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}
}

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References

SHOWING 1-10 OF 41 REFERENCES

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

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

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

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

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

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

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

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

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

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

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