Computer simulation of dislocations emitted from a crack

Scripta

METALLURGICA

V o l . 20, pp. 1 4 7 7 - 1 4 8 2 , 1986 Printed in the U . S . A .

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COMPUTER SIMULATION OF DISLOCATIONS EMITTED FROM A CRACK J.C.M. Li Department of Mechanical Engineering University of Rochester Rochester, NY 14627 ( R e c e i v e d J u l y 31, Introduction

1986)

The recent surge of interest in the interaction between a crack and nearby dislocations stems from the observation that there is a dislocation-free zone near the crack tip (i). Such a zone was not predicted by the BCS theory of dislocation distributions emitted from a crack (2). While some people think that the attractive region between a dislocation and the crack tip is the cause for the dislocation-free zone, an in-depth analysis (3) reveals that the real cause is the finite stress intensity factor required for dislocation emission at the crack tip. When such a factor is assumed zero ,as in the BCS theory, the dislocationfree zone becomes non-existent. A computer simulation (3) confirmed a direct relationship between the size of the dislocation-free zone and the stress intensity factor required for dislocation emission. Attempts to disprove the existence of the dislocation-free zone by direct experimental observation would not change the physics of this phenomena since the absence of the zone merely indicates a small value of the critical stress intensity for dislocation emission at the crack tip. Environmental effects such as those caused by hydrogen may introduce a chemical driving force (4) which may facilitate the dislocation emission process. When the stress intensity factor needed for dislocation emission is reduced, the dislocation-free zone diminishes also. The interaction of a single dislocation with a crack has been studied by many people. Louat (5) formulated the interaction between a finite crack and a screw dislocation which is parallel to the crack tip but not necessarily in the crack plane. The coplanar case was discussed by Smith (6). The interaction of a mixed dislocation with a semi-infinite crack was obtained by Rice and Thomson (7) for the isotropic medium and by Atkinson (8), Barnett and Asaro (9) and Asaro (10) for the anisotropic medium. Using a mapping technique, Majumdar and Burns (11) also examined the interaction between a screw dislocation and a seml-inflnlte crack. Chu (12) and Lee (13) used different methods but obtained the same interaction between a screw dislocation and a surface crack. The interaction between a crack and a single dislocation cannot be used directly to deal with the interaction of the crack with many dislocations. It is needed to work out the interaction between one dislocation and the image of another or to solve the elastic problem directly involving a crack and all the interacting dislocations. The latter scheme was used by BCS (2), Chang and Ohr (14) and Majumdar and Burns (15). The former scheme was used by the present author (16) and his coworkers (3,17-19) in many of the co~puter simulations and more recently by Lin and Thomson (20). While the latter scheme must deal with each problem separately, the former scheme is applicable to any distribution of dislocations around the crack tip. The present con~unication summarizes some results of the behavior of emitted dislocations from a crack tip. Most of them are from computer simulation. Criteria for Dislocation Emission from a Crack Tip When a dislocation is nucleated at the crack tip ,it has to travel through a region in which the dislocation is attracted toward the crack tip. For coplanar emission in Modes II or III loading, the size of such a region is given by

X0 = A2b2/402 £ = ~A2b2/2K 2

(i)

0036-9748/86 Copyright

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where A is ]J/2~ for Mode III or screw dislocations and ~/2~(I-u) for Mode II or edge dislocations, ~ is shear modulus, v is Poisson ratio, b is the Burgers vector of the dislocation, u is applied stress, ~ is the length of a semi-infinite crack and K is applied stress intensity factor before the dislocation is emitted.. Rice and Thomson (7) suggested that the applied stress should be large enough so that x 0 approaches the core radius r 0 of the dislocation. In other words, a critical stress intensity factor needed for dislocation emission is Ke = ~ q / ~ ' o

(2)

However,wlth a lattice friction stress TF for the motion of the dislocation, three more distances need to be considered X1 = ( K ~ ) 2 / 8 , ~

x2 = (K~TF) and

21+~T~

X3 = ( K ~ T F ) 2 1 8 U ~

(3)

(4) (5)

The dislocation is attracted toward the crack tip if the distance is less than Xl; it cannot move betweem x I and x 2 because the force is less than the lattice fractlonl it will move to x 3 if it is placed between x 2 and x31 and flnally, lt again cannot move if the distance is larger than x 3. When the lattice friction is small, it can be seen that both x I and x 2 approach x 0 and x 3 approaches infinity. Ohr (1) suggested that the applied stress should be large enough so that x 2 (his x 0) approaches the core radius of the dislocation. The argument is that the dislocation must be nucleated at a distance so that it can move away (actually it can move only to x3). On the other hand, one can argue also that, if the dislocation is nucleated sllghtly beyond Xl, it will be stablized by lattice friction so that it will not disappear into the crack. Based on this argument, the critical stress intensity factor needed for dislocatlon emission may be slightly less than K e of Eq. (2) taking into consideration the effect of lattice friction. In any case,the effect of lattice friction is usually small as can be seen from the ratio of the extra K (added to K e) to Ke (Eq. (13) of Ohr's paper (1) for coplanar emission under Mode III loading}. 41rr0~F/~b

(6)

Since r 0 is about b/2 and TF is about ~/100, this ratio gives only about a few percent correction to K e. The potential energy (16) of the dislocation around the crack tip under an applied stress (or K) has a maximum at: 2 b2[2-~Ab 2~ ~G = Ab [ l n ( _ - - - I - = : . ) - l ]

(7)

Per unit length of the dislocation. This is the activation energy needed for dislocation emission. Spontaneous emission should be possible if this energy is zero or negative. This argument leads to a critical stress intensity factor for dislocation amlsslon: (8) KD = (Able) where e is the base for natural Iogarithum. A comparison between Eqs. (2) and (8) shows KD/K e = 2/e. Since the emission process may be thermally activated, a stress intensity factor smaller than either KD or K e still may emit dislocations. Comparison with experiment is difficult, not only because time is involved but also due to possible environmental effects which will be discussed next. Dislocation Emission Enhanced b~ a Chemical Driving Force Hydrogen is a good example for the environmental effect. When the material such as iron is exposed to hydrogen at a fugacity f0 and mechanically loaded at the same time, hydrogen may facilitate its entry into the material by riding with the dislocation elaitted from a surface crack. If n hydrogen atoms per unit length of dislocation can lower their

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fugacities from f0 to some value f by residing in the emitted dislocation, the energy barrier for the emission process is lowered by nkTln(f0/f)

(9)

per unit length of dislocation. dislocations is lowered to:

K

=

Hence the critical K for spontaneous emission of

I_ nkT. f 0 ~ KDexp

(1o)

~b2.Ln~---j

To see the magnitude of the effect, K/KD is found to be 0.2 for f0 = 100f, T = 300°K, n = 1011/m, k = 1.38 x 10-23 N.m/°K, ~ = 83GPA, b = 2.49 x 10-10m and ~ = 0.3. A larger effect is expected in the case of high fugacity hydrogen such as employed by Tabata and Birnbaum (21). In their in-situ experiments in the electron microscope molecular hydrogen may be dissociated into atomic hydrogen in the high voltage electron beam. Atomic hydrogen will have a much higher fugacity than molecular hydrogen and can nucleate dislocations. D~namic Emission of Dislocations from a Crack Tip For a stationary crack, the number of dislocations which can be emitted in Mode IT or Mode III loading is given by (3) K 2 N = 1a2£ - ~ ) / 2 A B T F

(ii)

where KD is the stress intensity factor required for dislocation omission, TF is the lattice friction stress and other symbols are defined under Eq. (1). It is seen that the applied stress must exceed the following:

o > Ko/2,z~-~-~

(12)

before any dislocation can be emitted. Since the applied K a is oVr2~, the condition is simply K a > KD for dislocation emission. As mentioned earlier, a non-zero KD is the main reason for the existence of the dislocation-free zone in front of the crack ti . The number of dislocations which can be omitted at saturation is proportional to Ka2 - KD and is inversely proportional to the lattice friction. The emitted dislocations modify the stress of the crack tip such that the stress intensity factor is shielded (11,22-24) from the applied stress as follows

K = Ka + Ks

(13)

where K s is the stress intensity factor due to the emitted dislocations. As a result of negative K s ~the rate of emission of dislocations decreases (17) by orders of magnitude with number of dislocations omitted. The time needed to saturate the plastic zone is found to depend strongly on the lattice friction, weakly on the applied stress and almost independent of KD. For any given u and KD the size of the plastic zone varies with the square of the number of dislocations emitted independent of lattice friction. The dislocation density is very high in the beginning, being concentrated near the crack tip, although a dislocationfree zone exists as long as KD is finite. Then the dislocation distribution spreads out more uniformly into the plastic zone. Upon unloading, the dislocations are retracted into the crack. Complete retraction is possible if there is no lattice friction for dislocation motion. Otherwise ,only a part of the dislocations near the tip of the crack disappear and thereby increase the size of the dislocatlon-free zone. When the crack propagates while the dislocations are emitted, the steady of dislocations which move with the crack is smaller than that omitted from a crack. Such steady state number decreases with increasing crack velocity and lattice friction for dislocation motion. Before the steady state propagtion,

state number stationary increasing the crack

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stops momentarily after each dislocation emission because of the shielding effect. Then it accelerates when the dislocation moves away from the tip. Such acceleration continues but decreases in magnitude until another dislocation is emitted. The final steady state crack velocity approaches that corresponding to KD. Hence the shielding effect of emitted dislocations not only slows down the dislocation emission process but also slows down(rack propagation. The critical stress intensity factor KD not only controls dislocation emission but also controls crack growth. When KD is assumed zero as in the BCS (2) analysis, the crack would not be able to propagate. Dislocations Emitted from a Crack and Blocked by a Barrier The effect of a barrier such as a grain boundary on the dislocation distribution emitted from a crack tip in Mode II and III loading is as follows: let the crack occupy the -x half of the xz plane with the tip at the z axis. Let the barrier be located at x - d. Dislocations emitted from the crack may pile up against the barrier. The distribution of dislocations in terms of their total Burgers vectors, f(x)dx, between x and x + dx should satisfy the following singular integral equation d

x

~

£

TF

(14)

c

for all x' between c and d. The dislocation-free zone is between x = 0 and x = c. The symbols have been defined under Equations (1) and (2). Eq. (14) has the following solution [K(k), E(k), F(@,k) and E(@,k) are elliptic integrals]:

f(x) where

S1 J b ( x - c ) = ~-~

2TF

+ --AX 2 [K(k)E(8,k)-E(k)F(8,k)]

S = S 1 + S 2 = (O/A) J £ / b

(16)

S2/E(k) = 2TF /'~'--d - [ --KD~-

WA ~ b

(15)

S]/[K(k)-E(k)]

(17)

A2~

k2 = (d-c)/d

(18)

Sin@ - / (x-c)/x /k

(19)

For a given G, Kn, d and TF, k and hence the dlslocation-free zone, c, can be obtained from Eq. (17). The distribution of Eq. (15) gives the following stress concentration at the barrier

Tn = AS2k2/2

(20)

and the following total number of dislocations.

"-

2Sl /-/

db C,(k) - Cl-k 2)

2

Ck)]

~2

.K%.~!,2, Cl-(1-k 2) %(k)' J "

(21)

It is seen from Eq. (15) that the distribution is not bounded at x ffid. However, the stress concentration is found to decrease with increasing d contrary to the situation of a Frank-Read source. Furthermore, the stress concentration is independent of d when there is no lattice friction. These results question the usual practice of using the yield stress in the plastic gone for polycrystalllne materials. Certalnly,when the grain size is larger than the slngle-crystal plastic zone, single crystal yield stress should be used. For strong grain boundaries, the plastic zone is limited to the grain size independent of the yield stress.

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When the grain boundary is not very strong, the dislocations may spill over to the next grain and sometimes also pileup at the next grain boundary. The distriubtion of dislocations in the next grain is either concentrated in the middle of the grain sandwitched between two dislocatlon-free regions or piling up against the next grain boundary leaving a dlslocatlon-free zone attached to the first grain boundary. For weak boundaries, dislocations emitted from the crack can spread over many grains. The plastic zone size increases with the grain size until it reaches the single-crystal value. Then it remains constant insensitive to further increases of grain size. Crack Propa~atlon during Unloading The shielding effect as expressed by Eq. (13) has a grave implication; namely, unloading after loading can propagate the crack. While the ease of the emission of dislocations is an indication of toughness, a large amount of shielding (Ks being a substantial fraction of Ka) for Mode II and Mode III situations may cause a sudden crack surge upon unloading (19). The reason is that both positive and negative K propagates the crack in Mode II and Mode III situations. Hence during forward loading, the crack may have stopped because of the shielding effect of the emitted dislocations which exert a negative K s at the crack tip to reduce the effect of K a. Upon unloading (Ka = 0), the negative K s now can propagate the crack. Such u,loadlng behavior must play an important role in fatigue, because a subsequent loading in the reverse direction (Ka becomes negative) enhances the effect of negative K s so that the crack speeds up to swallow up as many as possible of the previously emitted dislocations. Then the crack will emit its own dislocations which produce a positive K s shielding the negative K a and eventually the crack propagation stops. Upon unloading, the crack moves again this time under the influence of a positive K s which is enhanced by a subsequent forward loading. Then the process repeats itself. It is seen that the crack propagation could be much more in fatigue than in static loading. Even in the case of Mode I loading, asymmetric dislocation emission along two inclined slip planes may cause crack branching to change to Mode II propagation upon unloading. Such branching will be enhanced by the compression cycle after the tension cycle to propagate the crack in a mixed mode (Modes I and II). This mixed mode propagation is caused by both the applied negative KII and the shielding negative KIT due to the dislocations emitted in the tension cycle. This propagation stops after the crack swallows up as many as possible of the previously emitted dislocations and emits its own shielding dislocations. It starts again upon unloading and continues in the tension cycle. However, during the tension cycle the crack may resume its Mode I propagation when the mixed mode is shielded by its own emitted dislocations. This may explain the zig-sag way of fatigue crack propagation and the linking of Mode I cracks with shear plane mixed mode cracks. Even without the externally applied stress, dislocations emitted from a crack tip by chemical means such as induced by hydrogen dissolution can cause crack propagation if the emitted dislocations are mostly of one sign. This possibility has been proposed as a cause for hydrogen embrlttlement (4). Although the crack propagates in Mode II (for edge dislocations) or Mode III (for screw dislocations), any applied or internal Mode I stress certainly will enhance the process. Since the crack propagates along a slip plane, it is often regarded as slip plane decoheslon (25-27) although the slip plane does not have to be weak for this mechanism to operate. In summary, while a plastic zone can shield the crack from the applied stress and is the major mechanism for plasticity toughening, it can also propagate the crack during unloading or reverse loading so that it plays the damaging role in fatigue. This possibility should be kept in mind when designing materials resistant to fatigue loading and chemical embrlttlement. Acknowledgements Discussions with S.M. Ohr and S.J. Burns are appreciated. The work was supported by D O E through DE-FG02-85ER45201. Some computer simulation was done by Wen-Lan Li who was a visiting scholar from Wuhan Techanical University, Wuhan, China during 1984-86.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

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