# Variational Evolution of Dislocations in Single Crystals

@article{Scala2019VariationalEO, title={Variational Evolution of Dislocations in Single Crystals}, author={Riccardo Scala and Nicolas Van Goethem}, journal={Journal of Nonlinear Science}, year={2019}, volume={29}, pages={319-344} }

In this paper, we provide an existence result for the energetic evolution of a set of dislocation lines in a three-dimensional single crystal. The variational problem consists of a polyconvex stored elastic energy plus a dislocation energy and some higher-order terms. The dislocations are modeled by means of integral one-currents. Moreover, we discuss a novel dissipation structure for such currents, namely the flat distance, that will serve to drive the evolution of the dislocation clusters.

## Topics from this paper

## 6 Citations

Energetic solutions to rate-independent large-strain elasto-plastic evolutions driven by discrete dislocation flow

- Mathematics
- 2021

This work rigorously implements a recent model, introduced in [34], of large-strain elasto-plastic evolution in single crystals where the plastic flow is driven by the movement of discrete…

A Variational Approach to Single Crystals with Dislocations

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2019

The graph boundary of maps whose curl is an integral 1-current with coefficients in $\Bbb Z^3$ is characterized under a suitable summa...

Damage model for plastic materials at finite strains

- PhysicsZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
- 2019

We consider a model for nonlinear elastoplasticity coupled with incomplete damage. The internal energy of the deformed elastoplastic body depends on the deformation y, on the plastic strain P , and…

Analytic and geometric properties of dislocation singularities

- PhysicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2020

Abstract 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…

Elasto-plastic evolution of single crystals driven by dislocation flow

- Physics
- 2021

This work introduces a model for large-strain, geometrically nonlinear elasto-plastic dynamics in single crystals. The key feature of our model is that the plastic dynamics are entirely driven by the…

Space-time integral currents of bounded variation

- Mathematics
- 2021

Motivated by a recent model for elasto-plastic evolutions that are driven by the flow of dislocations [12], this work develops a theory of space-time integral currents with bounded variation in time.…

## References

SHOWING 1-10 OF 54 REFERENCES

Existence of minimizers for a polyconvex energy in a crystal with dislocations

- Mathematics
- 2008

We provide existence theorems in nonlinear elasticity for minimum problems modeling the deformations of a crystal with a given dislocation. A key technical difficulty is that due to the presence of a…

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

- Mathematics, Physics
- 2014

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…

Introduction to Discrete Dislocation Dynamics

- Materials Science
- 2012

This chapter is a review of the dislocation dynamics method and its applications in solving and various problems in crystalline materials. In such materials, a dislocation can be easily understood by…

Currents and dislocations at the continuum scale

- 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…

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

- Physics, Computer ScienceSIAM J. Math. Anal.
- 2017

The main aim of the paper is to study the convergence of the evolution of the empirical measure as n-to-infty, and renormalize the elastic energy to remove the potentially large self- or core energy.

Molecular dynamics simulations of motion of edge and screw dislocations in a metal

- Materials Science
- 2002

Abstract Motions of a straight edge dislocation and a kinked screw dislocation in BCC Mo, described by the Finnis–Sinclair potential, are studied in periodic simulation cells subjected to an applied…

A distributional approach to the geometry of 2D dislocations at the continuum scale

- Mathematics
- 2012

This paper develops a geometrical model of dislocations and disclinations in single crystals at the continuum scale, with use of the distribution theory to represent concentrated effects in the…

Dynamics for Systems of Screw Dislocations

- Mathematics, Computer ScienceSIAM J. Appl. Math.
- 2015

The goal of this paper is the analytical validation of a model of Cermelli and Gurtin for an evolution law for systems of screw dislocations under the assumption of antiplane shear, leading to a system of differential inclusions.

Long-time behavior for crystal dislocation dynamics

- Physics, Mathematics
- 2016

We describe the asymptotic states for the solutions of a nonlocal equation of evolutionary type, which have the physical meaning of the atom dislocation function in a periodic crystal.
More…

Discrete-to-continuum limits of interacting dislocations

- Mathematics
- 2015

This thesis studies discrete-to-continuum limits of models for interacting dislocations. This analysis contributes to the ultimate goal of obtaining a system of equations which accurately describes…