a natural transition between equilibrium patterns of dislocation dipoles

J ElastDOI 10.1007/s10659-013-9464-z

A Natural Transition Between Equilibrium Patternsof Dislocation Dipoles

Yichao Zhu · Stephen Jonathan Chapman

Received: 30 October 2013© Springer Science+Business Media Dordrecht 2014

Abstract The equilibrium configurations of a row of uniformly distributed dislocationdipoles are first studied. The analysis is then generalised to study dipoles in two-dimensionalrectangular periodic lattices. By examining the stability of the equilibrium configurationswe find that the system may undergo a natural transition from the Taylor lattice to a rowof dipole walls. This bifurcation may be involved in the transition from channel-vein topersistent slip band (PSB) structures in the early stage of metal fatigue.

Keywords Periodic lattices · Dislocations · Equilibrium patterns · Channel-veins ·Persistent slip bands

Mathematics Subject Classification (2010) 74A60 · 74N15

1 Introduction

The dislocation structure under cyclic deformation in the early stage of metal fatigue hasbeen well-studied [1, 3, 8, 10]. Before reaching the saturation point, this structure can bedescribed as an assembly of veins of closely spaced long edge dislocations in multi-poleconfigurations. The veins are separated by relatively dislocation-free channels as shownin Fig. 1(a). Beyond the saturation point, a more regular “ladder” structure starts to form,known as persistent slip bands (PSBs). These are characterised by regularly-spaced high-density walls of edge dislocations, as shown in Fig. 1(b).

The mechanism of the transition from dislocation veins to PSBs is still unclear. Theproblem can be divided into two parts: (a) how do dislocations multiply in the early stage

Y. Zhu (B)The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong SAR,Chinae-mail: [email protected]

S.J. ChapmanMathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter,Woodstock Road, Oxford, OX2 6GG, UK

mailto:[email protected]

Y. Zhu, S.J. Chapman

Fig. 1 Dislocation structures in the early stage of metal fatigue (abstracted from [10])

of fatigue to lay down the channel-vein structure; and (b) what determines the arrangementof dislocations within a vein, and why does this evolve into a PSB. In [17] we make someattempt to address the first question mathematically. In this paper, we investigate the secondquestion by examining the equilibrium configurations of arrays of dislocations.

The study of such idealised dislocation structures dates back to the idea of the Taylorlattices [11, 13, 14]. In particular, there have been a number of investigations of periodic dis-location lattice structures. For example, analytical results for the stress field of a finite/semi-finite/infinite wall of dislocation monopoles have been derived [6, 9]. With the growth ofcomputing power, attempts to search for equilibrium patterns of a large finite number of dis-locations have also been made [2, 12]. Recently, the study of dislocations in periodic latticeshas received fresh attention for a number of reasons [4, 5, 15].

The dislocations in a channel-vein structure mainly occur in equal and opposite pairs:while not exactly dipoles, there are roughly the same number of “plus” and “minus” dis-locations, so that the net Burgers vector is close to zero. The equilibrium states of thesepositive and negative dislocation “poles” may reveal the mechanism of PSB formation.

Here we consider the equilibrium pattern of dislocations in rectangular two-dimensionalperiodic lattices. First, the equilibrium configurations of two rows of periodically laid op-posite monopoles are discussed, followed by the analysis of their stability. Then a two-dimensional lattice is considered, and three types of equilibria are found. Finally, a naturaltransition from a Taylor lattice to a row of dipole walls is found through examining thestability of each configuration.

2 Equilibria of One-Dimensional Rows of Oppositely Signed Monopoles

Throughout the paper we will consider configurations of parallel straight edge dislocationson a single slip system. This simplifies the problem enough to allow some analysis, whilemaintaining the essential features of dislocation veins.

We suppose that all dislocations are parallel to the z-direction, with Burgers vectors par-allel to the x-direction, so that the slip-planes are have normal parallel to the y-direction. Byconvention we denote those dislocations with Burgers vector in the x-direction as positive,and those with Burgers vector in the negative x-direction as negative.

A Natural Transition Between Equilibrium Patterns of Dislocation

Fig. 2 Configuration of twoone-dimensional rows ofmonopoles

We begin by considering just two slip planes, separated by a distance s. We suppose thatthe lower slip plane contains a row of equally spaced positive dislocations, while the upperslip plane contains a row of similarly spaced negative dislocations, as shown in Fig. 2. Wesuppose that the offset between the two families of dislocations is q , and that the configu-ration has period α, so that the positive dislocations are at positions (nα,0), n ∈ Z and thenegative dislocations are at positions (nα + q, s), n ∈ Z.

For such a configuration the dislocations will move in their glide plane (i.e., in the x-direction) in response to the stress component σ12. We normalise x and y by the period α,and σ12 by μb/(1 − ν)α, where μ and ν are shear modulus and Poisson’s ratio, respectively.Then the shear stress due to a single straight edge dislocation at (x ′, y ′) is [7, §5.2]

σ12(x, y) = (x − x ′)((x − x ′)2 − (y − y ′)2)

2π((x − x ′)2 + (y − y ′)2)2.

We suppose also that in addition to the stresses caused by the two rows of dislocations thereis an imposed (dimensionless) external stress σ e. The total shear stress may then be writtenas

σ t12(x, y) = 1

2π

∑

n∈Z

(x − n)((x − n)2 − y2)

((x − n)2 + y2)2

− 1

2π

∑

n∈Z

(x − Q − n)((x − Q − n)2 − (y − S)2)

((x − Q − n)2 + (y − S)2)2+ σ e,

where Q = q/α, S = s/α. In equilibrium the regular part of σ12, obtained by subtracting thedislocation self-stress, must be zero at each dislocation. For each positive dislocation thisimplies

1

2π

∑

n∈Z

(Q + n)((Q + n)2 − S2)

((Q + n)2 + S2)2+ σ e = 0. (1)

Note that there is no stress term due to the other positive dislocations; the stress due to thepositive dislocation n positions to the right is exactly cancelled by that due to the dislocationn positions to the left. Similarly for the negative dislocations we find

1

2π

∑

n∈Z

(Q − n)((Q − n)2 − S2)

((Q − n)2 + S2)2+ σ e = 0. (2)

In fact equations (1) and (2) are equivalent (as can be seen by relabelling n with −n in (2)),so that we only need to consider the solution of (1). Thus, given a slip plane separation S, (1)is the equation which determines Q, the relative x-separation of the positive and negativepairs. The series in (1) may be summed explicitly [6] (see also [7, §19.4]) to give

sin(2πQ)

2(cosh(2πS) − cos(2πQ))− πS sin(2πQ) sinh(2πS)

(cosh(2πS) − cos(2πQ))2+ σ e = 0. (3)


Fig. 3 The solid curves are setsof (Q,S) such that the system isin equilibrium under an externalstress σ e, which arecorrespondingly indicated on thecurve. The diamonds denote(Q,S) for some specificequilibrium states with σ e = 0.1or σ e = 0.01. We can also seethat for a given σ e, there exists amaximum Sc(σ

e), such that allS ≤ Sc(σ

e). Finally, the twodashed curves are the locus of themaximum values of S(Q) as σ e

varies, i.e., they are curves onwhich dS/dQ = 0; these are theboundaries between stable andunstable regions

This gives Q implicitly as a function of S.In Fig. 3 we show the relationship between Q and S for varying values of σ e. Also

shown in the figure are some direct numerical simulations. For these, following the idea in[12], we start with N pairs of dislocations within the interval [−N/2,N/2], with the slipplanes separated by a distance S. We then solve the system of algebraic equations

N−1∑

i=0,i �=k

1

2π(pk − pi)−

N−1∑

i=0

(pk − qi)((pk − qi)2 − S2)

2π((pk − qi)2 + S2)2+ σ e = 0 (4)

and

−N−1∑

i=0,i �=k

1

2π(qk − qi)+

N−1∑

i=0

(qk − pi)((qk − pi)2 − S2)

2π((qk − pi)2 + S2)2+ σ e = 0, (5)

for pk and qk , iteratively for all integer k that 1 ≤ k ≤ N − 1, where pk and qk are thex-positions of the k-th positive and negative dislocations, respectively. At each end, a pairof vertically aligned dislocation pair are locked, i.e., p0 = q0 = −N/2 and pN−1 = qN−1 =N/2. Providing N is sufficiently large, distributions obtained from the above scheme areapproximately the same as the equilibria of dipoles in periodic lattices [16, §5.5]. We seethat the direct simulations lie almost on top of their corresponding theoretical contour.

We see from Fig. 3 that for a given σ e, there exists a maximum Sc (attained at Qc say),such that all S ≤ Sc . Physically, Sc is the maximum spacing between slip planes at which arow of dipoles in equilibrium is possible: above this spacing the attraction between oppositedislocations is insufficient to overcome the separating external force, and the two rows willslide past one another. In Fig. 4, Sc is plotted against σ e.

Also in Fig. 3 we show the locus of the points (Qc,Sc) as σ e varies. Only equilibriabetween the two dashed curves are stable. Note that for each value of S there is a uniquestable value of Q, but for a given value of Q there may be more than one value of S.

As a special case, let us interpret the equilibrium configurations when there is no appliedstress. From Fig. 3 we see that there are three types of solution: either Q = 0; or Q = 1/2; orQ lies on a curve joining the point Q = 0, S = 0 to the point Q = 1/2, S = S∗. A schematicdiagram of the solutions in each is shown in Fig. 5. The value Q = 0 corresponds to the


Fig. 4 The maximum spacingSc to retain dipoles under σ e

Fig. 5 The three equilibrium configurations: (a) Type I for which Q = 0; (b) Type II for which Q = 1/2;(c) Type III, in which symmetry is broken, and for which Q = Q(S)

dislocations being aligned vertically above each other; such a configuration is unstable. Thevalue Q = 1/2 corresponds to the negative dislocations lying equidistant between the twonearest positive dislocations. In this case the dislocations are not forming dipoles, sinceeach negative dislocation is associated with two positive dislocations and vice versa. Thisconfiguration is stable if and only if the gap between slip planes S is greater than somecritical value S∗ indicated in Fig. 3, which is approximately 0.2445.

If S < S∗, a third type of equilibrium exists and is the stable configuration. In this thirdequilibrium each negative dislocation has one positive dislocation as its nearest neighbour,so that the dislocations are forming dipoles. As the separation between slip planes decreasesthe dislocations become tightly bound in dipoles, with Q → 0 as S → 0.

Thus the configuration undergoes a transition (bifurcation) from two interleaved rows ofmonopoles to a row of dipoles as S decreases through the critical value S∗. As we will seelater, such a transition also exists for two-dimensional periodic lattices.

3 Dislocation Dipoles in Lattices

We now consider two-dimensional rectangular lattices of dislocations. We consider the casein which each lattice cell contains one positive and one negative dislocation, as shownschematically in Fig. 6. For ease of exposition we consider the case in which σ e = 0. Wesuppose the period in the x-direction is α, and the period in the y-direction is β . Withineach cell we suppose the horizontal and vertical separation of the dislocation pair are q ands respectively, where, without loss of generality, we suppose 0 ≤ q ≤ α/2, 0 < s ≤ β/2.


Fig. 6 A cell from a rectangularlattice

As before, we normalise x by the x-period α, and σ12 by μb/(1 − ν)α; in addition wenow also normalise y by the y-period β . Then all positive dislocations are located at (n, k),and their negative counterparts are at (Q+n,S+k), where n, k ∈ Z, and Q = q/α, S = s/β .The total shear stress at (x, y) is then

σ t12(x, y) = 1

2π

∑

n,k∈Z

(x − n)((x − n)2 − (y − k)2/λ2)

((x − n)2 + (y − k)2/λ2)2

− 1

2π

∑

n,k∈Z

(x − n − Q)((x − n − Q)2 − (y − k − S)2/λ2)

((x − n − Q)2 + (y − k − S)2/λ2)2,

(6)

where λ = α/β . When the system is in equilibrium, the regular part of σ12 should vanish ateach dislocation, giving the force balance equation

1

2π

∑

n,k∈Z

(n − Q)((n − Q)2 − (k − S)2/λ2)

((n − Q)2 + (k − S)2/λ2)2= 0. (7)

Note that the lattice of positive dislocations is in equilibrium with itself, so that (7) representsthe force of the negative lattice on each positive dislocation. Thus we need to evaluate thestress due to a lattice of monopoles.

σ+12(x, y) = 1

2π

∑

n,k∈Z

(x − n)((x − n)2 − (y − k)2/λ2)

((x − n)2 + (y − k)2/λ2)2, (8)

after which the force balance equation is given by

σ+12(Q,S) = 0. (9)

There are two ways to estimate this sum. One is to first sum over n and then over k, sothat we calculate the stress for a row of monopoles and then sum over rows. The other is tofirst sum over k and then over n, so that we calculate the stress for a wall of monopoles andthen sum over walls.

If we first sum over n, we may use the result in Sect. 2 for the stress due to a row ofmonopoles. Thus

σ+12(x, y) =

∑

k∈Zσx

12(x, y; k), (10)

where

σx12(x, y; k) =

sin(2πx)

2(cosh(2π(y − k)/λ) − cos(2πx))− π(y − k) sin(2πx) sinh(2π(y − k)/λ)

λ(cosh(2π(y − k)/λ) − cos(2πx))2.

(11)


Fig. 7 Equilibrium configurations for rectangular lattices, shown for different values of the aspect ratio ofthe unit cell, λ

Note that σx12(x, y; k) decays exponentially with k, so that the sum in (11) is rapidly conver-

gent and easily truncated. Alternatively, if we sum first over k, we find [7, §19.4]

σ+12(x, y) =

∞∑

n=−∞σ

y

12(x, y;n), (12)

where

σy

12(x, y;n) = −π(x − n)(1 − cosh(2π(x − n)) cos(2πy/λ))

(cosh(2π(x − n)) − cos(2πy/λ))2. (13)

Again, σy

12(x, y;n) decays exponentially with n so that the sum is rapidly convergent. Whichapproach is better depends on the value of λ.

Although in the case of lattices we are unable to explicitly calculate the relationshipbetween Q and S, the rapid convergence of the sums (10) and (12) means it is easy tocalculate the relationship numerically. The equilibrium configurations are shown in Fig. 7for the case λ = 0.5, 1, and 2. As in Sect. 2 there are three types of equilibria. When Q iszero or 1/2 the system is always in equilibrium regardless the value of S. However, if S isless than some critical value S∗(λ) (which corresponds to the black dot in Fig. 7) there existsanother type of equilibrium in which Q varies with S.


Fig. 8 Equilibrium pattern in two-dimensional unit lattice

Fig. 9 S∗ against λ: S∗ has anupper limit of 0.25. On the otherhand, as λ → 0, the system isequivalent to the case ofone-dimensional array of dipoles,with the slope at the origin equalto the critical value S∗ in thatcase

When Q = 0 the positive and negative dislocations are aligned vertically, as shown inFig. 8a. When Q = 1/2 each negative dislocation lies equidistant from its neighbouringpositive dislocations, as shown in Fig. 8b. For the third type of equilibrium the symmetry isbroken, and the dislocations from dipole pairs, as shown in Fig. 8c.

In Fig. 9 we plot S∗(λ) against λ. We see that S∗ has an upper limit of 0.25 as λ → ∞.On the other hand, if we let λ → 0 (corresponding to β α), the rows of dipoles are wellseparated. Thus λ → 0 should take us back to the case of one-dimensional arrays of dipoles.However, we need to remember that in the present case s was scaled with β , while for therow of dipoles s was scaled with α. Thus the relationship between the two values is

S∗row = lim

λ→0

S∗lattice

λ;

thus for small λ, S∗(λ) should tend to zero as 0.2456λ, which it does.

3.1 Stability

The next question is whether these three types of equilibrium are stable or not. To answerthis question we consider the effect of perturbing the relative position of the two lattices (ofpositive and negative dislocations) while maintaining periodicity.


Fig. 10 A set of periodicallydistributed positive monopolesexert a stress field such thatσ+

12(x, y) > 0 in the shaded

regions and σ+12(x, y) < 0 in the

unshaded regions. Theequilibrium positions for a latticeof negative dislocationscorrespond to the boundaries thatseparate shaded and unshadedregions. A negative dislocation inthis system will be pushed to theleft in shaded regions and to theright in unshaded regions. Thusonly equilibrium states thatcorrespond to boundaries with anunshaded region on the left and ashaded region on the right arestable

Figure 10 illustrates the stress field of periodically distributed array of positivemonopoles; the shaded regions correspond to σ+

12(x, y) > 0 while the unshaded regionscorrespond to σ+

12(x, y) < 0. To find equilibrium state of our two-lattice configuration wesimply need to arrange the periodically distributed array of negative monopoles at posi-tions where σ+

12(x, y) = 0, corresponding to the boundaries between shaded and unshadedregions.

To determine the stability of such a configuration we perturb the negative monopolelattice by a small displacement δx in the x-direction and see how it responds to this pertur-bation. The horizontal force a negative dislocation at (x, y) feels is −σ+

12(x, y). Therefore ina shaded region these negative monopoles will move to the left and in an unshaded regionthey will move to the right. Thus only equilibrium states that correspond to boundaries withan unshaded region on the left and a shaded region on the right in Fig. 10 are stable. A com-parison with Fig. 7 shows that Type I is always unstable, Type III is always stable, and TypeII is stable if and only if when S > S∗.

More mathematically, the stability of a system is determined by expanding

−σ+12(Q + δx,S) = fr(Q,S) · δx +O

(δx2

), (14)

where

fr(Q,S) = −∂σ+12(x, S)

∂x

∣∣∣∣x=Q

. (15)

The equilibrium state is stable if and only if fr(Q,S) < 0. Moreover, the magnitude offr(Q,S) determines the degree of stability of the system: the more negative fr(Q,S) is, themore stable the system is (the quicker it returns to equilibrium).

In Fig. 11 we show how fr depends on S for all three equilibria. The existence of exactlyone stable equilibrium for each value of S is demonstrated.

4 Conclusion

We have examined the equilibria and stability of rows and lattices of dislocations of oppositesigns. We find that as the separation between slip planes is varied there is a bifurcation froma Taylor lattice to an array of dipoles.


Fig. 11 Stabilities for the equilibria of rectangular lattices. Type I is always unstable. Type II is conditionallystable. The most stable configuration of this type arises when S = 0.5. When S < S∗ (the black dot), TypeIII occurs. In this case, a dislocation will be coupled with one of its neibouring opposite counterpart to forma pair of dipole. Another key observation can be made from Fig. 11 is that for any given S, there exists onlyone stable state. When S > S∗ , it is of Type II; when S < S∗ , the stable system bifurcates to Type III

This change in configuration is suggestive of the observed change in dislocation densi-ties from channel-veins to PSBs. However, tempting as it might be, identifying our tightlybound dipoles with PSBs and our Taylor lattices with channel-veins is overly simplistic; itis unlikely that the rungs of PSB ladders are one dipole thick. Nevertheless, the bifurcationwhich is present in the local periodic structure may play some role in the larger scale patternformation. For example, the long range interactions may be significantly reduced for tightlybound dipoles by comparison to dislocations in a Taylor lattice, so that such dislocations aremore susceptible to localised pattern formation as in PSBs.

Acknowledgements YZ is supported by EPSRC grant EP/D048400/1.


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a natural transition between equilibrium patterns of dislocation dipoles

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