# Determination of Nonlinear Creep Parameters for Hereditary Materials
## 3 Methods of Determination of Linear Creep Parameters for Hereditary Materials
### Parameters \(\beta\) and \(\lambda\)  
Having divided both parts of Equation (3) into instantaneous elasticity modulus \(E_{0}\), for the case of linear strain of material, we obtain:  
\[\varepsilon(t)\ =\ \varepsilon_{0}\Bigg{[}1+\lambda\ \sum\limits_{n\ =\ 0}^{\infty}\ \frac{(-\beta)^{n}\ t^{(1-\alpha)(1+n)}}{\Gamma[(1-\alpha)(1+n)+1]}\Bigg{]}. \tag{10}\]  
Let us rewrite Equation (10) in the following form, inserting notation \(t_{k}\ =\ \beta^{-\ (\frac{1}{1-\alpha})}\):  
\[\varepsilon(t)\ =\ \varepsilon_{0}(1+\lambda\ F_{4i}), \tag{11}\]  
\[\text{where}\ F_{4i}\ =\ t_{k}^{(1-\alpha)}\sum\limits_{n\ =\ 0}^{\infty}\ \frac{(-1)^{n}\Big{(}\frac{t_{i}}{t_{k}}\Big{)}^{(1-\alpha)\ (1+n)}}{\Gamma\left[(1-\alpha)\ (1+n)+1 \right]}. \tag{12}\]  
Using the least squares method, we write the extremum condition, similar to condition of (6):  
\[S\ (\beta,\lambda)\ =\ \sum\limits_{i\ =\ 1}^{m}\left[\varepsilon_{0}(1+ \lambda\ F_{4i})-\varepsilon_{\varepsilon i}\right]^{2}\ \rightarrow\ \min. \tag{13}\]  
The following expressions are obtained for determination of parameters \(\beta\) and \(\lambda\) from two equations, composed on expressions \(\frac{\partial\mathcal{S}\ (\beta,\lambda)}{\partial\beta}\ =\ 0\) and \(\frac{\partial\mathcal{S}\ (\beta,\lambda)}{\partial\lambda}\ =\ 0\):  
\[\sum\limits_{i\ =\ 1}^{m}\Bigg{[}\left(1-\frac{\varepsilon_{i}}{ \varepsilon_{0}}\right)\sum\limits_{i\ =\ 1}^{m}\ F_{4i}^{2}+F_{4i}\sum\limits_{i\ =\ 1}^{m}\left(\frac{\varepsilon_{i}}{ \varepsilon_{0}}-1\right)\ F_{4i}\Bigg{]}\ F_{5i}\ =\ 0, \tag{14}\]\[\lambda\ =\ \ \frac{\sum\limits_{i\ =\ 1}^{m}\ \left(\frac{\varepsilon_{i}}{ \varepsilon_{0}}-1\right)\ F_{4i}}{\sum\limits_{i\ =\ 1}^{m}\ F_{4i}^{2}}, \tag{15}\]  
\[\text{where}\ F_{5i}\ =\ t_{k}^{2\ (1-\alpha)}\sum\limits_{n\ =\ 0}^{\infty}\ \frac{(-1)^{n}n\ \beta^{(n-1)}\left(\frac{t_{i}}{t_{k}}\right)^{(1- \alpha)(1+n)}}{\Gamma[(1-\alpha)(1+n)+1]}. \tag{16}\]  
The value of parameter \(\beta\) is determined from Equation (14). If the equation has an unambiguous solution, then it will be the target value for parameter \(\beta>0\). It is obvious that during determination of values for parameters \(\beta\) and \(\lambda\) from Equations (14) and (15), the previously calculated values are used for parameters \(\alpha\) and \(\varepsilon_{0}\).  
As it is known, one of the factors which limited the wide use of creep kernel with Rabotnov's fractional exponential function has been the fact that it was not an easy way to determine its parameters [22]. A special table has been developed [15], and with its use, the fractional exponential function has been calculated under the argument \(x=\beta\ t^{(1-\alpha)}\)_at_\(x\leq 4\). At \(x>4\) it is recommended to use asymptotic formulas of [PERSON] [29]. Calculations have shown that the infinite series in the right part of Equation (10) are poorly converged at small values of argument \(x<4\).  
Introduction of special notation \(t_{k}\ =\ \beta^{-(\frac{1}{1-\alpha})}\) and consideration in series (12) and (16) of not time \(t_{i}\), but relative time \((t_{i}/t_{k})\) have essentially increased the convergence of calculations and allowed the quick obtaining of more accurate values of the fractional exponential function practically at any values of argument.