Mathematical Problems in Engineering

Volume 2016, Article ID 4825709, 9 pages

http://dx.doi.org/10.1155/2016/4825709

## Neutrosophic Functions of the Joint Roughness Coefficient and the Shear Strength: A Case Study from the Pyroclastic Rock Mass in Shaoxing City, China

Key Laboratory of Rock Mechanics and Geohazards, Shaoxing University, Zhejiang, Shaoxing 312000, China

Received 29 February 2016; Accepted 26 April 2016

Academic Editor: Gregory Chagnon

Copyright © 2016 Jun Ye 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

Many studies have been carried out to investigate the scale effect on the shear behavior of rock joints. However, existing methods are difficult to determinate the joint roughness coefficient (JRC) and the shear strength of rock joints with incomplete and indeterminate information; the nature of scale dependency of rock joints is still unknown and remains an ongoing debate. Thus, this paper establishes two neutrosophic functions of the JRC values and the shear strength based on neutrosophic theory to express and handle the incomplete and indeterminate problems in the analyses of the JRC and the shear strength. An example, including four rock joint samples derived from the pyroclastic rock mass in Shaoxing city, China, is provided to show the effectiveness and rationality of the developed method. The experimental results demonstrate that the proposed neutrosophic functions can express and deal with the incomplete and indeterminate problems of the test data caused by geometry complexity of the rock joint surface and sampling bias. They provide a new approach for estimating the JRC values of the different-sized test profiles and the peak shear strength of rock joints.

#### 1. Introduction

In mining, civil, hydraulic, and petroleum engineering, engineers often face problems associated with jointed rock mass [1]. One of the most challenging tasks in engineering rock mechanics is comprehensively understanding the characteristics of rock joints, including orientation, extent, planarity, roughness, and strength of wall rock asperities. Roughness refers to the local departures from planarity at both small and large scale [2] and has a direct impact on the shear strength of rock joints. It has been researched in the past four decades since Barton introduced the joint roughness coefficient (JRC) [3].

A variety of researches have been carried out to quantitatively describe the surface roughness of fractured surfaces in an attempt to relate the conventional geometric parameters or fractal dimensions of surface roughness to the mechanical behavior of rock joints [4]. The most widely used formula for estimating the shear strength was proposed by Barton [5] as follows:where is the peak shear strength of rock joints, is the effective normal stress, is the basic friction angle of the discontinuity surface, and JCS is the joint wall compressive strength.

The variation in the mechanical behavior of joints with increasing scale is a well-known phenomenon which makes the mechanical parameters worked out in the laboratory unsuitable for the natural discontinuities located in real condition. To better understand the deformational behavior of rock systems, a large number of studies have been carried out over the last four decades for exploring the scale effect on the shear strength of rock joints [5–9]. However, many of these studies have produced conflicting results and the rock joints of the same lithology may exhibit an extremely large range in mechanical properties [10, 11]. The uncertainties in test results are mainly caused by the complex properties of surface roughness of rock joints. Du [12] found that the rock joint roughness in nature generally has the appearance-like nonuniformity, anisotropy, inhomogeneity, and scale effect. To describe the complex structure of joint surface, Chen et al. [13] proposed a geological statistics method to analyze the anisotropy and the size effects of structural surface of rocks. Meanwhile, the sampling bias and sampling disturbance effects may be responsible for incorrect conclusions concerning some of the apparent scale effects [14]. Time and budget constraints make it unfeasible to obtain sufficient data by lots of field experiments. Due to the special limitation in measurement accuracy and the variability of roughness, it is difficult to find a certain equation for describing the scale effect of different-sized rock joint samples.

Neutrosophy means the study of ideas and notions which are not true, nor false, but its study is between true and false, that is, neutral, indeterminate, unclear, vague, ambiguous, incomplete, contradictory, and so forth. Many types of indeterminacies are contained in many real situations. Therefore, the neutrosophic logic, neutrosophic set, neutrosophic probability, neutrosophic statistics, neutrosophic measure, neutrosophic precalculus, neutrosophic calculus, and so forth were born in neutrosophy [15]. Neutrosophy theory is very suitable for describing the incomplete and indeterminate information. Since there is the incomplete and indeterminate information in the JRC values of different-sized samples and the shear strength of rock joints indicated in previous researches, neutrosophic function is helpful to better express these incomplete and indeterminate phenomena and can express the relationship of the scale effect of JRC and the shear strength of rock joints. To do so, we use neutrosophic functions to describe the scale effect of JRC and shear strength and verify the effectiveness and rationality of the neutrosophic functions of the JRC values and the shear strength by real testing data for four large-scale rock joint samples in the field.

The rest of the paper is organized as follows. In Section 2, we introduce the some concepts of neutrosophic functions. In Section 3, the neutrosophic functions of JRC and the shear strength are established in different sampling lengths. In Section 4, a practical example based on real testing data for four test samples is given to illustrate the application and effectiveness of the developed approach. Finally, some final remarks and further work are given in Section 5.

#### 2. Some Concepts of Neutrosophic Functions

Smarandache [15] firstly proposed the neutrosophic functions and neutrosophic calculus to handle various problems of indeterminacy in real world.

Let us consider a neutrosophic function (thick function) , and thenwhere is all real numbers and is the set of all subsets of .

Thus, its graph is shown in Figure 1.