# Determination of Nonlinear Creep Parameters for Hereditary Materials
## 3 Methods of Determination of Linear Creep Parameters for Hereditary Materials
### Parameters \(\alpha\), \(\varepsilon_{0}\), and \(\delta\)  
As it is known, creep curves of materials, depending on stress level and temperature, can have two or three characteristic strain sites [26; 27; 28]: site I with unstabilized creep, site II with stabilized creep, and site III of accelerating creep. Then the method is proposed, according to which it will be sufficient to consider only site I of creep curve for determination of creep parameters of a material \(\alpha\), \(\varepsilon_{0}\), and \(\delta\). Having accepted \(n=0\), from Equation (3) we will find:  
\[\varphi[\varepsilon(t)]\ \ =\ \sigma\Bigg{[}1+\frac{\delta\ t^{(1-\alpha)}}{(1- \alpha)}\Bigg{]}. \tag{4}\]  
One can see that the right part of the obtained equation contains a well-known Abel's function with unknown parameters \(\alpha\) and \(\delta\). Having divided both parts of the Equation (4) into instantaneous elasticity modulus \(E_{0}\) for the case of linear deformed material, we will obtain:  
\[\varepsilon(t)\ =\ \varepsilon_{0}\bigg{[}1+\frac{\delta}{1-\alpha}t^{(1- \alpha)}\bigg{]}, \tag{5}\]  
where \(\varepsilon_{0}\) is conditionally instantaneous strain. Equation (5) contains three unknown parameters \(\varepsilon_{0}\), \(\alpha\), and \(\delta\). As it has been said above, the right part contains a known [PERSON]'s function with parameter of singularity \(\alpha\), which has the value within the interval \((0,1)\). Then parameter \(\alpha\) will be considered as known, and unknown parameters \(\varepsilon_{0}\) and \(\delta\) will be determined with the use of the least squares method. According to the least squares method, the values of parameters \(\varepsilon_{0}\) and \(\delta\) should meet the following condition:  
\[S(\varepsilon_{0},\delta)\ =\ \sum_{i\ =\ 1}^{m}\ \bigg{[}\varepsilon_{0} \bigg{(}1+\frac{\delta}{1-\alpha}t_{i}^{(1-\alpha)}\bigg{)}-\ \varepsilon_{ei}\bigg{]}^{2}\rightarrow\min, \tag{6}\]  
where \(S(\varepsilon_{0},\delta)\) is the sum of squares of deviations;  
\(\varepsilon_{ei}\) are values of creep strain determined experimentally;  
\(m\) is number of creep strain. From the following two partial derivatives \(\frac{\partial\mathcal{S}(\varepsilon_{0},\delta)}{\partial\varepsilon_{0}}\ =\ 0\) and \(\frac{\partial\mathcal{S}(\varepsilon_{0},\delta)}{\partial\delta}\ =\ 0\) we find expressions for determination of the parameters \(\varepsilon_{0}\) and \(\delta\):  
\[\varepsilon_{0}\ =\ \frac{\sum\limits_{i\ =\ 1}^{m}\ \varepsilon_{i}\sum\limits_{i\ =\ 1}^{m}t_{i}^{2(1-\alpha)}-\sum\limits_{i\ =\ 1}^{m}t_{i}^{(1-\alpha)}\sum\limits_{i\ =\ 1}^{m}\varepsilon_{i}t_{i}^{(1-\alpha)}}{m\sum\limits_{i\ =\ 1}^{m}t_{i}^{2(1-\alpha)}-\left[\sum\limits_{i\ =\ 1}^{m}t_{i}^{(1-\alpha)}\right]^{2}}, \tag{7}\]  
\[\delta\ =\ \frac{\sum\limits_{i\ =\ 1}^{m}\left(\frac{\varepsilon_{i}}{ \varepsilon_{0}}-1\right)t_{i}^{(1-\alpha)}}{\frac{1}{1-\alpha}\sum\limits_{i \ =\ 1}^{m}t_{i}^{2(1-\alpha)}}. \tag{8}\]  
Selecting sequentially the values for parameter \(\alpha\) from interval \((0,1)\) with specific step, we find the values for parameter \(\varepsilon_{0}\ =\ \varepsilon_{0}(\alpha)\) from Equation (7). Inserting the obtained values for parameter \(\varepsilon_{0}\) and corresponding values for parameter of singularity \(\alpha\) into expression (8), we determine the values for parameter \(\delta\ =\ \delta\ (\varepsilon_{0},\ \alpha)\). Then inserting sequentially the obtained values for parameters \(\alpha\), \(\varepsilon_{0}\), and \(\delta\) into Equation (5), we calculate the values for creep strain \(\varepsilon(t_{i})\ =\ \varepsilon_{i}\ (t_{i},\ \alpha,\ \varepsilon_{0},\ \delta)\).