## 4 Results
\begin{split}&\hat{\mathtt{c}}_{1}=1.261\,\mathrm{Pa}^{-1}\,,\,\, \hat{\mathtt{c}}_{1}=-4.331\cdot 10^{4}\,,\\ &\hat{\mathtt{c}}_{2}=1.780\cdot 10^{-2}\,\mathrm{s}\,\mathrm{Pa}^{-1} \,,\,\,\hat{\mathtt{c}}_{2}=1.051\,,\\ &\hat{\mathtt{c}}_{3}=3.588\cdot 10^{-5}\,\mathrm{s}^{2}\, \mathrm{Pa}^{-1}\,,\,\,\hat{\mathtt{c}}_{3}=-4.327\cdot 10^{4}\.\end{split}  \tag{39}\]  
The fit with the sigmoid model is shown in Figure 5.  
It is obvious that the fit with the sigmoid model is better than linear and cubic models. Especially in the variation of frequency, the sigmoid model performs an accurate fit at the lower, as well as the higher values. The linear model does not capture higher frequencies well. The cubic model is missing the expected accuracy. The sigmoid model performs an adequate fit in the whole interval of amplitude  
Figure 4: Cubic 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, \(\
u=1\,\mathrm{Hz}\). Figure 5: Sigmoid 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, \(\
u=1\,\mathrm{Hz}\). sweep. Linear and cubic models are simply not capable of fitting higher amplitudes; but the sigmoid model shows a good agreement with experimental data. It is important to address the lack of precision in the data. This fact is caused by extremely high stress amplitudes for this soft matter. Under 40 Pa of stress, depending on the frequency, the material can reach strain values around three (meaning 300% of stretching). Therefore, such measurements are challenging, and the lack of precision is fair. The quality of fit relies on the data's precision. We fit at once all different measurements and variations. In total, we use eight datasets: two energies from two different measurements for amplitude and frequency variations. Since we use linear regression minimizing the squared error between measurement and estimation, using two different measurements is equal to constructing a geometric mean of these measurements and then fit. We emphasize that for all datasets, the same material equation with the same material parameters is used to capture all of these effects. The sigmoid model performs with very good results only by using six parameters. The fit is in agreement for a large range of amplitudes and frequencies. As previously mentioned, for this kind of soft matter, the stress amplitude is large, providing a remarkable viscous character of the material, as well. In order to study this response, we exploit Lissajous-Bowditch plots. By applying the following harmonic shear stress:  
\[\sigma_{12}=\sigma=\sigma_{0}\sin(2\pi\upsilon t)\, \tag{40}\]  
we solve numerically the strain by using the linear model as in Equation (36), as well as the sigmoid model as in Equation (38). By using the fit parameters as in Equation (39), we obtain the response as shown in Figure 6.  
The linear model presents the same response function in lower, as well as higher frequencies and amplitudes. Conceivably, the sigmoid model presents different responses for the different amplitudes and frequencies. In the case of the high frequency and amplitude (see Figure 6h), the response is identical to the linear model. The value of the high frequency and amplitude depends on the material parameters; for the material used, 100 Hz and 100 Pa seem to be high enough. For lower frequencies and amplitudes, the response of the linear model remains the same; however, the sigmoid model shows a complex response. Indeed, we know that non-circular curves were reported previously by measuring the stress over shear; see for example [21] (Figure 9), [39] (Figure 6).