Messerschmidt U. Dislocation Dynamics During Plastic Deformation
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382 9 Intermetallic Alloys
Fig. 9.51. Stress relaxation curves of MoSi
2
single crystals deformed along 110.
Data from [15]
Fig. 9.52. Temperature dependence of the strain rate sensitivity of MoSi
2
sin-
gle crystals deformed along 110. Filled symbols: data from the beginning of the
relaxation tests belonging to the strain rate prior to the tests. Open symbols:for
relaxation curves with inverse curvature, data belonging to a lower strain rate of
about 10
−7
s
−1
(corresponding to about ln[− ˙σ(MPa s
−1
)] = −3). Data from [15]
as demonstrated for 698
◦
C. At high temperatures, the curves assume their
normal curvature again.
The dependence of the strain rate sensitivity belonging to the strain rate
before the tests measured from the beginning of the relaxation curves is plotted
in Fig. 9.52 as full symbols. The curve has exactly the same shape as that for
FeAl in Fig. 9.35 with very low values in the anomaly range. For relaxation
curves with an inverse curvature, r values measured at the lower strain rate
of about 10
−7
s
−1
are plotted as open symbols. Accordingly, these values are
greater than those corresponding to the initial higher strain rate.
For the deformation along 201, the temperature dependence of the strain
rate sensitivity is similar, the minimum is located at about 800
◦
C. At 850
◦
C,
strain rate cycling experiments yield negative values of the steady state strain
9.6 Molybdenum Disilicide 383
Fig. 9.53. Strain rate sensitivity of the flow stress of MoSi
2
single crystals deformed
along 201 in dependence on the logarithm of the strain rate. Parameter is the
temperature. Data from [621]
rate sensitivity. The dependence of the strain rate sensitivity r on the strain
rate ˙ε is demonstrated in Fig. 9.53. This dependence is normal for all temper-
atures except the two temperatures of 940 and 990
◦
C within the range of the
flow stress anomaly.
As described in Sect. 5.2.3, stress relaxation tests can also yield informa-
tion on the effective stress component τ
∗
. In Fig. 9.51, relaxation curves with
normal curvature are bent towards the stress axis, i.e., they converge against
certain stress values. In this case, the difference between the starting stress
of the relaxation test and the final stress may be called the relaxing stress
Δσ
rel
.ValuesofΔσ
rel
cannot be obtained from short-time stress relaxation
curves as that for room temperature, or from curves with inverse curvature
in the anomaly range as that for 638
◦
C. In the high-temperature range, the
internal stress can also relax owing to recovery. Nevertheless, some values of
Δσ
rel
for the deformation along 110 in the low-temperature range are plot-
ted in Fig. 9.54 as a function of temperature. The relaxing stress decreases
with increasing temperature. For the deformation along 110,afewtem-
perature change tests yielded activation enthalpies of about 0.4 eV at room
temperature and 2.5 eV at 800
◦
C.
9.6.4 Deformation Mechanisms
Different mechanisms control the plastic deformation of MoSi
2
in the three
temperature ranges. For a better understanding of the dislocation behavior,
the elastic properties of the dislocations of the two slip systems studied were
calculated in [15]. According to this, the elastic anisotropy of MoSi
2
does not
lead to elastic instabilities as, e.g., in NiAl (Fig. 9.20). The elastic equilibrium
shapes of the dislocations under stress are smooth, almost circular for the
{011}100 slip system.
384 9 Intermetallic Alloys
Fig. 9.54. Temperature dependence of the relaxing stress in MoSi
2
deformed along
110. Data from [15]
Low Temperatures
As in other materials, a remarkable part of the total flow stress is controlled by
long-range dislocation interactions leading to an athermal stress component τ
i
as described in Sect. 5.2.2. It may be estimated from the dislocation density
using the Taylor hardening model (5.11). For the numerical constant, a value
of α = π is used. The shear modulus μ is replaced by the anisotropic K
value for the screw dislocation, K
s
=4πE
0s
/b
2
= 198.4 GPa as introduced in
(3.40). With the dislocation density of =2.8 ×10
13
m
−2
from Sect. 9.6.2 for
the deformation at 380
◦
Calong110, the athermal stress component becomes
τ
i
≈ 170 MPa. At this temperature, the engineering flow stress σ is only about
200 MPa corresponding to a (shear) flow stress τ of only 92 MPa. Thus, the
estimation with the Taylor model greatly overestimates the athermal stress
component τ
i
. Nevertheless, this estimation shows that the athermal stress
component represents probably a large part of the total flow stress and cannot
be neglected. It may be assumed that the effective stress τ
∗
can completely
relax during long-time stress relaxation experiments. In this sense, the relaxing
stresses plotted in Fig. 9.54 may be considered lower limits of the effective
stress. Converted to shear stresses, τ
∗
may amount to about 40 MPa at room
temperature and decreases down to only about 5 MPa between 400 and 600
◦
C.
This is 12% or 5%, respectively, of the total shear flow stress. Thus, in the
low-temperature range, the athermal stress component is the dominating part
of the flow stress. In specimens deformed along 201, the dislocation densities
and accordingly also the internal stress component are lower [619].
As demonstrated in Fig. 9.38a, dislocations with 1/2111 Burgers vectors
are straight and oriented along 111 directions in the low-temperature range.
In the studies of the author and his coworkers, no micrographs are available
for dislocations with 100 Burgers vectors, but in [617] it is described that
these dislocations also tend to align along 111 at room temperature. These
features are explained by the action of the double-kink or Peierls mechanism
in the low-temperature range of the deformation of MoSi
2
.
9.6 Molybdenum Disilicide 385
According to Sect. 4.6, the dependence of the Gibbs free energy of activa-
tion on the effective stress τ
∗
for the double-kink mechanism can be expressed
by (4.66). From this potential, there follows the activation volume
V (τ
∗
)=−
∂ΔG
∂τ
∗
=
πΔF
fk
4τ
p
ln
16τ
p
πτ
∗
, (9.9)
with ΔF
fk
being the formation energy of a single kink, and τ
p
being the Peierls
stress at T =0K. As discussed above, the athermal stress component is large
but not well-known so that the analysis of V vs. τ
∗
is not possible. Therefore,
(4.66) and (9.9) together with the Arrhenius equation (4.8) were solved for T .
The result is
T = T
0
[1 − 2exp(−χV )(χV +1)], (9.10)
with
T
0
=
2ΔF
fk
k ln ( ˙γ
0
/ ˙γ)
and χ =
4τ
p
πΔF
fk
.
T
0
has a similar meaning as in Sect. 4.5.2. Figure 9.55 presents the respective
plots for temperatures where the Peierls mechanism is supposed to operate.
The data of the regression analysis are collected in Table 9.5. The kink forma-
tion energies are calculated using ln ( ˙γ
0
/ ˙γ) = 20. The kink energies assume
reasonable values. Using ΔF
fk
and τ
p
from the table, (4.66) and (4.8) yield an
activation energy is 0.5 eV for the deformation along 110, which is consis-
tent with the activation enthalpy of 0.4 eV determined from the temperature
change test (Sect. 9.6.3). The higher value for the deformation along 201
results from the larger absolute value of the Burgers vector. In accordance
with the higher temperature of the flow stress minimum, T
0
is also higher.
According to the elastic theory of the Peierls mechanism, the Peierls stress
τ
p
is related to the Peierls energy W
p
by (4.21), and this, in turn, to the kink
Fig. 9.55. Plot of temperature T versus activation volume V for deformation along
110 with data below 300
◦
C(squares) and deformation along 201 at and below
500
◦
C(fivefold stars). Theoretical curves after (9.10) and the data in the text. In
the regression curve for the deformation along 110, the room temperature point is
not considered. Data from [15]
386 9 Intermetallic Alloys
Table 9.5. Parameters of the regression curves in Fig. 9.55 characterizing the Peierls
mechanism in MoSi
2
. Data from [15]
Loading axis T
0
(K) χ (nm
−3
)ΔF
fk
(eV) τ
p
(MPa)
201 746 3.25 0.64 263
110 554 4.71 0.48 282
energy ΔF
fk
= W
fk
by (4.26). The width of the Peierls valleys may be set
h = b. The line energy is E
d
=ln(R/r
0
)E
0
=6.37E
0
,if1/
√
is set for
the outer cut-off radius R of the dislocations with =2.8 × 10
13
m
−2
from
Sect. 9.6.2 and r
0
= b for the inner one. With these data and the value of the
Peierls stress for the deformation along 110 from Table 9.5, the kink energy
would be ΔF
fk
=0.55 eV. This value is close to that from the regression
analysis of the activation volume. That means that the parameters of the
Peierls mechanism, ΔF
fk
and τ
p
, determined earlier are consistent within the
framework of the elastic theory of this mechanism.
The relatively low values of τ
p
(263 and 282 MPa) with respect to the
flow stresses at room temperature, τ = 336 MPa for the 110 deformation
axis, emphasize the large contribution of the long-range internal stress to
the total flow stress, as discussed above. If the dislocation multiplication and
consequently the dislocation density are triggered by the effective stress as
it is discussed in Sect. 5.1.1, a Cottrell–Stokes like behavior may result, i.e.,
the increasing effective stress from the Peierls mechanism with decreasing
temperature causes also an increasing athermal stress component, which may
be even greater than the thermal one.
Flow Stress Anomaly
It is suggested in [620] and [15] that the flow stress anomalies in the slip sys-
tems studied here are caused by the same mechanism as developed by Mills
et al. [583] for the dislocations with 110 Burgers vectors in NiAl (Sect. 9.4.4).
This mechanism was applied above to FeAl (Sect. 9.5.3), comprising the disso-
ciation of the dislocations along preferred orientations by glide and the motion
of the dissociated dislocations by combined glide and conservative climb. The
importance of diffusion processes in connection with the flow stress anomaly in
MoSi
2
was noted before not only in [618] but also in [628]. For the dislocations
with 1/2111 Burgers vectors the dissociation reactions may read
1/2[111](1
¯
10) → 1/6[331](1
¯
10) + 1/6[3
¯
31](013) + 1/6[
¯
331](103) (9.11)
for the preferred 331 edge orientation and
1/2[111](1
¯
10) → 1/6[331](1
¯
10) + 1/6[3
¯
31](
¯
116) + 1/6[
¯
331](1
¯
16) (9.12)
for the preferred mixed 110 orientation. Most of the references (e.g., [617,
629]) cite the dissociation of dislocations with 1/2111 Burgers vectors
according to
9.6 Molybdenum Disilicide 387
1/2[111] → 1/4[111] + 1/4[111].
The results in [617] confirm this reaction for screw dislocations but show a
dissociation outside the {110} plane for dislocations with an edge component.
The three slip planes involved in (9.11) and (9.12) intersect one another always
along the preference directions of the dislocations. In both reactions, the first
partial can glide on the glide plane of the parent dislocation. The {013} planes
of the product dislocations in (9.11) are considered before as slip planes in
MoSi
2
[617]. The dissociation according to (9.12) does not contradict the weak
beam images of Fig. 1 in [629] and of Fig. 7 in [617], which resolve two partials
of different contrast, but it does not agree with Fig. 10b in [616]. Calcula-
tions of the prelogarithmic factors of the dislocation energy using anisotropic
elasticity show that the dissociation is accompanied by a 3% gain in elastic
energy. The faults between the partial dislocations are APBs. They have high
energies [630] so that the dissociation width should be small. The latter two
partials cannot glide on the (1
¯
10) glide plane of the total dislocation. Thus,
the motion of the whole dislocation requires conservative climb in the dis-
location core, which is possible solely at sufficiently high temperatures. The
latter partials in (9.12) are pure edge components for the 60
◦
dislocations
along [110], which may explain that these dislocations experience the highest
glide resistance leading to their dominance in the dislocation structure in the
anomaly range.
For the dislocations with 100 Burgers vectors oriented along 111 a
dissociation reaction is suggested in [631]
[100](0
¯
11) → 1/6[301](12
¯
3) + 1/6[30
¯
1](1
¯
43). (9.13)
The three slip planes involved intersect along the [111] preference direction of
the dislocations. The reaction is accompanied with an energy gain of about
5%. The total dislocation can again move on its (0
¯
11) slip plane only by a
vacancy flux from one partial to the other. As suggested for FeAl in Sect. 9.5.3,
the dependence of the dissociation width on the dislocation velocity and
temperature is necessary to explain the occurrence of a flow stress anomaly.
The following experimental observations favor the proposed model:
• Within the anomaly range, dislocations tend to orient along specific
crystallographic directions as described earlier. The crystallographic orien-
tation is not a consequence of an anisotropy of the line tension. The elastic
equilibrium shapes are smoothly curved for both slip systems. The disloca-
tions are curved and not oriented at high temperatures above the anomaly
range as well as at the flow stress minimum below the anomaly range
(Fig. 9.42).
• Dislocations with 1/2111 Burgers vectors arranged along 110 are unsta-
ble and may dissociate according to (9.8) to form stacking faults on (001)
planes.
388 9 Intermetallic Alloys
• The dynamic properties of dislocations with rapid multiplication at high
dislocation velocities at first loading and viscous motion afterwards agree
with Cottrell effect-like behavior.
• The strain rate sensitivity is small in the anomaly range (Fig. 9.52), show-
ing an inverse dependence on the strain rate, i.e., the strain rate sensitivity
is higher for smaller strain rates or stresses (c.f., Fig. 9.53).
• At least for the deformation along 201, slip is localized in narrow bands
with plastic instabilities (serrated yielding) occurring (Fig. 9.49).
Since a theory of the conservative climb model with velocity and temper-
ature depending dissociation width is not yet available, some estimations are
made regarding the position and height of the flow stress maximum using
the theory of the Cottrell effect as it is described in Sect. 4.11 and applied
several times in this book for other intermetallics. The relations between the
temperature T
max
of the flow stress maximum, the diffusion coefficient D,the
elastic strength of the point defects β from (3.32), and the contribution to
the flow stress τ
max
are given by (4.99). The shear modulus μ is replaced by
the anisotropic K value of the edge dislocation K
e
. With elastic constants for
650
◦
, there follows K
e
= 187.2 GPa. Poisson’s ratio for dislocations with 100
Burgers vectors is very small in agreement with their almost circular equilib-
rium shape. By comparing (14–47) and (14–42) of [12], it may be concluded
that the relaxed volume of a vacancy should enter (3.32) for ΔΩ if the diffu-
sion takes place by the motion of vacancies. It is assumed here that it equals
about 30% of the atomic volume, i.e., ΔΩ =0.3a
2
c/6. a and c are the lattice
constants. Using these data yields β ≈ 2.8 ×10
−29
Nm
2
. Then, it follows from
(4.99) that the diffusion coefficient should be about D ≈ 5 × 10
−19
m
2
s
−1
to obtain the flow stress maximum at a strain rate of ˙γ =10
−5
s
−1
and a
temperature of T
max
= 1010 K for the deformation along 110 (Fig. 9.50).
Diffusion coefficients for both Mo and Si in MoSi
2
single crystals are pub-
lished in [632,633]. According to these measurements, the diffusion coefficients
are strongly anisotropic with higher diffusivities perpendicular to the c axis
than parallel to it. Besides, in the temperature range considered, only Si dif-
fuses. Comparable diffusion coefficients for Mo are observed only above about
1,500 K. The Si diffusion coefficients at 1,010 K are about 10
−16
m
2
s
−1
for
the diffusion perpendicular to the c axis and 3 × 10
−18
m
2
s
−1
for diffusion
parallel to it. Thus, the experimental diffusion coefficients of Si are consider-
ably higher than those required for the diffusion in the dislocation core for the
Cottrell effect. However, climb-controlled motion of the dislocation requires
the diffusion of both constituents, i.e., Si and Mo, so that the slower species
should control the total rate. Mo bulk diffusion is far too slow so that the
model requires strongly enhanced diffusion in the dislocation core.
The proposed mechanism should explain the different positions of the flow
stress maximum for the deformation along 110 and 201. For dislocations
with 1/2111 Burgers vectors on {110} planes, the value K
e
= 213.5GPa is
only slightly higher than the respective value for the { 011}100 slip system.
9.6 Molybdenum Disilicide 389
The remarkably larger Burgers vector cancels if (3.32) is inserted into (4.99).
However, an essential difference may arise from the anisotropy of the diffusion
coefficient. In each case, the transport of matter has to occur in the direction of
motion of the dislocation. For the dislocations with 100 Burgers vectors and
111 line directions, the transport occurs mainly perpendicular to the c axis.
This is the direction of a high diffusion coefficient. Alternatively, the direction
of motion of dislocations with 1/2111 Burgers vectors aligned along 110
has a large component parallel to the c axis. In this direction, the diffusion
coefficients of both Si and Mo are considerably lower than perpendicular to
it which, according to (4.99), results in a higher temperature T
max
of the
flow stress maximum as observed in Fig. 9.50. The maximum contribution
to the flow stress τ
max
in (4.99) is proportional to b
−3
.Thismayleadtoa
smaller value of τ
max
for the deformation along 201 under otherwise similar
conditions.
In addition to the mechanism causing the flow stress anomaly, there is
certainly also a large athermal flow stress component due to long-range dis-
location interactions. As described earlier, it dominates at the flow stress
minimum between 300 and 500
◦
C. It may even increase in the anomaly range
triggered by the increasing anomalous stress contribution.
High-Temperature Range
Above the flow stress peak, the flow stress decreases in the normal way for
both deformation axes (Fig. 9.50). As shown in [15], for the 110 compression
axis, the dislocations are no longer straight and oriented. Like in other mate-
rials, the high-temperature deformation can conveniently be treated in terms
of a power law equation for steady state creep (5.20). From a single experi-
ment in the high-temperature range, it follows that the experimental stress
exponent m
defined in (4.12) decreases from about 7.5 at 750
◦
Cdownto2.5
at 1,050
◦
C. In other materials, the creep exponents scatter between 2 and 6
(Sect. 5.2.4). Thus, the m
values are roughly in this range. The activation
enthalpy should correspond to the activation enthalpy for self-diffusion. The
experimental value is 2.5 eV at 800
◦
C (Sect. 9.6.3) and the diffusion enthalpies
are 1.93 and 2.33 eV for the diffusion of Si, and about 5.4 and 6.2 eV for the
diffusion of Mo, perpendicular and parallel to the c axis each [632,633]. Accord-
ingly, the situation is similar to that of the Cottrell effect interpretation of
the anomaly range: the diffusion data required for the respective model are
between the data for the diffusion of Si and Mo. At present, a convincing
interpretation of this fact cannot be given. For other intermetallic materials,
there is no such great difference between the diffusion data of the different
constituents.
9.6.5 Summary
The low-temperature increase of the flow stress of the two soft slip sys-
tems {110}1/2111 and {011}100 in MoSi
2
single crystals is most probably
390 9 Intermetallic Alloys
caused by the Peierls mechanism. In addition, a very great athermal part of
the flow stress originates from long-range dislocation interactions. Both slip
systems show an anomalous increase of the flow stress at intermediate tem-
peratures. For the {110}1/2111 slip system, the transition to the anomaly
range is accompanied with a change of the preferred orientation of straight
dislocations from the 60
◦
111 orientation to the 60
◦
110 orientation, which
are crystallographically not equivalent. The instantaneous formation of stack-
ing faults during in situ experiments in an HVEM from the dissociation of
dislocations with 1/2111 Burgers vectors oriented along 110 indicates that
these dislocations are not very stable. Their dynamic behavior is characterized
by rapid multiplication and glide at very high velocities during first loading
and later on by viscous motion. The occurrence of the flow stress anomaly
is explained by the same model as suggested for FeAl, i.e., a dissociation of
the dislocations oriented along crystallographic directions by glide out of the
glide plane of the total dislocation and the motion in the dissociated state by a
combination of glide and conservative climb. Respective dissociation reactions
are proposed for dislocations with both Burgers vectors studied. To obtain the
anomaly, it has to be assumed that the dissociation width depends on the dis-
location velocity and the temperature. The normal high-temperature decrease
of the flow stress is interpreted by recovery-controlled deformation.
9.7 Conclusions on Intermetallics
All the materials presented in this chapter show a normal temperature depen-
dence of the flow stress at low temperatures. The mechanisms controlling
the dislocation motion are the same as in other materials classes, i.e., the
Peierls mechanism (in MoSi
2
), localized obstacles like small precipitates (in
γ-TiAl and probably also in FeAl) or jogs resulting from cross slip (for the
{011}100 slip system in NiAl). Thus, they are not specific of the special
crystal structures in intermetallic alloys.
Except NiAl, all the materials exhibit a flow stress anomaly. In some mate-
rials, the anomaly is caused by special properties of the superdislocations
like the formation of Kear–Wilsdorf locks or similar reactions as in Ni
3
Al.
In TiAl, superdislocations also contribute to the plastic deformation. How-
ever, at all temperatures ordinary dislocations dominate so that the anomaly
has to be explained by processes connected with ordinary dislocations. In
several materials discussed here, the properties of the anomaly are not very
specific to the particular crystal structure. But the temperature of the flow
stress maximum seems to correlate with diffusion properties. This correlation
becomes obvious from the Cottrell effect-like behavior. The dislocations may
move in a viscous or unstable manner. The viscous motion is not typical of
the high-temperature deformation controlled by long-range dislocation inter-
actions as in pure metals where the dislocations move in a jerky way over
distances of the spacing between dislocations. It is also not characteristic of
9.7 Conclusions on Intermetallics 391
all locking–unlocking mechanisms, which are proposed for the deformation of
intermetallic materials. A particular feature of the Cottrell effect-like behavior
is the inverse dependence of the strain rate sensitivity r on the strain rate ˙γ
itself, i.e., decreasing r with increasing ˙γ in range A of the strain ageing curve
in Fig. 4.39. The inverse behavior is observed in all materials in which the
anomaly seems to be connected with diffusion, i.e., in TiAl, FeAl, and MoSi
2
.
Strain ageing with Cottrell effect-like behavior may cause the viscous
motion of ordinary dislocations in γ-TiAl where associates of antisite defects
and vacancies may give rise to the induced Snoek effect. Point defect atmo-
spheres around the dislocations may arise also from other existing lattice
defects like impurities or additions in low concentrations. The theory describes
the interaction of these defects with the dislocations, e.g., [634] for NiAl. Bind-
ing energies are calculated by determining the difference between the energy
of a dislocation containing the defect in its core and the sum of the energies
of the dislocation without the defect and the formation energy of the defect.
In most cases, positive or negative binding energies are obtained below 1 eV.
However, in the particular case of an Ni antisite defect in the core of a dis-
location on {01
¯
1} with a 111 Burgers vector, the dislocation reconstructs
locally, and the binding energy turns out to be remarkably higher than the
formation energy of the defect. This means that the energy of the dislocation
including the defect in its core is lower than the energy of the dislocation
without the defect. It is suggested in [618] that generally in intermetallics, the
lowest energy state of a dislocation may be a configuration containing a certain
concentration of intrinsic point defects in the dislocation core, independent of
further chemical or structural disorder of the material, i.e., the arrangement
of atoms near the core may differ from the regular arrangement in the ordered
structure. During motion, this low-energy state can follow the dislocation only
via diffusional processes including a frictional force, in close analogy to the
formation of atmospheres of existing defects like the Cottrell atmospheres.
Molecular dynamics simulations allowing for atomic mobility should be car-
ried out to prove the possibility of such a mechanism. The process is of intrinsic
nature, suggesting a viscous motion of the dislocation as observed during in
situ straining experiments on several intermetallic materials.
A special feature of dislocations with 110 Burgers vectors in NiAl and of
both Burgers vectors studied in FeAl and MoSi
2
at high temperatures is the
occurrence of straight dislocations arranged along preferred crystallographic
directions and moving viscously. The origin cannot be the usual lattice friction
as this operates at low temperatures, as observed for the 100 dislocations in
MoSi
2
. They are straight at low temperatures owing to the Peierls mechanism,
curved at intermediate temperatures and straight again at temperatures of the
flow stress anomaly. Besides, the low activation energy characteristic of the
Peierls mechanism is observed only at low temperatures. For the intermetallic
alloys with obstacle mechanisms at low temperatures, the Peierls mechanisms
is expected to operate at even lower temperatures. Thus, the straight oriented