Prediction of rock fracture toughness using a physics-integrated transformer model

Abstract

Accurately evaluating the temperature-dependent fracture toughness of rocks is essential for ensuring the long-term stability of deep underground engineering structures. However, conventional high-temperature fracture tests are operationally complex, time-consuming, and costly, which restricts their wide application in engineering design. To address these challenges, this study proposes a physics-informed Transformer framework designed to achieve high-accuracy predictions of Mode I, Mode II, and mixed-mode I–II fracture toughness under small-sample conditions. A temperature-dependent, multi-material database consisting of 120 Center-Cracked Brazilian Disk (CCBD) test cases was established for model training and validation. By embedding physically interpretable constraints into the Transformer model, the model effectively learns nonlinear couplings among input variables, enabling a synergistic integration of physical mechanisms and data-driven prediction. The proposed model maintains excellent accuracy despite limited data availability and demonstrates strong predictive performance across different temperatures and fracture modes. The results indicate that the method provides a reliable and interpretable approach for high-temperature fracture toughness prediction, offering an efficient, economical, and engineering-practical alternative in scenarios where experimental data are difficult to obtain.