Dynamic induced softening in frictional granular materials investigated by discrete-element-method simulation

A granular system composed of frictional glass beads is simulated using the discrete element method. The intergrain forces are based on the Hertz contact law in the normal direction with frictional tangential force. The damping due to collision is also accounted for. Systems are loaded at various stresses and their quasistatic elastic moduli are characterized. Each system is subjected to an extensive dynamic testing protocol by measuring the resonant response to a broad range of ac drive amplitudes and frequencies via a set of diagnostic strains. The system, linear at small ac drive amplitudes, has resonance frequencies that shift downward (i.e., modulus softening) with increased ac drive amplitude. Detailed testing shows that the slipping contact ratio does not contribute significantly to this dynamic modulus softening, but the coordination number is strongly correlated to this reduction. This suggests that the softening arises from the extended structural change via break and remake of contacts during the rearrangement of bead positions driven by the ac amplitude.

Figure 1

(Left) Representation of a sample, with large particles in a darker color. The packing is enclosed between two walls along the $y$ direction. (Right) The two horizontal sections in the sample enclose a local region where wall effects are minimal. The perturbation are movements of the top wall while the bottom wall adjusts to maintain constant loading stress.

Figure 2

Elastic and shear moduli $K$ and $G$ of the granular samples as a function of loading stress $\sigma$. The inset shows the compression and shear protocols.

Figure 3

Two series of frequencies sweeps obtained for $\sigma =55\text{kPa}$ (a) and $\sigma =695\text{kPa}$. Line color represents the drive strain of each sweep (from ${\varepsilon }_{\mathrm{drive}}=3.8\times {10}^{-7}$ in light green to ${\varepsilon }_{\mathrm{drive}}=4.8\times {10}^{-3}$ in dark blue). Harmonics are visible. Resonant frequencies decrease and peaks broaden as drive strains increase. As shown in Fig. 5, each figure corresponds to a different boundary condition.

Figure 4

The first modes are extracted from the frequency sweeps (Fig. 3) and reported in (a) as a function of local strain. When scaled according to predictions of Hertzian theory, resonances collapse onto primary curves (b). The two resonances correspond to different modes (Fig. 5). Resonant frequency remains constant for low drive strain, then decreases above a threshold value.

Figure 5

Each figure shows the strain profile of the sample vibrated at resonance. The first and second modes (a) and (b) of a low stress sample show a closed boundary at the top and open at the bottom. In the high stress sample (c), both boundaries are open.

Figure 6

Ratio of the average particle velocity $v_{\mathrm{local}}$ in the local layer to the maximal velocity of the wall $v_{\mathrm{drive}}$ under $\sigma =695\text{kPa}$. At low strain, the ratio of both velocities is roughly constant. At high strain, the local particle velocity decreases compared to drive velocity.

Figure 7

Coordination number $\langle Z_c\rangle$ (right-hand vertical axis, thick line), slipping contact ratio $\langle \text{SCR}\rangle$ (left-hand vertical axis, thin line), and normalized frequency spectrum (outlined in the background) in the linear (a), transition (b), and nonlinear (c) regions of the drive strain at loading stress $\sigma =695\text{kPa}$. Shifts in $\langle Z_c\rangle$ are below 0.01 for low drive strain. In the transition regime, Its average value increases as a result of perturbation up to a saturation value of 5.71. $\langle \text{SCR}\rangle$ increases peaks at resonant frequencies. In the nonlinear regime the system softens, and the coordination number decreases at resonance. Left axis shows the variation range for each case.

Figure 8

Coordination number at resonance for the first mode of the different packings. In all cases, the coordination number is constant at low strain and decreases at high strains.

Figure 9

(a) Resonant frequency of sample at $\sigma =695\text{kPa}$ for a range of local strains and three friction coefficients ${\mu }_s$. ${\mu }_s$ has no impact on the softening or the transition from linear to nonlinear. (b) Coordination number $\langle Z_c\rangle$ and (c) slipping contact ratio multiplied by the friction coefficient ${\mu }_s\langle \text{SCR}\rangle$ at resonance. The behavior of $\langle Z_c\rangle$ is similar and, at high friction, closely follows the shift in resonant frequency of (a). $\langle \text{SCR}\rangle$ is proportional to the local strain and to the friction coefficient both in the linear and nonlinear regimes.