Evolution of granular materials under isochoric cyclic simple shearing

By means of 3D particle dynamics simulations, we analyze the microstructure of granular materials subjected to isochoric (constant volume) cyclic shearing, which drives the system towards a liquefaction state characterized by loops of jamming-unjamming transition with periodic loss of strength and irreversible accumulation of shear strain. We first show that the macroscopic response obtained by these simulations agrees well with the most salient features of the well-known cyclic behavior of granular materials both before and after liquefaction. Then we investigate the evolution of particle connectivity, force transmission, and anisotropies of contact and force networks. The onset of liquefaction is marked by partial collapse of the force-bearing network with rapid drop of the coordination number and nonrattler fraction of particles, and significant broadening of the contact force probability density function, which begins in the preliquefaction period. We find that the jamming transition in each cycle occurs for a critical value of the coordination number that can be interpreted as the percolation threshold of the contact network and appears to be independent of the initial mean stress, void ratio, and cyclic shear amplitude. We show that upon unjamming in each cycle an isotropic loss of contacts occurs and is followed by the development of high contact anisotropy and a large proportion of particles with only two or three contacts. The higher mobility of the particles also involves a lower degree of frustration of particle rotations and thus lower friction mobilization and tangential force anisotropy. These findings are relevant to both undrained cyclic deformations of saturated soils and rheology of dense non-Brownian suspensions where volume change is coupled with pore liquid drainage conditions.

Figure 1

Illustration of particle arrangements and boundary conditions for a sample composed of 8000 particles: (a) at the end of sample preparation; (b) during constant height cyclic shearing. The gray particles are glued to the top and bottom walls of the simulation cell.

Figure 2

Macroscopic response of isochoric cyclic simple shear test T2 in Table 1: (a) stress path; (b) stress-strain curve; (c) normalized mean stress evolution; (d) shear strain development.

Figure 3

Snapshot of normal forces in the sheared sample for characteristic state: (a) ${\text{A}}_0$; (b) ${\text{A}}_1$; (c) ${\text{B}}_1$; (d) ${\text{C}}_1$; (e) ${\text{C}}_2$; (f) ${\text{C}}_{2^{\prime }}$. Line thickness is proportional to the normal force at each contact. Color code represents the mobilized friction index $I_m$ (see text) in the range between 0 and 1. The same camera view as Fig. 1 is used here.

Figure 4

Evolution of (a) coordination number $z_{\text{g}}$ and (b) nonrattler fraction $f_{\text{NR}}$ for simulation T2. The horizontal dashed line $z_{\text{g}}\simeq 3.6$ refers to the inflection point of $z_{\text{g}}$.

Figure 5

Detailed evolutions of $z_{\text{g}}$ and $f_{\text{NR}}$, together with those of the shear stress $\tau$, normalized mean stress $p/p_0$ and shear strain $\gamma$, for simulation T2 during three selected cycles: (a) cycle A; (b) cycle B; (c) cycle C.

Figure 6

Connectivity diagram expressing the fractions $P_c$ of particles with exactly $c$ contacts for simulation T2 at the characteristic states of (a) cycle A; (b) cycle B; (c) cycle C.

Figure 7

Evolution of proportion $P_c$ of particles with $c$ contacts at characteristic states of (a) $\tau ={\tau }^{\text{amp}}$ and (b) $\tau \simeq 0$ transitioning from unloading to loading (or ${\text{C}}_2$ in postliquefaction period) for simulation T2.

Figure 8

Probability density functions $P_n$ of normal forces $f_n$ normalized by the mean normal force in log-linear [(a), (b), (c)], and log-log scales [(d), (e), (f)] for simulation T2 at characteristic states of cycle A [(a), (d)]; cycle B [(b), (e)]; cycle C [(c), (f)]. Note that plots for cycle C share the legend, and the vertical dash-doted, dashed, and dotted lines there refer to $\langle f_n\rangle$ at ${\text{C}}_{0^{\prime }}$ or ${\text{C}}_{2^{\prime }}, {\text{C}}_0$ or ${\text{C}}_2$, and ${\text{C}}_1$ or ${\text{C}}_3$, respectively.

Figure 9

Probability density functions of friction mobilization index for simulation T2 at characteristic states of (a) cycle A; (b) cycle B; (c) cycle C.

Figure 10

Normal contact orientation given the azimuthal angle $\phi$ and the angle $\theta$ defined by the projection of the contact direction on the shear plane of $xz$. The double-headed arrow represents the cyclic shear direction.

Figure 11

Polar representation of the functions (a) $P_n\left(\theta \right)$, (b) $\langle f_n\rangle \left(\theta \right)$ and (c) $\langle f_t\rangle \left(\theta \right)$ at selected characteristic states for simulation T2.

Figure 12

(a) Evolutions of the contact and force anisotropies at $\tau =\pm {\tau }^{\text{amp}}$ for simulation T2 and deviatoric stress ratio $q/p$ both measured from the simulation data and expressed as a function of the anisotropies as in Eq. (24). (b) Contributing weights of fabric and force anisotropies to the deviatoric stress ratio $q/p$: all normalized by $(2/5)a_c+(2/5)a_n+(3/5)a_t$.

Figure 13

Snapshots of (a) particle connectivity diagram, probability density functions of (b) normal forces and (c) mobilized friction index when shear stress reaches its maximum amplitude in the last three cycles of simulation T2.

Figure 14

Evolutions of $S_{\left[\right]}$ and anisotropies, along with comparisons between calculated $q/p$ and $\tau /p$ and their predicted values by Eqs. (24) and (23) for simulation T2: (a) cycle A; (b) cycle B; (c) cycle C.

Figure 15

Fabric anisotropy $a_c$ versus coordination number $z_{\text{g}}$ during the cyclic shearing for simulation T2. The colors represent the stress level according to the color scale. The highlighted path is a full shear cycle in the postliquefaction period.

Figure 16

Effects of (a) initial void ratio $e$, (b) initial mean stress $p_0$ and (c) cyclic stress ratio (CSR) on the evolution of coordination number $z_{\text{g}}$.

Figure 17

Evolution of the respective contributions of anisotropies to the stress ratio $q/p$ at $\tau =\pm {\tau }^{\text{amp}}$ and the theoretical value of $q/p$ according to Eq. (24) for (a) T1, (b) T3, (c) T4, (d) T5, (e) T6, and (f) T7.

Figure 18

(a) Particle connectivity diagram, probability density functions of (b) normal forces and (c) mobilized friction index when the shear stress reaches its maximum amplitude in the last simulated cycle.