Buckling and metastability in membranes with dilation arrays

Abstract

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.

arXiv comment23 pages, 25 figures

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• Mechanics of pressurized cellular sheets
• Vibrations and tunneling of strained nanoribbons at finite temperature
• Chirality Effects in Molecular Chainmail
• Curvature as an external field in mechanical antiferromagnets
• Shape multistability in flexible tubular crystals through interactions of mobile dislocations
• Buckling in a rotationally invariant spin-elastic model