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Interactions Between Dislocations and Small-size Obstacles
Chapter 3 Interactions Between Dislocations and Small-size Obstacles This chapter describes the main properties of dislocations interacting with "small-size" obstacles such as "forest" dislocations cutting the slip plane or individual solute atoms or small clusters. For more details, especially concerning interactions with solute atoms, the reader can refer to Yoshinaga and Morozumi (1971), Hirth and Lothe (1982), Hirth (1983), Wille et al. (1987) and Neuhauser and Schwink (1993).
3.1.
THERMALLY ACTIVATED GLIDE ACROSS FIXED SMALL-SIZE OBSTACLES
The glide movement of dislocations across small-size obstacles, assumed here to be fixed, can be enhanced by thermal activation. The main equations describing this interaction are described below for several short-range phenomenological potentials. When a dislocation is pinned, the force exerted on the pinning point is, in the isotropic approximation (Figure 3.1): F-
2T sin a ~ p,b 2 sin a
(3.1)
where T is the line tension. (In order to take into account differences between edge and screw segments, the effective line tension defined by Kocks et al., 1975, must be used.) In the case of a low density of weak obstacles (small a) this force can also be written: (3.2)
F = "rbL F
where ~-is the effective stress and L F the distance between obstacles along the dislocation line. In this approximation by Fleischer (1961), the net force acting on LF is transferred to
F ...y'"'" '"-........ A .-'"'" "'".. A
T..........
T
Figure 3.1. Forces exerted by a dislocation on a pinning point. 57
58
Thermally Activated Mechanisms in Crystal Plasticity
the obstacle point. Combining Eqs. (3.1) and (3.2) yields: ~"=
/xb sin
a
(3.3)
LF This length L F has been estimated by Friedel (1964, p. 224) in the case of weak dislocation-obstacle interactions. It is defined as follows (see Figure 3.2): in a steady-state regime, each time one dislocation crosses a pinning point, another one is met. The area swept during the process is ~r = LFA, where A = R(1 - cos a) -~ (Ra2/2), R is the radius of curvature and a ~ (LF/R). Accordingly: 1 L3
--
2R
~"
-'- L 3 ~ ~b
(3.4)
The steady-state condition also implies: d
= d 2 = b2/Cb
(3.5)
where d is the in-plane average distance between obstacles and c b is their atomic concentration (Cb can be much smaller than the average concentration of solute atoms if the efficient obstacles are clusters of two or three atoms). Combining Eqs. (3.4) and (3.5) then yields the Friedel length: LF=d
~
=b
( ),,3 /z
TC b
LF
Figure 3.2. Dislocation escaping a pinning point in the Friedel approximation.
(3.6)
Interactions Between Dislocations and Small-size Obstacles
59
LF is found to be stress dependent because more strongly curved dislocations interact with a higher density of obstacles. Combining Eqs. (3.3) and (3.6), the effective stress becomes: b 003/2 r --/z ~(sin
(3.7a)
Or, using Eq. (3.1):
b(F)
r =/x ~
3/2
~
(3.7b)
In the case of a high density of stronger and more diffuse obstacles, the Mott-Labusch theory yields another stress-force relation (see, e.g. Labusch, 1970; Haasen, 1979, 1983; Neuhauser and Schwink, 1993): b 4/3
F
4/3
W
1/3
(3.8) where w is the width of the obstacles. Following Mott and Labusch, the validity limit between both the approximations is determined by the parameter: w(2T)
1/2
(3.9)
The Fleischer-Friedel approximation is valid for fl << 1 (w--- b) and the Mott-Labusch approximation must be used for/3 > 1. To predict the effect of temperature several approximations have been made for the short-range energy-distance and force-distance profiles. As discussed by Kocks et al. (1975, p. 141), the energy profile can be expressed as U(7")=Ema x 1 -
~
rmax where 0 < p < 1 and 1 < q < 2. "/'max is the maximum value of the effective stress r at which the obstacle is crossed without thermal activation. Several types of energy or force profiles have been proposed that correspond to specific values of p and q. They are described in the following. In each case, the stress versus temperature and activation volume versus temperature relations are established.
3.1.1 The rectangular force-distance profile The force of the obstacle is constant and equal to Fma x o v e r distance w (Figure 3.3a). The corresponding energy-distance profile is E(y) = FmaxY, and its maximum value is Emax -FmaxW (see Figure 3.3b).
Thermally Activated Mechanisms in Crystal Plasticity
60
U F, Fm~
E
F
Em~
Fm~ ~ I I I I I -W
0
I
w
0
w
Y
I I
~
ax
E m a ~ 1 2 ~ I
I |
I -w
0
w
Figure 3.3. Different obstacle profiles: (a) rectangular force-distance profile" (b) corresponding energy; (c) and (d) parabolic approximation; and (e) Cottrell-Bilby potential.
This profile is used to describe the forest mechanism when gliding dislocations have no long-range elastic interaction with the intersecting trees (Friedel, 1964, p. 221). In this case, Emax is the energy of the two jogs created on the two intersecting dislocations and w is the width of the dislocation cores. For attractive junctions close to their unzipping configuration, Emax may contain an additional term of dislocation line energy.
Interactions Between Dislocations and Small-size Obstacles
61
Under an applied force F, the height of the barrier is decreased by the quantity Fw, and the activation energy is:
U(F) = Emax - Fw = Emax 1 - Fmax
(3.10a)
In the Friedel approximation (Eq. (3.7b)) the activation energy becomes:
g('r)
= gmax 1 -
~
Tmax
(3.10b)
where %a~ = I~(bld)(FmaxOxb2)3/2. Eq. (3.10b) provides an energy of the type proposed by Kocks et al. (1975) with p = 2/3 and q = 1. The corresponding activation volume is: V.
.
0U . 0W
Emax( r ) -!/3 . Tmax '/'max
V = LFbw
(3.1 lb)
where LF is given by Eq. (3.6). Writing U = kTln(~/o/j~) and Emax = kTo ln(5'o/~/), where To is the temperature, the stress can be expressed as a function of temperature as:
T-- Tmax 1 -- To
3.1.2
(3.1 la)
athermal
(3.12)
The parabolic force-distance profile
The force-distance profile in the direction of motion is given by (Figure 3.3c):
max(1y 2)w
(3.13a)
The corresponding energy-distance profile is (Figure 3.3d):
E(y)
.
wFmax( . . y. w
I ( Y ) 3) -3 w
(3.13b)
Its maximum value is Emax -- 2 wFmax, for y = w, size of the obstacle. This profile is often used to describe interactions between dislocations and solute atoms. It is, however, thought to be valid only for F--~ Fmax, i.e. at high stresses and low temperatures (Wille et al., 1987). In the particular case of the size-effect interaction, which is discussed in Section 3.2.1, Fmax is approximately given by Eq. (3.25).
62
Thermally Activated Mechanisms in Crystal Plasticity
Under an applied force F, the saddle position is reached when the net force on the dislocation is zero, namely when (OE(y)/Oy) - F = 0. Using Eq. (3.13) this condition is satisfied for the critical value y = Yc given by:
yc _ (1_ w
Fmax
(3.14)
The corresponding activation energy is U(F) - E(yc) - Fyc, i.e.
g(f)--~Wfmax
l-fmax
= Emax 1
Fma~
(3.15a)
Using Eq. (3.7b), the activation energy can be written as:
U('r) - - Ema x
(
3/2 (3.15b)
1-7"max
where ~'max has the same value as for Eq. (3.10b). Again, U(z) is of the type proposed by Kocks et al. (1975) with p -- 2/3 and q = 3/2. The activation volume dependence on stress is:
-
V
-
0T
--
Tmax
~ Tmax
1-
~ Tmax
(3.16a)
As in the preceding case, the stress-temperature dependence is given by:
( (T2,3)
3/2
T ' - - "/'max 1 -
and the activation volume dependence on temperature is: V=
3.1.3
[
Tmax [ J T o
(3.17)
To
]-
1/2
1 -- ( T o )
(3.16b)
The Cottrell-Bilby potential (Cottrell and BUby, 1949)
The energy-distance profile is (Figure 3.3e):
E(y)
-
-
Emax
(3.18)
, +(y)2w This profile is used to describe dislocation-solute interactions, with less restrictions than the parabolic one (Wille et al., 1987). Here again, Fma x is approximately given by Eq. (3.25) below in the particular case of size-effect interaction.
Interactions Between Dislocations and Small-size Obstacles
63
Using the same procedure as above analytical solutions can be found. Following Wille et al. (1987), they can be approximated by: ) 3/2
U(F)=Emax 1 - (
F )~
(3.19a)
with Ema x --- (8/3xtr3)wFmax. This expression is similar to those given by Eqs. (3.10a) and (3.15a). In the Friedel approximation (Eq. (3.7b)), it can be expressed as:
( ( ~ )046)
3/2
U ( T ) --- Ema x
1--
(3.19b)
Tmax
namely 2.17 T = Tma x
(3.20)
1 --
where Zmaxhas the same value as for Eqs. (3.10b) and (3.15b). Eq. (3.20) can be compared with Eqs. (3.12) and (3.17) with p = 2/3 and q = 2.17. The activation volume is:
v 069~max(~) ~ -~max
' ( ~ )046)
(3.21a)
which is not too different from Eqs. (3.1 la) and (3.16a), or
v 069~max(~)'3(~ax~o ' (~)~3) ~o 7 3.2.
(3.21b)
DISLOCATIONS INTERACTING WITH MOBILE SOLUTE ATOMS
As the temperature rises solute atoms become sufficiently mobile to diffuse towards dislocations. The driving forces for this process are long-range interactions, different from the short-range ones described in the preceding section. They are estimated below and then used to compute dislocation mobility.
3.2.1 Long-range elastic interactions The most important interaction is the paraelastic one or size effect. In cylindrical coordinates, at a distance r from an edge dislocation and along a direction at an angle 0
64
Thermally Activated Mechanisms in Crystal Plasticity
from the Burgers vector, the hydrostatic pressure is p
O'rr
--
(TOO =
--
~ b sin 0 2at(1 - v)r
l(Orrr "~- (TOO+ Orzz), where
=
trzz = v(O-rr .ql_ (Too)
and
whence:
#b sin 0 1 + v
p =
37rr
1-
(3.22)
v
The interaction energy with a solute atom which induces a local change of atomic volume A O is accordingly (Haasen, 1979): Uint - - 3 1 - VpA ~ = _1/zbAg2 ~ s i n0 l+v "n" r
(3.23)
At given r and 0, Eq. (3.23) shows that the sign of Uint is directly connected to that of AO. The corresponding radial interaction force is: 1 Fin t "~ -- ~ "rr
AO sin 0 ~ r2
(3.24)
The m a x i m u m value of F is obtained when the slip plane is one interatomic distance from the obstacle, i.e. z ~ b (see Figure 3.4), whence:
Fmax
l ~AO
1 ~b2 AO
'
(3.25)
-
ar
b
3
~
O
,,,.
V
&
/k
Figure 3.4. Schematic description of the solute atom concentration around an edge dislocation. The size effect here corresponds to Ag2 > 0.
Interactions Between Dislocations and Small-size Obstacles
65
Uin t and F are often expressed in terms of the change of lattice parameter a with solute concentration: (~-- d In a/dc, taking into account A,Q/,O -----3(~. Screw dislocations are also subjected to the paraelastic interaction, provided they are dissociated in several mixed partials. The dielastic interaction, based on the so-called modulus effect, is weaker. Several other interactions can be considered, assuming that short-range atomic movements of solute atoms can take place in the vicinity of dislocation cores, e.g. shortrange ordering or local disordering in intermetallics (Haasen, 1983; Neuhauser and Schwink, 1993). The small-size obstacles considered in this chapter can be small aggregates of several atoms. Another important effect must be mentioned here, that is discussed in Chapter 6: Peierls forces in metals and alloys can substantially increase when the local concentration of solute atoms increases (e.g. oxygen in titanium). This effect is not a local pinning but it can also induce stress instabilities.
3.2.2
Static ageing, dynamic strain ageing and the Portevin-Lech~telier effect
This section introduces the main features of these complex phenomena. The static equilibrium concentration of solute atoms around a dislocation is
( gint)
c -- Co exp - ~-~--
(3.26)
where c o is the average solute concentration and Uin t is the dislocation-solute interaction energy. For a size-effect interaction, Uin t depends on r and 0, according to Eq. (3.23). The corresponding concentration is described schematically in Figure 3.4. Moving dislocations tend to drag their atmosphere of solute atoms and a dynamic equilibrium is established which depends on temperature and dislocation velocity. If all solute atoms are assumed to move only along the direction of dislocation motion (y > 0 in Figures 3.4 and 3.5) then their concentration obeys: -D
Oc Oy
Dc 0Uin t -- ( c - Co)V kT Oy
(3.27)
where D is the solute-diffusion coefficient, given by an expression similar to Eq. (8.5). The first left-hand-side term is the diffusion flux due to the concentration gradient Oc/Oy and the second one is the transport flux under the driving force 0 Uint/Oy. At steady state, the whole concentration profile around the dislocation is assumed to move at the dislocation velocity v. Consequently, the fight-hand-side term of Eq. (3.27) expresses the solute flux through the crystal. The resolution of this equation allows the dynamic equilibrium profiles around moving dislocations to be determined. Figure 3.5 shows the dynamic solute concentration along planes at various distances z from the dislocation core.
66
Thermally Activated Mechanisms in Crystal Plasticity T Z, and concentration
5%
Figure 3.5. Computed solute atmosphere around an edge dislocation, moving to the fight, along different planes above and below the slip plane. From Sakamoto (1981).
Figure 3.6a shows the dynamic concentration along the plane located at one interatomic distance from the slip plane for various dislocation velocities. The general shape and the maximum concentration value remain close to their equilibrium values at rest up to a velocity of 102 nrn/s. In addition, a depletion is formed just ahead of the moving dislocation. At higher velocities the maximum concentration decreases and a tail develops. ~
_J.. --
1 0 -1
O
a)
eol
-
"rl
-
l!
11
'I
I.
-
'I
I.
.
l I
I,
:I
1,~
o
102 [nms l]
10.2 , 25 [nm s- 1 _
"
0
-
9 :It "~i~'l
I.' II
I
103[nms-] 9
/9 ',:./10'~nms'l
. . . . .--..-'. -"
~"r
. . . . .., . . . .
~_~
1 0 .3
I
-15
I
-10
I
I
I
I
1
-5
0
5
10
15
ylb
Figure 3.6. Computed solute atmosphere around a moving edge dislocation (from Yoshinaga and Morozumi, 1971): (a) in a plane located one interatomic distance from the slip plane (the dislocation moves to the fight); (b) corresponding friction stress as a function of dislocation velocity; and (c) same as (b), for different increasing temperatures Tl < T2 < T3.
Interactions Between Dislocations and Small-size Obstacles
67
friction stress [9.8 MN'm'2]5I 1;1 " r
21,r
1;2
9
1 v1
0
v2
1
2
3
4
5
6
7
8
9
10 dislocation velocity [102nms-l]
friction stress
t T3 I
l
I
l
f I.
i
Figure 3.6. (continued)
The corresponding stress-velocity dependence is shown in Figure 3.6b. The stress increases up to ~'l as the velocity increases to v l - 70 nm/s. Cottrell and Jaswon (1949) showed that this critical velocity is of the order of 12 kTD/mbAl2, where D is the solutediffusion coefficient. This regime corresponds approximately to the translation of the equilibrium cloud discussed above, considering Figure 3.6a. Above Vl the friction stress decreases as the maximum concentration decreases. Above a critical velocity v2 the stress increases again because solute atoms can be considered as immobile with respect to the moving dislocation, and the conditions of Section 3.1 are satisfied. Similar stress-velocity
68
Thermally Activated Mechanisms in Crystal Plasticity
curves are shown at different temperatures in Figure 3.6c. It shows in particular that, as the temperature increases, v l and v2 increase as well and rl and 72 decrease. From the curves in Figure 3.6b and c, stress instabilities can be anticipated for stresses and velocities that correspond to dr/dv < 0. The origin and the properties of stress instabilities (or Portevin-Lechfitelier (PLC) effect) have been discussed in several articles (see, e.g. MacCormick, 1972; Van den Beukel, 1975; Mulford and Kocks, 1979; Strudel, 1980; Estrin and Kubin, 1989; Kubin and Estrin, 1990). Let us consider a sample containing a density of mobile dislocations, p, deformed at an imposed strain-rate, ~, ranging between pbv 1 and pbv 2. Figure 3.7a shows that when the stress increases to the critical value 7-1 dislocations suddenly accelerate from v~ to v ( r l ) > v2. Since pbv(7-1)> ~, the tensile machine relaxes and the applied stress decreases to the second critical value 7"2. The dislocation velocity then decreases instantaneously to v(7-2) < vl. Since pbv(7-2) < ~, the applied stress increases again and another cycle starts. This behaviour leads to stress instabilities or the PLC effect. These considerations show that the part of the curve of Figure 3.7a between Vl and v2 has no physical meaning. Increasing the applied strain-rate increases the total time spent by the dislocations in the high-velocity regime but the average flow stress keeps oscillating between 7-1 and 7-2.This corresponds to a zero stress-strain rate sensitivity. However, Figure 3.6c shows that this average flow stress decreases with increasing temperature. A slightly different behaviour is expected at low temperatures (lower than Tl in Figure 3.6c). Then, Vl is so small that dislocations moving at v < Vl can be considered as immobile (Figure 3.7b). Solute atoms then start to move to the dislocation core (static ageing). This results in an unpinning stress increasing with increasing waiting time (or decreasing strain-rate) 7-/1 > 7". Figure 3.7b shows that the average flow stress increases with decreasing strain-rate, which results in a negative stress-strain rate sensitivity. With increasing temperature, the average flow stress can also increase, which results in a yieldstress anomaly. The same behaviour is expected for larger values of v~, provided mobile dislocations are momentarily slowed down by extrinsic obstacles such as forest dislocations (Estrin and Kubin, 1989). Let us note that, in reality, the situation is still more complex because of strain localization.
3.2.3
Diffusion-controUed glide
Inspection of Figure 3.6b,c shows that, at a given dislocation velocity (given applied strainrate), deformation can take place in the low-velocity regime (v < Vl) provided the temperature is high enough (T --> T3). The aim of this section is to derive analytical expressions for this low-velocity/high-temperature regime. Consider the dislocation described schematically in Figure 3.4. When it moves along the y-direction, the solute atoms are assumed to move only in this direction.
69
Interactions Between Dislocations and Small-size Obstacles
L
"C2
! ! !
)
v(1:2)
vl
:
v2
v(q)
v
j/ I
V2
)
V(rl)
V(~al )
V
Figure 3.7. Origin of stress instabilities: (a) on the basis of Figure 3.6b and c; and (b) in the case vl ~ 0 (low temperature).
The dislocation velocity is estimated in the frame of two different approximations. For Uin t ~ kT, Eqs. (3.23) and (3.26) show that the cloud is highly asymmetrical, with a high concentration of solute atoms in the region corresponding to Uin t < 0 and a weak depletion in the opposite one (Uint > 0). This property results from the asymmetrical shape of the exponential function. According to Friedel (1964, p. 410), in the limit of unsaturated clouds the solute atoms are all very close to the dislocation core, forming a row of pinning points with average distance )t (Figure 3.8). The work done by the applied stress during the diffusion of one solute atom over one interatomic distance is ~-b2)t.
70
Thermally Activated Mechanisms in Crystal Plasticity
Figure 3.8. Schematic description of the diffusion-controlled glide of a dislocation pinned by a row of solute atoms.
The frequency of this event is VD ~-exp --
T
in the forward direction, and l b ( Ud-q-Tb2~) -- VD exp --
2
-A
kT
in the backward direction (cf. Section 7.2.2.2) (Vo is the Debye frequency and Ud is the solute-diffusion activation energy). The dislocation velocity is accordingly:
v = -~ Vo--~ exp -
kT
-
kT
or, assuming that rb2A << kT v=
vD~exp
--~
--Dk---T
This yields "r -
1 kT v b2 D
(3.28)
Under these circumstances, the friction force is independent of the solute concentration in the core. For Uin t < kT, the cloud is symmetrical. The local concentration (Eq. (3.26)) can indeed be approximated by c - Co ~ co(Uint/kT) so that c - Co has opposite values above and below the slip plane (points y, z and y, - z in Figure 3.4). The average concentration around the dislocation is thus Co, which a priori should yield no frictional force. However, a detailed description shows that this statement is erroneous. It is based on the works of Cottrell and Jaswon (1949) and Fuentes-Samaniego (1979), summarized by Hirth and Lothe (1982). It takes into account the energy dissipated by the solute atoms moving either in the direction of the dislocation or in the opposite one. The dynamic concentration profile
71
Interactions Between Dislocations and Small-size Obstacles
is as in Figure 3.5. For the sake of simplicity, the dislocation can be considered as surrounded by an excess of solute atoms on one side and by an excess of holes on the opposite one. The motion of a hole in one direction corresponds to the motion of a solute atom in the opposite one. All atoms and holes are assumed to move along the y-axis in such a way that Eq. (3.27) applies. The energy dissipated corresponds to the average work done by the solute atoms and holes in excess under the driving forces F = -(OUint[Oy) and - F , respectively. The elementary friction force in a strip of width 6z is thus equal to the sum of these forces, namely:
~Z ~+oo EF(6z)=
~
~)Uint
-oo - ( c - c ~
(3.29)
Oy dy
For z > 0, the expression in the integral is the work done by the solute atoms in excess. It is positive for y < 0 (OUint]Oy < 0) and negative for y > 0 (OUint]Oy > 0). The integral is, however, strictly positive because of the asymmetrical shape of c(y) shown in Figures 3.5 and 3.6a. For z < 0, the expression in the integral can be understood as the work done by the holes (concentration c - Co) under the driving force (OUint]Oy). The integral is also positive. Replacing (OUint]Oy) by its value deduced from Eq. (3.27) yields: ~Z E F ( r z ) = --~kT
[
V -~
+oo (C -- Co)2 dy -~ ~
-co
c
coc] Oy
+oo
The second integral in the brackets is I c - Co In cl_oo = 0. Since Uin t "~ kT we have c - c o ~ co Uint]kT and c ~ Co, whence: co
EF(rz)
-- D k T O v r z
Uin t dy
and, using Eq. (3.23) 1
v~z Co
rrkT
(z 2 -Jr-y2)2 dy
Then, EF(rz)=
kT
v r z Co
(/zbA[~ ) 2"tr 3kT 2z
1 A typing error in Eq. (18.53) of Hirth and Lothe (1982, 1992) has been corrected.
(3.30)
72
Thermally Activated Mechanisms in Crystal Plasticity
After integration along the z-direction, between a minimum value Zo equal to a few interatomic distances and a maximum value estimated as D/v, the resulting friction stress is: 1 vb t x A ~ A ~ D ~"= - ~ c ~lx -D k T 12 In VZo or
/xb /xAO 2 ~" "~ Co D k T
g2
v
(3.31)
The friction stress is thus similar to that given by Eq. (3.28), however, with a more complex constant factor. Taking Co -'~ 10 -2, ~ b 3 / k T -'- 102 and AD,/I2 --- 10 -l this factor has the same value as the one in the first approximation ((1/b2)(kT/D) in Eq. (3.28)). Another estimation proposed by Hirth and Lothe is based on a simplified interaction energy, varying as 1/r as in Eq. (3.2), but independent of 0. The result is twice the above estimation. Cottrell and Jaswon (1949) showed that taking into account diffusion along the z-direction introduces only a factor of 1/2 in Eq. (3.31). Therefore, it seems that Eq. (3.28) provides a satisfactory approximation, valid in the whole temperature range.
3.3.
COMPARISON WITH EXPERIMENTS
The f o r e s t mechanism Experimental evidence of the forest mechanism comes from the "Cottrell-Stokes" experiments in single and polycrystalline FCC metal (Cottrell and Stokes, 1955; Thornton et al., 1962), recently discussed by Nabarro (1990) and Saada (1999). The deformation curves exhibit strong linear and parabolic hardening stages (called stages II and III, respectively) which result from the storage of forest sessile dislocations. The flow stress is usually decomposed into an athermal or internal stress, z~,, and a temperature and strainrate dependent effective stress, r* (Section 2.1.4.1). The internal stress is irreversible upon strain-rate and temperature changes. It is due to the long-range stress field of forest dislocations and to the formation of attractive junctions. The effective stress is reversible upon strain-rate and temperature changes. Figure 3.9 shows that the ratio of flow stresses at high and low temperatures is almost independent of strain or total stress. This CottrellStokes law indicates that the effective stress remains approximately proportional to the internal stress during deformation. This shows that both stresses are related to the same obstacles, namely the forest dislocations. In addition, the strong decrease of the activation volumes with increasing strain and stress that they observe shows that the density of thermally activated obstacles increases with strain (Eqs. (3.6) and (3.11 b)) as expected for a forest mechanism.
3.3.1
73
Interactions Between Dislocations and Small-size Obstacles O'293/O'90
,J X
0.8
x
--C
0
0
0 0.7
0
I
I
I
I
I
lO
20
30
40
50
~"
e [%]
Figure 3.9. Ratio of flow stresses at room temperature and in liquid air, of an AI single crystal. Curve A was obtained by transitions from 293 to 90 K, and curve B by the reverse transitions. From Cottrell and Stokes (1955).
More quantitatively, the average distance between forest dislocations is related to the internal stress by the usual relation: d = / z b / r u . Using Eqs. (3.6) and (3.1 lb) then yields the activation volume: V = I~b2w/r~
Since for high strains we can assume ~'u ~ ~', we obtain" V r ~ tzb2w
which is a constant. This property is indeed satisfied, at least at large strains, as shown in Figure 3.10. Saada (1999), however, pointed out that this analysis does not take into account cross-slip that is also active in stage III. The exact origin of thermal activation can be the formation of jogs on intersecting dislocations or the recombination of short attractive junctions. For a very low forest density, the activation volumes are no longer proportional to the distance between adjacent trees. This behaviour has been interpreted as a breakdown of thermal activation considerations based on fully relaxed dislocation configurations when the mean spacings in the glide plane become very large (Argon and East, 1979). 3.3.2
D i s l o c a t i o n s - s o l u t e atoms interactions
The different situations described in Sections 3.1 and 3.2 are observed experimentally in many alloys as a function of temperature. Figure 3.11 shows the temperature dependence of the CRSS and of the stress-strain rate sensitivity in several CuA1 alloys. Four domains can be identified. In domain 1 (very low temperatures) the phonon frictional force becomes so low that dislocation velocities can reach very high values. The inertial effects are
74
Thermally Activated Mechanisms in Crystal Plasticity Vz e.v.
III 9
9
"7
9
9
9 9
9
9 9
yO O0
|
I
I
I
0
I
I
I
I
I
"
0.5
"
1.0
e
Figure 3.10. Curve V~" as a function of strain in a copper single crystal strained at 473 K. I, II, III refer
to the classical hardening stages. From Thornton et al. (1962).
then sufficiently important to help dislocations passing through solute atoms ("underdamped" dislocation motion, see Figure 3.12). This phenomenon has been described by Granato (1971). -
Domain 2 corresponds to the "overdamped" thermally activated motion of dislocations across fixed obstacles described in Section 3.1. Sometimes solute atoms can already start to move to dislocations (Flor and Neuhauser, 1980).
-
Domain 3, in which stress instabilities are associated to a small yield-stress anomaly and to a negative stress-strain rate sensitivity, corresponds to the situation described in Section 3.2.2.
-
In domain 4, dislocation motion is controlled by the drag of solute atmosphere (Section 3.2.3).
The corresponding dislocation kinetics have been clearly identified in the in situ TEM experiments of Monchoux and Neuhauser (1987) in CuGe alloys. In particular, in the domain of stress instabilities, sudden vigorous dislocation movements shook the specimen and made observations impossible. Domains 2 - 4 are described in more detail in what follows.
75
Interactions Between Dislocations and Small-size Obstacles CRSS [MPa] 35
30 Cu 15 at. % A1 25 ' 20
~
h k ~ v / C u 10 at. %AI 5 at. % A1
~, ~,/.
,
""
I
it I
10
i J
[]
0
l
l
I
l
200
400
600
800
1
2
3
|
9 w
1000 T[K] 4
b3 s(~) ~0~
b)
200
I
150
100
-1--
2
-'-"
3
-i-~4
--
pLc i
50
-50 0
i 200
i 400
i 600
i 800
" T [K]
Figure 3.11. Mechanical properties of CuAI alloys (from a review by Neuhauser and Schwink, 1993). Temperature domains 1 - 4 are described in the text. (a) CRSS as a function of temperature. Data from Suzuki and Kuramoto (1968), Startsev et al. (1979), Nixon and Mitchell (1981) and Neuhauser et al. (1990). The PLC regime is indicated by dotted lines and bars that refer to the amplitudes of the stress instabilities. (b) Stress-strain rate sensitivity of C u - 15% AI as a function of temperature and PLC domain. Data from Kopenaal and Fine (1962), Komnik and Demiskii (1981) and Neuhauser et al. (1990).
76
Thermally Activated Mechanisms in Crystal Plasticity
t,.)
with inertia T F i g u r e 3.12. Schematic description of the influence of inertia effects at low temperature.
3.3.2.1 Domain 2: thermally activated motion across fixed obstacles. As discussed by Wille et al. (1987), comparison between theory and experiment is complicated by the following points: (i)
Results in domain 2 must be extrapolated to 0 K in order to determine Ema x and "rmaxThis is rather questionable when the extension of domain 1 is large. (ii) There is usually a spectrum of obstacles of different strengths. (iii) The solute distribution is often not uniform and segregation can take place at stacking faults. Short-range ordering or clustering may also alter the mobility of dislocations. (iv) Deformation tends to be heterogeneous. In order to avoid the above effects, Wille et al. (1987) have selected the CuMn system, where the tendency to form short-range order is negligible, the stacking fault energy is almost independent of the Mn concentration and the size effect is very large. Figure 3.13a shows the temperature dependence of the CRSS. Domain 1 is smaller than in Figure 3.11a, which allows for an easier extrapolation to 0 K. Figure 3.13b shows that this stress varies with temperature according to Eq. (3.20) (Cottrell-Bilby potential). The activation volume in Figure 3.14a shows that domain 1 extends from 0 to 25 K. Above 25 K its stress dependence obeys Eq. (3.21a)if the variation of (1 - (T/7"max)0"46)1/2 is neglected (Figure 3.14b). It also varies with temperature according to Eq. (3.21b), as shown in Figure 3.14c. The values of "/'maxand Emax corresponding to the best fits are shown in Table 3.1. Emax ranges between 1.2 and 1.4 eV and "/'max increases with the solute concentration. The internal stress -r~, is assumed to be the high-temperature stress in Figure 3.13a. Fma x is deduced from Ema x included in Eq. (3.19a), assuming that w = 2.5b. Then, Eq. (3.16b) for
77
Interactions Between Dislocations and Small-size Obstacles CRSS 1'
[MPa]/
60
50
40
30
20
10
_•••,•,••'••"<•
7.6 at % Mn
-
~
2.0 at % Mn I
I
I
I
I
I
I
I
I
50
100
150
200
250
300
350
400
450
)
T[K]
1
1.0
0.8
0
7.6 at.% Mn
o
3.8 at.% Mn
v
2.0 at.% Mn
0.6
0.4
0
0.2
,ik
I
I
I
I
,
0.2
0.4
0.6
0.8
1.0
(T/To)2/3
Figure 3.13. CRSS in CuMn alloys (from Wille et al., 1987): (a) as a function of temperature; and (b) plotted so as to check the validity of Eq. (3.20).
78
Thermally Activated Mechanisms in Crystal Plasticity
V[nm3]l,T ~.~ I
o
7.6 at.% Mn
121 3.8 at.% Mn v 2.0 at.% Mn
10
0
I
I
I
I
I
50
1O0
150
200
250
T [K]
b3
--V--X103
T - 78K
T = 295K 9
CuMn CuGe
0 -
2'0
3'0
4:0
(rmax)0"46.rTM [MPa] Figure 3.14. Activation volumes in CuMn alloys (from Wille et al., 1987): (a) as a function of temperature; (b) plotted so as to check the validity of Eq. (3.21a), and (c) plotted so as to check the validity of Eq. (3.21b).
79
Interactions Between Dislocations and Small-size Obstacles V Vo
Vo = 0.69 max "t'max
0 7.6 at.% Mn n 3.8 at.% Mn v 2.0 at.% Mn
j
/O o 0;.1
0.2
0.3
014
0~
r/r o
Figure 3.14. (continued)
F = Fma x and 7" -- 7"max y i e l d s the a t o m i c c o n c e n t r a t i o n o f o b s t a c l e s c b
=
(b]d) 1/2. All these
p a r a m e t e r s are g i v e n in T a b l e 3.1. This table s h o w s that: -
T h e e s t i m a t i o n o f fl (Eq. (3.9)) e n s u r e s that the F l e i s c h e r - F r i e d e l a p p r o x i m a t i o n is valid.
-
The concentration
o f o b s t a c l e s Cb is 20 t i m e s s m a l l e r than the a v e r a g e solute
c o n c e n t r a t i o n c. In addition, the i n t e r a c t i o n e n e r g y Emax is f o u n d to be a b o u t 1.3 eV. Table 3.1. Mechanical parameters of CuMn alloys (from Wille et al., 1987). c (at.%) %/~/ "rmax (MPa) TO (K) Emax (eV) 'ru (MPa)
wlb Cb X 10-4
LF('Cmax)/b /3X 10 -2
0.4 Mn
1.2 Mn
2.0 Mn
3.8 Mn
7.6 Mn
23.5 10.3 658 1.34 2.0 _+ 0.2 2.5 1.33 324 2.9
25 20.7 631 1.36 5.7 +_ 0.3 2.5 5.1 164 5.6
23.5 25.0 610 1.21 10.3 _+ 0.6 2.5 10.8 120 8.1
25.8 40.2 569 1.28 14 + 1 2.5 23.5 79 12.0
26.5 58.2 533 1.23 20.4 + 1.5 2.5 53.8 53 18.3
80
Thermally Activated Mechanisms in Crystal Plasticity Since that expected for individual solute atoms is predicted to be smaller than 0.4 eV, thermal activation is thought to correspond to the crossing of doublets or triplets of solute atoms. The athermal stress z~ increases with the solute concentration. Since it has been observed that z~ is independent of the dislocation density, which can be changed by annealing, it seems accordingly that clusters involving more than three solute atoms, which cannot be overcome by thermal activation, are at the origin of this stress.
This analysis shows that calculations in Section 3.1 are reliable, at least as long as solute atoms are really immobile. In CuZn, on the contrary, Flor and Neuhauser (1980) measured non-logarithmic relaxations for which they assumed an increase of the activation energy with time due to solute segregation. In addition, Emax is observed to increase with increasing temperature, which indicates that the yield-stress is controlled by obstacles of different strengths at different temperatures. These results are of course much more difficult to compare with theoretical estimates.
3.3.2.2 Domain 3: stress instabilities and PLC effect. For a review of the experimental results in this temperature domain, the reader can refer to Strudel (1980) or Kubin and Estrin (1991). More recently, an exhaustive study of stress instabilities was made in binary CuMn and CuA1 alloys by Schwink and Nortmann (1997). The activation energies of the onset and of the end of the PLC effect are well below the volume-diffusion energy of Mn and A1 atoms in a Cu lattice, which suggests the occurrence of pipe diffusion in the core of dislocations. Surprisingly, plasticity in this domain may often be controlled by the movement of screw dislocations which are less subjected to pinning than edge ones. The release of screw segments generates fresh edge segments that can form dislocation sources (Neuhauser and Schwink, 1993; Suzuki, 1985). Since screw segments are also often subjected to Peierls-type friction forces, a combination of these two mechanisms (Peierls friction and solute effect) is expected in many cases. Note that solute atoms can also modify the core structure of sessile screw dislocations, e.g. in titanium and zirconium (Chapter 6). Such an interaction is different from that treated in this chapter. It may, however, contribute to enhance static and dynamic strain ageing when a Peierls mechanism acting on screw dislocations is rate controlling. The anomalous behaviour of several intermetallics, in which straight screw dislocations move in bursts, may be explained in this way (see Chapter 10).
3.3.2.3 Domain 4: glide controlled by solute.diffusion. Dislocation glide controlled by diffusion of solute atoms is often considered to explain the creep properties of "class I"
Interactions Between Dislocations and Small-size Obstacles
81
alloys (see, e.g. Takeuchi and Argon, 1976). Taking a dislocation density proportional to the stress squared, the resulting creep rate is indeed expected to vary as the third power of stress and the activation energy is expected to be that of solute-diffusion in agreement with some experimental results. In many cases, however, the problem is considered to be too complex to have simple solutions (see, e.g. Poirier, 1976).
REFERENCES
Argon, A.S. & East, G.H. (1979) in Strength of Metals and Alloys, Eds. Haasen, P., Gerold, V. & Kostorz G., Pergamon Press, Oxford, p. 9. Cottrell, A.H. & Bilby, B.A. (1949) Proc. Phys. Soc. London A, 62, 49. Cottrell, A.H. & Jaswon, M.A. (1949) Proc. Roy. Soc. A, 199, 104. Cottrell, A.H. & Stokes, R.J. (1955) Proc. Roy. Soc. A, 233, 17. Estrin, Y. & Kubin, L.-P. (1989) J. Mech. Behav. Mater., 2, 255. Fleischer, R.L. (1961) Acta Met., 9, 996. Flor, H. & Neuhauser, H. (1980) Acta Met., 28, 939. Friedel, J. (1964) Dislocations, Pergamon Press, Oxford. Fuentes-Samaniego, R. (1979) PhD thesis, Stanford University, California. Granato, A.V. (1971) Phys. Rev. B4, p. 2196; Phys. Rev. Lett. 27, p. 660. Haasen, P. (1979) in Dislocations in Solids, vol. 4, Ed. Nabarro, F.R.N., North Holland, Amsterdam, Chap. 15. Haasen, P. (1983) in Physical Metallurgy, Part II, vol. 8, Eds. Cahn, R.W. & Haasen P., 3rd Edition, North Holland Physics Publishing, Amsterdam, p. 1341. Hirth, J.P. (1983) in Physical Metallurgy, Part II, vol. 8, Eds. Cahn, R.W. & Haasen P., 3rd Edition, North Holland Physics Publishing, Amsterdam, p. 1223. Hirth, J.P. & Lothe, J. (1982) Theory of Dislocations, 2 nd Edition, Wiley Interscience, New York; (1992) second reprint edition, Krieger Pub. Comp., Malabar, Florida. Kocks, U.F., Argon, A.S. & Ashby, M.F. (1975) Thermodynamics and Kinetics of Slip, Pergamon Press, Oxford. Komnik, S.N. & Demiskii, V.V. (1981) Czech. J. Phys. B, 31, 187. Kopenaal, T.J. & Fine, M.E. (1962) Trans AIME, 224, 347. Kubin, L.P. & Estrin, Y. (1990) Acta Met. Mat., 38, 697. Kubin, UP. & Estrin, Y. (1991) J. Phys. III, 1,929. Labusch, R. (1970) Phys Stat. Sol., 41, 659. MacCormick, P.G. (1972) Acta Met., 20, 351. Monchoux, F. & Neuhauser, H. (1987) J. Mater. Sci., 22, 1443. Mulford, R.A. & Kocks, U.F. (1979) Acta Met., 27, 1125. Nabarro, F.R.N. (1990) Acta Metall. Mater., 38, 161. Neuhauser, H. & Schwink, C. (1993) in Material Science and Technology, vol. 6, Eds. Cahn, R.W., Haasen, P. & Kramer E.J., VCH Verlag, Weinheim, p. 191. Neuhauser, H., Plessing, J. & Schiilke, M. (1990) J. Mech. Behav. Met., 2, 231. Nixon, W.E. & Mitchell, J.W. (1981) Proc. Roy. Soc. London A, 376, 343. Poirier, J.P. (1976) Plasticit~ ?l Haute Tempdrature des Solides Cristallins, Eyrolles, Paris.
82
Thermally Activated Mechanisms in Crystal Plasticity
Saada, G. (1999) Deformation-Induced Microstructures: Analysis and Relation to Properties, Proceedings of 20th Ris~ International Symposium on Materials Science, Eds. Bilde-sorensen, J.B., Cartensen, J.V., Hansen, N., Jensen, D.J., Leffers, T., Pantleon, W., Pedersen, O.B. & Winther G., Rise National Laboratory, Roskilde, Denmark, p. 147. Sakamoto, M. (1981) Bull. Jpn. Inst. Met., 20, 912. Schwink, Ch. & Nortmann, A. (1997) Mat. Sci. Eng. A, 234-236, 1. Startsev, V.I., Demirskii, V.V. & Komnik, S.N. (1979) in Strength of Metals and Alloys, Eds. Haasen, P., Gerold, V. & Kostorz G., Pergamon Press, Oxford, p. 265. Strudel, J.L. (1980) in Dislocations et D~formation Plastique, Eds. Groh, P., Kubin, L.P. & Martin J.L., Les Editions de Physique, Les Ulis, p. 199. Suzuki, H. (1985) in Strength of Metals and Alloys, Eds. Mc Queen, H.J., Bailon, J.P., Dickson, J.L., Jonas, J.J. & Akben M.G., Pergamon, Toronto, p. 1727. Suzuki, H. & Kuramoto, E. (1968) Trans. JIM, 9(suppl.), 697. Takeuchi, S. & Argon, A.S. (1976) Acta Met., 24, 883. Thornton, P.R., Mitchell, T.E. & Hirsch, P.B. (1962) Phil. Mag., 7, 337. Van den Beukel, A. (1975) Phys. Star. Sol. A, 30, 197. Wille, T.H., Gieseke, W. & Schwink, C.H. (1987) Acta Met., 35, 2679. Yoshinaga, H. & Morozumi, S. (1971) Phil. Mag., 23, 1367.
3.1.
THERMALLY ACTIVATED GLIDE ACROSS FIXED SMALL-SIZE OBSTACLES
The glide movement of dislocations across small-size obstacles, assumed here to be fixed, can be enhanced by thermal activation. The main equations describing this interaction are described below for several short-range phenomenological potentials. When a dislocation is pinned, the force exerted on the pinning point is, in the isotropic approximation (Figure 3.1): F-
2T sin a ~ p,b 2 sin a
(3.1)
where T is the line tension. (In order to take into account differences between edge and screw segments, the effective line tension defined by Kocks et al., 1975, must be used.) In the case of a low density of weak obstacles (small a) this force can also be written: (3.2)
F = "rbL F
where ~-is the effective stress and L F the distance between obstacles along the dislocation line. In this approximation by Fleischer (1961), the net force acting on LF is transferred to
F ...y'"'" '"-........ A .-'"'" "'".. A
T..........
T
Figure 3.1. Forces exerted by a dislocation on a pinning point. 57
58
Thermally Activated Mechanisms in Crystal Plasticity
the obstacle point. Combining Eqs. (3.1) and (3.2) yields: ~"=
/xb sin
a
(3.3)
LF This length L F has been estimated by Friedel (1964, p. 224) in the case of weak dislocation-obstacle interactions. It is defined as follows (see Figure 3.2): in a steady-state regime, each time one dislocation crosses a pinning point, another one is met. The area swept during the process is ~r = LFA, where A = R(1 - cos a) -~ (Ra2/2), R is the radius of curvature and a ~ (LF/R). Accordingly: 1 L3
--
2R
~"
-'- L 3 ~ ~b
(3.4)
The steady-state condition also implies: d
= d 2 = b2/Cb
(3.5)
where d is the in-plane average distance between obstacles and c b is their atomic concentration (Cb can be much smaller than the average concentration of solute atoms if the efficient obstacles are clusters of two or three atoms). Combining Eqs. (3.4) and (3.5) then yields the Friedel length: LF=d
~
=b
( ),,3 /z
TC b
LF
Figure 3.2. Dislocation escaping a pinning point in the Friedel approximation.
(3.6)
Interactions Between Dislocations and Small-size Obstacles
59
LF is found to be stress dependent because more strongly curved dislocations interact with a higher density of obstacles. Combining Eqs. (3.3) and (3.6), the effective stress becomes: b 003/2 r --/z ~(sin
(3.7a)
Or, using Eq. (3.1):
b(F)
r =/x ~
3/2
~
(3.7b)
In the case of a high density of stronger and more diffuse obstacles, the Mott-Labusch theory yields another stress-force relation (see, e.g. Labusch, 1970; Haasen, 1979, 1983; Neuhauser and Schwink, 1993): b 4/3
F
4/3
W
1/3
(3.8) where w is the width of the obstacles. Following Mott and Labusch, the validity limit between both the approximations is determined by the parameter: w(2T)
1/2
(3.9)
The Fleischer-Friedel approximation is valid for fl << 1 (w--- b) and the Mott-Labusch approximation must be used for/3 > 1. To predict the effect of temperature several approximations have been made for the short-range energy-distance and force-distance profiles. As discussed by Kocks et al. (1975, p. 141), the energy profile can be expressed as U(7")=Ema x 1 -
~
rmax where 0 < p < 1 and 1 < q < 2. "/'max is the maximum value of the effective stress r at which the obstacle is crossed without thermal activation. Several types of energy or force profiles have been proposed that correspond to specific values of p and q. They are described in the following. In each case, the stress versus temperature and activation volume versus temperature relations are established.
3.1.1 The rectangular force-distance profile The force of the obstacle is constant and equal to Fma x o v e r distance w (Figure 3.3a). The corresponding energy-distance profile is E(y) = FmaxY, and its maximum value is Emax -FmaxW (see Figure 3.3b).
Thermally Activated Mechanisms in Crystal Plasticity
60
U F, Fm~
E
F
Em~
Fm~ ~ I I I I I -W
0
I
w
0
w
Y
I I
~
ax
E m a ~ 1 2 ~ I
I |
I -w
0
w
Figure 3.3. Different obstacle profiles: (a) rectangular force-distance profile" (b) corresponding energy; (c) and (d) parabolic approximation; and (e) Cottrell-Bilby potential.
This profile is used to describe the forest mechanism when gliding dislocations have no long-range elastic interaction with the intersecting trees (Friedel, 1964, p. 221). In this case, Emax is the energy of the two jogs created on the two intersecting dislocations and w is the width of the dislocation cores. For attractive junctions close to their unzipping configuration, Emax may contain an additional term of dislocation line energy.
Interactions Between Dislocations and Small-size Obstacles
61
Under an applied force F, the height of the barrier is decreased by the quantity Fw, and the activation energy is:
U(F) = Emax - Fw = Emax 1 - Fmax
(3.10a)
In the Friedel approximation (Eq. (3.7b)) the activation energy becomes:
g('r)
= gmax 1 -
~
Tmax
(3.10b)
where %a~ = I~(bld)(FmaxOxb2)3/2. Eq. (3.10b) provides an energy of the type proposed by Kocks et al. (1975) with p = 2/3 and q = 1. The corresponding activation volume is: V.
.
0U . 0W
Emax( r ) -!/3 . Tmax '/'max
V = LFbw
(3.1 lb)
where LF is given by Eq. (3.6). Writing U = kTln(~/o/j~) and Emax = kTo ln(5'o/~/), where To is the temperature, the stress can be expressed as a function of temperature as:
T-- Tmax 1 -- To
3.1.2
(3.1 la)
athermal
(3.12)
The parabolic force-distance profile
The force-distance profile in the direction of motion is given by (Figure 3.3c):
max(1y 2)w
(3.13a)
The corresponding energy-distance profile is (Figure 3.3d):
E(y)
.
wFmax( . . y. w
I ( Y ) 3) -3 w
(3.13b)
Its maximum value is Emax -- 2 wFmax, for y = w, size of the obstacle. This profile is often used to describe interactions between dislocations and solute atoms. It is, however, thought to be valid only for F--~ Fmax, i.e. at high stresses and low temperatures (Wille et al., 1987). In the particular case of the size-effect interaction, which is discussed in Section 3.2.1, Fmax is approximately given by Eq. (3.25).
62
Thermally Activated Mechanisms in Crystal Plasticity
Under an applied force F, the saddle position is reached when the net force on the dislocation is zero, namely when (OE(y)/Oy) - F = 0. Using Eq. (3.13) this condition is satisfied for the critical value y = Yc given by:
yc _ (1_ w
Fmax
(3.14)
The corresponding activation energy is U(F) - E(yc) - Fyc, i.e.
g(f)--~Wfmax
l-fmax
= Emax 1
Fma~
(3.15a)
Using Eq. (3.7b), the activation energy can be written as:
U('r) - - Ema x
(
3/2 (3.15b)
1-7"max
where ~'max has the same value as for Eq. (3.10b). Again, U(z) is of the type proposed by Kocks et al. (1975) with p -- 2/3 and q = 3/2. The activation volume dependence on stress is:
-
V
-
0T
--
Tmax
~ Tmax
1-
~ Tmax
(3.16a)
As in the preceding case, the stress-temperature dependence is given by:
( (T2,3)
3/2
T ' - - "/'max 1 -
and the activation volume dependence on temperature is: V=
3.1.3
[
Tmax [ J T o
(3.17)
To
]-
1/2
1 -- ( T o )
(3.16b)
The Cottrell-Bilby potential (Cottrell and BUby, 1949)
The energy-distance profile is (Figure 3.3e):
E(y)
-
-
Emax
(3.18)
, +(y)2w This profile is used to describe dislocation-solute interactions, with less restrictions than the parabolic one (Wille et al., 1987). Here again, Fma x is approximately given by Eq. (3.25) below in the particular case of size-effect interaction.
Interactions Between Dislocations and Small-size Obstacles
63
Using the same procedure as above analytical solutions can be found. Following Wille et al. (1987), they can be approximated by: ) 3/2
U(F)=Emax 1 - (
F )~
(3.19a)
with Ema x --- (8/3xtr3)wFmax. This expression is similar to those given by Eqs. (3.10a) and (3.15a). In the Friedel approximation (Eq. (3.7b)), it can be expressed as:
( ( ~ )046)
3/2
U ( T ) --- Ema x
1--
(3.19b)
Tmax
namely 2.17 T = Tma x
(3.20)
1 --
where Zmaxhas the same value as for Eqs. (3.10b) and (3.15b). Eq. (3.20) can be compared with Eqs. (3.12) and (3.17) with p = 2/3 and q = 2.17. The activation volume is:
v 069~max(~) ~ -~max
' ( ~ )046)
(3.21a)
which is not too different from Eqs. (3.1 la) and (3.16a), or
v 069~max(~)'3(~ax~o ' (~)~3) ~o 7 3.2.
(3.21b)
DISLOCATIONS INTERACTING WITH MOBILE SOLUTE ATOMS
As the temperature rises solute atoms become sufficiently mobile to diffuse towards dislocations. The driving forces for this process are long-range interactions, different from the short-range ones described in the preceding section. They are estimated below and then used to compute dislocation mobility.
3.2.1 Long-range elastic interactions The most important interaction is the paraelastic one or size effect. In cylindrical coordinates, at a distance r from an edge dislocation and along a direction at an angle 0
64
Thermally Activated Mechanisms in Crystal Plasticity
from the Burgers vector, the hydrostatic pressure is p
O'rr
--
(TOO =
--
~ b sin 0 2at(1 - v)r
l(Orrr "~- (TOO+ Orzz), where
=
trzz = v(O-rr .ql_ (Too)
and
whence:
#b sin 0 1 + v
p =
37rr
1-
(3.22)
v
The interaction energy with a solute atom which induces a local change of atomic volume A O is accordingly (Haasen, 1979): Uint - - 3 1 - VpA ~ = _1/zbAg2 ~ s i n0 l+v "n" r
(3.23)
At given r and 0, Eq. (3.23) shows that the sign of Uint is directly connected to that of AO. The corresponding radial interaction force is: 1 Fin t "~ -- ~ "rr
AO sin 0 ~ r2
(3.24)
The m a x i m u m value of F is obtained when the slip plane is one interatomic distance from the obstacle, i.e. z ~ b (see Figure 3.4), whence:
Fmax
l ~AO
1 ~b2 AO
'
(3.25)
-
ar
b
3
~
O
,,,.
V
&
/k
Figure 3.4. Schematic description of the solute atom concentration around an edge dislocation. The size effect here corresponds to Ag2 > 0.
Interactions Between Dislocations and Small-size Obstacles
65
Uin t and F are often expressed in terms of the change of lattice parameter a with solute concentration: (~-- d In a/dc, taking into account A,Q/,O -----3(~. Screw dislocations are also subjected to the paraelastic interaction, provided they are dissociated in several mixed partials. The dielastic interaction, based on the so-called modulus effect, is weaker. Several other interactions can be considered, assuming that short-range atomic movements of solute atoms can take place in the vicinity of dislocation cores, e.g. shortrange ordering or local disordering in intermetallics (Haasen, 1983; Neuhauser and Schwink, 1993). The small-size obstacles considered in this chapter can be small aggregates of several atoms. Another important effect must be mentioned here, that is discussed in Chapter 6: Peierls forces in metals and alloys can substantially increase when the local concentration of solute atoms increases (e.g. oxygen in titanium). This effect is not a local pinning but it can also induce stress instabilities.
3.2.2
Static ageing, dynamic strain ageing and the Portevin-Lech~telier effect
This section introduces the main features of these complex phenomena. The static equilibrium concentration of solute atoms around a dislocation is
( gint)
c -- Co exp - ~-~--
(3.26)
where c o is the average solute concentration and Uin t is the dislocation-solute interaction energy. For a size-effect interaction, Uin t depends on r and 0, according to Eq. (3.23). The corresponding concentration is described schematically in Figure 3.4. Moving dislocations tend to drag their atmosphere of solute atoms and a dynamic equilibrium is established which depends on temperature and dislocation velocity. If all solute atoms are assumed to move only along the direction of dislocation motion (y > 0 in Figures 3.4 and 3.5) then their concentration obeys: -D
Oc Oy
Dc 0Uin t -- ( c - Co)V kT Oy
(3.27)
where D is the solute-diffusion coefficient, given by an expression similar to Eq. (8.5). The first left-hand-side term is the diffusion flux due to the concentration gradient Oc/Oy and the second one is the transport flux under the driving force 0 Uint/Oy. At steady state, the whole concentration profile around the dislocation is assumed to move at the dislocation velocity v. Consequently, the fight-hand-side term of Eq. (3.27) expresses the solute flux through the crystal. The resolution of this equation allows the dynamic equilibrium profiles around moving dislocations to be determined. Figure 3.5 shows the dynamic solute concentration along planes at various distances z from the dislocation core.
66
Thermally Activated Mechanisms in Crystal Plasticity T Z, and concentration
5%
Figure 3.5. Computed solute atmosphere around an edge dislocation, moving to the fight, along different planes above and below the slip plane. From Sakamoto (1981).
Figure 3.6a shows the dynamic concentration along the plane located at one interatomic distance from the slip plane for various dislocation velocities. The general shape and the maximum concentration value remain close to their equilibrium values at rest up to a velocity of 102 nrn/s. In addition, a depletion is formed just ahead of the moving dislocation. At higher velocities the maximum concentration decreases and a tail develops. ~
_J.. --
1 0 -1
O
a)
eol
-
"rl
-
l!
11
'I
I.
-
'I
I.
.
l I
I,
:I
1,~
o
102 [nms l]
10.2 , 25 [nm s- 1 _
"
0
-
9 :It "~i~'l
I.' II
I
103[nms-] 9
/9 ',:./10'~nms'l
. . . . .--..-'. -"
~"r
. . . . .., . . . .
~_~
1 0 .3
I
-15
I
-10
I
I
I
I
1
-5
0
5
10
15
ylb
Figure 3.6. Computed solute atmosphere around a moving edge dislocation (from Yoshinaga and Morozumi, 1971): (a) in a plane located one interatomic distance from the slip plane (the dislocation moves to the fight); (b) corresponding friction stress as a function of dislocation velocity; and (c) same as (b), for different increasing temperatures Tl < T2 < T3.
Interactions Between Dislocations and Small-size Obstacles
67
friction stress [9.8 MN'm'2]5I 1;1 " r
21,r
1;2
9
1 v1
0
v2
1
2
3
4
5
6
7
8
9
10 dislocation velocity [102nms-l]
friction stress
t T3 I
l
I
l
f I.
i
Figure 3.6. (continued)
The corresponding stress-velocity dependence is shown in Figure 3.6b. The stress increases up to ~'l as the velocity increases to v l - 70 nm/s. Cottrell and Jaswon (1949) showed that this critical velocity is of the order of 12 kTD/mbAl2, where D is the solutediffusion coefficient. This regime corresponds approximately to the translation of the equilibrium cloud discussed above, considering Figure 3.6a. Above Vl the friction stress decreases as the maximum concentration decreases. Above a critical velocity v2 the stress increases again because solute atoms can be considered as immobile with respect to the moving dislocation, and the conditions of Section 3.1 are satisfied. Similar stress-velocity
68
Thermally Activated Mechanisms in Crystal Plasticity
curves are shown at different temperatures in Figure 3.6c. It shows in particular that, as the temperature increases, v l and v2 increase as well and rl and 72 decrease. From the curves in Figure 3.6b and c, stress instabilities can be anticipated for stresses and velocities that correspond to dr/dv < 0. The origin and the properties of stress instabilities (or Portevin-Lechfitelier (PLC) effect) have been discussed in several articles (see, e.g. MacCormick, 1972; Van den Beukel, 1975; Mulford and Kocks, 1979; Strudel, 1980; Estrin and Kubin, 1989; Kubin and Estrin, 1990). Let us consider a sample containing a density of mobile dislocations, p, deformed at an imposed strain-rate, ~, ranging between pbv 1 and pbv 2. Figure 3.7a shows that when the stress increases to the critical value 7-1 dislocations suddenly accelerate from v~ to v ( r l ) > v2. Since pbv(7-1)> ~, the tensile machine relaxes and the applied stress decreases to the second critical value 7"2. The dislocation velocity then decreases instantaneously to v(7-2) < vl. Since pbv(7-2) < ~, the applied stress increases again and another cycle starts. This behaviour leads to stress instabilities or the PLC effect. These considerations show that the part of the curve of Figure 3.7a between Vl and v2 has no physical meaning. Increasing the applied strain-rate increases the total time spent by the dislocations in the high-velocity regime but the average flow stress keeps oscillating between 7-1 and 7-2.This corresponds to a zero stress-strain rate sensitivity. However, Figure 3.6c shows that this average flow stress decreases with increasing temperature. A slightly different behaviour is expected at low temperatures (lower than Tl in Figure 3.6c). Then, Vl is so small that dislocations moving at v < Vl can be considered as immobile (Figure 3.7b). Solute atoms then start to move to the dislocation core (static ageing). This results in an unpinning stress increasing with increasing waiting time (or decreasing strain-rate) 7-/1 > 7". Figure 3.7b shows that the average flow stress increases with decreasing strain-rate, which results in a negative stress-strain rate sensitivity. With increasing temperature, the average flow stress can also increase, which results in a yieldstress anomaly. The same behaviour is expected for larger values of v~, provided mobile dislocations are momentarily slowed down by extrinsic obstacles such as forest dislocations (Estrin and Kubin, 1989). Let us note that, in reality, the situation is still more complex because of strain localization.
3.2.3
Diffusion-controUed glide
Inspection of Figure 3.6b,c shows that, at a given dislocation velocity (given applied strainrate), deformation can take place in the low-velocity regime (v < Vl) provided the temperature is high enough (T --> T3). The aim of this section is to derive analytical expressions for this low-velocity/high-temperature regime. Consider the dislocation described schematically in Figure 3.4. When it moves along the y-direction, the solute atoms are assumed to move only in this direction.
69
Interactions Between Dislocations and Small-size Obstacles
L
"C2
! ! !
)
v(1:2)
vl
:
v2
v(q)
v
j/ I
V2
)
V(rl)
V(~al )
V
Figure 3.7. Origin of stress instabilities: (a) on the basis of Figure 3.6b and c; and (b) in the case vl ~ 0 (low temperature).
The dislocation velocity is estimated in the frame of two different approximations. For Uin t ~ kT, Eqs. (3.23) and (3.26) show that the cloud is highly asymmetrical, with a high concentration of solute atoms in the region corresponding to Uin t < 0 and a weak depletion in the opposite one (Uint > 0). This property results from the asymmetrical shape of the exponential function. According to Friedel (1964, p. 410), in the limit of unsaturated clouds the solute atoms are all very close to the dislocation core, forming a row of pinning points with average distance )t (Figure 3.8). The work done by the applied stress during the diffusion of one solute atom over one interatomic distance is ~-b2)t.
70
Thermally Activated Mechanisms in Crystal Plasticity
Figure 3.8. Schematic description of the diffusion-controlled glide of a dislocation pinned by a row of solute atoms.
The frequency of this event is VD ~-exp --
T
in the forward direction, and l b ( Ud-q-Tb2~) -- VD exp --
2
-A
kT
in the backward direction (cf. Section 7.2.2.2) (Vo is the Debye frequency and Ud is the solute-diffusion activation energy). The dislocation velocity is accordingly:
v = -~ Vo--~ exp -
kT
-
kT
or, assuming that rb2A << kT v=
vD~exp
--~
--Dk---T
This yields "r -
1 kT v b2 D
(3.28)
Under these circumstances, the friction force is independent of the solute concentration in the core. For Uin t < kT, the cloud is symmetrical. The local concentration (Eq. (3.26)) can indeed be approximated by c - Co ~ co(Uint/kT) so that c - Co has opposite values above and below the slip plane (points y, z and y, - z in Figure 3.4). The average concentration around the dislocation is thus Co, which a priori should yield no frictional force. However, a detailed description shows that this statement is erroneous. It is based on the works of Cottrell and Jaswon (1949) and Fuentes-Samaniego (1979), summarized by Hirth and Lothe (1982). It takes into account the energy dissipated by the solute atoms moving either in the direction of the dislocation or in the opposite one. The dynamic concentration profile
71
Interactions Between Dislocations and Small-size Obstacles
is as in Figure 3.5. For the sake of simplicity, the dislocation can be considered as surrounded by an excess of solute atoms on one side and by an excess of holes on the opposite one. The motion of a hole in one direction corresponds to the motion of a solute atom in the opposite one. All atoms and holes are assumed to move along the y-axis in such a way that Eq. (3.27) applies. The energy dissipated corresponds to the average work done by the solute atoms and holes in excess under the driving forces F = -(OUint[Oy) and - F , respectively. The elementary friction force in a strip of width 6z is thus equal to the sum of these forces, namely:
~Z ~+oo EF(6z)=
~
~)Uint
-oo - ( c - c ~
(3.29)
Oy dy
For z > 0, the expression in the integral is the work done by the solute atoms in excess. It is positive for y < 0 (OUint]Oy < 0) and negative for y > 0 (OUint]Oy > 0). The integral is, however, strictly positive because of the asymmetrical shape of c(y) shown in Figures 3.5 and 3.6a. For z < 0, the expression in the integral can be understood as the work done by the holes (concentration c - Co) under the driving force (OUint]Oy). The integral is also positive. Replacing (OUint]Oy) by its value deduced from Eq. (3.27) yields: ~Z E F ( r z ) = --~kT
[
V -~
+oo (C -- Co)2 dy -~ ~
-co
c
coc] Oy
+oo
The second integral in the brackets is I c - Co In cl_oo = 0. Since Uin t "~ kT we have c - c o ~ co Uint]kT and c ~ Co, whence: co
EF(rz)
-- D k T O v r z
Uin t dy
and, using Eq. (3.23) 1
v~z Co
rrkT
(z 2 -Jr-y2)2 dy
Then, EF(rz)=
kT
v r z Co
(/zbA[~ ) 2"tr 3kT 2z
1 A typing error in Eq. (18.53) of Hirth and Lothe (1982, 1992) has been corrected.
(3.30)
72
Thermally Activated Mechanisms in Crystal Plasticity
After integration along the z-direction, between a minimum value Zo equal to a few interatomic distances and a maximum value estimated as D/v, the resulting friction stress is: 1 vb t x A ~ A ~ D ~"= - ~ c ~lx -D k T 12 In VZo or
/xb /xAO 2 ~" "~ Co D k T
g2
v
(3.31)
The friction stress is thus similar to that given by Eq. (3.28), however, with a more complex constant factor. Taking Co -'~ 10 -2, ~ b 3 / k T -'- 102 and AD,/I2 --- 10 -l this factor has the same value as the one in the first approximation ((1/b2)(kT/D) in Eq. (3.28)). Another estimation proposed by Hirth and Lothe is based on a simplified interaction energy, varying as 1/r as in Eq. (3.2), but independent of 0. The result is twice the above estimation. Cottrell and Jaswon (1949) showed that taking into account diffusion along the z-direction introduces only a factor of 1/2 in Eq. (3.31). Therefore, it seems that Eq. (3.28) provides a satisfactory approximation, valid in the whole temperature range.
3.3.
COMPARISON WITH EXPERIMENTS
The f o r e s t mechanism Experimental evidence of the forest mechanism comes from the "Cottrell-Stokes" experiments in single and polycrystalline FCC metal (Cottrell and Stokes, 1955; Thornton et al., 1962), recently discussed by Nabarro (1990) and Saada (1999). The deformation curves exhibit strong linear and parabolic hardening stages (called stages II and III, respectively) which result from the storage of forest sessile dislocations. The flow stress is usually decomposed into an athermal or internal stress, z~,, and a temperature and strainrate dependent effective stress, r* (Section 2.1.4.1). The internal stress is irreversible upon strain-rate and temperature changes. It is due to the long-range stress field of forest dislocations and to the formation of attractive junctions. The effective stress is reversible upon strain-rate and temperature changes. Figure 3.9 shows that the ratio of flow stresses at high and low temperatures is almost independent of strain or total stress. This CottrellStokes law indicates that the effective stress remains approximately proportional to the internal stress during deformation. This shows that both stresses are related to the same obstacles, namely the forest dislocations. In addition, the strong decrease of the activation volumes with increasing strain and stress that they observe shows that the density of thermally activated obstacles increases with strain (Eqs. (3.6) and (3.11 b)) as expected for a forest mechanism.
3.3.1
73
Interactions Between Dislocations and Small-size Obstacles O'293/O'90
,J X
0.8
x
--C
0
0
0 0.7
0
I
I
I
I
I
lO
20
30
40
50
~"
e [%]
Figure 3.9. Ratio of flow stresses at room temperature and in liquid air, of an AI single crystal. Curve A was obtained by transitions from 293 to 90 K, and curve B by the reverse transitions. From Cottrell and Stokes (1955).
More quantitatively, the average distance between forest dislocations is related to the internal stress by the usual relation: d = / z b / r u . Using Eqs. (3.6) and (3.1 lb) then yields the activation volume: V = I~b2w/r~
Since for high strains we can assume ~'u ~ ~', we obtain" V r ~ tzb2w
which is a constant. This property is indeed satisfied, at least at large strains, as shown in Figure 3.10. Saada (1999), however, pointed out that this analysis does not take into account cross-slip that is also active in stage III. The exact origin of thermal activation can be the formation of jogs on intersecting dislocations or the recombination of short attractive junctions. For a very low forest density, the activation volumes are no longer proportional to the distance between adjacent trees. This behaviour has been interpreted as a breakdown of thermal activation considerations based on fully relaxed dislocation configurations when the mean spacings in the glide plane become very large (Argon and East, 1979). 3.3.2
D i s l o c a t i o n s - s o l u t e atoms interactions
The different situations described in Sections 3.1 and 3.2 are observed experimentally in many alloys as a function of temperature. Figure 3.11 shows the temperature dependence of the CRSS and of the stress-strain rate sensitivity in several CuA1 alloys. Four domains can be identified. In domain 1 (very low temperatures) the phonon frictional force becomes so low that dislocation velocities can reach very high values. The inertial effects are
74
Thermally Activated Mechanisms in Crystal Plasticity Vz e.v.
III 9
9
"7
9
9
9 9
9
9 9
yO O0
|
I
I
I
0
I
I
I
I
I
"
0.5
"
1.0
e
Figure 3.10. Curve V~" as a function of strain in a copper single crystal strained at 473 K. I, II, III refer
to the classical hardening stages. From Thornton et al. (1962).
then sufficiently important to help dislocations passing through solute atoms ("underdamped" dislocation motion, see Figure 3.12). This phenomenon has been described by Granato (1971). -
Domain 2 corresponds to the "overdamped" thermally activated motion of dislocations across fixed obstacles described in Section 3.1. Sometimes solute atoms can already start to move to dislocations (Flor and Neuhauser, 1980).
-
Domain 3, in which stress instabilities are associated to a small yield-stress anomaly and to a negative stress-strain rate sensitivity, corresponds to the situation described in Section 3.2.2.
-
In domain 4, dislocation motion is controlled by the drag of solute atmosphere (Section 3.2.3).
The corresponding dislocation kinetics have been clearly identified in the in situ TEM experiments of Monchoux and Neuhauser (1987) in CuGe alloys. In particular, in the domain of stress instabilities, sudden vigorous dislocation movements shook the specimen and made observations impossible. Domains 2 - 4 are described in more detail in what follows.
75
Interactions Between Dislocations and Small-size Obstacles CRSS [MPa] 35
30 Cu 15 at. % A1 25 ' 20
~
h k ~ v / C u 10 at. %AI 5 at. % A1
~, ~,/.
,
""
I
it I
10
i J
[]
0
l
l
I
l
200
400
600
800
1
2
3
|
9 w
1000 T[K] 4
b3 s(~) ~0~
b)
200
I
150
100
-1--
2
-'-"
3
-i-~4
--
pLc i
50
-50 0
i 200
i 400
i 600
i 800
" T [K]
Figure 3.11. Mechanical properties of CuAI alloys (from a review by Neuhauser and Schwink, 1993). Temperature domains 1 - 4 are described in the text. (a) CRSS as a function of temperature. Data from Suzuki and Kuramoto (1968), Startsev et al. (1979), Nixon and Mitchell (1981) and Neuhauser et al. (1990). The PLC regime is indicated by dotted lines and bars that refer to the amplitudes of the stress instabilities. (b) Stress-strain rate sensitivity of C u - 15% AI as a function of temperature and PLC domain. Data from Kopenaal and Fine (1962), Komnik and Demiskii (1981) and Neuhauser et al. (1990).
76
Thermally Activated Mechanisms in Crystal Plasticity
t,.)
with inertia T F i g u r e 3.12. Schematic description of the influence of inertia effects at low temperature.
3.3.2.1 Domain 2: thermally activated motion across fixed obstacles. As discussed by Wille et al. (1987), comparison between theory and experiment is complicated by the following points: (i)
Results in domain 2 must be extrapolated to 0 K in order to determine Ema x and "rmaxThis is rather questionable when the extension of domain 1 is large. (ii) There is usually a spectrum of obstacles of different strengths. (iii) The solute distribution is often not uniform and segregation can take place at stacking faults. Short-range ordering or clustering may also alter the mobility of dislocations. (iv) Deformation tends to be heterogeneous. In order to avoid the above effects, Wille et al. (1987) have selected the CuMn system, where the tendency to form short-range order is negligible, the stacking fault energy is almost independent of the Mn concentration and the size effect is very large. Figure 3.13a shows the temperature dependence of the CRSS. Domain 1 is smaller than in Figure 3.11a, which allows for an easier extrapolation to 0 K. Figure 3.13b shows that this stress varies with temperature according to Eq. (3.20) (Cottrell-Bilby potential). The activation volume in Figure 3.14a shows that domain 1 extends from 0 to 25 K. Above 25 K its stress dependence obeys Eq. (3.21a)if the variation of (1 - (T/7"max)0"46)1/2 is neglected (Figure 3.14b). It also varies with temperature according to Eq. (3.21b), as shown in Figure 3.14c. The values of "/'maxand Emax corresponding to the best fits are shown in Table 3.1. Emax ranges between 1.2 and 1.4 eV and "/'max increases with the solute concentration. The internal stress -r~, is assumed to be the high-temperature stress in Figure 3.13a. Fma x is deduced from Ema x included in Eq. (3.19a), assuming that w = 2.5b. Then, Eq. (3.16b) for
77
Interactions Between Dislocations and Small-size Obstacles CRSS 1'
[MPa]/
60
50
40
30
20
10
_•••,•,••'••"<•
7.6 at % Mn
-
~
2.0 at % Mn I
I
I
I
I
I
I
I
I
50
100
150
200
250
300
350
400
450
)
T[K]
1
1.0
0.8
0
7.6 at.% Mn
o
3.8 at.% Mn
v
2.0 at.% Mn
0.6
0.4
0
0.2
,ik
I
I
I
I
,
0.2
0.4
0.6
0.8
1.0
(T/To)2/3
Figure 3.13. CRSS in CuMn alloys (from Wille et al., 1987): (a) as a function of temperature; and (b) plotted so as to check the validity of Eq. (3.20).
78
Thermally Activated Mechanisms in Crystal Plasticity
V[nm3]l,T ~.~ I
o
7.6 at.% Mn
121 3.8 at.% Mn v 2.0 at.% Mn
10
0
I
I
I
I
I
50
1O0
150
200
250
T [K]
b3
--V--X103
T - 78K
T = 295K 9
CuMn CuGe
0 -
2'0
3'0
4:0
(rmax)0"46.rTM [MPa] Figure 3.14. Activation volumes in CuMn alloys (from Wille et al., 1987): (a) as a function of temperature; (b) plotted so as to check the validity of Eq. (3.21a), and (c) plotted so as to check the validity of Eq. (3.21b).
79
Interactions Between Dislocations and Small-size Obstacles V Vo
Vo = 0.69 max "t'max
0 7.6 at.% Mn n 3.8 at.% Mn v 2.0 at.% Mn
j
/O o 0;.1
0.2
0.3
014
0~
r/r o
Figure 3.14. (continued)
F = Fma x and 7" -- 7"max y i e l d s the a t o m i c c o n c e n t r a t i o n o f o b s t a c l e s c b
=
(b]d) 1/2. All these
p a r a m e t e r s are g i v e n in T a b l e 3.1. This table s h o w s that: -
T h e e s t i m a t i o n o f fl (Eq. (3.9)) e n s u r e s that the F l e i s c h e r - F r i e d e l a p p r o x i m a t i o n is valid.
-
The concentration
o f o b s t a c l e s Cb is 20 t i m e s s m a l l e r than the a v e r a g e solute
c o n c e n t r a t i o n c. In addition, the i n t e r a c t i o n e n e r g y Emax is f o u n d to be a b o u t 1.3 eV. Table 3.1. Mechanical parameters of CuMn alloys (from Wille et al., 1987). c (at.%) %/~/ "rmax (MPa) TO (K) Emax (eV) 'ru (MPa)
wlb Cb X 10-4
LF('Cmax)/b /3X 10 -2
0.4 Mn
1.2 Mn
2.0 Mn
3.8 Mn
7.6 Mn
23.5 10.3 658 1.34 2.0 _+ 0.2 2.5 1.33 324 2.9
25 20.7 631 1.36 5.7 +_ 0.3 2.5 5.1 164 5.6
23.5 25.0 610 1.21 10.3 _+ 0.6 2.5 10.8 120 8.1
25.8 40.2 569 1.28 14 + 1 2.5 23.5 79 12.0
26.5 58.2 533 1.23 20.4 + 1.5 2.5 53.8 53 18.3
80
Thermally Activated Mechanisms in Crystal Plasticity Since that expected for individual solute atoms is predicted to be smaller than 0.4 eV, thermal activation is thought to correspond to the crossing of doublets or triplets of solute atoms. The athermal stress z~ increases with the solute concentration. Since it has been observed that z~ is independent of the dislocation density, which can be changed by annealing, it seems accordingly that clusters involving more than three solute atoms, which cannot be overcome by thermal activation, are at the origin of this stress.
This analysis shows that calculations in Section 3.1 are reliable, at least as long as solute atoms are really immobile. In CuZn, on the contrary, Flor and Neuhauser (1980) measured non-logarithmic relaxations for which they assumed an increase of the activation energy with time due to solute segregation. In addition, Emax is observed to increase with increasing temperature, which indicates that the yield-stress is controlled by obstacles of different strengths at different temperatures. These results are of course much more difficult to compare with theoretical estimates.
3.3.2.2 Domain 3: stress instabilities and PLC effect. For a review of the experimental results in this temperature domain, the reader can refer to Strudel (1980) or Kubin and Estrin (1991). More recently, an exhaustive study of stress instabilities was made in binary CuMn and CuA1 alloys by Schwink and Nortmann (1997). The activation energies of the onset and of the end of the PLC effect are well below the volume-diffusion energy of Mn and A1 atoms in a Cu lattice, which suggests the occurrence of pipe diffusion in the core of dislocations. Surprisingly, plasticity in this domain may often be controlled by the movement of screw dislocations which are less subjected to pinning than edge ones. The release of screw segments generates fresh edge segments that can form dislocation sources (Neuhauser and Schwink, 1993; Suzuki, 1985). Since screw segments are also often subjected to Peierls-type friction forces, a combination of these two mechanisms (Peierls friction and solute effect) is expected in many cases. Note that solute atoms can also modify the core structure of sessile screw dislocations, e.g. in titanium and zirconium (Chapter 6). Such an interaction is different from that treated in this chapter. It may, however, contribute to enhance static and dynamic strain ageing when a Peierls mechanism acting on screw dislocations is rate controlling. The anomalous behaviour of several intermetallics, in which straight screw dislocations move in bursts, may be explained in this way (see Chapter 10).
3.3.2.3 Domain 4: glide controlled by solute.diffusion. Dislocation glide controlled by diffusion of solute atoms is often considered to explain the creep properties of "class I"
Interactions Between Dislocations and Small-size Obstacles
81
alloys (see, e.g. Takeuchi and Argon, 1976). Taking a dislocation density proportional to the stress squared, the resulting creep rate is indeed expected to vary as the third power of stress and the activation energy is expected to be that of solute-diffusion in agreement with some experimental results. In many cases, however, the problem is considered to be too complex to have simple solutions (see, e.g. Poirier, 1976).
REFERENCES
Argon, A.S. & East, G.H. (1979) in Strength of Metals and Alloys, Eds. Haasen, P., Gerold, V. & Kostorz G., Pergamon Press, Oxford, p. 9. Cottrell, A.H. & Bilby, B.A. (1949) Proc. Phys. Soc. London A, 62, 49. Cottrell, A.H. & Jaswon, M.A. (1949) Proc. Roy. Soc. A, 199, 104. Cottrell, A.H. & Stokes, R.J. (1955) Proc. Roy. Soc. A, 233, 17. Estrin, Y. & Kubin, L.-P. (1989) J. Mech. Behav. Mater., 2, 255. Fleischer, R.L. (1961) Acta Met., 9, 996. Flor, H. & Neuhauser, H. (1980) Acta Met., 28, 939. Friedel, J. (1964) Dislocations, Pergamon Press, Oxford. Fuentes-Samaniego, R. (1979) PhD thesis, Stanford University, California. Granato, A.V. (1971) Phys. Rev. B4, p. 2196; Phys. Rev. Lett. 27, p. 660. Haasen, P. (1979) in Dislocations in Solids, vol. 4, Ed. Nabarro, F.R.N., North Holland, Amsterdam, Chap. 15. Haasen, P. (1983) in Physical Metallurgy, Part II, vol. 8, Eds. Cahn, R.W. & Haasen P., 3rd Edition, North Holland Physics Publishing, Amsterdam, p. 1341. Hirth, J.P. (1983) in Physical Metallurgy, Part II, vol. 8, Eds. Cahn, R.W. & Haasen P., 3rd Edition, North Holland Physics Publishing, Amsterdam, p. 1223. Hirth, J.P. & Lothe, J. (1982) Theory of Dislocations, 2 nd Edition, Wiley Interscience, New York; (1992) second reprint edition, Krieger Pub. Comp., Malabar, Florida. Kocks, U.F., Argon, A.S. & Ashby, M.F. (1975) Thermodynamics and Kinetics of Slip, Pergamon Press, Oxford. Komnik, S.N. & Demiskii, V.V. (1981) Czech. J. Phys. B, 31, 187. Kopenaal, T.J. & Fine, M.E. (1962) Trans AIME, 224, 347. Kubin, L.P. & Estrin, Y. (1990) Acta Met. Mat., 38, 697. Kubin, UP. & Estrin, Y. (1991) J. Phys. III, 1,929. Labusch, R. (1970) Phys Stat. Sol., 41, 659. MacCormick, P.G. (1972) Acta Met., 20, 351. Monchoux, F. & Neuhauser, H. (1987) J. Mater. Sci., 22, 1443. Mulford, R.A. & Kocks, U.F. (1979) Acta Met., 27, 1125. Nabarro, F.R.N. (1990) Acta Metall. Mater., 38, 161. Neuhauser, H. & Schwink, C. (1993) in Material Science and Technology, vol. 6, Eds. Cahn, R.W., Haasen, P. & Kramer E.J., VCH Verlag, Weinheim, p. 191. Neuhauser, H., Plessing, J. & Schiilke, M. (1990) J. Mech. Behav. Met., 2, 231. Nixon, W.E. & Mitchell, J.W. (1981) Proc. Roy. Soc. London A, 376, 343. Poirier, J.P. (1976) Plasticit~ ?l Haute Tempdrature des Solides Cristallins, Eyrolles, Paris.
82
Thermally Activated Mechanisms in Crystal Plasticity
Saada, G. (1999) Deformation-Induced Microstructures: Analysis and Relation to Properties, Proceedings of 20th Ris~ International Symposium on Materials Science, Eds. Bilde-sorensen, J.B., Cartensen, J.V., Hansen, N., Jensen, D.J., Leffers, T., Pantleon, W., Pedersen, O.B. & Winther G., Rise National Laboratory, Roskilde, Denmark, p. 147. Sakamoto, M. (1981) Bull. Jpn. Inst. Met., 20, 912. Schwink, Ch. & Nortmann, A. (1997) Mat. Sci. Eng. A, 234-236, 1. Startsev, V.I., Demirskii, V.V. & Komnik, S.N. (1979) in Strength of Metals and Alloys, Eds. Haasen, P., Gerold, V. & Kostorz G., Pergamon Press, Oxford, p. 265. Strudel, J.L. (1980) in Dislocations et D~formation Plastique, Eds. Groh, P., Kubin, L.P. & Martin J.L., Les Editions de Physique, Les Ulis, p. 199. Suzuki, H. (1985) in Strength of Metals and Alloys, Eds. Mc Queen, H.J., Bailon, J.P., Dickson, J.L., Jonas, J.J. & Akben M.G., Pergamon, Toronto, p. 1727. Suzuki, H. & Kuramoto, E. (1968) Trans. JIM, 9(suppl.), 697. Takeuchi, S. & Argon, A.S. (1976) Acta Met., 24, 883. Thornton, P.R., Mitchell, T.E. & Hirsch, P.B. (1962) Phil. Mag., 7, 337. Van den Beukel, A. (1975) Phys. Star. Sol. A, 30, 197. Wille, T.H., Gieseke, W. & Schwink, C.H. (1987) Acta Met., 35, 2679. Yoshinaga, H. & Morozumi, S. (1971) Phil. Mag., 23, 1367.