# Determination of Nonlinear Creep Parameters for Hereditary Materials
## 4 Algorithm for Description of Nonlinear Deformation of Hereditary Materials  
This section of the paper proposes a new method for description of nonlinear deformation of hereditary materials. This method is realized in several steps and performed in the following sequence:  
a. A new parameter is introduced called the experimental rheological parameter:  
\[k_{e}(t)\ =\ \ \frac{\varepsilon_{e}(t)}{\varepsilon_{0}^{e}}, \tag{17}\]  
where \(\varepsilon_{e}(t)\) is the value of creep strain at time moment \(t\), determined experimentally;  
\(\varepsilon_{0}^{e}\) is instantaneous strain at time moment \(t\) = 0, determined experimentally. Experimental rheological parameter \(k_{e}(t)\) represents by itself a normalized time function in relation to experimental instantaneous strain. It has the value equal to 1 at \(t\) = 0 and more than 1 at time values of \(t>0\). It shows in how many times the experimental values of creep strain are more at different time values compared with instantaneous strain, which has been also obtained experimentally. b. The values have been calculated for the experimental rheological parameter \(k_{e}(t)\) at different time values \(t\) and stresses \(\sigma\). The graphs for \(k_{e}(t)\) at various stresses \(\sigma\) are constructed according to the calculation results. All curves \(k_{e}(t)\) at various stresses coincide (for example, as in the work [30]) for physically linear material, i.e., we have only one mean curve \(k_{e}(t)\) for all stresses (Figure 3a). There are separate curves for various stresses \(k_{e}(t)\) (Figure 3b) for physically nonlinear material. Coincidence for some of them is not excluded. As it has been said above, the construction of isochronous creep curves is an illustrative way for the evaluation of physical nonlinearity (linearity) of the material. Therefore, we have two ways of visual evaluation for nonlinearity (linearity) of deformation for materials: first--through experimental rheological parameter \(k_{e}(t)\); second--by construction of isochronous creep curves. Using [PERSON]'s fractional exponential function (3) and [PERSON]'s function (4), we approximate the experimental values of creep strain for material, obtained at minimum stress \(\sigma_{1}\). Meanwhile, we find corresponding values of creep parameters \(\alpha\), \(\varepsilon_{0}\), \(\delta\), \(\beta\), and \(\lambda\) according to the abovementioned methods. d. In case of creep, i.e., at \(\sigma\) = const from integral Equation (1) we obtain:  
\[\varphi[\varepsilon(t)]\ =\ \ \sigma\ \bigg{[}1+\int_{0}^{t}\ K(t-\tau)d\tau \bigg{]}. \tag{18}\]  
The right part of this integral equation is denoted through \(k_{m}(t)\):  
\[k_{m}(t)\ =\ 1+\int_{0}^{t}K(t-\tau)d\tau \tag{19}\]  
and it is named as model (theoretical or calculation) rheological parameter. As it is seen from Equation (19), model rheological parameter \(k_{m}(t)\) depends only on time \(t\). Similar to experimental rheological parameter \(k_{e}(t)\), the model rheological parameter has the value equal to 1 at \(t\) = 0 and more than 1 at \(t\) > 0. It shows in how many times the calculated values of creep strain are bigger compared with the instantaneous strain, determined by calculations. Whereas the experimental rheological parameter can be determined for all stresses, the model rheological parameter is determined only for minimum stress \(\sigma_{1}\). e. According to [PERSON]'s isochronous creep curves method, we find the calculated values of instantaneous strains at various stresses under the following formula:  
\[\varepsilon_{0}^{m}(\sigma)\ =\ \frac{\varepsilon_{e}(t_{s})}{k_{m}(t_{s})}, \tag{20}\]  
where \(\varepsilon_{e}(t_{s})\) is the experimental value of creep for material at fixed time \(t_{s}\) at stress \(\sigma\);  
\(k_{m}(t_{s})\) is the value of model rheological parameter at fixed time \(t_{s}\) (similarity coefficient of isochronous creep curves according to [PERSON]).