We will present a theoretical analysis in a one-dimensional setting of a multiscale method coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). Such methods have been recently applied for the study of e.g. nanoindentation in materials.
We study both the general case of a convex energy and a specific example of a nonconvex energy, the Lennard-Jones case.
In the latter situation, we prove that the coupled model needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious effects may appear. We also consider the effect of the discretization of the continuum model on the behaviour of the coupled model.