Continuum models in dynamics of granular media. Review

Mechanical models and equations of state used for solving dynamic problems in mechanics of granular media are reviewed. We study elastic, hypoelastic, hyperelastic, elastoplastic, and hydrodynamic models. Within the framework of hyperelasticity several generalized models, including the modified Arruda-Boyce model and some other hyperelastic models, are examined to investigate the formation of hysteresis loops during cyclic loading. We find that due to the switching mechanism the hysteresis loops can also be studied using the concept of hyperelasticity. The elastic-plastic models with one or several plastic flow surfaces, as well as the models with continuously distributed flow surfaces (microplasticity), are analyzed. Some newly developed models without plastic flow surfaces are considered (hypoplasticity, barodesy). Special attention is given to the elastic-plastic models with isotropic hardening and the models used to analyze the formation of hysteresis loops. The elastic-plastic models for studying non-degenerate hysteresis loops under cyclic loading conditions are examined as well. The comparative analysis of elastic, hyperelastic, hypoelastic and various elastic-plastic models reveals that under quasistatic cyclic loading conditions the cam-clay and other critical state models that take into account both the hardening and softening phenomena in real granular materials appear to be the most suitable ones for modeling hysteresis loops in cohesionless granular materials. However, the study of non-stationary dynamical processes requires finding appropriate models. For such problems, the combination of hyperelastic and hypoelastic models provides a simple adequate tool for taking into account the nucleation effects associated with the formation of local discontinuities. This is mainly due to the capabilities of hyper- and hypoelastic models to account for variations in elastic parameters and, consequently, in elastic wave velocities under wave propagation. In the future we plan to combine hyper- or hypo-elastic models with cam-clay or other critical state models.