Buckling and Metastability in 2D Impurity Arrays

We study a periodic array of impurities that produce local dilations, embedded in a two-dimensional crystalline solid that can buckle out of the plane. These arrays provide a simple elastic model of shape memory. As the size of each impurity increases (or the relative cost of bending to stretching decreases), it becomes energetically favorable for the impurities to buckle either up or down, allowing for a vast number of metastable states. Using discrete simulations and continuum theory, we consider the buckling of isolated impurities as well as impurity arrays, guided by an analogy to the Ising antiferromagnet. We characterize the buckling transition and conjecture ground states for systems with triangular and square lattice microstructures.