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

Authors: Rui Yong; Leiyu Gu; Jun Ye; Shi-Gui Du; Man Huang; Gaojian Hu; Jie Liu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

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 variousJRC values.The parametric equations forJRC anisotropic ellipse were presented based on theJRC statistical assessment of joint profiles of various orientations. TheJRC 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.


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## Body

## 1. Introduction

Joint roughness is one of the most important parameters for understanding the shear behavior of rock joints [1–5]. 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 [7–12]. 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 directionalJRC 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 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 [17–19] seems appropriate for expressing them because a NN (z=a+bI) consists of a determinate part a and an indeterminate part bI for a,b∈R, where the symbol I denotes indeterminacy and R 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 decision-making 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 functionh: R→H(R), where R represents all real numbers and H(R) is all interval functions, denoted by the form of an interval function h(x)=h1x,h2x for x∈R. 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 variousJRC 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 forJRC anisotropic ellipse to characterize the joint profiles in various orientations; (3) presenting a neutrosophic function with NNs to describe theJRC values of different sized joint samples; and (4) developing a neutrosophic parameter ψ to evaluate the scale effect on the surface roughness anisotropy using the ratio of maximum directional roughness to minimum directional roughness.


## 2. NNs and Neutrosophic Functions

### 2.1. Some Basic Operations of NNs

In indeterminate environments, Smarandache [17–19] presented the concept of NN. It consists of a determinate part a and an indeterminate part bI. Its mathematical expression is z=a+bI for a,b∈R, where I is indeterminacy and R represents all real numbers. It has been successfully applied to represent determinate and/or indeterminate information in real problems.When NN is considered asz=2+2I, its determinate value is 2 and its indeterminate value is 2I. In application, a possible interval range of indeterminacy I is often specified to satisfy concrete application requirements. For example, if the indeterminacy I is considered to fall within an possible interval [0, 0.1], then it is equivalent to z=[2,2.2], where z is within the interval [2,2.2]. If I∈[0.1,0.2] is the interval range, thenz = [2.2, 2.4].Yet,z=a+bI may be a possible interval number, where z=[a+bIL,a+bIU] for z∈Z (Z is all NNs) and I∈[IL,IU]. Specially, if bI=0 for the best case, then z equals the determinate part z=a. But if a=0 is the worst case, then z is the same as the indeterminate partz =bI.For two NNsz1 =a1 +b1I andz2 =a2 +b2I forz1,z2 ∈ Z andI∈ [IL, IU], the basic operational laws are derived as follows:(1)z1+z2=a1+a2+b1+b2I=a1+a2+b1IL+b2IL,a1+a2+b1IU+b2IU;(2)z1-z2=a1-a2+b1-b2I=a1-a2+b1IL-b2IL,a1-a2+b1IU-b2IU;(3)z1×z2=a1a2+a1b2+a2b1I+b1b2I2=min⁡a1+b1ILa2+b2IL,a1+b1ILa2+b2IU,a1+b1IUa2+b2IL,a1+b1IUa2+b2IU,max⁡a1+b1ILa2+b2IL,a1+b1ILa2+b2IU,a1+b1IUa2+b2IL,a1+b1IUa2+b2IU;(4)z1z2=a1+b1Ia2+b2I=a1+b1IL,a1+b1IUa2+b2IL,a2+b2IU=min⁡a1+b1ILa2+b2IU,a1+b1ILa2+b2IL,a1+b1IUa2+b2IU,a1+b1IUa2+b2IL,max⁡a1+b1ILa2+b2IU,a1+b1ILa2+b2IL,a1+b1IUa2+b2IU,a1+b1IUa2+b2IL.


## 2.1. Some Basic Operations of NNs

In indeterminate environments, Smarandache [17–19] presented the concept of NN. It consists of a determinate part a and an indeterminate part bI. Its mathematical expression is z=a+bI for a,b∈R, where I is indeterminacy and R represents all real numbers. It has been successfully applied to represent determinate and/or indeterminate information in real problems.When NN is considered asz=2+2I, its determinate value is 2 and its indeterminate value is 2I. In application, a possible interval range of indeterminacy I is often specified to satisfy concrete application requirements. For example, if the indeterminacy I is considered to fall within an possible interval [0, 0.1], then it is equivalent to z=[2,2.2], where z is within the interval [2,2.2]. If I∈[0.1,0.2] is the interval range, thenz = [2.2, 2.4].Yet,z=a+bI may be a possible interval number, where z=[a+bIL,a+bIU] for z∈Z (Z is all NNs) and I∈[IL,IU]. Specially, if bI=0 for the best case, then z equals the determinate part z=a. But if a=0 is the worst case, then z is the same as the indeterminate partz =bI.For two NNsz1 =a1 +b1I andz2 =a2 +b2I forz1,z2 ∈ Z andI∈ [IL, IU], the basic operational laws are derived as follows:(1)z1+z2=a1+a2+b1+b2I=a1+a2+b1IL+b2IL,a1+a2+b1IU+b2IU;(2)z1-z2=a1-a2+b1-b2I=a1-a2+b1IL-b2IL,a1-a2+b1IU-b2IU;(3)z1×z2=a1a2+a1b2+a2b1I+b1b2I2=min⁡a1+b1ILa2+b2IL,a1+b1ILa2+b2IU,a1+b1IUa2+b2IL,a1+b1IUa2+b2IU,max⁡a1+b1ILa2+b2IL,a1+b1ILa2+b2IU,a1+b1IUa2+b2IL,a1+b1IUa2+b2IU;(4)z1z2=a1+b1Ia2+b2I=a1+b1IL,a1+b1IUa2+b2IL,a2+b2IU=min⁡a1+b1ILa2+b2IU,a1+b1ILa2+b2IL,a1+b1IUa2+b2IU,a1+b1IUa2+b2IL,max⁡a1+b1ILa2+b2IU,a1+b1ILa2+b2IL,a1+b1IUa2+b2IU,a1+b1IUa2+b2IL.


## 3. 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 fine-grained, 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° (Figure1). 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].Figure 1
Arrangement of measuring profiles on the natural slate joint surface. (a) Measurement orientation; (b) examples of the arrangement of the test profiles in the orientations of 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°.
(a) (b)Figure 2
Mechanical hand profilograph used for measuring the profiles of rock joints.Many studies regardingJRC evaluation have been carried out over the past three decades. The most commonly used methods include visual comparison, root mean square method (Z2), roughness profile index method (Rp), 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 inJRC evaluation via the comparisons with different kinds of test results. Consequently, it was used in this study to calculate theJRC 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(5)hav=∫x=0x=LydxL=∑i=1Nyi+1+yixi+1-xi2LwhereL is the length of a digitized roughness profile andx andy 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 squareZ2′ taking into account the positive dilation angles was determined by (6)Z2′=1L∫x=0x=Lmax⁡0,dydx2dx=∑i=1Mmax⁡0,yi+1-yi2xi+1-xiL1/2A roughness indexλ was introduced by Zhang et al. [27] and written as(7)λ=hLα·Z2′1-α,α=13TheJRC value was then calculated based on the following logistic correlation withλ, as(8)JRC=401+e-20λ-20This procedure was repeated to determine theJRC evaluation for all profiles.


## 4. Neutrosophic Function with NNs for Anisotropy Characteristics of Joint Surface Roughness

TheJRC values of joint samples in different orientations were calculated and displayed on the polar plots. For example, Figure 3 shows theJRC values of 10 cm joint samples with various orientations from 0° to 345° at a 15° interval. As can be observed, theJRC values in some orientations (e.g., 0° and 180°) were greater, while the values in the orientations of 90° and 270° were relatively smaller. TheJRC values in all orientations were randomly distributed in a range from 0 to 20. TheJRC distributions of 10cm joint samples in the orientations of 0°, 90°, 180°, and 270° are illustrated in Figure 4. These frequency distributions can be represented graphically by the characteristic bell-shaped curves, which are close to the “Normal Distribution” or the “Right-skewed Distribution.” The mean value JRCmean and the standard deviationσ ofJRC values in the 0° orientation were 10.536 and 2.233. For those joint samples in the 180° orientation, JRCmean andσwere 9.850 and 2.144, respectively. TheJRC mean values in the 90° and 270° orientations were 7.047 and 6.802, respectively, and the standard deviations were 2.405 and 2.117.Figure 3
Polar plots of directionalJRC values of 10cm joint samples on the slate fracture surface.Figure 4
Frequency distributions of 10cm joint samples in the orientations of (a) 0°, (b) 90°, (c) 180°, and (d) 270°.
(a) (b) (c) (d)A low standard deviation indicates that the data points tend to approximate the mean value. Here, theJRC values in a range from JRCmean-σ (lower limit) to JRCmean+σ (upper limit) were chosen to describe the roughness anisotropy characteristics. As shown in Figure 4, a total of 68.62%, 73.37%, 72.41%, and 71.19% of the directional joint roughness samples fell within the range [JRCmean-σ, JRCmean+σ]. The polar plots of the upper and lower limits ofJRC values showed a pattern similar to an ellipse (Figure 5). The parametric equations for each anisotropic ellipse are(9)x=acos⁡θy=bsin⁡θ,wherex,y are the coordinates of any point on the ellipse;a andb are the radius on thex andy axes, respectively, andθ is the orientation, which ranges from 0 to 2π in radians.Figure 5
Upper and lower limits ofJRC values and the corresponding ellipse fitting envelopes.For joint samples of each size, two ellipse envelopes can be established to fit the upper and lower limits ofJRC values based on the least square method (LSM). TheJRC values indicating the anisotropy characteristics should be spread among the two ellipse envelopes. According to the concept of NNs, the parametric equations with NNs forJRC are(10)x=n1+μn1Icos⁡θy=n2+μn2Isin⁡θ,I∈0,0.5,wheren1,n2 are the radius of the inside ellipse envelope;μn1,μn2 are twice the difference in radius of the outside and inside ellipse envelopes. For example,n1,n2 of the inside ellipse envelope were 8.655 and 5.808, andn3,n4 of the outside ellipse envelope were 13.074 and 9.924 (Figure 5). Then, theμn1,μn2 terms can be calculated as follows:(11)μn1=2×13.074-8.655=8.838μn2=2×9.924-5.808=8.232.Thus, the neutrosophic functions based on the parametric equations with NNs for describing the anisotropy characteristic of 10cm joint samples were(12)x=8.655+8.838·Icos⁡θy=5.808+8.232·Isin⁡θ,I∈0,0.5.The fitting results of two NN radiuses (a=n1+μn1I andb=n2+μn2I forI∈[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.Table 1
Fitting results of NNs for theJRC values of different sized joint samples.
Scale (cm) n 1 μ n 1 n 2 μ n 2 10 8.655 8.838 5.808 8.232 20 8.332 7.009 5.710 6.467 30 8.028 6.455 5.548 5.477 40 7.902 5.978 5.479 4.668 50 7.638 5.925 5.343 4.349 60 7.476 5.639 5.524 4.976 70 7.503 5.229 5.246 4.602 80 7.401 4.693 5.137 4.179 90 7.271 4.256 5.097 3.735 100 7.178 3.946 5.045 3.319By using (3) and (10), the directional JRC values with NN functions, JRC(θ), withinI∈[0, 0.5] can be derived by (13)JRCθ=x2+y2=n1+μn1Icos⁡θ2+n2+μn2Isin⁡θ2=n1+μn1I·n1+μn1Icos2θ+n2+μn2I·n2+μn2Isin2θ=n12,n1+μn122cos2θ+n22,n2+μn222sin2θ=n12cos2θ+n22sin2θ,n1+μn122cos2θ+n2+μn222sin2θAccording to the fitting results of NNs of different sized joint samples in Table1, theJRC neutrosophic functions of different sized joint profiles in each orientation can be obtained by using (13). The calculated results are tabulated in Table 2.Table 2
Neutrosophic function values ofJRC(θ) within I∈[0,0.5] (indeterminate intervals) for different sized joint samples in various orientations.
Orientation J R C (°) 10 cm 20 cm 30 cm 40 cm 50 cm 0 [ 8.655,13.074 ] [ 8.332,11.837 ] [ 8.028,11.255 ] [ 7.902,10.891 ] [ 7.638,10.600 ] 15 [ 8.494,12.887 ] [ 8.183,11.665 ] [ 7.886,11.081 ] [ 7.763,10.712 ] [ 7.506,10.422 ] 30 [ 8.038,12.362 ] [ 7.760,11.184 ] [ 7.485,10.591 ] [ 7.371,10.209 ] [ 7.134,9.920 ] 45 [ 7.370,11.607 ] [ 7.143,10.491 ] [ 6.900,9.883 ] [ 6.799,9.478 ] [ 6.591,9.189 ] 60 [ 6.635,10.798 ] [ 6.466,9.748 ] [ 6.261,9.120 ] [ 6.174,8.685 ] [ 6.000,8.395 ] 75 [ 6.041,10.165 ] [ 5.922,9.166 ] [ 5.748,8.518 ] [ 5.674,8.056 ] [ 5.527,7.763 ] 90 [ 5.808,9.924 ] [ 5.710,8.944 ] [ 5.548,8.287 ] [ 5.479,7.813 ] [ 5.343,7.518 ] 105 [ 6.041,10.165 ] [ 5.922,9.166 ] [ 5.748,8.518 ] [ 5.674,8.056 ] [ 5.527,7.763 ] 120 [ 6.635,10.798 ] [ 6.466,9.748 ] [ 6.261,9.120 ] [ 6.174,8.685 ] [ 6.000,8.395 ] 135 [ 7.370,11.607 ] [ 7.143,10.491 ] [ 6.900,9.883 ] [ 6.799,9.478 ] [ 6.591,9.189 ] 150 [ 8.038,12.362 ] [ 7.760,11.184 ] [ 7.485,10.591 ] [ 7.371,10.209 ] [ 7.134,9.920 ] 165 [ 8.494,12.887 ] [ 8.183,11.665 ] [ 7.886,11.081 ] [ 7.763,10.712 ] [ 7.506,10.422 ] 180 [ 8.655,13.074 ] [ 8.332,11.837 ] [ 8.028,11.255 ] [ 7.902,10.891 ] [ 7.638,10.600 ] 195 [ 8.494,12.887 ] [ 8.183,11.665 ] [ 7.886,11.081 ] [ 7.763,10.712 ] [ 7.506,10.422 ] 210 [ 8.038,12.362 ] [ 7.760,11.184 ] [ 7.485,10.591 ] [ 7.371,10.209 ] [ 7.134,9.920 ] 225 [ 7.370,11.607 ] [ 7.143,10.491 ] [ 6.900,9.883 ] [ 6.799,9.478 ] [ 6.591,9.189 ] 240 [ 6.635,10.798 ] [ 6.466,9.748 ] [ 6.261,9.120 ] [ 6.174,8.685 ] [ 6.000,8.395 ] 255 [ 6.041,10.165 ] [ 5.922,9.166 ] [ 5.748,8.518 ] [ 5.674,8.056 ] [ 5.527,7.763 ] 270 [ 5.808,9.924 ] [ 5.710,8.944 ] [ 5.548,8.287 ] [ 5.479,7.813 ] [ 5.343,7.518 ] 285 [ 6.041,10.165 ] [ 5.922,9.166 ] [ 5.748,8.518 ] [ 5.674,8.056 ] [ 5.527,7.763 ] 300 [ 6.635,10.798 ] [ 6.466,9.748 ] [ 6.261,9.120 ] [ 6.174,8.685 ] [ 6.000,8.395 ] 315 [ 7.370,11.607 ] [ 7.143,10.491 ] [ 6.900,9.883 ] [ 6.799,9.478 ] [ 6.591,9.189 ] 330 [ 8.038,12.362 ] [ 7.760,11.184 ] [ 7.485,10.591 ] [ 7.371,10.209 ] [ 7.134,9.920 ] 345 [ 8.494,12.887 ] [ 8.183,11.665 ] [ 7.886,11.081 ] [ 7.763,10.712 ] [ 7.506,10.422 ]  360 [ 8.655,13.074 ] [ 8.332,11.837 ] [ 8.028,11.255 ] [ 7.902,10.891 ] [ 7.638,10.600 ]  Orientation J R C (°) 60 cm 70 cm 80 cm 90 cm 100 cm  0 [ 7.476,10.295 ] [ 7.503,10.117 ] [ 7.401,9.747 ] [ 7.271,9.399 ] [ 7.178,9.151 ] 15 [ 7.361,10.158 ] [ 7.373,9.966 ] [ 7.271,9.599 ] [ 7.146,9.256 ] [ 7.055,9.008 ] 30 [ 7.039,9.774 ] [ 7.007,9.540 ] [ 6.905,9.182 ] [ 6.793,8.853 ] [ 6.708,8.605 ] 45 [ 6.573,9.224 ] [ 6.473,8.925 ] [ 6.370,8.580 ] [ 6.279,8.272 ] [ 6.204,8.021 ] 60 [ 6.071,8.639 ] [ 5.892,8.265 ] [ 5.787,7.932 ] [ 5.718,7.646 ] [ 5.654,7.392 ] 75 [ 5.676,8.185 ] [ 5.426,7.746 ] [ 5.319,7.422 ] [ 5.270,7.153 ] [ 5.215,6.895 ] 90 [ 5.524,8.012 ] [ 5.246,7.547 ] [ 5.137,7.227 ] [ 5.097,6.964 ] [ 5.045,6.704 ] 105 [ 5.676,8.185 ] [ 5.426,7.746 ] [ 5.319,7.422 ] [ 5.270,7.153 ] [ 5.215,6.895 ] 120 [ 6.071,8.639 ] [ 5.892,8.265 ] [ 5.787,7.932 ] [ 5.718,7.646 ] [ 5.654,7.392 ] 135 [ 6.573,9.224 ] [ 6.473,8.925 ] [ 6.370,8.580 ] [ 6.279,8.272 ] [ 6.204,8.021 ] 150 [ 7.039,9.774 ] [ 7.007,9.540 ] [ 6.905,9.182 ] [ 6.793,8.853 ] [ 6.708,8.605 ] 165 [ 7.361,10.158 ] [ 7.373,9.966 ] [ 7.271,9.599 ] [ 7.146,9.256 ] [ 7.055,9.008 ] 180 [ 7.476,10.295 ] [ 7.503,10.117 ] [ 7.401,9.747 ] [ 7.271,9.399 ] [ 7.178,9.151 ] 195 [ 7.361,10.158 ] [ 7.373,9.966 ] [ 7.271,9.599 ] [ 7.146,9.256 ] [ 7.055,9.008 ] 210 [ 7.039,9.774 ] [ 7.007,9.540 ] [ 6.905,9.182 ] [ 6.793,8.853 ] [ 6.708,8.605 ] 225 [ 6.573,9.224 ] [ 6.473,8.925 ] [ 6.370,8.580 ] [ 6.279,8.272 ] [ 6.204,8.021 ] 240 [ 6.071,8.639 ] [ 5.892,8.265 ] [ 5.787,7.932 ] [ 5.718,7.646 ] [ 5.654,7.392 ] 255 [ 5.676,8.185 ] [ 5.426,7.746 ] [ 5.319,7.422 ] [ 5.270,7.153 ] [ 5.215,6.895 ] 270 [ 5.524,8.012 ] [ 5.246,7.547 ] [ 5.137,7.227 ] [ 5.097,6.964 ] [ 5.045,6.704 ] 285 [ 5.676,8.185 ] [ 5.426,7.746 ] [ 5.319,7.422 ] [ 5.270,7.153 ] [ 5.215,6.895 ] 300 [ 6.071,8.639 ] [ 5.892,8.265 ] [ 5.787,7.932 ] [ 5.718,7.646 ] [ 5.654,7.392 ] 315 [ 6.573,9.224 ] [ 6.473,8.925 ] [ 6.370,8.580 ] [ 6.279,8.272 ] [ 6.204,8.021 ] 330 [ 7.039,9.774 ] [ 7.007,9.540 ] [ 6.905,9.182 ] [ 6.793,8.853 ] [ 6.708,8.605 ] 345 [ 7.361,10.158 ] [ 7.373,9.966 ] [ 7.271,9.599 ] [ 7.146,9.256 ] [ 7.055,9.008 ] 360 [ 7.476,10.295 ] [ 7.503,10.117 ] [ 7.401,9.747 ] [ 7.271,9.399 ] [ 7.178,9.151 ]


## 5. Neutrosophic Function with NNs for Scale Effect of Joint Surface Roughness

The meanJRC values in various orientations were tabulated (Table 3). For each orientation, as the mean JRC value for each sample size decreased, it showed a negative scale effect. For example, theJRC values of different sized joint samples in the orientation of 0° were illustrated in Figure 6, according to the calculatedJRC intervals. The upper and lower limits ofJRC values were shown in Figure 6. Both the upper and lower limits decreased as the joint sample length increased. These decreasing trends are fitted by logarithmic functions: (14)JRCdown=m1+m2·ln⁡lJRCup=m3+m4·ln⁡lwhere JRCdown and JRCup are the fitting values of the joint roughness coefficient of different sized test profile;m1,m2,m3, andm4 are fitting parameters of the logarithmic functions revealing the indeterminacy.Table 3
Summary of meanJRC values obtained in different orientations and their variations with scale.
Orientations MeanJRC values (°) J R C 10 J R C 20 J R C 30 J R C 40 J R C 50 J R C 60 J R C 70 J R C 80 J R C 90 J R C 100 0 10.536 9.653 9.273 8.975 8.822 8.802 8.682 8.601 8.568 8.463 15 10.711 9.968 9.343 9.271 9.230 8.973 8.833 8.587 8.360 8.140 30 10.594 9.929 9.572 9.121 9.092 8.601 8.760 8.571 8.293 8.104 45 9.924 9.201 9.008 8.508 8.334 8.624 8.326 8.077 7.846 7.573 60 9.025 8.405 7.884 7.794 7.187 8.268 7.360 7.138 6.872 6.713 75 7.935 7.460 6.773 6.306 6.545 6.499 6.244 6.093 5.950 5.832 90 7.047 6.691 6.338 5.999 6.148 6.089 5.954 5.893 5.826 5.822 105 7.777 7.222 6.677 6.232 6.463 6.461 6.167 5.992 5.890 5.836 120 9.132 8.521 8.200 7.967 7.321 7.824 7.247 7.065 6.854 6.691 135 9.226 8.567 8.090 7.819 7.374 7.666 7.385 7.087 6.915 6.759 150 10.467 9.565 8.910 8.906 8.393 8.816 8.206 8.015 7.656 7.444 165 10.604 9.965 9.532 8.876 8.612 8.646 8.393 8.111 7.905 7.717 180 9.850 9.098 8.757 8.600 8.297 8.127 7.965 7.898 7.834 7.829 195 9.938 9.230 8.678 8.799 8.531 8.102 7.922 7.656 7.461 7.313 210 9.590 8.941 8.653 8.260 8.207 7.383 7.753 7.505 7.249 7.048 225 8.917 8.255 8.133 7.701 7.680 7.437 7.318 7.131 6.865 6.674 240 7.858 7.303 6.824 6.743 6.325 6.818 6.353 6.152 5.914 6.685 255 7.217 6.864 6.335 6.105 6.033 6.069 5.892 5.712 5.780 5.395 270 6.802 6.312 6.006 5.882 5.787 5.971 5.853 5.738 5.826 5.586 285 7.006 6.494 6.111 5.845 5.756 6.061 5.840 5.639 5.472 5.363 300 8.472 7.812 7.530 7.281 6.953 7.067 6.800 6.641 6.446 6.310 315 10.143 9.455 8.964 8.570 8.122 8.686 8.366 7.658 7.464 7.354 330 9.829 9.001 8.326 8.329 7.871 8.053 7.913 7.650 7.347 7.093 345 9.683 9.176 8.773 8.474 7.860 7.848 7.727 7.467 7.178 7.004 360 10.536 9.653 9.273 8.975 8.822 8.802 8.682 8.601 8.568 8.463Figure 6
JRC values of different sized joint samples in the orientation of 0°.According to the concept of NNs, the neutrosophic functionJRC(l) forI∈[0, 0.5] is written as(15)JRCl=m1+μm1I+m2+μm2I∗ln⁡l=m1+m2ln⁡l+μm1+μm2ln⁡lI=m1+m2ln⁡l,m1+m2ln⁡l+μm1+μm2ln⁡l2,I∈0,0.5.wherem1,m2 are the fitting parameters for the logarithmic function of the lower bound values andμm1,μm2 are taken as twice the difference in the fitting parameters for the logarithmic function between the lower and upper bound values. For example,m1,m2 of different sized joint samples in the orientation of 0° were 10.224 and -0.652, andm3,m4 of the outside ellipse envelope were 16.793 and -1.615 (Figure 5). Then,(16)μm1=2×16.793-10.224=13.138μm2=2×-1.615+0.652=-1.926.Thus, the NN functionJRC(l) withI∈[0, 0.5] for describing theJRC values of different sized joint samples in the orientation of 0° was(17)JRCl=10.224+13.138I+-0.652-1.926I∗ln⁡l=10.224-0.652ln⁡l,16.793-1.615ln⁡lThe fitting results of two NNs (z=m1+μm1I andz=m2+μm2I for I∈[0,0.5]) in the neutrosophic functionJRC(l) for anisotropy characteristics of different sized joint samples were tabulated in Table 4.Table 4
Fitting results of NNs inJRC(l) for I∈[0,0.5] corresponding to the JRC values of joint samples in different orientations.
Orientation (°) m 1 μ m 1 m 2 μ m 2 0 10.224 13.138 -0.652 -1.926 15 10.025 13.057 -0.635 -1.918 30 9.459 12.850 -0.586 -1.902 45 8.629 12.586 -0.514 -1.886 60 7.710 12.360 -0.431 -1.880 75 6.964 12.239 -0.362 -1.885 90 6.669 12.215 -0.334 -1.892 105 6.964 12.239 -0.362 -1.885 120 7.710 12.360 -0.431 -1.880 135 8.629 12.586 -0.514 -1.886 150 9.459 12.850 -0.586 -1.902 165 10.025 13.057 -0.635 -1.918 180 10.224 13.138 -0.652 -1.925 195 10.025 13.057 -0.635 -1.918 210 9.459 12.850 -0.586 -1.902 225 8.629 12.586 -0.514 -1.886 240 7.710 12.360 -0.431 -1.880 255 6.964 12.239 -0.362 -1.885 270 6.669 12.215 -0.334 -1.892 285 6.964 12.239 -0.362 -1.885 300 7.710 12.360 -0.431 -1.880 315 8.629 12.586 -0.514 -1.886 330 9.459 12.850 -0.586 -1.902 345 10.025 13.057 -0.635 -1.918 360 10.224 13.138 -0.652 -1.926


## 6. 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 parameterψ =a/b, i.e., the ratio of major and minor principal axis, was utilized to indicate the anisotropic degree. However, botha andb are NNs, which consist of determinate and indeterminate parts, as shown below:(18)a=n1+μn1Ib=n2+μn2I,I∈0,0.5.Using (4), a neutrosophic parameterψ for evaluating the roughness anisotropy can be obtained as follows: (19)ψ=ab=n1+μn1In2+μn2I,I∈0,0.5.=min⁡n1+μn1×0n2+μn2×0.5,n1+μn1×0n2+μn2×0,n1+μn1×0.5n2+μn2×0.5,n1+μn1×0.5n2+μn2×0,max⁡n1+μn1×0n2+μn2×0.5,n1+μn1×0n2+μn2×0,n1+μn1×0.5n2+μn2×0.5,n1+μn1×0.5n2+μn2×0=2n12n2+μn2,2n1+μn12n2Figure7 illustrates the upper and lower bounds of the parameterψ for different sized joint samples. The difference between the ψup and ψdown decreased with the sample size and approached a certain value when the sample size exceeded 60 cm.Figure 7
Upper and lower bounds of the parameterψ of different sized joint samples.


## 7. Conclusion

The statistical assessment ofJRC 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 JRC(θ) andJRC(l) were developed for analyzing and expressing the anisotropy characteristics and scale effect of joint surface roughness. The neutrosophic function JRC(θ) for anisotropic ellipse was derived according to the statistical assessment data ofJRC of joint profiles in various orientations. The upper and lower limits ofJRC values decreased with the increasing sampling length, showing a negative scale effect on joint roughness. The neutrosophic functionJRC(l) was then acquired to determine theJRC values of different sized joint samples. The neutrosophic parameterψ 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.



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