Volume 51, Number 5, September-October 2017
|Page(s)||1903 - 1929|
|Published online||15 November 2017|
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.
Received: 3 October 2016
Revised: 13 March 2017
Accepted: 14 March 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 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.
Mathematics Subject Classification: 49J45
Key words: Variational theory of fracture / discrete-to-continuum analysis / Γ-convergence / Lennard−Jones potentials
© EDP Sciences, SMAI 2017
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