## 4 Results  
We have presented the measurements and the energy-based method for determining the material constants of a nonlinear material equation. From the measurements, we construct the \(\mathbf{b}\) vector. The matrix of integrals, \(\mathbf{A}\), is generated by computing the integrals numerically with Simpson's rule. We use standard numerical techniques accessible via open-source packages in Python language. Concretely, we employ NumPy [37] for computation and MatPlotLIB [38] for visualization. It is of paramount importance to recall that the solution of the material constants are unique since we solve a linear regression problem. The computation is only some linear algebra steps lasting no more than a minute (Python 2.7 running in an UbuntuTM 14.04 working on a single IntelTM Core i7-4650U with a 1700-MHz processor). We start with a linear model and increase the complexity as explained in the latter section. Every trial is accomplished quickly owing to the energy-based method. In order to present the followed strategy clearly, we start with the linear material equation,  
\[d_{ij}=\,^{\circ}\!\!c_{i}\sigma_{ij}+\,^{\circ}\!\!c_{z}\sigma_{ij}^{*}+\,^{ \circ}\!\!c_{y}\sigma_{ij}^{*}\, \tag{36}\]  
where the rate of stress is involved such that we may name it the Maxwell model. We apply the code for determining \(\,^{\circ}\!\!c_{1},\,^{\circ}\!\!c_{2},\,^{\circ}\!\!c_{3}\) and plot experiments as dots and the material model as continuous lines in Figure 3.  
The lower frequencies and small amplitudes can be approximated with this model. For higher frequencies and amplitudes, this model fails to approximate the behavior. Therefore, we try a more complicated model and determine the material parameters for the following cubic model:  
\[ \begin{split}& d_{ij}=\big{(}\,^{\circ}\!\!c_{1}+\,^{\circ}\!\!c_{1} II_{\mathbf{\sigma}}+\,^{\circ}\!\!c_{1}II_{\mathbf{\sigma}^{*}}+\,^{\circ}\!\!c_{1}II_{\mathbf{ \sigma}^{*}}\big{)}\sigma_{ij}\\ &\quad+\big{(}\,^{\circ}\!\!c_{2}+\,^{\circ}\!\!c_{2}II_{\mathbf{ \sigma}}+\,^{\circ}\!\!c_{2}II_{\mathbf{\sigma}^{*}}+\,^{\circ}\!\!c_{1}II_{\mathbf{ \sigma}^{*}}\big{)}\sigma_{ij}^{*}\\ &\quad+\big{(}\,^{\circ}\!\!c_{3}+\,^{\circ}\!\!c_{3}II_{\mathbf{ \sigma}}+\,^{\circ}\!\!c_{3}II_{\mathbf{\sigma}^{*}}+\,^{\circ}\!\!c_{3}II_{\mathbf{ \sigma}^{*}}\big{)}\sigma_{ij}^{*}\.\end{split}  \tag{37}\]  
This time, 12 material constants are obtained by using the code. Although the material equation is highly nonlinear, by using the energy-based method, we determine all of these material constants uniquely. With this cubic model, higher frequencies are modeled better; however, the general accuracy is limited, and actually, amplitudes cannot be modeled at all; see Figure 4.  
Figure 3: Linear model, loss (red circles) and stored (blue squares) energies obtained from the measurement; corresponding fit curves as continuous lines. Left: Frequency variation by fixed amplitude, \(\sigma_{0}=40\,\mathrm{Pa}\). Right: Amplitude variation by fixed frequency, \(\
u=1\,\mathrm{Hz}\). Finally, we have applied the sigmoid model in different configurations and chosen the simplest model with only six material constants,  
\[ \begin{split}& d_{ij}=\bigg{(}\,\hat{\mathtt{c}}_{1}+\,\hat{\mathtt{c}}_ {1}\frac{2}{\pi\sqrt{II_{\sigma}}}\arctan\Big{(}\frac{\sqrt{II_{\sigma}}}{B_{ 1}}\Big{)}\,\bigg{)}\sigma_{ij}\\ &+\bigg{(}\,\hat{\mathtt{c}}_{2}+\,\hat{\mathtt{c}}_{2}\frac{2}{ \pi\sqrt{II_{\sigma^{*}}}}\arctan\Big{(}\frac{\sqrt{II_{\sigma^{*}}}}{B_{2}} \Big{)}\,\bigg{)}\sigma_{ij}^{*}\\ &+\bigg{(}\,\hat{\mathtt{c}}_{3}+\,\hat{\mathtt{c}}_{3}\frac{2}{ \pi\sqrt{II_{\sigma^{*}}}}\arctan\Big{(}\frac{\sqrt{II_{\sigma^{*}}}}{B_{3}} \Big{)}\,\bigg{)}\sigma_{ij}^{*}\.\end{split}  \tag{38}\]  
This model is capable of capturing the materials responses as accurately as possible. By setting \(B_{1}=0.01\,\mathrm{Pa}\), \(B_{2}=0.01\,\mathrm{Pa}\,\mathrm{s}^{-1}\), \(B_{3}=0.01\,\mathrm{Pa}\,\mathrm{s}^{-2}\), we have obtained the following material constants:  
\[