# Hybrid Finite-Discrete Element Modeling of the Mode I Tensile Response of an Alumina Ceramic

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Computational Approach

#### 2.1. The Cohesive Law

#### 2.2. The Microscopic Stochastic Fracture Model

## 3. The Hybrid Finite-Discrete Element Method

#### 3.1. Modeling Mode I Failure Considering Distributed Flaws

#### 3.2. The Effect of Flaw Distribution

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The mode I constitutive behavior of the cohesive element with ${K}_{1}=4.6$ × ${10}^{7}$ N/mm${}^{3}$, ${\sigma}_{0}$ = 440 MPa, and ${G}_{1}^{c}=0.04$ N/mm. In the inset figure, a cohesive (crack) element is interspersed throughout two tetrahedral elements.

**Figure 2.**The green line is Weibull’s statistical strength distribution of cohesive elements obtained from Equation (8) with ${m}_{0}=11.0$, ${m}_{a}=-11.0$, ${\sigma}_{0}=440.0$, $A=0.01268$ and ${A}_{0}=0.013$. The orange bar is the statistics of the facet strength of the cohesive element with random flaws generated by Monte Carlo simulations. The percentage of low-strength cohesive elements (below 350 MPa) is around 7.9%, which is associated with the big flaws in the material. The rest of the cohesive elements (around 92.1%) have strong strength (between 350 and 530 MPa), which corresponds to the material with smaller flaws.

**Figure 3.**(

**a**) Variation of engineering stress-strain response of the CeramTec 98% alumina during the direct tension simulations with four kinds of mesh sizes. (

**b**) The shaded region is the tensile strength of the CeramTec 98% obtained by Brazilian disk experiments [29], and the dots are the simulation results with four kinds of mesh sizes.

**Figure 4.**The failure pattern of the samples with different mesh sizes. The legends in the figure correspond to the displacement of the samples in the z-direction (U${}_{3}$). It is observed that a horizontal crack perpendicular to the loading direction is generated during the loading process. The single main crack causes catastrophic failure. In some cases, the main crack may split into multiple branches. Some fragments appear near the crack surfaces due to crack branching.

**Figure 5.**Microscopic strength distributions with different Weibull modulus (${m}_{0}$ = 9, 10, 11, 12, and 13).

**Figure 6.**(

**a**) The engineering stress-strain response of the CeramTec 98% alumina during the direct tension simulations with different Weibull modulus (${m}_{0}$ = 9, 10, 11, 12, and 13). (

**b**) The full dots are the tensile strength and the hollow dots are the elastic modulus obtained by simulation with five different Weibull modulus.

**Figure 7.**The failure pattern of the simulation results with different Weibull modulus (${m}_{0}=$ 9, 10, 11, 12, and 13). The U${}_{3}$ legend in the figure indicates the displacement of the samples in the z-direction. The failure pattern is consistent with Figure 4, a horizontal crack forms perpendicularly to the loading direction during the loading process, ultimately leading to a catastrophic failure caused by the single main crack. In certain instances, the main crack can divide into multiple branches, resulting in fragments near the crack surfaces due to crack branching.

The Mechanical Properties of the CeramTec 98% Alumina | |
---|---|

Porosity | <2% |

Hardness | 13.5 (GPa) |

Density | 3.8 (g/cm${}^{3}$) |

Young’s modulus | 335 (GPa) |

Poisson’s ratio | 0.23 |

The Properties for the Microscopic Stochastic Fracture Model | |

Weibull modulus of the strength distribution | 11 |

Weibull modulus for the effective area modification | −11 |

Weibull characteristic strength | 440 (MPa) |

Characteristic area | 0.013 (mm${}^{2}$) |

Mode I fracture energy | 0.04 (N/mm) |

Validation Models | Average Mesh Size (mm) | Average Facet Area (mm${}^{2}$) |
---|---|---|

Case 1 | 0.25 | 0.038 |

Case 2 | 0.15 | 0.01268 |

Case 3 | 0.1 | 0.00611 |

Case 4 | 0.075 | 0.00336 |

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**MDPI and ACS Style**

Zheng, J.; Li, H.; Hogan, J.D.
Hybrid Finite-Discrete Element Modeling of the Mode I Tensile Response of an Alumina Ceramic. *Modelling* **2023**, *4*, 87-101.
https://doi.org/10.3390/modelling4010007

**AMA Style**

Zheng J, Li H, Hogan JD.
Hybrid Finite-Discrete Element Modeling of the Mode I Tensile Response of an Alumina Ceramic. *Modelling*. 2023; 4(1):87-101.
https://doi.org/10.3390/modelling4010007

**Chicago/Turabian Style**

Zheng, Jie, Haoyang Li, and James D. Hogan.
2023. "Hybrid Finite-Discrete Element Modeling of the Mode I Tensile Response of an Alumina Ceramic" *Modelling* 4, no. 1: 87-101.
https://doi.org/10.3390/modelling4010007