<p>This article explores the hysteretic behavior and the damping features of sheared granular media using discrete element method (DEM) simulations. We consider polydisperse non-cohesive frictional spherical particles, enclosed in a container with rigid but moving walls, subjected to a cyclic simple shear superimposed on a confining pressure. The mechanical response of the grains is analyzed in the permanent regime, by fitting the macroscopic stress–strain relation applied to the box with a Dahl-like elasto-frictional model. The influence of several parameters such as the amplitude of the strain, the confining pressure, the elasticity, the friction coefficient, the size and the number of particles are explored. We find that the fitted parameters of our macroscopic Ansatz rely qualitatively on both a well-established effective medium theory of confined granular media and a well-documented rheology of granular flow. Quantitatively, we demonstrate that the single degree-of-freedom elasto-frictional reduced model reliably describes the nonlinear response of the granular layer over a wide range of operating conditions. In particular, we show that the mechanical response of a granular slab under simple shear depends on a unique dimensionless parameter akin to an effective Coulomb threshold at low shear/high pressure. Furthermore, exploring higher shear/lower pressure, we evidence optimal damping at the crossover between a loose unjammed regime and a dense elastic regime.</p> Graphical Abstract <p></p>