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Dislocation loops at crack tips: control and analysis of sources in silicon
Materials Science and Engineering, A170 (1994) 99-109
99
Dislocation loops at crack tips: control and analysis of sources in silicon G6rard Michot, M. Angela Loyola de Oliveira and Amand George Laboratoire M~tallurgie Physique et Science des Mat&iaux, CNRS (URA 155), Ecole des Mines de Nancy, Institut National Polytechnique de Lorraine, Nancy (France) and LURE, CNRS-CEA, Universit~ de Paris-Sud, Orsay (France)
Abstract New observations of the nature and configuration of the very first dislocations emitted at crack tips in single-crystal silicon double-centilever-beam samples loaded in mode I are reported. An estimate of the shear stress acting in dislocation arrays was derived from the etch pit pattern on the crack surface. The activated Burgers vectors could be assessed by X-ray topography. The emission of dislocations with Burgers vectors parallel to the crack plane from the beginning of plastic relaxation is confirmed. Such non-blunting dislocations are shown to be important for crack opening.
1. Introduction The brittle-ductile transition (BDT) in silicon is obviously related to the formation and growth of a plastic zone (PZ) around the crack tip [1-8], but there is still some controversy on whether the BDT temperature at a constant loading rate is solely determined by the dislocation mobility or depends also on the rate of dislocation nucleation at or near the crack tip. In a recent paper [8] two of the present authors reviewed the characteristics of PZs (activated Burgers vectors, slip planes and shape of dislocations loops), which were studied in single crystals with four different orientations of the crack plane and crack front. The possibility that bulk microdefects could act as nucleation centres of dislocation loops was ruled out and evidence was provided that dislocations nucleate at the crack tip, but in a very inhomogeneous manner, from special spots along the crack front. Such information about dislocation sources was obtained through the etch pit distribution over the crack plane in samples which had failed very near the BDT or whose loading had been stopped just below Kxc so that only a few loops had been emitted. (In the latter case samples were subsequently broken at room temperature to allow the crack plane to be etched.) A strong limitation is that chemical etching does not allow the Burgers vectors to be assessed. The first purpose of this work was to perform X-ray topography (XRT) observations of the very firstly emitted dislocations in order to determine whether they have the same Burgers vectors as those identified in well-developed PZs. The difficulty comes from the poor spatial resolu(1921-5093/94/$7.00 SSl)l {)921-5093(93)(12551 -D
tion of XRT, several microns, which prevents too small loops from being resolved. A two-step loading treatment was used to obtain a small number of loops of sufficient size. First results are reported here. Other important and usually missing information about crack tip plasticity is the stress level within the PZ. In previous work [3] we could only determine the shear stress acting on leading dislocations, i.e. at the border of the PZ, from direct measurements of their velocity, thanks to the known stress and temperature dependences of dislocation mobilities in Si [9]. The dislocation density within the PZ was too large for the motion of a given loop to be followed by XRT. The second purpose of this paper is to suggest that useful information can be derived from etch pit patterns when dislocations form well-defined and isolated arrays. The third and last question that will be addressed here deals with the slip systems that are activated at crack tips. It was pointed out in previous work that considerations on the crack tip stress field did not suffice to explain the observations satisfactorily. In particular, the frequent observation of dislocations with their Burgers vector parallel to the crack plane was surprising to many. We propose a tentative justification for the emission of such loops under mode I loading.
2. Experimental details Tapered double-cantilever-beam (DCB) samples were used [1, 8]. The sample thickness was about 0.6 ram. Crack initiation and arrest were achieved at room © 1994 Elsevier Sequoia. All rights reserved
1O0
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Dislocation loops at crack tips
temperature using St. John's wedge technique [1]. For a given applied load P the stress intensity factor K is obtained through the calibration curve given by St. John. Samples studied here were cleaved along {111} planes, with the crack propagating in a (110) direction for the so-called a orientation or in a (112) direction for the fl orientation. Precracked samples were loaded in mode I in the high temperature deformation stage described by George and Michot [10], with an improved temperature control and a new loading system which allows the applied K to be increased at a constant rate. This stage was mounted on the two-axis spectrometer of the topography beam port of the synchrotron facility LURE-DCI. Experiments were conducted under a reducing atmosphere ( 10% He, 90% N2). Unless they had broken during in situ loading, samples were cooled with the load applied. Their PZ was characterized in further detail by XRT. Then samples were loaded up to fracture at room temperature on an Instron machine. Broken half-samples were immersed for a few seconds in Sirtl etchant and examined in a scanning electron microscope after gold metallization to complete the determination of the actual crack tip geometry and dislocation distribution. The PZs of sufficient size to be investigated by XRT usually have too many dislocations which cannot be resolved. A two-step loading sequence at high temperature was used: a small number of dislocation loops were created at K l =(0.7-0.8)Kic and then expanded at smaller stress (K l = (0.4-0.5)Kxc) in order to prevent further nucleation. It would be desirable to have both XRT and etch pit information on the same sample. This was difficult to achieve with fl samples, which usually break on inclined planes and not along the precrack plane once dislocations have been emitted.
3. Source location and activated slip systems at the onset of crack tip plasticity 3.1. General remarks Some information about the location of the first sources activated along the crack front can in specific cases be extracted from the observation of well-defined PZs whose external parts should contain those dislocations which have travelled the longest distance and have been emitted first. Figure 1 shows a partial view of the etch pit distribution on the large (5]2) free surface of an a-oriented sample. External dislocations form well-defined arrays which in principle allow the activated slip planes to be traced back to the sources. Within the PZ at a shorter distance from the crack tip,
-...j Fig. 1. Dislocationetch pits at the border of the plastic zone in an ct sample loaded at T=1073 K under K~=0.71 MPa m~/2 (approximately0.76 K~c) for 30 min (marker: 100/~m).
evidence for such planar slip on a few distinct planes is lost, as if the first sources have stopped being active and have been replaced by others located in between along the crack front. This can be readily explained by considering the modification of the local stress intensity factor after dislocation emission. In a 1D or 2D picture the dislocations have a shielding effect and reduce the local K at the source, but in a realistic 3D picture they can also -- and in fact do -- have an antishielding effect on parts of the crack front between the sources [11], allowing dislocation nucleation at other places. A second reason for the apparently more homogeneous dislocation distribution near the crack tip in large PZs stems from the interactions and reactions between different slip systems which could favour other processes of dislocation multiplication such as double cross-slip at Lomer barriers, etc. Finally, it must be stressed that it is very difficult in practice to identify the exact source point on the crack front of a dislocation array seen on lateral sample free surfaces. Therefore dislocation sources can be identified and characterized only in samples containing a small number of dislocations. 3.2. Typical source configuration in a-oriented samples 3.2.1. Description The orientation is given in Fig. 2. Figure 3 shows typical etch pit distributions on the {111} cleavage surfaces around an internal dislocation source in a sample loaded at/£ = 62 Pa m 1/2 s- 1 and T= 885 K up to fracture, which occurred at a load of 1.1K~c. Five such configurations appeared along the crack front as marked by the letters O and S. In this and all other figures in this paper the crack was propagated from top to bottom. Dislocation rows have been numbered for ease of description. It can be seen that there are probably two source points at the crack tip. The main one has given the long dislocation arrays and the secondary has emitted row 4 and perhaps some dislocations in row 1. Etch pit patterns must obey simple rules: (i) since dislocation loops are too small to reach the sample free
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Dislocation loops at crack tips
surfaces, each loop must form two etch pits on a given half-sample; (ii) below the crack tip the material was continuous during dislocation emission and dislocation segments developing ahead of the tip had to cut the {111 } plane in the prolongation of the cleavage surface and therefore must form pits on the two facing surfaces. Indeed, rows 1-4 are seen on (111) and (] i i ) surfaces. A close examination, however, reveals some differences in the dislocation locations. An obvious example is seen along row 2, where the fourth dislocation from the head of the row is not at the same distance from the crack tip on the two surfaces. This can be explained by some backward motion of dislocations just after fracture, when the temperature has not yet been sufficiently decreased. Observations of etch patterns on broken samples were sometimes criticized because of this possible relaxation of dislocation configurations. We feel, on the contrary, that the present example indicates that such backward motion is rather
111
Fig. 2. Stereographic projection of a samples (crack surface (111), propagation direction [] 10], sample free surface (i ] 2)). The shaded half-sample was used for XRT.
5 s
i
[III]
(~[~1O] (ii I I
"'3. 4
(TTI) [I12~ 7 .8
[iT]]
~[iio] ~0
(lil)
"~ (Tll) J I I I
(ill)
2
1O1
,m
[iT2]
Fig. 3. Etch patterns around a dislocation source observed on two opposite crack su~aces (marker: 10/~m). See text for details.
102
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limited: in the main dislocation arrays, loop locations differ by less than 12%. Above the crack tip, i.e. behind it, dislocations emerge at the two crack surfaces and behave independently on each side of the crack. Etch pits need not coincide, which gives some useful indication of the shape of the dislocation loops. Rule (i) must be regarded with caution when the dislocation density is large. Very near the crack tip, linear etch figures (grooves) suggest that a number of near dislocations have remained attached to the crack. Of course, they cannot be counted. An attempt was made to determine the Burgers vectors of these dislocations by XRT. A topograph of one half of the sample is shown in Fig. 4. The dark spot marked by an arrow corresponds to the groups of dislocations shown in Fig. 3. Clearly, resolving individual dislocations is beyond the capacity of the Lang technique and even the main dislocation families could no be identified. In more detail the different dislocation families can be described as follows. Loops expanding in (i11) planes. Ahead of the crack tip, row 1 has about 15 dislocations distributed over at least two parallel planes. On the rear side of the tip, on the (111) crack surface, row 5 contains at least 23 dislocations in two groups on a first (i 11) plane and about 14 dislocations on a second parallel plane. In both rows accumulated pits form a nearly continuous trace very near the source point. On the opposite ( 11 ] ) surface, behind the tip, only two isolated pits appear
(row 7). Loops expanding in (lil) planes. Ahead of the tip, two distinct parallel arrays are seen on both surfaces. The longer row 3 has about 37 loops in one plane and the smaller row 4 about nine loops also in one and the same plane, to the accuracy of etch pit observations. Behind the tip, no dislocations emerge on the (111) surface, while row 8 is formed on (1 i ]). This last row consists of of six well-shaped pits and at least 25 others
Fig. 4. X-ray topograph of the half-sample shown in the upper part of Fig. 3: g = 400 (marker: 200/xm).
of various smaller sizes. A few pits are seen in the same plane as row 4. Loops expanding in (11i) planes. The situation here is apparently much simpler. Four aligned dislocation pits appear ahead of the tip (row 2). Behind the tip, seven aligned loops emerge on the (111) surface (row 6) and none, except maybe very near the crack front, on(lll). It may be observed, especially on the etched (i ] i) surface, that the pits have different shapes and contrast. Usually they look similar in parallel rows ahead of and behind the crack front, with exceptions in row 8 and perhaps row 1. It may be checked also that the two pits in row 7 are different from those in row 1 but look very similar to those in rows 2 and 6. Figures 5-7 propose a possible 3D reconstruction of the configuration based on etch pit patterns and considerations of the crack tip stress fields. The resolved shear stresses r acting on a given slip system can be derived from the mode I stress tensor [12] in the form
KiH( cp) r = (2xR)l/2 where R and q~ are the polar coordinates in the slip plane (with cp= 0 along OX, the intersection between this plane and the crack plane). Results are expressed in the form of equal-stress contours G(qg) = [H( q~)]2[8]. The sketches of Figs. 5 and 6 are reconstructions of what should appear on a 440 topograph taken with Mo K a radiation. The orientations shown are projections of crystallographic directions on the photographic plate.
3.2.2. Possible Burgers vectors Row 5 contains more loops than row 1. According to the stress contour, it is likely that most dislocations behind the tip on ( i l l ) have b=(a/2)[101]. These dislocations, when the loops are small compared with the sample thickness, are confined behind the tip on one side of the cracked sample and should appear on ( 111 ) only. Dislocations ahead of the crack front can have either (a/2)[011] or (a/2)[l10] Burgers vector. Both stress contours are compatible with etch pit distributions on cracked surfaces. (a/2)[l10] has the larger stress on average and should be more constrained to remain attached to the tip on the ( i ] ]) side than (a/2)[011]. However, (a/2)[110] is very seldom detected in well-developed PZs, in contrast with (a/2)[011], and direct Burgers vector assessment by XRT would be necessary. On (1 ] 1) the smaller row 4 and the longer row 3 probably have (a/2)[110] or (a/2)[10i] Burgers vector, the former being more probable because no emerging loops appear on ( 111 ) behind the crack front. Disloca-
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Dislocation loops at crack tips
103
d
# //
J
.................... D io] ......... [oli] Boq
/j5
[itl]
7//
+
/r. /
r
,(
&
(a)
'
+
(b)
Ii
"'1,,.
/
1
!
I,°ll
/
III
ii O.I
$X [01]]
Fig. 5. (a) Possible dislocation configuration in the ( ] 11 ) plane from Figs. 3 and 4. Dislocations drawn as full lines are those seen in Fig. 4. Dashed lines represent the parts of loops belonging to the other half-sample. (b) Equal-resolved-shear-stress contours in the (111) plane for available Burgers vectors (a orientation). The stress acting on full-line dislocations corresponds to the y> 0 domain, that on dashed-line dislocations to y < 0.
II
d
J~ ijip
[ito]
...................
[oii]
........
[ioi]
,fl-i; JI IIl[ii~ JJJ
;;" ~;
J
[o,q
-.... 2
1
(a)
+~1.1 I/'/
t
-4-
(b)
[ioq
[i,o]
l[i lO1
x
0.1
Fig. 6. (a) Possible dislocation configuration in the (111 ) plane. Same convention as in Fig. 5a. See text for details. (b) Equal-resolvedshear-stress contours in the (111 ) plane. Same convention as in Fig. 5b.
tions with b = (a/2)[011] are excluded in r o w 4 but can exist in r o w 8. D e p e n d i n g on the shapes of pits, the Burgers vectors in rows 1 and 3 should be different, but a different orientation of e m e r g i n g loops can also be invoked.
O n (111) the o b s e r v a t i o n s are not easy to u n d e r stand. Clearly only four loops d e v e l o p a h e a d of the tip while seven a p p e a r behind it on ( 111 ). It is unlikely that such loops run a longer distance behind the crack than a h e a d of it in view of the stress contours. We believe
104
G. Michot et al.
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/
Dislocation loops at crack tips
,'
A 111
/\k tl¢
/
b':
~lI •
Fig. 7. 3D reconstruction of the arrangement of dislocations around the source shown in Fig. 3. Same convention as in Figs. 5a and 6a.
that the two isolated pits forming row 7 belong to loops of row 6 which have cross-slipped on to (111 ). Those seven loops probably have b=(a/2)[lO1] and equally probably from stress contours as (a/2)[0]i] (which could cross-slip only on to ( 111 )). Cross-slipping must have occurred at or very near the crack tip, since rows 2 and 7 converge at the crack front. Figure 6 shows the possible shape of such a loop expanding on two planes. Etch pits should, of course, also appear ahead of the crack front along row 2, but cross-slipping has probably prevented this part of the loop from expanding freely far from the source and pits cannot be resolved because of the high dislocation density at the source point. Alternatively, it is possible that two loops with b = (a/2)[101] started to develop on ( ] 11) behind the tip and then cross-slipped on to (11 i) from the crack point where they had to remain attached on ( i 11 ). 3.3. Preliminary results on source configurations in fl samples The two-step loading technique described above was applied to six samples. The/3 orientation (Fig. 8), for which results are still scarce, is interesting because a glide plane contains the [011] crack tip (which realizes a 2D situation as in current models) and because the Burgers vector analysis is simpler than in a samples. Results are summarized in Table 1. In agreement with former observations in well-developed PZs, it seems from XRT that nearly all dislocations lie on ( 1 i 1 ) and ( 11 i ) and not on ( i 11 ), which contains the crack front. It is also confirmed that the (a/2)[011] Burgers vector is never observed and that Burgers vectors parallel to the crack plane can be emitted from the very beginning of dislocation emission. Figure 9 shows about 12 identical dislocations still attached to one point of the crack front at or very near
"'"'--.
,"
s"
I/
•
-r.
ii~ ,,,,," .........
Fig. 8. Stereographic projection of fl samples (crack surface ( 111 ), propagation direction [211 ], sample free surface (0[ 1)).
TABLE 1. Dislocation source analysis:number of sources, glide planes and Burgers vectors of dislocations, and number of dislocations per family Sample
Plane (1 i 1)
Plane (1 i 1)
Number of sources
1ol EA3 EA7 EA9 EA10 EA12 EA13
iio
2 6 10-12 3 4 6 -60--4 12
oil
lOl
liO
oii
-------
4-6 1012 2
167 ---16
-------
1(2) -2 2 1 1
the sample surface. They glide on (111) and have b---(a/2)[lO1]. This confirms the importance of the sample surface in nucleation. In the sample shown in Fig. 10 it was possible to observe the etch pit distribution on one half-sample and a striking feature was revealed. The etch pattern proves that 35 or more dislocations were emitted in the (i 11 ) plane parallel to the ideal crack tip. These dislocations have been emitted behind the crack tip, which was slightly off [01]], and the leading one runs for several hundred microns. X-ray topographs, on the other hand, reveal that most dislocations have b = (a/2)[110] but glide on (1 i 1). This strongly suggests that dislocations are first emitted on ( 111 ), run more or less parallel to the crack front over some distance and cross-slip on to the ( 111 ) planes where they form large loops. The detailed geometry of this cross-slip process is still under study and this observation must be confirmed in other fl samples before any conclusion can be drawn.
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Dislocation h)ops at crack tips
105
Fig. 9. X-ray topographs of a fl sample loaded at T = 1023 K under KI=0.71 MPa m ~:-'(approximately 0.76Kic) for 1 min and then under KI= 0.31 MPa m ~/2for 30 min: a, g= 22(); b, g= 202; c, g = 202, (marker: 30(t t~m).
4. Stress field within the plastic zone Row 3 of Fig. 3 consists of about 37 dislocations distributed between point I at 15.8 ~ m from the emission point E and point O at 70/~m from E. There is a dislocation-free zone between ! and J (at 11.9/~m from E). Such a dislocation-free zone resulted from a break in the source activity in this plane. Assuming a negligible motion of dislocations after fracture, the stress acting on each dislocation j can be calculated by summing the applied crack tip stress field Kl(H(cfl)} r a - (2xR)t/: and the interaction stress due to other dislocations,
"~i
aktb V
1
27r i~] Rj -- R
i
Using (H(q))}=H(0)=0.23 [11], a-~1.25 for 60 ° dislocations, K I -~ 1 MPa m m (derived from the fracture stress) and Rj measured on the micrograph and neglecting corrections due to the vicinity of the crack [13], one finds the results of Fig. 11, which show that
the effective stress is roughly equal to 20 MPa and fairly constant within the PZ except close to the dislocation-free zone. This estimate is consistent with the stress measured at the border of the PZ under similar loading conditions [3].
5. Crack opening 5.1. Macroscopic observations
During high temperature testing under a constant opening rate 6, plastic opening is assumed because of the departure from linearity at P,, of the load vs. deflection curve, leading to a larger compliance than that expected from an elastic sample, even if temperature corrections of the elastic constants are introduced. As could be expected from differences in source activities from sample to sample, a large scatter in P,, values is noticed. After removal of the load a permanent opening of a few microns is detected. Measurements taken at room temperature indicate smaller compliance values for plastically deformed samples (Fig. 4 of ref. 14), which can be explained by the static compressive stresses induced by the plastic zone: this fact expresses crack shielding.
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Dislocation loops at crack tips
Thus we must conclude that dislocations account for two opposite mechanisms, i.e. crack opening and crack closure. The question is: are antishielding dislocations required for plastic opening? It will be shown from experimental results obtained on the fl orientation that this condition is not always necessary.
5.2. Microscopic observations Observation of the full width of the sample (Fig. 12) shows that in addition to the PZ developed around the crack tip, a few dislocations are seen coming from the crack surfaces above the PZ, while a strong plastic deformation is observed on the right-hand edge of the sample in the continuation of the main PZ growth direction [0ii] (see Fig. 8). Such dislocations can be found at sources located on the sample edges. For a diffraction vector perpendicular to the crack plane, most dislocations of the PZ and those coming from the sample edge as well are out of contrast: this means that the possible Burgers vectors, (a/2)[1 i0] or
/ ~ xl I
I
! / ! / / / I
/
Q [iii]
(c)
[11] I
Fig. 10. X-ray topographs and etch patterns of a fl sample loaded at T = 1023 K under K I =0.66 MPa m 1/z for 3 min and then under KI = 0.38 MPa m u2 for 30 min: a, g = 1 i i; b, g = i i i (same magnification as Fig. 9); c, etch distribution observed on the cleavage surface (marker: 50 #m).
G. Michot et al. /
Dislocation loops tit crack tips
107
: l d 4 ( c m -1 )
X2
-I
-" X 1
v P
I
25
,
50
b2
b -
I
!
j,
/ /
"r( M P a )
./"
40
\,\
~'~.
C
d
:_: ::.- T:-T-:=......... ,'"-
|/-b I
, R lpml
50
". /
Fig. 11. Evolution of the dislocation density (a), internal stress (b), applied stress (c) and effective stress (d)in dislocation array 3 of Fig. 3 as a function of the distance from the source.
d~ •_
. . . . . . ~ .............. ,,! lT I
I
..
I
Fig. 13. Definitions (a), shielding dislocation (b_,> 0 according to ref. 14)(b). shielding dislocation (b~ >0 according to ref. 14, 0 < s~/3)(c) and shielding dislocation (b I < 0, 0 > x/3) (d).
the crack tip has a potential energy Ep (per unit thickness of sample) given by
Ep = E~,+ Ed °~ + Ed ~- P(6, + rid)
Fig. 12. X-ray topograph (g= 202) showing the full width of a/~ sample (inverse contrast). See text for details.
(a/2)[]01], are parallel to the crack plane and not expected from mode I loading. The use of other diffraction vectors has confirmed this result.
5.3. Closure vs. opening (i) Thermodynamical considerations are required in order to determine whether the emission of a dislocation will lead to crack closure or crack opening [15]. A semi-infinite crack, loaded at x a by line forces P (N m-l)(Fig. 13(a)), with a dislocation located at (r, 0) of
where the first term on the right-hand relates to the elastic energy induced by the applied load P, the second to the dislocation self-energy in an infinite medium, the third to its variation due to neighbouring surfaces and the fourth to the external work when the applied force is displaced by 6,~ (due to the force-P) plus 6 d (due to the dislocation). Shielding occurs when the product K , K d is negative, i.e. when the effective stress intensity factor K, + K d is smaller than K,. (ii) Emission from the crack tip requires the radial force acting on the dislocation, J;= -(OEp/Or)o to be positive:
The first term on the right-hand side, the image force, is always negative, so the derivative of the second term must necessarily be positive, but the function Pdd, itself can be either positive or negative. For a semiinfinite crack, because 6d(Xl) is a function of 0 and ~p only, it can be shown that ( Odd/Or)o = - x I ( Odd/OX l)/r. Thus emission requires positive values for Odd/dX 1. This condition depends on the Burgers vector.
G. Michot et al. / Dislocation loops at crack tips
108
For the positive dislocation b 2 of Fig. 13(b) it is shown in ref. 15 that the crack opening varies from - b 2 at infinity to zero at the crack tip. Of course, negative 6~ values under mode I loading only make sense if a larger Oa opening is superimposed. This dislocation, which obviously has a shielding effect, must, according to 6 d variations with x~, sustain a positive force and so is likely to be emitted. The situation is more complex for dislocation b~. For 0 smaller than Jr/3 the crack opening varies from zero at infinity to zero at the crack tip (Fig. 13(c)), going through a negative extremum. The sign of (06JOx~)o depends on the location of the point x a with respect to the extremum Xm: as far as emission is concerned, the small-scale yielding condition (Xa'> r) holds, i.e. such a shielding dislocation is attracted. In other words, antishielding- b~ dislocations are emitted for 0 < :r/3. For larger 0 values a - bl dislocation leads to both crack tip closure (crack shielding) and crack opening at a given distance x a from the tip (Fig. 13(d)): under the small-scale yielding condition this shielding dislocation is likely to be emitted. Figure 14(a) shows that an array of such dislocations is equivalent to a tilt subboundary, confirming the assumption set forth in ref. 16. There are differences between bl and b 2 behaviours: when the dislocation moves away from the crack tip, the closure effect induced by a dislocation b2 at x a decreases (Fig. 13(b)), while the crack opening due to a dislocation - b ~ increases slowly up to a maximum
- b 1 T~'~b -,T\ T T\ .~.T "r\'
value for Xm/r (Fig. 13(c)) for which the sign of the radial force on the dislocation reverses.
5.4. Application to fl orientation 5.4.1. The (1]1) and (ii1) planes (see Fig. 9) In a real 3D geometry the usual slip systems are ( a / 2 ) [ i l 0 ] ( ] i l ) and (a/2)[i01](]li). Because their combination gives an equivalent (a/2)[211] Burgers vector parallel to the crack propagation direction, the situation is equivalent to a 2D problem. However, we must bear in mind that emission does not take place along planes which contain the crack tip and that the sketch of Fig. 14(a) is just an equivalent representation. A shielding dislocation is likely to be "emitted" along the [011] direction for 0 = 125 °. Because an antishielding - b l would be emitted for 0 = 55 ° (according to Section 5.3), antishielding bl dislocations are expected to develop in the negative sector (0 = - 5 5 °) along the [011] direction because of mirror symmetry with respect to the crack plane. This means that the crack can emit an equivalent full dislocation in both directions. This loop globally shields the crack, because the influence of the antishielding component bl is seven times smaller than that of the shielding one - b~ [15], but the full loop contributes to crack opening. For such a loop, using the relations given in ref. 15 for a semi-infinite crack, it is possible to evaluate the opening to be about 0.1 b per loop ( r - 1 mm, x a = 13 mm, 0 = 125 °, -55°). Under subcritical creep conditions (KI=0.8KIc) etch pit counting [11] indicates about 7000 loops inside a "saturated" PZ. This should lead to an opening of about 0.2/~m, which is one order of magnitude smaller than that observed experimentally. However, because of the finite size of our samples, some elastic parameters (compliances, stress intensity factors, etc.) are one order of magnitude larger than those of the semi-infinite medium for the same applied load and crack length: therefore the "bending" dislocations could account for crack opening.
5.4.2. The (l ] ]) plane
/ / / .......
b
t/
/
4/
/ Fig. 14. 2D representation of the fl orientation: a, "bending" dislocations responsible for crack opening; b, "blunting" dislocations.
The dislocation b emitted for 0 = 70 ° (Fig. 14(b)) shields the crack, because the b2 component has the largest effect. Because emission of a shielding dislocation is expected for 0 = 110 °, from mirror symmetry a b dislocation is likely to be emitted for 0 = - 1 1 0 °. These two dislocations have a shielding and a blunting effect which both tend to decrease the stress concentration. However, because closure due to shielding tends to cancel the influence of blunting and because dislocation activity is very limited in that plane, we conclude that such dislocations do not contribute markedly to crack opening.
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Dislocation loops at crack tips
References 1 C.F. St. John, Philos. Mag., 32 (1975) 1193. 2 G. Michot, K. Badawi, A. R. Abd E1 Halim and A. George, Philos. Mag., 42 (1980) 195. 3 G. Michot and A. George, Inst. Phys. Conf. Ser., 104 (1989) 385. 4 M. Brede and R Haasen, Acta Metall., 36 (1988) 2003. 5 M. Brede, Acta Metall. Mater., 41 (1993) 211. 6 R B. Hirsch, S. G. Roberts and J. Samuels, I¥oc. R. Soc. A, 421 (1989)25. 7 P.B. Hirsch and S. G. Roberts, Philos. Mag. A, 64 (1991) 55. 8 A. George and G. Michot, Mater. Sci. Eng. A, 164 (1993) 118.
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9 A. George and G. Champier, Phys. Status Solidi A, 53 (1979) 529. 10 A. George and G. Michot, J. AppL Crv.stallogr., 15 (1982) 412. 11 G. Michot, Th~se d'Etat, INPL, Nancy, 1982. 12 P. C. Paris and G. C. Sih, Fracture Toughness Testing and Its Applications, ASTM Spec. Tech. Publ. 381, ASTM, Philadelphia~ PA, 1965, p. 32. 13 B.S. Majumdar and S. Burns, Acta Metall., 29 (1981 ) 579. 14 G. Michot, A. George and G. Champier, in K. L. Maurer and F. E. Matzer (eds.), Proc. 4th Eur. Conf. on Fracture, Leoben, 1982, EMAS, 1982, p. 30. 15 M. A. de Oliveira and G. Michot, Mater. Sci. Eng. A, 176 (1994) 135-141. 16 G. Michot and A. George, Proc. ICSMA 7, Montreal, 1985, Pergamon, New York, 1985, p. 1187.
99
Dislocation loops at crack tips: control and analysis of sources in silicon G6rard Michot, M. Angela Loyola de Oliveira and Amand George Laboratoire M~tallurgie Physique et Science des Mat&iaux, CNRS (URA 155), Ecole des Mines de Nancy, Institut National Polytechnique de Lorraine, Nancy (France) and LURE, CNRS-CEA, Universit~ de Paris-Sud, Orsay (France)
Abstract New observations of the nature and configuration of the very first dislocations emitted at crack tips in single-crystal silicon double-centilever-beam samples loaded in mode I are reported. An estimate of the shear stress acting in dislocation arrays was derived from the etch pit pattern on the crack surface. The activated Burgers vectors could be assessed by X-ray topography. The emission of dislocations with Burgers vectors parallel to the crack plane from the beginning of plastic relaxation is confirmed. Such non-blunting dislocations are shown to be important for crack opening.
1. Introduction The brittle-ductile transition (BDT) in silicon is obviously related to the formation and growth of a plastic zone (PZ) around the crack tip [1-8], but there is still some controversy on whether the BDT temperature at a constant loading rate is solely determined by the dislocation mobility or depends also on the rate of dislocation nucleation at or near the crack tip. In a recent paper [8] two of the present authors reviewed the characteristics of PZs (activated Burgers vectors, slip planes and shape of dislocations loops), which were studied in single crystals with four different orientations of the crack plane and crack front. The possibility that bulk microdefects could act as nucleation centres of dislocation loops was ruled out and evidence was provided that dislocations nucleate at the crack tip, but in a very inhomogeneous manner, from special spots along the crack front. Such information about dislocation sources was obtained through the etch pit distribution over the crack plane in samples which had failed very near the BDT or whose loading had been stopped just below Kxc so that only a few loops had been emitted. (In the latter case samples were subsequently broken at room temperature to allow the crack plane to be etched.) A strong limitation is that chemical etching does not allow the Burgers vectors to be assessed. The first purpose of this work was to perform X-ray topography (XRT) observations of the very firstly emitted dislocations in order to determine whether they have the same Burgers vectors as those identified in well-developed PZs. The difficulty comes from the poor spatial resolu(1921-5093/94/$7.00 SSl)l {)921-5093(93)(12551 -D
tion of XRT, several microns, which prevents too small loops from being resolved. A two-step loading treatment was used to obtain a small number of loops of sufficient size. First results are reported here. Other important and usually missing information about crack tip plasticity is the stress level within the PZ. In previous work [3] we could only determine the shear stress acting on leading dislocations, i.e. at the border of the PZ, from direct measurements of their velocity, thanks to the known stress and temperature dependences of dislocation mobilities in Si [9]. The dislocation density within the PZ was too large for the motion of a given loop to be followed by XRT. The second purpose of this paper is to suggest that useful information can be derived from etch pit patterns when dislocations form well-defined and isolated arrays. The third and last question that will be addressed here deals with the slip systems that are activated at crack tips. It was pointed out in previous work that considerations on the crack tip stress field did not suffice to explain the observations satisfactorily. In particular, the frequent observation of dislocations with their Burgers vector parallel to the crack plane was surprising to many. We propose a tentative justification for the emission of such loops under mode I loading.
2. Experimental details Tapered double-cantilever-beam (DCB) samples were used [1, 8]. The sample thickness was about 0.6 ram. Crack initiation and arrest were achieved at room © 1994 Elsevier Sequoia. All rights reserved
1O0
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Dislocation loops at crack tips
temperature using St. John's wedge technique [1]. For a given applied load P the stress intensity factor K is obtained through the calibration curve given by St. John. Samples studied here were cleaved along {111} planes, with the crack propagating in a (110) direction for the so-called a orientation or in a (112) direction for the fl orientation. Precracked samples were loaded in mode I in the high temperature deformation stage described by George and Michot [10], with an improved temperature control and a new loading system which allows the applied K to be increased at a constant rate. This stage was mounted on the two-axis spectrometer of the topography beam port of the synchrotron facility LURE-DCI. Experiments were conducted under a reducing atmosphere ( 10% He, 90% N2). Unless they had broken during in situ loading, samples were cooled with the load applied. Their PZ was characterized in further detail by XRT. Then samples were loaded up to fracture at room temperature on an Instron machine. Broken half-samples were immersed for a few seconds in Sirtl etchant and examined in a scanning electron microscope after gold metallization to complete the determination of the actual crack tip geometry and dislocation distribution. The PZs of sufficient size to be investigated by XRT usually have too many dislocations which cannot be resolved. A two-step loading sequence at high temperature was used: a small number of dislocation loops were created at K l =(0.7-0.8)Kic and then expanded at smaller stress (K l = (0.4-0.5)Kxc) in order to prevent further nucleation. It would be desirable to have both XRT and etch pit information on the same sample. This was difficult to achieve with fl samples, which usually break on inclined planes and not along the precrack plane once dislocations have been emitted.
3. Source location and activated slip systems at the onset of crack tip plasticity 3.1. General remarks Some information about the location of the first sources activated along the crack front can in specific cases be extracted from the observation of well-defined PZs whose external parts should contain those dislocations which have travelled the longest distance and have been emitted first. Figure 1 shows a partial view of the etch pit distribution on the large (5]2) free surface of an a-oriented sample. External dislocations form well-defined arrays which in principle allow the activated slip planes to be traced back to the sources. Within the PZ at a shorter distance from the crack tip,
-...j Fig. 1. Dislocationetch pits at the border of the plastic zone in an ct sample loaded at T=1073 K under K~=0.71 MPa m~/2 (approximately0.76 K~c) for 30 min (marker: 100/~m).
evidence for such planar slip on a few distinct planes is lost, as if the first sources have stopped being active and have been replaced by others located in between along the crack front. This can be readily explained by considering the modification of the local stress intensity factor after dislocation emission. In a 1D or 2D picture the dislocations have a shielding effect and reduce the local K at the source, but in a realistic 3D picture they can also -- and in fact do -- have an antishielding effect on parts of the crack front between the sources [11], allowing dislocation nucleation at other places. A second reason for the apparently more homogeneous dislocation distribution near the crack tip in large PZs stems from the interactions and reactions between different slip systems which could favour other processes of dislocation multiplication such as double cross-slip at Lomer barriers, etc. Finally, it must be stressed that it is very difficult in practice to identify the exact source point on the crack front of a dislocation array seen on lateral sample free surfaces. Therefore dislocation sources can be identified and characterized only in samples containing a small number of dislocations. 3.2. Typical source configuration in a-oriented samples 3.2.1. Description The orientation is given in Fig. 2. Figure 3 shows typical etch pit distributions on the {111} cleavage surfaces around an internal dislocation source in a sample loaded at/£ = 62 Pa m 1/2 s- 1 and T= 885 K up to fracture, which occurred at a load of 1.1K~c. Five such configurations appeared along the crack front as marked by the letters O and S. In this and all other figures in this paper the crack was propagated from top to bottom. Dislocation rows have been numbered for ease of description. It can be seen that there are probably two source points at the crack tip. The main one has given the long dislocation arrays and the secondary has emitted row 4 and perhaps some dislocations in row 1. Etch pit patterns must obey simple rules: (i) since dislocation loops are too small to reach the sample free
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Dislocation loops at crack tips
surfaces, each loop must form two etch pits on a given half-sample; (ii) below the crack tip the material was continuous during dislocation emission and dislocation segments developing ahead of the tip had to cut the {111 } plane in the prolongation of the cleavage surface and therefore must form pits on the two facing surfaces. Indeed, rows 1-4 are seen on (111) and (] i i ) surfaces. A close examination, however, reveals some differences in the dislocation locations. An obvious example is seen along row 2, where the fourth dislocation from the head of the row is not at the same distance from the crack tip on the two surfaces. This can be explained by some backward motion of dislocations just after fracture, when the temperature has not yet been sufficiently decreased. Observations of etch patterns on broken samples were sometimes criticized because of this possible relaxation of dislocation configurations. We feel, on the contrary, that the present example indicates that such backward motion is rather
111
Fig. 2. Stereographic projection of a samples (crack surface (111), propagation direction [] 10], sample free surface (i ] 2)). The shaded half-sample was used for XRT.
5 s
i
[III]
(~[~1O] (ii I I
"'3. 4
(TTI) [I12~ 7 .8
[iT]]
~[iio] ~0
(lil)
"~ (Tll) J I I I
(ill)
2
1O1
,m
[iT2]
Fig. 3. Etch patterns around a dislocation source observed on two opposite crack su~aces (marker: 10/~m). See text for details.
102
G. Michotet al. / Dislocationloops at cracktips
limited: in the main dislocation arrays, loop locations differ by less than 12%. Above the crack tip, i.e. behind it, dislocations emerge at the two crack surfaces and behave independently on each side of the crack. Etch pits need not coincide, which gives some useful indication of the shape of the dislocation loops. Rule (i) must be regarded with caution when the dislocation density is large. Very near the crack tip, linear etch figures (grooves) suggest that a number of near dislocations have remained attached to the crack. Of course, they cannot be counted. An attempt was made to determine the Burgers vectors of these dislocations by XRT. A topograph of one half of the sample is shown in Fig. 4. The dark spot marked by an arrow corresponds to the groups of dislocations shown in Fig. 3. Clearly, resolving individual dislocations is beyond the capacity of the Lang technique and even the main dislocation families could no be identified. In more detail the different dislocation families can be described as follows. Loops expanding in (i11) planes. Ahead of the crack tip, row 1 has about 15 dislocations distributed over at least two parallel planes. On the rear side of the tip, on the (111) crack surface, row 5 contains at least 23 dislocations in two groups on a first (i 11) plane and about 14 dislocations on a second parallel plane. In both rows accumulated pits form a nearly continuous trace very near the source point. On the opposite ( 11 ] ) surface, behind the tip, only two isolated pits appear
(row 7). Loops expanding in (lil) planes. Ahead of the tip, two distinct parallel arrays are seen on both surfaces. The longer row 3 has about 37 loops in one plane and the smaller row 4 about nine loops also in one and the same plane, to the accuracy of etch pit observations. Behind the tip, no dislocations emerge on the (111) surface, while row 8 is formed on (1 i ]). This last row consists of of six well-shaped pits and at least 25 others
Fig. 4. X-ray topograph of the half-sample shown in the upper part of Fig. 3: g = 400 (marker: 200/xm).
of various smaller sizes. A few pits are seen in the same plane as row 4. Loops expanding in (11i) planes. The situation here is apparently much simpler. Four aligned dislocation pits appear ahead of the tip (row 2). Behind the tip, seven aligned loops emerge on the (111) surface (row 6) and none, except maybe very near the crack front, on(lll). It may be observed, especially on the etched (i ] i) surface, that the pits have different shapes and contrast. Usually they look similar in parallel rows ahead of and behind the crack front, with exceptions in row 8 and perhaps row 1. It may be checked also that the two pits in row 7 are different from those in row 1 but look very similar to those in rows 2 and 6. Figures 5-7 propose a possible 3D reconstruction of the configuration based on etch pit patterns and considerations of the crack tip stress fields. The resolved shear stresses r acting on a given slip system can be derived from the mode I stress tensor [12] in the form
KiH( cp) r = (2xR)l/2 where R and q~ are the polar coordinates in the slip plane (with cp= 0 along OX, the intersection between this plane and the crack plane). Results are expressed in the form of equal-stress contours G(qg) = [H( q~)]2[8]. The sketches of Figs. 5 and 6 are reconstructions of what should appear on a 440 topograph taken with Mo K a radiation. The orientations shown are projections of crystallographic directions on the photographic plate.
3.2.2. Possible Burgers vectors Row 5 contains more loops than row 1. According to the stress contour, it is likely that most dislocations behind the tip on ( i l l ) have b=(a/2)[101]. These dislocations, when the loops are small compared with the sample thickness, are confined behind the tip on one side of the cracked sample and should appear on ( 111 ) only. Dislocations ahead of the crack front can have either (a/2)[011] or (a/2)[l10] Burgers vector. Both stress contours are compatible with etch pit distributions on cracked surfaces. (a/2)[l10] has the larger stress on average and should be more constrained to remain attached to the tip on the ( i ] ]) side than (a/2)[011]. However, (a/2)[110] is very seldom detected in well-developed PZs, in contrast with (a/2)[011], and direct Burgers vector assessment by XRT would be necessary. On (1 ] 1) the smaller row 4 and the longer row 3 probably have (a/2)[110] or (a/2)[10i] Burgers vector, the former being more probable because no emerging loops appear on ( 111 ) behind the crack front. Disloca-
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Dislocation loops at crack tips
103
d
# //
J
.................... D io] ......... [oli] Boq
/j5
[itl]
7//
+
/r. /
r
,(
&
(a)
'
+
(b)
Ii
"'1,,.
/
1
!
I,°ll
/
III
ii O.I
$X [01]]
Fig. 5. (a) Possible dislocation configuration in the ( ] 11 ) plane from Figs. 3 and 4. Dislocations drawn as full lines are those seen in Fig. 4. Dashed lines represent the parts of loops belonging to the other half-sample. (b) Equal-resolved-shear-stress contours in the (111) plane for available Burgers vectors (a orientation). The stress acting on full-line dislocations corresponds to the y> 0 domain, that on dashed-line dislocations to y < 0.
II
d
J~ ijip
[ito]
...................
[oii]
........
[ioi]
,fl-i; JI IIl[ii~ JJJ
;;" ~;
J
[o,q
-.... 2
1
(a)
+~1.1 I/'/
t
-4-
(b)
[ioq
[i,o]
l[i lO1
x
0.1
Fig. 6. (a) Possible dislocation configuration in the (111 ) plane. Same convention as in Fig. 5a. See text for details. (b) Equal-resolvedshear-stress contours in the (111 ) plane. Same convention as in Fig. 5b.
tions with b = (a/2)[011] are excluded in r o w 4 but can exist in r o w 8. D e p e n d i n g on the shapes of pits, the Burgers vectors in rows 1 and 3 should be different, but a different orientation of e m e r g i n g loops can also be invoked.
O n (111) the o b s e r v a t i o n s are not easy to u n d e r stand. Clearly only four loops d e v e l o p a h e a d of the tip while seven a p p e a r behind it on ( 111 ). It is unlikely that such loops run a longer distance behind the crack than a h e a d of it in view of the stress contours. We believe
104
G. Michot et al.
\
/
Dislocation loops at crack tips
,'
A 111
/\k tl¢
/
b':
~lI •
Fig. 7. 3D reconstruction of the arrangement of dislocations around the source shown in Fig. 3. Same convention as in Figs. 5a and 6a.
that the two isolated pits forming row 7 belong to loops of row 6 which have cross-slipped on to (111 ). Those seven loops probably have b=(a/2)[lO1] and equally probably from stress contours as (a/2)[0]i] (which could cross-slip only on to ( 111 )). Cross-slipping must have occurred at or very near the crack tip, since rows 2 and 7 converge at the crack front. Figure 6 shows the possible shape of such a loop expanding on two planes. Etch pits should, of course, also appear ahead of the crack front along row 2, but cross-slipping has probably prevented this part of the loop from expanding freely far from the source and pits cannot be resolved because of the high dislocation density at the source point. Alternatively, it is possible that two loops with b = (a/2)[101] started to develop on ( ] 11) behind the tip and then cross-slipped on to (11 i) from the crack point where they had to remain attached on ( i 11 ). 3.3. Preliminary results on source configurations in fl samples The two-step loading technique described above was applied to six samples. The/3 orientation (Fig. 8), for which results are still scarce, is interesting because a glide plane contains the [011] crack tip (which realizes a 2D situation as in current models) and because the Burgers vector analysis is simpler than in a samples. Results are summarized in Table 1. In agreement with former observations in well-developed PZs, it seems from XRT that nearly all dislocations lie on ( 1 i 1 ) and ( 11 i ) and not on ( i 11 ), which contains the crack front. It is also confirmed that the (a/2)[011] Burgers vector is never observed and that Burgers vectors parallel to the crack plane can be emitted from the very beginning of dislocation emission. Figure 9 shows about 12 identical dislocations still attached to one point of the crack front at or very near
"'"'--.
,"
s"
I/
•
-r.
ii~ ,,,,," .........
Fig. 8. Stereographic projection of fl samples (crack surface ( 111 ), propagation direction [211 ], sample free surface (0[ 1)).
TABLE 1. Dislocation source analysis:number of sources, glide planes and Burgers vectors of dislocations, and number of dislocations per family Sample
Plane (1 i 1)
Plane (1 i 1)
Number of sources
1ol EA3 EA7 EA9 EA10 EA12 EA13
iio
2 6 10-12 3 4 6 -60--4 12
oil
lOl
liO
oii
-------
4-6 1012 2
167 ---16
-------
1(2) -2 2 1 1
the sample surface. They glide on (111) and have b---(a/2)[lO1]. This confirms the importance of the sample surface in nucleation. In the sample shown in Fig. 10 it was possible to observe the etch pit distribution on one half-sample and a striking feature was revealed. The etch pattern proves that 35 or more dislocations were emitted in the (i 11 ) plane parallel to the ideal crack tip. These dislocations have been emitted behind the crack tip, which was slightly off [01]], and the leading one runs for several hundred microns. X-ray topographs, on the other hand, reveal that most dislocations have b = (a/2)[110] but glide on (1 i 1). This strongly suggests that dislocations are first emitted on ( 111 ), run more or less parallel to the crack front over some distance and cross-slip on to the ( 111 ) planes where they form large loops. The detailed geometry of this cross-slip process is still under study and this observation must be confirmed in other fl samples before any conclusion can be drawn.
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Dislocation h)ops at crack tips
105
Fig. 9. X-ray topographs of a fl sample loaded at T = 1023 K under KI=0.71 MPa m ~:-'(approximately 0.76Kic) for 1 min and then under KI= 0.31 MPa m ~/2for 30 min: a, g= 22(); b, g= 202; c, g = 202, (marker: 30(t t~m).
4. Stress field within the plastic zone Row 3 of Fig. 3 consists of about 37 dislocations distributed between point I at 15.8 ~ m from the emission point E and point O at 70/~m from E. There is a dislocation-free zone between ! and J (at 11.9/~m from E). Such a dislocation-free zone resulted from a break in the source activity in this plane. Assuming a negligible motion of dislocations after fracture, the stress acting on each dislocation j can be calculated by summing the applied crack tip stress field Kl(H(cfl)} r a - (2xR)t/: and the interaction stress due to other dislocations,
"~i
aktb V
1
27r i~] Rj -- R
i
Using (H(q))}=H(0)=0.23 [11], a-~1.25 for 60 ° dislocations, K I -~ 1 MPa m m (derived from the fracture stress) and Rj measured on the micrograph and neglecting corrections due to the vicinity of the crack [13], one finds the results of Fig. 11, which show that
the effective stress is roughly equal to 20 MPa and fairly constant within the PZ except close to the dislocation-free zone. This estimate is consistent with the stress measured at the border of the PZ under similar loading conditions [3].
5. Crack opening 5.1. Macroscopic observations
During high temperature testing under a constant opening rate 6, plastic opening is assumed because of the departure from linearity at P,, of the load vs. deflection curve, leading to a larger compliance than that expected from an elastic sample, even if temperature corrections of the elastic constants are introduced. As could be expected from differences in source activities from sample to sample, a large scatter in P,, values is noticed. After removal of the load a permanent opening of a few microns is detected. Measurements taken at room temperature indicate smaller compliance values for plastically deformed samples (Fig. 4 of ref. 14), which can be explained by the static compressive stresses induced by the plastic zone: this fact expresses crack shielding.
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Dislocation loops at crack tips
Thus we must conclude that dislocations account for two opposite mechanisms, i.e. crack opening and crack closure. The question is: are antishielding dislocations required for plastic opening? It will be shown from experimental results obtained on the fl orientation that this condition is not always necessary.
5.2. Microscopic observations Observation of the full width of the sample (Fig. 12) shows that in addition to the PZ developed around the crack tip, a few dislocations are seen coming from the crack surfaces above the PZ, while a strong plastic deformation is observed on the right-hand edge of the sample in the continuation of the main PZ growth direction [0ii] (see Fig. 8). Such dislocations can be found at sources located on the sample edges. For a diffraction vector perpendicular to the crack plane, most dislocations of the PZ and those coming from the sample edge as well are out of contrast: this means that the possible Burgers vectors, (a/2)[1 i0] or
/ ~ xl I
I
! / ! / / / I
/
Q [iii]
(c)
[11] I
Fig. 10. X-ray topographs and etch patterns of a fl sample loaded at T = 1023 K under K I =0.66 MPa m 1/z for 3 min and then under KI = 0.38 MPa m u2 for 30 min: a, g = 1 i i; b, g = i i i (same magnification as Fig. 9); c, etch distribution observed on the cleavage surface (marker: 50 #m).
G. Michot et al. /
Dislocation loops tit crack tips
107
: l d 4 ( c m -1 )
X2
-I
-" X 1
v P
I
25
,
50
b2
b -
I
!
j,
/ /
"r( M P a )
./"
40
\,\
~'~.
C
d
:_: ::.- T:-T-:=......... ,'"-
|/-b I
, R lpml
50
". /
Fig. 11. Evolution of the dislocation density (a), internal stress (b), applied stress (c) and effective stress (d)in dislocation array 3 of Fig. 3 as a function of the distance from the source.
d~ •_
. . . . . . ~ .............. ,,! lT I
I
..
I
Fig. 13. Definitions (a), shielding dislocation (b_,> 0 according to ref. 14)(b). shielding dislocation (b~ >0 according to ref. 14, 0 < s~/3)(c) and shielding dislocation (b I < 0, 0 > x/3) (d).
the crack tip has a potential energy Ep (per unit thickness of sample) given by
Ep = E~,+ Ed °~ + Ed ~- P(6, + rid)
Fig. 12. X-ray topograph (g= 202) showing the full width of a/~ sample (inverse contrast). See text for details.
(a/2)[]01], are parallel to the crack plane and not expected from mode I loading. The use of other diffraction vectors has confirmed this result.
5.3. Closure vs. opening (i) Thermodynamical considerations are required in order to determine whether the emission of a dislocation will lead to crack closure or crack opening [15]. A semi-infinite crack, loaded at x a by line forces P (N m-l)(Fig. 13(a)), with a dislocation located at (r, 0) of
where the first term on the right-hand relates to the elastic energy induced by the applied load P, the second to the dislocation self-energy in an infinite medium, the third to its variation due to neighbouring surfaces and the fourth to the external work when the applied force is displaced by 6,~ (due to the force-P) plus 6 d (due to the dislocation). Shielding occurs when the product K , K d is negative, i.e. when the effective stress intensity factor K, + K d is smaller than K,. (ii) Emission from the crack tip requires the radial force acting on the dislocation, J;= -(OEp/Or)o to be positive:
The first term on the right-hand side, the image force, is always negative, so the derivative of the second term must necessarily be positive, but the function Pdd, itself can be either positive or negative. For a semiinfinite crack, because 6d(Xl) is a function of 0 and ~p only, it can be shown that ( Odd/Or)o = - x I ( Odd/OX l)/r. Thus emission requires positive values for Odd/dX 1. This condition depends on the Burgers vector.
G. Michot et al. / Dislocation loops at crack tips
108
For the positive dislocation b 2 of Fig. 13(b) it is shown in ref. 15 that the crack opening varies from - b 2 at infinity to zero at the crack tip. Of course, negative 6~ values under mode I loading only make sense if a larger Oa opening is superimposed. This dislocation, which obviously has a shielding effect, must, according to 6 d variations with x~, sustain a positive force and so is likely to be emitted. The situation is more complex for dislocation b~. For 0 smaller than Jr/3 the crack opening varies from zero at infinity to zero at the crack tip (Fig. 13(c)), going through a negative extremum. The sign of (06JOx~)o depends on the location of the point x a with respect to the extremum Xm: as far as emission is concerned, the small-scale yielding condition (Xa'> r) holds, i.e. such a shielding dislocation is attracted. In other words, antishielding- b~ dislocations are emitted for 0 < :r/3. For larger 0 values a - bl dislocation leads to both crack tip closure (crack shielding) and crack opening at a given distance x a from the tip (Fig. 13(d)): under the small-scale yielding condition this shielding dislocation is likely to be emitted. Figure 14(a) shows that an array of such dislocations is equivalent to a tilt subboundary, confirming the assumption set forth in ref. 16. There are differences between bl and b 2 behaviours: when the dislocation moves away from the crack tip, the closure effect induced by a dislocation b2 at x a decreases (Fig. 13(b)), while the crack opening due to a dislocation - b ~ increases slowly up to a maximum
- b 1 T~'~b -,T\ T T\ .~.T "r\'
value for Xm/r (Fig. 13(c)) for which the sign of the radial force on the dislocation reverses.
5.4. Application to fl orientation 5.4.1. The (1]1) and (ii1) planes (see Fig. 9) In a real 3D geometry the usual slip systems are ( a / 2 ) [ i l 0 ] ( ] i l ) and (a/2)[i01](]li). Because their combination gives an equivalent (a/2)[211] Burgers vector parallel to the crack propagation direction, the situation is equivalent to a 2D problem. However, we must bear in mind that emission does not take place along planes which contain the crack tip and that the sketch of Fig. 14(a) is just an equivalent representation. A shielding dislocation is likely to be "emitted" along the [011] direction for 0 = 125 °. Because an antishielding - b l would be emitted for 0 = 55 ° (according to Section 5.3), antishielding bl dislocations are expected to develop in the negative sector (0 = - 5 5 °) along the [011] direction because of mirror symmetry with respect to the crack plane. This means that the crack can emit an equivalent full dislocation in both directions. This loop globally shields the crack, because the influence of the antishielding component bl is seven times smaller than that of the shielding one - b~ [15], but the full loop contributes to crack opening. For such a loop, using the relations given in ref. 15 for a semi-infinite crack, it is possible to evaluate the opening to be about 0.1 b per loop ( r - 1 mm, x a = 13 mm, 0 = 125 °, -55°). Under subcritical creep conditions (KI=0.8KIc) etch pit counting [11] indicates about 7000 loops inside a "saturated" PZ. This should lead to an opening of about 0.2/~m, which is one order of magnitude smaller than that observed experimentally. However, because of the finite size of our samples, some elastic parameters (compliances, stress intensity factors, etc.) are one order of magnitude larger than those of the semi-infinite medium for the same applied load and crack length: therefore the "bending" dislocations could account for crack opening.
5.4.2. The (l ] ]) plane
/ / / .......
b
t/
/
4/
/ Fig. 14. 2D representation of the fl orientation: a, "bending" dislocations responsible for crack opening; b, "blunting" dislocations.
The dislocation b emitted for 0 = 70 ° (Fig. 14(b)) shields the crack, because the b2 component has the largest effect. Because emission of a shielding dislocation is expected for 0 = 110 °, from mirror symmetry a b dislocation is likely to be emitted for 0 = - 1 1 0 °. These two dislocations have a shielding and a blunting effect which both tend to decrease the stress concentration. However, because closure due to shielding tends to cancel the influence of blunting and because dislocation activity is very limited in that plane, we conclude that such dislocations do not contribute markedly to crack opening.
G. Michot et al.
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Dislocation loops at crack tips
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