  We study periodic arrays of impurities that create localized regions of
expansion, embedded in two-dimensional crystalline membranes. These arrays
provide a simple elastic model of shape memory. As the size of each dilational
impurity increases (or the relative cost of bending to stretching decreases),
it becomes energetically favorable for each of the $M$ impurities to buckle up
or down into the third dimension, thus allowing for of order $2^M$ metastable
surface configurations corresponding to different impurity "spin"
configurations. With both discrete simulations and the nonlinear continuum
theory of elastic plates, we explore the buckling of both isolated dilations
and dilation arrays at zero temperature, guided by analogies with Ising
antiferromagnets. We conjecture ground states for systems with triangular and
square impurity superlattices, and comment briefly on the possible behaviors at
finite temperatures.
