Advances in Materials Science and Engineering

Volume 2016, Article ID 2181438, 10 pages

http://dx.doi.org/10.1155/2016/2181438

## Fractal Characteristics of Rock Fracture Surface under Triaxial Compression after High Temperature

^{1}School of Architecture and Civil Engineering, Nantong University, Nantong 226019, China^{2}State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China

Received 15 June 2016; Accepted 8 September 2016

Academic Editor: Mikhael Bechelany

Copyright © 2016 X. L. Xu and Z. Z. Zhang. 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

Scanning Electron Microscopy (SEM) test on 30 pieces of fractured granite has been researched by using S250MK III SEM under triaxial compression of different temperature (25~1000°C) and confining pressure (0~40 MPa). Research results show that () the change of fractal dimension (FD) of rock fracture with temperature is closely related to confining pressure, which can be divided into two categories. In the first category, when confining pressure is in 0~30 MPa, FD fits cubic polynomial fitting curve with temperature, reaching the maximum at 600°C. In the second category, when confining pressure is in 30~40 MPa, FD has volatility with temperature. () The FD of rock fracture varies with confining pressure and is also closely related to the temperature, which can be divided into three categories. In the first category, FD has volatility with confining pressure at 25°C, 400°C, and 800°C. In the second category, it increases exponentially at 200°C and 1000°C. In the third category, it decreases exponentially at 600°C. () It is found that 600°C is the critical temperature and 30 MPa is the critical confining pressure of granite. The rock transfers from brittle to plastic phase transition when temperature exceeds 600°C and confining pressure exceeds 30 MPa.

#### 1. Introduction

In 1982, French mathematician Mandelbrot [1] created the fractal geometry theory, which provided a new method for describing irregularities of distribution of nature. Xie [2, 3] first introduced fractal theory into the study of rock mechanics and opened a new way for the research of rock mechanics. After that, domestic scholars began to study fractal characteristics of rock fracture [4–7]. Peng et al. [8, 9] discussed the method of calculating FD of rock using box-counting dimension according to the characteristics of two-dimensional digital image. At the same time, relationship between porosity and FD was discussed by combining fractal theory with rock slice images obtained by CT scanning. Zuo et al. [10] developed analysis program of FD and lacunarity indices of rock SEM image and discussed the influence of magnification ratio and threshold size of the image on results.

In the aspect of studying the variation of FD of rock fracture with the change of confining pressure, Ni et al. [11] researched the fractal on fracture surface of granite under different confining pressures. The study showed that, with the increase of confining pressure, FD increased along the direction of failure shear and was basically unchanged perpendicular to the direction. Chen et al. [12] defined the FD based on rock damage in order to describe the evolution of specimen under various loading conditions. The results showed that FD and damage variable increased gradually with the increase of load, and the initial damage of rock was delayed with the increase of confining pressure. Huang et al. [13] researched the distribution law of rock fragmentation and its correlation with energy dissipation and release combined with fractal theory and energy principle, based on triaxial high stress and unloading confining pressure tests of marble. Yang et al. [14] investigated triaxial compression test and CT scanning test under various confining pressure of tight sandstone; it was found that FD of crack showed exponential decline with the increase of confining pressure.

In the aspect of FD of rock fracture with temperature, Xu et al. [15] systematically researched macro mechanical properties and micro pore structure characteristics of granite under temperature effect; the study found that FD of rock pore distribution decreased with temperature. Cao et al. [16] showed that, in 20~60°C, relative changes of FD of rock surface topography increased with the rise of temperature after the action of water and decreased in 60~80°C; the relative FD and water interaction temperature showed a parabolic function. Su et al. [17] found FD increased with the rise of temperature and loading rate; the damage form of sandstone transformed gradually from mixed tensile and shear failure to a single oblique shear failure; the extent of the damage was more and more serious. Zhang et al. [18] conducted statistical analysis of rock fragments under uniaxial compression of various temperature effect; it showed that FD of limestone decreased with temperature, which was a proper characteristic statistic that reflected the degree of fragmentation of limestone after high temperature. Zhang et al. [19, 20] calculated the correlation fractal dimensions (CFDs) of AE counts series at different stress level using Grassberger-Procaccia algorithm; as the heat temperature rises, the maximum CFD value and the corresponding stress level both increase from 25°C to 200°C and decrease from 200°C to 800°C and then increase again from 800°C to 1200°C; the CFD value at the failure point shows polynomial decline with rising heat temperature.

Based on above research results, the FD of rock fracture mostly reflected the state of uniaxial compression under high temperature and triaxial compression at normal temperature; it is rarely reported in the condition of triaxial compression under high temperature. But in the development of geothermal resources, nuclear waste treatment, underground energy reserves, and other projects, rock is always in the coupling environment of temperature and confining pressure, so the study of deformation and failure mechanism of rock under different temperature and confining pressures has become a hot and frontier issue in the study of rock mechanics [21–24]. The macroscopic mechanical properties of granite in the experimental temperature (25~1000°C) and confining pressure (0~40 MPa) had been studied in detail in the early stage [25]. Therefore, in this paper, the microscopic characteristics of rock after macro test are studied by Scanning Electron Microscope (SEM); the quantitative description of image obtained by SEM is analyzed by using box-counting method. Variation of FD of rock fracture with temperature and confining pressure is calculated in order to obtain some meaningful conclusions. The research results provide reference value for the study of macro fracture mechanism of rock under various temperature and pressure by using fractal theory.

#### 2. SEM Test

##### 2.1. Test Instrument

S250MKIII was used for SEM test, which is in the Analysis and Testing Center of China University of Mining and Technology. Its technical indicators are as follows: resolution is 60 Å, amplification factor is 200 thousand times, and accelerating voltage is 40 KV. AN10000 energy spectrometer and WDX-2A spectrometer are the main accessories. The application is quantitative micro analysis of solid material, surface structure, and element distribution.

##### 2.2. Test Procedure

This test is mainly on the SEM analysis of granite under triaxial compression after high temperature. Firstly, cut the fracture of sample into appropriate size; clean the sample with acetone or alcohol to prevent illusion or with ultrasonic cleaning oscillator if necessary. Secondly, because rock minerals are nonconducting samples, charge accumulation can be generated under the action of electron beam; this will affect the trajectory of two electron motions of incident electron beam spot and sample emission; then the image quality will decline. Therefore, conductive layer should be sprayed before observation; usually using gold or silver or carbon film with the secondary electron emission coefficient is higher; the film thickness is controlled at about 20 *μ*m. Finally, the sample is bonded to the sample holder with a conductive adhesive, and then put it into vacuum chamber to observe.

The test has six temperature points of 25 (normal temperature), 200, 400, 600, 800, and 1000°C and five confining pressures of 0 (uniaxial compression), 10, 20, 30, and 40 MPa; it consists of 30 samples; each rock sample scans 8–10 pieces of images.

#### 3. Fractal Theory

In fractal theory, the common FD has capacity dimension, information dimension, correlation dimension, general dimension, self-similar dimension, and so on. FD of Cantor set, Koch curve, Sierpinski set, and Menger sponge can be obtained by using self-similar dimension, because they have strict self-similarity, which can be estimated according to the generating element and structural rules. Fractal in nature is statistical self-similarity and unable to determine the generation of a natural fractal; therefore, in the study of rock mechanics, the box-counting dimension is widely used by many researchers, mainly because the box-counting dimension can be a good description and characterization of crack morphology of irregular characteristics, and the mathematical calculation and estimation are relatively easy. In this paper, FD of rock fracture is calculated by box-counting method. The box-counting dimension is calculated using cover method. Because the crack of rock has the characteristics of fractal distribution, it can be regarded as a fractal set of ; cover the cracks of rock fracture surface with a square lattice; the size of grid is changing. Given as the size of box, the total number () of boxes covered with cracks can be counted. Assuming using the grid of to cover the crack at the step, is the required number of boxes; then the formula of box-counting dimension is as follows [8]:where is the length of side grid covered rock cracks and is the intersection grid number with intersection of fractal set of .

Because the fractal set of is composed of points, when reduces to a certain value of , will be a fixed value and no longer varies with the change of . Therefore, in practical applications, the usual practice is to seek a series of and ; then the linear slope of the graph in the double logarithm coordinate is the FD.

Based on above basic theory, FD of rock fracture can be obtained by SEM images. Calculation step includes importing image, processing image, calculating area, covering number of statistics box, calculating box-counting dimension, and deriving calculation results. Figure 1 is an introduction of calculating FD using SEM images at the temperature of 25°C and confining pressure of 10 MPa.