# Discrete Element Simple Shear Test Considering Particle Shape

^{1}

^{2}

^{*}

## Abstract

**:**

^{3D}) to simulate the simple shear test, and ellipsoidal particles with different aspect ratios were prepared to study the effects of particle shape on the mechanical behavior and fabric evolution of granular materials under complex stress paths. The numerical results show that the particle shape has a significant effect on the peak strength, dilatancy, non-coaxiality, and other mechanical properties of granular materials. The contact fabric evolves from orthotropy to transverse isotropy under the principal stress axes rotation. This paper will provide a reference for natural granular materials with different shapes in the study of mechanical behavior and the micro-constitutive model.

## 1. Introduction

## 2. Numerical Simulation

#### 2.1. Simple Shear Test Model

#### 2.2. Sample Preparation and Parameter Setting

^{3D}was employed to generate particles of various shapes with different aspect ratio AR. The smaller AR is closer to the rotation of the sphere, and the effect of the particle shape seems to not be obvious, while the larger AR of the particle will lead to the self-locking of the particle. Therefore, concerning the relatively reliable design, [18] we set the AR = a:b:c to 1.3:1:1, 1.6:1:1, and 1.9:1:1, respectively. The particle size ranges from 1 to 3 mm, and the sample’s height and radius are 17 mm and 30 mm, respectively. Figure 2a,b shows the particle’s geometric shape and internal filling. Ellipsoidal particles pack various sizes of spherical particles. The linearpbond contact model employed between the spherical particles and the microscopic mechanism is shown in Figure 2c. Pieces 1 and 2 in the contact model can bear and transmit shear stress, tension, and torque by parallel contact, which may represent the actual stress in granular material particles. Compared to the linear contact model, the stiffness of the parallel bond model is divided into two parts: the linear normal stiffness ${K}_{n}$ and tangential stiffness ${K}_{s}$, and the bond stiffness $\overline{{K}_{n}}$ and $\overline{{K}_{\mathrm{s}}}$. When the external stress surpasses the given limit, the parallel bond model will degenerate into a linear model. As a result, several researchers utilize the model to simulate particle breakage [29,30,31]. The simulation in this paper concentrates on particle shape’s effect and does not consider particle breakage, there is no relative deformation between the balls inside the particles, and only a linear contact model between the particles is set.

^{2}, and the particle’s natural deposition process to reach the particles balanced. During sample preparation, the particles naturally deposit horizontally. The long axis of the particles is randomly dispersed in the horizontal direction, and some particles are not horizontal. As a result, constant vertical pressure is supplied to the model to replicate the consolidation of geomaterials under hydrostatic pressure, as shown in Figure 2d. After the sample particles are balanced under the consolidation stress conditions, the rigid wall and lade plane in Figure 2d will be deleted, to generate the laminar sidewall and top wall in Figure 1c at the same position. The sample will be sheared at a specific horizontal speed until the shear strain reaches 25%; to reduce the dynamic influence between model elements [25], the horizontal speed of the laminar sidewall is set according to the method described in Figure 1c, and is converted to the angular speed of the rotation of the vertical wall shown in Figure 1a, which is 2 × 10

^{−4}rad/s.

#### 2.3. Test Scheme and Variable Monitoring

^{3D}.

^{3D}. The measurement of contact fabric is complicated, which can be quantified by fabric tensor and used to describe anisotropy. As in Oda [32], given the fabric tensor expression as shown in Equation (2), $n$ is the unit vector of the contact normal on the integral interval $\mathsf{\Omega}=4\pi $, the physical meaning is shown in Figure 3a. ${n}_{i}$ and ${n}_{j}$ are the components of the unit vector $n$ along three axes, $E(n)$ is the probability density function of the spatial distribution of unit vector $n$, and it satisfies ${\int}_{\mathsf{\Omega}}E(n)\mathrm{d}\mathsf{\Omega}=1$.

_{1}, F

_{2}, and F

_{3}represent the fabric components in three orthogonal directions, it represents the difference of geomaterial properties in various directions. When F

_{1}= F

_{2}= F

_{3}, it shows that the material is isotropic and exhibits the same stress–strain characteristics when loaded in different directions. When the three components are different, the deformation in various directions is different under the same stress condition. To characterize this difference in theoretical calculation, different strain components are often obtained by combining with the fabric tensor, which reflects the anisotropy of geomaterials. Where a

_{1}and a

_{2}are the anisotropic amplitude parameters on the two orthogonal planes in the Cartesian coordinate system, and the values range from 0 to 1, the value of ${a}_{i(i=1,2,3)}$ can be used to represent the difference of material properties in different directions and can measured through experiment. a is the anisotropic parameter of the transverse-isotropic fabric (Equation (4)), and the value ranges from 0 to 1. The fabric is isotropic when a = 0 and anisotropic when a ≠ 0. The expression of ${a}_{i(i=1,2,3)}$ is shown in Equation (5), N is the number of unit vectors and $n$ in the measurement range. ${\theta}_{1}^{(k)}$ and ${\alpha}^{(k)}$ are the angles between the k th unit vector $n$ and the x

_{1}and x

_{3}axes, as shown in Figure 3b. In the shear process of discrete element samples, the unit vector of normal contact between particles is obtained at each timestep. The evolution of particle contact fabric can be investigated with Equations (3) and (5), which simplifies the complex process of fabric calculation and makes the physical meaning clearer.

## 3. Results and Analysis

#### 3.1. Stress–Strain Relationship

_{xy}-γ curves with different vertical stress and particle shapes that indicate strain hardening, in the legend, a, b, and c represent the axial lengths of the three orthogonal directions of the particles, as shown in Figure 2a. Figure 5 shows volumetric compression, which is similar to the result observed in Li’s [34] simple shear tests on loose and medium-density sand. Peak stress and shear modulus increase as the particle aspect ratio increases, which has a noticeable effect on stress–strain. The shear stress σ

_{xy}and shear modulus are maximal when AR = 1:1:1.9. The effect of particle shape on shear modulus becomes increasingly apparent as confining pressure increases, and the disparity in shear modulus of samples is higher. Under low pressure, particle shape has little effect on the volumetric strain. As confining pressure increases, the effect of particle shape is more prominent, and the volumetric contraction becomes increasingly apparent as the particle aspect ratio AR increases.

#### 3.2. Micro Mechanical Properties

_{ij}as confining pressures and aspect ratios alter. When combined with Figure 8, it is clear that ellipsoidal particles are orthotropic initially. F

_{1}and F

_{3}increasingly coincide as the principal stress rotates, suggesting that the particles are rearranged after being subjected to force and are close to isotropy on the action surface with major principal stress. Compared to the T01–T03 (or T04–T06, T07–T09) test group, inherent anisotropy is more apparent as the aspect ratio increases, and the evolution of fabric is slightly slower. When the aspect ratio AR is constant, the evolution of the fabric is delayed with increasing vertical pressure, as illustrated in Figure 8a,d,g, a similar evolution pattern can be seen in Figure 8b,e,h.

_{1}and F

_{2}(F

_{3}) reduces when the principal stress rotates, similarly with ellipsoidal particles. The stress–strain relationship is noncoaxial due to fabric anisotropy [35]. However, the anisotropy of the fabric after evolution is no longer noticeable in the later stages of loading, and the influence on the macroscopic mechanical characteristics is no longer significant [36]. The stress and strain rate directions are coaxial in the high stress ratio area.

#### 3.3. Noncoaxiality

## 4. Conclusions

^{3D}to model a three-dimensional simple shear test and integrated particle morphologies. In the loading, the principal stress axes are rotated, the aim is to investigate the influence of particle morphologies on the stress–strain relationship, noncoaxiality response, and fabric evolution. The following is a summary of the key outcomes of the simulation:

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Stress element of simple shear test and model of DEM: (

**a**) lode of simple shear test, (

**b**) the boundary of sample, and (

**c**) model of DEM.

**Figure 2.**Particle shape and internal filling: (

**a**) ellipsoid particles, (

**b**) particle filled, (

**c**) microscopic contact of spherical particles, and (

**d**) sample preparation of simple shear test.

**Figure 3.**Contact normal unit vector diagram: (

**a**) normal contact unit vector and (

**b**) the angle between unit vector and coordinate axis.

**Figure 4.**Shear stress and shear strain of different particle shapes: (

**a**) σv = 200 kPa, (

**b**) σv = 400 kPa, (

**c**) σv = 600 kPa, and (

**d**) σv = 200 kPa, AR = 1:1:1.

**Figure 5.**Volumetric strain of different particle shapes: (

**a**) σ

_{v}= 200 kPa, (

**b**) σ

_{v}= 400 kPa, (

**c**) σ

_{v}= 600 kPa, and (

**d**) σ

_{v}= 200 kPa, AR = 1:1:1.

**Figure 6.**Stress–strain relationship comparison with reference [34]: (

**a**) σ

_{v}= 200 kPa and (

**b**) σ

_{v}= 400 kPa.

**Figure 7.**Distribution of contact fabric: (

**a**) normal contact number of vertical pressure 200 kPa, (

**b**) normal contact number of vertical pressure 400 kPa, (

**c**) normal contact number of vertical pressure 600 kPa, and (

**d**) normal contact force.

**Figure 8.**The distribution of contact fabric: (

**a**) contact fabric of T01, (

**b**) contact fabric of T02, (

**c**) contact fabric of T03, (

**d**) contact fabric of T04, (

**e**) contact fabric of T05, (

**f**) contact fabric of T06, (

**g**) contact fabric of T07, (

**h**) contact fabric of T08, (

**i**) contact fabric of T09, and (

**j**) contact fabric of T10.

**Figure 9.**The rule of stress direction and strain rate direction angle: (

**a**) direction angle of T01, (

**b**) direction angle of T02, (

**c**) direction angle of T03, (

**d**) direction angle of T04, (

**e**) direction angle of T05, (

**f**) direction angle of T06, (

**g**) direction angle of T07, (

**h**) direction angle of T08, (

**i**) direction angle of T09, (

**j**) direction angle of T10, (

**k**) comparison with test data in references [23], and (

**l**) comparison with simulation data in references [23].

Parameters | Values |
---|---|

Clump_normal stiffness/(kPa) | 1 × 10^{7} |

Clump_tangential stiffness/(kPa) | 1 × 10^{7} |

Porosity/(%) | 0.45 |

Density/(Kg/m^{3}) | 2.7 × 10^{3} |

Elastic modulus/(kPa) | 1 × 10^{7} |

Coefficient of friction | 0.5 |

Damp/(Pa∙s) | 0.7 |

Test Number | AR | Cell Pressure (kPa) | Clump Number |
---|---|---|---|

T01 | 1.3:1:1 | 200 | 16,536 |

T02 | 1.6:1:1 | 200 | 22,088 |

T03 | 1.9:1:1 | 200 | 27,310 |

T04 | 1.3:1:1 | 400 | 16,536 |

T05 | 1.6:1:1 | 400 | 22,088 |

T06 | 1.9:1:1 | 400 | 27,310 |

T07 | 1.3:1:1 | 600 | 16,536 |

T08 | 1.6:1:1 | 600 | 22,088 |

T09 | 1.9:1:1 | 600 | 27,310 |

T10 | 1:1:1 | 200 | 1011 |

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**MDPI and ACS Style**

Zhu, H.; Li, X.; Lv, L.; Yuan, Q.
Discrete Element Simple Shear Test Considering Particle Shape. *Appl. Sci.* **2023**, *13*, 11382.
https://doi.org/10.3390/app132011382

**AMA Style**

Zhu H, Li X, Lv L, Yuan Q.
Discrete Element Simple Shear Test Considering Particle Shape. *Applied Sciences*. 2023; 13(20):11382.
https://doi.org/10.3390/app132011382

**Chicago/Turabian Style**

Zhu, Houying, Xuefeng Li, Longlong Lv, and Qi Yuan.
2023. "Discrete Element Simple Shear Test Considering Particle Shape" *Applied Sciences* 13, no. 20: 11382.
https://doi.org/10.3390/app132011382