Variational Evolution of Dislocations in Single Crystals

Scala, Riccardo; Goethem, Nicolas Van

doi:10.1007/s00332-018-9488-4

Cited by 11 publications

(22 citation statements)

References 36 publications

(86 reference statements)

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“…Specifically, we consider energy densities which also depend on div F and, relying on Helmholtz decomposition F = Du + F 0 with div F 0 = 0, we consider the cases where (i) div F = 0, and (ii) div F ∈ L q (Ω) for some appropriate exponent q. Obviously, these results must be considered as a milestone toward more general results to appear in future works [40,41]. Variational problems of this kind have been already treated by several authors but without the generality of unfixed dislocation loops.…”

Section: Scope and Structure Of The Workmentioning

confidence: 99%

Analytic and geometric properties of dislocation singularities

Scala¹,

Goethem

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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This paper deals with the analysis of the singularities arising from the solutions of the problem ${-}\,{\rm Curl\ } F=\mu $, where F is a 3 × 3 matrix-valued Lp-function ($1\les p<2$) and μ a 3 × 3 matrix-valued Radon measure concentrated in a closed loop in Ω ⊂ ℝ3, or in a network of such loops (as, for instance, dislocation clusters as observed in single crystals). In particular, we study the topological nature of such dislocation singularities. It is shown that $F=\nabla u$, the absolutely continuous part of the distributional gradient Du of a vector-valued function u of special bounded variation. Furthermore, u can also be seen as a multi-valued field, that is, can be redefined with values in the three-dimensional flat torus 𝕋3 and hence is Sobolev-regular away from the singular loops. We then analyse the graphs of such maps represented as currents in Ω × 𝕋3 and show that their boundaries can be written in term of the measure μ. Readapting some well-known results for Cartesian currents, we recover closure and compactness properties of the class of maps with bounded curl concentrated on dislocation networks. In the spirit of previous work, we finally give some examples of variational problems where such results provide existence of solutions.

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Section: Scope and Structure Of The Workmentioning

confidence: 99%

Analytic and geometric properties of dislocation singularities

Scala¹,

Goethem

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Let us emphasize that the external fieldF is not necessarily in equilibrium. Indeed this boundary condition is equivalent to a Dirichlet boundary condition, since we can always write the external fieldF as the gradient of a suitable torus-valued mapũ, and then fixinĝ F corresponds to fixũ, as done in [34]. It was shown in [31,Section 5.4] that (essentially due to the solenoidal property of the dislocation density) the solution is not the trivial one (i.e., the absence of dislocations in Ω).…”

Section: The Minimization Settingmentioning

confidence: 99%

“…This analysis is a necessary prerequisite to study the evolution of dislocation clusters, in particular at the quasistatic regime. A first contribution as a sequel of this work has been given in [34].…”

Section: Introductionmentioning

confidence: 99%

A Variational Approach to Single Crystals with Dislocations

Scala¹,

Goethem²

2019

SIAM J. Math. Anal.

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We study the graphs of maps u : Ω → R 3 whose curl is an integral 1-current with coefficients in Z 3. We characterize the graph boundary of such maps under suitable summability property. We apply these results to study a three-dimensional single crystal with dislocations forming general one-dimensional clusters in the framework of finite elasticity. By virtue of a variational approach, a free energy depending on the deformation field and its gradient is considered. The problem we address is the joint minimization of the free energy with respect to the deformation field and the dislocation lines. We apply closedness results for graphs of torus-valued maps, seen as integral currents and, from the characterization of their graph boundaries we are able to prove existence of minimizers.

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“…Let ∈ ( ) such that → and ⇀ in . Then for everŷ∈  there exists a mutual recovery sequencê in the sense of (49).…”

Section: Closedness Of Stable States (C5)mentioning

confidence: 99%

“…It is versatile, as it has been employed in many different settings, like in problems of nonlinear plasticity (e.g., []), damage, cracks growth (e.g. []), delamination, dislocations evolution, and many others (see [] and references therein for a more detailed discussion and a more exhaustive bibliography).…”

Section: Introductionmentioning

confidence: 99%

Damage model for plastic materials at finite strains

2019

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We introduce a model for elastoplasticity at finite strains coupled with damage. The internal energy of the deformed elastoplastic body depends on the deformation, the plastic strain, and the unidirectional isotropic damage. The main novelty is a dissipation distance allowing the description of coupled dissipative behavior of damage and plastic strain. Moving from time‐discretization, we prove the existence of energetic solutions to the quasistatic evolution problem.

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Variational Evolution of Dislocations in Single Crystals

Cited by 11 publications

References 36 publications

Analytic and geometric properties of dislocation singularities

Analytic and geometric properties of dislocation singularities

A Variational Approach to Single Crystals with Dislocations

Damage model for plastic materials at finite strains

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