# Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity

@article{Conti2014ModelingOD, title={Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity}, author={Sergio Conti and Adriana Garroni and Annalisa Massaccesi}, journal={Calculus of Variations and Partial Differential Equations}, year={2014}, volume={54}, pages={1847-1874} }

In the modeling of dislocations one is led 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…

## 38 Citations

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- Mathematics
- 2016

A striking geometric property of elastic bodies with dislocations is that the deformation tensor cannot be written as the gradient of a one-to-one immersion, its curl being nonzero but equal to the…

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This paper provides an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal and discusses a novel dissipation structure for such currents, namely the flat distance, that will drive the evolution of the dislocation clusters.

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Considering the existence of solutions to a minimum problem for dislocations in finite elasticity (21), in the present paper we analyze the first variation of the energy at the minimum points with…

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- 2015

This paper deals with a variational problem for dislocations in which the curl of the deformation tensor is constrained by a concentrated measure in a set of lines, called the dislocation density,…

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- Materials Science
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Abstract Dislocations in crystals can be studied by a Peierls–Nabarro type model, which couples linear elasticity with a nonconvex term modeling plastic slip. In the limit of small lattice spacing,…

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- 2020

In this paper we study the homogenization of a class of energies concentrated on lines. In dimension $2$ (i.e., in codimension $1$) the problem reduces to the homogenization of partition energies…

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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…

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- Physics
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Abstract We develop a semi-discrete particle method for Volterra dislocation currents in which the particles, or monopoles, represent an element of line and carry a Burgers vector. The monopoles move…

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- Materials Science, Mathematics
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Abstract We study dislocation networks in the plane using the vectorial phase-field model introduced by Ortiz and coworkers, in the limit of small lattice spacing. We show that, in a scaling regime…

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