Exact solutions for the vibration of finite granular beam using discrete and gradient elasticity cosserat models

Abstract

The present study theoretically investigates the free vibration problem of a discrete granular system. This problem can be considered as a simple model to rigorously study the effects of the microstructure on the dynamic behavior of the equivalent continuum structural model. The model consists of uniform grains confined by discrete elastic interactions, to take into account the lateral granular contributions. This repetitive discrete system can be referred to discrete Cosserat chain or a lattice elastic model with shear interaction. First for the simply supported granular beam resting on Winkler foundations, due to the critical frequencies which concern the nature of the dynamic results, the natural frequencies are exactly calculated, starting from the resolution of the linear difference eigenvalue problem. The natural frequencies of such a granular model are analytically calculated for whatever modes. It is shown that the difference equations governed to the discrete system converge to the differential equations of the Bresse-Timoshenko beam resting on Winkler foundation (also classified as a continuous Cosserat beam model) for an infinite number of grains. A gradient Bresse-Timoshenko model is constructed from continualization of the difference equations. This continuous gradient elasticity Cosserat model is obtained from a polynomial or a rational expansion of the pseudo-differential operators, stemming from the continualization process. Scale effects of the granular chain are captured by the continuous gradient elasticity model. The natural frequencies of the continuous gradient Cosserat models are compared with those of the discrete Cosserat model associated with the granular chain. The results clarify the dependency of the beam dynamic responses to the beam length ratio.