The Forces Exerted on Dislocations and the Stress Fields Produced by Them

It is shown that the force $d\mathbf{F}$ exerted on a line element $\ensuremath{\nu}d\ensuremath{\sigma}$ of a dislocation with Burgers vector f by a stress $\ensuremath{\tau}$ is given by $d\mathbf{F}=\ensuremath{-}\ensuremath{\nu}\ifmmode\times\else\texttimes\fi{}(\mathbf{f}\ifmmode\cdot\else\textperiodcentered\fi{}\ensuremath{\tau})d\ensuremath{\sigma}$. An analogy is drawn between the behavior of a closed line dislocation in a stress field and the behavior of a closed current-carrying loop in a field of magnetic induction. Then formulas for the stress components caused at any point of an infinite elastically isotropic crystal by a line element of a general Burgers dislocation are deduced from Burger's expressions for the displacements (see Sec. III(C)). These formulas bear a close analogy to the Biot-Savart formula of electromagnetic theory. Both of these results taken together constitute a complete system for the investigation of the mutual interaction of dislocations in an infinite elastically isotropic crystal.