Modeling surface roughness degradation of rock joint wall during monotonic and cyclic shearing

Abstract

Based on previously proposed surface roughness description parameters (k _a, θ_s, SR_s, DR_r, a ₀), two generalized rock joint surface roughness degradation models were proposed to predict the variation of joint surface degradation during shearing under both constant normal stress (CNS) and constant normal stiffness (CNK) loading conditions. The first model was developed based on the evolution of secondary roughness (an extension of an existing model) and the second was developed based on the concept of “average asperity probable contact angle.” Model variables can be initial normal stress (σ_n0; k _n ≥ 0), normal stiffness (k _n; σ_n0 ≥ 0), accumulated shear displacement u _s-tot (monotonic or cyclic shearing), and surface roughness amplitude a ₀. Good agreement between experimental and predicted degradation was observed. The models also allow prediction of surface degradation in large-scale shear fractures. Both models are semi-incremental, readily implemented in a numerical code, and adaptable to existing elastoplastic joint behavior models.

Appendix 1: Estimating the proposed roughness parameters

The minimum requirement for estimating the proposed roughness parameters is the value of the Z2 parameter, which is given by:

$$ Z2_{{x,y}} = {\sqrt {\frac{1}{{(N - 1)\Updelta x^{2}}}{\sum\limits_{i = 1}^{N - 1} {{\left({z_{{i + 1}} - z_{i}} \right)}^{2}}}}} $$

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where subscripts x and y denote the directions supporting the profile data acquisition; N = number of discrete measurements of height; Δx = sampling step; and z _i = discrete height of the point on the profile.

The Z2 value can also be derived from the empirical relationship proposed by Tse and Cruden [58], as follows:

$$ \log_{10}(Z2) \approx {{{{\frac{{\rm JRC - 32.20}}{{32.47}}} }}}, \quad r = 0.986 $$

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where JRC = joint roughness coefficient defined in [6].

The procedure to estimate proposed roughness parameters is as follows:

• (a) estimation of the roughness profile mean angles θ _p(x) and θ _p(y), starting from Z2_(x) and Z2_(y)

• (b) calculation of the pseudo-surfacial mean angle $\theta_{p\_xy} (= [\theta_{p(x)} + \theta_{p(y)}]/2)$ and k _a (Eq. 2)

• (c) estimation of the surface mean angle θ_s from the pseudo-surfacial mean angle $\theta_{p\_xy}$

• (d) calculation of R ^*_s = cos⁻¹(θ_s) before estimating R _s from R ^*_s

• (e) calculation of surface roughness coefficient R _s

• (f) calculation of the degree of surface relative roughness, DR_r = (R _s−1)/R _s

To obtain different empirical equations, the main morphological parameters (θ _p(x), θ _p(y), k _a, θ_s, DR_r) were calculated for the four different joint types: two man-made granite joints with a sanded surface (a ₀ = 0.552 mm) and a hammered surface (a ₀ = 1.742 mm), and two mortar joints with a corrugated surface (a ₀ = 2.000 mm) and an irregular rough surface (a ₀ = 8.103 mm). The man-made joints represented “pure” second-order roughness, while the mortar joints represented a combination of second-order and first-order roughness. The resultant equations are given as follows:

• (a)
  $$\left\{\begin{aligned}& \theta_{{p(x,y)}} = a_{0}^{3} {\left({46.5135{\left({\frac{{Z2_{{x,y}}}}{{a_{0}^{3}}} + 0.0145} \right)}^{{1.1389}}} \right)}\quad{\hbox{for}}\quad a_{0} < 2 \\& \\& \theta _{{p(x,y)}} = a_{0}^{3} {\left({42.9125{\left({\frac{{Z2_{{x,y}}}}{{a_{0}^{3}}}} \right)} + 0.0216} \right)}\quad{\hbox{for}}\quad a_{0} \geq 2 \\ \end{aligned} \right.$$

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• (b)
  $$\theta_{p\_xy} = \frac{\theta_{p(x)} + \theta _{p(y)}}{2}$$

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• (c)
  $$\left\{\begin{aligned}& \theta_{s} = a_{0}^{3} {\left({1.5171{\left({\frac{{\theta_{{p\_xy}}}}{{a_{0}^{3}}}} \right)} - 0.0688} \right)}\quad{\hbox{for}}\quad a_{0} < 2 \\& \\& \theta_{s} = a_{0} ^{3} {\left({1.7081{\left({\frac{{\theta_{{p\_xy}}}}{{a_{0}^{3}}}} \right)}^{{1.089}}} \right)}\quad{\hbox{for}}\quad a_{0} \geq 2 \\ \end{aligned} \right.$$

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• (d) (According to Riss et al. [50])

  $$R^{{{*}}}_{\rm s} = \frac{1}{{\cos (\theta_{\rm s})}}$$

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• (e)
  $$\left\{\begin{aligned}& R_{\rm s} = a_{0} \times 1.0081{\left({\frac{{R^{*}_{\rm s}}}{{a_{0}}}} \right)}^{{0.9899}}\quad {\hbox{for}}\quad a_{0} < 2 \\& \\& R_{\rm s} = a_{0} \times 0.9971{\left({\frac{{R^{*}_{\rm s}}}{{a_{0}}}} \right)}^{{0.9878}}\quad {\hbox{for}}\quad a_{0} \geq 2 \\ \end{aligned} \right.$$

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• (f)
  $${\rm DR}_{\rm r} = {\left({1 - \frac{1}{{R_{\rm s}}}} \right)}$$

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