Models of defects in atomistic systems

Abstract

We analyze the overall behavior of discrete systems in the presence of defects (modeled by truncated quadratic potentials for some of the interactions) by exhibiting approximate free-discontinuity continuous energies. We give bounds on the limit surface energy densities, and we prove that these bounds are sharp in the classes of energy densities depending only on the normal to the discontinuity set, or concave and depending only on the jump across the interface. As a preliminary result we give a continuous descriptions of the ‘discrete Neumann sieve’

Quantitative analysis of finite-difference approximations of free-discontinuity problems

“…The main difference between the case n = 2 and n = 3 (and higher) is related to the problem of describing the structure of the sets of lattice sites where the parameter v is close to 0, which approximates the set jump S u . In principle, if that discrete set presents "holes" the limit surface energy may depend on the values u ± of u on both sides of S u (see [29]). In two dimensions this is ruled out by showing that such lattice sets can be locally approximated by a continuous line.…”

A Bridging Mechanism in the Homogenization of Brittle Composites with Soft Inclusions

“…Indeed, explicitly depending on the crack-opening [u], they take into account that fracture is a gradual process due to the fact that atomic bonds stretch before breaking. In a discrete-to-continuum setting, cohesive surface energies have been derived by means of Γ-convergence in [10] starting from one-dimensional discrete systems and in [12] by mixing quadratic and defected springs. Moreover, cohesive-type models have been obtained in e.g.…”