## 4 The Crack Initiation Threshold as an Extension Strain Criterion in Uniaxial Compression Tests
### Discussion of the Results  
A summary of the laboratory results for the UCS and the extension strain criterion corresponding to the crack initiation stress is given in Table 2 and Figure 6. As can be noted, regardless of the method for determining the CI threshold, the obtained values are similar. The corresponding values of the extension strain \(\varepsilon_{\text{ext}}\) reflect the crack initiation criterion for the sandstone under analysis. According to the literature [5; 19], the CI range related to UCS usually ranges between 0.3-0.5 UCS. In the case of the sandstone under analysis, this value corresponds to 0.2 UCS for Poisson's ratio method, to 0.21 UCS for the lateral strain method, and to 0.25 UCS for the crack volumetric strain method, respectively.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Sample** & \multicolumn{2}{c}{**Lateral Strain**} & \multicolumn{2}{c}{**Diltancy Method**} & \multicolumn{2}{c}{**Crack Volumetric Strain**} & \multicolumn{2}{c}{**Poisson’s Ratio**} \\ \hline  & \(\varepsilon_{\text{ext}}\) [] & **CI [MPa]** & \(\varepsilon_{\text{ext}}\) [] & **CI [MPa]** & \(\varepsilon_{\text{ext}}\) [] & **CI [MPa]** & \(\varepsilon_{\text{ext}}\) [] & **CI [MPa]** \\ \hline p\_j\_1 & \(-0.000207\) & \(13.84\) & \(-0.000063\) & \(8.28\) & \(-0.000271\) & \(17.99\) & \(-0.000228\) & \(15.00\) \\ \hline p\_j\_2 & \(-0.000174\) & \(16.10\) & \(-0.000092\) & \(9.52\) & \(-0.000250\) & \(17.53\) & \(-0.000141\) & \(13.36\) \\ \hline p\_j\_3 & \(-0.000204\) & \(17.99\) & \(-0.000081\) & \(9.18\) & \(-0.000218\) & \(19.06\) & \(-0.000160\) & \(14.97\) \\ \hline p\_j\_4 & \(-0.000091\) & \(11.83\) & \(-0.000039\) & \(7.39\) & \(-0.000169\) & \(16.21\) & \(-0.000099\) & \(12.09\) \\ \hline  & \(-0.000095\) & \(12.36\) & \(-0.000079\) & \(9.85\) & \(-0.000204\) & \(16.89\) & \(-0.000103\) & \(12.94\) \\ \hline  & \(-0.000154\) & \(14.42\) & \(-0.000071\) & \(8.85\) & \(-0.000222\) & \(17.54\) & \(-0.000146\) & \(13.67\) \\ \hline  & \(-0.000051\) & \(2.32\) & \(0.000018\) & \(0.89\) & \(0.000035\) & \(0.97\) & \(0.000047\) & \(1.15\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The CI and extension strain criterion values determined using different UCS methods.  
Figure 5: The CI threshold determined for a sandstone sample p_j_3 using the methods of (**a**) crack volumetric strain, and (**b**) volumetric strain and the lateral strain–axial stress relationship.  
The extension strain criterion values obtained using the lateral strain method, the crack volumetric method, or Poisson's ratio method are similar and within the range of the results given by [PERSON] [1] for various rocks. The dilatancy method for the sandstone under analysis yielded the lowest value of the extension strain criterion and differed from the other results. It should be noted that the crack volumetric method is most often used in the literature on the subject to determine the CI threshold [17,18].  
Referring to the crack volumetric strain method, the CI thresholds for triaxial test results were also determined. They were compared with compressive strength for all test circular pressures and UCS (Figure 7). The slope of the best-fit CI was determined using the least squares approximation method:  
\[\sigma_{1}=0.26\sigma_{c}+1.83\sigma_{3} \tag{5}\]  
Equation (5) is basically consistent with the results obtained by other researchers [5] based on which the damage initiation surface in the \(\sigma_{1}\)-\(\sigma_{3}\) space can be approximated with Equation (6):  
\[\sigma_{1}=0.4\sigma_{c}+(1.5\div 2.0)\sigma_{3} \tag{6}\]  
The value of \(0.26\sigma_{c}\) for the tested sandstone under UCS conditions deviates from the literature results, but the dependence on \(\sigma_{3}\) is within the specified range.