LAUSR

Deep buried tunnels often suffer brittle instability and failure during the excavation process under high geological stress conditions (Liu et al. 2018). During this process, the surrounding rocks partially begin to break with crack initiation and growth. The deformation and failure of brittle rocks is a gradual process involving the initiation, growth and aggregation of cracks (Martin and Chandler 1994). The deformation and failure process of brittle rocks has been widely concerned by many researchers (Dai et al. 2015; Yang 2016; Yang and Hu 2019; Yang et al. 2018; Zhang and Wong 2013; Zhang et al. 2017; Zhou et al.2015). Researchers studied the failure of brittle rocks and divided the stress–strain curves of brittle rocks into five stages: stage I-crack closure; stage II-linear elastic deformation; stage III-crack initiation and stable crack growth; stage IV-unstable crack growth; and stage V-failure and post-peak behavior (Wawersik and Brace 1971; Eberhardt et al. 1999; Hoek and Bieniawski 1965; Scholz 1968; Zhou et al. 2014; Cheng et al. 2016; Zhang et al. 2011; Hallbauer et al. 1973; Tapponnier and Brace 1976). These five stages of the stress–strain curve are divided by four stress thresholds: crack closure stress (σcc), crack initiation stress (σci), crack damage stress (σcd) and peak strength (σf). The crack closure stress (σcc) indicates the closing of the preexisting microcracks within the rock; the crack initiation stress (σci) indicates the beginning of newborn microcracks within the rock; the crack damage stress (σcd) characterizes the coalescing of the nascent microcracks, which means the specimen loads into the unstable crack growth stage. To accurately obtain the stress thresholds, many methods have been proposed by researchers. Brace et al. (1966) * Xiao-Ping Zhang [email protected]