Dispersion and wave propagation in discrete and continuous models for granular materials

A generalised continuum model for granular media is derived jy direct homogenisation of the discrete equations of motion. In contrast to previous works on this topic, continuum concepts such as stress and moment stress are introduced after homogenisation. First, a very simple one-dimensional model is considered and the continuum version For this model is derived by replacing the difference quotients of the discrete model by differential quotients. The dispersion relations of the discrete and the continuous model are derived and compared. Variational boundary conditions for the continuous model are deduced from the stationarity of the corresponding Lagrangian. The three-dimensional case is treated in an essentially similar fashion. The resulting continuum theory is a combination of a Cosserat Continuum and a higher-order deformation gradient continuum. The salient features of the theory are illustrated by means of the dispersion relations for planar wave propagation. Copyright (C) 1996 Elsevier Science Ltd.