Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1903-1929.

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.

DOI : 10.1051/m2an/2017011
Classification : 49J45
Mots clés : Variational theory of fracture, discrete-to-continuum analysis, Γ-convergence, Lennard−Jones potentials
Braides, Andrea 1 ; Gelli, Maria Stella 2

1 Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy.
2 Dipartimento di Matematica, Università di Pisa, Pisa, Italy.
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Braides, Andrea; Gelli, Maria Stella. Analytical treatment for the asymptotic analysis of microscopic impenetrability constraints for atomistic systems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 5, pp. 1903-1929. doi : 10.1051/m2an/2017011. http://www.numdam.org/articles/10.1051/m2an/2017011/

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