## 3 Nonlinear Material Equations and the Energy-Based Method for the Inverse Analysis
### Constructing Nonlinear Material Equations  
For generating material equations, we use tensor representation theorems. For a more detailed introduction, we refer to [28] (Section 4). Since stress is the controlled parameter, we search for a functional relation, where \(d_{ij}\) depends on \(\sigma_{ij}\), in other words, we aim for \(d_{ij}=d_{ij}(\sigma_{kl})\). For an isotropic material, the most general form of such an equation reads:  
\[d_{ij}=c_{0}\delta_{ij}+c_{1}\sigma_{ij}+c_{2}\sigma_{ik}\sigma_{kj}. \tag{19}\]  
The material coefficients, \(c_{0}\), \(c_{1}\), \(c_{2}\), are scalar functions depending on the stress via its invariants,  
\[ \begin{split} c_{\alpha}=c_{\alpha}(I_{\sigma},II_{\sigma},III_{ \sigma})\,\ \alpha\in\{0,1,2\}\,\\ I_{\sigma}=\sigma_{ij}\delta_{ij}\,\ II_{\sigma}=\sigma_{ij} \sigma_{jk}\delta_{ik}\,\ III_{\sigma}=\sigma_{ij}\sigma_{jk}\sigma_{kl}\delta_{il}\.\end{split}  \tag{20}\]  
The first invariant, \(I_{\sigma}\), is linear in \(\sigma_{ij}\); the second invariant, \(II_{\sigma}\), is quadratic in \(\sigma_{ij}\); the third invariant, \(III_{\sigma}\), is cubic in \(\sigma_{ij}\). The relation in Equation (19) represents the general equation in three-dimensional space. We can construct material equations in any complexity. In order to present it, we generate first a linear equation:  
\[d_{ij}=\big{(}\,\hat{c}_{\circ}+\,\hat{c}_{\circ}I_{\sigma}\big{)}\delta_{ij}+ \,\hat{c}_{\circ}\sigma_{ij}\, \tag{21}\]  
where \(\,\hat{c}_{\circ}\,\hat{c}_{\circ}\,\hat{c}_{\circ}\) are material constants to be determined by an inverse analysis. If \(\,\hat{c}_{\circ}\) and \(\,\hat{c}_{\circ}\) vanish for a material, the result is often called a Newtonian or Stokes fluid. Analogously, we can generate a quadratic equation:  
\[d_{ij}=\big{(}\,\hat{c}_{\circ}+\,\hat{c}_{\circ}I_{\sigma}+\,\hat{c}_{\circ} I_{\sigma}^{2}+\,\hat{c}_{\circ}II_{\sigma}\big{)}\delta_{ij}+\big{(}\,\hat{c}_{ \circ}+\,\hat{c}_{\circ}I_{\sigma}\big{)}\sigma_{ij}+\,\hat{c}_{\circ}\sigma_ {ik}\sigma_{kj}\, \tag{22}\]  
Figure 2: Loss (red circles) and stored (blue squares) energy densities obtained from the measurement: Left: Frequency variation by fixed amplitude, \(\sigma_{0}=40\,\mathrm{Pa}\). Right: Amplitude variation by fixed frequency, \(\
u=1\,\mathrm{Hz}\). or even a cubic equation:  
\[ \begin{split} d_{ij}&=\big{(}\,\hat{\texttt{c}}_{ \texttt{o}}+\,\hat{\texttt{c}}_{\texttt{o}}I_{\boldsymbol{\sigma}}+\,\hat{ \texttt{c}}_{\texttt{o}}I_{\boldsymbol{\sigma}}^{2}+\,\hat{\texttt{c}}_{ \texttt{o}}II_{\boldsymbol{\sigma}}+\,\hat{\texttt{c}}_{\texttt{o}}I_{ \boldsymbol{\sigma}}^{3}+\,\hat{\texttt{c}}_{\texttt{o}}I_{\boldsymbol{ \sigma}}II_{\boldsymbol{\sigma}}+\,\hat{\texttt{c}}_{\texttt{o}}III_{ \boldsymbol{\sigma}}\big{)}\delta_{ij}\\ &\quad+\big{(}\,\hat{\texttt{c}}_{\texttt{i}}+\,\hat{\texttt{c}} _{\texttt{i}}I_{\boldsymbol{\sigma}}+\,\hat{\texttt{c}}_{\texttt{i}}I_{ \boldsymbol{\sigma}}^{2}+\,\hat{\texttt{c}}_{\texttt{i}}II_{\boldsymbol{ \sigma}}\big{)}\sigma_{ij}+\big{(}\,\hat{\texttt{c}}_{\texttt{i}}+\,\hat{ \texttt{c}}_{\texttt{i}}I_{\boldsymbol{\sigma}}\big{)}\sigma_{ik}\sigma_{kj} \,.\end{split}  \tag{23}\]  
The nonlinearity can be increased by using a polynomial decomposition. Instead of a polynomial function, we can apply a trigonometric function, capturing the nonlinearities successfully. We propose to use a sigmoid model:  
\[d_{ij}=\,\hat{\texttt{c}}_{\texttt{o}}\delta_{ij}+\bigg{(}\,\hat{\texttt{c}} _{\texttt{i}}+\,\hat{\texttt{c}}_{\texttt{i}}\frac{2}{\pi\sqrt{II_{\boldsymbol {\sigma}}}}\arctan\Big{(}\frac{\sqrt{II_{\boldsymbol{\sigma}}}}{B_{1}}\Big{)} \,\bigg{)}\sigma_{ij}\,. \tag{24}\]  
The idea of using such a model is based on [33]; some applications and analysis about the behavior of such a material equation can be found in [34; 35; 36]. The sigmoid model allows a sharp change between positive and negative values of \(d_{ij}\) with a differentiable function \(arctan()\). Hence, it is an adequate model for computations, as well as it captures high nonlinearities in materials' response, as well.