Network disorder and nonaffine deformations in marginal solids

The most profound effect of disorder on the elastic response of solids is the nonaffinity of local displacements whereby the atoms (particles, network junctions) do not simply follow the macroscopic strain, as they do in perfect crystals, but undergo additional displacements which result in a softening of response. Whether disorder can produce further effects has been an open and difficult question due to our poor understanding of nonaffinity. Here we present a systematic analysis of this problem by allowing both network disorder and lattice coordination to vary continuously under account of nonaffinity. In one of its limits, our theory, supported by numerical simulations, shows that in lattices close to the limit of mechanical stability the elastic response stiffens proportionally to the degree of disorder. This result has important implications in a variety of areas: from understanding the glass transition problem to the mechanics of biological networks such as the cytoskeleton.