Macroscopic dislocation modelling
Work-hardened metals typically possess large numbers of dislocations in complex three-dimensional configurations about which little is known theoretically. Here these large numbers of dislocations are accounted for by means of a dislocation density tensor, which is obtained by applying an averaging process to families of discrete dislocations. Some simple continuous distributions are examined and an analogy is drawn with solenoids in electromagnetism before the question of the equilibrium of dislocation configurations is studied. It is then proved that the only finite, simply-connected distribution of dislocations in equilibrium in the absence of applied stresses are ones in which all components of stress vanish everywhere. Some examples of these zero stress everywhere (ZSE) distributions are then given, and the concept of 'plastic distortion' is used to facilitate their interpretation as rotations of the crystal lattice. Plastic distortion can also be understood as a distribution of infinitesimal dislocation loops ('Kroupa loops'), and this idea is used in Chapter 4 to investigate the dislocation distributions which correspond to elastic inclusions. The evolution, under an applied stress, of some simple ZSEs is analysed, and the idea of 'polarisation' is introduced, again in analogy with electromagnetism. Finally, a mechanism is conjectured for the onset of plastic flow.