Motion of discrete interfaces in low-contrast periodic media

We study the motion of discrete interfaces driven by ferromagnetic interactions in a two-dimensional low-contrast periodic environment, by coupling the minimizing movements approach by Almgren, Taylor and Wang and a discrete-to-continuum analysis. As in a recent paper by Braides and Scilla dealing with high-contrast periodic media, we give an example showing that in general the effective motion does not depend only on the Γ-limit, but also on geometrical features that are not detected in the static description. We show that there exists a critical value δ˜ of the contrast parameter δ above which the discrete motion is constrained and coincides with the high-contrast case. If δ<δ˜ we have a new pinning threshold and a new effective velocity both depending on δ. We also consider the case of non-uniform inclusions distributed into periodic uniform layers.