Formal justification of a continuum relaxation model for one-dimensional moiré materials

Mechanical relaxation in moiré materials is often modeled by a continuum model where linear elasticity is coupled to a stacking penalty known as the Generalized Stacking Fault Energy (GSFE). We review and compute minimizers of a one-dimensional version of this model, and then show how it can be formally derived from a natural atomistic model. Specifically, we show that the continuum model emerges in the limit $\epsilon \downarrow 0$ and $\delta \downarrow 0$ while holding the ratio $\eta :=\frac{{\epsilon }^2}{\delta }$ fixed, where $\epsilon$ is the ratio of the monolayer lattice constant to the moiré lattice constant and $\delta$ is the ratio of the typical stacking energy to the monolayer stiffness.