## 1 Introduction  
Especially in polymer science and biological tissue modeling, the material response fails to be modeled accurately by a linear material equation. For a fluid-like material, Newtonian material model establishes a linear connection between stress and velocity gradient. The material coefficient, called _viscosity_, is a constant, and since a constant fails to attain an accurate modeling, there are various approaches to increase the complexity by choosing a viscosity depending on the velocity gradient [1]. There are several proposals about developing such nonlinear material (constitutive) equations starting from a microstructure or using chain interactions, as well as exploiting thermodynamical reasoning. There are various nonlinear models in the literature. None of them are delivering accurate results for all applications. For example, widely-used models as in [2; 3; 4] have been modified by [5; 6] in order to obtain models providing results for different scenarios. Even the modified version was criticized in [7], and a possible modification was proposed in [8]. In order to include a rate, the objective time derivative was proposed by [9]. Instead of a material equation with a rate, integral equations were proposed in order to include the full history on deformation as in [10; 11]. These equations were used as models in [12], and measurements were undertaken in [13]. However, in numerical mechanics, integral equations are more challenging to implement than differential equations. Hence, in this paper, we (author and reader) will benefit from representation theorems to generate nonlinear material equations; as already used in [14; 15; 16; 17]. As stated in [18], the nonlinear modeling in rheology is still an open question, and we cannot expect to acquire a consensus; but we need tools making the inverse analysis for different suggestions possible. Such tools may lead to easier comparisons of different models. In this work, we present and use an approach capable of delivering coefficients for nonlinear material equations. Cyclic measurements by using a rotary viscometer (rheometer) are optimized for linear rheology, where the material coefficients are constants. The output of such a measurement is given as two moduli, one modulus for elastic behavior and another modulus for viscous behavior. This interpretation is valid and appropriate for a linear material model. In the case of a nonlinear material behavior, we fail to investigate the elastic and viscous effects separately in these moduli. Thus, we need to extract the data in such a way that the coefficients in a nonlinear material model can be determined. A considerable amount of literature has been published in the last two decades. Among others, for example as in [19; 20; 21; 22; 23; 24; 25], various polynomial expansion techniques have been proposed in order to fit the experimental data. As shown in [26], they have mathematical similarities. In order to apply them, a revision is necessary of the control unit of a conventional rheometer. Newer devices deliver such functionality, so their approach is possible, but in a way also limited to the existing measurement device. An energy-based method was proposed in [27] (Section 9) and applied in [28]. By using the energy-based method, we generate a linear regression problem out of the conventional output of an oscillatory measurement in a rheometer even in the case of nonlinear material equations. In a linear regression problem, the solution is unique and computationally far easier than any nonlinear regression problem. Hence, by using the energy-based method, an experimentalist can quickly try different material equations by increasing the complexity step by step. We emphasize that the inverse analysis lasts less than a minute on a moderate computer. The optimization is fast simply because of the linear regression. Hence, the whole method can be described as the approach for how to create a linear regression problem in the case of a nonlinear material equation.