# Numerical Characterization of Cohesive and Non-Cohesive ‘Sediments’ under Different Consolidation States Using 3D DEM Triaxial Experiments

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Discrete Element Method—Granular and Bonded Approach

^{TM}to investigate the mechanical behavior of numerical ’sediments’. The software implements the DEM technique following principles defined by Cundall and Strack [12] and offers several contact models to generate different mechanical behaviors [31]. We selected the Hertz–Mindlin contact model to generate a cohesionless granular material (sand-like; see Section 2.1) and the linear parallel-bond contact model to generate a cohesive, elastoplastic behavior (clay-like, see Section 2.2). Both contact models were previously applied using PFC and other discrete element software to generate a range of geo-materials such as soils and rocks [17,29,32,33].

#### 2.1. Granular Approach: The Hertz–Mindlin Contact Model (Cohesionless, Elastoplastic)

#### 2.2. Granular Cohesive Approach: The Linear Parallel-Bond Contact Model (Cohesive, Elastoplastic)

## 3. Experimental Setup

#### 3.1. Model Geometry

#### 3.2. Particle and Bond Micro-Properties

#### 3.2.1. Hertz–Mindlin Contact Model—Granular ‘Sand-Like’ Materials

#### 3.2.2. Linear Contact Bond Model—Cohesive ‘Clay-Like’ Materials

#### 3.3. Model Run Stages of the Numerical Triaxial Tests

_{z}= 20%), similar to laboratory triaxial tests, e.g., [56].

#### 3.4. Model Interpretation and Calculations

_{z}, these stress–strain curves give an insight into the deformation behavior, including determination of the peak strength, strain hardening and softening effects, etc. (Figure 3a and Figure 4a).

## 4. Results

^{11}, 1e

^{10}, 1e

^{8}Pa; ‘clay’: PB

_{coh}= 55e

^{3}, 110e

^{3}, 210e

^{3}Pa). These 12 materials were deformed under different loading conditions of 100, 250 and 500 (kPa), simulating different burial depths. In total, 36 experimental material settings were tested, 18 for each material type (Table 1).

#### 4.1. Stress–Strain Behavior

^{11}Pa samples reached a peak at around 15% strain, whereas samples with ${G}_{p}$ = 1e

^{8}Pa reached a peak only at around 17–18% strain).

^{8}Pa). Samples with a high micro-shear modulus show a high rate of increasing stress (Figure S2 in the Supplementary Materials), whereas samples of ${G}_{p}$ = 1e

^{8}Pa micro-shear modulus presented a moderate rate of change in stress and reached peak deviator stress at around 2–7.5% strain.

^{3}Pa) showed a decrease in the residual strength (for confining stresses of 100 and 250 kPa) or a decrease followed by a slight increase up to the previous peak value (confining stress of 500 kPa). In loose ’clay’ samples, by increasing the micro-cohesive bond strength ($P{B}_{\mathit{coh}}$), a lower peak strength was achieved under lower axial strain (rapid failure under lower stresses).

#### 4.2. Volumetric Strain, Porosity and Coordination Number

^{3}Pa). The loose ’clay’ showed a slight initial increase in volumetric strain, yet after 10% of strain the trend changed and showed a decrease in volumetric strain to 1–2% (Figure 4b).

^{8}Pa), where a volumetric decrease was seen up to 2% of strain, followed by a volumetric strain increase.

^{8}Pa), showed a different trend where a porosity decrease of 2% of strain was followed by a porosity increase of up to 4%. ‘Clay’ samples showed a general trend of increasing porosity. Loose ’clay’ samples showed a slight change in porosity, increasing by about 1% from the initial value. Densely packed ’clay’ samples showed a larger increase in porosity of 4% from the initial value.

_{z}= 20%), most samples, regardless of their initial average coordination number, reached an average value of four contacts per particle. Loose ’sand’ samples showed a slight averaged increase in the mean coordination number of 0.2 contacts per particle. Densely packed ’sand’ samples showed a decrease in the mean coordination number of one to two contacts per particle. Exceptional behavior was observed for both loose and densely packed ‘sand’ samples with a low micro-shear modulus (${G}_{p}$ = 1e

^{8}Pa), where the initial and the final mean coordination number were higher than in other samples. ‘Clay’ samples showed a similar trend for both loose and densely packed samples, which varied only in their rate of change at similar strain values. An increase in the mean coordination number (from 0.2 to 0.8 average contacts per particle) was observed under low strain (<5%). The average increase in the number of contacts per particle is inverse to the samples’ micro-cohesive bond strength ($P{B}_{\mathit{coh}}$). The stronger the micro-cohesive bond strength ($P{B}_{\mathit{coh}}$), the smaller the change in the mean number of contacts per particle. As strain increases (>5%), the mean coordination number decreases down to an average of four contacts per particle. The range of the decrease in the mean number of contacts per particle is of two contacts per particle for dense samples and 0.2 to 0.8 contacts per particle for loose samples. The decrease in the coordination number of both loose and densely packed ‘clays’ to a similar averaged coordination number value, is a result of bond breakage, leading to local dilation even in the loose samples, along a with similar coefficient of friction.

#### 4.3. Strain Localization

_{z}= 12%) and are limited to the perimeter of the sample (Figure 5d). As bonds break, the mode of failure alternates between slip along shear planes and focused areas of compression (Supplementary Video S3). In densely packed ’clay’ samples, strain was widely distributed within the sample with both maximum positive and negative gradient values (Figure 5e). The number of shear planes is highest in this material.

## 5. Discussion

#### 5.1. Parametrization of Numerical ‘Sediments’

#### 5.2. Classification of the Granular Assemblage

#### 5.3. Application

_{coh}) and an initial consolidation state via the initial micro- coefficient of friction (${\mu}_{p}$). The new material can simulate the numerical behavior of sediments that have undergone strengthening through various actions such as waves or seismic activity, and their consolidation changes as a result of the process.

## 6. Conclusions

_{coh}for clays). This will enable the simulation of shallow sediments, for which only shear strength has changed—such as inherited shear strength due to original deposited sediment micro-fabric, or shear strength altered by bioturbation or early digenesis cementation.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. DEM Force–Displacement Calculation

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**Figure 1.**(

**a**) An illustration of particles interaction according to the Hertz–Mindlin contact model. (Left) The contact point is illustrated as a red sphere and the contact stiffness as a spring. R

_{1}and R

_{2}represent the radii of particles P

_{1}and P

_{2}, accordingly. (Right) A 2D cross section of the acting normal (${F}_{n}$) and shear (${F}_{s}$) forces and the particle’s overlapping δ. (

**b**) An illustration of particles interaction according to the linear parallel-bond contact model. (Left) In addition to a contact at the interacting point of two particles (contact stiffness presented as a spring), a bond is implemented as a cylindrical disk (in red), and its interaction is illustrated by two parallel springs. R

_{1}and R

_{2}represent the radius of particles P

_{1}and P

_{2}, accordingly. (Right) A 3D illustration of the bond’s normal (B

_{n}) and shear (B

_{s}) forces and the moments (M

_{n}and M

_{s}) that result from the applied force. The size of the applied bond is according to the average radius of the two interacting particles and represented as an average 2R.

**Figure 2.**The 3D Discrete Element Method (DEM) cubic isotropic sample. The initial particle setup for the triaxial test are under strain (ε

_{z}) = 0%. The blue sphere in the center of the pre-tested cubic sample presents the location of the measurement sphere. The main stress (σ

_{1}) is acting parallel to the z-axis; ${\sigma}_{2}$ and ${\sigma}_{3}$ are equal and act parallel to axes x and y, respectively.

**Figure 3.**Triaxial tests results for ‘sand’ samples presented as (

**a**) the deviator stress, (

**b**) the volumetric strain, (

**c**) the porosity and (

**d**) the coordination number over the axial strain; the micro-shear modulus (${G}_{p}$) is increasing from left to right. Black lines represent loose packed samples and red lines represent dense packed samples. Each confining stress is represented by a different symbol for 100, 250 and 500 kPa.

**Figure 4.**Triaxial tests results for ‘clay’ samples presented as (

**a**) the deviator stress, (

**b**) the volumetric strain, (

**c**) the porosity and (

**d**) the coordination number over the axial strain; the micro-cohesive bond strength ($P{B}_{\mathit{coh}}$) is increasing from left to right. Black lines represent loose packed samples and red lines represent dense packed samples. Each confining stress is represented by a different symbol for 100, 250 and 500 kPa.

**Figure 5.**Strain localization visualized for deformation stages. Deformation stages are defined between points (1) and (3) along a typical stress–strain curve, where (1) is the yield stress, (2) the peak stress and (3) the post-peak situation. Deformation is imaged along a vertical cross section of the 3D test (location indicated by the grey cross section). (

**a**) Initiation of strain localization highlighted by oval circles (between stages 1 and 2, Experiment DC-2). Localization is presented for four samples tested under confining stress σ

_{2,3}= 250 kPa. (

**b**) Experiment LS-2, loose ’sand’ (${G}_{p}$ = 1e

^{10}Pa) taken after 17% of strain (

**c**) Experiment DS-2, dense ’sand’ (${G}_{p}$ = 1e

^{10}Pa) taken after 17% of strain. (

**d**) Experiment LC-2, loose ’clay’ ($P{B}_{\mathit{coh}}$ = 110e

^{3}Pa) taken after 12% of strain. (

**e**) Experiment DC-2, dense ’clay’ ($P{B}_{\mathit{coh}}$ = 110e

^{3}Pa) taken after 12% of strain. Black lines indicate selected areas of strain localization further explained in the text.

**Figure 6.**Modified Mohr–Coulomb failure envelopes for (

**a**) loose ’sand’ (

**b**) dense ’sand’ (

**c**) loose ’clay’ and (

**d**) dense ’clay’ samples. Circles denote the differential peak shear strength measured in each test under the appropriate normal stress.

**Figure 7.**A comparison between the mechanical behavior (stress–strain) of the numerical results and laboratory experiments of sediments. Laboratory experiments of loose and dense sands and normally consolidated and overconsolidated clays are respectively compared to the numerical results. (

**a**) loose ‘sand’ (

**b**) dense ‘sand’ (

**c**) loose ‘clay’ (

**d**) dense ‘clay’. Solid black lines are results from the DEM simulations, found to resemble empirical lab results. Above each line, the micro-parameter size and confining stress are detailed. Dashed and dotted red lines are results of empirical triaxial tests on natural sediments. Data from Guo and Su [3] were recalculated to fit the deviatoric stress axis. For other data ([65,66]), the original y-axis is present on the right-hand side (red labels).

**Table 1.**Fixed and tested micro-parameters applied in each of the triaxial tests according to the contact model applied during the test.

Parameters | Unit of Measure | Symbol | Value |
---|---|---|---|

Fixed Properties | |||

Sample dimensions: width; height; length | (m) | 220; 220; 220 | |

Total number of particles in a sample | 21,172 | ||

Radius (particles) | (m) | R_{p} | 3.7; 3.9; 4.6; 5.5 |

Particle density | (kg/m^{3}) | ρ_{p} | 2650 |

Damping coefficient | damp | 0.7 | |

Particle friction coefficient (during triaxial test) | µ_{test} | 0.5 | |

Particle friction coefficient (initial) | µ_{setup} | 0.1 (dense) 0.5 (loose) | |

Wall properties | |||

Wall friction coefficient | µ_{(wall)} | 0.0 | |

Wall normal stiffness | (Pa) | ${k}_{n\left(wall\right)}$ | 1e^{12} |

Hertz–Mindlin micro-properties (‘sand’) | |||

Poisson’s ratio | dimensionless | ν | 0.25 |

Shear modulus (small; medium; large) | (Pa) | ${G}_{p}$ | 1e^{11}; 1e^{10}; 1e^{8} |

Parallel-bond micro-properties (‘clay’) | |||

Parallel-bond contact normal and shear stiffness | (Pa) | ${k}_{n,s}$ | 1e^{10} |

Bond radius multiplier | λ | 1 | |

Bond friction coefficient | ${\mu}_{bond}$ | 0.54 | |

Bond stiffness | (Pa) | ${B}_{n,s}$ | 1e^{5} |

Bond cohesive strength (small; medium; large) | (Pa) | $P{B}_{coh}$ | 210e^{3}; 110e^{3}; 55e^{3} |

Bond tensile strength (small; medium; large) | (Pa) | $P{B}_{ten}$ | 110e^{3}; 55e^{3}; 25e^{3} |

**Table 2.**Calculated macro-properties of different material samples: LS: loose ‘sand’, DS: dense ‘sand’, LC-loose ‘clay’, DC-dense ‘clay’. Numbers in brackets are cohesion values, as measured from the linear regression of the failure envelope.

Experiment | Mean Normal Stress (kPa) | Peak Shear Strength (τ_{max}) (kPa) | Macro-Friction Coefficient (µ_{M}) | Bulk Cohesion C (kPa) | Experiment | Mean Normal Stress σ_{3} (kPa) | Peak Shear Strength (τ_{max}) (kPa) | Macro-Friction Coefficient (µ_{M}) | Bulk Cohesion (kPa) |
---|---|---|---|---|---|---|---|---|---|

‘Sand’ | |||||||||

LS-3 (${G}_{p}$ = 1e^{8} Pa) | 153 | 53.3 | 0.35 | (3.1) | DS-3 (${G}_{p}$ = 1e^{8} Pa) | 241 | 141.9 | 0.64 | (33.3) |

382 | 132.4 | 624 | 374.0 | ||||||

754 | 254.5 | 1280 | 780.0 | ||||||

LS-2 (${G}_{p}$ = 1e^{10} Pa) | 191 | 91.2 | 0.37 | (26.1) | DS-2 (${G}_{p}$ = 1e^{10} Pa) | 224 | 124.1 | 0.52 | (21.2) |

425 | 175.2 | 504 | 254.1 | ||||||

808 | 308.7 | 976 | 475.9 | ||||||

LS-1 (${G}_{p}$ = 1e^{11} Pa) | 190 | 90.6 | 0.47 | (23.3) | DS-1 (${G}_{p}$ = 1e^{11} Pa) | 166 | 66.6 | 0.40 | (4.1) |

513 | 263.4 | 407 | 157.2 | ||||||

903 | 403.7 | 808 | 308.2 | ||||||

‘Clay’ | |||||||||

LC-3 (PB_{coh} = 55e^{3} Pa) | 287 | 187.6 | 0.38 | 92.6 | DC-3 (PB_{coh} = 55e^{3} Pa) | 531 | 431.7 | 0.53 | 199 |

528 | 278.4 | 801 | 551.8 | ||||||

911 | 411.4 | 1291 | 791.5 | ||||||

LC-2 (PB_{coh} = 110e^{3} Pa) | 226 | 126.2 | 0.44 | 52.8 | DC-2 (PB_{coh} = 110e^{3} Pa) | 390 | 290.4 | 0.55 | 112 |

537 | 287.2 | 679 | 429.9 | ||||||

906 | 406.4 | 1174 | 674.4 | ||||||

LC-1 (PB_{coh} = 210e^{3} Pa) | 223 | 123.0 | 0.43 | 32 | DC-1 (PB_{coh} = 210e^{3} Pa) | 315 | 215.5 | 0.50 | 81 |

452 | 202.8 | 597 | 347.5 | ||||||

884 | 384.8 | 1050 | 550.7 |

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## Share and Cite

**MDPI and ACS Style**

Elyashiv, H.; Bookman, R.; Siemann, L.; ten Brink, U.; Huhn, K.
Numerical Characterization of Cohesive and Non-Cohesive ‘Sediments’ under Different Consolidation States Using 3D DEM Triaxial Experiments. *Processes* **2020**, *8*, 1252.
https://doi.org/10.3390/pr8101252

**AMA Style**

Elyashiv H, Bookman R, Siemann L, ten Brink U, Huhn K.
Numerical Characterization of Cohesive and Non-Cohesive ‘Sediments’ under Different Consolidation States Using 3D DEM Triaxial Experiments. *Processes*. 2020; 8(10):1252.
https://doi.org/10.3390/pr8101252

**Chicago/Turabian Style**

Elyashiv, Hadar, Revital Bookman, Lennart Siemann, Uri ten Brink, and Katrin Huhn.
2020. "Numerical Characterization of Cohesive and Non-Cohesive ‘Sediments’ under Different Consolidation States Using 3D DEM Triaxial Experiments" *Processes* 8, no. 10: 1252.
https://doi.org/10.3390/pr8101252