An analytical method based upon parametric derivation, for solving the Peierls-Nabarro equation, was investigated by using a more general case of the Peierls-Nabarro integral equation with a non-sinusoidal law of force between the atoms above an edge dislocation line and below it in a simple cubic lattice. The result for the displacement field function of the basic Peierls-Nabarro model, with a sinusoidal interatomic force law, was taken to be a special case of the equation with a non-sinusoidal force law. The method involved first differentiating the displacement field function, as derived for the special case, in order to obtain the corresponding Maclaurin series. Parametric modifications were then performed on this series. A general solution to the Peierls-Nabarro equation with non-sinusoidal force was then derived from the first-order approximation to the parametrically modified series. The corresponding shear stress distribution on the slip plane was also obtained from the general solution to the displacement field function. The present general solution to the displacement function was found to be in agreement with the results obtained by using other methods.

A Parametric Derivation Method for Solving the Peierls-Nabarro Dislocation Equation with a Non-Sinusoidal Law of Interatomic Force. Y.X.Gan, B.Z.Jang: Journal of Materials Science Letters, 1996, 15[6], 2044-7