Forces B/w Dislocations

1. Forces B/w Dislocations • Consider two parallel (//) edge dislocations lying in the same slip plane. • The two dislocations can be of same sign or different signs (a) Same Sign (on same slip plane)

2. (14.34) • When the two dislocations are separated by large distance: The total elastic energy per unit length of the dislocation is given by: • When dislocations are very close together: The arrangement can be considered approximately a single dislocation with Burgers vector = 2b In order to reduce the total elastic energy, same sign dislocations will repel each other (i.e., prefer large distance separation). (14.35)

3. Dislocations of opposite sign (on same slip plane) • If the dislocations are separated by large distance: • If dislocations are close together: Burgers vector = b - b = 0 Hence, in order to reduce their total energy, dislocations ofopposite signs will prefer to come together and annihilate (cancel) each other.

4. The same conclusions are obtained for dislocation of mixed orientations • (a) and (b) above can be summarized as: • Like dislocations repel and • unlike dislocations attract

5. x2 x1 x3 Dislocations Not on the Same Slip Plane • Consider two dislocations lying parallel to the z (x3) -axis: • In order to solve this: (a) We assume that dislocation “I” is at the origin (b) We then find the interaction force on dislocation “II” due to dislocation “I” II I

6. 14.29 • Recall Eqn. 14.29 • Note that the dislocation at the origin (dislocation I) provides the stress field, while the Burgers vector and the dislocation length belongs to dislocation II • Since is edge: • Also bII is parallel to x1: Therefore, This means that b2 = b3 = 0 and b1 = b

7. Since tII is parallel to x3, then This means that t1 = t2 = 0 and t3 = 1 • From Eqn. 14.31, we can write: Therefore

8. But Therefore, F along the x1 Direction is given as: 21b This component of force is responsible for dislocation glide motion - i.e., for dislocation II to move along x1 axis. 14.30

9. 14.31 F along the x2 Direction is given as: - 11b This component of force is responsible for climb (along x2). • At ambient (low) temperature, Fx2 is not important (because, no climb). • For edge dislocation, movement is by slip & slip occurs only in the plane contained by the dislocation line & its Burgers vector.

10. Consider only component Fx1 For x1>0: Fx1 is negative (attractive) when x1<x2 for same sign, or x1>x2 for opposite sign. For x1<0: Fx1 is positive (repulsive) when x1>-x2same sign disl. or x1<-x2for opposite sign disl. Fx1=0 when x1 = 0,  x2, For edge dislocations of opposite signs Usually for edge dislocations of same sign

11. Hence • Stable positions for two edge dislocations. 900 450

12. Equations 14-30 and 14-31 can also be obtained by considering both the radial and tangential components. The force per unit length is given by: • Because edge dislocations are mainly confined to the plane, the force component along the the x direction, which is the slip direction, is of most interest, and is given by: 14.32 14.33

13. 14.34 Eqn. 14-34 is same as 14-30. Figure 14-5 is a plot of the variation of Fx with distance x, using equation 14-34. Where x is expressed in units of y. Curve A is for dislocations of the same sign; curve B is for dislocations of opposite sign. Note that dislocations of the same sign repel each other when x > y, and attract each other when x < y.

14. Figure 14-5. Graphical representation of Eq. (14-21). Solid curve A is for two edge dislocations of same sign. Dashed curve B is for two unlike two dislocations.

15. x2 x1 x3 S • Example: A dislocations lies parallel to [100] with Burgers vector b<110>. Compute the force acting on the dislocation due to the stress field of a neighboring screw dislocation lying parallel to [001]. Assume that for the screw dislocations Solution:

16. Let the screw dislocation be dislocation I at the origin. The stress field for screw dislocation is given by: based on the assumption, we have

17. For the other dislocation

18. (continued)

19. (b) (a) Figure 14-6. (a) Diffusion of vacancy to edge dislocation; (b) dislocation climbs up one lattice spacing