Neutrosophic Function with NNs for Analyzing and Expressing Anisotropy Characteristic and Scale Effect of Joint Surface Roughness

The shear behavior of rock mass significantly depends upon the surface roughness of rock joints which is generally characterized by the anisotropy characteristic and the scale effect. The large-scale natural rock joint surfaces, at Qingshi Town, southeast of Changshan County, Zhejiang Province, China, were used as a case study to analyze the roughness characteristics. A statistical assessment of joint roughness coefficient (JRC) indicated the roughness anisotropy of different sized rock joints. The lower limit (JRCmean-σ) was regarded as the determinate information, and the difference between lower and upper limits represented indeterminate information. The neutrosophic number (NN) was calculated to express the various JRC values. The parametric equations for JRC anisotropic ellipse were presented based on the JRC statistical assessment of joint profiles of various orientations. The JRC values of different sized joint samples were then quantitatively described by the neutrosophic function. Finally, a neutrosophic parameter ψ for evaluating the scale effect on the surface roughness anisotropy was introduced using the ratio of maximum directional roughness to minimum directional roughness. The case study indicates that the proposed method has the superiority in moving forward from subjective assessment to quantitative and objective analysis on anisotropy characteristic and scale effect of joint surface roughness.


Introduction
Joint roughness is one of the most important parameters for understanding the shear behavior of rock joints [1][2][3][4][5]. Yet the irregularity or roughness of rock surfaces is difficult to estimate. Thus, during the past few decades, considerable effort has been devoted to estimating the roughness properties of rock joint surfaces. Since Barton [6] first introduced the joint roughness coefficient (JRC) for quantifying rock joint roughness, various roughness parameters have been established for roughness characterization [7][8][9][10][11][12]. The scale effect on surface roughness has been proven to be an inherent property of rock joints. Consequently, a large number of researchers have attempted to determine the scale dependency of joint surface roughness by relating the roughness parameters to the joint sample size. Du et al. [13] found that the negative scale effect exists in rock joint roughness by statistical analysis of asperity amplitudes. Furthermore, anisotropy is everywhere in rock engineering, and the joint roughness varies directionally, which is an important source of anisotropic behavior of rock joints. Chen et al. [14] used geological statistics to analyze the anisotropy and the scale effect of rock joints using the sill value and the range of variogram. The statistical analysis of the directional JRC values of different sized rock joint samples may disclose the scale effect of joint roughness. Fardin et al. [15] investigated the scale effect on the surface roughness of rock joints using fractal concepts. The fractal parameters remained almost constant when the sampling windows were larger than the stationarity threshold. Yong et al. [16] analyzed the sampling problem in studying the scale effect of rock 2 Mathematical Problems in Engineering joints and proposed a sampling method based on a systematic sampling technique for quantitatively evaluating roughness characteristics and representativeness of different sized rock joints. However, the roughness characteristics are difficult to quantify because of the irregularity and inconsistency of joint roughness of different sized samples in various measurement orientations. Moreover, none of the previous studies considered the scale effect on the surface roughness anisotropy of rock joints.
The real world consists of determinate and/or indeterminate information. A neutrosophic number (NN) presented by Smarandache [17][18][19] seems appropriate for expressing them because a NN ( = + ) consists of a determinate part and an indeterminate part for , ∈ , where the symbol denotes indeterminacy and represents real numbers. Recently, Ye [20] developed a bidirectional projection model of NNs for a multiple attribute group decision-making problem. Then, Ye [21] proposed a deneutrosophication method and a possibility degree ranking method for neutrosophic numbers and applied these methods to a group decisionmaking problem under a NN environment. Kong et al. [22] introduced a cosine similarity measure of NNs and applied it to the misfire fault diagnosis of gasoline engine. Further, Ye [23] presented an exponential similarity measure of NNs for the fault diagnosis of steam turbine in a NN environment. To express indeterminate functions in indeterminate problems, Smarandache [24] introduced the interval function (thick function), which is defined as a neutrosophic function h: → ( ), where represents all real numbers and ( ) is all interval functions, denoted by the form of an interval function ℎ( ) = [ℎ 1 ( ), ℎ 2 ( )] for ∈ . Smarandache's neutrosophic function, however, cannot express and deal with functions involving NNs. Ye et al. [25] expressed the joint roughness coefficient and the shear strength in rock mechanics by means of neutrosophic functions (interval functions). It is necessary to introduce new neutrosophic functions containing NNs for statistical assessment of joint roughness, containing both determinate information and indeterminate information. Consequently, NN is appropriate for expressing various JRC values, because it consists of determinate and indeterminate parts.
The objectives of this paper are as follows: (1) presenting some basic operations of NNs and new neutrosophic function with NNs; (2) developing the parametric equations for JRC anisotropic ellipse to characterize the joint profiles in various orientations; (3) presenting a neutrosophic function with NNs to describe the JRC values of different sized joint samples; and (4) developing a neutrosophic parameter to evaluate the scale effect on the surface roughness anisotropy using the ratio of maximum directional roughness to minimum directional roughness.

Some Basic Operations of NNs.
In indeterminate environments, Smarandache [17][18][19] presented the concept of NN. It consists of a determinate part and an indeterminate part . Its mathematical expression is = + for , ∈ , where is indeterminacy and represents all real numbers. It has been successfully applied to represent determinate and/or indeterminate information in real problems.
When NN is considered as = 2 + 2 , its determinate value is 2 and its indeterminate value is 2 . In application, a possible interval range of indeterminacy is often specified to satisfy concrete application requirements. For example, if the indeterminacy is considered to fall within an possible interval [0, 0 = 0 for the best case, then equals the determinate part = . But if = 0 is the worst case, then is the same as the indeterminate part z = bI. For two NNs z 1 = a 1 + b 1 I and z 2 = a 2 + b 2 I for z 1 , z 2 ∈ Z and I ∈ [I L , I U ], the basic operational laws are derived as follows: (1) (2) (3) ] . (4)

Data Acquisition
A 141.4 cm ×141.4 cm joint surface sample was collected from a large sized, slightly weathered, slate rock surface with an orientation of N40W, 35NW at Qingshi Town, southeast of Changshan County, Zhejiang Province, China. The grayish-green colored slate rock is foliated, very finegrained, and formed by the metamorphosis of intermediate tuff, and the joint surface is relatively smooth and planar. The surface profiles were measured in 25 different orientations at an interval of 15 ∘ (Figure 1). A simple mechanical hand profilograph ( Figure 2) was employed to measure the rock joint profiles. The profilograph consists of feeler, drawing pen, balance block, fixed board, bubble levels, and drawing paper. It is easy to operate and correctly records a large number of joint traces with minimal cost. In this study, the number of measured roughness profiles with different orientations and sample sizes was 19,616. These recorded joint profiles were scanned by a large format scanner. On the basis of pixel  analysis, the joint profiles were then digitized with a sampling interval of 0.5 mm, which is fairly often used in previous studies on joint roughness determination [26]. Many studies regarding JRC evaluation have been carried out over the past three decades. The most commonly used methods include visual comparison, root mean square method (Z 2 ), roughness profile index method (R p ), and fractal approaches (D). However, none of these studies considered the combined effects of shear direction, the scale of joint surfaces, the inclination angle, and the amplitudes of asperities in joint roughness calculations. To overcome these shortcomings, Zhang et al. [27] developed a logistic function between JRC and the ratio of asperity amplitude to profile length. The inclination angle, the amplitude of asperities, and their directions were considered in this method. This procedure was proved to be accurate and efficient in JRC evaluation via the comparisons with different kinds of test results. Consequently, it was used in this study to calculate the JRC values for different sized and orientated profiles. It is briefly introduced as follows.
First, an arbitrary horizontal reference line was assumed to pass through the digitized profile. The mean vertical distance between the reference line and the points along the roughness profile was calculated by Mathematical Problems in Engineering where L is the length of a digitized roughness profile and x and y are the horizontal and vertical coordinates of the points on the profile, respectively. Those profile segments whose dip-direction is opposite to the shear direction were significant in determining joint roughness in the direction of shearing. The modified root mean square 2 taking into account the positive dilation angles was determined by A roughness index was introduced by Zhang et al. [27] and written as The JRC value was then calculated based on the following logistic correlation with , as This procedure was repeated to determine the JRC evaluation for all profiles.

Neutrosophic Function with NNs for Anisotropy Characteristics of Joint Surface Roughness
The JRC values of joint samples in different orientations were calculated and displayed on the polar plots. For example, Figure 3 shows A low standard deviation indicates that the data points tend to approximate the mean value. Here, the JRC values in a range from mean -(lower limit) to mean + (upper limit) were chosen to describe the roughness anisotropy characteristics. As shown in Figure 4, a total of 68.62%, where x, y are the coordinates of any point on the ellipse; a and b are the radius on the x and y axes, respectively, and is the orientation, which ranges from 0 to 2 in radians. For joint samples of each size, two ellipse envelopes can be established to fit the upper and lower limits of JRC values based on the least square method (LSM). The JRC values indicating the anisotropy characteristics should be spread among the two ellipse envelopes. According to the concept of NNs, the parametric equations with NNs for JRC are where n 1 , n 2 are the radius of the inside ellipse envelope; n1 , n2 are twice the difference in radius of the outside and inside ellipse envelopes. For example, n 1 , n 2 of the inside ellipse envelope were 8.655 and 5.808, and n 3 , n 4 of the outside ellipse envelope were 13.074 and 9.924 ( Figure 5). Then, the 1 , 2 terms can be calculated as follows: The fitting results of two NN radiuses (a=n 1 + n1 I and b=n 2 + n2 I for I∈[0, 0.5]) using the neutrosophic functions based on the parametric equations for the anisotropy characteristics of different sized joint samples are tabulated in Table 1.
By using (3) and (10) According to the fitting results of NNs of different sized joint samples in Table 1, the JRC neutrosophic functions of different sized joint profiles in each orientation can be obtained by using (13). The calculated results are tabulated in Table 2.

Neutrosophic Function with NNs for Scale Effect of Joint Surface Roughness
The mean JRC values in various orientations were tabulated (Table 3). For each orientation, as the mean JRC value for Mathematical Problems in Engineering 7 The fitting results of two NNs (z=m 1 + m1 I and z=m 2 + m2 I for ∈ [0,0.5]) in the neutrosophic function JRC(l) for anisotropy characteristics of different sized joint samples were tabulated in Table 4.

Neutrosophic Parameter for Evaluating the Roughness Anisotropy
Tatone [28] quantified the roughness anisotropy using the ratio of the maximum and minimum directional roughness according to the polar plots of the roughness parameters. The parameter = a/b, i.e., the ratio of major and minor principal axis, was utilized to indicate the anisotropic degree. However, both a and b are NNs, which consist of determinate and indeterminate parts, as shown below: Using (4), a neutrosophic parameter for evaluating the roughness anisotropy can be obtained as follows: Figure 7 illustrates the upper and lower bounds of the parameter for different sized joint samples. The difference between the up and down decreased with the sample size and approached a certain value when the sample size exceeded 60 cm.

Conclusion
The statistical assessment of JRC values was applied to analyze the roughness characteristics considering both the sample size and the measurement orientation. Basic operations of NNs were first introduced. The neutrosophic functions ( ) and JRC(l) were developed for analyzing and expressing the anisotropy characteristics and scale effect of joint surface roughness. The neutrosophic function ( ) for anisotropic ellipse was derived according to the statistical assessment data of JRC of joint profiles in various orientations. The upper and lower limits of JRC values decreased with the increasing sampling length, showing a negative scale effect on joint roughness. The neutrosophic function JRC(l) was then acquired to determine the JRC values of different sized joint samples. The neutrosophic parameter was developed according to the ratio of maximum and minimum directional roughness. This was proved to be a new parameter for indicating the relationship between the scale effect and the anisotropy of joint roughness. The neutrosophic functions efficiently and conveniently showed both the anisotropy characteristic and the scale effect of the surface roughness of rock joints.

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that they have no conflicts of interest.