Dislocation queueing and fracture in an elastically anisotropic material

$Page 1: Dislocation queueing and fracture in an elastically anisotropic material$
DISLOCATION QUEUEING AND FRACTURE IN AN ELASTICALLY ANISOTROPIC MATERIAL*

R. W. ARMSTRONGt$ and A. K. HEADP

A dislocation queueing model has been investigated whereby the dislocation, driven by an applied shear stress, are repelled from the environment of the crystal boundary confronting them because the material has a certain elastic anisotropy. The model has beeu applied to determining the fracture stress of a polycrystalline aggregate subject to the following cumulative condit,ions: (1) the leading dislocation of the queue is pushed into the boun&ry and becomes a locked dislocation; and (2) the fraoture stress obtains when the next dislocation is brought to within a Burgers vector distance of the locked dislocation.

This study reveals that the fracture stress of a material should become more strongly dependent on grain size as the d~l~ations are increasingly repelled from the boundaries. Also it appears that a portion of the experimental friction &rem, rB, associated with the Petch fracture stress-grain size analysis (7 fit: r,, + iEE-I’*) is a fictitious stress increment produced by fitting an inverse square root grain size dependence to actual curves of the form herein calculated.

FORMATION D’EMPILEMENTS DE DISLOCATIONS ET RUPTURE DANS UN MATERIAU ELASTIQUEMENT ANISOTROPE

Les auteurs ont Btudie un modele d’empilements de dislocations, dans lequel les dislocations, pous&es par une certaine eontrainte ~ngentielle, sent repoussees du voisinage imm&liat du joint de gram formant obstacle, parce qua le m&al possede une oertaine anisotropie Blastique. Ce modele a et& utilid pour determiner la tension de rupture d’un polycristal repondant aux conditions suivrmtes: (1) la dislocation situ& en t&e de l’empilement est pous&e dans le joint et devient une dislocation bloquee; et (2) la contrainte de rupture est atteinte quand la dislocation voisine est amen&e a une distance d’un vecteur de Burgers de la dislocation bloq&e.

Cette Etude montre que la tension de rupture d’un mate&u doit dependre plus fortement de la dimension du grain si les dislocations sont plus fortement repoussees loin des joints. que la composante ?. sssoci$e a f’analyse de Petch (7 w r.

11 a;parait egalement f kl-‘I*) comporte une partie fictive obtenue

en adaptant une 101 faisant intervenir l’inverse de la racine ear&e de la dimension du grain, loi tenant oompte de I’anisotropie Blastique.

VERSETZUNGSAUFSTAUUNG UND BRUCH IN EINEM ELASTISCH ANISOTROPEN MATERIAL

Es wird ein Model1 der Versetzungsaufstauung untersucht, bei dem die Versetzungen, auf die eine Schubspannung wirkt, von der Umgebung der Kristallgrenze infolge der elastischen Anisotropie des Materials abgestol3en werden. Mit Hilfe dieses Modells wurde die Bruehspannung eines polykrist~l~en Materials unter folgenden Bedingungen ermittelt: (1) die Verstzung am Kopf der Auf~tu~g wird in die Grenzflache gedriiokt und hierin festgehalten, (2) die Bruchspannung ist erreicht, wenn die n&hate Versetzung bis auf sine Entfernung ist einem Burgersvektor an die blockierte Versetzung herangebracht worden ist.

Diem Untersuohung ergibt, da13 die Bruchspannung mit zunehmender AbstoBung der Versetzungen von der Grenzflache eiue starkere Abhiingigkeit von der KorngroDe zeigen sollte. Auch sohemt es, da8 ein Teil der experimentell gefundenen Reibungsspanuung rO, die in der Petch-Analyse des Zusammen- hanges Bruchspa~~g-Ko~gr~~e (r w r. + kt--lf*) auftritt, fiktiv ist und nur von der Anpassung der ta~h~ichen Kurven, die von der Form der hier berechneten Kurven sind, an die Z-l/%“Abh~ngigkeit herriihrt.

INTRODUCTION

The equilibrium of queueing dislocations subjected to various systems of stress has been studied by several methods: (l), the transformation technique of StieItjes;Q*2) (2), integration over an assumed contin- uous dislocation density;(ssQ and, (3), numerical (computer) oalculations.(5*6) An important advantage of the first two methods has been that, if a solution is at all obtainable, it may generally be expressed in

The theory of dislocation queueing has been notably applied to explaining the brittle fracture of polycrystalline aggregates.(7-g) The dislocations are assumed to he in a single slip plane and are piled up against a locked (fixed) dislocation by a constant applied stress. The queueing dislocations at once apply the stress concentration required to produce a crack and the atomic mechanism by which it forms, Taking the polycrystal average grain diameter, 1, as a measure of the queueing length and applying the

* Received November 30, 1364. t Commonwealth Scientific and Industrial Research

Organization, Division of Tribophysics, University of Mel- bourne, Parkville, Victoria, Australia.

$ Formerly on academic leave programme from Westing- house Research Laboratories, Pittsburgh 35, Pennsylvania, U.S.A. Now at: Brown University, Providence, R.I.

ACTA METALLURGICA, VOL. 13, JULY 1965 75%

analytic form and holds for a large number of dislocations. The need for dealing with quite a few dislocations, say 50 or more, is necessary for application of the results to the observed nature of slip processes in bulk single crystals and polycrystals.

760 ACTA METALLURGICA, VOL. 13, 1965

theory leads to the prediction that the tensile fracture stress ought to vary as iY112. The same form of rela- tionship has been applied in a less detailed manner to the yield and plastic flow stresses observed for polycrystals.(lO)

There are theoretical and practical reasons for questioning whether the fracture stress, or any of these other stresses, is actually proportional to 1e1J2 over the widest possible range of grain size. For ex- ample, any additional force which a dislocation may experience as it- approaches closer to the locked dislocation and the grain boundary at the end of its slip plane has been completely neglected in applying the preceding. theories. In experiments, the range in grain diameters employed for testing has often been quite restricted, perhaps, mainly for fear that the exceptional treatment required to obtain the largest possible range would result in complicating internal structural changes additional to the desired variation in grain diameter.

In the present study, a computer program was employed to investigate dislocation queueing against a barrier in a situation where the stress field of the barrier, thougka very particular one, was taken into account. It was hoped to express the results in an analytic form and apply them to the fracture process mentioned above.

DISLOCATION QUEUEING AGAINST ELASTIC ANISOTROPY

The dislocation queueing model which was studied is shown in Fig. 1 and is an extension of a previous model described by Head. (11~2) The plane z = 0 is a grain boundary across which the orientation of the crystal changes. If the crystal is elastically anisotropic, then the stress field of a dislocation is modified by the change in elastic properties across the grain boundary. This can be described by saying that each

T -

c Fro. 1. Dislocations at a grain boundary, with images.

real dislocation in the right hand grain generates a stress field as if there were an image dislocation at the image point in the left hand grain, the strength of the image dislocation being a fraction K of the strength of the real dislocation. Thus, the equilibrium arrange- ment of a queue of n dislocations will be influenced by the stress fields of the image dislocations. The actual value of K depends on the elastic properties of the crystal and on the misorientation across the boundary. For iron K ranges from 0 to $0.29 depending on the misorientation and for copper from -0.11 to +0.28.(12)

The equations of equilibrium for n dislocations acted on by an applied shear stress 7 are

-g!- +Kijl& - 1 = 0 ; j = 1,2, . . n. iC1Xj - xi ) I i#J

(1)

where xj is the distance of thejth dislocation from the boundary measured in units of ,u6/2rr, ,U being the equivalent shear modulus C,,H.(ll) These equations have been solved by computer for n = 1, 2, 4, 8, 16, 32, and 64 and K = 0.05, 0.10, 0.15, 0.20, 0.30, 0.40 and 0.50. The model applies strictly to screw dislocations, however, it should also give a good approximation for edge dislocations to the extent that it may be assumed image forces solely act in the same way upon them.03) For edge dislocations the distance units should be increased by a factor l/(1 - Y) where v is Poissons ratio.

The positions of the dislocations were obtained with the view of specifying, as a convenient limiting con- dition, the shear stress required to bring the leading dislocation of the queue to a certain distance, x1, from the crystal boundary. Following preceding investi- gators, the position, x,, of the last dislocation in the queue was to be taken equal to the slip length, thus, giving a maximum stress concentration and a mini- mum applied stress to accomplish the event to be prescribed.

An explicit value of r obtains for each K if the func- tional dependence on n of x1 and x, are determined. Figure 2 shows, in the range 0 < K < 0.50, that, for increasing values of K, x1 varies inversely as n taken to a decreasing power. From equation (1) and the curves shown in Fig. 2, the value of x1 may be expressed to an accuracy better than f5% by the equation

x1 = (K/2)/n’l-l.ix) (2)

At values of K > 0.50, equation (2) gives an under- estimate of x1. Figure 3 reveals, for K = 0.15 and 0.50, the dependence of x, on n, in comparison with the value of x, determined for K = 0 but with a locked

ABMSTRONG AND HEAD: DISLOCATION QUEUEING AND FRACTURE 761

.

I I I , I I I

2 5 IO 20 50 100 a0

n, number of mobile dislocations

FIG. 2. The position, z*, of the first mobile dislocation 8s a function of n for various values of the elastic anisotropy,

specified by K.

dislocation at x = 0 from the analysis of Eshelby et 02.‘~) Prom the comparison, the values of x, for the case of anisotropy only are observed at large n to approach the form

x,B = 2n(l + 0.9X) (3)

The value of r required to bring a dislocation to a

distance x1 of the boundary may be expressed, from equations (2) and (3) in ordinary units, in the convenient dimensionless form

7 -zzz

QUEUEING AGAINST A LOCKED DISLOCATION AND ANISOTROPY

Stroh(s*14) has previously suggested that the tensile brittle fracture stress of a polycrystal may be determined by the stress required to force the first mobile dislocation of a queue to within a Burgers vector distance, b, of a looked dislocation placed at the tip of the queue. His analysis is an application of the queueing theory described by Eshelby et aZ.(l) Now,

at all values of K > 0, the leading dislocation being

0.1 I I I I , I I

I 2 5 IO 20 50 IO0 200

n, numbrr of mobile dislocations

Fro. 3. The position, r*, of the last mobile dislocation as a function of 92 and K and, also, for the case of Eshelby

et dfx)

repelled by its own image will be opposed by a barrier of sufficient strength to allow coalescence of the following dislocations. Consider, therefore, the following extension of the qneueing model proposed for eIastic anisotropy: (l), the leading dislocation of the queue is forced to within a very small distance, say, less than b, from the crystal boundary, thereafter becoming a locked dislocation at x = 0; and, (2), the larger stress is determined which is required to force the next dislocation to within a distance b of the locked dislocation. Placing the former leading dislocation at the position x = 0 and locking it proves to be a good approximation to the actual model for the range of anisotropy specified by 0 < K Q 0.50. The original leading dislocation may also be viewed as becoming physically locked at the interface x = 0 because of the ~rystallographjc misorientation which occurs across the boundary.

For this situation, the equilibrium of the remaining mobile dislocations is determined by the equations



FIG. 4. q for the cases: (a), anisotropy only; (b), a locked dislocation only; and, (c), anisotropy coupled with

a locked dislocation.

In (5), another dislocation has been added to the back of the queue SO that the distance x1 is still taken to measure the distance from the interface (where the locked dislocation is) to the leading mobile dislocation of the queue.

The new values of zr determined for dislocations being repelled by a locked dislocation at x = 0 and either of two conditions of elastic anisotropy, K = 0.05 and 0.50, are shown in Fig. 4. For the purpose of comparison, the results for these two values of anisotropy only and for a locked dislocation only are also shown. At large B, the new values of x1 appear to asymptotically approach the same dependence on n as for anisotropy only. In the range 0 < K < 0.50, a single value of x1 at n = 1 is obtained by linear extrapolation of the curves drawn at large n. The extrapolated value of x1 = 1.9 may be compared with the value x1 = 1.84 obtained for (n - 1) dislocations by Eshelby et al. as the first zero of the Bessel function, J,. On the basis of this comparison and the results shown in Fig. 4, the analytic form for x1 in the extended model is taken as

5r = l.g[+i.=) (6)

The new result for x, appears more simple. Figure 5 indicates that the value of x, given by (3) for anisotropy only applies equally well for large n to the extended model.

The extended model may be applied to determining the brittle fracture stress of a polyerystal if the new values of x1 and x, are taken equal to b and 1 (the grain diameter), respectively. Thus, the value of r obtained from (3) and (6) may be expressed as

1 ,-,I??

z 2 i &[1.9][1 + ()9f(](l-1.1”)

(Z--;.lK) 1 --=

l-7 1 2-1.1K (7) P r i;

Figure 6 shows r/p versus (Z/b)-1/2 for various values of K. For K = 0, the Eshelby et al. analysis applies and an W2 dependence is obtained. The dashed lines superposed on the curves for other values of K indicate that the curves may be approximated over a limited range of 1 by straight lines giving a positive stress intercept. This applies even though at K = 0.50, the equation gives nearly an l-II3 dependence for T. The scale on the top of Fig. 6 was obtained by taking 6 = 2 x lo-* cm and indicates that the fitting of straight lines to the calculated curves applies to a reasonably practical range of grain diameters.

For the situation in which a locked dislocation is present at the origin, it is also interesting to enquire about the influence of negative values of K in the pre- oeding analysis, i.e. one may consider the case where the influence of elastic anisotropy is to attract dislocations toward a boundary and, yet, the leading dislocation still becomes locked at the boundary so that

L I I f I f 5 IO 20 50 100 200

n, number 01 mobile dislocations

FIQ. 5. 2, for the cases: (a), anisotropy only; and (b), anisotropy coupled with a locked dislocation.

ARMSTRONG AND HEAD: DISLOCATION QUEUEING AND FRACTURE 763

2.0 -

FIG. 6. The theoretical brittle fracture stress as a function of grain size for the case of anisotropy coupled

with a locked dislocation.

it repells the remaining ones which are trying to ap-

proach closer. A comparison of values for x1 and x, is

shown in Figs. 7 and 8, respectively, for K = 0 and

f0.30. Figure 7 reveals, if the sign of K is taken into

account, a similar dependence on n of xi for positive

and negative values of K but that the intercept at n =

1 decreases with decreasing (negative) values of K. Figure 8 shows a similar dependence on n of x, for

positive and negative values of K. The results indicate

1

2 I I # I I I

5 IO 20 50 I00 200


FIQ. 7. z1 for zero, positive, and negative values of K coupled with a locked dislocation.

5

I I I I 1 I I

2 5 IO 20 50 100 200

n, number oi mobile dislocations

FIG. 8. 2, for zero, positive and negative values of K coupled with a locked dislocation.

that equation (7) may be used, also, to qualitatively

estimate the dependence of r on 1 for negative values

of K.

DISCUSSION

A model for dislocation queueing has been developed

which incorporates one of the types of barrier forces-

that due to elastic anisotropy-which dislocations

may experience as they approach closer to a crystal

boundary.

In applying the model to a determination of the

polycrystalline fracture stress, the barrier force pre-

sented by elastic anisotropy may be considered either

as the locking force for the leading dislocation or as a

force additional to the structural process by which a

dislocation may become locked at the tip of a queue.

Because the forces between dislocations are over-

estimated at distances of the order of their Burgers

vector, the criterion employed for determining the

fracture stress should lead to an overestimate of the

applied stress. However, a recent evaluationos) of this

criterion for the situation in which the forces between

dislocations are underestimated, gives a fracture stress

approximately 13% less than that obtained from

Eshelby et al. Whereas the accuracy of the approximations in-

volved in determining the original positions, zl, of the

leading dislocations has been described (equation (2)),

those approximations involved in determining the


positions, x,, given by (3) and the values of xi incorporating a locked dislocation, (6) are more difficult to assess. If it is assumed that the barrier force opposing the queueing dislocations is due solely to elastic anisotropy, then the error involved in replacing the original leading dislocation by a locked dislocation at the origin produces an overestimate of x1 ranging between < 1 y0 for K = 0.05 and 10% for K = 0.50. A comparison of all the results with the analytic results obtained by Eshelby et al. indicates that the new results should have a similar validity at the large values of n where they would normally be applied.

The model for dislocation queueing against a barrier incorporating elastic anisotropy indicates that an increasing repulsion from the barrier because of elastic anisotropy enhances the polycrystal fracture stress dependence on grain size. In the range of anisotropy specified by 0 < K < 0.50, the exponent of 1 lies between -$ and -4. As mentioned earlier, the polycrystal yield and flow stresses generally show a similar dependence on grain size to that of the fracture stress; and, an l-* dependence has been suggested by Baldwin to apply to the experimental data for the yield stress of a number of polycrystals. For a negative value of K coupled with a locked dislocation at the barrier, an exponent of 1 less than -+ is obtained.

A positive stress increment is obtained by fitting an 1-i dependence to curves of the form herein calculated for positive values of K and, as well, for negative K values coupled with a locked dislocation. On this basis, it must be presumed that, at least, a portion of the fracture stress intercept value which is experimen- tally measured for an elastically anisotropic material is a fictitious contribution to this friction stress. It is interesting that the magnitude of the fictitious stress increment is related to the slope of the line fitted to the theoretical curve and may, possibly, be estimated in

this way. However, a quantitative comparison of theory and experiment is imagined difficult at this stage because, on the theoretical side, other barrier forces remain to be taken into account, and, on the experimental side, auxiliary plastic flow, impurity seg- regation, and inclusions obviously play a role in determining the measurements.

SUMMARY

A dislocation queueing model which incorporates a particular type of barrier force due to elastic anisotropy has been investigated by a numerical method and analytic results have been obtained. In applying the model to the brittle fracture of a polycrystalline aggregate, it occurs that the dependence of the theoretical fracture stress on the queueing length varies according to the elastic anisotropy characteristic of the material. Over a practical range of grain size the curves may be fitted by the Petch fracture stress- grain size analysis (T w T,, + kW2) and, then r0 and k vary according to the nature of the elastic anisotropy.

REFERENCES

1. J. D. ESHELBY, F. C. FRANK and F. R. N. NABARRO, Phil. Mug. 42, 351 (1951).

2. A. K. HEAD and P. F. THOMSON, Phil. Mag. 7,439 (1962). 3. G. LEIBFRIED, 2. Phvsik 130, 214 (1951). 4. A. K. HEAD, AZ&. J. Phys. 13, 613 (1960). 5. A. K. HEAD, Phil. Mag. 4, 295 (1959). 6. Y. T. CHOU, F. GAROFALO and R. W. WHITMORE, A&

Met. 8, 480 (1960). 7. N. J. PETCH, J. Iron St. Inst. 174, 25 (1953). 8. A. N. STROH, Adwanc. Phys. 6,418 (1957). 9. A. H. COTTRELL, Trans. Amer. Inst. Min. (Metall.)

Engr8 212, 192 (1958). 10. R. W. ARMSTRONG, I. CODD, R. M. DOUTHWAITE and

N. J. PETCH, Phil. Mag. 7, 45 (1962). 11. A. K. HEAD, Aust. J. Phys. 13, 278 (1960); Phys. Stat.

Sol. 5, 51 (1964); ibid 0, 461 (1964). 12. A. K. HEAD, to be published. 13. A. K. HEAD, Proc. Whys. Sot. Lond. B66, 793 (1953). 14. A. N. STROH, Proc. Roy. Sot. A232, 548 (1955). 15. R. W. ARMSTRONG, to be published. 16. W. M. BALDWIN, Jr., Acta Met. 6, 139 (1958).

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Dislocation queueing and fracture in an elastically anisotropic material