# Coarse-Graining of a Discrete Model for Edge Dislocations in the Regular Triangular Lattice

@article{Alicandro2022CoarseGrainingOA, title={Coarse-Graining of a Discrete Model for Edge Dislocations in the Regular Triangular Lattice}, author={Roberto Alicandro and Lucia De Luca and Giuliano Lazzaroni and Mariapia Palombaro and Marcello Ponsiglione}, journal={Journal of Nonlinear Science}, year={2022}, volume={33} }

We consider a discrete model of planar elasticity where the particles, in the reference configuration, sit on a regular triangular lattice and interact through nearest-neighbor pairwise potentials, with bonds modeled as linearized elastic springs. Within this framework, we introduce plastic slip fields, whose discrete circulation around each triangle detects the possible presence of an edge dislocation. We provide a Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym…

## One Citation

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

- Mathematics
- 2022

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;R) such a notion extends the standard definition of Jacobian determinant…

## References

SHOWING 1-10 OF 29 REFERENCES

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

- Materials Science
- 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…

### Discrete Crystal Elasticity and Discrete Dislocations in Crystals

- Materials Science, Mathematics
- 2005

This article is concerned with the development of a discrete theory of crystal elasticity and dislocations in crystals. The theory is founded upon suitable adaptations to crystal lattices of elements…

### Gradient theory for plasticity via homogenization of discrete dislocations

- Materials Science
- 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…

### Long range order in atomistic models for solids

- Materials ScienceJournal of the European Mathematical Society
- 2021

The emergence of long-range order at low temperatures in atomistic systems with continuous symmetry is a fundamental, yet poorly understood phenomenon in Physics. To address this challenge we study a…

### Metastability and Dynamics of Discrete Topological Singularities in Two Dimensions: A Γ-Convergence Approach

- Mathematics
- 2014

This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ-convergence. We consider discrete systems, described by scalar…

### On the derivation of linear elasticity from atomistic models

- MathematicsNetworks Heterog. Media
- 2009

This approach generalizes a recent result of Braides, Solci and Vitali (2) and studies mass spring models with full nearest and next-to-nearest pair interactions, and drops the assumption that atoms are allowed to interact only along the associated minimum problems.

### A derivation of linear elastic energies from pair-interaction atomistic systems

- MathematicsNetworks Heterog. Media
- 2007

It is shown that the derivation of linear theories by $\Gamma$-convergence can be obtained directly from lattice interactions in the regime of small deformations.

### Finite difference approximation of energies in Fracture Mechanics

- Mathematics
- 2000

We provide a variational approximation by finite-difference energies of functionals of the type defined for u E SBD(Q), which are related to variational models in ’ fracture mechanics for…

### An Energy Estimate for Dislocation Configurations and the Emergence of Cosserat-Type Structures in Metal Plasticity

- Materials Science
- 2016

We investigate low energy structures of a lattice with dislocations in the context of nonlinear elasticity. We show that these low energy configurations exhibit in the limit a Cosserat-like behavior.…

### Variational equivalence between Ginzburg-Landau, XY spin systems and screw dislocations energies

- Physics
- 2009

We introduce and discuss discrete two-dimensional models for XY spin systems and screw dislocations in crystals. We prove that, as the lattice spacing e tends to zero, the relevant energies in these…