The dynamics of edge dislocation generation along a plane orthogonal to a mode I crack

Scripta

METALLURGICA

Vol. 23, pp. Printed in

925-930, 1989 the U . S . A .

Pergamon Press plc All rights reserved

THE DYNAMICS OF EDGE DISLOCATION GENERATION ALONG A PLANE 0RTHOGONAL TO A MODE I CRACK P.B. Hirsch and S.G. Roberts Department of Metallurgy and Science of Materials University of Oxford, Parks Road, Oxford 0XI 3PH, U.K. (Received

March

24,

1989)

Introduction In a recent study of the brittle-ductile transition (BDT) of precracked crystals of Si (i-4), a model for the BDT has been developed in which dislocation loops generated and moving away from the crack tip shield it so rapidly that the local stress intensity factor Kie remains below the critical stress intensity factor K1c for all values of the applied stress intensity factor K>KIc up to a value at which general yielding occurs. In order to make the modelling tractable mode III deformation is assumed; the dislocation loops have Burgers vectors parallel to the crack front, and are elongated in the Burgers vector direction. Under these conditions the dominant dislocation-dislocation interaction is assumed to be that between straight parallel screws, and the curvature at the ends of the loops is taken into account by a line tension term. The model then becomes one-dimensional (see fig. la), and the shear stress at any dislocation x i is given by K • xi = ( 2 ~ i ) ½

_ ~gb ~b x xl + ~ j~i Z (-J)½ 1 xl (x i-xJ)

(i)

where xj is the position of the jth dislocation, b the Burgers vector, ~ the shear modulus, Poisson's ratio, and ~ the line tension/image stress parameter which depends on the shape of the loop (taken as ¼ in most of the calculations). The first term in eqn. (1) is the crack tip field stress, the second a combined configurational/image stress term, the third the dislocation-dislocation interaction term. The dislocation velocity v may be written in the form v = A Tm exp-U/kT m ~m vo

(2)

where U is the activation energy controlling the dislocation velocity, m is a parameter which varies only slowly in the temperature regime considered (5), A is a constant, k the Boltzmann constant, T the temperature and v o is the temperature dependent part of v, for a constant strain-rate test (K = constant). Equation (i) then becomes

(

)I/~ (

i ) i / ~ : (2~.xi)½

xl

~

Z j~i

xl

(xi-xJ)

(3)

It is then assumed that dislocation loops are nucleated at the tip when the net stress at a critical distance x c is sufficient to expand the loop, i.e. the right-hand side of eqn. (I) is equal to zero at xi=x c. The first dislocation is emitted at a critical value of K m K N, where KN = ~b(2~/xc)½

(4)

For a given K the evolution of the dislocation array can then be calculated from eqn. (3), as a function of K. Since the dislocation positions are known, the dislocation shielding term KDIII can be calculated (6) as a function of K. In the experiments considered, dislocation loops are emitted only from particular points on the crack front. The local stress intensity factor Kez at a point Z remote from the source is assumed to be Kez

= K

- KDIII

= K

- X

~b

J>Jo (2:rxl)½

925 0036-9748/89 $3.00 + Copyright (c) 1 9 8 9 P e r g a m o n

(5)

.00 Press

plc

926

DISLOCATION

GENERATION

AT A C R A C K

Vol.

summing only over all dislocation loops which have moved past this point Z. K=_KF at which brittle fracture occurs is given by the condition Kez

=

Kic

23, No.

The value of

(6)

,

where Kzc is the critical stress intensity factor for cleavage below the transition temperature (Tc). The model is found to account satisfactorily for the observed transition temperatures and their dependence on strain rate and predicts that abrupt transitions of the type observed in Si occur only if crack tip sources are nucleated at an applied stress intensity factor KmK o just below Kic, and if these sources then continue to operate at a stress condition at x c , equivalent (for the first dislocation emitted) to K N << K o . In the actual experiments, however, the dislocations generated have edge and screw components and move on an inclined plane passing through the crack tip (3). The object of the present paper is to report on the development of a mode I model involving pure edge dislocations, and to compare the results with those from the mode III model. Mode I model The interaction between edge dislocations on inclined planes is complex and involves image dislocations on a plane inclined at an angle to the crack plane equal and opposite to that of the actual glide plane (6). The dislocation-dislocation interactions therefore involve distances in two dimensions. However, for pure edge dislocations gliding on a plane passing through the crack front at 90 ° to the crack plane, the dislocations and their images are coplanar, and the problem becomes again one-dimensional and greatly simplified; the geometry is shown in fig. lb. Using the expression for the crack tip stress field, the forces between edge dislocations and the image forces given by Lin and Thomson (6,7) and in a different form by Lakshmanan and Li (8), we find that the shear stress on an edge dislocation at Yl in the slip plane (at right angles to the crack plane) in the Burgers vector (b) direction is given by KI

~Yi = ( ~ 1 7 [

_

a~b

~b

)½

4=(1-~)Yl + ~

j,iZ {(

1

(Yl-Yj)

+

2

(yi+yj~ (yi~_y~)~

(7)

where yj is the position of the jth dislocation, and K I the applied mode I stress intensity factor.

Using eqn. (2), as before, ~Yi is replaced by (~I)I/mvc (~i)I/m__.

Comparison of

eqn. (7) with eqn. (i) shows that apart from numerical factors, which tend to make the dislocation interactions stronger relative to the crack tip stress field, there is also one additional interaction term. The value of K at which the first dislocation is emitted becomes ~b (l_~)(~q%) ,

KN =

(8)

and the local mode I stress intensity factor (from eqn. 38.1 in (6)) becomes K.z = K I - KDI = K~ - Z

4(l_~j)~_..

(9)

J>Jo where KDX is the mode I shielding term. Comparedwith eqn. (5), for the same dislocation position the shielding (for mode I) from edge dislocations in the orthogonal plane is about 40% more e f f i c i e n t than that from screws (for mode I I I ) in the crack plane. The edge dislocations in the orthogonal plane also induce a mode II anti-shielding term KIID, which, from eqn. 38.1 in (6), is found to be KDII

=

~

-

4(i-v)~b(~j)~

(10)

J>Jo The total local force fcl on the crack tip, acting in the crack plane, is therefore (6)

fcl Sinclair

and Finnis

: ~ 2~

(9) have

{(K~ - KDI) 2 ÷ K ~ I }

argued

that

the Griffith

(11) criterion

should

be written as

6

Vol.

23,

No.

6

DISLOCATION

GENERATION

AT

A CRACK

927

fcl = 27 where ? is the surface energy. However, Lin and Thomson (7) take the view that the separation of the crack faces rather than their relative shear is the important factor in crack propagation, and that the critical condition for cleavage fracture is K I - KD[ = KIc. In practice, in the experiments on Si (3), K I < 2KIt, and since KDI I = I/3KDI, under these conditions the effect of the KDI ~ term in (ii) is small and can be neglected. We have therefore retained the fracture criterion in eqn. (6), where Kez is given by eqn. (9). For this condition the instability criterion dKe/da >0 is also satisfied. Results X"ne method of computation is similar to that for the mode III case (l-g), except that equations (7), (8), (9) replace (3), (~), (5) and the condition for nucleation that the right hand side of eqn. (7) is zero at Yl = Yc replaces the equivalent condition from eqn. (i); in addition the dislocation velocity~stress~temperature data used are those for 60 ° rather than for screw dislocations (5). Fig. 2 compares mode III and mode I simulations for K o = 0.95 KIc and K N = 0.2KIt. The critical distance (dcrlt) which the dislocations have to move before causing shielding at the point Z away from the source is chosen as 7.5~m; this corresponds to the experimental situation (3, 4). It should be noted that there is little difference in the transition temperature T c predicted by the mode III and I calculations. However, the height of the step in K F at T c is considerably lower for the mode I case, and fewer dislocations (NF) are emitted. The dislocations also travel less far than in the corresponding mode III simulation; this is shown by the extent of the dislocation array at fracture (dF). Fig. 3 shows the variation of Kez with time at T c for both mode III and mode I. The mode I curve is much flatter, and does not show the pronounced dip below KIc and subsequent slow rise to K1c, which causes the large step in K F for the mode III simulations. The height of the step in K F increases as K o approaches Kic; fig. 4 shows KF, d F and N F as a function of temperature for K o = 0.98Kic , and K~ = 0.2KIt. The number of dislocations expected in the array when the specimen fractures at the transition at ~570°C, and K F ~ 1.5 MPam½ is about 150, and d F is ~40gm. This compares with the experimentally observed values of ~i00 and ~lOOgm respectively at T c = 540°C and K F = 1.6 MPam½ (see 3, 4). Generally the dislocations in the mode I arrays are more strongly coupled, and it is this fact which causes the differences in the simulations, rather than any differences between the velocities of screw and 60 ° dislocations. Using the Mode I equations but with screw dislocation velocity data produces results very similar to those from Mode 1/60 ° dislocation velocity data calculations. Fig. 5 maps the overall behaviour of the mode III and mode I models. The triangles cover all combinations of Ko,K N for which Ko>K N . The points marked are those conditions for which simulations over a range of temperatures around T c have been carried out. For each such point the value of T c is marked; it is clear that T c increases gradually with increasing K ° and K~. On each diagram the region within which the transition is predicted to be sharp, i.e. where there is a step in K F at Tc, is indicated. This is very much smaller for mode I simulations (and for a given Ko/K ~ combination the step in K F is smaller); the overall conclusion from the mode III model that K o must be close to KN and Ko>Kic is also confirmed for the mode I model. For a given value of K o, the height of the step increases with increasing Kc/K N . In conclusion, the predictions of the mode I simulation, for edge dislocations moving in a plane at right angles to the crack plane, and containing the crack front, are broadly similar to those of the mode III simulation. We expect similar conclusions to apply for mixed dislocations on inclined planes, corresponding to the experimental situation in (3)References I. 2. 5. 4. 5. 6. 7. 8. 9.

P.B. Hirsch, S.G. Roberts and J. Samuels, Scripta Metall. 21, 1523 (1987). P.B. Hirsch, S.G. Roberts and J. Samuels, Revue Phys. Appl?-2_~, 409 (1988). J. Samuels and S.G. Roberts, Proc.Roy.Soc. A421, 1 (1989). P.B. Hirsch, S.G. Roberts and J. Samuels, Proc.Roy.Soc. A421, 25 (1989). A. George and G. Champier, Phys.Stat.Solidi (a) 53, 529 (1979). R. X"nomson, Solid State Physics, eds. H. F~renreich and D. Turnbull, (Academic Press, New York) 1986, Vol. 39, p.l. I-H. Lin and R. Thomson, Acta Metall. 34, 187 (1986). V. Laksb_manan and J.C.M. Li, Acta Metall. in press. J.E. Sinclair and J. Finnis, in "Atomistios of Fracture", eds. R. Latanision and J. Pickens, (John Wiley, New York, 1983), p.i047.

928

DISLOCATION GENERATION AT A CRACK

Vol. 23, No. 6

-4

t

-I

MODE Ill

MODE I

xi

@ ~ O

~

®~

y,

A

>

eeoc

•

,,

•

T

•

•

y

image

•

~

|-

@ 0 O o

emitted screw dislocations

dislocations

-'1

>-

o

emitted

edge

dislocations

image

dislocations

o o

(a)

/

(b)

FIG. 1

Geometry of (a) Mode III and (b) Mode I simulations of dislocation arrays and cracks. The simulations are for expanding arrays, to which new dislocations ere continually added.

KF (M Po~rm)

dF(pm)

NF

I

20

O0

I

103 f~ 0

~0

10 . . . . . . . . . . 500

0 550

10~ 10

600

FIG. 2 Results of Mode III and Mode I simulations for

Ko=O.95Kzc, K~=0.2KIc.

10 ~ 2"0

I00 •103

I.l.J

0

50

NF .....

..............Y-' F r"

10

50O

.10 z

s"

0 550 600 Temperature (°C)

-10

The

variation with temperature of applled K at Fracture (KF) , length of dislocation array at fracture (dF) and number of emitted dislocations at fracture (N~) are shown.

Vol.

23,

No.

6

DISLOCATION

GENERATION

K (MPoJ'm)

AT

A CRACK

929

K (MPa'/'m)

K I--'4

~ - - ~

K

KIc Kez

I-0'

2-0.

L~

c1 0

1.0-

--.----Kez

Kes o

o

16o

Kes ol

260

o

time (s) K

(MPoV'm)

1o'00

20"00

time (s) K (MPaJ'm)

K

Kez KIc

1-0

2"0'

ULJ

n o

Kes

o

16o time (s)

Fig. 3

260

1'0"

KIc

°o

io o

20"00

time (s)

Variation of K, K,,, K,z at the transition temperature for Mode III (540°C) and Mode I (5~5°C) (K o = 0 . 9 5 K , c, K. = 0.2KI¢). For the Mode III case, there is an initial drop in Kez (left), which rises to equal Klc only after a much longer time (right). For the Mode I case, K,, remains almost constant for a short time (left), and then rises slowly (right). This accounts for the different sizes of the steps in K F, d F and N F at the brlttle-ductile transition.

930

DISLOCATION

GENERATION

KF MPaV'm)

AT

A

CRACK

Vol.

23,

No.

6

d F (pm) N F

NF

"200

,,dF 150

I

2"0

i0

• i I

.100

/KF

SO

14 soo

0

600

0

Temperature (°C) FIG. 4

Mode I simulation with increased K o. the steps in K F, N F and d F increase.

MODE III

(K o = 0.98Kzc,

K s = 0.2Kzc ).

,,~,

/ m•l.O 0-8

The sizes of

1.o

MODE I

0.8

~,~ Kxc

0"4.-~Xc •2 (

0

.

0.2

.

.

.

O.Z~ 0-6

K°/Kk FZG. 5

0.8

"~0

1.0

~

0

"

2

0

0.2

0-/,

0-6

0-8

1.0

0

K°,/Kl¢

Maps o f b r i t t l e - d u c t i l e transition temperatures f r o m Mode I I I a n d Mode I simulations, showing the results o f v a r y i n g K~ a n d Ko . The shaded regions are where the transitions are "sharp" - l.e. there is a step in K F.