Derivation of a Line-Tension Model for Dislocations from a Nonlinear Three-Dimensional Energy: The Case of Quadratic Growth

Garroni, Adriana; Marziani, Roberta; Scala, Riccardo

doi:10.1137/20m1330117

Cited by 11 publications

(12 citation statements)

References 42 publications

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“…In this paper, we provide a unified treatment of the three-dimensional continuum models which covers many different regularizations, as for example the one with mixed growth conditions, and clarifies the equivalence of all these possible approximations and corresponding regularizations. Our general approach produces also a simpler and more direct proof of some results from the literature [CGO15,GMS21]; we expect that this unified approach will prove helpful in further future generalizations. We assume that dislocations are dilute, in the sense that the curve γ on which µ is supported has some regularity, which may however degenerate in the limit.…”

Section: Introductionmentioning

confidence: 68%

Line-tension limits for line singularities and application to the mixed-growth case

Conti¹,

Garroni²,

Marziani³

2022

Preprint

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We study variational models for dislocations in three dimensions in the line-tension scaling. We present a unified approach which allows to treat energies with subquadratic growth at infinity and other regularizations of the singularity near the dislocation lines. We show that the asymptotics via Gamma convergence is independent of the specific choice of the energy and of the regularization procedure.

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Section: Introductionmentioning

confidence: 68%

Line-tension limits for line singularities and application to the mixed-growth case

Conti¹,

Garroni²,

Marziani³

2022

Preprint

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“…As we see, Equation ( 21) differs from relations (17) and (19), given in [12] as Equations ( 13) and (14). It is easy to notice that ( 17) and (19) are correct only for one dimensional planar case and not for the axisymmetric one as it is stated in [12].…”

Section: Analysis Of the Modelmentioning

confidence: 83%

“…To determine the profile of the phase field described by (14), consider an unbounded domain where the breakdown channel axis is a smooth curve Λ. Assuming the vanishing electric field in the medium, we see that φ satisfies…”

Section: Formal Description Of the Modelmentioning

confidence: 99%

“…The model [12] is considered as a specific example of a phase field model of "codimension two". Nevertheless, we believe that the presented considerations are of the general interest and importance and potentially can be applied to the construction of phase field models for a such effectively mixed-dimensional problems as elastic multi-structures [13], dislocation lines dynamics [14], coupled 1D/3D tissue perfusion models [15] and well models in geophysics [16], and numerous electrostatic problems which requires analysis of thin charged or conducting lines, which is a classical problem of electrostatics with numerous application, see, e.g., [17].…”

Section: Introductionmentioning

confidence: 99%

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On the Diffuse Interface Models for High Codimension Dispersed Inclusions

Zipunova

Савенков

2021

Mathematics

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Diffuse interface models are widely used to describe the evolution of multi-phase systems of various natures. Dispersed inclusions described by these models are usually three-dimensional (3D) objects characterized by phase field distribution. When employed to describe elastic fracture evolution, the dispersed phase elements are effectively two-dimensional (2D) objects. An example of the model with effectively one-dimensional (1D) dispersed inclusions is a phase field model for electric breakdown in solids. Any diffuse interface field model is defined by an appropriate free energy functional, which depends on a phase field and its derivatives. In this work we show that codimension of the dispersed inclusions significantly restricts the functional dependency of the free energy on the derivatives of the problem state variables. It is shown that to describe codimension 2 diffuse objects, the free energy of the model necessarily depends on higher order derivatives of the phase field or needs an additional smoothness of the solution, i.e., its first derivatives should be integrable with a power greater than two. Numerical experiments are presented to support our theoretical discussion.

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“…This last issue is much more delicate, since it requires the extension of our approach to maps with vectorial lifting. Finally, an extension of our approach to the 3d setting [21] deserves future investigations; in such a case, techniques coming from the theory of integral and Cartesian currents could be exploited, as successfully done in [34,20,50,51,52,30].…”

mentioning

confidence: 99%

A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation

Luca¹,

Scala²,

Goethem³

2022

Preprint

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We introduce a weak notion of 2 × 2-minors of gradients of a suitable subclass of BV functions. In the case of maps in BV (R 2 ; R 2 ) such a notion extends the standard definition of Jacobian determinant to non-Sobolev maps.We use this distributional Jacobian to prove a compactness and Γ-convergence result for a new model describing the emergence of topological singularities in two dimensions, in the spirit of Ginzburg-Landau and core-radius approaches. Within our framework, the order parameter is an SBV map u taking values in S 1 and the energy is made by the sum of the squared L 2 norm of ∇u and of the length of (the closure of) the jump set of u multiplied by 1 ε . Here, ε is a length-scale parameter. We show that, in the | log ε| regime, the Jacobian distributions converge, as ε → 0 + , to a finite sum µ of Dirac deltas with weights multiple of π, and that the corresponding effective energy is given by the total variation of µ .

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Derivation of a Line-Tension Model for Dislocations from a Nonlinear Three-Dimensional Energy: The Case of Quadratic Growth

Cited by 11 publications

References 42 publications

Line-tension limits for line singularities and application to the mixed-growth case

Line-tension limits for line singularities and application to the mixed-growth case

On the Diffuse Interface Models for High Codimension Dispersed Inclusions

A new approach to topological singularities via a weak notion of Jacobian for functions of bounded variation

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