# Experimental and Analytical Study on Residual Stiffness/Strength of CFRP Tendons under Cyclic Loading

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## Abstract

**:**

## 1. Introduction

## 2. Experiment

#### 2.1. Material Properties

#### 2.2. Tension–Tension Fatigue Tests

## 3. Fatigue Damage Model

#### 3.1. Residual Stiffness Model

- H. A. Whitworth (1987) [5]:$$\frac{S\left(n\right)}{S\left(0\right)}={e}^{\frac{\mathrm{ln}\left[1-\frac{\beta n}{N}\right]}{\alpha}}$$
- J. N. Yang et al. (1990) [6]:$$\frac{S\left(n\right)}{S\left(0\right)}=-\alpha \xb7{\left(n/N\right)}^{\beta}+\gamma $$
- A. T. Echtermeyer et al. (1995) [8]:$$\frac{S\left(n\right)}{S\left(0\right)}=\mathsf{\alpha}-\mathsf{\beta}\xb7\mathrm{log}n$$
- T. P. Philippidis and V. A. Passipoularidis (2000) [9]:$$\frac{S\left(n\right)}{S\left(0\right)}=1-\alpha \xb7\frac{n}{N}$$
- W. X. Yao et al. (2012) [11]:$$\frac{S\left(n\right)}{S\left(0\right)}=1-\left(1-\frac{S\left(cr\right)}{S\left(0\right)}\right)\left(1-\frac{1-{\left(n/N\right)}^{\alpha}}{{\left(1-n/N\right)}^{\beta}}\right)$$

#### 3.2. Redisual Strength Model

- L. J. Brountman and S. Sahu (1972) [22]:$$\frac{S\left(n\right)}{S\left(0\right)}=1-(1-\frac{{S}_{max}}{S\left(0\right)})\frac{n}{N}$$
- J. R. Scaff and B. D. Davidson (1997) [24]:$$\frac{S\left(n\right)}{S\left(0\right)}=1-(1-\frac{{S}_{max}}{S\left(0\right)}){\left(n/N\right)}^{\alpha}$$
- W. X. Yao and N. Himmel (2000) [23]:$$\frac{S\left(n\right)}{S\left(0\right)}=1-\left(1-\frac{{S}_{max}}{S\left(0\right)}\right)\left[\frac{\mathrm{sin}(\beta n/N)\mathrm{cos}\left(\beta -\alpha \right)}{\mathrm{sin}\beta \mathrm{cos}(\beta n/N-\alpha )}\right]$$
- T. P. Philippidis and V. A. Passipoularidis (2007) [40]:$$\frac{S\left(n\right)}{S\left(0\right)}=1-(1-\frac{{S}_{max}}{S\left(0\right)}){\left(n/N\right)}^{\mathrm{\alpha exp}\left(\beta n/N\right)}$$
- N. Stojković et al. (2017) [25]:$$\frac{S\left(n\right)}{S\left(0\right)}=1-\left(1-\frac{{S}_{max}}{S\left(0\right)}\right)[1-{\left(1-{\left(\frac{n}{N}\right)}^{\alpha}\right)}^{\beta}]$$

## 4. Discussion of Experimental Results

## 5. A New Proposed Fatigue Model

- (a)
- Under cyclic loading with various stress ranges, both the stiffness and strength of CFRP tendon have three-stage degradations throughout the fatigue life cycle.
- (b)
- Stiffness/strength degradation rate of the CFRP tendon increases with the growth of stress range. This means that if the stress range gets larger, stiffness/strength degradation of the CFRP tendon become more obvious.
- (c)
- When the stress range gets smaller, the stiffness/strength degradation rates of initial stage and final stage become closer to that of the second stage.

## 6. Conclusions

- (a)
- Both the stiffness and strength of CFRP tendon degrade during the fatigue loading process. Also, it can be observed that as the stress range increases, stiffness and strength of CFRP tendons degrade more obviously.
- (b)
- The three-stage regularity can be observed from degradation processes of stiffness and strength when CFRP tendons and other FRP composites subjected to fatigue loading. In the first stage, transverse cracks become more saturated when CFRP tendons subjected to lower stress range. Therefore, in the first stage, mechanical property degrades more obviously at lower range. The damage mechanism in the first stage is matrix cracking and is matrix/fiber debonding in the second stage at lower stress range. However, the damage mechanism in the first and second stage becomes the mixing of matrix cracking, matrix/fiber debonding and fiber breakage at higher stress range. Therefore, the boundaries between adjacent stages become more obvious when stress range decreases.
- (c)
- The proposed fatigue damage model is applicable to predict both residual stiffness and residual strength throughout fatigue life cycle. This new proposed model has a better accuracy than the models from the literature based on the experimental results of CFRP tendons and results from the literature.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Predicted and measured residual stiffness of CFRP tendons at different stress ranges: (

**a**) ∆σ = 900MPa, (

**b**) ∆σ = 800MPa, (

**c**) ∆σ = 600MPa, (

**d**) ∆σ = 500MPa, (

**e**) ∆σ = 400MPa.

**Figure 5.**Predictions of residual strength models and measurements of CFRP tendons at different stress ranges: (

**a**) ∆σ = 900MPa, (

**b**) ∆σ = 800MPa, (

**c**) ∆σ = 600MPa, (

**d**) ∆σ = 500MPa.

**Figure 7.**Residual strength predictions of new proposed model: (

**a**) prediction of residual strength, (

**b**) strength degradation rate.

**Figure 8.**Mechanical property degradations of composite materials under different stress levels: (

**a**) residual stiffness, (

**b**) residual strength.

Specimen | Young’s Modulus, E (GPa) | Ultimate Strength, (MPa) | ${\mathit{E}}_{\mathit{a}\mathit{v}\mathit{e}}$ (Gpa) | ${\mathit{\sigma}}_{\mathit{u},\mathit{a}\mathit{v}\mathit{e}}$ (Mpa) |
---|---|---|---|---|

S1 | 155.4 | 2064.9 | 155.9 | 2084.0 |

S2 | 162.1 | 2104.4 | ||

S3 | 159.5 | 2142.4 | ||

S4 | 149.7 | 2049.2 | ||

S5 | 152.6 | 2059.3 |

Stress Range | Residual Stiffness | Residual Strength | ||||||
---|---|---|---|---|---|---|---|---|

$\mathsf{\alpha}$ | $\mathsf{\beta}$ | $\mathsf{\gamma}$ | ${\mathbf{R}}^{2}$ | $\mathsf{\alpha}$ | $\mathsf{\beta}$ | $\mathsf{\gamma}$ | ${\mathbf{R}}^{2}$ | |

900 | −0.00605 | 0.98912 | 0.91945 | 0.862 | −0.07095 | 0.99923 | 0.81094 | 0.823 |

800 | −0.00276 | 0.98483 | 0.97684 | 0.945 | −0.06266 | 0.99941 | 0.80541 | 0.776 |

600 | −0.00165 | 0.98125 | 0.99557 | 0.721 | −0.06482 | 0.99968 | 0.79876 | 0.768 |

500 | −0.00175 | 0.96750 | 0.99769 | 0.746 | −0.05227 | 0.99971 | 0.76475 | 0.794 |

400 | −0.00042 | 0.94790 | 0.99454 | 0.693 | N/A |

Residual Stiffness | Stress Level | Residual Strength | Stress Level | ||||
---|---|---|---|---|---|---|---|

Ⅰ | Ⅱ | Ⅲ | Ⅰ | Ⅱ | Ⅲ | ||

Whitworth | 0.9674 | 0.9799 | 0.9814 | Brountman | 0.9975 | 0.9849 | 0.7920 |

Yang | 0.9917 | 0.9841 | 0.9934 | Scaff | 0.9859 | 0.9921 | 0.6712 |

Echtermeyer | 0.9377 | 0.9223 | 0.9203 | Yao | 0.9714 | 0.9888 | 0.6005 |

Philippidis | 0.9946 | 0.9586 | 0.9675 | Philippidis | 0.9849 | 0.9109 | 0.5772 |

Yao | 0.9958 | 0.9845 | 0.9897 | Stojković | 0.9856 | 0.9224 | 0.5820 |

New model | 0.9961 | 0.9897 | 0.9937 | New model | 0.9886 | 0.9995 | 0.8648 |

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**MDPI and ACS Style**

Wang, C.; Zhang, J.
Experimental and Analytical Study on Residual Stiffness/Strength of CFRP Tendons under Cyclic Loading. *Materials* **2020**, *13*, 5653.
https://doi.org/10.3390/ma13245653

**AMA Style**

Wang C, Zhang J.
Experimental and Analytical Study on Residual Stiffness/Strength of CFRP Tendons under Cyclic Loading. *Materials*. 2020; 13(24):5653.
https://doi.org/10.3390/ma13245653

**Chicago/Turabian Style**

Wang, Chao, and Jiwen Zhang.
2020. "Experimental and Analytical Study on Residual Stiffness/Strength of CFRP Tendons under Cyclic Loading" *Materials* 13, no. 24: 5653.
https://doi.org/10.3390/ma13245653