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

@article{Conti2015TheLA, title={The Line-Tension Approximation as the Dilute Limit of Linear-Elastic Dislocations}, author={Sergio Conti and Adriana Garroni and Michael Ortiz}, journal={Archive for Rational Mechanics and Analysis}, year={2015}, volume={218}, pages={699-755} }

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…

## 41 Citations

Gradient Theory for Geometrically Nonlinear Plasticity via the Homogenization of Dislocations

- Mathematics
- 2015

This article gives a short description and a slight refinement of recentwork [MSZ15], [SZ12] on the derivation of gradient plasticity models fromdiscrete dislocations models.We focus on an array of…

On the Motion of Curved Dislocations in Three Dimensions: Simplified Linearized Elasticity

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2021

It is shown that in core-radius cutoff regularized simplified elasticity, the force on a dislocation curve by the negative gradient of the elastic energy asymptotically approaches the mean curvature of the curve as the cutoff radius converges to zero.

Dislocation microstructures and strain-gradient plasticity with one active slip plane

- Materials Science, Mathematics
- 2016

Abstract We study dislocation networks in the plane using the vectorial phase-field model introduced by Ortiz and coworkers, in the limit of small lattice spacing. We show that, in a scaling regime…

A Variational Model for Dislocations at Semi-coherent Interfaces

- Physics, MathematicsJ. Nonlinear Sci.
- 2017

It is proved that, for minimizers, the former scales like the surface area of the interface, the latter like its diameter, and the proposed continuum model is built on some explicit computations done in the framework of the semi-discrete theory of dislocations.

Plasticity as the Γ -Limit of a Two-dimensional Dislocation Energy: The Critical Regime Without the Assumption of Well-Separateness

- 2019

In this paper, a strain-gradient plasticity model is derived from a mesoscopic model for straight parallel edge dislocations in an infinite cylindrical crystal. The main difference to existing work…

Strain-Gradient Plasticity as the $\Gamma$-Limit of a Nonlinear Dislocation Energy with Mixed Growth

- Mathematics
- 2018

In this paper a we derive by means of $\Gamma$-convergence a macroscopic strain-gradient plasticity from a semi-discrete model for dislocations in an infinite cylindrical crystal. In contrast to…

A line-tension model of dislocation networks on several slip planes

- Materials Science
- 2015

Abstract Dislocations in crystals can be studied by a Peierls–Nabarro type model, which couples linear elasticity with a nonconvex term modeling plastic slip. In the limit of small lattice spacing,…

Plasticity as the $${\Gamma}$$Γ-Limit of a Two-dimensional Dislocation Energy: The Critical Regime Without the Assumption of Well-Separateness

- Physics, MathematicsArchive for Rational Mechanics and Analysis
- 2019

In this paper, a strain-gradient plasticity model is derived from a mesoscopic model for straight parallel edge dislocations in an infinite cylindrical crystal. The main difference to existing work…

A semi-discrete line-free method of monopoles for dislocation dynamics

- Physics
- 2021

Abstract We develop a semi-discrete particle method for Volterra dislocation currents in which the particles, or monopoles, represent an element of line and carry a Burgers vector. The monopoles move…

Constraint reaction and the Peach–Koehler force for dislocation networks

- Physics
- 2016

In the presence of dislocations, the strain field F is not a gradient, but satisfies the condition Curl F = Λ T L (with Λ L a measure concentrated on the dislocation L). Then F ∈ L p with 1 ≤ p < 2.…

## References

SHOWING 1-10 OF 95 REFERENCES

Line-Tension Model for Plasticity as the Gamma-Limit of a Nonlinear Dislocation Energy

- Computer Science, MathematicsSIAM J. Math. Anal.
- 2012

In this paper we rigorously derive a line-tension model for plasticity as the $\Gamma$-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off…

Line-tension model for plasticity as the -limit of a nonlinear dislocation energy

- 2012

In this paper we rigorously derive a line-tension model for plasticity as the Γ-limit of a nonlinear mesoscopic dislocation energy, without resorting to the introduction of an ad hoc cut-off radius.…

Renormalized Energy and Forces on Dislocations

- Mathematics, Computer ScienceSIAM J. Math. Anal.
- 2005

This work gives a sharpasymptotic estimate of the minimum energy as the core radius tends to zero, which allows one to eliminate this internal length scale from the problem.

Gradient theory for plasticity via homogenization of discrete dislocations

- Mathematics, Physics
- 2008

We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so…

a Phase-Field Theory of Dislocation Dynamics, Strain Hardening

- Materials Science
- 2001

A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for: an arbitrary number and arrangement of dislocation…

Γ-Convergence Analysis of Systems of Edge Dislocations: the Self Energy Regime

- Mathematics
- 2012

This paper deals with the elastic energy induced by systems of straight edge dislocations in the framework of linearized plane elasticity. The dislocations are introduced as point topological defects…

A Variational Model for Dislocations in the Line Tension Limit

- Mathematics
- 2006

We study the interaction of a singularly-perturbed multiwell energy (with an anisotropic nonlocal regularizing term of H1/2 type) and a pinning condition. This functional arises in a phase field…

An Exactly Solvable Phase-Field Theory of Dislocation Dynamics, Strain Hardening and Hysteresis in Ductile Single Crystals

- Physics
- 2001

An exactly solvable phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystals is developed. The theory accounts for: an arbitrary number and arrangement…

A line-tension model of dislocation networks on several slip planes

- Materials Science
- 2015

Abstract Dislocations in crystals can be studied by a Peierls–Nabarro type model, which couples linear elasticity with a nonconvex term modeling plastic slip. In the limit of small lattice spacing,…

DISLOCATIONS AT THE CONTINUUM SCALE: FUNCTIONAL SETTING AND VARIATIONAL PROPERTIES

- Mathematics
- 2014

Considering the existence of solutions to a minimum problem for dislocations in finite elasticity (21), in the present paper we analyze the first variation of the energy at the minimum points with…