# An Anisotropic Damage Model of Quasi-Brittle Materials and Its Application to the Fracture Process Simulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Anisotropic Damage Model

#### 2.1. Failure Criterion

_{t0}is the uniaxial tensile strength, and σ

_{c0}is the uniaxial compressive strength. In the absence of compressive stress, this failure criterion is reduced to $\tau =\sqrt{n{\sigma}_{\mathrm{t}0}}$, where $\sqrt{n{\sigma}_{\mathrm{t}0}}$ is the shear strength of the failure plane without normal stress.

_{1}and σ

_{3}are the first and third principal stresses, respectively. Moreover, the compressive stresses are positive.

_{1}≤ n − σ

_{t0}, the material fails in tension. When σ

_{1}> n − σ

_{t0}, it fails in shear. Therefore, σ

_{1}= n − σ

_{t0}is the critical principal stress for switching the failure mode. In addition, for shear failure, $\frac{{\sigma}_{1}+{\sigma}_{3}}{2}-\frac{n}{2}\ge 0$ corresponds to compression–shear failure, whereas $\frac{{\sigma}_{1}+{\sigma}_{3}}{2}-\frac{n}{2}<0$ corresponds to the tension–shear failure.

_{eq}is the equivalent shear strain on the failure plane, deduced using Equation (6), as follows:

#### 2.2. Stiffness Degradation Model

_{ijkl}as follows:

_{ijkl}is expressed as

#### 2.3. Damage Evolution Equations

#### 2.3.1. Tension Damage

_{t}= σ

_{tr}/σ

_{t0}is the residual tensile strength coefficient; σ

_{tr}is the residual tensile strength; ε

_{t0}= σ

_{t0}/E

_{0}is the elastic ultimate tensile strain; ε

_{tu}is the ultimate tensile strain before complete failure; $\overline{\epsilon}=\sqrt{\langle {\epsilon}_{1}\rangle {}^{2}+\langle {\epsilon}_{2}\rangle {}^{2}+\langle {\epsilon}_{3}\rangle {}^{2}}$ is the equivalent tensile strain and ε

_{1}, ε

_{2}, ε

_{3}are the first, second, third principal strains, respectively.

#### 2.3.2. Shear Damage

_{s}= τ

_{r}/τ

_{0}is the residual shear strength coefficient, τ

_{r}is the shear residual strength, and τ

_{0}is the shear strength.

_{eq}and principal strain component, the nonlinearity of the damage evolution equations is further increased compared with the conventional damage mode. Moreover, the complexity and uncertainty of the damage problem are also significantly increased.

## 3. Simulation of the Fracture Process

_{0}to 0 before loading. Then, in each loading step determine the elastic tensor according to the initial damage variable D

_{0}, and calculate the stress, strain, and new damage variable D

_{1}. Judge whether the difference between D

_{1}and D

_{0}meets the specified tolerance. If yes, go to the next loading step. If not, assign D

_{1}to D

_{0}, update the elastic tensor, and recalculate the stress, strain, and new damage variable D

_{1}. Repeat the above process until the last loading step.

## 4. Test of the Anisotropic Damage Model

_{c0}= 200 MPa, and λ

_{s}= 0.1. The damage variable D was zero everywhere except within the crack band, where D was computed using Equation (12). Figure 6 shows the distribution of the damage variable and elastic modulus after failure. The proposed anisotropic damage model with two cases of confining pressure, p

_{c}= 0 MPa and p

_{c}= 10 MPa, was compared with the conventional isotropic damage model in the literature [44,45]. In the conventional isotropic damage model, the failure criterion is the Mohr–Coulomb strength criterion, and the shear damage evolution equation is based on compressive strain.

## 5. Uniaxial and Biaxial Compression Tests

_{0}is the scale parameter related to the average value of variable u. m is the shape parameter, the homogeneity index, which reflects the degree of material homogeneity.

_{0}= 60 GPa, σ

_{t0}= 20 MPa, n

_{0}= 107, and λ

_{t}= λ

_{s}= 0.1. Moreover, four confining pressures were studied: pc = 0 MPa, 10 MPa, 20 MPa, and 40 MPa. Confining pressure was imposed on the specimen in the first loading step, and displacement increment was applied to the top boundary in the subsequent loading steps. A numerical sensitivity analysis of the loading step length on the uniaxial compression test indicated that the loading step length in a reasonable quasi-static range did not affect the macroscopic strength and the main crack propagation significantly [31]. Considering calculation accuracy and efficiency, a constant displacement increment of 0.005 mm/step was used here.

#### 5.1. Damage Evolution

#### 5.2. Fracture Modes

#### 5.3. Stress–Strain Curves

_{c}is 74, the fitting parameter m is 0.9, and s is 1 or 0.

## 6. Brazilian Splitting Test

## 7. Conclusions

- A novel anisotropic damage model is proposed to improve the conventional isotropic damage model. It avoids the interpenetration of crack surfaces and unifies three forms of damage, which include pure tension, tension–shear, and compression–shear, as one smooth criterion. In addition, the shear damage evolution equation considers the equivalent shear strain on the failure face as the independent variable, which has a clearer physical meaning.
- The proposed anisotropic damage model is able to simulate the failure behavior of quasi-brittle materials more effectively. It can effectively simulate not only the mesoscale pure tension, tension shear, and compression shear damages but also the macroscale fracture mode and its evolution. It can also simulate the strength characteristics, including the peak and residual strengths, approximately.
- The simplified two-dimensional geometries were used to verify the proposed anisotropic damage model by comparison with a conventional damage model. However, the proposed anisotropic damage model and finite element implementation strategy proposed in this study also apply to three-dimensional, multi-scale, and multi-field coupling problems.
- Numerical tests of the failure behavior of quasi-brittle materials have the advantages of strong universality, convenience, flexibility, and repeatability compared with laboratory tests. In addition, they can provide mesoscale mechanical information that is difficult to observe in laboratory tests. However, owing to the limitations of the nonlinear solution method, numerical tests will take longer if there are more loading steps. The combination of numerical simulation and machine learning algorithms will help improve the efficiency of numerical tests better to serve the scientific research and teaching of quasi-brittle materials, which is also a follow-up research goal of the authors.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Comparison of failure modes (

**a**) anisotropic damage model, p

_{c}= 0 MPa; (

**b**) anisotropic damage model, p

_{c}= 10 MPa; (

**c**) isotropic damage model, p

_{c}= 0 MPa; (

**d**) isotropic damage model, p

_{c}= 10 MPa.

**Figure 7.**Finite element model of the rock specimen in compression tests, (

**a**) model geometry; (

**b**) mesh grids and initial elastic modulus.

**Figure 8.**Failure process of rock specimen under axial compression, p

_{c}= 0 MPa, (

**a**) damage variable; (

**b**) damage mode.

**Figure 9.**Failure process of rock specimen under biaxial compression, p

_{c}= 10 MPa, (

**a**) damage variable; (

**b**) damage mode.

**Figure 10.**Failure process of rock specimen under biaxial compression, p

_{c}= 20 MPa, (

**a**) damage variable; (

**b**) damage mode.

**Figure 11.**Failure process of rock specimen under biaxial compression, p

_{c}= 40 MPa, (

**a**) damage variable; (

**b**) damage mode.

**Figure 12.**Fracture modes of rock specimens under different confining pressures, (

**a**) simulated results of elastic modulus; (

**b**) experimental results, reprinted with permission from [46], 2016 Elsevier.

**Figure 15.**Finite element model of the rock specimen in Brazilian tensile tests, (

**a**) model geometry; (

**b**) mesh grids and initial elastic modulus.

**Figure 18.**Fracture mode of disc specimen, (

**a**) simulated result of elastic modulus; (

**b**) experimental result, reprinted with permission from [52], 2021 Elsevier.

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**MDPI and ACS Style**

Wang, H.; Zhou, B.; Xue, S.; Deng, X.; Jia, P.; Zhu, X.
An Anisotropic Damage Model of Quasi-Brittle Materials and Its Application to the Fracture Process Simulation. *Appl. Sci.* **2022**, *12*, 12073.
https://doi.org/10.3390/app122312073

**AMA Style**

Wang H, Zhou B, Xue S, Deng X, Jia P, Zhu X.
An Anisotropic Damage Model of Quasi-Brittle Materials and Its Application to the Fracture Process Simulation. *Applied Sciences*. 2022; 12(23):12073.
https://doi.org/10.3390/app122312073

**Chicago/Turabian Style**

Wang, Haijing, Bo Zhou, Shifeng Xue, Xuejing Deng, Peng Jia, and Xiuxing Zhu.
2022. "An Anisotropic Damage Model of Quasi-Brittle Materials and Its Application to the Fracture Process Simulation" *Applied Sciences* 12, no. 23: 12073.
https://doi.org/10.3390/app122312073