## 3 Nonlinear Material Equations and the Energy-Based Method for the Inverse Analysis
### Inverse Analysis  
The observables from a measurement are \(e_{1}\) and \(e_{2}\) for a set of frequencies and amplitudes. Suppose we want to try to fit the experimental data by using the sigmoid model as posed in Equation (24). Stress is the controlled variable:  
\[\sigma_{12}=\sigma_{0}\sin(\omega t)\,\ \sigma_{12}^{\star}=\sigma_{0}\omega \cos(\omega t)\, \tag{26}\]  
and even normal stresses occur; we assume that the stress is deviatoric such that its trace (sum of diagonal components) vanishes. The sigmoid model has the following shear response:  
\[d_{12}=\bigg{(}\,\hat{\texttt{c}}_{\texttt{i}}+\,\hat{\texttt{c}}_{\texttt{i} }\frac{\sqrt{2}}{\pi|\sigma_{12}|}\arctan\Big{(}\frac{\sqrt{2}|\sigma_{12}|}{ B_{1}}\Big{)}\,\bigg{)}\sigma_{12}. \tag{27}\]  
With the help of Equation (18), we can now calculate the energies, \(e_{1}\) and \(e_{2}\). We start with:  
\[e_{1}=\int_{0}^{2\pi/\omega}\sigma_{ij}d_{ji}\,\mathrm{d}t=\int_{0}^{2\pi/ \omega}2\sigma_{12}d_{12}\,\mathrm{d}t=A_{11}\,\hat{\texttt{c}}_{\texttt{i}} +A_{12}\,\hat{\texttt{c}}_{\texttt{i}}\, \tag{28}\]  
with:  
\[ \begin{split} A_{11}=\int_{0}^{2\pi/\omega}2\sigma_{12}\sigma_{1 2}\,\mathrm{d}t\,\\ A_{12}=\int_{0}^{2\pi/\omega}2\sigma_{12}\frac{\sqrt{2}}{\pi| \sigma_{12}|}\arctan\Big{(}\frac{\sqrt{2}|\sigma_{12}|}{B_{1}}\Big{)}\sigma_{ 12}\,\mathrm{d}t\,\end{split}  \tag{29}\]since \(v\), are constants. Analogously, we can calculate:  
\[e_{2}=\int_{0}^{\pi/(2\omega)}\sigma_{ij}d_{ji}\,\mathrm{d}t=\int_{0}^{\pi/(2 \omega)}2\sigma_{12}d_{12}\,\mathrm{d}t=A_{21}\,\mathrm{\char 37}_{1}+A_{22}\, \mathrm{\char 37}_{1}\,, \tag{30}\]  
with:  
\[A_{21}=\int_{0}^{\pi/(2\omega)}2\sigma_{12}\sigma_{12}\,\mathrm{d}t\, \tag{31}\]  
We can rearrange \(A_{\mathrm{xx}}\) in the form of a matrix \(A\). Every component of \(A\) can be calculated for a measurement with the known \(\sigma_{12}\) function. The matrix \(A\) is \(2\times 2\) for a specific frequency, \(\omega\), and amplitude, \(\sigma_{0}\). Unknowns are the two material constants, \(v_{\mathrm{c}}\). We already have discussed the measurement data in Equations (16) and (17), so we may rewrite a measurement for a specific frequency, \(v\), and amplitude, \(\sigma_{0}\), as follows:  
\[ \begin{pmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{pmatrix}\begin{pmatrix}v_{\mathrm{c}}\\ \mathrm{\char 37}_{1}\\ \mathrm{\char 37}_{1}\\ \mathrm{\char 37}_{1}\end{pmatrix} = \begin{pmatrix}e_{1}=\sigma_{0}^{2} \dfrac{2\pi\mathrm{G}^{\prime\prime}}{(G^{\prime})^{2}+(G^{\prime\prime})^{2} }\\ e_{2}=\sigma_{0}^{2}\dfrac{G^{\prime}+\frac{\pi}{2}G^{\prime\prime}}{(G^{\prime} )^{2}+(G^{\prime\prime})^{2}}\end{pmatrix} . \tag{32}\]  
Suppose we have \(m\) measurements with different frequencies, viz., \(v^{1}\), \(v^{2}\), \(v^{3}\), \(\dots\), \(v^{m}\), at a specific amplitude. Moreover, we have \(n\) measurements with different amplitudes, \(\sigma_{0}^{1}\), \(\sigma_{0}^{2}\), \(\sigma_{0}^{2}\), \(\dots\), \(\sigma_{0}^{n}\), at a specific frequency. Then, the matrix \(A\) will be \(2(m+n)\times 2\) as follows:  
\[ \begin{pmatrix}A_{11}|_{v^{1}}&A_{12}|_{v^{1}}\\ A_{21}|_{v^{1}}&A_{22}|_{v^{1}}\\ \vdots&\vdots\\ A_{11}|_{v^{m}}&A_{12}|_{v^{m}}\\ A_{21}|_{v^{m}}&A_{22}|_{v^{m}}\\ A_{11}|_{v^{1}_{0}}&A_{12}|_{v^{1}_{0}}\\ A_{21}|_{v^{1}_{0}}&A_{22}|_{v^{1}_{0}}\\ \vdots&\vdots\\ A_{11}|_{\sigma_{0}^{n}}&A_{12}|_{\sigma_{0}^{n}}\\ A_{21}|_{\sigma_{0}^{n}}&A_{22}|_{\sigma_{0}^{n}}\end{pmatrix}\begin{pmatrix}v_{ \mathrm{c}}\\ \mathrm{\char 37}_{1}\\ \mathrm{\char 37}_{1}\\ \mathrm{\char 37}_{1}\end{pmatrix} = \begin{pmatrix}e_{1}|_{v^{1}}\\ e_{2}|_{v^{1}}\\ \vdots\\ e_{1}|_{v^{m}}\\ e_{2}|_{v^{m}}\\ e_{1}|_{v^{1}_{0}}\\ e_{2}|_{v^{1}_{0}}\\ \vdots\\ e_{1}|_{v^{m}_{0}}\\ e_{2}|_{v^{m}_{0}}\end{pmatrix} . \tag{33}\]  
By using linear algebra, we write all measurements:  
\[A\mathbf{u}=\mathbf{b}\, \tag{34}\]  
where the unknowns, \(\mathbf{u}\), are collected as above and the experimental results, \(\mathbf{b}\), for each frequency and amplitude are written among each other.