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The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations

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Abstract

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 \({\psi(b, t)}\), which depends on the local Burgers vector and orientation of the dislocation, does not, however, necessarily coincide with the self-energy per unit length \({\psi_0(b, t)}\) 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 \({\mathcal H^1}\)-elliptic envelope.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, 53115, Bonn, Germany

    Sergio Conti

  2. Dipartimento di Matematica, Università di Roma, Sapienza, 00185, Rome, Italy

    Adriana Garroni

  3. Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA, 91125, USA

    Michael Ortiz

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  1. Sergio Conti
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  2. Adriana Garroni
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  3. Michael Ortiz
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Conti, S., Garroni, A. & Ortiz, M. The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations. Arch Rational Mech Anal 218, 699–755 (2015). https://doi.org/10.1007/s00205-015-0869-7

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