# Asymptotic analysis of microscopic impenetrability constraints for atomistic systems

@article{Braides2015AsymptoticAO, title={Asymptotic analysis of microscopic impenetrability constraints for atomistic systems}, author={Andrea Braides and Maria Stella Gelli}, journal={Journal of The Mechanics and Physics of Solids}, year={2015}, volume={96}, pages={235-251} }

## 7 Citations

### Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems

- Mathematics
- 2017

In this paper we provide rigorous statements and proofs for the asymptotic analysis of discrete energies defined on a two-dimensional triangular lattice allowing for fracture in presence of a…

### On Lennard-Jones systems with finite range interactions and their asymptotic analysis

- MathematicsNetworks Heterog. Media
- 2018

This work provides an explicit expression for the continuum limit in the case of finite range interactions of Lennard-Jones type by means of $\Gamma$-convergence techniques and studies suitably rescaled energies in which bulk and surface contributions scale in the same way.

### Surface energies emerging in a microscopic, two-dimensional two-well problem

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2017

In this paper we are interested in the microscopic modelling of a two-dimensional two-well problem that arises from the square-to-rectangular transformation in (two-dimensional) shape-memory…

### Interactions beyond nearest neighbours and rigidity of discrete energies: a compactness result and an application to dimension reduction

- Mathematics
- 2016

We analyse the rigidity of discrete energies where at least nearest and next- to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest…

### On the effect of interactions beyond nearest neighbours on non-convex lattice systems

- MathematicsCalculus of Variations and Partial Differential Equations
- 2017

We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond…

### Motion of Discrete Interfaces on the Triangular Lattice

- MathematicsMilan Journal of Mathematics
- 2020

We study the motion of discrete interfaces driven by ferromagnetic interactions on the two-dimensional triangular lattice by coupling the Almgren, Taylor and Wang minimizing movements approach and a…

### On the effect of interactions beyond nearest neighbours on non-convex lattice systems

- Mathematics
- 2017

We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond…

## References

SHOWING 1-10 OF 41 REFERENCES

### Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems

- Mathematics
- 2017

In this paper we provide rigorous statements and proofs for the asymptotic analysis of discrete energies defined on a two-dimensional triangular lattice allowing for fracture in presence of a…

### Separation of Scales in Fracture Mechanics: From Molecular to Continuum Theory via Γ Convergence

- Physics
- 2004

We propose a procedure to obtain a consistent, mesh-objective, continuous model starting from chains composed of discrete springs exhibiting strain softening. Observing the size-dependent response of…

### 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.

### An Atomistic-to-Continuum Analysis of Crystal Cleavage in a Two-Dimensional Model Problem

- PhysicsJ. Nonlinear Sci.
- 2014

It is rigorously proved that, in the discrete-to-continuum limit, the minimal energy of a crystal under uniaxial tension leads to a universal cleavage law and energy minimizers are either homogeneous elastic deformations or configurations that are completely cracked and do not store elastic energy.

### ENERGIES IN SBV AND VARIATIONAL MODELS IN FRACTURE MECHANICS

- Mathematics
- 2008

We describe some applications of special functions of bounded variation to problems in fracture mechanics. 1. Free Discontinuity Problems. In the framework of Griffith’s theory of fracture mechanics,…

### 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.

### Continuum surface energy from a lattice model

- MathematicsNetworks Heterog. Media
- 2014

The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions using a new bond counting approach, which reduces the problem to the lattice point problem of number theory.

### Asymptotic analysis of Lennard-Jones systems beyond the nearest-neighbour setting: A one-dimensional prototypical case

- Mathematics
- 2016

We consider a one-dimensional system of Lennard-Jones nearest- and next-to-nearest-neighbour interactions. It is known that if a monotone parameterization is assumed then the limit of such a system…

### On a discrete-to-continuum convergence result for a two dimensional brittle material in the small displacement regime

- PhysicsNetworks Heterog. Media
- 2015

We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in…

### Microscopic fracture studies in the two-dimensional triangular lattice

- Physics
- 1976

In order to understand the static and dynamic bases of macroscopic fracture mechanics, we study flawed microscopic crystals obeying Newton's equations of motion. The particles in these crystals…