## 3 Establishment of Constitutive Rubber Model of Elastic Cam Based on the Phenomenological Theory  
### Establishment of Nominal Stress and Elongation Function Based on Large Deformation of Rubber  
The statistical thermodynamic constitutive model of rubber only applies to the slight deformation of rubber. However, rubber's most important mechanical feature is that it can produce large elastic deformation. In order to describe the stress-strain behavior of rubber under considerable strain, most research and engineering applications have focused on using continuum mechanics methods to conduct phenomenological analysis of rubber under considerable strain [25]. The phenomenological theory does not study the microstructure and molecular mechanism of rubber but rather observes the macroscopic stress-strain relationship of rubber by the experimental method, finding the expression of strain energy stored in the rubber elastomer, and on this basis, establishing the theory. Based on the continuum mechanics of phenomenological theory, two primary expressions exist for the strain energy function of rubber's constitutive model. One is the strain energy function expressed by the principal elongation, such as the Ogden model, the other is the strain energy function expressed by the strain invariant, such as the Mooney-Rivlin (M-R) model. 1. Nominal Stress and Elongation Function based on the Ogden Rubber Constitutive Model Ogden strain energy function W is expressed as principal elongation [26]: \[\text{W}=\sum_{i=1}^{N}\frac{\mu_{i}}{\alpha_{i}}(\Lambda_{1}^{a_{i}}+\Lambda _{2}^{a_{i}}+\Lambda_{3}^{a_{i}}-3)\] (10)where \(\lambda_{i}\) is the elongation in three directions, and \(\alpha_{i}\) and \(\mu_{i}\) are material constants and are not limited to integers. According to the incompressible characteristics of rubber volume, there is \(\lambda_{1}\lambda_{2}\lambda_{3}=1\). The nominal stress can be obtained directly by the deflection of strain energy function W with respect to the elongation ratio:  
\[\sigma_{nom\_i}=\frac{\partial W}{\partial\lambda_{i}}-p\lambda_{i}^{-1} \tag{11}\]  
where \(p\) is the undetermined Lagrange multiplier, which can be determined by the boundary conditions. Based on rubber uniaxial tensile test:  
\[\left\{ \begin{array}{l}\lambda_{1}=\lambda\\ \lambda_{2}=\lambda^{-\frac{1}{2}}\\ \lambda_{3}=\lambda^{-\frac{1}{2}}\\ \sigma_{2}=\sigma_{3}=0\end{array} \right. \tag{12}\]  
By substituting the Equation (10) into Equation (11) and combined with Equation (12), we derive:  
\[\sigma_{\text{nom}}=\sum_{i=1}^{N}\mu_{i}(\lambda^{\alpha_{i}-1}-\lambda^{- \frac{1}{2}\alpha_{i}-1}) \tag{13}\]  
Expanding Equation (13), when \(N=1\), the strain energy function is called the Ogden1-order model, and the nominal stress \(\sigma_{nom}\):  
\[\sigma_{\text{nom}}=\mu_{1}(\lambda^{\alpha_{1}-1}-\lambda^{-\frac{1}{2} \alpha_{1}-1}) \tag{14}\]  
When \(N=2\), the strain energy function is called the Ogden2-order model, and the nominal stress \(\sigma_{nom}\):  
\[\sigma_{\text{nom}}=\mu_{1}(\lambda^{\alpha_{1}-1}-\lambda^{-\frac{1}{2} \alpha_{1}-1})+\mu_{2}(\lambda^{\alpha_{2}-1}-\lambda^{-\frac{1}{2}\alpha_{2} -1}) \tag{15}\]  
When \(N=3\), the strain energy function is called the Ogden3-order model, and the nominal stress \(\sigma_{nom}\):  
\[\sigma_{\text{nom}}=\mu_{1}(\lambda^{\alpha_{1}-1}-\lambda^{-\frac{1}{2} \alpha_{1}-1})+\mu_{2}(\lambda^{\alpha_{2}-1}-\lambda^{-\frac{1}{2}\alpha_{2} -1})+\mu_{3}(\lambda^{\alpha_{3}-1}-\lambda^{-\frac{1}{2}\alpha_{3}-1}) \tag{16}\]  
Through uniaxial tensile test, coefficients \(\alpha_{i}\) and \(\mu_{i}\) can be obtained respectively. 1.