The Peach-Koehler Formula

The last two terms, which are equal, represent the interaction energies, and are the only terms that will be affected by the excursion of the dislocation. Hence, we may write 

The result of this integration may be derived after some calculations demanding algebraic care, with the result that 

We see that in the case in which the Burgers vectors are equal, the force of interaction is repulsive, whereas in the case in which they are equal and opposite, the resulting force is attractive. 

The result derived above is but a special case of a more general formula which can be derived from eqn (8.44). In order to compute the more general result, we imagine a small excursion of the segment of dislocation as shown in fig. 8.21. In 

Recall that the definition of the Burgers vector is bi = Mj(+) — Ui(-), which has been used in writing the above expression in this final form. 

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By exercising the instructions dictated by the Peach-Koehler formula using the state of stress given above, the force on dislocation 2 is given by... 

Note that we denote the distance of closest approach of the dislocations (i.e. along the Xi-axis) as s. Carrying out the relevant algebra in conjimction with the Peach-Koehler formula leads to the result that... 

The physical interpretation of this result is that it is the force per unit length of dislocation acting on a segment by virtue of the presence of the stress field a. This expression is the famed Peach-Koehler formula and will be seen to yield a host of interesting insights into the interactions between dislocations. 

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