Granular flow models: Analysis and numerical simulations

We study elastoplastic transitions in solutions of the antiplane shear model of granular flow, and describe a time-periodic solution that arises when the antiplane shear model is discretized in space. The antiplane shear model is a simplification of the continuum equations representing the flow of granular materials. The modeling of granular flow has many applications, from agricultural silos to geomechanics: improved accuracy in modeling will lead to safer and more economical designs for silos and industrial hoppers, and make oil drilling a more efficient process.;We construct approximate solutions to the antiplane shear model with piecewise linear initial data, which feature transitions between elastic and plastic states. These transitions travel with fixed speed. Numerical simulations demonstrate that the same elastoplastic transitions are the prominent features of the numerical solution.;The periodic solution of discretized antiplane shear appears at a critical value of the elasticity parameter for antiplane shear. The bifurcation to a periodic solution appears to be a Hopf bifurcation. The periodic solution contains elastoplastic transitions, as well as a shear band that appears over part of the period. Away from the shear band, the periodic solution has four distinct regions, three elastic and one plastic. Refinement of the spatial discretization further resolves these states.