Formal justification of a continuum relaxation model for one-dimensional moir\'e materials

Mechanical relaxation in moir\'e 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 moir\'e lattice constant and $\delta$ is the ratio of the typical stacking energy to the monolayer stiffness.