Discrete and gradient Elasticity Cosserat Modeling of Granular Chains

International audienceThe present study theoretically investigates the free vibration problem of a discrete granular system. This micro structured system consists of uniform grains elastically connected by shear and rotation springs. Such a granular structural system is confined by discrete elastic interactions, to take into account the lateral granular contributions. This discrete repetitive system could be considered as a discrete Cosserat chain or a lattice elastic model with shear interaction. First, from the vibration equation governing the model, the natural frequencies are exactly calculated for the simply supports boundary conditions. Then it is shown that the discrete equations of this granular system, for an infinite number of grains, converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations of the granular system. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. Scale effects and wave dispersion characteristics of the granular chain are clearly captured by the continuous gradient elasticity model