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Stability of dislocation arrays
ScriptaMaterialia, Vol. 37, No. 9, pp. 1407-1408,1997 Elsevier Science Ltd Com’kht 0 1997 Acta Metalluwica kc. G&d in the USA. All rightskerwd 1359~6462/97 $17.00 + .OO
Pergamon
PI1 S1359-6462(97)00244-3
STABILITY OF DISLOCATION ARRAYS G. Schoeck Institute of Materials Physics, University of Vienna, A- 1090 Vienna, Boltzmanngasse 5 (Received March 10, 1997) (Accepted June 27,1997)
In a frictionless medium it is impossible to have a stable array of a set (i) of parallel dislocations which have same common components of the Burgers vector unless the dislocations are somehow constrained. When the components are parallel the dislocations will repel each other and escape at the surface. When the components are antiparallel they will attract each other and annihilate their components. A common stabilisation exists becabse at low temperatures the dislocations are constrained to their glide planes (and in addition are usually coMected in a 3-dimensional network). At high temperature, when the dislocation can readily climb and undergo cross-slip, this geometrical constraint is removed. Undler this condition they can be stabilised when they are repulsive and when they are connected by planar faults such as stacking faults or antiphase boundaries, counteracting their repulsion. Let us now assume the dislocation bi may be connected to the dislocation bk at distance dit by a planar fault of specific energy yk. It can now be shown by quite general arguments (1) that when an array of parallel dislocations in a linear elastic anisotropic medium is in equilibrium, the total energy of the general stacking fault area must be equal to the total prelogarithmic interaction energy coefficients of the dislocations
where fi is the usual Stroh tensor of the elastic constants. To simplify the argument we assume now that we have z dislocations and all Burgers vectors have the same direction and length. Then the interaction terms bi fi bk on the right side of eqn. (1) all have the same magnitude but are positive when the dislocations are parallel and hence repulsive, and negative when they are antiparallel and hence attractive. Since the left side of eqn. (1) is positive, from the total number z of dislocations only a certain fraction n = fz are allowed to be antiparallel in order that the total interaction stays repulsive. Intuitively, one would at first think that attraction will dominate for f > l/2. This is, however, true only for z 4 00and for small z the situation is different. With n antiparallel Burgers vectors we have Z. = n(z-n) attractive interactions and Z, = z*/2 - 2/2 n(z-n) repulsive interactions. Since we must have Z, > Z, for a stable configuration we find n < (z - 4z)/2 1407
Vol. 37, No. 9
STABILITY OF DISLOCATION ARRAY
1408
where n of course is limited to integer values. We have listed below the number n of Burgers vector allowed to be antiparallel in order that the total array can be stable. 2
n
2 0
3 0
4 0
5 1
6 1
7 2
8 2
9 2
10 3
11 3
12 4
100 44
This implies that in a total array of 4 no dislocation, and in a total array of 9 just two dislocations can be antiparallel in order that equilibrium without constraints can be reached. References 1. G. Schoeck, Phil. Mag. Lett. 69, 131 (1994).
Pergamon
PI1 S1359-6462(97)00244-3
STABILITY OF DISLOCATION ARRAYS G. Schoeck Institute of Materials Physics, University of Vienna, A- 1090 Vienna, Boltzmanngasse 5 (Received March 10, 1997) (Accepted June 27,1997)
In a frictionless medium it is impossible to have a stable array of a set (i) of parallel dislocations which have same common components of the Burgers vector unless the dislocations are somehow constrained. When the components are parallel the dislocations will repel each other and escape at the surface. When the components are antiparallel they will attract each other and annihilate their components. A common stabilisation exists becabse at low temperatures the dislocations are constrained to their glide planes (and in addition are usually coMected in a 3-dimensional network). At high temperature, when the dislocation can readily climb and undergo cross-slip, this geometrical constraint is removed. Undler this condition they can be stabilised when they are repulsive and when they are connected by planar faults such as stacking faults or antiphase boundaries, counteracting their repulsion. Let us now assume the dislocation bi may be connected to the dislocation bk at distance dit by a planar fault of specific energy yk. It can now be shown by quite general arguments (1) that when an array of parallel dislocations in a linear elastic anisotropic medium is in equilibrium, the total energy of the general stacking fault area must be equal to the total prelogarithmic interaction energy coefficients of the dislocations
where fi is the usual Stroh tensor of the elastic constants. To simplify the argument we assume now that we have z dislocations and all Burgers vectors have the same direction and length. Then the interaction terms bi fi bk on the right side of eqn. (1) all have the same magnitude but are positive when the dislocations are parallel and hence repulsive, and negative when they are antiparallel and hence attractive. Since the left side of eqn. (1) is positive, from the total number z of dislocations only a certain fraction n = fz are allowed to be antiparallel in order that the total interaction stays repulsive. Intuitively, one would at first think that attraction will dominate for f > l/2. This is, however, true only for z 4 00and for small z the situation is different. With n antiparallel Burgers vectors we have Z. = n(z-n) attractive interactions and Z, = z*/2 - 2/2 n(z-n) repulsive interactions. Since we must have Z, > Z, for a stable configuration we find n < (z - 4z)/2 1407
Vol. 37, No. 9
STABILITY OF DISLOCATION ARRAY
1408
where n of course is limited to integer values. We have listed below the number n of Burgers vector allowed to be antiparallel in order that the total array can be stable. 2
n
2 0
3 0
4 0
5 1
6 1
7 2
8 2
9 2
10 3
11 3
12 4
100 44
This implies that in a total array of 4 no dislocation, and in a total array of 9 just two dislocations can be antiparallel in order that equilibrium without constraints can be reached. References 1. G. Schoeck, Phil. Mag. Lett. 69, 131 (1994).