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 b
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G. A. Maugin Institut Jean Le Rond d’Alembert Universit´e Pierre et Marie Curie, Paris, France Abstract 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.
C. Sansour et al. (eds.), Generalized Continua and Dislocation Theory © CISM, Udine 2012
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G. A. Maugin
1 Lecture 1: The Eshelby–Kroener View of Defects and their Driving Force: Peach–Koehler Force, Incompatibility, Eshelby Stress 1.1
The Case of a Dislocation Line (Peach–Koehler Force; Eshelby’s Derivation)
We start with the evaluation of a material force acting on one singular line. This was first developed by Peach and Koehler (1950)) in a celebrated paper, a true landmark in dislocation theory. A dislocation line L is seen in continuum physics as a line along which the displacement vector of elasticity suffers, in a certain sense, a finite discontinuity, called the Burgers vector, ˜ in order to avoid any confusion with the Eshelby stress that we shall note b (although there exists a relation between these two notions). The magni˜ characterize the different types of dislocations (see tude and direction of b ˜ can only be equal to a finite number Lardner, 1974). In discrete crystals b of the vectors of the lattice. What exactly occurs is
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