## 2 Materials and Methods  
### Multi-Stress Acceleration Model Considering Generalized Coupling  
#### 2.1.1 Arrhenius-Based Multi-Stress Generalized Coupling Acceleration Model  
The Arrhenius model originated in the field of chemistry and is widely used in accelerated test modeling in papers [11; 12; 16; 17; 18; 19; 20]. It is generally used to describe the relationship between the product life characteristic quantity and the applied temperature stress, which can be expressed as Equation (1). \[k(T)=\alpha_{0}\exp(\frac{E_{a}}{k_{B}T}) \tag{1}\]  
where \(k(T)\) represents the life characteristics and \(\alpha_{0}\) is the product coefficient, which is a constant and is only related to its failure mechanism. \(E_{a}\) is the activation energy of the chemical reaction, \(k_{B}\) is the Boltzmann constant, and \(T\) is the temperature stress in Kelvin temperature. Equation (1) shows that life characteristics are positively correlated with chemical reaction speed. Since \(k_{B}\) is a constant, for the sake of generalization, \(E_{a}/k_{B}\) is abbreviated to \(\alpha_{1}\) and \(1/T\) is abbreviated to \(X_{1}\); that is, Equation (1) can be simplified to Equation (2). \[k(T)=\alpha_{0}\exp(\alpha_{1}X_{1}) \tag{2}\]  
where \(X_{1}\) represents the effect of temperature on life and \(\alpha_{1}\) represents the influence coefficient of temperature. Scientists studied the life-related model under dual stress, including temperature and voltage, as well as temperature and humidity, based on the traditional Arrhenius chemical model, as shown in Equation (3) [21; 22; 23; 24; 25]. \[k(T,X_{2})=\alpha_{0}\exp(\frac{E_{a}}{k_{B}T})\cdot\exp(\alpha_{2}X_{2}+\frac {\alpha^{\prime}{}_{3}X_{2}}{k_{B}T}) \tag{3}\]  
where \(X_{2}\) represents any non-temperature stress, \(\alpha^{\prime}{}_{3}X_{2}/k_{B}T\) represents other stress options such as humidity and current that may be coupled with temperature, and \(\alpha_{2}\) and \(\alpha_{3}\) are substitutes for unknowns in different material environments. Among them, \(\alpha^{\prime}{}_{3}/k_{B}\) can be simplified to \(\alpha_{3}\). This formula can be simplified to Equation (4). \[k(T,X_{2})=\alpha_{0}\exp(\alpha_{1}X_{1})\cdot\exp(\alpha_{2}X_{2})\cdot\exp (\alpha_{3}X_{1}X_{2}) \tag{4}\]  
Similar to the dual-stress reaction rate model, the N stress reaction rate models based on the Arrhenius model will consist of single-stress coupling terms, dual-stress coupling terms, triple-stress coupling terms, quadruple-stress coupling terms, etc., where m stress coupling terms contain N stress coupling terms, a total of N terms, where m stress coupling terms contain \(C_{N}^{m}=N!/[(N-m)!m! ]\) elements, for example, the dual-stress coupling term will contain 3 elements, and the triple-stress coupling term will contain 9 elements. In summary, the above model can be expanded to N different stress life models, which are Equation (5):  
\[k(\alpha_{1},\alpha_{2},\ldots\alpha_{N})=\alpha_{0}\prod\limits_{m=1}^{N} \exp(\alpha_{m}X_{m})\times\prod\limits_{m=1,n=1,m\
eq n,m<n,}^{N}\exp(\alpha _{a}X_{m}X_{n})\times\]  
\[ \begin{array}{c}\prod\limits_{m=1}^{N}\exp(\alpha_{b}X_{m}X_{n}X_{p})\times \cdots\exp(\alpha_{N}X_{1}X_{2}\cdots X_{N})\\ m=1,n=1,p=1,m\
eq n\
eq p,m<n<p,\\ b=N+C_{N}^{2}+1,N+C_{N}^{2}+2,\ldots,N+C_{N}^{3}\end{array}  \tag{5}\]  
\(X_{*}\) represents \(N\) types of stress, such as temperature-humidity stress, and \(N\) stress triple-stress coupling terms, such as temperature-humidity-current stress, until all \(N\) stresses are coupled. Here, \(\alpha_{*}\) represents the unknown parameter of the model, and \(n>m\) is to ensure that the same term is not included in the consecutive multiplication terms. It should be noted that when \(N\leq 2\), the second and subsequent multiplication terms do not exist in Equation (5), and the double-stress coupling term is represented by the last term in Equation (5).