caa a22 4500 605516014 CHVBK 20210128100712.0 cr unu---uuuuu 210128e20151101xx s 000 0 eng 10.1007/s00205-015-0869-7 doi (NATIONALLICENCE)springer-10.1007/s00205-015-0869-7 The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations [Elektronische Daten] [Sergio Conti, Adriana Garroni, Michael Ortiz] 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 Γ-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 $${\psi(b, t)}$$ ψ ( b , t ) , which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length $${\psi_0(b, t)}$$ ψ 0 ( b , t ) 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 $${\mathcal H^1}$$ H 1 -elliptic envelope. Springer-Verlag Berlin Heidelberg, 2015 Conti Sergio Institut für Angewandte Mathematik, Universität Bonn, 53115, Bonn, Germany aut Garroni Adriana Dipartimento di Matematica, Università di Roma, Sapienza, 00185, Rome, Italy aut Ortiz Michael Division of Engineering and Applied Science, California Institute of Technology, 91125, Pasadena, CA, USA aut Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/2(2015-11-01), 699-755 0003-9527 218:2<699 2015 218 205 https://doi.org/10.1007/s00205-015-0869-7 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00205-015-0869-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Conti Sergio Institut für Angewandte Mathematik, Universität Bonn, 53115, Bonn, Germany aut NATIONALLICENCE 700 1- Garroni Adriana Dipartimento di Matematica, Università di Roma, Sapienza, 00185, Rome, Italy aut NATIONALLICENCE 700 1- Ortiz Michael Division of Engineering and Applied Science, California Institute of Technology, 91125, Pasadena, CA, USA aut NATIONALLICENCE 773 0- Archive for Rational Mechanics and Analysis Springer Berlin Heidelberg 218/2(2015-11-01), 699-755 0003-9527 218:2<699 2015 218 205