Elasto-frictional reduced model of a cyclically sheared container filled with particles

TL;DR

This work addresses how cyclic simple shear of confined granular media dissipates energy and exhibits hysteresis. By coupling a Dahl elasto-frictional term with a linear elastic component and performing dimensional analysis, the authors show that the collective response reduces to two non-dimensional parameters, $\xi$ and $\Psi$, with a single dominant threshold $\xi$ governing the hysteretic regime. The reduced description closely matches DEM results across a wide range of particle properties, pressures, and strain amplitudes, and reveals that damping is maximized at intermediate $\xi$ values, making confinement pressure a practical tuning knob. The findings connect microscopic frictional dissipation and Hertzian elasticity to macroscopic damping performance via effective-medium theory, enabling efficient design of granular dampers and granular-structure devices.

Abstract

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, submitted to a cyclic simple shear superimposed to 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 qualitatively rely 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 an unique dimensionless parameter, akin to an effective Coulomb threshold, at low shear/high pressure. Further, exploring higher shear/lower pressure, we evidence an optimal damping at the crossover between a loose unjammed regime and a dense elastic regime.

Figures (12)

• Figure 1: (top) Granular sample during (left) initial compression and (right) cyclic shear. (a) Imposed shear strain, (b) shear stress and (c) hydrostatic pressure, as functions of time for two values of the volume fraction $\phi = 0.64$ (red) and $\phi = 0.70$ (blue) at $\gamma_m=0.15$. Stress-strain curves for (d)$\phi = 0.70$ and (e)$\phi = 0.64$ at various amplitudes: $\gamma_m=0.05$, $0.10$ and $0.15$.
• Figure 2: (a,b) Spectra of the shear strain and the pressure fluctuation and (c,d) pressure fluctuation as a function of the shear strain, for the two examples shown in Fig. \ref{['fig:1_sketchs_examples']}: (a,c)$\phi = 0.70$ and (b,d)$\phi = 0.64$, both at at $\gamma_m=0.15$. Data in (c,d) are compared to a quadratic behavior, with (c)$R = 7.54$ GPa and (d)$R = 4.49$ GPa, where $\Delta P=R\gamma_m^2/2$ is the peak amplitude of the pressure fluctuations, see Eq. \ref{['eq:Reynolds_pressure']}.
• Figure 3: (a,b,c) Maps of the RMSE between $\tau(\mu_D,G_D,G_E)$ and $\tau_\mathrm{DEM}$ plotted in the three planes $(\mu_D,G_E)$, $(\mu_D,G_D)$ and $(G_D,G_E)$, for the example $(\phi,\gamma_m)=(0.70,0.15)$ shown in Fig. \ref{['fig:1_sketchs_examples']}d. The parameters are scaled by the optimal triplet $\mu_D^\mathrm{opt}=0.24$, $G_D^\mathrm{opt}=9.7$ GPa and $G_E^\mathrm{opt}=2.3$ GPa. (d) Stress-strain relation from DEM simulation superimposed on the fits. The purple dotted line represents the hysteretic Dahl contribution $\tau_D(\gamma)$, the solid blue line represents the linear contribution $G_E\gamma$, see Eq. \ref{['eq:model_reduit']}, and the red dashed line is the sum of the two.
• Figure 4: (a): Dimensionless fitted parameters $G_D/k_n$ (open symbols) and $G_E/k_n$ (solid symbols) for various volume fractions $\phi$ and elasticities $E_p$ as functions of the dimensionless pressure $P/k_n$ at $\gamma_m=0.15$. (b): Dahl friction coefficient versus dimensionless pressure for the same data set. (c): Dissipated energy per cycle (open symbols) and maximum stored energy (solid symbols) computed from the hysteresis response using Eqs. \ref{['eq:E_diss']} and \ref{['eq:E_stored']} respectively, as functions of the dimensionless pressure. (d): Loss factor computed from \ref{['eq:loss_factor']}, as a function of the dimensionless pressure. Legend symbols are common for (a-d).
• Figure 5: (a): Total shear modulus $G_T=G_D+G_E$ versus effective shear modulus $G_\mathrm{EMT}^\infty$ (Eq. \ref{['eq:EMT_Ginfty']}) with a linear fit of slope $\alpha = 0.687 \pm 0.073~(10.6\%)$. The same legend as in Fig. \ref{['fig:4_parametric_results']}. (b,c): identified Dahl friction coefficient $\mu_D$ and coefficient $\alpha$ as functions of the inter-particle friction $\mu_p$.