A universal elliptical disc(UED)model to represent natural rock fractures

Since natural fractures are often non-equidimensional,the circular disc model still has great limitations.By contrast,the elliptical disc model is more applicable to representing natural fractures,especially for slender ones.This paper developed a universal elliptical disc(UED)model by incorporating the center point,size,and azimuth of fractures as variables.Specifically,with respect to the azimuth of elliptical fractures in three-dimensional(3D)space,we proposed a paradigm to construct its probability density function(PDF)by coupling the orientation and rotation angle of long axis based on three coordinate transformations.To illustrate the construction process of the PDF of the fracture azimuth,we took the orientation following the Fisher distribution and the rotation angle following Von Mises distribution as an example.A rock slope is used to show the use of the developed UED model,and the 3D DFNs for the slope rock mass are generated by Monte Carlo simulation.In addition,the DFNs for the rock mass are also generated based on the existing circular disc model and non-universal elliptical disc model.The comparison results from the three models clearly illustrate the superiority of the UED model over the existing circular and non-universal elliptical disc models.

Estimation of fracture size and azimuth in the universal elliptical disc model based on trace information

The geometric characteristics of fractures within a rock mass can be inferred by the data sampling from boreholes or exposed surfaces.Recently,the universal elliptical disc(UED)model was developed to represent natural fractures,where the fracture is assumed to be an elliptical disc and the fracture orientation,rotation angle,length of the long axis and ratio of short-long axis lengths are considered as variables.This paper aims to estimate the fracture size-and azimuth-related parameters in the UED model based on the trace information from sampling windows.The stereological relationship between the trace length,size-and azimuth-related parameters of the UED model was established,and the formulae of the mean value and standard deviation of trace length were proposed.The proposed formulae were validated via the Monte Carlo simulations with less than 5%of error rate between the calculated and true values.With respect to the estimation of the size-and azimuth-related parameters using the trace length,an optimization method was developed based on the pre-assumed size and azimuth distribution forms.A hypothetical case study was designed to illustrate and verify the parameter estimation method,where three combinations of the sampling windows were used to estimate the parameters,and the results showed that the estimated values could agree well with the true values.Furthermore,a hypothetical three-dimensional(3D)elliptical fracture network was constructed,and the circular disc,non-UED and UED models were used to represent it.The simulated trace information from different models was compared,and the results clearly illustrated the superiority of the proposed UED model over the existing circular disc and non-UED models。

A Method for Evaluating the Maximum Bending Degree of Flexural Toppling Rock Masses Based on the Rock Tensile Strain-Softening Model

Flexural toppling occurs when a series of layered rock masses bend towards their free face.It is important to evaluate the maximum bending degree and the requirement of supports of flexural toppling rock mass to prevent rock mass cracking and even failure leading to a landslide.Based on the rock tensile strain-softening model,this study proposes a method for calculating the maximum curvature(C_(ppmax))of flexural toppling rock masses.By applying this method to calculate Cppmax of 9 types of rock masses with different hardness and rock layer thickness,some conclusions are drawn:(1)the internal key factors affecting C_(ppmax)are E^(⋆)(E^(⋆)=E_(ss)/E_(0),where E_(0)and E_(ss)are the mean deformation moduli of the rock before and after reaching its peak tensile strength,respectively),the strainεt corresponding to the tensile strength of rock,and the thickness(h)of rock layers;(2)hard rock layers are more likely to develop into block toppling than soft rock layers;and(3)thin rock layers are more likely to remain in flexural toppling state than thick rock layers.In addition,it is found that C_(ppmax)for flexural toppling rock masses composed of bedded rocks such as gneiss is related to the tensile direction.