Generalized Continua and Dislocation Theory

The present lecture notes have for main purpose to introduce the reader to the notion of driving forces acting on defects in various classes of materials. These classes include elasticity, the standard case in its pure homogeneous form, and more complex behaviors including inhomogeneous and dissipative materials. A typical such driving force is the Peach-Koehler force acting on a dislocation line. More generally, these forces of a non-Newtonian nature are so-called material or configurational forces which are contributors to the canonical equation of momentum, here the momentum equation completely, canonically projected onto the material manifold. The latter indeed is the arena of all material defects and the essential ingredient then becomes the so-called material Eshelby stress tensor. This stress is the driving force behind various types of local matter rearrangements such as plasticity, damage, growth, and phase transformations. Its material divergence provides the sought driving force on different types of “defects” such as, dislocations, disclinations, point defects, cracks, phase-transition fronts and shock waves. Here the emphasis is placed on defects more particularly related to materials science and for materials presenting a microstructure such as polar materials and micromorphic ones. Of importance is the fact that the concept of driving force is always accompanied by a parallel energy approach, so that the dissipation (energy release rate) occurring during the progress of a defect is exactly the non-negative product of the driving force by the velocity of progress. Modern notions of mathematical physics (Noether’s theorem, Lie groups, Cartan geometry) as well as efficiently adapted mathematical tools (e.g., generalized functions or “distributions”) are exploited where necessary. The three great heroes of the reported story are J. D. Eshelby, E. Kroener and J. Mandel.

The classical theory of continuum crystal plasticity is first recalled and then generalized to incorporate the effect of lattice curvature on material hardening. The specific notations used in this chapter are summarized in section 1.6.

This chapter is a review of the dislocation dynamics method and its applications in solving and various problems in crystalline materials. In such materials, a dislocation can be easily understood by considering that a crystal can deform irreversibly by slip, i.e. shifting or sliding along one of its atomic planes. If the slip displacement is equal to a lattice vector, the material across the slip plane will preserve its lattice structure and the change of shape will become permanent. However, rather than simultaneous sliding of two half-crystals, slip displacement proceeds sequentially, starting from one crystal surface and propagating along the slip plane until it reaches the other surface. The boundary between the slipped and still unslipped crystal is a dislocation and its motion is equivalent to slip propagation. In this picture, crystal plasticity by slip is a net result of the motion of a large number of dislocation lines, in response to applied stress. It is interesting to note that this picture of deformation by slip in crystalline materials was first observed in the nineteenth century by (1883) and Ewing and (1899). They observed that deformation of metals proceeded by the formation of slip bands on the surface of the specimen. Their interpretation of these results was obscure since metals were not viewed as crystalline at that time.