# Review of Methods to Solve Desiccation Cracks in Clayey Soils

## Abstract

**:**

## 1. Introduction

## 2. Methods to Simulate Desiccation Cracks in Clayey Soils

#### 2.1. The Finite Element Method (FEM)

#### 2.1.1. Equilibrium Equation (Cauchy Equation of Motion)

#### 2.1.2. Balance Equation (Continuity Equation also Known as Richards’ Equation)

#### 2.1.3. Conservation of Energy Equation (First Law of Thermodynamics)

#### 2.1.4. Stress–Strain Thermos-Mechanical Constitutive Law

#### 2.1.5. Generalized Darcy’s Law for Unsaturated Soils and Permeability Tensor

#### 2.1.6. Water Retention Curve

#### 2.1.7. Fourier’s Law

#### 2.1.8. Hydro-Mechanical Formulation to Resolve Desiccation Cracks in Clayey Soils

**u**are the displacements and

**p**is the suction.

#### 2.2. Lattice Boltzmann Method (LBM)

#### 2.2.1. Formulation of LBM

#### 2.2.2. A Multi-Material Multiple-Relaxation-Time LBM

#### 2.3. Phase Field Method (PFM)

#### 2.3.1. General Phase Field for Fracture Model

**u**, and the phase field, $d$. The contour conditions are the usual Dirichlet boundary $\partial {\mathsf{\Omega}}_{u}$ and Neumann boundary $\partial {\mathsf{\Omega}}_{t}$ conditions.

#### 2.3.2. Govern and Constitutive Equations

#### 2.4. Discrete Element Method (DEM)

#### DEM Overview

#### 2.5. Cellular Automaton Method (CAM)

#### 2.5.1. Formulation of CAM

- (1)
- Strong discontinuity;
- (2)
- Tracking a moving crack;
- (3)
- Cellular automaton updating rule;
- (4)
- Yield criteria and crack growth laws.

- Strong discontinuity

- Tracking a moving crack

- Cellular automaton updating rule

- Yield criteria and crack growth laws

#### 2.5.2. Governing Equations and Local Stiffness Matrix

**u**is the displacement;

**b**is the body force per unit volume;

**n**is the unit outward normal; $\overline{\mathit{t}}$ and $\overline{\mathit{u}}$ are the prescribed traction and displacement, respectively; $\epsilon $ is the strain tensor; and

**C**is the Hooke tensor.

## 3. Integration of Methods to Improve Simulations and Analysis

## 4. Finite Element and Cellular Automaton Method (FEM-CAM)

#### Integration of FEM with CAM to Simulate Desiccation Cracks in Clayey Soils

## 5. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Drying, wetting, flooding, and drying 36-day tests on cylindrical clays sample, 80 cm in diameter and 10 cm high, in an environmental chamber. From Dr. Hector U. Levatti—Ph.D. [34].

**Figure 2.**Soil–atmosphere and soil–structure (tray) interactions can significantly affect the behavior of the cracks.

**Figure 3.**Lattice Boltzmann method. (

**a**) Algorithm 1—Streaming step. Particles travel to connected neighbors. (

**b**) Algorithm 2—Collision step. Particles travel to lattice node.

**Figure 4.**(

**a**) An intact solid body with Dirichlet boundary $\partial {\mathsf{\Omega}}_{u}$ and Neumann boundary $\partial {\mathsf{\Omega}}_{t}$. (

**b**) A solid body with a crack represented by the crack set Γ. (

**c**) A solid body with a crack represented by the crack-density function γ (d).

**Table 1.**Methods that effectively tackle challenges when simulating desiccation cracks in clayey soils, and the scale levels they work into. The methods are classified into non-physical-based (nPb) and physical-based (Pb) methods into columns for every scale level.

Common Challenges | Non-Physical-Based (nPb) Physical-Based (Pb) | Scale Level | |||||
---|---|---|---|---|---|---|---|

Microscale | Mesoscale | Macroscale | |||||

nPb | Pb | nPb | Pb | nPb | Pb | ||

Heterogeneity | DEM CAM | FEM | CAM | FEM | CAM | FEM | |

Multiphase medium | DEM CAM | PFM | CAM | PFM | CAM | FEM | |

Coupled nonlinear THM problem | DEM CAM | PFM | CAM | LBM PFM | CAM | FEM | |

Effect of the soil composition, mineralogy, pore structure, initial moisture content | DEM CAM | PFM | CAM | LBM PFM | CAM | FEM | |

Dealing efficiently with computationally intensive methods at large-scale simulations | CAM | CAM | CAM | ||||

Large deformations | DEM CAM | PFM | CAM | LBM PFM | CAM | FEM | |

Capture shrinkage and cracking using advanced constitutive equations | DEM CAM | PFM | CAM | LBM PFM | CAM | FEM | |

Complex crack patterns | CAM | CAM | CAM |

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**MDPI and ACS Style**

Levatti, H.U.
Review of Methods to Solve Desiccation Cracks in Clayey Soils. *Geotechnics* **2023**, *3*, 808-828.
https://doi.org/10.3390/geotechnics3030044

**AMA Style**

Levatti HU.
Review of Methods to Solve Desiccation Cracks in Clayey Soils. *Geotechnics*. 2023; 3(3):808-828.
https://doi.org/10.3390/geotechnics3030044

**Chicago/Turabian Style**

Levatti, Hector U.
2023. "Review of Methods to Solve Desiccation Cracks in Clayey Soils" *Geotechnics* 3, no. 3: 808-828.
https://doi.org/10.3390/geotechnics3030044