# 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  
For the study of the creep characteristics and the mathematical establishment of the law of deformation in time of the andesite rocks type, the rheological tests were performed on intensively hydrothermal metamorphosed pyroxene andesite; these being the most unfavorable in terms of their stability. The creep tests were performed on uniaxial compression, with different stresses corresponding to the load degree of the andesite samples, according to the data in Table 1. The obtained results are shown in Table 2 and Figures 2 and 3. Through analyzing the experimental creep curve obtained by the tests, based on the data provided by the specialized literature [3; 4; 5; 10; 15; 19; 24; 25; 30; 32; 26; 33; 40; 41; 42], it is found that the Poynting-Thomson model is the model with which we could assimilate the deformation in time of strongly hydrothermal metamorphosed pyroxene andesite rock types. Because the assumption implies a confirmation in this regard, we took the model with the creep equation and analyzed it based on the obtained data from the tests through  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Rock Type**} & \multicolumn{2}{c}{**Samples Dimensions (mm)**} & \multirow{2}{*}{**Load Stress, \(\sigma_{\omega}\)**} & \multicolumn{2}{c}{**Tests Duration, t**} & \multirow{2}{*}{**Mean Breaking**} & \multirow{2}{*}{**Load Degree,**} \\ \cline{2-2} \cline{5-6}  & **d** & **h** & & **(MPa)** & & **Hours** & **Days** \\ \hline \multirow{3}{*}{Altered pyroxene} & \multirow{3}{*}{42} & \multirow{3}{*}{42} & 11.55 & 1198 & 50 & \multirow{3}{*}{28.00} & 0.41 \\  & & & 14.44 & 1198 & 50 & & 0.51 \\ \cline{1-1} \cline{5-6}  
an analytical processing. The functional equation of the rock deformation in time can be expressed in the following form:  
\[\epsilon=\text{f(t)} \tag{2}\]  
The explanation of this function according to the Poynting-Thomson model for a uniaxial compression test becomes:  
\[\epsilon=\frac{\sigma_{0}}{\text{E}}-\left(\frac{\sigma_{0}}{\text{E}}-\epsilon_ {0}\right)\text{e}^{-\frac{\text{E}}{\hbar}\t} \tag{3}\]  
Figure 3: Deformation rate and load degree correlation for the strongly altered analyzed andesite. Figure 2: Creep tests of the analyzed andesite subjected to different loads. where \(\sigma_{0}\) is the load stress, MPa; E is the modulus of elasticity, MPa; \(\epsilon_{0}\) is the instantaneous elastic deformation; \(\eta=T\cdot E\) is the coefficient of viscosity, \(Ns/m^{2}\); and t is the time. Or, Equation (3) can be written in the form:  
\[\varepsilon=a-be^{-ct} \tag{4}\]  
in which the following substitutions were made:  
\[a=\frac{\sigma_{0}}{E}\ ;\ b=\frac{\sigma_{0}}{E}-\varepsilon_{0}\ ;\ c=\frac{E}{\eta}=\frac{E}{TE}=\frac{1}{T} \tag{5}\]  
Based on the fact that from the experimental data we can choose two arbitrary points with the abscissa \(t_{1}\) and \(t_{2}\), and having the obtained values from the creep test for a stress equal to \(\sigma_{0}=11.55\) MPa, then:  
\[t_{3}=\frac{t_{1}+t_{2}}{2}\ ;\ a=\frac{\varepsilon_{1}\varepsilon_{2}- \varepsilon_{3}^{2}}{\varepsilon_{1}+\varepsilon_{2}-2\varepsilon_{3}} \tag{6}\]  
which become:  
\[t_{3}=\frac{7.135+51.5}{2}\ \Rightarrow\ t_{3}=29.3125\ days \tag{7}\]  
and:  
\[a=\frac{3637\times 10^{-6}\times 5291\times 10^{-6}-\left(5180\times 10^{-6} \right)^{2}}{3637\times 10^{-6}+5291\times 10^{-6}-2\times 5180\times 10^{-6}} \ \Rightarrow\ a=5299.50\times 10^{-6} \tag{8}\]  
where the value 0.005180 represents \(\varepsilon_{3}\), which corresponds to \(t_{3}\) and which was determined by interpolation.