An Equilibrium Theory of Dislocation Continua

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Abstract
A homogenised model for elastic media containing large numbers of dislocations is described. First, discrete dislocations are discussed from the mathematical and crystallographical points of view, and their stress fields are calculated. These building blocks are averaged to construct a homogenised model in which the dislocation distribution is averaged to a number density tensor. Equilibrium configurations are then considered and it is proved that, in the absence of externally applied stresses, the only possible finite, simply connected distributions are ones in which all components of stress vanish everywhere. Some examples are given of these zero-stress everywhere (ZSE) distributions, and their geometrical interpretation is considered in terms of the plastic distortion tensor, which shows that they are equivalent to local rotations of the crystal lattice. Finally, some conjectures are made about the response of a cellular ZSE distribution to an applied stress, introducing the idea of 'polarization' by analogy with electromagnetism, and a possible scenario is sketched for the large-scale deformations associated with plastic flow