The particle size ratio (PSR) is an important parameter for binary granular materials, which may affect the microstructure and macro behaviors of granular materials. However, the effect of particle ratio on granular assemblies with different arrangements is still unclear. To explore and further clarify the effect of PSR in different packing structures, three types of numerical samples with regular, layered, and random packing are designed. Numerical results show that PSR has significant effects on binary granular samples with regular packing. The larger the PSR, the stronger the strength, the larger the modulus, and the smaller the angle between the shear band and the load direction. And a theoretical solution of the peak stress ratio

A granular material generally refers to a collection of a large number of discrete solid particles larger than 1 μ m in size, such as sand and soil. The unique mechanical properties of granular materials have attracted the attention of a large number of researchers. In the past two decades, a considerable number of studies have been carried out to investigate and predict the mechanical behaviors of granular materials [

Binary granular materials are a kind of common granular materials composed of particles of two different sizes. Generally, the size ratio of small particles to large particles (PSR) and the volume fraction of small particles (VS) are key factors affecting the properties of binary granular materials [

For regular packing, binary granular materials are called binary granular crystals or metamaterials. At present, the properties of wave or acoustics in binary granular crystals are mainly focused [

In this study, the strength and deformation characteristics of binary granular materials with different packing structures in a quasi-static state are concerned. Which is of great significance for the design of metamaterials and the study of well-graded mixtures. The PSR ranges from 0.5 to 0.9 in binary granular materials according to the actual situation in binary granular crystals. Binary granular materials with different packing structures are focused on, which are regular packing, layered packing, and random packing. Their quasi-static mechanical properties are investigated based on DEM under biaxial loading in 2-dimensional. The effect of PSR on their strength and deformation characteristics at different confining stress are discussed.

Discrete element method (DEM) was initiated by Cundall et al. in 1979 [

The motion equation of a particle A can be written according to Newton’s second law as:

In this study, Hertz-Mindlin contact law is used to calculate the contact force between particles. As shown in

The normal force is related to the overlap of two particles:

The calculation of

In this section, Numerical samples of binary granular materials with three packing structures were established. The size of the samples is about 1.5 m × 2 m. In all samples, the radius of large particles is R = 0.01 m, and the radius of small particles increases from r = 0.5 R to r = 0.9 R. With the decrease in the regularity of the packing structure, three different packing arrangements are generated, which are regular packing, layered packing, and random packing, as shown in

The biaxial test is widely used in the study of granular materials, especially in DEM numerical simulation, as it is easier to visualize particle interactions in two dimensions, and the results are valid for 3-dimensional analysis. The biaxial test is regarded as a simplification of the triaxial test, so a series of biaxial tests for granular samples is performed, as shown in

Parameters | ^{3}) |
E (GPa) | ν | μ |
---|---|---|---|---|

Value | 2000 | 50 | 0.3 | 0.3 |

The numerical results for each granular sample, including the axial stress-strain curve, axial-volumetric strain curve, and the distribution of the effective strain [

A series of DEM simulations were carried out for regular granular samples with different PSR at different confining stress. The deviatoric stress-axial strain curves of regular granular samples with different PSR under different confining stress levels are shown in

For the biaxial tests, there is an analytical solution for the strength of the regular packing structure. Utilizing the condition for sliding at contact points and the symmetry of the packing structure, the analytical solution of the maximum stress ratio was given by Rowe [

It can be seen from

Layered granular samples whose PSR increases from 0.5 to 0.9 are considered, as shown in

It can be seen that the normal contact stiffness is related to particle radius, the larger the particle radius, the greater the normal contact stiffness. With the increase of PSR, the particle radius of small particle increases, and the normal contact stiffness between small particles increase, resulting in a slight increase in modulus. To demonstrate this explanation, we use the linear contact model for layered granular samples with the same contact stiffness kn = 3e7 N/m and ks = 1e7 N/m. The layered granular samples are loaded in the same way at the confining stress of 100 kPa. The result is shown in

In summary, the PSR has little influence on the strength and deformation characteristics of layered granular samples. It may be because although the PSR of the layered granular samples is different, they have very similar microstructures, resulting in similar strength and deformation properties.

Random granular samples with the same porosity whose PSR increases from 0.5 to 0.9 are considered, as shown in

It can be seen that within the range of 0.5–0.9, PSR has little influence on the mechanical and deformation properties of random granular samples. It may be related to the similar microstructure. As indicated by Zhu et al. [

Although the mechanical and deformation properties of random granular samples are similar within such a small range of PSR, the contact behavior is changed to some extent. For binary mixture with random packing, the contacts in a particle assembly can be classified as big particle-big particle (

Within a limited range of 0.5–0.9, the microstructure changes caused by PSR are very limited. We generated random granular samples with a larger range of PSR with VS of 50% in the same way, in which PSR is 1/2, 1/4, 1/8, and 1/16, and performed the same loadings at confining stress of100 kPa. If the radius of large particles is fixed, the PSR will cause a large computational cost when the PSR is relatively small, so we fix the radius of small particles to 0.008 m. The deviatoric stress-axial strain curves of random granular samples with different PSR at confining stress of 100 kPa are shown in

The effect of PSR on the strength and deformation of binary granular materials with different packing methods was studied. PFC2D was used to simulate the influence of PSR on basic mechanical properties such as shear strength, volumetric strain, and so on. Besides, the effective strain distribution and the fabric tensors of the granular materials are analyzed. The main conclusions are as follows:

The macroscopic mechanical properties of binary granular materials with regular packing are greatly affected by the PSR. It is mainly because the effect of PSR mainly comes from the changes in microstructure induced by PSR. The theoretical formula for the peak stress ratio and PSR is obtained, which is in good agreement with the DEM simulation. The strength and modulus of binary granular materials with regular packing increase with increasing PSR, which is mainly due to the change in the structural slip angle. Clear shear bands appear in binary granular materials with regular packing after peak deviatoric stress. With the increase of PSR, the angle between the main shear band and the loading direction decreases.

The PSR has little influence on the macroscopic mechanical properties of binary granular materials with layered packing, which is caused by similar microstructure. The modulus of binary granular materials with layered packing increases with increasing PSR, and we suspect that it is due to the Hertz-Mindlin contact law used in the DEM simulations, which results in greater contact stiffness with an increase of PSR.

For random binary granular materials with VS of 50%, PSR has little effect on mechanical and deformation properties within the range of 0.5–0.9. In the larger range of PSR, with the increase of PSR, the modulus of the random binary granular materials increases. Strong force chains are mainly distributed in

The coordinates of particles A and B at time

In the local coordinate system (

Refer to local coordinate system (

Then the displacement derivative matrix can be defined as