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

¹ Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy.
² Dipartimento di Matematica, Università di Pisa, Pisa, Italy.
gelli@dm.unipi.it

Received: 3 October 2016
Revised: 13 March 2017
Accepted: 14 March 2017

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.