Variational equivalence between Ginzburg-Landau, XY spin systems and screw dislocations energies

Alicandro, Roberto; Cicalese, Marco; Ponsiglione, Marcello

doi:10.1512/iumj.2011.60.4339

Cited by 46 publications

(99 citation statements)

References 29 publications

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“…Secondly, we wanted to push the methods developed for nonlocal energies beyond the case of radially symmetric potentials, still retaining the critical, logarithmic singularity at zero. Finally, we wanted to explore the connection between the theory of vortices and the theory of dislocations, which has been successfully exploited so far in the case of discrete and screw dislocations (see, e.g., [1,2]). The literature on nonlocal interaction energies is vast.…”

Section: )mentioning

confidence: 99%

The Equilibrium Measure for a Nonlocal Dislocation Energy

Mora

Rondi

Scardia³

2018

Comm Pure Appl Math

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In this paper we characterize the equilibrium measure for a nonlocal and anisotropic weighted energy describing the interaction of positive dislocations in the plane. We prove that the minimum value of the energy is attained by a measure supported on the vertical axis and distributed according to the semicircle law, a well‐known measure that also arises as the minimizer of purely logarithmic interactions in one dimension. In this way we give a positive answer to the conjecture that positive dislocations tend to form vertical walls. This result is one of the few examples where the minimizer of a nonlocal energy is explicitly computed and the only one in the case of anisotropic kernels. © 2018 Wiley Periodicals, Inc.

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Section: )mentioning

confidence: 99%

The Equilibrium Measure for a Nonlocal Dislocation Energy

Mora

Rondi

Scardia³

2018

Comm Pure Appl Math

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“…In [18], it has been proved that the screw dislocations functionals 4π 2 SDε | log ε| Γ-converge to π|µ|(Ω), where µ is the limiting vorticity measure and is given by a finite sum of Dirac masses. In [2], it has been shown that the energies SDε | log ε| h and GLε | log ε| h (with h ≥ 1) are variationally equivalent, which means, roughly speaking, that they have the same Γ-limit (up to a factor) and share the same equicoerciveness property with respect to the same convergence. The Γ-limit |µ|(Ω) is not affected by the position of the singularities and hence does not account for their interaction, which is an essential ingredient in order to study the dynamics.…”

Section: Overview Of the Model For Discrete Screw Dislocationsmentioning

confidence: 99%

Dynamics of discrete screw dislocations via discrete gradient flow

Luca

2014

Proc Appl Math and Mech

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We present variational approaches (developed in [3,4,11]) to the study of statics and dynamics of screw dislocations in crystals.We model the crystal as a cubic lattice and we give the asymptotic Γ-convergence expansion of the elastic energy induced by a finite family of screw dislocations as the lattice spacing goes to zero. We show that the effective energy associated to the presence of a finite system of screw dislocations coincides with the renormalized energy, studied within the Ginzburg-Landau framework and ruling the interactions between the dislocations. As a byproduct of this analysis, we show the existence of many metastable configurations of dislocations pinned by energy barries. Using the minimizing movement approach á la De Giorgi, we introduce a discrete-in-time variational dynamics, referred to as Discrete Gradient Flow, which allows to overcome these energy barriers. More precisely, we show that lettting first the lattice spacing and then the time step of minimizing movements tend to zero, dislocations move accordingly with the gradient flow of the renormalized energy.

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“…A linear-elastic model of a finite number of wellseparated dislocations with a core-radius regularization was studied by De Luca et al [22] and extended by Scardia and Zepperi [58] to nonlinear elasticity with subquadratic growth. The relation of the model to Ginzburg-Landau and spin models was discussed by Alicandro et al [3]. Second-order expansions of the energy were employed by Leoni and Cermelli [17] to derive leading-order approximations of the interaction energies.…”

Section: Introductionmentioning

confidence: 99%

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

Conti

Garroni

Ortiz

2015

Arch Rational Mech Anal

119

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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 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 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 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 H 1 -elliptic envelope.

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Variational equivalence between Ginzburg-Landau, XY spin systems and screw dislocations energies

Cited by 46 publications

References 29 publications

The Equilibrium Measure for a Nonlocal Dislocation Energy

The Equilibrium Measure for a Nonlocal Dislocation Energy

Dynamics of discrete screw dislocations via discrete gradient flow

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

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