期刊
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
卷 218, 期 2, 页码 699-755出版社
SPRINGER
DOI: 10.1007/s00205-015-0869-7
关键词
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资金
- Deutsche Forschungsgemeinschaft [Sonderforschungsbereich 1060]
- U.S. National Science Foundation through the Partnership for International Research and Education (PIRE) on Science at the Triple Point Between Mathematics, Mechanics and Materials Science [0967140]
- Office Of Internatl Science &Engineering
- Office Of The Director [967140] Funding Source: National Science Foundation
We prove that the classical line-tension approximation for dislocations in crystals, that is, the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the I-limit of regularized linear-elasticity as the lattice parameter becomes increasingly small or, equivalently, as the dislocation measure becomes increasingly dilute. We consider two regularizations of the theory of linear-elastic dislocations: a core-cutoff and a mollification of the dislocation measure. We show that both regularizations give the same energy in the limit, namely, an energy defined on matrix-valued divergence-free measures concentrated on lines. The corresponding self-energy per unit length , which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length obtained from the classical theory of the prelogarithmic factor of linear-elastic straight dislocations. Indeed, microstructure can occur at small scales resulting in a further relaxation of the classical energy down to its -elliptic envelope.
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