# Crack Propagation in the Tibia Bone within Total Knee Replacement Using the eXtended Finite Element Method

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## Abstract

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## 1. Introduction

## 2. Materials and Methodology

#### 2.1. Modeling and Simulation Workflow

#### 2.2. Patient Data and Finite Element Model Set-Up

^{2}Body Mass Index (BMI)) with a knee implant instrumented on his right knee side (Figure 2a).

#### 2.3. Fracture Mechanics Analysis under Monotonic and Cyclic Loading Conditions

#### 2.3.1. Under Monotonic Loading Conditions

#### 2.3.2. Under Cyclic Loading Conditions

^{−8}$\left(\frac{m}{cycle}\frac{1}{{\left(MPa\sqrt{m}\right)}^{n}}\right)$ and n = 2.8 were considered from the literature [31,32,33]. The crack propagation was modeled using eXtended Finite Element Method (X-FEM), in combination with the automatic mesh adaptivity through the Salome-Meca software.

## 3. Computational Results

#### 3.1. Mesh Sensitivity Study

#### 3.2. Crack Propagation Analysis

#### 3.2.1. Fixed Initiation Location Strategy

#### 3.2.2. Critical Point Identification Process and Age-Related Effect Analysis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Overview of the developed modeling and simulation workflow for analyzing the tibia crack propagation.

**Figure 2.**CT images of the human knee with total knee replacement (TKR) implant of the patient (

**a**) and associated meshed models of the tibia bone (cortical and cancellous bones corresponding to the green and yellow parts) and implant (the red part) (

**b**).

**Figure 3.**Location of the crack initiation based on the clinical observation (

**a**) and the Stress-Cycles to fatigue diagram (the so-called SN curve) from the accumulation of fatigue micro-damage in the human cortical bone of two different ages used in the age-related critical point identification process extracted from [20] (

**b**).

**Figure 5.**Four meshes of the tibia model without implant—coarse (

**a**), medium (

**b**), fine (

**c**), and very fine (

**d**) meshes.

**Figure 6.**Four meshes of the tibia model with implant (TKR model)—coarse (

**a**), medium (

**b**), fine (

**c**), and very fine (

**d**) meshes.

**Figure 7.**Mesh size effect on the values of ${K}_{I}$ and of G versus the curvilinear coordinate along the crack tip—the tibia model without the implant (

**a**,

**b**) and the tibia model with the implant (

**c**,

**d**).

**Figure 9.**Crack propagation in the tibia model with implant—different steps (

**a**) and a slicing surface along the implant in the last step (

**b**).

**Figure 10.**Maximum stress location (red arrow) of the 56-aged model, (

**a**) the distribution of the stress component SIGYY of the 56-aged model (

**b**), and the history of SIGYY_max versus the displacement at the maximum stress location (

**c**).

**Figure 11.**The fatigue damage curve (

**a**) and S–N curve (

**b**) for two age-related tibia models with implant.

Elastic Modulus (GPa) | Poisson’s Ratio | |
---|---|---|

Cortical bone (24-year-old) [29] | 16.6 | 0.3 |

Cortical bone (27-year-old) [19] | 15.3 | 0.3 |

Cortical bone (56-year-old) [19] | 13.4 | 0.3 |

Cancellous bone [29] | 2.4 | 0.3 |

Tibial prosthesis (Implant) [29] | 117 | 0.3 |

The tibia model without the implant | ||

Mesh type | Mesh type | Mesh type |

Coarse | 21,677 | 108,168 |

Medium | 27,141 | 140,153 |

Fine | 39,210 | 211,162 |

Very fine | 41,649 | 224,590 |

The tibia model with the implant | ||

Coarse | 38,359 | 174,303 |

Medium | 43,429 | 203,865 |

Fine | 51,671 | 252,138 |

Very fine | 61,607 | 310,445 |

$\mathbf{Stress}\text{}\mathbf{Intensity}\text{}\mathbf{Factor}\text{}{\mathit{K}}_{\mathit{I}}\left(\mathbf{MPa}\sqrt{m}\right)$ | $\mathbf{Energy}\text{}\mathbf{Release}\text{}\mathbf{Rate}\text{}\mathit{G}\text{}\left(\mathbf{J}{m}^{-2}\right)$ | |
---|---|---|

27-year-old bone model | 1.66 ÷ 12.3 | 746 ÷ 11089 |

56-year-old bone model | 1.65 ÷ 10.8 | 768 ÷ 9832 |

Experiment [16,31] | 2.2 ÷ 6.3 [31] | 780 ÷ 1120 [16] |

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**MDPI and ACS Style**

NGUYEN, H.-Q.; NGUYEN, T.-N.-T.; PHAM, T.-Q.-D.; NGUYEN, V.-D.; TRAN, X.V.; DAO, T.-T.
Crack Propagation in the Tibia Bone within Total Knee Replacement Using the eXtended Finite Element Method. *Appl. Sci.* **2021**, *11*, 4435.
https://doi.org/10.3390/app11104435

**AMA Style**

NGUYEN H-Q, NGUYEN T-N-T, PHAM T-Q-D, NGUYEN V-D, TRAN XV, DAO T-T.
Crack Propagation in the Tibia Bone within Total Knee Replacement Using the eXtended Finite Element Method. *Applied Sciences*. 2021; 11(10):4435.
https://doi.org/10.3390/app11104435

**Chicago/Turabian Style**

NGUYEN, Ho-Quang, Trieu-Nhat-Thanh NGUYEN, Thinh-Quy-Duc PHAM, Van-Dung NGUYEN, Xuan Van TRAN, and Tien-Tuan DAO.
2021. "Crack Propagation in the Tibia Bone within Total Knee Replacement Using the eXtended Finite Element Method" *Applied Sciences* 11, no. 10: 4435.
https://doi.org/10.3390/app11104435