Advances in Civil Engineering

Volume 2019, Article ID 2048958, 17 pages

https://doi.org/10.1155/2019/2048958

## Numerical Simulation of Particle Breakage of Granular Assemblies in Discrete Element Analyses

State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China

Correspondence should be addressed to Tao Zhang; moc.361@818080oatgnahz

Received 30 July 2019; Revised 11 September 2019; Accepted 27 September 2019; Published 18 November 2019

Academic Editor: Chiara Bedon

Copyright © 2019 Tao Zhang and Chi 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

This paper presents numerical simulations of high-pressure biaxial tests on breakable granular soils with the discrete element method. The 2D setting is more economic in terms of computational cost, which allows simulation with a larger number of particles with a wider size distribution. The results of breakable and unbreakable agglomerates show that particle breakage has a significant influence on the macro- and micromechanical behaviors of the assembly. Higher confining pressure and larger axial strain result in the variation of particle grading and agglomerate numbers. The evolution of bond breakage during shearing makes it possible to trace the failure process and breakage mechanism at the microlevel. The breakage energy is found to account for a small fraction of total energy input compared with friction energy. A hyperbolic correlation between relative particle breakage and total energy input per unit volume was established regardless of the influence of confining pressure.

#### 1. Introduction

Particle breakage frequently occurs in granular materials as the exterior energy acting on granular particle exceeds its strength. This phenomenon prevalently encountered in civil, petroleum, and mining engineering, such as crushing in end-bearing piles, rock-fill dams, oil wells, and mine construction [1–5]. Even some natural soils composed of high-strength mineral materials, such as deep underground granular soils, may undergo serious breakage under high pressure [6]. Up until now, more than 500 vertical shafts exceeding 500 meters were constructed in northern China. It is inevitable that the disturbance in the process of construction dramatically changes the stress field in soils, including serious particle breakage. As a cohesionless granular soil, the property of sand particle is relatively simpler than clay or silt, such as its coarser particle size and poor plasticity. Thus, this paper makes a deep and systematical research on particle breakage under high-pressure shear test, taking sand particle as a research object.

Grading change due to particle breakage has a great influence on the macroscopic behavior of a granular soil. This is true because particle breakage may create a more drastic change in the internal structure than can be achieved by rearrangement [7]. For conventional triaxial tests, crushing at sliding contacts or breakage of particles affects stress-strain behavior and decreases the rate of dilation [8–13]. Bolton [14] showed that strength and dilatancy of sand are influenced by relative density and stress level, relating to particle breakage. Coop et al. found that a constant grading can be reached at very large strains on carbonate sands, even quartz sands at low stress levels, were subjected to small amounts of particle breakage [15, 16]. Wu et al. [17] and Hyodo et al. [18] found that the large amount of particle breakage resulting from the high-level strain was induced by particle transformation and rotation during shearing. Yu [12] conducted a great deal of triaxial tests under various influence factors (e.g., confining pressure, initial void ratio, and drainage condition) and discussed the effects of these factors on particle breakage and whole volume change. From what have been investigated above, particle breakage is principally influenced by particle strength and effective stress state, especially at high pressures. Despite these considerable studies, the micromechanics of particle breakage in granular soils is not well understood because it is difficult to observe the crack propagation, movement, and internal interaction at the particle level. Alikarami et al. [11] successfully applied X-ray microtomography to the crushing process at the scale of particle and described the evolution of shear bands. Hall et al. [19] quantified the onset and evolution of localized deformation processes in sand with particle-scale resolution and confirmed the importance of particle rotations associated to strain localization with X-ray microtomography imaging. However, the grading change during shearing was not considered in these studies in which limited loads were permitted in this environment.

Discrete element method (DEM) has proved to be a powerful numerical methodology developed and applied for simulating granular materials. For the crushable soils, different methods have been used in DEM: (1) forming agglomerates by cementing elementary balls (rigid and unbreakable). The bonds between elementary balls are deformable and breakable [7, 20–24], and (2) replacing large particles by a group of small ones when a certain strength is reached [25–28]. In addition, combined DEM and FEM method was adopted as an effective method for simulation of polygon-shaped particles [29, 30]. Raisianzadeh et al. [31] proposed a combined DEM and XFEM approach to simulate particle interaction by DEM and the breakage analysis by XFEM of angular particles. The crushing mechanism of two-dimensional circular particles has been studied with the first method. Moreno et al. [32] investigated the influence of impact angle on the breakage characteristics of circular agglomerates. Kun and Herrmann [33] adopted PFC^{2D} to simulate the crushing process and obtained size distribution of fragments for an agglomerate under impact. Ueda et al. [23] investigated the effects of particle shape on the breakage behavior of granular materials. The researches mentioned above have greatly improved our understanding of the crushing mechanics in granular materials, but a limited number of particles with a poor size distribution were allowed.

The main purpose of this study is to simulate the process of particle breakage in high-pressure biaxial tests on sands with large amounts of particles, using PFC^{2D} with the first method. The following study provides both macro- and micromechanical behaviors for an assembly of DEM agglomerates in the sand samples. First, the macromechanical behaviors are discussed to prove the correctness and feasibility of this method, in which particle breakage is verified as an indispensable phenomenon in high-pressure shear tests. Then, the micromechanical behaviors, such as anisotropy of force distribution, crack evolution, and energy dissipation due to particle breakage, are investigated at great length. Unbreakable agglomerates are also examined in a parallel study to investigate the effects of particle breakage on the strength, dilatancy, and anisotropy.

#### 2. Numerical Simulation Methodology

##### 2.1. DEM Method for a Sand Sample

The contact models used in this study are the same with those in Xu et al. [24]. The linear parallel bond model provides the behavior of two interfaces: (1) an infinitesimal, linear elastic (no-tension), and frictional interface that carries a force and a finite-size, linear elastic, and (2) bonded interface that carries a force and moment. The first interface is equivalent to the linear model: it does not resist relative rotation, and slip is accommodated by imposing a Coulomb limit on the shear force. The second interface is called a parallel bond because when bonded, it acts in parallel with the first interface. When the second interface is bonded, it resists relative rotation, and its behavior is linear elastic until the strength limit is exceeded and the bond breaks, making it unbonded [34]. In the linear model, the force-displacement law for the normal and tangential components is given bywhere *k*_{n} and *k*_{s} are the constant normal and shear stiffnesses, is the surface gap, and Δ*δ*_{s} is the adjusted relative increment of shear displacement. The linear stiffnesses, *k*_{n} and *k*_{s}, can be inherited from the contacting particles; thus, the linear stiffnesses are converted into the following equations:where the superscripts (1) and (2) denote the properties of particle 1 and 2, respectively. Moreover, the linear stiffnesses can be described by the elastic constants of Young’s modulus (*E*) and Poisson’s ratio (*ν*). *E* and *ν* are emergent properties that can be related to the effective modulus (*E*∗) and the normal-to-shear stiffness ratio (*κ*∗) at the contact. *E*∗ and *κ*∗, as the arguments of the linear model method, are given bywhere *L* is the distance between the centers of contacting particles or the distance between the center of the particle and wall and *A* is the projection area of the smaller diameter of contacting particles in PFC^{2D}, as shown in Figure 1.