## Abstract

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 pile-up. The results are applied to Fe close to the temperature of the α–γ phase transition, where the inclusion of anisotropy leads to a strikingly different prediction from that of the isotropic approximation.

## 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 *c*_{ijkl}. In the cubic symmetry case, this tensor is fully described by defining three independent stiffness constants *C*_{11}, *C*_{12} and *C*_{44} (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 550°C, 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 950°C (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, 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 α–γ 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 configurations of bcc Fe: , , 〈100〉{011} and 〈100〉{001}.1 A similar immobile dislocation lies at the origin, and a source at some distant point *x*_{0} produces dislocations under the influence of an applied stress *τ*_{0}. Although dislocations in real crystals are in general curvilinear, this effectively two-dimensional 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 α–γ phase transition at *T*_{α–γ}=912°C.

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 〈100〉{001} dislocations, it takes the form(2.1)where *C*′=(*C*_{11}−*C*_{12})/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(2.2)where the roots *x*_{i} are the positions of *n* dislocations in units of |** b**| (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 (

*τ*

_{0}), a differential equation for

*f*can be derived, with the solution

*L*′

_{n+1}, the first derivative of the (

*n*+1)th Laguerre polynomial (see e.g. Boas 2005), defined by(2.3)The positions of the

*n*dislocations are given by the zeros of

*f*: for

*n*=20 say, we have(2.4)The scale factor

*A*/2

*τ*

_{0}, 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 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 *n*+1 dislocations situated at the origin. This can be confirmed by expanding about *x*=∞:(2.5)so the total field ∼(*n*+1)/*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 10^{4}*b*, so this treatment is reasonable) and use the same solution as in §2. The solution for *n* mobile dislocations is now , and the positions are given as before by the scaled zeros of *f*(3.1)for *n*+2=22. The above polynomial has *n*+1 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 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 *τ*_{0} and *n*.

The greatest root *y*_{0} of the scaled equation grows approximately linear in *n*, and a good (1%) fit for the first 120 is given by(3.2)Returning to normal units *x*=(*A*/2*τ*_{0})*y* gives the linear relationship(3.3)Furthermore, we can determine the total plastic displacement at a given *τ*_{0} by summing the distances travelled by the *n* dislocations to their equilibrium positions. Though the positions are in principle known, solving an *n*th order polynomial can be numerically inconvenient as *n* gets large. Fortunately, we need only the sum of the roots, which is given by(3.4)so for the total distance travelled by the dislocations we have(3.5)where in the second line we used equations (2.3) and (3.4). The crystal's plastic displacement is then given by |** b**|

*d*/

*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

*τ*

_{0}increasing incrementally, and assume a quasistatic process where the dislocations instantaneously reach their updated equilibrium positions with each increment in

*τ*

_{0}, we can calculate

*τ*

_{0}and

*d*at each

*n*and determine the relationship between the two.

## 4. Application to *α*-Fe at high temperatures

Figure 2*a* shows the elastic constants of single-crystal α-Fe, measured at temperatures from 25 to 900°C; 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 *C*_{44}/*C*′, equal to 1 in the isotropic limit, is approximately 2.4 at 25°C and rises to 7.4 at 900°C. Moreover, *C*′ is approaching zero, which indicates the instability of the bcc crystal structure at these high temperatures. Calculations (Hasegawa *et al*. 1985) predict this effect as the α–γ bcc–fcc phase transition temperature *T*_{α–γ} is approached. Of course, iron remains solid beyond *T*_{α–γ}, but there is still a marked effect on the crystal properties as *T*_{α–γ} is approached from below.

The data in figure 2 show a deviation from smooth behaviour from approximately 750°C, which is associated with spin fluctuations in iron above the Curie temperature; figure 2*b* shows the data for *C*′ together with a function of the form(4.1)which is reasonable near the point of this phase transition. The values of parameters *μ*_{1}=0.497, *μ*_{2}=0.478 were obtained by fitting equation (4.1) to the data below 750°C only, and by adding the artificial point (0, 912°C). Equation (4.1) gives an economical fit to the data from the most trusted region, consistent with the assumption that *C*′ becomes small as *T* approaches 912°C. Figure 3 shows the value of *A* versus temperature for four configurations of edge dislocation. The data points were calculated numerically, except for the 〈100〉[001] case, for which the analytic expression equation (2.1) is available. The magnitude of the Burgers' vector enters into the expression for *A* as |** b**|

^{2}; we work in units of the lattice period, where |

*b*_{100}|=1, . In the isotropic approximation, this implies that the interaction force for 〈111〉-type dislocations is always 0.75 times that for the 〈100〉-type. The inclusion of anisotropy demonstrates that this is not satisfied at high temperatures, where the strength of interaction between 〈100〉[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 〈100〉[001] configuration,(4.2)when

*C*′ is small, as the α–γ transition is approached, and so it falls towards zero as . This behaviour cannot be understood within isotropic elasticity, since that approximation uses only one shear modulus

*C*

_{44},

*C*′→

*μ*. Taking

*μ*=(

*C*

_{44}+

*C*′)/2, it is clear that the soft mode is overlooked, since although

*μ*would fall to half the value of

*C*

_{44}, it would not approach zero, and the extreme softening behaviour would not appear. Figure 4 shows the variation with temperature of

*A*

_{〈100〉[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 *A*^{−1}). 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 α–γ transition, while the latter does not, maintaining a γ-type structure across the temperature range 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 pile-ups in iron at high temperatures. The variation with temperature of inter-dislocation 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 〈100〉{001} dislocations was derived. The increasing discrepancy between the anisotropic and isotropic approximations as temperature increases was established.

Essentially, the behaviour of the *C*′ elastic modulus leads to a severe plastic softening of the crystal as the α–γ 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 〈100〉{001}-type dislocations will provide the dominant sources of plasticity as they begin to collapse at high temperatures. Even if not present initially, pinned 〈100〉{001} segments can form via the reaction of two 〈111〉 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 α–γ phase transition, and absent in those that do not. This is quite apart from any impact the behaviour of *C*′ 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.

## Acknowledgments

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 NMP3-CT-2004-500253.

## Footnotes

↵Only the latter configuration allows the derivation of compact formulae for the quantities involved. However, as the temperature increases towards

*T*_{α–γ}, the 〈100〉{001} configuration becomes the most important, because the strain energy of 〈100〉{001} dislocation segments falls sharply (see later). This is consistent with electron microscope observations of dislocation loops, which show the dominant occurrence of 〈100〉-type orientations of the dislocation segments at elevated temperatures (Masters 1963; Little & Eyre 1973).↵Neglecting anisotropy overestimates

*A*by approximately 14% at 25°C, rising to 63% at 900°C.- Received March 17, 2008.
- Accepted April 16, 2008.

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