# Parameter Identification Methods for Hyperelastic and Hyper-Viscoelastic Models
## 3 Parameter Identification Method for the Hyper-Viscoelastic Model
### Theories of the Hyper-Viscoelastic Model  
The integration of the relaxation modulus for viscoelasticity is expressed in Equation (12)  
\[\sigma(t)=\int_{-\infty}^{t}Y(t-\xi)d\varepsilon(\xi) \tag{12}\]  
Figure 2: The FE (finite element) models for numerical verification: (a)initial model of uniaxial tension experiment; (b)deformed model of uniaxial tension experiment (\(\lambda=5\)); (c) initial model of planar tension experiment; (d) deformed model of planar tension experiment (\(\lambda=6.164\)); (e) initial model of biaxial tension experiment; (f) deformed model of biaxial tension experiment (\(\lambda=5.04\)).  
where \(Y(t-\xi)\) is the relaxation modulus function for characterizing the viscoelastic properties, and \(d\epsilon(\xi)\) is the micro-strain of materials. Generally, when \(\xi<0\), the outer action is assumed to be 0, so Equation (12) will be changed to  
\[\sigma(t)=\int_{0}^{t}Y(t-\xi)d\epsilon(\xi) \tag{13}\]  
Equation (13) can be calculated by dividing the time \(t\) into \(n\) equal portions, then we will get  
\[\sigma(t)=\sum_{i=1}^{n}\int_{t_{i}}^{t_{i+1}}Y(t-\xi)d\epsilon(\xi) \tag{14}\]  
When \(\xi\in(t_{i},t_{i+1})\), the Young modulus of materials is assumed to be a constant \(E_{i}\), and the Prony series is introduced to characterize the relaxation properties. Besides, the strain loading speed is also deemed as a constant \(\lambda_{i}\), then Equation (14) will become  
\[\sigma(t)=\sum_{i=1}^{n}\int_{t_{i}}^{t_{i+1}}\lambda_{i}E_{i}(\mu_{0}+\sum_{ k=1}^{m}\mu_{k}e^{-(t-\xi)/t_{k}})d\xi, \tag{15}\]  
where \(\mu_{k}\) is the weight of modulus corresponding to the relaxation time \(t_{k}\), \(m\) is the number of terms considered for the Prony series, \(\mu_{0}\) is the weight of the equilibrium modulus. It is obvious for the equation \(\sum\limits_{k=0}^{m}\mu_{k}=1\) to be true, and the physical model of Equation (15) is presented in Figure 3.  
In Figure 3, \(E_{i}\), \(\mu_{0}\), \(\mu_{k}\) shares the same meaning with that in Equation (15), \(\eta_{k}\) represents the viscosity coefficient and \(\eta_{k}=\mu_{k}E_{i}t_{k}\), \(t_{k}\) is the relaxation time included in Equation (15).  
This physical model includes two main features, the first one is that it substitutes the stretch-dependent modulus spring for the constant modulus spring of the generalized Maxwell model, which is based on an assumption that the modulus corresponded to the viscosity coefficient \(\eta_{k}\) is obtained by multiplying the instantaneous modulus \(E_{i}\) and the weight \(u_{k}\). The second one is that the instantaneous modulus \(E_{i}\) can be decided by the hyperelasticity properties of materials, and the weight \(u_{k}\) is the undetermined parameters which are time independent.  
If we stipulate that \(t_{k}=10^{k-1}t_{1}\), which is adopted by the FE software LS-DYNA [33], then, \(\sigma(t)\)can be expressed as  
\[\sigma(t)=\sum_{i=1}^{n}\int_{t_{i}}^{t_{i+1}}\lambda_{i}E_{i}(\mu_{0}+\sum_{ k=1}^{m}\mu_{k}e^{-(t-\xi)/(10^{k-1}t_{1})})d\xi, \tag{16}\]  
Figure 3: Physical model of the hyper-viscoelastic model.  
As mentioned before, \(E_{i}\) represents the instantaneous modulus of materials within the time range from \(t_{i}\) to \(t_{i+1}\). It can be solved by the following procedures. Firstly, the fast stretch-stress experiment data is obtained by conducting experiment at sufficiently high speed to exclude the viscous effect. Then they will be fitted by the Ogden model to decide the undetermined parameters, which has been explained in the previous section. After acquiring the stretch-stress curve based on the Ogden model, \(E_{i}\) will be solved by the derivation of the stress \(\sigma\) with respect to the principal stretch \(\lambda\), namely, \(\left(\frac{\partial\sigma}{\partial\lambda}\right)_{t=t_{i}}\), or taking the secant modulus as a substitution for the tangent modulus, namely, \(\left(\sigma_{i+1}-\sigma_{i}\right)/\left(\lambda_{i+1}-\lambda_{i}\right)\).