Advances in Materials Science and Engineering

Volume 2015 (2015), Article ID 537692, 10 pages

http://dx.doi.org/10.1155/2015/537692

## Fractal and Morphological Characteristics of Single Marble Particle Crushing in Uniaxial Compression Tests

^{1}Department of Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China^{2}Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200240, China

Received 12 September 2015; Revised 24 November 2015; Accepted 25 November 2015

Academic Editor: Pavel Lejcek

Copyright © 2015 Yidong Wang et al. 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

Crushing of rock particles is a phenomenon commonly encountered in geotechnical engineering practice. It is however difficult to study the crushing of rock particles using classical theory because the physical structure of the particles is complex and irregular. This paper aims at evaluating fractal and morphological characteristics of single rock particle. A large number of particle crushing tests are conducted on single rock particle. The force-displacement curves and the particle size distributions (PSD) of crushed particles are analysed based on particle crushing tests. Particle shape plays an important role in both the micro- and macroscale responses of a granular assembly. The PSD of an assortment of rocks are analysed by fractal methods, and the fractal dimension is obtained. A theoretical formula for particle crushing strength is derived, utilising the fractal model, and a simple method is proposed for predicting the probability of particle survival based on the Weibull statistics. Based on a few physical assumptions, simple equations are derived for determining particle crushing energy. The results of applying these equations are tested against the actual experimental data and prove to be very consistent. Fractal theory is therefore applicable for analysis of particle crushing.

#### 1. Introduction

Granular materials are widely used in rock fill dams, highways, railways, and dykes, due to their engineering properties such as high hydraulic permeability, high density, high shear strength, and low settlement. Coarse particles are susceptible to particle crushing at a high compressive strength, which directly modifies their structure, influencing dilatancy, friction angle, strength, and permeability [1]. The strength of granular material decreases during particle crushing as its compression increases, which may eventually lead to significant deformations and ultimately to structural instability [2]. Determining the exact mechanics of particle crushing, or particle breakage of granular material, is one of the most intractable problems in the geosciences. This topic is of interest to many subfields of research including powder technology, minerals and mining engineering, geology, geophysics, and geomechanics [3].

The problems associated with particle crushing in geomechanics began to attract attention in the early 1960s and from then on gradually developed into a topic of significant study. Research on particle crushing has since been carried out via four approaches. Firstly, researchers have experimented with various artificial materials. Takei et al. [4] carried out single particle compressive strength testing and one-dimensional compression testing with plaster and talc sticks, glass beads, and quartz, discussing the fragmentation mechanism of these four different materials in detail. Secondly, studies have attempted to mathematically describe the crushing characteristics of particles, which is of particular interest to the field of geomechanics. They linked the crushing behaviour of particles and their mechanical response through the use of behavioural constitutive models. Existing constitutive models are based on simple curve-fitting parameters, which are determined in isolation by discrete stress-strain tests. Nakata et al. [5] conducted single particle crushing tests on three kinds of particles and analysed the results using particle survival probability curves. Matsui et al. [6] investigated methods for estimating the ratio of net work input to crushing and specific surface area produced based on load conditions and material properties. Rozenblat et al. [7] expressed particle strength distributions based on an extended logistic function of the crushing force of a number of individual particles. Thirdly, the mechanical properties of crushed particles on the macroscopic level were determined. The macroscopic mechanical properties of particle crushing have been studied through direct shear tests, ring shear tests, uniaxial compression tests, and triaxial shear tests, which mainly characterised the strength and the deformation [8] characteristics of the particles. Lastly, mesoexperiments on particle crushing were carried out through the use of some advanced instruments and methods such as electron microscopy and X-ray [9] in order to obtain detailed structural information.

The majority of analyses focused on experimentally determining the physical characteristics of single particle crushing. Russell et al. [10–12] analysed breakage behaviour of characteristic elastic-plastic granules using compression tests. The study describes the influences of granule size, moisture content, and loading intensity on the energy absorption and recovery at stressing. Mader-Arndt et al. [13] investigated the particle contacts by means of atomic force microscopy (AFM), nanoindentation, and shear tests. Ribas et al. [14] and Portnikov et al. [15] designed compression testers to accurately measure the force-displacement curves, the distribution of strengths, and the fracture energies of single particles.

Mandelbrot [16] has however observed that several natural phenomena can be accurately described by fractal theory. The fractal dimension, , is a fraction with a value between 0 and 3.0. Fractals are self-similar objects, allowing the fractal fragment size distribution at any scale to be predicted [17]. If the shape of some rock fragments is fractal, then their fractal dimension may hence be estimated from their size distribution [18]. Prior studies have generally been aimed towards characterising the particle size distribution (PSD). Many statistical methods have been proposed for describing the PSD of comminuted materials, and the most significant of such PSD functions, including normal, log-normal, Gates-Gaudin-Schumann, and Rosin-Rammler distribution functions, have been reviewed in detail by Allen [19]. The use of fractal size distribution analysis is another approach for linearizing the size distribution curve [20–22]. The concept of utilising probability for studying particle crushing was introduced by McDowell et al. [23], who described a method whereby the compression behaviour of particles could be expressed using a probabilistic approach based around the mathematics of fractals. Xu et al. [24] expressed a significant “size effect” in ice failure strength and used the fractal model for studying ice particle fragmentation by employing modified Weibull [25] statistics. Combining traditional experimental methods with insights from fractal theory is hence an effective method for determining the morphology and mechanical characteristics of particle crushing.

This paper aims at exploring fractal crushing characteristics of single particle. Crushing tests on various sized particles are carried out to evaluate the crushing characteristics of each individual particle. Particle shape plays an important role in both the micro- and macroscales responses of a granular assembly [26]. The relationships between failure modes and particle shapes are analysed, and the fractal model is employed to describe the particle crushing behaviour. The fractal dimension of the particle size distribution is determined and found to be equal to 2.48 for marble particles. The crushing strength is then connected to the particle size combining with the fractal fragmentation of marble particles. The Weibull statistics was modified to estimate the probability of fracture for marble particles, and the Weibull modulus was determined using the fractal dimension of particle fragmentation. The formula of the size effect on the crushing energy is also analysed using fractal model for particle fragmentation.

#### 2. Fractal Model for Particle Crushing

A fractal is a shape consisting of parts similar to the whole in some manner [16]. A certain value is usually used to express the similarity between the parts and the entirety, which is termed the fractal dimension. The fractal dimension attempts to objectively represent how densely a fractal occupies the metric space in which it lies. Fractal dimensions are important because they can be defined in relation to real-world material behaviour and can hence be measured experimentally.

A fractal particularly suited to the analysis of particle crushing is that defined as a box-counting measure , the number of particles having diameters greater than or equal to , which displays scale invariance with a noninteger exponent [27]:In (1), the fractal dimension can be estimated from the slope of a straight line in the log-log plot. The number of particles is very difficult to count accurately, but the mass of the particles may easily be measured. A simple approach to calculating the fractal dimension of particle crushing is the following algorithm [18]:where is the total mass of the particles.

If the particles are screened after crushing with a sieve of aperture , the total mass of the particles under the sieve of aperture is and the total mass of all the particles is .

The particle density is a constant value, so the mass of the particles of diameter less than can be expressed as where is a shape factor. Equation (3) normalized by the total mass of all the particles givesFollowing (4), the fractal dimension of the particle-size distribution can be determined from the slope of versus .

#### 3. Single Particle Crushing Tests

##### 3.1. Material and Apparatus

The material used for the experiment is marble pebble, an actual rock, the main components of which are CaCO_{3}, MgCO_{3}, and SiO_{2}, with impurities of Al_{2}O_{3} and Fe_{2}O_{3}. The characteristic diameters of the particles used for the experiment range from 6.0 mm to 32.0 mm. The shapes of the particles are not perfectly spherical, so it has no way to using “diameter” in the strict sense to describe particle size. Then characteristic diameter is employed to express the size of the particles, which is defined as , where , , and are the lengths of each particle measured along three different axes.

Figure 1 contains an apparatus of particle crushing test. The test is carried out by transducer placing the particle between two hardened platens and then bringing the upper platen downwards at a constant velocity in order to crush the particle. The transducer diameter of the removable hardened platens is 150 mm, and the load-measuring capacity of the apparatus is 50 kN, with a resolution of 0.01 kN. The transducers measuring the displacement and the force are arranged on the instrument, which can collect and record data automatically.