# Fatigue Life Appraisal and Its Corrected Stress Intensity Factor for Repaired Off-CentrallyCracked Aluminum Plates

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## Abstract

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## 1. Introduction

## 2. Experimental Procedure and Results

## 3. Finite Element Analysis

## 4. Results and Discussion

#### 4.1. Summary of K_{I} and F_{w} Theoretical Equations for Off-Central, Central and Edge Cracks

_{w}. For the surface crack scenario studied herein, suitable parameters are needed for the rational assessment of F

_{w}. On this basis, two analytical equations are compared as below.

#### 4.1.1. Crack Line Stress Field Method for Off-Centrally Cracked Case

_{I}dominated plastic segments at the crack tips (A and B) and outside plastic segments. Based on the basic theory for a crack opening symmetrically with respect to the undeformed crack plane, the corresponding normal stress σ

_{x}along the x axis is written as:

_{WA}and F

_{WB}are the correction factors, d is the diameter of pseudo plastic zone at the crack front. When the distances from the centers of the pseudoplastic zone are greater than 0.5d

_{0A}or 0.5d

_{0B}, the square F

_{WA}or F

_{WB}is chosen due to an allowance of finite width and the corresponding normal stress σ

_{x}is expressed as:

_{0}

_{A}and d = d

_{02}

_{B}, it can be obtained as d

_{0}

_{A =}0.5c/F

_{WA}

^{2}and d

_{0}

_{B =}0.5c/F

_{WB}

^{2}. Equating the tensile stress along the crack extended line to the remote tension stress, σ, along the y axis yields:

_{WA}

^{2}-1/F

_{WB}

^{2}) approaching zero when s = 0 (no eccentricity), solving Equations (4) and (5) gives:

#### 4.1.2. The Westergaard Function Based Method for Off-Centrally Cracked Case

#### 4.1.3. Comparison of Referred Calculations]

_{w}is increased which indicates a much higher stress concentration of the stress field at the crack tip. Calculated F

_{w}for crack tip A is consistently lower than that for crack tipB and seemingly less influenced by the eccentricity of the crack. Such an underestimation can be owed to the different plastic zones at the crack fronts of tip A and B, which was simplified as the same in referred methods. For design purpose, however, only the larger F

_{w}, i.e., for crack tip B, related to greater stress concentration is considered in the following analysis.

#### 4.2. Finite Element Analysis Based Evaluation of F_{w} for Off-Centrally Cracked Plates

_{w}for off-centrally cracked plates is evaluated based on a finite element parametric study. One hundred and twelve finite element models—which are permutations and combinations of seven sets of 2s/b ranging from 0.17 to 0.7, and fourteen sets of c/b ranging from 0.1 to 0.5—are built for the stress intensity factor analysis. Based on the finite element parametric study, the calculation for F

_{w}can be developed as the following poly-fit equation, including the ratios of 2s/b and c/b.

_{w}predicted from modelling agrees well with those from referred calculation. The cases with central cracks and edge cracks are also referred to further the comparison with analytical results. For the centrically cracked plate of finite width, the stress intensity factor at the crack tip was originally modified from that for an infinite sheet with a periodic array of through-thickness cracks under uniformly distributed stress. The formation is deemed accurate for up to c/b = 0.5, and the tangent correction as documented in Reference [3] is given by:

_{w}is more or less involved with the edge cracks or the central cracks when the 2s/b is greater than 0.55 or less than 0.17.

#### 4.3. Fatigue Life Evaluation

_{W}, and the mode I elastic stress intensity factor, the fatigue life of the test specimens can be evaluated. As per the suggestion by Su et al. [28], the aluminum material was defined following a rate-independent isotropic elastic-plastic relation using the classical J

_{2}-flow theory of the plasticity. It fits the experimentally measured stress-strain curve with the extraction of strain-hardening features. When above discussed boundary correction coefficient, F

_{W}, is determined, the mode I elastic stress intensity factor can be calculated using Equation (1), and then the fatigue life of test specimens can be evaluated. It is generally recognized that three regions exist for the relationship between the fatigue crack growth rate and the stress intensity factor; threshold, linear growth, and accelerated growth. To calculate the fatigue life, the crack propagation during the latter two regions can be analyzed using the concept of fracture mechanics [29] and the well-known Forman equation [3], including the stress ratio effect and F

_{W}aforementioned in the Section 4.2, as:

_{c}is the fracture toughness of the test material, which is equal to 1.181 × 10

^{3}MPa·mm

^{0.5}. C

_{F}is the material constant converted from its counterpart in Paris equation as 0.174 × 10

^{−10}. Assuming m = 3, the integration of Equation (19) results in the fatigue life, N, of the cracked aluminum plate as:

_{0}and c

_{1}are the initial and final crack lengths, respectively. As the magnitude of ΔK

_{I}is determined by c

_{0}and c

_{1}, N can be obtained via the numerical analysis and corresponding data for S-N relation can be deduced. Figure 10 shows a good correlation of fatigue lives between experimental test results and finite element modelling based analytical results using the Forman equation when the centrally cracked and off-centrally cracked aluminum plates are concerned. This also indicates that F

_{w}defined by the proposed poly-fit equation would be a good choice in representing stress intensity factors, and thus the fatigue life.

## 5. Concluding Remarks

- The developed finite element parametric study-based poly-fit equation for the boundary correction coefficient incorporating the eccentricity ratio and the crack size ratio is demonstrated to agree well with referred calculations. Moreover, it fits in well between centrally cracked cases and edge crack casesfor the off-centrally cracked aluminum plates.
- The fatigue life prediction on the basis ofthe Forman equation accounting for the crack in linear growth and accelerated growth is shown to correlate well with the test results of aluminum plates with central cracks and off-central cracks. The corresponding S-N curves are also comparable to those suggested by codified curves.
- The beneficial effect of patched laminate repair can be identified from the increase in test fatigue lives by 30–60% and 90–120% for centrally cracked and off-centrally cracked aluminum plates, respectively. Thus, the repair at the crack tip close to the plate edge is deemed to be effective in the fatigue life improvement for off-centrally crack aluminum plates.The strengthening effect of patched laminate repair, however, was not modelled in detail, and therefore requires more experimental work in furthering the correction of defined parameters in the fatigue life prediction.

## Author Contributions

## Funding

## Conflicts of Interest

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Material | Yield Strength (MPa) | Ultimate Strength (MPa) | Elastic Modulus (MPa) | Elongation (%) | Critical Stress Intensity Factor (MPa·mm^{0.5}) |
---|---|---|---|---|---|

Aluminum | 307 | 445 | 7.2 × 10^{5} | 15 | 1181 |

Carbon Fiber | - | 4216 | 2.52 × 10^{5} | 1.76 | - |

Adhesive | - | 30 | 4.5 × 10^{3} | 0.9 | - |

Specimen Series | Test Number | Crack Mode | Repair | 2s/b | c/b |
---|---|---|---|---|---|

T1 | 9 | Centrally | × | 0 | 0.15–0.45 |

T2 | 4 | Centrally | √ | 0 | 0.3 |

T3 | 4 | Off-Centrally | × | 0.5 | 0.3 |

T4 | 4 | Off-Centrally | √ | 0.5 | 0.3 |

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**MDPI and ACS Style**

You, X.; Wang, Z.; Zhou, X.; Liu, Z.; Jiang, R.; Gai, W.
Fatigue Life Appraisal and Its Corrected Stress Intensity Factor for Repaired Off-CentrallyCracked Aluminum Plates. *Materials* **2020**, *13*, 4014.
https://doi.org/10.3390/ma13184014

**AMA Style**

You X, Wang Z, Zhou X, Liu Z, Jiang R, Gai W.
Fatigue Life Appraisal and Its Corrected Stress Intensity Factor for Repaired Off-CentrallyCracked Aluminum Plates. *Materials*. 2020; 13(18):4014.
https://doi.org/10.3390/ma13184014

**Chicago/Turabian Style**

You, Xiang, Zhiyu Wang, Xiafang Zhou, Zifeng Liu, Ruijuan Jiang, and Weiming Gai.
2020. "Fatigue Life Appraisal and Its Corrected Stress Intensity Factor for Repaired Off-CentrallyCracked Aluminum Plates" *Materials* 13, no. 18: 4014.
https://doi.org/10.3390/ma13184014