Separation of scales in Fracture mechanics. From molecular to continuum theory via Gamma convergence

We propose a procedure to obtain a consistent, mesh-objective, continuous model starting from chains composed of discrete springs exhibiting strain softening. Observing the size-dependent response of tensile chains and the corresponding scaling law, recent results for the variational convergence of discrete functionals (F convergence) are used to pass from a molecular to a continuum theory. The limit model, where softening and fracture are interpreted by the dichotomy of bulk and surface energies, reproduces the same overall properties of the discrete system. In particular, fracture energy does not vanish in the limit and the discrete approximations of the resulting continuum model are mesh objective.