LAUSR

The hydro-mechanical behavior of rock masses is mainly dominated by the behavior of the contained rock joints (Cook 1992; Grasselli and Egger 2003; Bahaaddini et al. 2014). It is well-recognized that studying the closure behavior of a rock joint at small scale in laboratory is a prerequisite to comprehensively understand the in situ hydro-mechanical behavior of a jointed rock mass (Bandis et al. 1983). The aperture distribution of rock joints in rock masses, which usually behaves as the major pathways of underground water or air, is directly affected by its closure behavior under the action of compressive loading. An understanding of how flow evolves in such aperture is significant to many engineering applications (Hopkins 2000), such as geothermal heat extraction, geologic disposal of high-level nuclear wastes, and oil/gas exploration, and is also relevant in the prediction of natural phenomena including earthquakes (Cook 1992; Hopkins 2000; Nemoto et al. 2009; Zhu et al. 2019; Zou et al. 2019). In addition, wave propagation in rock masses is also closely related to the rock-joint closure, and studying the interaction of stress wave and rock joints is of significance to evaluate the stability of rock engineering structures under dynamic loading, such as tunnels, foundations, and rock slopes (Li et al. 2014a, b, 2017; Chen et al. 2015; Han et al. 2020). The closure displacement of a rock joint under the action of compressive loading is usual non-linear (Kulhawy 1975; Bandis et al. 1983). The relation between closure displacement and normal stress could be empirically described by the initial normal stiffness and the maximum closure displacement. The hyperbolic model, first proposed by Goodman (1976) and further developed by Bandis et al. (1983) and Barton et al. (1985), provides a good fit to experimental data in a wide range of normal stresses for both matching and un-matching rock joints. The empirical model proposed by Bandis et al. (1983) is widely used in practice and academia. There are also several other models available in the literature, such as semi-logarithmic model (Xia et al. 2003), power-law model (Swan 1983; Xia et al. 2003), and exponential model (Malama and Kulatilake 2003). An empirical excess stress model was developed by Li et al. (2020) to capture the rate-dependent compressive behavior of rock joint. Nevertheless, the parameters used in those empirical models should be determined by closure tests. To predict the closure of rock joints, theoretical models with the inclusion of Hertz contact theory have been developed in the past decades. The pioneering work was done by Greenwood and Williamson (1966), followed by more elaborated models with various geometrical shapes of asperity (Swan 1983; Brown and Scholz 1985; Sun et al. 1985; Lanaro and Stephansson 2003; Matsuki et al. 2008) and the statistical approach for describing the morphology (Misra 1999; Beeler and Hickman 2001). The basic idea behind most of the Hertz-based theoretical models is that the surface roughness can be treated as a random process with spherical asperities of equal radius (Fig. 1). Greenwood and Wu (2001) pointed out that the notion of asperity in itself was questionable, but there was a lack of criterion to separate these “asperities” due to the fractal nature of roughness that each “asperity” has other “asperities” at its tip and so on. Because of the relative simplicity and computational attractiveness, Greenwood and Williamson theory survived severe criticism and became a widely accepted theory for the contact of two rough surfaces. Different from the steel surface used by Greenwood and Williamson, rock joints are usually composed of small-scale asperity and large-scale * Qing Zhao Zhang [email protected]