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

Conti, Sergio; Garroni, Adriana; Massaccesi, Annalisa

doi:10.1007/s00526-015-0846-x

Cited by 46 publications

(85 citation statements)

References 19 publications

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“…This reduces the problem to a two-dimensional domain Ω ⊂ R 2 . So far most rigorous results have been restricted to this setting, but in recent work of Conti, Garroni & Ortiz [CGO15] the line tension limit has been derived in a full three-dimensional setting for linear elasticity and in the dilute limit (see also [CGM14], [SvG14]). Thus an extension of the results below to a more general three-dimensional setting might be possible, but is far from obvious.…”

Section: Introductionmentioning

confidence: 99%

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Müller

Scardia

Zeppieri

2015

Analysis and Computation of Microstructure in Finite Plasticity

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Abstract. This article gives a short description and a slight refinement of recent work [MSZ15], [SZ12] on the derivation of gradient plasticity models from discrete dislocations models. We focus on an array of parallel edge dislocations. This reduces the problem to a two-dimensional setting. As in the work Garroni, Leoni & Ponsiglione [GLP10] we show that in the regime where the number of dislocation N ε is of the order log 1 ε (where ε is the ratio of the lattice spacing and the macroscopic dimensions of the body) the contributions of the self-energy of the dislocations and their interaction energy balance. Upon suitable rescaling one obtains a continuum limit which contains an elastic energy term and a term which depends on the homogenized dislocation density. The main novelty is that our model allows for microscopic energies which are not quadratic and reflect the invariance under rotations. A key mathematical ingredient is a rigidity estimate in the presence of dislocations which combines the nonlinear Korn inequality of IntroductionClassical theories of plasticity are scale independent. Nonetheless experiments show a notable size dependence of plastic behaviour in the micron and submicron range.

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Section: Introductionmentioning

confidence: 99%

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Müller

Scardia

Zeppieri

2015

Analysis and Computation of Microstructure in Finite Plasticity

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“…For static Γ-convergence the very first results have recently been proved by Conti, Garroni, and Massaccesi [9], but the mathematical understanding of the three-dimensional situation is still very much in its infancy.…”

Section: Extensions and Open Questionsmentioning

confidence: 99%

Convergence of Interaction-Driven Evolutions of Dislocations with Wasserstein Dissipation and Slip-Plane Confinement

Mora

Peletier

Scardia

2017

SIAM J. Math. Anal.

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We consider systems of n parallel edge dislocations in a single slip system, represented by points in a two-dimensional domain; the elastic medium is modelled as a continuum. We formulate the energy of this system in terms of the empirical measure of the dislocations, and prove several convergence results in the limit n → ∞.The main aim of the paper is to study the convergence of the evolution of the empirical measure as n → ∞. We consider rate-independent, quasi-static evolutions, in which the motion of the dislocations is restricted to the same slip plane. This leads to a formulation of the quasistatic evolution problem in terms of a modified Wasserstein distance, which is only finite when the transport plan is slip-plane-confined.Since the focus is on interaction between dislocations, we renormalize the elastic energy to remove the potentially large self-or core energy. We prove Gamma-convergence of this renormalized energy, and we construct joint recovery sequences for which both the energies and the modified distances converge. With this augmented Gamma-convergence we prove the convergence of the quasi-static evolutions as n → ∞.

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“…Here, following again standard terminology and notation for R 3 , div μ is the distributional, row-wise divergence of μ. The measure μ can also be interpreted as a 1-current, as was done for example in [18], then the condition div μ = 0 implies that μ has no boundary.…”

Section: The Geometrical Theory Of Dislocationsmentioning

confidence: 99%

“…Whereas parallel straight dislocations and co-planar dislocations may be characterized as jump sets of BV functions, in three dimensions dislocations are rectifiable curves with vector-valued multiplicity and, therefore proving compactness is more challenging. The compactness and relaxation of functionals defined on curves has been studied by Conti et al [18] within the framework of integral vector-valued currents (cf. [57] for a similar approach).…”

Section: Introductionmentioning

confidence: 99%

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

Conti

Garroni

Ortiz

2015

Arch Rational Mech Anal

Self Cite

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

Cited by 46 publications

References 19 publications

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

Convergence of Interaction-Driven Evolutions of Dislocations with Wasserstein Dissipation and Slip-Plane Confinement

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

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