## 3 Establishment of Constitutive Rubber Model of Elastic Cam Based on the Phenomenological Theory
### Determination of Different Rubber Constitutive Model Coefficients  
The stress and strain results of the rubber test were input into the Abaqus software material physical properties interface in the form of an array. The Ogden1-order, Ogden2-order, Ogden3-order, and M-R methods were selected for fitting evaluation, and the corresponding solving coefficients based on the different evaluation methods were displayed on the evaluation interface, as shown in Table 2, where the variance S\({}^{2}\) is the average square of the difference between the fitted and test values. The solution coefficients based on the model coefficients in Table 2 were substituted into Equations (14)-(16), and Equation (18), respectively. The stress-strain curves of the different rubber constitutive models are presented in Figure 8.  
The test data are shown in Figure 8. By comparing the results of the different rubber constitutive models with test data, the relationship between strain and stress in the M-R constitutive model tends to increase linearly. The relationship between strain and stress of the Ogden1-order, Ogden2-order, and Ogden3-order constitutive models is nonlinear under the condition of large deformation. With the increasing rubber strain, the stress of  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Constitutive Model Name** & **Solution Coefficients** & **Variance S\({}^{2}\)** \\ \hline Ogden1-order & \(\mu=1.097\), \(\alpha=2.508\) & 0.296 \\ \hline Ogden2-order & \(\mu_{1}=1.552\), \(\alpha_{1}=2.608\), \(\mu_{2}=-0.814\), \(\alpha_{2}=-5.734\) & 0.11 \\ \hline Ogden3-order & \(\mu_{1}=1.524\), \(\alpha_{1}=2.528\), \(\mu_{2}=5.16\times 10^{-8}\), \(\alpha_{2}=9.22\), \(\mu_{3}=-0.757\), \(\alpha_{3}=-5.256\) & 0.0246 \\ \hline M-R & \(C_{10}=1.26\), \(C_{01}=-1.04\) & 2.786 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Coefficients under different constitutive models.  
Figure 8: The results of different rubber constitutive models compared with test data.  
the Ogden3-order is closer and closer to the test value, and the deviation of M-R stress from the test value increases. As shown in Figure 9, when the strain is less than 4, the fitting error of the Ogden1-order is lower than that of M-R, Ogden2-order, and Ogden3-order, and is less than 5%. The one with the greatest error is the M-R stress fitting value, whose error reaches up to 23%. When the strain is greater than 4, the Ogden3-order stress value has the lowest error, and the error is less than 3%. Therefore, the function of nominal stress and strain is characterized by strain equal to 4 as the dividing point, in other words, the abscissa coordinates corresponding to the point where the four curves intersect, and the error is the least at the dividing point of the piecewise function.  
The nominal stress-strain of the rubber constitutive model is simulated as a piecewise function:  
\[\sigma_{\text{nom}}=\begin{cases}\mu_{1}\Big{[}(1+\varepsilon)^{\alpha_{1}-1} -(1+\varepsilon)^{-\frac{1}{2}\alpha_{1}-1}\Big{]}\\ \mu_{1}\Big{[}(1+\varepsilon)^{\alpha_{1}-1}-(1+\varepsilon)^{-\frac{1}{2} \alpha_{1}-1}\Big{]}+\mu_{2}\Big{[}(1+\varepsilon)^{\alpha_{2}-1}-(1+ \varepsilon)^{-\frac{1}{2}\alpha_{2}-1}\Big{]}+\mu_{3}\Big{[}(1+\varepsilon)^ {\alpha_{3}-1}-(1+\varepsilon)^{-\frac{1}{2}\alpha_{3}-1}\Big{]}\end{cases} \quad(\varepsilon\leq 4) \tag{21}\]  
Introduce a step function:  
\[\text{H}(\text{t})=\left\{\begin{array}{l}1(\text{t}>0)\\ 0(\text{t}\leq 0)\end{array}\right. \tag{22}\]  
Let t = \(\varepsilon-4\), substitute it into Equation (22), and then substitute the result into Equation (21):  
\[\sigma_{\text{nom}}=\mu_{1}\Big{[}(1+\varepsilon)^{\alpha_{1}-1