Quasi-continuum (QC) methods are computational techniques, which reduce the complexity of atomistic simulations in a static setting while keeping information on small-scale structures and effects. The main idea is to couple atomistic and continuum models and thus to obtain quite detailed but still not too expensive numerical simulations.
We aim at a mathematically rigorous verification of QC methods by means of discrete to continuum limits. In this article we present our first results for the so-called quasi-nonlocal QC method in the context of fracture mechanics. To this end we start from a one-dimensional chain of atoms with nearest and next-to-nearest neighbour interactions of Lennard-Jones type. This is considered as a fully atomistic model of which the Γ-limits (of zeroth and first order) for an infinite number of atoms are known .
The QC models we construct are equal to this fully atomistic model in the atomistic region; in the continuum regime we approximate the next-to-nearest neighbour interactions by some nearest neighbour potential which is related to the so-called Cauchy-Born rule. Further we choose certain representative atoms in order to coarsen the mesh in the continuum region. It turns out that the selection of the representative atoms is crucial and influences the Γ-limits.
We regard a QC model as good if the Γ-limits of zeroth and first order or at least their minimal values and minimizers are the same as those of the fully atomistic model. Our analysis shows that, while in an elastic regime only the size of the atomistic region matters, in the case of fracture a proper choice of the representative atoms is an essential ingredient.