Ordered patches cap the stability of disordered frameworks

Abstract

Structural disorder can stabilize frameworks against pressure-induced failure, yet the ordered regions it contains limit this gain. We show that, in rigid-jointed frames under hydrostatic pressure, an ensemble containing arbitrarily large patches of a crystal at positive density, whose centres carry a scaled crystal prestress, cannot stay linearly stable once those patches are stressed beyond the crystal's threshold for localizable instabilities. A homogeneous loss of positive definiteness while all Bloch modes, including the acoustic limit, remain stable cannot localize, leaving a window for stabilization. Extending an acoustic criterion to prestress, we show for all point groups that such a window requires a strain that keeps the crystal's symmetry. Unless the patches are shielded from stress, disorder cannot protect a crystal against a failure that breaks its symmetry, and protects a cubic crystal at most against volume collapse. In two-dimensional Truchet frames, planted ordered domains bring the threshold within 1% of the bound, and correlations short of long-range order localize failure on ordered clusters. Across 876 slender-strut nets, 798 fail first by a localizable instability; the other 78 have a median window of 7.6%. Stability gained through disorder is capped by the localizable instabilities of the local order it permits.