# A Three-Dimensional Elastoplastic Constitutive Model for Geomaterials
## 5 Strength Parameters Determined on Meridian Plane
### Determination and Verification of Strength Parameters  
The GMC-L criterion can be established only using triaxial compression and extension tests. We selected the triaxial compression and extension test data in the literature [40] to verify the GMC-L criterion's applicability. Figure 4: GMC-L in the \(p\)-\(q\) plane with triaxial compression and extension. Figure 5 shows the strength test data of Monterey sand on the \(p\)-\(q\) plane and the corresponding GMC-L failure surfaces. It can be seen from Figure 5 that the GMC-L criterion's fitting effect on Monterey sand is satisfactory, and that the correlation coefficient R\({}^{2}\) > 0.96. According to Equations (19) and (20), the internal friction angle under triaxial compression is 37\({}^{\circ}\) and under triaxial extension is 46\({}^{\circ}\). We also calculated the internal friction angle of triaxial compression and extension of other geotechnical materials [41; 42; 43] using the GMC-L criterion, as shown in Table 1.  
Table 1 shows that the triaxial compression friction angle \(\phi_{0}\) of the geotechnical materials is smaller than that of the triaxial extension friction angle \(\phi_{1}\), which is consistent with the results observed in other studies [44; 45]. With the increase in \(\sigma_{2}\), the internal friction angle of the geotechnical materials also gradually increases. Therefore, the influence of \(\sigma_{2}\) on the soils' strengths can be referred to as \(\phi_{0}\) and \(\phi_{1}\). For different soil types, the influence of \(\sigma_{2}\) is different. For example, for Santa Monica sand, the internal friction angle of the triaxial extension is about 14.8% higher than that of the compression, but, for Fujinomori clay, this value is about 6%. The GMC-L criterion has good applicability to geotechnical materials. We also verified the GMC criterion's applicability for rock materials by calculating the failure surfaces corresponding to the GMC-L criterion of different rocks on the \(p\)-\(q\) plane and comparing them with the experimental data in the literature [35; 46; 47]. The strength test data of different rocks on the \(p\)-\(q\) plane and the corresponding GMC-L failure surfaces are shown in Figure 6. The GMC-L criterion clearly has good applicability to rocks. Table 2 displays the GMC-L strength parameters for rocks. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**S.No** & **Soil Name** & \(\phi_{0}\) (\({}^{\circ}\)) & \(\phi_{1}\) (\({}^{\circ}\)) & **Source** \\ \hline
1 & Monterey sand & 37 & 46 & [PERSON] and [PERSON] (1993) \\
2 & Santa Monica sand & 40.92 & 46.96 & [PERSON] and [PERSON] (2001) \\
3 & Shanghai fine sand & 45.26 & 52.25 & [PERSON] et al. (2011) \\
4 & Normally-consolidated & 35.19 & 37.31 & [PERSON] et al. (1986) \\  & Fujinomori clay & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Strength parameters of soils under GMC-L criterion. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**S.No** & **Rock Name** & \(\phi_{0}\) (\({}^{\circ}\)) & \(\phi_{1}\) (\({}^{\circ}\)) & \(c_{0}\) (MPa) & \(c_{1}\) (MPa) & **Source** \\ \hline
1 & Darley sandstone & 35.3 & 39.6 & 26.8 & 31.3 & [PERSON] (1965) \\
2 & Westyryl granite & 53.4 & 59.7 & 41.4 & 52.7 & [PERSON] (1967) \\
3 & Duham dolomite & 46.8 & 51.2 & 49.5 & 57.6 & [PERSON] (1967) \\
4 & Red Wildmoor & 28.5 & 35.6 & 6.0 & 7.9 & [PERSON] et al. (2000) \\  & sandstone & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Strength parameters of soils under GMC-L criterion. Figure 5: Test data and GMC-L criteria failure surfaces on \(p\)-\(q\) plane [40]. The internal friction angle and rock cohesion are smaller in triaxial compression than they are in extension, as can be determined from Table 2; therefore, as \(\sigma_{2}\) rises, the parameters governing the strength will gradually increase. The experimental outcomes produced by [48] were consistent. We could also observe that the intermediate principal stress's effects on the strength metrics of various rocks vary.