# Fatigue Reliability Analysis of Composite Material Considering the Growth of Effective Stress and Critical Stiffness

^{*}

## Abstract

**:**

_{2S}laminates are utilized to verify the proposed probabilistic models. Finally, the effective stress growth mechanism and its influence on the failure threshold are elaborated, and a pair of fatigue reliability models for composite materials are developed. Moreover, the differences between the strength-based and stiffness-based reliability analysis results of composite materials are compared and discussed.

## 1. Introduction

## 2. Fatigue Damage Characterized by Performance Degradation

_{2S}laminates are utilized to evaluate the fitting accuracy of different fatigue damage models.

#### 2.1. Fatigue Damage Characterized by Strength Degradation

_{2S}laminates in Ref. [38] are utilized to evaluate the fitting accuracy of the above four strength-based fatigue damage models. Note from Ref. [38] that both the static and fatigue tests of the Gr/PEEK [0/45/90/−45]

_{2S}laminates are conducted on the MTS 810 machine. Based on the strength degradation data from Ref. [38], the nonlinear least square method is applied to estimate the parameters (i.e., $a$, $b$ and $c$) of the above four models. The parameter estimation results are shown in Table 2.

_{2S}laminates with good agreement. By comparison, the fitting accuracy of Yao’s model is higher than that of other three models. Specifically, the goodness of fit values of Yao’s model under three different stress levels are 0.9992, 0.9991 and 0.9993, respectively. Based on the parameter estimation results listed in Table 2, the fitting curves of these four strength-based fatigue damage models are plotted, as shown in Figure 1.

_{2S}laminates exhibits a slow–fast trend during fatigue failure. Fatigue damage accumulates slowly when the cycle ratio is $0<\lambda <0.8$, while it accumulates rapidly when the cycle ratio is $0.8<\lambda <1$. The later stage is always regarded as the ‘sudden death’ behavior of a composite material under high cyclic loading. In this study, Yao’s model is introduced to quantify the strength-based fatigue damage of the Gr/PEEK [0/45/90/−45]

_{2S}laminates because of its superior fitting accuracy. Therefore, the strength-based fatigue damage model of the composite material can be expressed as

#### 2.2. Fatigue Damage Characterized by Stiffness Degradation

_{2S}laminates from Ref. [38] are utilized to examine the fitting accuracy of these four stiffness-based fatigue damage models. Based on the stiffness degradation data in Ref. [38], the nonlinear least square method is used to calculate the parameters of the above four models. The parameter calculation results are listed in Table 4.

_{2S}laminates. By comparison, Gao’s model has higher fitting accuracy than other three models. For three different stress levels, the goodness of fit values of Gao’s model are 0.9999, 0.9987 and 0.9944, respectively. Based on the parameter calculation results shown in Table 4, the fitting curves of these four stiffness-based fatigue damage models are plotted, as shown in Figure 2.

_{2S}laminates exhibits a fast–slow–fast trend during fatigue failure. Fatigue damage grows quickly when the cycle ratio is $0<\lambda <0.2$, then it accumulates slowly when the cycle ratio is $0.2<\lambda <0.9$ and finally increases rapidly when the cycle ratio is $0.9<\lambda <1$. This phenomenon is consistent with the existing studies, such as Refs. [19] and [22]. In this study, Gao’s model is introduced to characterize the stiffness-based fatigue damage of the Gr/PEEK [0/45/90/−45]

_{2S}laminates due to its excellent fitting accuracy. Under such circumstances, the stiffness-based fatigue damage model of a composite material can be formulated as

## 3. Performance Degradation of Composite Material

_{2S}laminates are adopted to verify the proposed probabilistic models.

#### 3.1. Probabilistic Model of Residual Strength

_{2S}laminates is plotted, as shown in Figure 3.

_{2S}laminates. However, due to the random distribution of original defects (such as voids, bubbles and broken fibers) in the composite material, both initial strength ${S}_{0}$ and residual strength $S\left(\lambda \right)$ exhibit significant dispersion. It is generally considered that the dispersions of initial strength and residual strength are dominated by identical original defects inside the composite material [43]. The initial strength data (i.e., ultimate strength data from Ref. [38]) of the Gr/PEEK [0/45/90/−45]

_{2S}laminates are shown in Table 5.

_{2S}laminates are plotted, as shown in Figure 4.

_{2S}laminates. As mentioned above, the dispersions of initial strength and residual strength depend on the same original defects inside a composite material. Therefore, it is reasonable to deduce the PDF of residual strength from the PDF of initial strength. Based on the probability theory, the CDF of residual strength is formulated as

_{2S}laminates is a random variable.

#### 3.2. Probabilistic Model of Residual Stiffness

_{2S}laminates is obtained, as shown in Figure 6.

_{2S}laminates with good agreement. As a result of the influence of uncertain factors, various original defects will be produced in a composite material. The randomness of original defects has a significant influence on initial stiffness ${E}_{0}$ and residual stiffness $E\left(\lambda \right)$. Therefore, it is necessary to explore the stiffness degradation rule of a composite material from the probability point of view. The initial stiffness data (i.e., elastic modulus data from Ref. [38]) of the Gr/PEEK [0/45/90/−45]

_{2S}laminates are listed in Table 6.

_{2S}laminates. It is generally considered that the dispersions of initial stiffness and residual stiffness are dominated by the same original defects. Therefore, the PDF of residual stiffness can be derived from the PDF of initial stiffness. According to the probability theory, the CDF of residual stiffness is expressed as

_{2S}laminates. In addition, the residual stiffness is a random variable under a given cycle ratio.

## 4. Effective Stress Growth and Critical Stiffness Increase

#### 4.1. Effective Stress Growth Caused by Fatigue Damage

#### 4.2. Critical Stiffness Increase Caused by Fatigue Damage

## 5. Fatigue Reliability Analysis Considering Effective Stress Growth

_{2S}laminates are analyzed based on the proposed models. Moreover, the differences between the strength-based and stiffness-based reliability approaches are compared and discussed.

#### 5.1. Improved Strength-Based Fatigue Reliability Model

_{2S}laminates is obtained, as shown in Figure 11.

_{2S}laminates, the effective stress grows nonlinearly with the increase in the cycle ratio. In other words, the nominal stress and the effective stress are equal when there is no fatigue damage in the composite material. When fatigue damage occurs and accumulates, the effective stress is larger than the nominal stress.

_{2S}laminates is plotted, as shown in Figure 12.

_{2S}laminates presents two distinct stages. At the first stage, the reliability remains at a high level without significant changes. At the second stage, the reliability decreases rapidly, which is consistent with the ‘sudden death’ behavior of a composite material. By comparison, the calculation results of the traditional strength-based reliability model are conservative. Based on Equations (24) and (25), the strength-based failure rate model of a composite material can be expressed as Equation (26). Therefore, the corresponding failure rate curve of the Gr/PEEK [0/45/90/−45]

_{2S}laminates is obtained, as shown in Figure 13.

_{2S}laminates exhibits two distinctly different periods. For a given cycle ratio, the failure rate calculated by the improved model is higher than that calculated by the traditional model. This is because the improved model considers both effective stress growth and strength degradation, while the traditional model only considers the latter.

#### 5.2. Improved Stiffness-Based Fatigue Reliability Model

_{2S}laminates is plotted, as shown in Figure 15.

_{2S}laminates, the critical stiffness grows nonlinearly with the cycle ratio. Therefore, the traditional method holds that the critical stiffness remains unchanged, which is inconsistent with the actual situation.

_{2S}laminates is obtained, as shown in Figure 16.

_{2S}laminates is plotted, as shown in Figure 17.

_{2S}laminates is similar to but different than the bathtub shape. For a specific cycle ratio, the failure rate calculated by the improved model is higher than that calculated by the traditional model. This is because the improved model considers both critical stiffness growth and stiffness degradation, while the traditional model only considers the latter.

#### 5.3. Comparison between Different Reliability Analysis Approaches

## 6. Conclusions

- (1)
- The fatigue damage accumulation of a composite material is quantified from the perspective of performance degradation. The fitting accuracy of some representative fatigue damage models is compared based on the fatigue damage data of Gr/PEEK [0/45/90/−45]
_{2S}laminates. Yao’s model and Gao’s model are adopted to characterize the strength-based and stiffness-based fatigue damage, respectively. - (2)
- The Weibull distribution is applied to depict the probability distributions of initial strength and initial stiffness. A pair of probabilistic models of residual strength and residual stiffness are developed to characterize the performance degradation of a composite material. The strength and stiffness degradation data of Gr/PEEK [0/45/90/−45]
_{2S}laminates are used to verify the developed probabilistic models. - (3)
- The effective bearing area of a composite material is treated as damage metric, and the growth mechanisms of effective stress and critical stiffness caused by fatigue damage accumulation are elaborated. A pair of strength-based and stiffness-based fatigue reliability models are proposed. The fatigue reliability and failure rate of Gr/PEEK [0/45/90/−45]
_{2S}laminates under different conditions are compared and discussed. - (4)
- The strength degradation and stiffness degradation are dominated by the same damage state of composite materials. There is a certain correlation between these two degradation behaviors. In the future, the induced mechanisms and coupling effects between strength degradation behavior and stiffness degradation behavior will be further studied and discussed.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Fitting curves of strength-based fatigue damage models: (

**a**) S = S

_{2}; (

**b**) S = S

_{3}; (

**c**) S = S

_{4}.

**Figure 2.**Fitting curves of stiffness-based fatigue damage models: (

**a**) S = S

_{2}; (

**b**) S = S

_{3}; (

**c**) S = S

_{4}.

**Figure 4.**Probability distribution of initial strength of Gr/PEEK [0/45/90/−45]

_{2S}laminates: (

**a**) PDF of initial strength; (

**b**) PDF of initial strength.

**Figure 5.**Evolution rule of PDF of residual strength under different stress levels: (

**a**) PDF of residual strength under S

_{2}; (

**b**) PDF of residual strength under S

_{3}; (

**c**) PDF of residual strength under S

_{4}.

**Figure 7.**Probability distribution of initial stiffness of Gr/PEEK [0/45/90/−45]

_{2S}laminates: (

**a**) PDF of initial stiffness under S

_{2}; (

**b**) CDF of initial stiffness under S

_{2}; (

**c**) PDF of initial stiffness under S

_{3}; (

**d**) CDF of initial stiffness under S

_{3}; (

**e**) PDF of initial stiffness under S

_{4}; (

**f**) CDF of initial stiffness under S

_{4}.

**Figure 8.**Evolution rule of PDF of residual stiffness under different stress levels: (

**a**) PDF of residual stiffness under S

_{2}; (

**b**) PDF of residual stiffness under S

_{3}; (

**c**) PDF of residual stiffness under S

_{4}.

**Figure 10.**Schematic diagram of traditional and improved strength-based fatigue reliability models: (

**a**) traditional strength-based fatigue reliability model; (

**b**) improved strength-based fatigue reliability model.

**Figure 14.**Schematic diagram of traditional and improved stiffness-based fatigue reliability model: (

**a**) traditional stiffness-based fatigue reliability model; (

**b**) improved stiffness-based fatigue reliability model.

**Figure 18.**Comparison between strength-based and stiffness-based approaches: (

**a**) fatigue reliability curve; (

**b**) failure rate curve.

Researchers | Models | Parameters |
---|---|---|

Gao [34] | ${D}_{S}\left(\lambda \right)=1-{\left(1-\lambda \right)}^{a}$ | $a$ |

Reifsnider [35] | ${D}_{S}\left(\lambda \right)=1-{\left(1-{\lambda}^{b}\right)}^{a}$ | $a$, $b$ |

Yao [36] | ${D}_{S}\left(\lambda \right)=\frac{\mathrm{sin}(a\lambda )\mathrm{cos}(a-b)}{\mathrm{sin}(a)\mathrm{cos}(a\lambda -b)}$ | $a$, $b$ |

Mu [37] | ${D}_{S}\left(\lambda \right)=1-\frac{1-{\lambda}^{a}}{1+c{\lambda}^{b}}$ | $a$, $b$, $c$ |

Stress Levels/MPa | Models | Parameters | Goodness of Fit | ||
---|---|---|---|---|---|

a | b | c | R^{2} | ||

S_{2} = 646.31 | Gao’s model | 0.0577 | — | — | 0.9771 |

Reifsnider’s model | 0.0755 | 1.433 | — | 0.997 | |

Yao’s model | 1.603 | −9.358 | — | 0.9992 | |

Mu’s model | 16.1 | 1.51 | 0.1027 | 0.9983 | |

S_{3} = 623.50 | Gao’s model | 0.0683 | — | — | 0.9793 |

Reifsnider’s model | 0.0883 | 1.411 | — | 0.9977 | |

Yao’s model | 1.611 | -9.344 | — | 0.9991 | |

Mu’s model | 15.48 | 1.518 | 0.1237 | 0.9988 | |

S_{4} = 600.68 | Gao’s model | 0.0787 | — | — | 0.9784 |

Reifsnider’s model | 0.1025 | 1.422 | — | 0.9975 | |

Yao’s model | 1.584 | 0.06 | — | 0.9993 | |

Mu’s model | 14.7 | 1.51 | 0.1417 | 0.9989 |

Researchers | Models | Parameters |
---|---|---|

Shiri [27] | ${D}_{E}\left(\lambda \right)=\frac{\mathrm{sin}(\alpha \lambda )\mathrm{cos}(\alpha -\beta )}{\mathrm{sin}(\alpha )\mathrm{cos}(\alpha \lambda -\beta )}$ | $\alpha $, $\beta $ |

Gao [19] | ${D}_{E}\left(\lambda \right)=\frac{\mathrm{sin}(\alpha \lambda )\mathrm{cos}(\beta )}{\mathrm{sin}(\alpha )\mathrm{cos}(\beta {\lambda}^{\alpha})}$ | $\alpha $, $\beta $ |

Mu [37] | ${D}_{E}\left(\lambda \right)=1-\frac{1-{\lambda}^{\alpha}}{1+\gamma {\lambda}^{\beta}}$ | $\alpha $, $\beta $, $\gamma $ |

Mao [39] | ${D}_{E}\left(\lambda \right)=\alpha {\lambda}^{\beta}+\left(1-\alpha \right){\lambda}^{\gamma}$ | $\alpha $, $\beta $, $\gamma $ |

Stress Levels/MPa | Models | Parameters | Goodness of Fit | ||
---|---|---|---|---|---|

α | β | γ | R^{2} | ||

S_{2} = 646.31 | Shiri’s model | 2.906 | 1.348 | — | 0.7566 |

Gao’s model | 2.896 | 1.556 | — | 0.9999 | |

Mu’s model | 94.56 | 0.5177 | 0.087 | 0.887 | |

Mao’s model | 0.0783 | 0.437 | 82.8 | 0.8954 | |

S_{3} = 623.50 | Shiri’s model | 2.868 | 1.316 | — | 0.8398 |

Gao’s model | 2.831 | 1.546 | — | 0.9987 | |

Mu’s model | 75.46 | 0.5184 | 0.1161 | 0.9492 | |

Mao’s model | 0.1044 | 0.4804 | 83.67 | 0.9461 | |

S_{4} = 600.68 | Shiri’s model | 2.844 | 1.3 | — | 0.8606 |

Gao’s model | 2.799 | 1.537 | — | 0.9944 | |

Mu’s model | 75.09 | 0.5569 | 0.1541 | 0.9685 | |

Mao’s model | 0.8658 | 80.78 | 0.5055 | 0.9651 |

Initial Strength Data/MPa | Mean Value/MPa | Standard Deviation/MPa |
---|---|---|

723.13, 725.19, 743.20, 748.85, 754.77, 770.98, 774.84, 776.90, 788.99, 796.70 | 760.36 | 25.42 |

Stress Level/MPa | Initial Stiffness Data/MPa | Mean value/MPa | Standard Deviation/MPa |
---|---|---|---|

S_{2} = 646.31 | 52.78, 54.29, 55.43, 56.42, 56.86, 57.63, 58.43, 58.95, 59.27, 59.84, 60.48, 61.11, 61.52, 62.16, 63.22 | 58.56 | 3.00 |

S_{3} = 623.50 | 52.83, 54.15, 55.31, 56.18, 57.15, 57.47, 57.96, 58.29, 58.76, 59.53, 59.98, 60.62, 61.38, 62.21, 63.27 | 58.34 | 2.95 |

S_{4} = 600.68 | 52.69, 54.09, 55.12, 55.78, 56.86, 57.55, 58.02, 58.63, 59.12, 59.78, 60.11, 60.98, 61.58, 62.84 | 58.08 | 2.94 |

Stress Level/MPa | Shape Parameter C | Scale Parameter D |
---|---|---|

S_{2} = 646.31 | 23.8008 | 59.9005 |

S_{3} = 623.50 | 22.9969 | 59.6813 |

S_{4} = 600.68 | 23.5257 | 59.4076 |

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Gao, J.-X.; Heng, F.; Yuan, Y.-P.; Liu, Y.-Y.
Fatigue Reliability Analysis of Composite Material Considering the Growth of Effective Stress and Critical Stiffness. *Aerospace* **2023**, *10*, 785.
https://doi.org/10.3390/aerospace10090785

**AMA Style**

Gao J-X, Heng F, Yuan Y-P, Liu Y-Y.
Fatigue Reliability Analysis of Composite Material Considering the Growth of Effective Stress and Critical Stiffness. *Aerospace*. 2023; 10(9):785.
https://doi.org/10.3390/aerospace10090785

**Chicago/Turabian Style**

Gao, Jian-Xiong, Fei Heng, Yi-Ping Yuan, and Yuan-Yuan Liu.
2023. "Fatigue Reliability Analysis of Composite Material Considering the Growth of Effective Stress and Critical Stiffness" *Aerospace* 10, no. 9: 785.
https://doi.org/10.3390/aerospace10090785