P. Ariza, S. Conti, A. Garroni and M. Ortiz Variational modeling of dislocations in crystals in the line-tension limit In V. Mehrmann and M. Skutella, editor, European Congress of Mathematics, Berlin, 2016
page 583-598.
Publisher: EMS
2018 10.4171/176-1/27
2017
Andrea Braides, Sergio Conti and Adriana Garroni Density of polyhedral partitions Calc. Var. Partial Differential Equations, 56(2): Art. 28, 10 2017 10.1007/s00526-017-1108-x
Sergio Conti, Adriana Garroni and Stefan Müller Homogenization of vector-valued partition problems and dislocation cell structures in the plane Boll. Unione Mat. Ital., 10(1): 3--17 2017 10.1007/s40574-016-0083-z
2016
Sergio Conti, Adriana Garroni and Stefan Müller Dislocation microstructures and strain-gradient plasticity with one active slip plane J. Mech. Phys. Solids, 93: 240-251 2016 10.1016/j.jmps.2015.12.008
2015
Sergio Conti, Adriana Garroni and Michael Ortiz The line-tension approximation as the dilute limit of linear-elastic dislocations Arch. Ration. Mech. Anal., 218(2): 699-755 2015 http://dx.doi.org/10.1007/s00205-015-0869-7
Abstract: We prove that the classical line-tension approximation for dislocations in crystals, i.e., the approximation that neglects interactions at a distance between dislocation segments and accords dislocations energy in proportion to their length, follows as the $\Gamma$-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 the classical energy down to its $\calH^1$-elliptic envelope.
Sergio Conti, Adriana Garroni and Annalisa Massaccesi Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity Calc. Var. PDE, 54(2): 1847-1874 2015 http://dx.doi.org/10.1007/s00526-015-0846-x
Abstract: In the modeling of dislocations one is lead naturally to energies concentrated on lines, where the integrand depends on the orientation and on the Burgers vector of the dislocation, which belongs to a discrete lattice. The dislocations may be identified with divergence-free matrix-valued measures supported on curves or with 1-currents with multiplicity in a lattice. In this paper we develop the theory of relaxation for these energies and provide one physically motivated example in which the relaxation for some Burgers vectors is nontrivial and can be determined explicitly. From a technical viewpoint the key ingredients are an approximation and a structure theorem for 1-currents with multiplicity in a lattice.