Vibrations of Dislocation Lines with Movable Pinning Points. III

Abstract

This work is motivated by problems arising in dislocation theory: A dislocation is represented by a line and defects as points along the line which exert a drag proportional to their velocity. We therefore study the motion of a string that is fixed at the endpoints, is driven by a harmonic force, and is drag-free except at $n$ distinct "pinning points." The mass term is neglected throughout. By treating the drag term as an inhomogeneity, the appropriate differential equation is converted into an integral equation. Solutions with the drag coefficient $\gamma$ either 0 or $\infty$ form the basis for solutions for the intermediate cases: Two series expansions for the displacement $y\left(x\right)\mathrm{}$ as a function of position $x$ are obtained (one in $n\gamma$, the other in $\frac{n}{\gamma }$, valid, respectively, for small and large $\gamma$). In both series, real terms (in phase with the applied force) and purely imaginary ones (out of phase) alternate. Some results are obtained for arbitrary locations of pinning points, and subsequently averaged over pinning-point locations; more detailed results are obtained for the situation in which all pinning points are required to be at their average position (viz., equally spaced), and it is shown that the latter results provide a good approximation to the former when $n$ is large. For even larger $n$, the details between pinning points are no longer of interest, and the replacement of discrete drag points by continuous drag provides an even simpler approximation.