# Shear-Jamming in Two-Dimensional Granular Materials with Power-Law Grain-Size Distribution

## Abstract

**:**

## 1. Introduction

**Figure 1.**Fragment of a satellite image of fragmented sea ice in the marginal ice zone off the Antarctic Peninsula (source: Landsat [18]).

## 2. Model Description

#### 2.1. Model Equations

#### 2.2. Model Configuration and Simulations

**Table 1.**Physical and numerical model parameters used in the simulations. GSD, grain-size distributions.

Parameter | Symbol | Value | Units |
---|---|---|---|

Grain density | ${\rho}_{i}$ | 910 | kg/m${}^{3}$ |

Friction coefficient | ${C}_{f}$ | 1.025 | kg/m${}^{2}$/s |

Disk thickness | ${h}_{i}$ | 1.5 | m |

Mean grain radius | $\overline{r}$ | 4.0 | m |

Exponent of the power-law GSD | α | 1.8 | — |

Elastic modulus | E | 9.0$\xb7{10}^{9}$ | Pa |

Poisson’s ratio | ν | 0.33 | — |

Static yield criterion | μ | 0.7 | — |

No. of grains | N | 20,000 | — |

Speed at the upper boundary | ${u}_{b}$ | 0.2–1.0 | m/s |

Time step | $\Delta t$ | $5\xb7{10}^{-4}$ | s |

## 3. Results and Discussion

**Figure 2.**Snapshots of contact networks in the modeled system in the unjammed (

**left**; $A=0.890$, ${u}_{b}=0.5$ m/s) and jammed (

**right**; $A=0.905$, ${u}_{b}=1.0$ m/s) state. For each grain, i, a line is drawn from its center to the center of the neighboring grain, j, if $j\in {C}_{i}\left(t\right)$. Grains belonging to the ‘frozen’ and moving boundaries are not shown.

#### 3.1. Shear Jamming: General Characteristics

**Figure 3.**Time series of the average contact number, ${\eta}_{c}$ (

**a**), force-network anisotropy ${\eta}_{a}$ (

**b**), pressure p (

**c**), shear stress τ (

**d**) and the principal angle, ${\theta}_{p}$ (

**e**), during simulations with: $A=0.908$ and ${u}_{b}=1.0$ m/s (blue); $A=0.905$ and ${u}_{b}=0.5$ m/s (black); $A=0.905$ and ${u}_{b}=0.2$ m/s (magenta).

**Figure 4.**Profiles of the average (

**a**) and standard deviation (

**b**) of the x-component of grain velocity in the function of the normalized y-distance ($y=0$ at the “frozen” boundary and $y=1$ at the moving boundary). Results obtained with ${u}_{b}=1.0$ m/s.

**Figure 5.**Normalized entropy E of the anomalies, ${\mathbf{u}}_{i}^{\prime}(\mathbf{x},t)$, in the function of the grain packing fraction, A (

**a**), and grain size (

**b**). On each box, the central mark is the median, the edges of the box are the 25th and 75th percentiles and the whiskers extend to the most extreme data points not considered outliers. In (b), for the two selected values of A (0.890 and 0.905), the statistics are calculated three times: for all N grains and for the subsets of the 10% largest and 10% smallest grains, respectively. Results obtained with ${u}_{b}=1.0$ m/s.

**Figure 6.**Snapshots of the modeled system for ${u}_{b}=1.0$ m/s and the packing fraction $A=0.809$ (

**a**) and $A=0.905$ (

**b**), showing the linear correlation coefficient, C, between the velocity anomalies, ${\mathbf{u}}_{i}^{\prime}(\mathbf{x},t)$, of a selected grain (dark brown, $C=1$) and all other grains in the system. C was calculated for a period of time equal to 100 minutes. Grains belonging to the “frozen” and moving boundaries are not shown.

**Figure 7.**The correlation coefficient, C, between the velocity anomalies, ${\mathbf{u}}_{i}^{\prime}(\mathbf{x},t)$, calculated for pairs of grains from a subset of the 10% largest (continuous lines) and 10% smallest (dashed lines) grains in the whole ensemble, in the function of the grain-grain distance. Results of simulations with ${u}_{b}=1.0$ m/s and $A=0.890$ (blue), $A=0.905$ (red).

#### 3.2. The Role of Polydispersity

**Figure 8.**pdfsof the correlation coefficient, C, between the velocity anomalies, ${\mathbf{u}}_{i}^{\prime}(\mathbf{x},t)$, calculated for pairs of grains from a subset of the 10% largest (blue) and 10% smallest (red) grains in the whole ensemble. Results of simulations with ${u}_{b}=1.0$ m/s and $A=0.890$ (

**a**), $A=0.905$ (

**b**).

**Figure 9.**Selected properties of the contact networks in the modeled system for a number of packing fractions, A: number of contacts of individual non-rattler grains, ${n}_{c,i}$ (

**a**); ${n}_{c,i}$ scaled with grain perimeter $2\pi {r}_{i}$ (

**b**); percentage of the simulation time when individual grains were non-rattler grains (

**c**); percentage of grains with at least three contacts (

**d**); average contact number ${\eta}_{c}$ (

**e**); and contact-network anisotropy ${\eta}_{a}$ (

**f**). In (

**d**–

**f**), the elements of the box symbols are the same as in Figure 5; they reflect the time variability of the analyzed variables during the simulation.

**Figure 10.**Selected properties of the contact networks in the bidisperse (BD) model case for a number of packing fractions, A: mean number of contacts of individual non-rattler grains, ${n}_{c,i}$, from the finer and coarser fractions (

**a**); percentage of the simulation time when individual grains from the finer (

**b**) and coarser (

**c**) fraction were non-rattler grains; percentage of grains with at least three contacts (

**d**); average contact number ${\eta}_{c}$ (

**e**); and contact-network anisotropy ${\eta}_{a}$ (

**f**). In (

**b**–

**f**), the elements of the box symbols are the same as in Figure 5; they reflect the time variability of the analyzed variables during the simulation.

**Figure 11.**Zoomed fragments of the modeled system (top: $A=0.890$; bottom: $A=0.905$) corresponding to the situations shown in Figure 2. Color scale: number of contacts, ${n}_{c,i}$, of individual grains (dark blue: zero; light blue: one; yellow: two; brown: three or more). Red lines: forces between grains with ${n}_{c}\ge 3$; black lines: the remaining forces.

**Figure 12.**Exceedance probability of the contact lifetime (in seconds) for two values of the packing fraction ($A=0.890$, dashed lines; $A=0.905$, continuous lines), calculated for contacts between the 10% largest (blue) and 10% smallest (red) grains.

**Figure 13.**pdfs of the maximum contact angle, ${\alpha}_{max}$, for two model runs: with bidisperse (BD; $A=0.816$) and power-law (PL; $A=0.905$) grain-size distribution. For the BD case, the pdfs are calculated separately for the coarser and finer grain fraction; for the PL case—for the subsets of 10% largest and 10% smallest grains. The vertical dotted and dashed lines mark the values ${\alpha}_{max}={120}^{\circ}$ and ${\alpha}_{max}={180}^{\circ}$, respectively (see the text for a description).

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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Herman, A.
Shear-Jamming in Two-Dimensional Granular Materials with Power-Law Grain-Size Distribution. *Entropy* **2013**, *15*, 4802-4821.
https://doi.org/10.3390/e15114802

**AMA Style**

Herman A.
Shear-Jamming in Two-Dimensional Granular Materials with Power-Law Grain-Size Distribution. *Entropy*. 2013; 15(11):4802-4821.
https://doi.org/10.3390/e15114802

**Chicago/Turabian Style**

Herman, Agnieszka.
2013. "Shear-Jamming in Two-Dimensional Granular Materials with Power-Law Grain-Size Distribution" *Entropy* 15, no. 11: 4802-4821.
https://doi.org/10.3390/e15114802