Shear vibration modes of granular structures: Continuous and discrete approaches

This work is dedicated to the study of shear vibration modes of one-dimensional granular microstructured beams using a discrete Cosserat model. The dynamic response of the one-dimensional granular beam (discrete beam) is investigated for various boundary conditions, through the analytical resolution of an exact discrete eigenvalue problem. For an infinite number of grains, the dynamic behavior of the discrete problem converges towards a Bresse-Timoshenko continuum beam model. First, the effective modeling parameters, the constitutive equations and the governing equations of motion are derived in a difference form. For simply supported granular beams, the appearance of pure shear modes of vibration is highlighted (vibration modes without deflection), a phenomenon which is already known in the free vibration of a continuous Bresse-Timoshenko beam problem. The eigenfrequencies of these pure shear modes for the discrete granular beam coincide with the critical frequencies of the discrete system. These pure shear modes may appear for granular beams composed of few grains, which give a physical support of a phenomenon well studied in the literature from a continuum beam model. The eigenfrequencies of the discrete granular beam are also calculated for various boundary conditions, including clamped, hinge and free end boundary conditions. Based on the discrete element method (DEM), several numerical simulations of dynamic tests in some representative examples have been performed. The responses of the numerical approach are accurately compared against the exact results of the analytical solutions based on the difference eigenvalue problem, which illustrates the relevance of the proposed approach.