# Parameter Identification Methods for Hyperelastic and Hyper-Viscoelastic Models
## 2 Parameter Identification Method for the Hyperelastic Model
### Theories of the Ogden Model  
According to the theories of the continuum mechanics, there exists a strain energy function \(W\) for the hyperelasticity properties of rubbers. The stresses can be obtained by the partial derivative of the variable \(W\) with respect to the strain.  
\[S=2\frac{\partial W}{\partial C} \tag{1}\]  
where \(C\) is the right Cauchy-Green deformation gradient tensor, \(S\) is the second Piola-Kirchhoff stress.  
After transforming the second Piola-Kirchhoff stress to the Cauchy stress [30], we will get  
\[T=-pI+2\frac{\partial W}{\partial I_{1}}B-2\frac{\partial W}{\partial I_{2}}B^ {-1} \tag{2}\]  
where F is the deformation gradient tensor, \(B\) is the left Cauchy-Green deformation gradient tensor, \(I_{1},I_{2},I_{3}\) are the three strain invariants of \(B\), and \(p\) is the undetermined hydrostatic pressure which can be decided by the underlying equilibrium and boundary conditions of the particular problem.  
In view of the nearly incompressible properties of the volume of rubbers, the strain energy of the Ogden model can be defined with Equation (3) without regard for the contribution of the volume strain.  
\[W=\sum_{n}\frac{\mu_{n}}{\alpha_{n}}(\lambda_{1}{}^{\alpha_{n}}+\lambda_{2}{} ^{\alpha_{n}}+\lambda_{3}{}^{\alpha_{n}}-3) \tag{3}\]  
In Equation (3), \(n\) is the number of terms considered for the Ogden model, \(\mu_{n}\), \(\alpha_{n}\) are the undetermined parameters of the model, \(\lambda_{i}\) is the principal stretch.  
In the uniaxial tension experiment, the deformation gradient tensor F and the left Cauchy-Green deformation gradient tensor B are described as  
\[B=\left(\begin{array}{ccc}\lambda_{1}{}^{2}&0&0\\ 0&1/\lambda_{1}&0\\ 0&0&1/\lambda_{1}\end{array}\right) \tag{4}\]  
Taking the boundary condition \(T_{22}=T_{33}=0\) into consideration, the corresponding nominal stress for uniaxial tension is  
\[f_{1}=T_{11}/\lambda_{1}=\sum_{n}\mu_{n}(\lambda_{1}{}^{\alpha_{n}-1}-\lambda_ {1}{}^{-\alpha_{n}/2-1}) \tag{5}\]  
For the biaxial tension experiment, the boundary conditions are \(T_{22}=T_{33}\), \(T_{11}=0\), \(\lambda_{2}=\lambda_{3}\), and the nominal stress is  
\[f_{2}=T_{22}/\lambda_{2}=\sum_{n}\mu_{n}(\lambda_{2}{}^{\alpha_{n}-1}-\lambda_ {2}{}^{-2\alpha_{n}-1}) \tag{6}\]  
The boundary condition of the planar tension experiment is \(T_{33}=0,\lambda_{2}=1\), and the nominal stress is  
\[f_{3}=T_{11}/\lambda_{1}=\sum_{n}\mu_{n}(\lambda_{1}{}^{\alpha_{n}-1}-\lambda_ {1}{}^{-\alpha_{n}-1}) \tag{7}\]  
As for the simple shear experiment, the direction of the principal stretch is not consistent with the direction of the applied deformation, rather it involves a rotation of axes [1]. The left Cauchy-Green deformation tensor \(B\) is expressed as  
\[B=\left(\begin{array}{ccc}1+\gamma^{2}&\gamma&0\\ \gamma&1&0\\ 0&0&1\end{array}\right) \tag{8}\]and the nominal shear stress is shown in Equation (9).  
\[f_{4}=T_{12}=\frac{\sum\limits_{n}\mu_{n}({\lambda_{1}}^{\alpha_{n}}-{\lambda_{1} }^{-\alpha_{n}})}{\lambda_{1}+1/\lambda_{1}} \tag{9}\]  
where the principal stretch \(\lambda_{1}\) can be obtained by calculating the eigenvalues of the tensor \(B\).  
\[\lambda_{1}=\sqrt{1+\frac{\gamma^{2}}{2}+\gamma\sqrt{1+\frac{\gamma^{2}}{4}}} \tag{10}\]