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 microscopic impenetrability constraint. As the lattice parameter goes to 0, we prove that any limit deformation with finite energy is piecewise rigid and we prove a general lower bound with a suitable Griffith-fracture energy density which reflects the anisotropies of the underlying triangular lattice. For such a continuum energy we also provide a class of (piecewise rigid) deformations satisfying "opening-crack" conditions on which the lower bound is sharp. Relying on these results, some consequences have been already presented in the companion paper [A. Braides et al., J. Mech. Phys. Solids 96 (2016) 235-251] to validate models in Computational Mechanics in the small-deformation regime.

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

Gelli, Maria Stella
2017-01-01

Abstract

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 microscopic impenetrability constraint. As the lattice parameter goes to 0, we prove that any limit deformation with finite energy is piecewise rigid and we prove a general lower bound with a suitable Griffith-fracture energy density which reflects the anisotropies of the underlying triangular lattice. For such a continuum energy we also provide a class of (piecewise rigid) deformations satisfying "opening-crack" conditions on which the lower bound is sharp. Relying on these results, some consequences have been already presented in the companion paper [A. Braides et al., J. Mech. Phys. Solids 96 (2016) 235-251] to validate models in Computational Mechanics in the small-deformation regime.
2017
Braides, Andrea; Gelli, Maria Stella
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/891313
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