The normal stiffness of rock joints is decreased substantially by the opening between joint walls. This paper presents a modified semi-logarithmic relationship to describe the normal deformability of irregular rock joints under varying contact states. The effect of unevenness on the joint closure on opened joint profiles is considered by introducing a dimensionless coefficient. The magnitude of the coefficient highly depends on the joint roughness degree. Experimental data from normal compressive tests on several natural rock joints match well with the analytical results. The constitutive law, after being embedded in a distinct element method code, is able to analyse the stability of rock masses where opened joints exist due to stress relief, nearby blasting and earthquake vibration.


The closure of rock joints governs the overall deformability of rock masses. Critical factors dominating the joint normal deformation include joint wall material, surface roughness and joint matching state. Initially closed rock joints can become dislocated or opened due to underground excavation, nearby blasting and earthquake vibration (Li, 2016, Li et al., 2014, Tang et al., 2016). Under identical degree of normal compressive force, opened rock joints exhibit considerably softer normal stiffness with much higher joint closure, compared to those of closed ones (Li et al., 2016a, Li et al., 2016b, Tang et al., 2013). To predict the movement of rock blocks with opened joint walls, a constitutive law representing the normal deformability of rock joints under distinct contact conditions is required.

It has long been well-recognised that the normal stress-joint closure relationship is typically non-linear. Extensive studies indicated that the normal stiffness of closed rock joints can be shaped by empirical formulations, including hyperbolic equation (Bandis et al., 1983, Goodman, 1976), power equation (Swan, 1983) and logarithmic equation (Evans et al., 1992, Li et al., 2016a, Malama and Kulatilake, 2003, Zangerl et al., 2008, Zhao and Brown, 1992). Investigations into the normal deformation of opened joint walls are much less reported. Bandis et al. (1983) suggested that a semi-logarithmic function could be suitable to interpret the non-linearity of the normal stiffness of unmated rock joints. However, the semi-logarithmic model is incapable to consider the closure under varying degrees of joint opening. Saeb and Amadei (1992) took into account the effect of dislocation on the joint deformation by introducing a dilative component due to opening into the hyperbolic relationship of Bandis et al. (1983). Nevertheless, the assumption that the eventual joint closure equals the summation of the initial aperture created by mismatching and the maximum joint closure of closed joints, conflicts with the experimental observations that mismatched rock joints would never reach the maximum closure state (Bandis et al., 1983, Li et al., 2016a). On the other hand, Xia et al. (2003) and Tang et al. (2013) described the normal closure of mismatched rock joints based on the Hertzian contact theory. Their predictive models involve overly many parameters, making it impossible for practical use. Recently, Li et al. (2016a) proposed a semi-logarithmic formulation that related the normal stiffness variation with the degree of joint opening, the performance of which has been validated against experimental curves of normal compression tests on opened rock joints. The semi-logarithmic equation is merely applicable for idealised rock joints with various initial openings.

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