Modelling crack tip plastic zones and brittle-ductile transitions

ELSEVIER

Materials

Modelling

Science

and Engineering

A234+236

(1997)

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crack tip plastic zones and brittle-ductile

transitions

S.G. Roberts * Department

of Materials,

University

of Oxford,

Received

Parks

12 February

Road,

Oxford

OX1

3PH,

UK

1997

Abstract Plasticity at crack tips may be modelled as self-organising arrays of dislocations emitted from a source near the crack tip. For materials where dislocation motion is slow and any friction stress is low, the modelled arrays are far from equilibrium. If dislocation motion is fast, and there is a substantial dislocation pinning stress, zf, the arrays are quasi-static and the model predicts behaviour very similar to those of earlier static models. A power-law relation connects the shielded crack tip stress intensity, k, to the pinning stress, TV,the length of the dislocation array, d and the size of the ‘dislocation free zone’ near the crack tip. Using either of two fracture criteria, (a) k = K,,, or (b) the stress at some point ahead of the crack tip exceeds a local fracture stress, the variation of stress intensity at fracture KF with temperature is predicted to be controlled by the temperature variation of the yield stress: KF x a; o.57.0 1997 Elsevier Science S.A. Keywords:

Crack

tip; Plastic

zones;

Brittleeductile

transitions;

Fracture;

1. Introduction Many crystalline solids fail by cleavage at low temperatures and by plastic processes at high temperatures. In the transition region, cleavage occurs at stresses which increase with increasing temperature. The most frequently observed transition is of a gradual nature, where the fracture stress increases over a range extending over the order of 100 K or more [l]. BCC metals [2,3], intermetallics [4-61, MgO [7] and many other materials exhibit this type of transition. In some cases, e.g. for Si [S-l l] and A&O, (sapphire) single crystals [12], the transition is very sharp, occurring over a temperature range of < 10 K. The transitions are strain-rate dependent; for BCC metals the strain-rate dependence is small [2,3]; for Si single crystals, increasing the strain-rate by a factor of ten increases the temperature of the transition typically by 100 K [8- 111. The increase in fracture toughness with temperature is associated with a increase in the dislocation mobility resulting in an increasing amount of plasticity around the crack tip. A plastic zone is formed, which reduces * Tel.: + 44 1865 273775: [email protected] 0921-5093/97/$17.00 PUSO921-5093(97)00180-9

fax:

Q 1997 Elsevier

+ 44

Science

1865

273164;

S.A. All

rights

e-mail:

reserved.

Modelling

the high stresses in front of the crack tip through elastic interactions, and which also blunts the crack. Models of crack tip plastic zones in terms of dislocation distributions fall into several categories: (1) Continuous distributions of dislocation density, with a constant shear (yield) stress inside the plastic zone (e.g. that of Bilby, Cottrell and Swinden, ‘BCS’ [13]); (2) distributions of discrete dislocations (possibly including superdislocations), statically ‘balanced’ against a shear yield stress [14-201; and (3) dislocations emitted from a source and moving outwards at velocities determined by the stress on each one; the arrays are not at equilibrium [1,21L231. This paper describes recent developments of a model of type (3), with a dislocation velocity law which includes a minimum stress for dislocation motion. The predictions of dislocation distributions from this model are shown to give results equivalent to those from models of types (1) and (2). The model therefore can be used to describe a large range of dislocation behaviour near crack tips, independent of whether dislocations are far from or close to equilibrium positions. The model also computes the shielding effects of dislocation arrays on the crack tip and can be used to make predictions of brittle-ductile transition behaviour.

S.G. Roberts

/Materials

Science

and Engineering

2. The basic model

We simplify the crack tip plastic zone as a single slip plane which intersects the crack plane along the line of the crack tip (Fig. 1). The angle between the slip plane and the crack plane, 0, in the modelling reported here was ?O.Y, this being the plane subjected to the maximum shear stress in mode I loading. A dislocation source is positioned on the slip plane at a fixed distance, n,, ahead of the crack tip (50b-5000b for these simulations). The source emits a single edge dislocation when the stress on a dislocation at the source is sufficient to move it away from the crack tip. The stress z, on a dislocation at ni is given by:

The first term is the crack tip field stress(a depends on 0) at an applied stress intensity K, the second term is the image stress(given by Lakshmanan and Li [15]), the third term is the stressarising from dislocation-dislocation/image interactions [15], ,u is the shear modulus, b the Burgers vector. The model requires a dislocation velocity law to function: z’i = f’(%,, T)

C-9

The model then runs at a fixed dK/dt (k), and in each program cycle (equivalent to a short time interval dt) examines the source to see if a dislocation should be nucleated (i.e. whether Z, > 0), calculates the stresson all dislocations and move’s them a distance 6t up K is increased by dtl?, and the next cycle begins. At each cycle, the model usesthe positions of the dislocations to calculate the local crack tip stressintensity, k, which is lower than the applied K because of the ‘shielding’ effect of the emitted dislocations and their images. k=KtK,

(3)

Expressions for KD (which is negative for dislocations emitted outwards from the source into the ‘plastic zone’) are given by Lakshmanan and Li [I 51. As time and applied K increase in the model, k increases, as the increasing back stressfrom the already emitted dislocations restricts further emission from the source (this is effectively equivalent to work-hardening). Eventually k reaches K,, (the normal low temperature fracture stress intensity); this is used as the model’s fracture criterion, defining a fracture stressintensity KF. At any temperature where there is significant dislocation activity, at fracture, KF > K,,. Predicted plots of KF as a function of T have been shown to give a good fit to experimental K,/T curves for a wide variety of materials, e.g. MO and Ge [1,2,24].

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This simple model’ is r’ound to fir reasonably well fo experiment when, in the real test material, there is a high density of dislocation sources along the crack tip; an implied assumption of the model outlined above, that the shielding is uniform along the crack tip, is then true. This is the case for most materials. In such materials, KF rises gradually from K,, as the test ot simulation temperature increases, and thus as the amount of dislocation activity around the crack tip gradually increases. Eventually the stressapplied on the specimen as a whole to achieve KF exceeds the yield stress of the material, and the transition from (semi) brittle to ductile behaviour has occurred. In some materials (notably silicon), the transition is not of this form; rather, KF rises abruptly from K,, at the transition temperature. The experimental evidence indicates that this is a result of a very low density of dislocation sources in the material and especially along the crack tip [25,26]. Models taking this into account produce KJT curves that closely fit the experimental results. This paper is not concerned with this rare type of behaviour; further details are to be found in [1,21,25-281. In all earlier simulations of the BDT behaviour of single crystals we used [I]: v, = A(%rexp(-$)

(4)

The stress dependence of dislocation velocity is such that at any non-zero stress the dislocations move at some velocity. Further, the dislocation source emits a dislocation once the total stresson a dislocation at the source position is above zero (though in some versions of the model, p in Eq. (1) has been used to mimic the effects of a finite source size on the dislocation emission criterion [21]). In most materials, a critical level of stresswill be needed to activate dislocation sources and there is a level of shear stressbelow which dislocations will not move at all. In our early studies on silicon, the

Fig. 1. Configuration of models. An edge dislocation may be emitted from a source onto a slip plane inclined to the crack plane when the stress at the source is high enough to move a dislocation away from the crack.

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I2r----- -.~-.-----l_.^--- --T 2500

effects of a ‘dislocation pinning stress’ on the behaviour of the model were investigated. The conclusion was that the low values of pinning stress caused by typical impurities in silicon (l-20 MPa) did not significantly affect the behaviour of the model. Because of their relatively low mobility around T,, the stresses on dislocations emitted from crack tip sources were always much higher than the pinning stress and the dislocation arrays were always very far from equilibrium [21].

/=I

10

J

2000

2

0 5000

0

3. Model with pinning stress Here, we investigate the effects of pinning stresses on near-crack tip dislocation array evolution for materials, such as BCC metals, where the pinning stresses (from interactions with interstitial atoms, etc.) may be relatively high [29]. The dislocation velocity relation (Eq. (4)) was modified to include a dislocation pinning stress:

1r.J> zf+ui=

(y)A(?)‘“,,p($)

(6)

Above the pinning stress (rf) the velocity rises rapidly. The modification is a purely empirical way of ensuring that the dislocation velocity rises smoothly above rf from zero towards the value given by Eq. (4), so that the simulation process is stable (a steplike rise in velocity would require a very small 6t to ensure that dislocations did not substantially ‘overshoot’ in a single program cycle; the ratio of CPU time to simulated time would become unusably large). For convenience, the velocity parameters used were those previously used for simulations of the BDT in molybdenum: A = 8.61 x 1O-3 ms-r, r0 = 1 MPa, U= 0.491 eV and m = 3.4 +333.6/T, giving the best fit to the screw dislocation velocity data of Imura and Saka [30]. It should be emphasised that this does not mean that the results presented here apply only, or indeed particularly, to molybdenum. As will be seen, the behaviour of the model is controlled predominantly by zf and the applied K. The dislocation velocity parameters used are simply to provide a self-consistent and typical range of mobilities; the ‘temperature’ used was always 180 K (this is close to the normal BDTT for molybdenum, so the velocities at a given stress are in the range of velocities in the transition temperature range). 3.1. Dislocation

10000 Time

arrays

15000

(s)

Fig. 2. Results of model using rr= 100 MPa, dislocation mobility parameters as for MO at 180 K. K, applied stress intensity; k, crack tip stress intensity. The dislocation array, length d, containing n dislocations is in quasistatic equilibrium at all stages of the simulation.

the array is clearly in equilibrium: there is no further change in array length (d) or number of dislocations emitted from the source (n). Also, the crack tip stress intensity, k, does not change once K reaches 10 MPa m’j2, indicating that the positions of all dislocations within the array stay fixed. We would expect from the results of continuum fracture mechanics [31] that, for a constant yield stress aY (i.e. if all dislocations in the array are in equilibrium against a friction stress tf = a,/J3), then: (a) The crack opening displacement, COD cc n K (K’/a,); and (b) the plastic zone size = d E (K/o;)~. Fig. 3 shows typical data from the model with zf = 100 MPa; d and n are proportional to K2 at all stages of the growth of the dislocation array. The values for d and n produced by the model vary only slightly with x,: e.g. for K = 10 MPa mii2 and r,-= 100 MPa, with x, = 5000b, d = 595 urn (625 urn for X, = 50b) and n = 1925 (2170 for X, = 50b). For x, = 50b and zf between 100 and 600 MPa (i.e. aev between 173 and 1040 MPa) d/(K/ gY)’ = 0.188 + 3%; if we take COD = n x b x sin 70.5”, __I-__-.

- -. ..I-. .._..^-

I--

T-I-2000

-

1000

-

1000

1

c

.- 200 O0

20

40

60

80

100

K2 (MPa ml’*) *

Fig. 2 shows the results of a typical simulation, for a source distance x, = 50b, and rf = 100 MPa, with R = 1000 Pa m1j2 s - ’ up to K = 10 MPa rn”‘, with K then being held steady. At K = 10 MPa m’12, i = 0,

Fig. 3. Evolution of dislocation array for rr = 100 MPa, up to K = 10 MPa ml/‘. Both the number of dislocations in the array, n and the array length, d are proportional to K’.

S.G. Roberts/Materials

Science

and Engineering

A234-236

(1997)

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55

(7)

01

c 0

100

200

300 Distance

along

400

500

slip plane

(pm)

600

t 700

Fig. 4. Dislocation distribution along the slip plane, for x, = Sob, rr= 100 MPa, K= 10 MPa rn1j2,

then COD/(K2/EaY) = 0.345 + 4%. The corresponding values from finite element analyses [32,33] are d/(K/ cr,,)2= 0.175 and COD/(K2/EgY) = 0.425. This quasi-static behaviour was shown by all the simulations at T = 180 K for zr > = 80 MPa. For lower values of rr, n and d are lessthan the quasi-static values at a given applied stressintensity, K, and at a constant K, both n and d increase with time to the eventual equilibrium value. As rf increases, a lower dislocation mobility suffices for the model to behave quasi-statically (e.g. if rr = 400 MPa, T can be as low as 115 K at this k) Fig. 4 shows a typical dislocation distribution along the slip plane, for x, = 50b, r,= 100 MPa, K = 10 MPa rn1j2; it is in the form of an inverted pile-up strongly peaked near the crack tip, as expected from previous studies of arrays of dislocations in equilibrium near a crack tip [13,15]. In the BCS treatment [13], solutions were found for specific numbers of dislocations only, and in the treatment of Lakshmanan and Li [15], this approach was extended by finding solutions for successively greater numbers of dislocation in the array until no valid solution could be found, defining the final values of d and n for easy dislocation emission from the crack tip. In our method, unlike these earlier treatments, the number of dislocations in the array follows automatically from the characteristics of the dislocation source. The result is rather insensitive to the crack-tip dislocation source distance x,, provided that the source is close enough to the crack tip for dislocation emission to start at relatively low stress intensity values.

For the case of shielding by edge dislocations on an inclined slip plane, no such solution exists. We examine here the form of the shielding, using the equations of Lakshmanan and Li [15] to calculate the crack tip stress intensity, k, from the dislocation distributions resulting from the model for each value of rr and K and thus with varying resultant values of d and c. Results are shown in Fig. 5, plotted as k/(z, ~‘1’) as a function of ln(d/c). If Eq. (7) is followed, as for mode III, then the data would lie on a straight line; instead, for these edge dislocations on an inclined slip plane in mode I loading, the data are found to lie on a single curve (only slight deviations from the curve result from using different dislocation source positions, x3. Data derived from the calculations of Lakshmanan and Li [15] of equilibrium dislocation arrays on the 70.5” slip plane are also shown. Fig. 6 replots these data as ln(d/c) against ln(k/(r,c”2)): The data lie on a straight line indicating a power law dependence. A best fit line through these data give a relation as follows: o.317

-=

Thus, given that d/(K/uJ2 = 0.188, with gY= J3 rf, k = 1 82 ~0.634 aO.366 cO.l83 (9) Y The dislocation free zone size, c is found to be relatively insensitive to zf and K for a given value of x,; see Fig. 7. For other values of x,, the DFZ size is similarly always slightly greater than x,. Thus from Eq. (9), for a material with a given (near) crack-tip dislocation source configuration, there is a l *

4

3.2. Shielding The stress fields produced by the dislocations in the plastic zone reduce the crack tip stressintensity, k (and thus raise the applied stress intensity, K, required to produce fracture at k = K,,). In mode III loading an approximate solution exists for k as a function of the friction stress, TV,the plastic zone size, d and the ‘dislocation free zone’ size, c (the distance between the crack tip and the innermost dislocation in the array):

(8)

n OC 0

Lakshmanan & Li

100

I

t

200

300

k / (tt cl’*)

Fig. 5. Dislocation array size data for simulations with 100 MPa I q I 600MPa, 10 MPa ml’* I K < 40 MPa m1i2, Data from Lakshmanan and Li [15] are also shown. d, array length; c, dislocation free zone size; x,, distance of dislocation source from crack tip.

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x/(K/Q2

-

0.01

0 15

.# 4 104

-

I

0.02 I

I

I

I\

a+**

0

I

10

l %

G

.d xx b

Pa 10’

-

0 0

1 r

10

I

I

100

1000

0

-10

to show power-law

relation

dislocations images

-5

k I (z$‘*) Fig. 6. Data of Fig. 5 replotted d/c and k/(tfc’i2).

5

between

I 0

I

I

0.05

0.1 X/d

Fig. 8. CT,,),stresses on the crack tip plane, ahead of a crack to 20 MPa ml:*, with q= 100 MPa (a,. = 173 MPa).

well-defined relation between crack tip stress intensity, k, applied stress intensity, K and the yield stress, g,,, or minimum stress to move dislocations, Z~ If we assume that fracture occurs when at k = K,,, then the fracture K, KF, is given by: K

= F

BK’.5Xa Ic

- 0.57 y

(10)

where B depends weakly on c and is effectively a constant. The values of the constants and the exponents in Eqs. (8)-(10) will vary slightly with the elastic constants and Burgers vector of the material (values for molybdenum are used here), since these will affect the strength of dislocation interactions. However, they will not depend on the details of the dislocation velocity law, as long as the dislocations move quickly enough (and a,, is high enough) for the arrays to be close to their static equilibrium configurations.

“..-l”-_l.^--~.-l~l-_.~...“..““.---

25

is E. 0 .-w In 2 n

.._.-“l”l”^ _..,^,.__-..^,

l

20

8

l

El

. . xc . . .. .. .. . .. . .. . .. .. -1

10 0

100

200

300

400

500

600

~~ WW Fig. 7. Variation m’?

of dislocation

tip loaded

3.3. Stresses ahead of the crack Fig. 8 shows the form of the stress fields ahead of a crack tip loaded to 20 MPa mli2, with zf= 100 MPa (i.e. g-b.= 173 MPa). The figure shows the stress CJ~?, along the continuation of the crack plane, from (a) the applied stress intensity, K; (b) the shielding caused by the dislocations on the slip plane; (c) the images of the dislocations; and (d) the resulting total stress gY,.. The dislocations themselves actually increase the stress g.“.,, on this plane ahead of the crack; it is their image dislocations that provide the shielding which reduces k below the applied stress intensity. The forms and relative magnitudes of the stress/position curves are the same for all zf and K. The stress field near the crack tip no longer has C--‘J CCX-“~. If we examine the variation of oYJ on the continuation of the crack plane, taking uYJ oc x ~ “, the exponent n varies with x. Near to the crack tip, where the dislocation image stress field varies rapidly (x < z d/30), N is very close to l/3. With increasing x, the local value of y1 increases to a value exceeding l/2, before reaching l/2 asymptotically as x increases further. In the near crack tip region g.VY(j = 0) is proportional to k, and so is of the form:

0

A 15

I 0.15

free zone size, c with

zr and K (MPa

(11) The X-I/~ variation of c-,,~ is the same as that from the Hutchinson, Rice and Rosengren (HRR) continuum model [34,35] for a workhardening exponent of two. The exponent of the (K/o,)~ term is also very close to l/3, the value expected from continuum mechanics for the same workhardening exponent. In our model, the workhardening arises because of the influence of the dislocations in the array on the operation of the source;

S.G. Roberts/Materials

Science

and Engineering

as more dislocations are emitted, their back-stresses raise the K at which the next dislocation can be nucleated. This is a consequence of the criterion for dislocation emission being that a critical stress must be exceeded at a source positioned ahead of the crack tip, rather than that a critical level of crack tip k must be reached (for the latter emission criterion, either k will never exceed K,, and fracture will never occur, or k > K,,, and brittle fracture always occurs). If fracture is controlled by a critical micro-fracture event ahead of the main crack tip, (e.g. nucleation of cracks at carbides), which occurs at a fixed distance xr., from the crack tip, as proposed by Ritchie, Knott and Rice (RKR) [36], when ayJi = rrr, a critical local fracture stress, the condition for fracture becomes K-

xxo.5o E

R

al.57 f

oTo.57 ”

(12)

The form of the variation of KF with 6y is the same as that where the fracture criterion is k = K,, (Eq. (10)). Fig. 9 compares the variation of K, with temperature predicted from our model with the RKR model [36], with the proportionality constant in Eq. (12) fitted to the RKR data at - 140°C. The temperature variation of a,, is taken from the same paper [36]. The result is similar to that obtained with the continuum mechanics solutions.

4. Summary Plasticity at crack tips may be modelled as self-organising arrays of dislocations emitted from a source near the crack tip. For materials where dislocation motion is slow, and any friction stress is low, the modelled arrays are far from equilibrium (measured dislocation array lengths in, e.g. silicon, correspond

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closely to these predicted lengths [37]). If dislocation motion is fast, and there is a substantial dislocation pinning stress, rf, the arrays are quasi-static, and the model predicts behaviour very similar to those of earlier static models. The behaviour of the model is relatively insensitive to the characteristics of the dislocation source. The dislocation array length, d, was found to be given by: d = 0.188 (K/o,)‘, where K is the applied stress intensity and 4,, is the yield stress of the material (taken to be ,/3r,). The form of the stress fields ahead of the crack tip was investigated. Close to the crack tip, g,,r cc z x ~ ‘/3, characteristic of a ‘plastic zone’ with a work-hardening exponent of two. The shielding arises from the action of dislocation images; the dislocations themselves (on a 70” slip plane) increase the stresses ahead of the crack tip. A power-law relation has been found connecting the shielded crack tip stress intensity, k, to the pinning stress, rf, the length of the dislocation array, d, and the size of the ‘dislocation free zone’ near the crack tip, c: k -= TfCQ2

5 38 1’ o.3’7 . 0c

The value of c is relatively insensitive to K and rr. Thus, using a fracture criterion either k = K,,, or that the stress at some point ahead of the crack tip exceeds at local fracture stress, the variation of stress intensity at fracture KF with temperature is predicted to be: KF x g-; ‘.” This type of model is currently being extended, to model fatigue [38] as well as behaviour in monotonic loading.

Acknowledgements This work could not have been done without the constant interest, insight and encouragement of Professor Sir Peter Hirsch. The initial impetus for studying the effects of a large rr was the result of discussions with Peter Gumbsch.

References [l]

100 -150

-130

-110

-90

-70

Temperature ("C) Fig. 9. Comparison of fracture K (KF) of mild steel predicted by the model presented here (dashed line) and that of RKR [30] (solid line).

[2] [3] [4] [5]

S.G. Roberts, in: H.O. Kirchner, L.P. Kubin, V. Pontikis (Ed.), Computer Simulation in Materials Science-nano/meso/macroscopic Space and Time Scales, NATO ASI Series, Series E (Applied Sciences), 308, Kluwer, Dordrecht, 1996, p. 409. M. Ellis, D.Phil Thesis, University of Oxford, UK, 1991. D. Hull, P. Beardmore, A.P. Valintine, Phil. Mag. 14 (1965) 1021. S.G. Roberts, A.S. Booth, Acta Mater., 45 (1997) 1017. F.C. Serbena. D.Phil Thesis, University of Oxford, UK, 1995.

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[6] H. Vehoff, in: C.T. Liu, R.W. Cahn, G. Sauthoff (Eds.), Ordered Intermetallics-Physical Metallurgy and Mechanical Behaviour, NATO Advanced Science Institutes Series vol 213 (Series E, Applied Sciences), Kluwer, Dordrecht, 1992, p. 299. [7] A.S. Booth, S.G. Roberts, J. Am. Ceram. Sot. 77 (1994) 1457. [8] S.G. Roberts, J. Samuels, Proc. R. Sot. Lond. A 421 (1989) 1. [9] G. Michot, A. George, Scripta Metall. 16 (1982) 519. [lo] M. Brede, P. Haasen, Acta Metall. 36 (1988) 2003. [11] C. StJohn, Phil. Mag 32 (1975) 1193. [12] S.G. Roberts, H.S. Kim, P.B. Hirsch, in: D.G. Brandon, R. Chaim, A. Rosen (Eds.), Proceedings Ninth International Conference on the Strength of Metals and Alloys, Haifa, July 1991, Freund, London, 1991, p. 783. [13] B.A. Bilby, A.H. Cottrell, K.H. Swinden, Proc. Roy. Sot. A272 (1963) 304. [14] M.J. Lii, X.-F. Chen, Y. Katz, W.W. Gerberich, Acta Metall. Mater. 38 (1990) 2435. [15] V. Lakshmanan, J.C.M. Li, Mat. Sci. Eng. Al04 (1988) 95. [16] B.S. Majumdar, S.J. Burns, Acta Metall. 29 (1981) 579. [17] S.-J. Chang, S.M. Ohr, J. Appl. Phys. 52 (1981) 7174. [18] I.-H. Lin, R. Thomson, Acta Metall. 34 (1986) 187. [19] R. Thomson, in: H. Ehrenreich, D. Turnbull (Eds.), Physics of Fracture in Solid State Physics, Academic Press, London, 39, 1986, p. 1. [20] H. Huang, W.W. Gerberich, Acta Metall. Mater. 40 (1992) 2873. [21] P.B. Hirsch, S.G. Roberts, J. Samuels, Proc. R. Sot. Lond. A 421 (1989) 25.

[22] [23] [24] [25]

[26] [27] [28] [29] [30] [31] [32] [33]

[34] [35] [36] [37]

[38]

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M. Brede, Acta Metall. Mater. 41 (1993) 211. V.R. Nitzsche, K.J. Hsia, Mat. Sci. Eng. Al76 (1994) 155. F.C. Serbena, S.G. Roberts, Acta Metall. Mater. 42 (1994) 2505. H. Azzouzi, G. Michot, A. George, in: D.G. Brandon, R. Chaim, A. Rosen (Eds.), Proceedings 9th Conference on Strength of Metals and Alloys, Freund, London, 1991 p. 783. P.D. Warren, Scripta Metall. 23 (1989) 637. P.B. Hirsch, S.G. Roberts, Phil. Mag. A 64 (1991) 55. P.B. Hirsch, S.G. Roberts, Acta Mater. 44 (1996) 2361. A.H. Cottrell, B.A. Bilby, Proc. Phys. Sot. A62 (1949) 49. T. Imura, H. Saka, Mem. of Fat., Nagoya University 28 (1976) 55. J.F. Knott, Fundamentals of Fracture Mechanics, Butterworths, London, 1973. N. Levy, P.V. Marcal, W.J. Ostergren, J.R. Rice, Int. J. Fracture Mech. 7 (1971) 143. J.R. Rice, M.A. Johnson, in: M.F. Kanninen et al. (Eds.), Inelastic Behaviour of Solids. McGraw-Hill, New York, 1970, p. 641. J.R. Rice, G.F. Rosengren, J. Mech. Phys. Solids 16 (1968) 1. J.W. Hutchinson, J. Mech. Phys. Solids 16 (1968) 337. R.O. Ritchie, J.F. Knott, J.R. Rice, J. Mech. Phys. Solids 21 (1973) 395. P.B. Hirsch, S.G. Roberts, J. Samuels, P.D. Warren, Structure and properties of dislocations in semiconductors, Inst. Phys. Conf. Ser. 104 (1989) ~373. A.J. Wilkinson, S.G. Roberts, Scripta Mater. 35 (1996) 1365.