# A Non-Arrhenius Model for Mechanism Consistency Checking in Accelerated Degradation Tests

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## Abstract

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## 1. Introduction

#### 1.1. A Brief Introduction to the Accelerated Lifetime and Degradation Tests

#### 1.2. Literature Review

- A non-Arrhenius model is proposed to portray the degradation behavior of electromagnetic relays under ADT. The model is based on the theory of the crystal vibration energy of the material of the spring. As compared to the conventional Arrhenius-based degradation model, EFM is able to better capture the temperature characteristics of coefficients in the degradation model (TCCDM) over a much wide temperature range (see Figure 3).
- A procedure for degradation mechanism consistency checking is devised. The procedure leverages the statistical hypothesis (i.e., the ${\chi}^{2}$ test), and checks whether the parameter characterizing the degradation mechanism changes or not over temperature. Analytic expressions of the partial derivatives to the likelihood function are derived, and the Bayesian information criterion is employed to compare EFM- and Arrhenius-based degradation models.
- The proposed model can be used to explain the degradation of a wide range of materials and components, such as the capacitor or rubber.

## 2. Mathematical Description of the Arrhenius Model

- The temperature characteristics of degradation parameters (TCDPs) (e.g., the force of the spring over temperature) align with the shape that the Arrhenius model prescribes.
- The temperature characteristics of coefficients in the degradation model (TCCDM) (e.g., d only changes with temperature) align with the shape of TCDP.

## 3. Theory of the EFM

#### 3.1. Temperature Characteristics Based on Crystal Vibration Energy

#### 3.2. Comparison of the Arrhenius Model and EFM

## 4. Procedure for Mechanism Consistency Checking

- Establish a degradation model;
- Define a criterion for mechanism consistency checking.

#### 4.1. Degradation Model

#### 4.2. Criteria for Mechanism Consistency Checking

#### 4.3. Parameter Estimation

## 5. Case Study on the Degradation of Electromagnetic Relays

#### 5.1. Stress Relaxation as the Degradation Parameter

#### 5.2. Loss of Spring Force as the Degradation Parameter

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**An illustration of the lifetime prediction of the spring under the accelerated lifetime test (ALT). The time span from the beginning of the test to the spring breakage varies with temperature. If the times-to-breakage at different temperatures are connected with a curve, the curve shows an Arrhenius relationship.

**Figure 2.**An illustration of the lifetime prediction of the spring under the accelerated degradation test (ADT). The variation of the force (i.e., the degradation parameter of the springs) with time, and at four different temperatures, are shown by the four colored curves. The intersections of these degradation curves with a pre-specified degradation threshold, as marked by the green squares, show an Arrhenius relationship.

**Figure 5.**Vibration energy of crystals, temperature, and the force of the spring. A spring that has a considerable number of crystals. The vibrational energy of the crystals shows a normal distribution. The sum of each crystal’s vibrational energy greater than the threshold p is proportional to the force of the spring. As the temperature T increases, the threshold ${p}^{*}-{c}^{*}T$ in Equation (9) decreases, and the force of the spring gradually becomes smaller, as shown by the solid line in Figure 5.

**Figure 6.**A platform for theory testing. The mechanical sensor is placed inside a temperature-controlled chamber to measure how the spring force changes with temperature. The power supply provides energy to the mechanical sensor while the processing unit analyzes its signal. The display unit shows the processed results of the test information. The temperature-controlled chamber is used to alter the spring’s temperature environment.

**Figure 7.**Temperature characteristics of the force of the spring. The shape of the solid blue line is similar to that in Figure 5. This is evidence that the proposed model is reasonable.

**Figure 8.**Lifetime prediction based on the Arrhenius model and EFM. The sum of squares error (SSE) of the Arrhenius model is 98,985.29 and that of EFM is 2.31. The smaller the SSE, the higher the fitting accuracy.

**Figure 10.**Release time of electromagnetic relays under ADT. The colored data are the results of the measurements and the black data are the mean values of the samples at one temperature.

**Figure 13.**Fitting results of TCCDM in electromagnetic relays. In ${H}_{1}$, ${g}^{*}$ is not limited by additional conditions; in ${g}^{*}$, the results of EFM and the Arrhenius model in ${H}_{1}$ are the same.

Accelerated Model | Parameters | ${\mathit{H}}_{0}$ | ${\mathit{H}}_{1}$ | ${\mathit{H}}_{0}$ | ${\mathit{H}}_{1}$ |
---|---|---|---|---|---|

Arrhenius model | A | - | - | $268.77$ | - |

${E}_{a}/R$ | - | - | $2941.92$ | - | |

${\kappa}_{1}$ | - | - | - | $0.07$ | |

${\kappa}_{2}$ | - | - | - | $0.08$ | |

${\kappa}_{3}$ | - | - | - | $0.08$ | |

${\kappa}_{4}$ | - | - | - | $0.18$ | |

${\kappa}_{5}$ | - | - | - | $0.23$ | |

EFM | ${a}^{*}$ | $0.08$ | - | - | - |

${p}^{*}$ | $24.96$ | - | - | - | |

${c}^{*}$ | $0.06$ | - | - | - | |

b | $0.08$ | - | - | - | |

${\mathcal{F}}_{1}$ | - | $0.07$ | - | - | |

${\mathcal{F}}_{2}$ | - | $0.08$ | - | - | |

${\mathcal{F}}_{3}$ | - | $0.08$ | - | - | |

${\mathcal{F}}_{4}$ | - | $0.18$ | - | - | |

${\mathcal{F}}_{5}$ | - | $0.23$ | - | - | |

${\theta}_{1}^{*}$ | $1.09$ | $1.11$ | $1.17$ | $1.11$ | |

${\theta}_{2}^{*}$ | $1.09$ | $1.06$ | $1.05$ | $1.06$ | |

${\theta}_{3}^{*}$ | $1.05$ | $1.06$ | $0.90$ | $1.06$ | |

${\theta}_{4}^{*}$ | $0.55$ | $0.54$ | $0.66$ | $0.54$ | |

${\theta}_{5}^{*}$ | $0.42$ | $0.43$ | $0.26$ | $0.43$ | |

$\sigma $ | $0.07$ | $0.07$ | $0.07$ | $0.07$ |

Accelerated Model | $ln{\mathcal{L}}_{0}\left({\widehat{\mathsf{\Theta}}}_{0}^{\mathbf{MLE}}\right)$ | $ln{\mathcal{L}}_{1}\left({\widehat{\mathsf{\Theta}}}_{1}^{\mathbf{MLE}}\right)$ | $\mathsf{\Lambda}$ | ${\mathsf{\chi}}_{\mathsf{\alpha}}^{2}\left(\left|{\mathsf{\xi}}_{{\mathit{H}}_{0}}-{\mathsf{\xi}}_{{\mathit{H}}_{1}}\right|\right)$ | Outcome |
---|---|---|---|---|---|

Arrhenius model | $568.55$ | $582.64$ | $28.18$ | ${\chi}_{0.05}^{2}\left(3\right)=7.815$ | Reject ${H}_{0}$ |

EFM | $582.30$ | $582.64$ | $0.68$ | ${\chi}_{0.05}^{2}\left(1\right)=3.841$ | Retain ${H}_{0}$ |

Temperature | Time | Force Loss | Degradation Rate | Arrhenius | EFM |
---|---|---|---|---|---|

(K) | (h) | (N) | (${10}^{-6}$N/h) | $\mathsf{\u03f5}$ | $\mathsf{\u03f5}$ |

$298.15$ | 3456 | $0.06$ | $15.77$ | $-0.05$ | $-2.35$ |

$353.15$ | 3216 | $0.15$ | $47.67$ | $12.30$ | $19.05$ |

$373.75$ | 3456 | $0.43$ | $123.47$ | $-34.01$ | $-21.25$ |

$396.95$ | 3456 | $0.47$ | $136.95$ | $-3.40$ | $5.66$ |

$423.15$ | 3456 | $0.61$ | $176.51$ | $22.67$ | $0.74$ |

Model | $\mathsf{\sigma}$ | $ln\widehat{\mathcal{L}}$ | BIC |
---|---|---|---|

Arrhenius | $2.14\times {10}^{-5}$ | $47.16$ | −91.11 |

EFM | $1.46\times {10}^{-5}$ | $49.08$ | −91.72 |

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**MDPI and ACS Style**

You, J.; Fu, R.; Liang, H.; Lin, Y.
A Non-Arrhenius Model for Mechanism Consistency Checking in Accelerated Degradation Tests. *Actuators* **2023**, *12*, 319.
https://doi.org/10.3390/act12080319

**AMA Style**

You J, Fu R, Liang H, Lin Y.
A Non-Arrhenius Model for Mechanism Consistency Checking in Accelerated Degradation Tests. *Actuators*. 2023; 12(8):319.
https://doi.org/10.3390/act12080319

**Chicago/Turabian Style**

You, Jiaxin, Rao Fu, Huimin Liang, and Yigang Lin.
2023. "A Non-Arrhenius Model for Mechanism Consistency Checking in Accelerated Degradation Tests" *Actuators* 12, no. 8: 319.
https://doi.org/10.3390/act12080319