Let the subscripts o and c define quantities relevant to the octahedral and the cube planes respectively. The ratio of the antiphase boundary (APB) energies is chosen as Z=7o/7c
(la)
which equivalently can be defined as z = ~,c/Ao
(lb)
where the distances ;t o and ~,c correspond to )~ox= E/7o,c
(2)
2E is the prelogarithmic term in the expression of the self-energy of an undissociated screw superdislocation. We also define the following elasticity parameter A+2 a -A31/2
(3)
where A is the usual Zener anisotropy coefficient. © 1993 - Elsevier Sequoia. All rights reserved
282
G. Saada, P. Veyssi&e
/
We consider the transition path in which superpartial SO is ascribed to slip in the octahedral plane while S~ is confined to the cube plane (Fig. 1) and we limit our analysis to that of the forces, F o and Fc, that are applied to superpartials So and Sc respectively:
Flow stress in L12 structure
and the companion superpartials experience the following force in the octahedral plane Fo = y~(3 -~/2 - z)
(8)
Then, when z > 3 - ]/2, the superdislocation is stable in the cube plane. It follows from comparison of eqns. (6) and (8), that the nucleation of a KW configuration (transition C--" O) is not submitted to the same driving forces as its destruction (transition C --, O). In the following, we take a = 0.93
(9a)
and we assume with
z> a
fJ = alo z + 3]/21¢z + 21o1¢
(4c)
When allowed to dissociate in the octahedral plane only, a screw superdislocation is at equilibrium (Fo = 0) when the separation between its superpartials satisfies
lo=,~o
(5)
However, each superpartial is submitted to a net force in the cube plane that writes Fc = y¢(1 - z/a)
(6)
when z < a, the superdislocation is at equilibrium although F c is not cancelled. Conversely, when z > a, Fc tends to promote the extension of the superdislocation in the cube plane. In the latter plane both superpartials are then submitted to a force couplet which is the torque term introduced by Yoo [3]. Similarly, when equilibrium is ascribed to the cube plane (Fc = 0), then
xa troll
tf~l/ ~o (7 > 0
0
X3 [101]
Xl
[0101
(C |
q)ca> Ol~) $c / Fig. 1. The geometrical conditions under which an infinitely long screw superdislocation, dissociated in the (1 l i ) primary octahedral plane, may transform into a KW configuration. Superpartials SO and S¢ border an APB on the octahedral (width lo) and on the cube (width lc) planes respectively.
(9b)
which is the case for almost every documented L12 alloy [2]. However, since the difference ( z - a ) varies between almost zero (0.07) to 0.87 [2], it should be realized that the uncertainty on z is such that F¢ may be mistaken by one order of magnitude. 2.2. The role o f an appliedstress The properties of the dihedral configuration of Fig. 1 are formally the same as under no applied stress (Section 2.1 ), provided now that Z where Z = Yo - cPoab
(10)
7c+~cob
is substituted for z in the discussion, q% and q0c are the Schmid factors on the octahedral and the cube planes respectively. Upon application of an external load, the relative stability of a screw superdislocation, depending on whether it is dissociated in a cube or in an octahedral plane, can be modified by appropriate adjustment of Z with respect to the above critical values (3 ]/z and a). Finally, it is shown in ref. 2 that, irrespective of the applied load, a dihedral configuration is never under stable equilibrium. When it is assumed for the sake of simplicity that lattice friction can block occasionally the leading superpartial as it moves on the cube plane, then one finds that incomplete dihedral locks can be stabilized. 2.3. Stress-induced destruction of Kear- Wilsdorf locks From eqns. (8)-(10), the critical applied stress above which a KW lock is transformed into an O configuration is given by the condition (11)
Z<3-U2
which, in the absence of friction stresses, is equivalent tO
z31/2- 1 s { z ' N } = N + 3 ~/~
s{z, 0} l + N 3 1/2
(12)
G. Saada, P. Veyssi~re
/
where N =cpc
(13a)
~o
ab
s=q~o 7c
(13b)
N, the usual stress orientation parameter [4], is positive; s can be regarded as a dimensionless quantity, expressed in 7c/b units, which is proportional to the level of shear stress resolved in the octahedral plane.
3. Determination of saturation stresses
3.1. Hypotheses In this discussion, we make the three following assumptions. (a) The rate of creation of KW locks increases both with temperature and with the force acting in the cube plane. This force is the sum of F c (eqn. (6)) and of the shear stress in the cube plane. Accordingly, the larger N, the easier the formation of KW segments. (b) Once nucleated, the process of transformation of screw segments yields fully formed sessile KW locks. (c) KW locks control dislocation multiplication in the primary octahedral plane (see the dynamical simulations of Mills and Chrzan [5]). It is the density of KW locks relative to non-screw segments that limits the rate of dislocation expansion. The larger this relative density, the shorter the average mobile mixed segments and the larger the resistance of the crystal to deformation [2]. No measurement of appropriate dislocation densities has been made available so far in order to check point (a). This point is however consistent with the generally accepted property that KW formation is a thermally aided process. An important consequence of point (a) is that the rate at which the flow stress increases with temperature is an increasing function of N. By contrast, assumptions (b) and (c) differ from those made classically to explain the positive temperature dependence of the flow stress [6, 7], but they are in good agreement with the following experimental properties: (i) the existence of KW locks at low temperature or in the absence of any applied stress on the cube plane [8, 9]; (ii) the fact that the flow stress anomaly disappears at very low levels of permanent strain, that is in the microdeformation domain [10]; (iii) very large strain hardening rates [9-11]; (iv) a low strain rate sensitivity [10, 12]. The relevance of point (i) to the present hypotheses is obvious. Points (ii) and (iii) are a consequence of the fact that the more the sample is deformed, the more
Flow stress in L I~ structure
283
KW locks are created. Moreover, the larger the restriction to slip, the larger the strain hardening. However, since the dislocations which produce deformation in the upper temperature range of flow stress anomaly are non-screw in character, their intrinsic slip motion is that of regular glissile and the strain rate sensitivity should not be large. In other words, the activation area, which is the average area swept by dislocations which contribute to plastic strain, relates to those dislocations that are not sitting in a locked configuration. These views are in addition supported by creep experiments which show that primary creep is anomalous and yields a large number of KW segments and that cube glide operates on the cross-slip plane at temperatures below the peak Tp [13]. Consistently, KW configurations are observed to undergo a local bending in the cube plane when tests are conducted under constant strain rate and the amplitude of bending increases with temperature [9]. The bending in the cube plane may hinder further motion of mixed segments when regarded as regular kinks [14]. Under these conditions, the yield stress should increase with temperature until it reaches the fracture stress. However, before this upper limit is attained, two alternative mechanisms may operate which can oppose stress augmentation. (1) Dislocations may glide in the cube plane resulting in the elimination of KW segments by expansion in this plane. We assume that the rate-limiting factor in this process is the existence of a friction force fc( T ) in the cube plane that decreases with temperature. For consistency with eqns. (12) and (13b), re(T) is expressed in 7c/b units. (2) KW locks may be destroyed under an appropriately applied stress (Section 2.2) by transformation into a screw dislocation glissile in the octahedral plane. We recall that, consistent with experimental measurements [4], point (a) implies that when N2 > N1, then ds( T, N 2)/d T > ds( T, N 2)/d T and
s( T, N2)> s( T,N,) The variations of the critical resolved shear stress s as a function of temperature, for two such values of N, are plotted in Fig. 2 in order to study the implications of the two saturation mechanisms above.
3.2. Saturation by destruction of Kear-Wilsdorf locks In this case the flow stress ceases to increase when it reaches the level defined by eqn. (12). It is clear from Fig. 2(a), that the intersection of the horizontal lines, which represent eqn. (12) for each value of N, and of
284
G. Saada, P. Veyssi~re
/
Flow stress in L12 structure 2
s
tjr)m
S
s(z,NI)
*
s(z,N)
I I (a)
TP2
IT Tp1
I
(b)
I
r
Tp2 Tpl
Fig. 2. Graphical determinations of the flow stress peak as the intersection of the flow stress curves (plotted for two values of N
such that N2>N1) with (a) the resolved shear stress that destroys KW locks by a cross-slip process back to the primary octahedral plane s(z,N), and (b) the thermally activated friction on the cube plane that opposes dislocationexpansionon this plane.
the flow stresses s( T, N2) and s( T, N 1), define two saturation stresses s* such that s2*(N2) < sl*(N 1)
(14)
Furthermore, it is obvious that Tp2 is smaller than Tp1.
3.3. Saturation by cube slip In contrast, the elimination of KW locks by slip on the cube plane occurs when the stress resolved in the cube plane is larger or equal to lattice friction in this plane. Hence, the peak corresponds in this case to
Ns*( N)= fc( T )
(15)
This is determined graphically in Fig. 2(b), where the flow stresses s(T, Nz) and s(T, N1) are now superimposed with the curves f~/N 1 and fc/Nz, again defining two saturation points s2* and Sl*. Under the hypothesis of the peak being controlled by slip on the cube plane, one again obtains the relation sz*(Nz)< sl*(N1), with
(a)
(b)
Fig. 3. Prediction of the variation of flow stress with temperature depending on whether the stress level at which KW segments may further expand on the cube plane is located (a) above s(z, N) or (b) below s(z,N). In the former case, a plateau is predicted whose extension cannot be quantified. Experimentally, the plateau should correspond to a wide rounded peak and there are such experimental peaks in the literature.
that yields KW destruction by cross-slip is lower than the critical cube friction stress. Figure 3(b) corresponds to the opposite case. In the first case the saturation process is the destruction of the KW locks, in the second case the saturation process is cubic glide. It should be emphasized that in the real deformation experiments the temperature at which elimination occurs, both by back cross-slip or by cube slip, is not defined accurately, essentially because of stress concentration and by virtue of thermally activated processes. Consequently, the real macroscopic behaviour of L12 crystals should be significantly rounded with respect to the present theoretical curves. This effect may contribute to the progressive decrease in the large activation volumes that are measured in the upper temperature range of the anomalous domain [12], when the flow stress exhibits its inflexion towards its eventual saturation.
Tpl> Tp2. 4. Comparison with experiments and conclusion
3.4. Comments In fact, the above two mechanisms for elimination of KW locks should occur in domains of temperatures and stresses which are not predictable from simple arguments. They both predict an increase in do/dT with increasing N and this is observed experimentally [4]. However, there are no available measurements of the friction stress on the cube plane that would enable us to conduct a more accurate comparison between the present predictions and experiments. We can nevertheless consider schematically the nature of the peak under two distinct situations which are plotted in Fig. 3. Figure 3(a) corresponds to the case where the stress
We have not considered in detail the mechanism of KW creation, and it is thus clear that the particular properties of tension compression asymmetry are beyond the scope of the present analysis. Regarding a comparison with experimental determinations of peak flow stresses and peak temperatures as a function of load orientation, the present hypotheses yield a good qualitative agreement. However, the role of friction, both on the cube and the octahedral planes, together with its response to several dislocation velocities remains one of the major unknowns in this problem [14]. This is important because Ni3A1 has been pre-
G. Saada,P. Veyssi~re
/
Flow stress in L1, structure
285
pared in non-stoichiometric compositions and most generally with the help of ternary addition, sometimes in considerable amounts, in order to facilitate the growth of single crystals. Consequently, solution strengthening or other composition related phenomena are liable to hinder analysis of the peak stress significantly. Equation (12) and in particular the relation
Acknowledgments
s{z,N} = slz,O}/(1 + N f 3 )
1 P.B. Hirsch, Prog. Mater. Sci., 36 (1992) 63. P. B. Hirsch, Philos. Mag. A, 65(1992) 569. 2 G. Saada and P. Veyssi6re, Philos. Mag. A, 66 (1992) 6, 1081. 3 M.H. Yoo, Scripta Metall., 20(1986) 915. 4 D.P. Pope and S. S. Ezz, Int. Met. Rev., 29 (1984) 136. 5 M. J. Mills and D. C. Chrzan, Acta Metall. Mater., (1992) in press. 6 S. Takeuchi and E. Kuramoto, Acta Metall., 21 (1973) 415. 7 V. Paidar, D. E Pope and V. Vitek, Acta Metall., 32 (1984) 435. 8 A. Korner, Philos. Mag. A, 58(1988) 507. 9 C. Bontemps-Neveu, Th~se, l'Universit6 de Paris-Sud, 1991. 10 P.H. Thornton, R. Davies and T. L. Johnston, Metall. Trans., 1 (1970)207. 11 J. Oliver, Th~se, Universit6 de Paris-Sud, 1992. 12 J. Bonneville and J. L. Martin, in L. A. Johnson, D. P. Pope and J. O. Stiegler (eds.), High Temperature Ordered Intermetallics Alloys IV, Vol. 213, The Materials Research Society, Pittsburgh, PA, 1991, p. 629. 13 K. Hemker, M. J. Mills, K. R. Forbes, D. D. Stenbergh and W. D. Nix, in T. C. Lowe, A. D. Rollet, P. S. Follanabee and G. E. Daehn, Modelling the Deformation of Crystalline Solids, The Minerals, Metals & Materials Society, 1991, p. 411. 14 G. Saada and P. Veyssi~re,Philos. Mag. Lett., 64( 1991 ) 365. 15 Y.Q. Sun, PhD Thesis, University of Oxford, 1990.
has been checked for consistency with experimental results in rather pure L12 alloys, such as Ni3AI(Hf ) [9] and Ni3Ga [15]. In both cases good quantitative agreement is obtained on the orientation dependence of the peak stress, even when the load is oriented in the vicinity of the [001] axis. The present analysis does not account for the fact that, under certain load orientations, deformation above the peak temperature is dominated by slip that differs from the cube cross-slip plane. It is clear that this should modify the conclusions of the present analysis, since the flow stress is no longer associated with the value of q~c but with the lowest schmid factor on the cube plane. However, whereas it is true that above the peak temperature there is considerable evidence of slip on the cube plane with lowest Schmid factor, the situation is much less clear in the immediate vicinity of the peak where evidence of slip traces on the cube cross-slip plane have been reported [9]. These properties will be addressed more in more detail in a forthcoming paper.
The authors are happy to thank Professor P. B. Hirsch and Dr. D. Dimiduk for stimulating discussions.
References