Expression and Analysis of Joint Roughness Coefficient Using Neutrosophic Number Functions

Abstract

Abstract:In nature, the mechanical properties of geological bodies are very complex, and its various mechanical parameters are vague, incomplete, imprecise, and indeterminate. In these cases, we cannot always compute or provide exact/crisp values for the joint roughness coefficient (JRC), which is a quite crucial parameter for determining the shear strength in rock mechanics, but we need to approximate them. Hence, we need to investigate the anisotropy and scale effect of indeterminate JRC values by neutrosophic number (NN) functions, because the NN is composed of its determinate part and the indeterminate part and is very suitable for the expression of JRC data with determinate and/or indeterminate information. In this study, the lower limit of JRC data is chosen as the determinate information, and the difference between the lower and upper limits is chosen as the indeterminate information. In this case, the NN functions of the anisotropic ellipse and logarithmic equation of JRC are developed to reflect the anisotropy and scale effect of JRC values. Additionally, the NN parameter ψ is defined to quantify the anisotropy of JRC values. Then, a two-variable NN function is introduced based on the factors of both the sample size and measurement orientation. Further, the changing rates in various sample sizes and/or measurement orientations are investigated by their derivative and partial derivative NN functions. However, an actual case study shows that the proposed NN functions are effective and reasonable in the expression and analysis of the indeterminate values of JRC. Obviously, NN functions provide a new, effective way for passing from the classical crisp expression and analyses to the neutrosophic ones.

Scale Effect and Anisotropy Analyzed for Neutrosophic Numbers of Rock Joint Roughness Coefficient Based on Neutrosophic Statistics

“…In other words, the larger the length L is, the smaller the average value (range) of JRC-NNs is. Thus, the scale effect in different lengths is consistent with that of the literature [10].…”

“…What's more, when the sample length is large enough, the anisotropy of the JRC values may decrease to some stable tendency. This situation is consistent with the tendency in [10].…”

“…(1) The neutrosophic statistical analysis method without fitting functions is more feasible and more reasonable than the existing method [10]. From what has been discussed above, the proposed neutrosophic statistical analysis method of JRC-NNs provides a more convenient and feasible new way for the scale effect and anisotropic characteristic analysis of JRC values in rock mechanics.…”

“…. , 10), where a ij = µ ij − σ ij is the lower bound of the JRC value and z ij may choose a robust range/confidence interval [µ ij − σ ij , µ ij + σ ij ] for the symmetry about the average value µ ij (see the references [10,11] in detail), and then based on µ ij and σ ij in Table 1 a ij and b ij in z ij (i = 1, 2, . .…”

Neutrosophic Linear Equations and Application in Traffic Flow Problems

“…However, the neutrosophic functions introduced in [12,13] are interval functions (thick function), but they cannot express and handle actual problems containing NN information. Furthermore, NNs were also applied to expressions and analyses of rock joint roughness coefficient in rock mechanics [14][15][16]. Further, Ye [17] put forth an NN linear programming method to obtain optimal solutions of NN linear programming problems under NN environments.…”