Elastic anomalies in disordered square networks. (22nd April 2015) - Moukarzel, Cristian F
Abstract: The compressive elastic modulus B of a square network with an amount ϵ of positional disorder, which is a simple structural model of isostatic networks such as glasses, is studied numerically under fixed (FBC) and periodic (PBC) boundary conditions. Under PBC, anomalous properties are found and compared with results for FBC. It is already known for isostatic networks, that B is finite and size-independent when ϵ  = 0, but goes to zero with increasing size for nonzero disorder, in a manner that depends on boundary conditions. It is reported here that, under FBC, B is constant for L  <  L 0 ( ϵ ) and decays as 1/ L for L  >  L 0 . For PBC, B  ∼ 1/ L when L  <  L 0 and B  ∼ 1/ L 2 for L  >  L 0 . It is shown how these large-size behaviors for both FBC and PBC can be understood using an extension of previously published arguments. The crossover length L 0 ( ϵ ) is found to behave in both cases as 1/ ϵ 2 and a justification for this behavior is provided. Additionally, the case of PBC shows surprising properties, which do not admit a simple explanation, such as: (a) B (PBC) ( ϵ, L ) is a discontinuous function of disorder strength ϵ, for all sizes L, since it is constant for zero disorder but decays as 1/ L in the limit ϵ  → 0 and (b) the amount of site-displacement due to compression, while being exactly zero for ϵ  = 0 (ordered square networks), behaves as 1/ ϵ 2 for nonzero disorder. These puzzling properties are due to the existence of degenerate flexes in the undistorted network with PBC, which do not exist for FBC. The undistorted square network with PBC (but not with FBC) is thus unstable under compression, within nonlinear elasticity, which makes it inappropriate as a model to study the static and/or dynamic properties of disordered isostatic networks. Key ideas to understand these anomalies are advanced, leaving a detailed analytical treatment for a forthcoming publication.