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Über dieses Buch

Defects, dislocations and the general theory.- Approaches to generalized continua.- Generalized continuum modelling of crystal plasticity.- Introduction to discrete dislocation dynamics.

The book contains four lectures on generalized continua and dislocation theory, reflecting the treatment of the subject at different scales. G. Maugin provides a continuum formulation of defects at the heart of which lies the notion of the material configuration and the material driving forces of in-homogeneities such as dislocations, disclinations, point defects, cracks, phase-transition fronts and shock waves. C. Sansour and S. Skatulla start with a compact treatment of linear transformation groups with subsequent excursion into the continuum theory of generalized continua. After a critical assessment a unified framework of the same is presented. The next contribution by S. Forest gives an account on generalized crystal plasticity. Finally, H. Zbib provides an account of dislocation dynamics and illustrates its fundamental importance at the smallest scale. In three contributions extensive computational results of many examples are presented.



Defects, Dislocations and the General Theory of Material Inhomogeneity

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.
G. A. Maugin

Approaches to Generalized Continua

Decades ago, it has been recognized that for some materials the kinematics on meso- and micro-structural scale needs to be considered, if the material’s resistance to deformation exhibits a finite radius of interaction on atomic or molecule level, e.g. (1963); (1964) outlined that this is the case if the deformation wave length approaches micro-structural length scale. Differently said, if the external loading corresponds material entities smaller than the representative volume element (RVE), then the statistical average of the macro-scopical material behaviour does not hold anymore. In this sense the fluctuation of deformation on micro-structural level as well as relative motion of micro-structural constituens, such as granule, crystalline or other heterogeneous aggregates, influence the material response on macro-structural level. Consequently, field equations based on the assumption of micro-scopically homogeneous material have to be supplemented and enriched to also include non-local and higher-order contributions.
Carlo Sansour, Sebastian Skatulla

Generalized Continuum Modelling of Crystal Plasticity

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.
Samuel Forest

Introduction to Discrete Dislocation Dynamics

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.
Hussein M. Zbib
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