# Mathematical Model for Establishing the Time-Dependent Behavior of Rocks by the Gradient Method
## 3 Deformation Characteristics in Time of Altered Andesite
### Deformation Characteristics in Time of Altered Andesite Established in Laboratory
Knowing that:  
\[\epsilon=\text{f(t)} \tag{2}\]0  
then:  
\[\epsilon=\text{f(t)} \tag{2}\]1  
Equation (3) can also be written in the form:  
\[\epsilon=\text{f(t)} \tag{2}\]2  
making the logarithm of Equation (11), it results in:  
\[\epsilon=\text{f(t)} \tag{2}\]3  
noting:  
\[\epsilon=\text{f(t)} \tag{2}\]4  
then, we get the relationship:  
\[\epsilon=\text{f(t)} \tag{2}\]5  
the value of B is determined according to the relation (13), resulting:  
\[\epsilon=\text{f(t)} \tag{2}\]6  
knowing the value of B, from Equation (13), we can obtain the value of C:  
\[\epsilon=\text{f(t)} \tag{2}\]7from where:  
\[\epsilon=\text{f(t)} \tag{2}\]8  
therefore, the value of C will be given by the relation:  
\[\epsilon=\text{f(t)} \tag{2}\]9  
where \(\sum\limits_{i=1}^{n}p_{i^{\prime}}\)\(\sum\limits_{i=1}^{n}t_{i}\) were determined based on resulted data from the creep test and n = 25 is the number of obtained data. Namely:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]0  
whence:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]1  
it is known that:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]2  
and:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]3  
once the value of \(\overline{c}\) (the first approximation) is established, Equation (3) becomes:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]4  
which the derivative with respect to \(\overline{c}\) will be:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]5  
from the processing of the obtained experimental data, it can easily determine the value of (s), that is:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]6  
in this case, it turns out:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]7  
where \(\overline{\overline{c}}\) represents the second approximation. On the basis of \(\overline{\overline{c}}\), the accuracy increases from 0.520404 to 0.097772. Equation (3), respectively (23), becomes:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]8  
returning to the substitutions made, the relation (5), we can now determine the search parameters of the creep curve. So:  
\[ \begin{array}{l}E=\frac{\sigma_{0}}{a}\\ E=\frac{11.55}{0.0052995}\ \Rightarrow\ E=2179.4\ MPa\end{array}  \tag{28}\]this value confirms the good results of the elasticity module determined in the laboratory. \[\mathrm{T}=\frac{1}{\mathrm{\varepsilon}}\ \Leftrightarrow\ \mathrm{T}=\frac{1}{0.0935}\ \Rightarrow\ \mathrm{T}=10.7\ \mathrm{days} \tag{29}\]  
and:  
\[\eta=\mathrm{T}\times\mathrm{E}\ \Leftrightarrow\ \eta=2179.4\times 10.7\ \Rightarrow\ \eta=23319.6\ \mathrm{MPa}\times\mathrm{day} \tag{30}\]  
therefore, Equation (6) becomes:  
\[\mathrm{\varepsilon}=0.00529816-0.00317316\mathrm{e}^{-0.0935\mathrm{t}} \tag{31}\]  
Equation (31) represents the analytical equation established for the strongly metamorphosed andesite that has been analyzed and which confirms that this rock can be assimilated as a manner of behavior in time with the model given by the Poynting-Thomson model (see Figure 4).