Shock and Vibration

Volume 2015 (2015), Article ID 678573, 12 pages

http://dx.doi.org/10.1155/2015/678573

## Numerical Tests on Failure Process of Rock Particle under Impact Loading

^{1}Mining College, Guizhou University, Guiyang, Guizhou 550025, China^{2}Guizhou Key Laboratory of Comprehensive Utilization of Non-Metallic Mineral Resources, Guizhou University, Guiyang, Guizhou 550025, China^{3}Guizhou Engineering Lab of Mineral Resources, Guiyang, Guizhou 550025, China^{4}Engineering Center for Safe Mining Technology under Complex Geologic Condition, Guiyang, Guizhou 550025, China^{5}School of Resources & Civil Engineering, Northeastern University, Shenyang, Liaoning 110004, China

Received 14 July 2014; Accepted 28 September 2014

Academic Editor: Caiping Lu

Copyright © 2015 Yu-Jun Zuo 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

By using numerical code RFP (dynamic version), numerical model is built to investigate the failure process of rock particle under impact loading, and the influence of different impact loading on crushing effect and consumed energy of rock particle sample is analyzed. Numerical results indicate that crushing effect is good when the stress wave amplitude is close to the dynamic strength of rock; it is difficult for rock particle to be broken under too low stress wave amplitude; on the other hand, when stress wave amplitude is too high, excessive fine particle is produced, and crushing effect is not very good on the whole, and more crushing energy is consumed. Secondly, in order to obtain good crushing effect, it should be avoided that wavelength of impact load be too short. Therefore, it is inappropriate to choose impact rusher with too high power and too fast impact frequency for ore particle.

#### 1. Introduction

The purpose of crushing is to reduce the particle size of rock materials or to liberate valuable minerals from ores. In order to obtain higher efficiency of crushing, it is very important to select the appropriate crusher and crushing circuit. Research shows that the breakage of a brittle material is a very complex process, in which the results are influenced by loading conditions and the rock properties [1]. First of all, breakage of particles can be either single particle or interparticle [2]. The breakage behavior of single particle without constraint and stressing a large number of particles are not fully identical, because loading conditions on the surface of particles can be more complicated when stressing a large number of particles [3]. For this reason, when studying particle breakage behavior, single particle breakage conditions need to be adequately considered. If single particle is suffering loading from other adjacent particles, constraint condition in particle beds should be taken into consideration. What is more, the particle can not only be crushed under static loading but also be broken under dynamic loading. Therefore, particle breakage will require consideration of material properties, particle shape, and particle size. With such complex requirements, it is unlikely that any analytical approach would be adequate. The use of computer simulations seems to be the appropriate tool to obtain some clarification [4]. If we can get the rock crushing mechanism, according to numerical tests, and design or choose different crusher to control crushing effect, then technological process of coarse crushing and fine crushing can be optimized.

So far, it is seldom to investigate the mechanism on particle crushing from point of mechanical analysis. Traditionally, the breakage of material in crushing is regarded as relying upon single particle breakage without considering the confinement condition, and the physical point of departure is the Griffith theory of brittle fracture. The understanding of this breakage behaviour of a particle is based on the knowledge obtained in indirect tensile strength tests of rock, such as the Brazilian test or compression of an irregular specimen [5]. As for the breakage of material subjected to dynamic loading, researchers mainly focused on damage mechanism, fracture process, and dynamic fracture criterion of various materials in the past few decades [6]. Besides, the relationship between parameters such as modulus of elasticity, strength, and deformation rate is elucidated; therefore strength criterion and constitutive relation of rock particles are summarized under dynamic loading [7–9]. The finite element method and the finite difference method based on the traditional continuum mechanics are suitable for prediction of damages and failure, but they are difficult to be used for calculation and simulation of the complete failure process directly. Based on mesoscopic damage mechanics, numerical code (dynamic version) is developed by Tang and Kou [2, 10] to simulate the failure process of a single particle breakage and interparticle breakage subjected to an unconfined or confined static compressive load, which provides a reasonable description of fundamental mechanisms of particles breakage. In fact, particle breakage is usually subjected to dynamic loading. In order to study the damage and breakage mechanism of rock particles under dynamic loading, (dynamic version) was used to study dynamic fragmentation of rock, and the effect of loading rate on failure characteristics was discussed sketchily [11].

In this paper, by using numerical code (dynamic version), the failure process of ore particle under impact loading is simulated, and the influence of different impact loading on crushing effect and consumed energy of ore particle sample is analyzed.

#### 2. Numerical Simulator Description

The newly developed (dynamic edition) is a two-dimensional code that can simulate the fracture and failure process of rock under static or dynamic loading conditions. To model the failure of rock material or rock mass, the rock medium is assumed to be composed of many mesoscopic elements whose material properties are different from one to another and are specified according to a Weibull distribution [12]. The finite element method is employed to obtain the stress fields in the mesoscopic elements. Elastic damage mechanics is used to describe the constitutive law of the mesoscale elements when the maximum tensile strain criterion and the Mohr-Coulomb criterion that incorporate the effect of stress rate are utilized as damage thresholds [13].

##### 2.1. Assignment of Material Properties

In RFPA, the solid or structure is assumed to be composed of many mesoscopic elements with the same size, and the mechanical properties of these elements are assumed to conform to a given Weibull distribution as defined by the following probability density function (PDF):where is the mechanical parameter of the element (such as strength or elastic modulus), the scale parameter is related to the average of the element parameters, and the parameter defines the shape of the distribution function. From the properties of the Weibull distribution, a larger value of implies a more homogeneous material and vice versa. Therefore, the parameter is called the homogeneity index in our numerical simulations. In the definition of the Weibull distribution, the value of the parameter must be larger than 1.0. Using the PDF, in a computer simulation of a medium composed of many mesoscopic elements, one can produce numerically a heterogeneous material. The computationally produced heterogeneous medium is analogous to a real specimen tested in the laboratory, so in this investigation it is referred to as a numerical specimen.

##### 2.2. The Constitutive Model for the Mesoscopic Element

Initially an element is considered elastic, with elastic properties defined by Young’s modulus and Poisson’s ratio. The stress-strain relation for an element is considered linear elastic until the given damage threshold is attained and then is modified by softening. Under a dynamic stress state, the elements undergo damage when one of the following damage criteria is satisfied at the element level: where is the dynamic uniaxial compressive strength of the element, which is closely related to the strain rate (or stress rate) of dynamic loading condition, is initial elastic modulus of the element that is assumed to be not affected by the stress rate, is the ratio of compressive and tensile strength, and is the internal frictional angle of the element. and are the major and minor principal stresses of the element.

In this study, the following relation between dynamic uniaxial compressive strength and loading rate, which has been proposed by Zhao [7], is used to reflect the effect of stress loading rate on the dynamic strength:where is also the dynamic uniaxial compressive strength (MPa), is the stress rate (MPa/s), is the uniaxial compressive strength at the quasistatic stress rate that is approximately 5 × 10^{−2} MPa/s, and is a parameter depending on the material. In addition, the experimental results of Zhao [7] also indicated that the ratio of tensile and compressive strength () and internal frictional angle are not influenced by the stress rate. In this respect, when the obtained from (3) is substituted into (2), the effect of strain rate on the strength of elements will be incorporated.

The first part of (2) is the maximum tensile strain criterion, whilst the second part is the classical Mohr-Coulomb failure criterion for tensile and shear damage thresholds, respectively. Thus, an element may be damaged in either tension (corresponding to the maximum tensile strain criterion) or shear (corresponding to the Mohr-Coulomb criterion). Once (2) is satisfied at the element level, the elastic modulus of the element is reduced according to the following expression:where represents a damage variable, and are the elastic modulus of the damaged and the undamaged element, respectively. In the current method, the element and its damage are assumed isotropic, so the , , and are all scalar quantities. The sign convention used throughout this paper is that compressive stress and strain are positive.

When the mesoscopic element is in a uniaxial stress state (both uniaxial compression and uniaxial tension), the constitutive relation of elements is as illustrated in Figure 1. Initially, the stress-strain curve is linear elastic and no damage exists; that is, . When the maximum tensile strain criterion is satisfied, damage occurs in the element in a brittle mode.