  The aim of this work is to provide further insight into the qualitative
behavior of mechanical systems that are well described by Lennard-Jones type
interactions on an atomistic scale. By means of $\Gamma$-convergence
techniques, we study the continuum limit of one-dimensional chains of atoms
with finite range interactions of Lennard-Jones type, including the classical
Lennard-Jones potentials. So far, explicit formulae for the continuum limit
were only available for the case of nearest and next-to-nearest neighbour
interactions. In this work, we provide an explicit expression for the continuum
limit in the case of finite range interactions. The obtained homogenization
formula is given by the convexification of a Cauchy-Born energy density.
  Furthermore, we study suitably rescaled energies in which bulk and surface
contributions scale in the same way. The related discrete-to-continuum limit
yields a rigorous derivation of a one-dimensional version of Griffith' fracture
energy and thus generalizes earlier derivations for nearest and next-to-nearest
neighbors to the case of finite range interactions.
  A crucial ingredient to our proofs is a novel decomposition of the energy
that allows for refined estimates.
