Polyaxial Rock Failure Criteria: Insights from Explainable and Interpretable Data-Driven Models

Abstract

The estimation of rock strength is one of the most important basic requirements in rock engineering design and analysis. In this study, six different data-driven machine learning (ML) models to predicate the major principal stress (${\sigma }_{1}$) at the failure of intact rock material under the polyaxial (true triaxial) stress condition are introduced and investigated. In this regard, multiple linear regression, support vector machine, random forest (RF), extreme gradient boosting (XGBoost), K-nearest neighbors, and multivariate adaptive regression splines (MARS) methods are developed based on uniaxial compressive strength (${\sigma }_{c}$), minor principal stress (${\sigma }_{3}$), and intermediate principal stress (${\sigma }_{2}$) as the input of the proposed models. 80% out of 930 polyaxial compiled datasets of 29 rocks from the literature are used to train the data-oriented rock failure models and the remaining, i.e., 20% datasets are considered to test the performance of established polyaxial rock failure models. Comparison of the performance results indicates XGBoost model outperforms other models with ${R}^{2}$ = 1, RMSE = 4.698 MPa, and AAREP = 0.460% in the training phase. However, RF model with ${R}^{2}$ = 0.992, RMSE = 29.620 MPa, and AAREP = 7.906% and XGBoost model with ${R}^{2}$ = 0.991, RMSE = 31.081 MPa, and AAREP = 7.462% perform superior than other models in the testing phase. Accordingly, feature importance, interaction, and visualization methods are used to perform XGboost model explanation and scrutinize the effect of ${\sigma }_{c}$, ${\sigma }_{3}$, and ${\sigma }_{2}$ on polyaxial rock failure strength, i.e., ${\sigma }_{1}$. Moreover, based on the interpretable MARS method and a wide-ranging database of laboratory tests, a new comprehensive and generalize equation for estimating polyaxial rock failure strength with ${R}^{2}$ = 0.98 is proposed. In general, employed ML techniques demonstrated their higher performance accuracy and generalization ability in predicting the failure strength of different intact rocks under polyaxial conditions compared with conventional failure criteria in the form of ${\tau }_{\mathrm{oct}}=f({\sigma }_{\mathrm{oct}})$ and ${\tau }_{\mathrm{oct}}=f({\sigma }_{m,2})$.