Rigidity and transient wave dynamics of random networks

Abstract

This thesis work deals with two major topics: the rigidity of random line networks and transient wave-front propagation in random discrete media. Rigidity means the ability of a mechanical system to store elastic energy when it is deformed. Random line networks composed of springs are found to be nonrigid at their basic configuration. This is confirmed both by a relaxation method and by a topological algorithm. These networks are found to become rigid at large strains for which the nonlinear stress-strain behaviour is analysed. A reinforced random line network is found to have a transition between rigid and nonrigid phases. The numerical evidence does not support a pure second-order phase transition. The velocity and amplitude of the leading front of elastic waves propagating in various one- and two-dimensional networks are studied. The leading front is defined in this work as the first displacement maxima of the points initially at rest in the network. In perfect lattices the amplitude first oscillates and then decays following a power law. With increasing disorder the early-time behaviour of the front changes to an exponential decay. The decay coefficient is a universal power-law as a function of the dilution parameter. Two limits of wave-front propagation dynamics are found. If the longitudinal and transverse velocities are equal, disorder is small, and wavelengths are large, the effective-medium approximation correctly estimates the wave-front behaviour. Conversely, if the two modes have different velocities, the system is more disordered, and the wavelengths are small, the propagation of the leading front takes place along effectively one-dimensional paths of propagation. In this limit the amplitude usually decays faster and the velocity is higher than in the effective-medium limit. Roughening of the leading wave front is analysed in both elastic percolation lattices and TLM lattices. Roughening means increase of the width of an interface as a function of time and the scale of observation. The kinetics of the roughening of the leading front is found to be partly described by a scaling argument based on Huygens' principle. Finally, conclusions of the results of this thesis are drawn and their implications discussed.