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Proc. R. Soc. A (2008) 464, 2549–2559
doi:10.1098/rspa.2008.0116
Published online 13 May 2008
Dislocation pile-ups in Fe
at high temperature
B Y S. P. F ITZGERALD *
AND
S. L. D UDAREV
EURATOM/UKAEA Fusion Association, Culham Science Centre,
Abingdon OX14 3DB, UK
Dislocation ‘pile-ups’ occur in crystals when a number of similar dislocations, gliding in a
common slip plane, are driven by an applied stress towards an obstacle that they cannot
overcome. In contrast to dislocation walls, pile-ups give rise to a long-range stress field,
and their properties strongly influence the plastic behaviour of the crystal as a whole. In
this paper, we apply the analytic model of a pile-up (due to Eshelby, Frank and Nabarro)
to a cubic crystal. Full anisotropic elasticity is used, and the model is extended to predict
the plastic displacement generated by a dislocation source during the formation of a pileup. The results are applied to Fe close to the temperature of the a–g phase transition,
where the inclusion of anisotropy leads to a strikingly different prediction from that of
the isotropic approximation.
Keywords: dislocations; anisotropic elasticity; iron
1. Introduction
In this paper, we investigate dislocation pile-ups within the framework of
anisotropic elasticity theory, characterized by the full tensor of stiffness constants
cijkl. In the cubic symmetry case, this tensor is fully described by defining three
independent stiffness constants C11, C12 and C44 (Nye 1985), rather than two as
in the case of the isotropic elasticity approximation (Hirth & Lothe 1991).
Our interest in the problem stems from the need to understand the plastic
behaviour of iron and ferritic steels at elevated temperatures, stimulated by the
development of ferritic–martensitic and oxide dispersion-strengthened steels for
fusion and nuclear applications. While the current designs of fusion power plants
involve reduced-activation ferritic–martensitic steels as the first wall structural
materials and assume operating temperatures below 5508C, considerable
efficiency gains can be made by raising this temperature (Zinkle 2005). Another
example where understanding high-temperature plasticity is essential is the case
of a sulphur–iodine hydrogen plant driven by a very high-temperature nuclear
reactor operating at 9508C (Hoffelner 2005).
Iron is the most elastically anisotropic of all the body-centred-cubic (bcc)
metals (Reid 1966; Bacon 1985). On the fundamental level, the anisotropy of
iron is related to the fact that the bcc crystal phase is stabilized by magnetism,
* Author for correspondence (steve.fi[email protected]).
Received 17 March 2008
Accepted 16 April 2008
2549
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2550
S. P. Fitzgerald and S. L. Dudarev
and in a non-magnetic configuration iron would adopt a hexagonal close-packed
crystal structure (Guo & Wang 2000). At elevated temperatures, the magnetic
fluctuations erode the stability of the bcc phase, giving rise to the a–g bcc–fcc
phase transformation (Hasegawa & Pettifor 1983). This transition is associated
with the softening of one of the elastic constants of the material in the direction
of the pathway of the phase transformation (Hasegawa et al. 1985). This
softening has a strong effect on the self-energies of dislocations in iron at high
temperatures (Dudarev et al. 2008), as well as on the dislocation–dislocation
interactions and high-temperature plasticity.
When dislocations pile up against an obstacle, driven by an applied stress,
the effect at a point beyond the obstacle is to amplify the applied stress, by a
factor proportional to the number of dislocations involved (Eshelby et al. 1951;
Bacon & Hull 2001). If the barrier is a grain boundary, the increased stress can
activate dislocation sources in neighbouring grains and thus trigger yielding
(Hall 1951; Petch 1953). The elastic properties of pile-ups, as derived by Eshelby
et al. lead to the experimentally verified Hall–Petch relationship between yield
stress and grain size (Armstrong et al. 1962). Dislocation pile-ups under stress have
been observed by electron microscopy (see, e.g. Mughrabi 1968) and in molecular
dynamics simulations (Schiøtz 2004), and although alternative interpretations of
the Hall–Petch equation are possible (Li & Chou 1970), the notion of pile-ups
provides the link between the microscopic properties of dislocations and the
macroscopic yield behaviour of polycrystals (Armstrong 2005).
In this study, we consider a set of infinitely long straight edge dislocations with a
common Burgers’ vector, lying in a common slip plane, in the principal dislocation
h100i{011} and h100i{001}.1
configurations of bcc Fe: h111if110g,
h111if112g,
A similar immobile dislocation lies at the origin, and a source at some distant
point x0 produces dislocations under the influence of an applied stress t0. Although
dislocations in real crystals are in general curvilinear, this effectively twodimensional idealization is analytically tractable and as such provides insight that
a more realistic numerical calculation cannot. Also, if the dislocations are being
produced as growing loops by a Frank–Read (Frank & Read 1950) source, segments
piling up at the immobile dislocation can be reasonably approximated as straight
and parallel. We restrict our attention to edge dislocations, as they are the most
relevant to pile-ups in Fe. In a bcc lattice, screw dislocations can cross-slip at
sufficiently high temperatures, and hence are not obliged to form such linear arrays.
In their paper in 1951, Eshelby, Frank and Nabarro (EFN) presented a
detailed analytical model of dislocation pile-ups in various situations (Eshelby
et al. 1951). We apply their analysis to the above situation in a cubic crystal,
using the integral formulation of anisotropic elasticity theory (see, e.g. Bacon
et al. 1979) to calculate the forces between dislocations involved in the pile-up,
which are required as an input to the EFN model. The results are then applied to
Fe at temperatures approaching that of the a–g phase transition at Ta–gZ9128C.
1
Only the latter configuration allows the derivation of compact formulae for the quantities
involved. However, as the temperature increases towards Ta–g, the h100i{001} configuration
becomes the most important, because the strain energy of h100i{001} dislocation segments falls
sharply (see later). This is consistent with electron microscope observations of dislocation loops,
which show the dominant occurrence of h100i-type orientations of the dislocation segments at
elevated temperatures (Masters 1963; Little & Eyre 1973).
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Dislocation pile-ups in Fe at high temperature
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Most investigations of dislocation behaviour use elasticity theory in the
isotropic approximation, as it affords a considerable simplification for both
analytic and numerical studies. However, at high temperatures, Fe displays
unusual elastic properties related to phonon softening (Hasegawa et al. 1985),
that the isotropic elasticity approximation cannot take into account, and the
effect of this softening on dislocation pile-ups is significant. As well as being of
basic interest in its own right, this behaviour demonstrates the importance
of including anisotropic effects in any calculation or simulation concerning Fe, or
ferritic steels, at these elevated temperatures.
2. Dislocation interactions and pile-ups
The magnitude of the repulsive force between two similar dislocations is given by
A/r, where r is their separation and A depends on the material’s elastic moduli
and the particular geometrical configuration. This parameter represents the
‘strength’ of a pile-up, i.e. its resistance to compression by external forces. It can
be calculated in elasticity theory, though in the general anisotropic case there is
no simple closed-form expression. However, in the simplest case of h100i{001}
dislocations, it takes the form
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C12 C C 0
C44 C 0
AZ
;
ð2:1Þ
4p
ðC12 C C44 C C 0 ÞðC12 C 2C 0 Þ
where C 0 Z(C11KC12)/2 (see appendix A). The other configurations involve
integral expressions which we evaluate numerically.
The starting point of the EFN model (Eshelby et al. 1951) is a polynomial of
order n defined as
f ðxÞ Z ðx K x 1 Þðx K x 2 Þ.ðx K xn Þ;
ð2:2Þ
where the roots xi are the positions of n dislocations in units of jbj (since the
dislocations are straight, parallel and confined to a slip plane, the problem is
essentially one-dimensional). By balancing the inter-dislocation forces with those
due to an applied shear (t0), a differential equation for f can be derived, with the
0
solution L nC1
, the first derivative of the (nC1)th Laguerre polynomial (see e.g.
Boas 2005), defined by
Lm ðzÞ Z
m
X
1 z dm m Kz
m!
zi
i
ðz
e
Þ
Z
ðK1Þ
e
:
m! dz m
i!ðmKiÞ! i!
i Z0
ð2:3Þ
The positions of the n dislocations are given by the zeros of f: for nZ20 say,
we have
fxi g Z
A
ð0:1749; 0:5873; .; 68:38Þ:
2t0
ð2:4Þ
The scale factor A/2t0, entering via the derivative, encodes the relative
magnitudes of the applied and interaction forces. It is a length, corresponding to
half the separation required for the interaction force between two dislocations
to equal the applied force. Figure 1 shows the equilibrium positions of the
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2552
S. P. Fitzgerald and S. L. Dudarev
0
20
40
60
80
100
120
140
160
180
Figure 1. Equilibrium positions for nZ5, 10, 15, 20, 30, 50.
dislocations for various values of n. As n increases, the length of the glide plane
occupied by the pile-up increases, but the dislocations nearer the source become
more closely spaced, since they feel the combined repulsion of an increasing
number of dislocations acting with the applied stress. Conversely, the outermost
dislocations become more widely spaced, as the increasing repulsion counteracts
rather than amplifies the applied stress.
At large distances from the pile-up, the field approaches that of nC1
dislocations
situated at the origin. This can be confirmed by expanding f 0 =f Z
Pn
K1
Z force=A about xZN:
iZ1 ðx K x i Þ
00
LnC1
ðxÞ n nðn C 1Þ
Z C
C Oðx K3 Þ;
0
2
LnC1 ðxÞ
x
x
ð2:5Þ
so the total field w(nC1)/x when the field of the immobile dislocation is included.
3. Dislocation sources and plastic displacement
The Frank–Read source can be thought of as a segment of dislocation line pinned
at both ends. If the sides of the loops emitted by the source pile up against an
immobile straight dislocation, as discussed in the introduction, their stress fields
acting back on the source segment will eventually cancel the applied stress, and
the source will stop producing dislocations. Neglecting the small threshold stress
required to initiate the multiplication, the condition of this equilibrium is simply
that of zero force, which is identical to that used to derive the equilibrium
positions in the EFN model.
Hence, we can regard the source as another dislocation with the same Burgers’
vector as the others (a typical source has initial length of order 104b, so this
treatment is reasonable) and use the same solution as in §2. The solution for n
0
mobile dislocations is now f Z LnC2
, and the positions are given as before by the
scaled zeros of f
fxi g Z
A
ð0:1669; 0:5605; .; 61:01; 72:11Þ;
2t0
ð3:1Þ
for nC2Z22. The above polynomial has nC1 zeros, the first n of which are the
equilibrium coordinates of the mobile dislocations. Note how these are reduced
compared with equation (2.4), owing to the source’s own repulsion. The greatest
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Dislocation pile-ups in Fe at high temperature
root is the position of the source, which is by definition fixed at some x 0, so
for given A, we need to find a consistent relation between the imposed force
t0 and n.
0
ðyÞZ 0 grows approximately
The greatest root y0 of the scaled equation LnC2
linear in n, and a good (1%) fit for the first 120 is given by
y0 Z n1 nK n2 ;
n1 Z 3:830;
ð3:2Þ
n2 Z 4:02:
Returning to normal units xZ(A/2t0)y gives the linear relationship
t0 ðnÞ Z
A
y ðnÞ:
2x 0 0
ð3:3Þ
Furthermore, we can determine the total plastic displacement at a given t0 by
summing the distances travelled by the n dislocations to their equilibrium
positions. Though the positions are in principle known, solving an nth order
polynomial can be numerically inconvenient as n gets large. Fortunately, we need
only the sum of the roots, which is given by
N
N
X
X
a
fyi jf ðyi Þ Z 0g Z nK1 for polynomial f ðyÞ Z
ap y p ;
a
n
i Z1
pZ1
so for the total distance travelled by the dislocations we have
n
nC1
X
X
A
xi Z ðn C 1Þx 0 K
xi Z ðn C 1Þx 0 K
d Z nx 0 K
2t0
i Z1
i Z1
nC1
X
ð3:4Þ
!
yi
iC1
A
3x 2 t
ðn C 2Þ z 0 0 for large n;
Z ðn C 1Þ x 0 K
2t0
8A
ð3:5Þ
where in the second line we used equations (2.3) and (3.4). The crystal’s plastic
displacement is then given by jbjd/x 0 (Bacon & Hull 2001), and depends only on
n, as it should; for given n, the equilibrium positions of the dislocations are
dictated by the scaling of the roots of f, which is fixed by x 0. The force which
must be applied to reach n depends on the elastic properties of the crystal,
encoded in A. If we imagine the applied force t0 increasing incrementally, and
assume a quasistatic process where the dislocations instantaneously reach
their updated equilibrium positions with each increment in t0, we can calculate
t0 and d at each n and determine the relationship between the two.
4. Application to a-Fe at high temperatures
Figure 2a shows the elastic constants of single-crystal a-Fe, measured at
temperatures from 25 to 9008C; the data are taken from Dever (1972), see also
Fisher & Dever (1970). Even at room temperature, iron is far from being
elastically isotropic; the so-called anisotropy ratio C44/C 0 , equal to 1 in the
isotropic limit, is approximately 2.4 at 258C and rises to 7.4 at 9008C. Moreover,
C 0 is approaching zero, which indicates the instability of the bcc crystal structure
at these high temperatures. Calculations (Hasegawa et al. 1985) predict this
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2554
S. P. Fitzgerald and S. L. Dudarev
(b)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
elastic constants (100 GPa)
elastic constants (100 GPa)
(a)
100 200 300 400 500 600 700 800 900
temperature (°C)
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0 100 200 300 400 500 600 700 800 900
temperature (°C)
Figure 2. (a) Elastic constants of a-Fe versus temperature (Dever 1972). Pluses, C 0 ; crosses, C44;
asterisks, C12. (b) Fit of the data for C 0 below 7508C, plus artificial point at (0,9128C). Pluses, data;
dashed line, fit.
effect as the a–g bcc–fcc phase transition temperature Ta–g is approached. Of
course, iron remains solid beyond Ta–g, but there is still a marked effect on the
crystal properties as Ta–g is approached from below.
The data in figure 2 show a deviation from smooth behaviour from
approximately 7508C, which is associated with spin fluctuations in iron above
the Curie temperature; figure 2b shows the data for C 0 together with a function
of the form
T m2
0
;
ð4:1Þ
C Z m1 1K
Ta–g
which is reasonable near the point of this phase transition. The values of
parameters m1Z0.497, m2Z0.478 were obtained by fitting equation (4.1) to the
data below 7508C only, and by adding the artificial point (0, 9128C). Equation
(4.1) gives an economical fit to the data from the most trusted region, consistent
with the assumption that C 0 becomes small as T approaches 9128C. Figure 3
shows the value of A versus temperature for four configurations of edge
dislocation. The data points were calculated numerically, except for the
h100i[001] case, for which the analytic expression equation (2.1) is available.
2
The magnitude of the Burgers’ vector enters into the expression
pffiffiffi for A as jbj ; we
work in units of the lattice period, where jb100jZ1, jb111 jZ 3=2. In the isotropic
approximation, this implies that the interaction force for h111i-type dislocations
is always 0.75 times that for the h100i-type. The inclusion of anisotropy
demonstrates that this is not satisfied at high temperatures, where the strength
of interaction between h100i[001] dislocations falls steeply towards zero,
becoming weaker than that for all other configurations. The leading logarithmic
term in the expression for the elastic self-energy per unit length of an infinite
straight dislocation is proportional to A, and so the energies for the different
configurations behave as figure 3 as well, in agreement with Dudarev et al.
(2008). The key point is that for the h100i[001] configuration,
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
C 0 C12 C44
Ah100i½001 z
;
ð4:2Þ
4p C12 C C44
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2555
Dislocation pile-ups in Fe at high temperature
A (100 GPa (lattice const.)2/unit length)
0.18
0.17
0.16
0.15
0.14
0.13
0.12
0.11
0.10
0.09
0.08
0
100
200
300
400 500 600
temperature (°C)
700
800
900
Figure 3. Values taken by the dislocation interaction force constant A versus temperature for the
principal bcc dislocation configurations. Dot dashed line, h100i{011}; solid line, h100i{001}; dotted
line, h111if110g;
dashed line, h111if112g.
when C 0pis
ffiffiffiffiffiffismall, as the a–g transition is approached, and so it falls towards
zero as C 0 . This behaviour cannot be understood within isotropic elasticity,
since that approximation uses only one shear modulus C44, C 0 /m. Taking
mZ(C44CC 0 )/2, it is clear that the soft mode is overlooked, since although m
would fall to half the value of C44, it would not approach zero, and the extreme
softening behaviour would not appear. Figure 4 shows the variation with
temperature of Ah100i[001] calculated using anisotropic elasticity, and in the
isotropic approximation resulting from replacing the two shear moduli by their
average (quantities are relative to room temperature values2). The curves
marked ‘theoretical’ show the same quantities calculated with the fitted values
for the moduli, as described above. These indicate the expected behaviour if the
true values of the moduli more closely followed the theoretical predictions of
Hasegawa et al. (1985).
At lower temperatures, other mechanisms will probably bring about tensile
failure, including the motion of lower-mobility screw dislocations. However,
plastic flow will proceed via the path of least resistance, and at higher temperatures, the edge dislocation pile-ups will provide this. As the parameter A gets
smaller, the sources emit more dislocations, and the plastic displacement grows
faster (since it is proportional to AK1). Above some critical temperature, the
edge pile-ups will become the dominant source of plastic flow, and if we associate
tensile failure with some maximum plastic strain, the applied stress at which this
maximum strain is reached will be proportional to A.
The inset in figure 4 shows the variation with the temperature of the ultimate
tensile strength of three steels: Eurofer 97 (Mergia & Boukos 2008), a similar
ferritic–martensitic steel (Zinkle & Ghoniem 2000), and a type 310 stainless steel
(Shi & Northwood 1995). The first two steels undergo an a–g transition, while
the latter does not, maintaining a g-type structure across the temperature range
2
Neglecting anisotropy overestimates A by approximately 14% at 258C, rising to 63% at 9008C.
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2556
S. P. Fitzgerald and S. L. Dudarev
1.0
0.9
0.7
UTS/UTS (25°C)
A /A (25°C)
0.8
0.6
0.5
0.4
EF97
F / M steel
310SS
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0 100 200 300 400 500 600 700 800 900
0.3
0
100
200
300
400
500
temperature (°C)
600
700
800
900
Figure 4. Values taken by the dislocation interaction force constant A versus temperature for
h100i{001}-type dislocations (squares); solid line (theoretical) show the result if the adjusted fits
for the elastic moduli are used. Isotropic predictions are shown for both cases: circles, isotropic;
dashed line, theoretical isotropic. Inset: ultimate tensile strength of three steels versus temperature.
shown. Although the data in the inset and the main figure are not directly
comparable, qualitative conclusions can be drawn. In the region of their phase
transitions (which occur over a different temperature range than in iron), the two
ferritic–martensitic steels exhibit a sharp drop in tensile strength, the curves
resembling the square-root shape of the calculations of A for iron. The 310
stainless steel shows a far more gradual, approximately linear, decrease in
strength with temperature. This is more reminiscent of the isotropic calculation,
which does not capture the effect of the phase transition.
5. Conclusions
In this paper, we have studied the effect of crystal anisotropy on dislocation pileups in iron at high temperatures. The variation with temperature of interdislocation forces has been investigated for the principal configurations in bcc
iron, and an exact (as far as linear elasticity is concerned) analytical formula for
the strength of the interaction force between h100i{001} dislocations was
derived. The increasing discrepancy between the anisotropic and isotropic
approximations as temperature increases was established.
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Dislocation pile-ups in Fe at high temperature
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Essentially, the behaviour of the C 0 elastic modulus leads to a severe plastic
softening of the crystal as the a–g transition temperature is approached, which is
only revealed by elasticity theory when anisotropic effects are taken into account.
Since plastic deformation will proceed via the path of least resistance, pile-ups
involving h100i{001}-type dislocations will provide the dominant sources of
plasticity as they begin to collapse at high temperatures. Even if not present
initially, pinned h100i{001} segments can form via the reaction of two h111i
dislocations during plastic deformation. This may offer an explanation for the
precipitous fall in tensile strength at high temperature, which is observed in
steels that display the a–g phase transition, and absent in those that do not. This
is quite apart from any impact the behaviour of C 0 might have on creep. While
weakening repulsion between similar dislocations will undoubtedly influence
creep failure, the developing plastic instability will manifest itself as loss of
strength on much shorter time scales.
Although the situation considered was an idealized one, involving infinite
straight parallel dislocations, the elastic properties underlying the behaviour may
manifest themselves in more realistic situations. In particular, line dislocation
dynamics simulations, which currently offer the most promising means to
accurately model mesoscale crystal plasticity, treat curvilinear dislocations as
networks of straight segments (see, e.g. Bulatov & Cai 2006). Elasticity theory is
used to calculate the forces of interaction between them, and the results of §4
demonstrate the importance of including anisotropy when considering iron at
high temperatures.
We would like to thank Prof. R. Bullough and Prof. D. J. Bacon for their stimulating discussions.
Work at UKAEA was supported by the UK Engineering and Physical Sciences Research
Council, by EURATOM, and by EXTREMAT integrated project under contract number NMP3CT-2004-500253.
Appendix A. Calculation of A
The stress field of an infinitely long straight dislocation is given by Bacon
et al. (1979)
s1mn Z
1
c
b1 ½KMp Sis C Np ðNN ÞikK1 f4pBks C ðNM Þkr Srs g;
2pjrj mnip s
ðA 1Þ
where cmnip is the elastic constant tensor; r is the vector from the dislocation to
the field point at which the stress is evaluated; b1 is the dislocation’s Burgers’
vector; MZr/jr j; and N Z to M . The bracketed quantities are defined as
ðABÞjk Z cijkl Ai Bl ;
for any two vectors A, B, and in a cubic crystal
cijkl Z C12 dij dkl C C44 dik djl C dil djk C 2ðC 0 K C44 Þdijkl ;
ðA 2Þ
ðA 3Þ
where C 0 Z(C11KC12)/2 and the symbol dijkl is equal to one if all the suffices are
equal, and zero otherwise.
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S. P. Fitzgerald and S. L. Dudarev
Figure 5. The geometry under consideration.
The two matrices B and S depend only the elastic constants and the line
direction t (figure 5).
ð
1 2p
Sks ZK
ðnnÞkjK1 ðnmÞjs du;
2p 0
ð
1 2p
K1
½ðmmÞjs KðmnÞjr ðnnÞrk
ðnmÞks du:
ðA 4Þ
Bjs Z 2
8p 0
The force experienced by a second dislocation, due to the stress field of the first,
is given by the Peach–Koehler formula (Peach & Koehler 1950)
1 2
F PK
12 Z ðs $b Þo t;
ðA 5Þ
where b2 is the dislocation’s Burgers’ vector and t is its line direction. A is
defined via jF jhA/jrj, so is given by
jeamb tb b2n
1
K1
c
b1 ½KMp Sis C Np ðNN Þik
f4pBks C ðNM Þkr Srs gj:
2p mnip s
ðA 6Þ
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