## 5 Conclusions  
1. Based on the classical MC criterion, an improved nonlinear MC criterion was proposed by introducing correction functions for the temperature and confining pressure. This criterion considers the critical confining pressure condition when the compressive strength remains constant as \(n\sigma_{\rm c}\) rather than \(\sigma_{\rm c}\). Additionally, it assumes a linear decrease in rock compressive strength with the temperature, thus establishing a linear relationship model between temperature and rock strength.  
By comparing the strength prediction accuracy of the HB criterion, Bieniawski criterion, and HTP-MC criterion for granite, shale, and carbonate rock, it was found that, under low confining pressure conditions, all three failure criteria show high prediction accuracy. However, under high confining pressure conditions, the HTP-MC criterion better captures the nonlinear characteristics of the rock failure envelope, significantly outperforming the other two failure criteria in terms of strength prediction accuracy.
3. The HTP-MC criterion overcomes the limitation of the Hoek-Brown and Bieniawski criteria, which cannot account for the effect of temperature on rock strength. The average \(MRE\) value for the prediction of granite, shale, and carbonate rock under high-temperature and high-pressure conditions using the HTP-MC criterion is 5.25%, demonstrating that this criterion provides high prediction accuracy for rock strength under high-temperature and high-pressure conditions.
4. All rock types exhibit the general trend that the predicted \(\sigma_{1}\) increases with the increasing parameter \(n\) and decreases with the increasing parameter \(B\). However, the sensitivity of predicted \(\sigma_{1}\) to these parameters varies significantly among different rock types. In particular, granite shows the lowest sensitivity to both \(n\) and \(B\), indicating better adaptability of the prediction model for granite strength. Furthermore, the predicted \(\sigma_{1}\) is generally more sensitive to changes in parameter \(B\) than in \(n\).  
Conceptualization, Q.Z.; Methodology, Z.L., Q.Z. and C.M.; Project administration, Z.L. and H.L.; Validation, J.W. and P.Z.; Formal analysis, P.Z.; Investigation, Q.Z. and C.M.; Resources, Z.L. and H.L.; Data curation, Z.L. and J.W.; Writing--original draft, Q.Z. and C.M.; Writing--review and editing, X.F.; Funding acquisition, Z.L. and X.F.; Supervision, H.L. and X.F. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Nos. 42172313 and 42402164), the Natural Science Foundation of Sichuan Province (No. 2024 NSFSC0203), and the CNPC Chuanqing Drilling Engineering Company Limited Program (No. CQXN-2023-10). Not applicable. Not applicable. Not applicable. Not applicable. Data Availability Statement: The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding authors. The authors would like to express their gratitude to the editors and anonymous reviewers for their constructive comments on the draft paper. The authors are also grateful to [PERSON], [PERSON], [PERSON], and [PERSON] for their contributions to data curation, investigation, and reviewing. The authors declare no conflicts of interest.