Asymptotic analysis of microscopic impenetrability constraints for atomistic systems

In this paper we analyze a two-dimensional discrete model of nearest-neighbour Lennard-Jones interactions under the microscopical constraint that points on a lattice triangle maintain their order. This can be understood as a microscopical non-interpenetration constraint and amounts to the positiveness of the determinant of the gradient of the piecewise-affine interpolations of the discrete displacement. Under such a constraint we examine the continuum fracture energy deriving from a discrete-to-continuum analysis at a scaling where surface energy is preponderant. We give a lower bound by an anisotropic Griffith energy. This bound is optimal if the macroscopic displacement satisfies some opening-crack conditions on the fracture site. We show that if such conditions are not satisfied then the computation of the energy due to continuum cracks may involve non-local effects necessary to bypass the positive-determinant constraint on crack surfaces and at points where more cracks meet. Even when the limit fracture energy may be described by a surface energy density, this may depend on the crack orientation both in the reference and in the deformed configuration. While these effects lead to very interesting analytical issues, they call into question the necessity of the determinant constraint for fracture problems.