Mathematical Problems in Engineering

Volume 2018, Article ID 9835341, 14 pages

https://doi.org/10.1155/2018/9835341

## A Method for Estimating the Surface Roughness of Rock Discontinuities

Correspondence should be addressed to Hui-ming Tang; nc.ude.guc@mhgnat

Received 19 October 2017; Revised 26 January 2018; Accepted 8 February 2018; Published 18 March 2018

Academic Editor: Fabrizio Greco

Copyright © 2018 Yi Cai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The primary objective of this study is to develop a parameter with a clear physical meaning to estimate the surface roughness of rock discontinuities. This parameter must be closely related to the shear strength of rock discontinuities. The first part of this study focuses on defining and computing this parameter. The estimation formula for the shear strength of a triangle within a discontinuity surface is derived based on Patton’s model. The parameter, namely, the index of roughness (), is then proposed to quantitatively estimate discontinuity roughness. Based on laser scanning techniques, digital models of discontinuities and discontinuity profiles are constructed, and then their corresponding values are computed. In the second part of this study, the computational processes and estimated effects of the two-dimensional (2D) and three-dimensional (3D) values of the discontinuities are illustrated through several applications. Results show that the 2D and 3D values of these discontinuities indicate anisotropy and sampling interval effects. In addition, a strong linear correlation is detected between and the joint roughness coefficient (JRC) for seventy-four profiles and eleven discontinuity specimens, respectively. Finally, the proposed method, back analysis method, root mean square () method, and Grasselli’s method are compared to study the use of the parameter .

#### 1. Introduction

Rock masses usually contain significant discontinuities such as joints, faults, bedding planes, fractures, and other mechanical defects. The surface roughness of rock discontinuities plays an important role in the mechanical properties of rocks, including the shear strength and deformation [1–4], and can also directly affect the seepage behaviors of discontinuities [5, 6]. One of the purposes of studying discontinuities is to accurately and quickly estimate their quantitative roughness to then assess their shear strength. Thus, researchers have attempted to develop roughness estimation methods for discontinuities.

Myers [7] proposed the root mean square parameter* Z*_{2} to quantitatively describe the roughness of discontinuity profiles:where the* x*- and* y*-axes are the extension and fluctuation directions of the profile, respectively;* L* is the length of the horizontal projection of the profile;* D* is the horizontal sampling interval;* M *is the number of discrete line segments of the profile; and* θ *is the dip angle of a line segment between two adjacent points. Barton [1] proposed the joint roughness coefficient (JRC) to describe the surface roughness of a discontinuity, and ten typical profiles were presented by Barton and Choubey [2]. The structure function (SF) and roughness profile index () were also proposed to quantitatively evaluate surface roughness [8, 9]. Later, correlations between the JRC and other parameters, such as

*Z*

_{2}, SF, and , have been investigated by researchers [10–13]. Recently, several other methods have been proposed to quantify discontinuity roughness. Grasselli and his colleagues proposed a roughness estimation parameter based on the statistical analysis of the potential contact areas [14–17]. Ye et al. [18] proposed a JRC estimation method based on neutrosophic functions. Moreover, according to previous studies, several improved roughness parameters have been proposed. For example, regarding

*Z*

_{2}, Belem et al. [19] proposed a formula for calculating the 3D root mean square (

*Z*

_{2}), and Zhang et al. [20] suggested a modified root mean square () that can describe the anisotropies of profile roughness. Additionally, fractal theory was used to estimate the roughness of discontinuities [21–23]. Meanwhile, the scale effect, sampling interval effect, and anisotropy of discontinuity roughness were studied by researchers. For example, Barton and Choubey [2] first emphasized the existence of a scale effect during discontinuity roughness estimations, and Yu and Vayssade [12] noted that roughness parameters, such as

*Z*

_{2}and SF, were sensitive to the sampling interval.

In conclusion, existing discontinuity roughness estimation methods can be mainly subdivided into three categories: empirical methods [1, 2], statistical methods [7–20], and fractal analysis methods [21–23]. Although these methods can effectively describe the morphological features of discontinuities, they may have the following limitations: the roughness estimated by an empirical method may be subjective and most statistical methods and fractal analysis methods estimate the roughness without closely incorporating their shear failure mechanisms, making the physical meanings of these roughness parameters unclear and resulting in a weak relationship between the roughness parameters and shear strength of the discontinuities. For example,* Z*_{2}, which was proposed by Myers [7], can represent the root mean square of the tangential values of all of the tooth dip angles (i.e., ) without considering the shear directions (see (1)). However, the relationship between the tooth dip angle and the tooth shear strength does not conform to an exact tangential function (see (2) proposed by Patton [24]):where *σ* and are the normal stress and the basic friction angle, respectively. Therefore, the shear strength of discontinuities estimated with* Z*_{2} cannot directly reflect the influence of the basic friction angle (). In other words, the relationship between* Z*_{2} and the shear strength of discontinuities is not well established. As another example, Grasselli’s method can effectively estimate the 3D discontinuity roughness using the parameter ; however, the physical meaning of this parameter is unclear [25, 26].

A strong connection between the roughness parameter and the shear strength of discontinuities should be established and could contribute to building a shear strength model of discontinuities. Therefore, based on the shear failure mechanism of discontinuities, a formula for the shear strength of a triangle within a discontinuity surface is derived from Patton’s model, and the roughness estimation parameter, index of roughness (), and its calculation are then presented. In addition, the calculation procedure and effects of the proposed method are presented using applied examples. Finally, the rationality of the new method is analyzed based on comparisons with other methods.

#### 2. Mechanical Basis

The normal stress directly affects the shear failure mechanism of discontinuities. Jaeger [27] noted that some of the asperities on a discontinuity can be sheared off under low normal stress and that failures under only dilation or shear are both extreme cases. Pereira and de Freitas [28] drew similar conclusions in their study of the shear failure mechanism of sandstone with teeth-discontinuities. Therefore, for natural discontinuities, when the normal stress is low, the dilation effect is dominant, and when the normal stress is high, shear failure of asperities is dominant over dilation.

In addition, when materials fill the discontinuity gap, the shear resistance of the discontinuity can be affected by the properties of these filling materials. To simplify the discontinuities in this study, we hypothesize that the discontinuities are well-mated and that no gaps are present along them.

##### 2.1. Shear Failure Mechanism of a Regular-Teeth-Discontinuity

The teeth-discontinuity model proposed by Patton [24] is shown in Figure 1(a). The asperity features include the number of teeth and their dip angles and heights . In this paper, Patton’s teeth-discontinuity model, which uses the same parameters and for all of the teeth, is defined as the regular-teeth-discontinuity (RTD) model. Based on the RTD specimens, Patton studied the relation between the shear strength (* τ*) and the dip angle (

*). Figure 1(b) shows a force analysis of a single tooth from the RTD model under a normal stress . The basic friction angle of the discontinuity is , and the normal and shear stresses on a tooth surface are and , respectively.By assuming that the shear strength*

*θ**of the RTD is consistent with the Coulomb-Navier criterion (4) and inserting the corresponding parameters of (3) into (4), (2) can be obtained.Patton studied the shear strength envelopes of RTD specimens with different teeth inclinations, as illustrated in Figure 2(a), where is the residual friction angle of the RTD specimen. Additional studies indicate that (2) can be applied to the shear strength calculations of RTD models only at low normal stress. When the normal stress*

*τ**exceeds (Figure 2(b)), the shear strength envelope will experience a transition. After the transition, the shear strength*

*σ**τ*of the RTD is calculated bywhere

*C*is the cohesion. Therefore, a bilinear model (Patton’s model) for calculating the shear strength of the RTD model is given as