Local fluctuations and spatial correlations in granular flows under constant-volume quasistatic shear

We investigate the local fluctuations in dense granular media subjected to athermal, quasistatic shearing, based on three-dimensional discrete element method simulations. By shearing granular assemblies of different polydispersities under constant-volume constraint, we quantify the characteristics of local structures (in terms of local volume and local anisotropy) and local deformation (using local shear strain and nonaffine displacement). The distribution of the local volume in a granular medium is found unchanged during the entire shearing process, which indicates a constant temperaturelike compactivity for the material. The compactivity is not, however, equilibrated among different particle groups in a polydisperse assembly. The local structures of a disordered granular assembly are inherently anisotropic. The fluctuations in local anisotropy can be well captured by a gamma or mixed-gamma distribution function, which is also unchanged during the shear. The local anisotropic orientation evolves towards the coaxial direction of the stress anisotropy with shear. The deformation characteristics of a jammed granular medium have their origins in the structural amorphousness. The local shear strain field depicts clear shear transformation zones which act as plasticity carriers. The spatial correlation of the local shear strains exhibits a fourfold pattern which is stronger in the stress deviatoric planes than in the stress isotropic plane. The fluctuations of nonaffine displacement suggest an isotropic granular temperature and an isotropic spatial correlation independent of the stress state. Both the local strain and the nonaffine displacement exhibit a power-law decayed distribution with a long-range correlation. We further modify the shear-transformation-zone theory to predict the pressure-dependent constitutive behavior of a sheared granular material and compare its prediction with our simulation data.

Figure 1

Illustration of constant-volume triaxial shearing: (a) before and (b) after shearing.

Figure 2

Stress evolution of the four assemblies under constant-volume triaxial shearing. The inset is a zoom-in at small strains, where the star marks the initial yield point.

Figure 3

Radical Voronoi tessellation in (a) two and (b) three dimensions. Radical planes are shown as red lines in (a) and transparent glasses in (b). $\Delta \mathbf{x}$ and $\Delta \mathbf{u}$ are the relative position and displacement between the two neighboring particles, respectively. (c) A Voronoi cell $K$ with surface $\partial K$ and outward normal $\mathbf{n}$.

Figure 4

Radial distribution function for the four assemblies at three different shearing levels. The measured radial distance interval is 0.1 mean particle diameter. The insets show the color map resolving of $g\left(\mathbf{r}\right)$ onto three orthogonal planes at ${\epsilon }_y=45\%$. The presented distance range in the insets is $x,y,z/\langle 2R\rangle =\pm 2$.

Figure 5

Local density distribution for the four assemblies at different shearing levels. GDF is used to fit the data in MN; while in polydisperse assemblies, a mixed-GDF is used [44]. The dotted curves show the contributions from individual groups. In the insets, the quantity is normalized by its mean value. And the distributions for different groups are distinguished. ${\varphi }_L^{-1}\approx 1.325$ corresponds to the theoretical largest volume fraction achievable in a random monodisperse assembly [43].

Figure 6

(a) $k$-gamma fitting of normalized local density. The collapsed symbols include data from all the eight groups in the four assemblies and at three different shearing levels (${\epsilon }_y=0,22.5\%$, and $45\%$). (b) Evolution of $\chi$ measured in the four assemblies for different groups, and the mean and standard deviation (shown as error bar) of $\chi$ throughout the shearing process. $\chi$ is in the unit of volume and is normalized by the volume of the smallest particle $V_s\approx 0.0335{\mathrm{mm}}^3$.

Figure 7

Probability distribution of the local anisotropic intensities for the four assemblies at different shearing levels. GDF is used to fit the data in MN, while mixed-GDF is used for the other cases. The insets distinguish the distributions for different groups.

Figure 8

Rose diagram of the spatial distribution of the local anisotropic orientations in TRI at the initial stress isotropic state (${\epsilon }_y=0$) and the sheared steady state (${\epsilon }_y=45\%$). The overlaid transparent membranes are the corresponding Fourier approximations.

Figure 9

Distribution of local deviatoric strain ${\epsilon }_L^q$ in TRI within the applied deformation interval $\Delta {\epsilon }_y=0.9\%$. ${\epsilon }_y=0,22.5\%$, and $45\%$ are three reference states. The color maps show the $x$-$y$ plane resolving of ${\epsilon }_L^q$ for a slice with a thickness of four mean particle diameter. Circled regions are identified as STZs.

Figure 10

Spatial correlation of local shear strains at steady state in BI21 resolved onto three orthogonal cross sections. The global strain interval is $0.9\%$. The cross-sectional slices are reconstructed with a thickness of three mean particle diameter. The measured distance interval is 0.5 mean particle diameter.

Figure 11

Distribution of local nonaffine displacement $\left|\tilde{\mathbf{u}}\right|$ in TRI at different reference states within a same global deformation interval $\Delta {\epsilon }_y=0.9\%$. The color maps show the $x$-$y$ plane resolving of $\left|\tilde{\mathbf{u}}\right|$. The slice is the same as that in Fig. 9. Spots undergoing large nonaffine displacements are highlighted with arrows.

Figure 12

Probability distribution of the nonaffine displacements for the four assemblies in the $x$, $y$, and $z$ directions and at different shearing levels. The $q$-Gaussian distribution is used to fit the data with the probability density function $\propto {[1-b(1-q){\left(\frac{{\tilde{u}}_{*}-\langle {\tilde{u}}_{*}\rangle }{{\sigma }_{{\tilde{u}}_{*}}}\right)}^2]}^{1/(1-q)}$. The fitted values of the two parameters $q$ and $b$ are given in the figure.

Figure 13

Spatial correlation of nonaffine displacements at steady state in BI21 resolved onto three orthogonal cross sections. The measurement is the same as that described in Fig. 10.

Figure 14

Pearson correlation among different local measurements in the four assemblies.

Figure 15

Comparison of the STZ model predictions of (a) deviatoric stress and (b) mean stress against the DEM data. The curves for the different cases are shifted upwards with an interval of 200 kPa for clarity of presentation.