  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.
