In this paper, the densely arrayed bonded particle model is proposed for simulation of granular materials with discrete element method (DEM) considering particle crushing. This model can solve the problem of pore calculation after the grains are crushed, and reduce the producing time of specimen. In this work, several one-dimensional compressing simulations are carried out to investigate the effect of particle crushing on mechanical properties of granular materials under a wide range of stress. The results show that the crushing process of granular materials can be divided into four different stages according to

Particle crushing plays an important role in the variation of granular material properties, such as gradation, compressibility and hydraulic conductivity [

Laboratory experiment is the most direct way to study the mechanical response of granular materials. Since Terzaghi et al. [

Numerical experiment is an important supplemental method for laboratory tests. DEM, proposed by Cundall et al. [

In the simulation of particle crushing with DEM, the key is to build crushable particle models reasonably. Two types of crushable particle models have been proposed [

The problem of void ratio, which is often ignored in the present replacement and agglomerate methods, is of great importance in particle crushing simulation with DEM. In replacement method, if a crushed particle is replaced by circular fragments within its outer boundary, the conservation of mass will fail, and specimen void ratio will change abruptly. Inversely, if the law of mass conservation is kept, large unbalanced contact force will be introduced due to the inevitable overlaps between replacing fragments [

In this paper, a special BPM, densely arrayed BPM, is proposed to simulate the crushing of particles. This method can calculate the releasing volume of inner voids more accurately and accelerate the generation of granular samples. Basic features of this model, especially the correction of fragment equivalent diameter and internal pore, are firstly introduced in detail. Then one-dimensional compression tests are simulated with densely arrayed BPM to characterize the development of particle crushing and its effects on mechanical properties granular materials in a wide stress range. The simulation results show that, the compression tests can be divided into four stages with different characteristics, and specimens with different initial states present similar mechanical properties under high stress level conditions. The influence of particle crushing on granular materials mainly focuses on fragment gradation, lateral pressure coefficient, compressibility and extra deformation.

Densely arrayed BPM is formed by the same sized micro-elements arrayed in the densest way. The unified and specific geometrical relationship between those micro-elements makes it possible to estimate the volume of internal pores. How to stack balls in the densest way is known as “Kepler’s Conjecture,” which has been strictly proved that triangular lattice is the densest stack of equal-sized disks [

Parallel bond is widely utilized in BPM, and the behavior of bonds can be defined by five parameters [

In order to solve the problem of calculation error of equivalent grain size, correction of equivalent particle diameter is introduced in this work. In the present numerical study, the statistics of fragment size is based on calculation of diameters of the equivalent volume or circle area (DECA). The equivalent diameter

where

In densely arrayed BPM, formula

Densely arrayed BPM can avoid the problem of internal pore releasing. Due to the dense arrangement of elements, the released inner voids will not be filled by any fragments after crushing, as

It should be noted that, for the absence of theoretical dense arrangements of polydisperse spherical granular systems and non-spherical granular systems, the analyses about DECA correction and void ratio, as well as the problem of void releasing in this work, are only appropriate for densely arrayed BPM. The correction of polydisperse spherical and non-spherical granular systems still demands further research.

For densely arrayed BPM, the average stress of a particle can be defined in Ehlers’ way [

where

where

To investigate the crushing rules of granular materials, DEM simulations of silica one-dimension compression tests were carried out. The contact and bond parameters were calibrated by a single particle crushing test, and for the purpose of saving computing resource, all the numerical tests were modeled in 2D condition.

In this section, a particle was placed and crushed between two rigid parallel plates, as shown in ^{3}, and the friction factor is 0.8. The particle was crushed into two large and several small fragments as shown in

48.0 | 30.0 | 12.0 | 8.0 | 5.0 ^{−3} |

The reaction force

In this section, two numerical tests were designed to study the evolution of particle crushing in granular specimens under one-dimensional compression, as shown in

On account of the different definition of void ratio in 2D and 3D models, and for the convenience of study, the concept of relative void ratio

where _{0} is the initial void ratio. The change of

The compression process can be divided into four stages with distinctive features, as shown in

The upper limit of loading stress in laboratory tests is always no more than 100 MPa [

The form of particle crushing in each stage differs from each other, and can be mainly classified into three categories [

The change of specimen gradation is one of the most significant influence of particle crushing. For the convenience of analysis, a normalized dimensionless parameter

in which the yield stress

In spite of the difference of initial gradings, the gradations of two specimens evolve in similar rules, and the maximum DECAs of two specimens keep almost unchanged. This phenomenon also can be found in

Lateral pressure coefficient _{0} is an important mechanical parameter of geotechnical materials. In this work, _{0} was calculated as the ratio of mean normal stresses acting on lateral and axial walls. _{0} curves of two specimens. It can be seen that both curves oscillate randomly, and present inconspicuous evolution rule in the first two loading stages. Evolution of lateral pressure coefficient in Stage III is much obvious and can be divided into two parts. Firstly, particle crushing lowers the stress level acting on sidewalls. When relative void ratio is greater than about 0.3, _{0} keeps between 0.2 and 0.4. Secondly, in the latter half of Stage III, the lateral pressure coefficients of two specimens show a general increasing trend and remain at about 0.7.

Due to particle crushing and specimen densification, _{0} curves keep fluctuating in a small range throughout the test. In spite of the difference under low stress level, _{0} curves of two specimens still show some common characteristics. Evolution laws of _{0} under high stress level prove again that properties of granular materials with different initial states become similar after being fully crushed.

The change of lateral pressure coefficient results from three aspects. Firstly, high pressure and particle crushing make specimen denser than initial state, and the improvement of relative compaction can influence specimen lateral pressure coefficient. Secondly, particle crushing changes the size distribution of grains, which is closely associated with specimen lateral pressure coefficient. Thirdly, particle crushing and structure rebuilding will induce a change in specimen frictional angle, which is one of the determinants of specimen lateral pressure coefficient.

Particle crushing can impact the compressibility and resiliency of granular materials, and induce extra deformation, which may bring safety threat to rockfill. The issue of compressibility can be investigated by comparison between unloading and reloading experiments, and the problem of extra deformation can be investigated by comparison between crushable and uncrushable compression simulations.

In this section, an unloading and reloading test was carried out based on specimen 1to study the impact of particle crushing on compressibility and resiliency of granular materials. The relative void ratio and bond failure curves under cycle loading are marked in green in

To study the extra deformation caused by particle crushing, a compression test of uncrushable specimen was carried out based on specimen 1, and the bond strength parameters of uncrushable specimen were set to be extremely high values to avoid any bond failures. The

_{0} curves of uncrushable and crushable specimens are drawn in _{0} curves overlap with each other in the early stage, and exhibit different increasing features after the appearance of bond failures. The _{0} curve of uncrushable specimen keeps an increasing trend in general, and doesn’t present obverse changing stages throughout the tests, while the curve of crushable specimen shows a moderate growth, especially in the third stage. This phenomenon indicates that particle crushing slows the growth of stress acting on side walls.

The gap between crushable and uncrushable curves represents the extra deformation brought by particle crushing. The extra decrement of relative void ratio with vertical stress less than 1,000 MPa is shown in

This paper mainly focuses on DEM simulations of particle crushing in a wide stress range. As a customized bonded-particle model, the densely arrayed BPM can solve the issue of inner-pore releasing and equivalent diameter calculating by introducing simple and reasonable correction. Evolution of granular material properties under compression is analyzed, and the following conclusions can be drawn:

The compressing process of granular materials can be divided into four different stages. Particle crushing mainly occurs after the yield point in the third stage. Hence, the change of material properties in this stage is more significant. Particle crushing and its effects become insignificant when

Under high stress level, properties of granular materials become stabilized and convergent, and the information of initial void ratio and gradation will be wiped by extreme high-stress. The simulation results indicate a possible existence of theoretically ultimate state for fully crashed granular materials.

Particle crushing has great influence on lateral pressure coefficient of granular materials, and reduces the level of stress acting on side walls. _{0} curves of two specimens share similar characteristics when extensive crushing occurs. _{0} curves show an upward trend generally after the yield point, and reach a stable state at the last stage when particles are fully crushed.

Particle crushing brings considerable extra and unrecoverable deformation to granular materials. The value of extra deformation firstly increases and then decreases with the crushing process, and reaches its peak at the terminus of the third stage. The evolution law of extra crushing deformation may help the constitutive modeling of granular materials considering particle crushing.

In brief, with the modification of granular specimen generating, as well as the correction of void ratio and DECA statistics, densely arrayed BPM is an appropriate method for numerical analyses of particle crushing. But the computational cost will be higher as more micro-elements are employed to form a macro crushable specimen. However, the simulation results and conclusions will bring inspiration to the constitutive modelling of crushable granular materials.