The stress field of an infinite set of discrete dislocations

Abstract: Computer experiments using particle models A one-dimensional plasma model The simulation program Time integration schemes The particle-mesh force calculation The solution of field equations Collisionless particle models Particle-particle/particle-mesh algorithms Plasma simulation Semiconductor device simulation Astrophysics Solids, liquids and phase changes Fourier transforms Fourier series and finite Fourier transforms Bibliography Index

Abstract: The effective interactions of ions, dipoles and higher-order multipoles under periodic boundary conditions are calculated where the array of periodic replications forms an infinite sphere surrounded by a vacuum. Discrepancies between the results of different methods of calculation are resolved and some shape-dependent effects are discussed briefly. In a simulation under these periodic boundary conditions, the net Hamiltonian contains a positive term proportional to the square of the net dipole moment of the configuration. Surrounding the infinite sphere by a continuum of dielectric constant e.9 changes this positive term, the coefficient being zero as e9 ->∞ . We report on the simulation of a dense fluid of hard spheres with embedded point dipoles; simulations are made for different values of showing how the Kirkwood gr-factor and the long-range part of hA (r) depend on e9 in a finite simulation. We show how this dependence on e9 nonetheless leads to a dielectric constant for the system that is independent of e . In particular, the Clausius-Mosotti and Kirkwood formulae for the dielectric constant e of the system give consistent e values.

Abstract: If, in a multiply‐connected elastic solid, discontinuities are permitted across a stationary barrier in either the strain or its first derivatives or both, dislocations of a more general type than encountered in classical theory are possible. A number of these more general dislocations have been obtained for states of plane and anti‐plane strain in a hollow right circular cylinder when the surface of discontinuity is a single stationary plane barrier. Some of the dislocations found possess the characteristic that although the strain is continuous across the barrier the displacement discontinuity is not one which would be possible in a rigid body. Examination of the conditions for the uniqueness of solution of the boundary value problems of elasticity reveals that when dislocations of the more general type are admitted appropriate data must be given at each point on the specified barrier in addition to the usual information.

Abstract: Various problems concerning infinitely many, infinitely small, parts, had been solved before the infinitesimal calculus was invented; for example, Archimedes on the circumference of the circle. The essence of the invention of the calculus appears to be that the passage to the limit was thereby taken at the earliest possible stage, where diverse problems had operations like d / dx in common. Although the infinitesimal calculus has been a splendid success, yet there remain problems in which it is cumbrous or unworkable. When such difficulties are encountered it may be well to return to the manner in which they did things before the calculus was invented, postponing the passage to the limit until after the problem had been solved for a moderate number of moderately small differences. For obtaining the solution of the difference-problem a variety of arithmetical processes are available. This memoir deals with central differences arranged in the simplest possible way, namely, that explained by the writer in the papers cited in the footnote. Advancing differences are ignored, and so are the varieties of central-difference-process in which accuracy is gained by complicating the arithmetic at an early stage.