A parametric derivation method for solving the Peierls-Nabarro dislocation equation with a non-sinusoidal law of interatomic forces

摘要

An analytical method, based on parametric derivation, for solving the Peierls-Nabarro (P-N) dislocation equation was investigated. Treated by this method was a more general case of the P-N integral equation with a non-sinusoidal law of force between the atoms above an edge dislocation line and those below it in a simple cubic lattice. The result of the displacement field function from the basic P-N model with a sinusoidal law of interatomic force was taken as a special solution to the equation with a non-sinusoidal law of force. The method involved first differentiating the displacement field function derived in the special case to obtain the corresponding Maclaurin series. This was followed by performing parametric modifications on this series. A general solution to the P-N dislocation equation with a non-sinusoidal force was then derived from the first-order approximation of the parainetrically modified series. The corresponding shear stress distribution on the slip plane was also obtained from the general solution of displacement field function. The presently obtained general solution of the displacement function has been found to be in agreement with the result obtained by the deduction method proposed by Foreman et cil [1].