DEM study on the effect of particle shape on the shear behaviour of granular materials

Abstract

This study investigates the effects of particle convexity, sphericity and aspect ratio (AR) on the behaviour of sheared granular materials using two-dimensional discrete element method simulations. Isotropic, dense and loose assemblies with different particle shapes were prepared and subjected to drained shearing via biaxial compression until the critical state was reached. Macroscopic characteristics such as strength and dilatancy are presented. The factors underlying the macroscopic behaviour are then investigated by considering the coordination number, fabric anisotropy, particle moment, friction mobilisation at contacts and particle rotation. For the range of shapes considered here, the data indicate that the shear strength decreases as particle convexity and sphericity increases while the shear strength increases with increasing AR. The shear strength and convexity are weakly correlated, however a stronger correlation is observed between AR and strength. The volumetric strain at large strains tends to increase with increasing AR. There is a stronger correlation between the critical state strength and both the critical state coordination number and the critical state mechanical void ratio than there is between the critical state void ratio and the critical state strength. The contact fabric anisotropy, the magnitude of the moment transmitted by particles and the friction mobilised at the contacts are important factors underlying strength. The critical state strength increases as both the mean particle moment and the mean mobilised friction increased. Analysis of particle rotation provides insights into the response of the granular materials to shearing.

1 Introduction

It is well established that particle shape influences the macroscopic behaviour of granular materials. For example, Cho et al. [1], in their experimental study on the effect of particle shape on the packing density, stiffness and strength of natural and crushed sands, showed that critical state strength decreased as particle roundness and sphericity increased. Altuhafi et al. [2] showed that the critical state friction angle for natural sands obtained experimentally decreased as the particles became more spherical (as convexity increased and as aspect ratio, AR, decreased). While studies on the effect of particle shape in laboratory experiments are useful in that they showed how particle shape influences the behaviour of sand, the microscale mechanics which underly the macroscopic behaviour are not readily accessible in experiments. In essence, conventional experimental research studies cannot answer the question of why particle shape influences the behaviour of sand in the way it does.

The discrete element method (DEM) which was proposed by Cundall and Strack [3] is an effective numerical tool which has been used to access and interrogate the micromechanics of granular material behaviour. A number of researchers have considered the effects of particle shape on the behaviour of sheared granular materials in both 2-D and 3-D DEM simulations. While many of this numerical studies focused on the effect of particle elongation (i.e., AR) [4,5,6,7,8,9,10,11,12], a few have considered additional shape measures such as angularity, roundness and convexity [13,14,15,16,17,18]. On the other hand, experimental studies on the effect of particle shapes generally consider multiple shape measures including aspect ratio, sphericity, convexity and roundness [1, 2, 19,20,21,22,23]. Apart from a few DEM studies including Yang et al. [24] few prior DEM studies have systematically considered the effects of multiple particle shape measures. Since real particles are not characterised by a single particle shape, it is important to consider the effects of multiple particle shape measures to better understand laboratory data. Here, the effects of particle convexity, sphericity, and AR on the macroscopic and microscopic behaviour of granular materials are considered. We aim to provide a mechanistic explanation to link the influence of shape on the overall behaviour.

Several experimental and numerical studies on particle shape effects have been conducted in 2-D [4, 5, 10, 11, 14,15,16,17, 21, 25,26,27,28,29]. These studies demonstrate that the complex mechanical phenomenon obtained in 3-D models and in real granular materials have been adequately captured by two-dimensional representations of particle shape. While certain phenomenon may be peculiar to the dimension (2-D or 3-D) adopted in an experiment or simulation as shown by Rothenburg and Bathurst [11], 2-D packings of granular assemblies are expected to behave in many ways as 3-D packings [4]. Here, we modelled particle shapes in 2-D to systematically study shape effect. Although three-dimensional representation of particle shape allows for a more realistic modelling of sands, the entire phase space covering particles in 3-D makes it more difficult to control shape (i.e., varying a shape descriptor while keeping another constant). Modelling particle shape in 2-D provides the opportunity to control shape more easily, thereby furnishing a systematic study of shape effect.

In this paper first, the particle shapes studied are quantified and an overview of the numerical simulation approach is provided. Then the data are presented to show how convexity, sphericity, and AR influence the strength and dilatancy of the model granular materials considered. To understand the factors underlying the behaviour observed at the macroscale, the particle-scale interactions and their variation with particle shape were quantified. The correlations between the microscale measures and the macroscale characteristics are discussed before the concluding remarks.

2 Particle shape modelling and quantification

Clusters of discs and spheres have been extensively used in the DEM literature to model realistic particle shapes [30,31,32,33,34]. The level sets method [35] which provides a mathematical representation of the surface of arbitrarily-shaped particles is an alternative to the clump logic for modelling particle shapes. Superquadrics have also been used in the literature to model smooth convex-shaped particles [36, 37]. The particles considered here were made of 2-disc or 3-disc clusters (clumps). These 2-disc and 3-disc clusters are the simplest way to study shape effects using a disc/sphere-based DEM code and this type of particle has been used in many DEM studies including Yan and Zhang [38], Powrie et al. [39], Jiang et al. [40], Salot et al. [12] and Maeda et al. [15]. The variation in shape achieved through use of the clump particles varied the overall form of the particle shape but did not influence the surface texture or roughness. Also, the critical state friction angle for granular assemblies of particles with simplified shapes was insensitive to the geometry idealization/simplification implemented by Jerves et al., [27]. The particle generation method adopted here offers a simplified approach to capturing and systematically studying shape without compromising the characteristics expected of realistic particle shapes as evident in the results obtained here. Shape measures characterising the surface morphologies of sand particles at lower scale such as roundness, surface asperities studied by Pan et al., [41] and roughness covered by Cavarretta et al., [19] are beyond the scope of this study.

Here, the sphericity, $S$ and the convexity, $C$ values of the clumps were calculated following Altuhafi et al. [42] as:

$$ S = \frac{{2\sqrt {{\uppi }A_{{\text{p}}} } }}{P} $$

(1)

$$ C = A_{{\text{p}}} /A_{{{\text{ch}}}} $$

(2)

where $A_{{\text{p}}}$ represents the imaged area of a particle, P is the projected perimeter of the particle and $2\sqrt{\pi A}$ is the perimeter of a circle having equal area with the projected particle area. ${A}_{\mathrm{ch}}$ is the area of the convex hull that encloses the particle. Altuhafi et al., [2] considered 3-D particles with sphericity and AR ranging from S = 0.8280–0.9450 and AR = 1.07–1.5, respectively.

In this study, the clump sphericity, S, was systematically varied between 0.9516 and 1.000 (Fig. 1). The aspect ratio, AR, of the clumps which ranged from 1.0–1.5 was determined as shown in Fig. 2. Table 1 details the number of discs used for each particle, the convexity, sphericity and the AR values of the particles studied here.

Fig. 1

Particle types used in the DEM study with their sphericity, S, values

Fig. 2

Illustration of particle aspect ratio (AR)

Table 1 Shape properties of the particles in the samples

3 Numerical simulation approach

All samples studied here were generated to have a linear grading with $Cu$=${d}_{60}/{d}_{10}$=1.2 (60% of the particles by area are smaller than ${d}_{60}$ and 10% of the particles by area are smaller than${d}_{10}$) and minimum and maximum particle diameters ${d}_{\mathrm{min}}$ = 0.173 mm and ${d}_{\mathrm{max}}$ = 0.25 mm, respectively. The effect of particle shape on granular assemblies with different $Cu$ values is beyond the scope of this study. The particle size distribution data here used the major axis length, A, of the particles (Fig. 2). Each sample in this study contained 5004 particles. Adesina et al. [43], in their 2-D DEM study on RVE for DEM simulations of non-circular particles, showed that the granular assemblies generated to have $Cu$ = 1.2 as in the present study were representative when they contained particles ≥ 2,500 and periodic boundary conditions are applied. Prior studies on systems with similar narrow distributions in 3D, e.g., by Barreto and O’Sullivan [44] found that in shear deformation circa 4000 particles of spheres are adequate to simulate triaxial compression when periodic boundaries are used. A modified version of LAMMPS [45,46,47] was used to conduct the simulations. Using an in-house algorithm, a non-contacting cloud of particles enclosed within square periodic boundaries were generated. The samples were then isotropically compressed to a stress state of 100 kPa. The simplified Hertz–Mindlin contact model was used in the simulations. The parameters used in the simulations are particle shear modulus,$G=27 \mathrm{GPa}$, Poisson’s ratio, $\nu =0.3$, and particle density of 2670 kg/m³. Following earlier DEM studies including Keishing [48] and Hanley et al., [49], in the simulations here, damping was not applied during shearing. Energy dissipation in our simulations is therefore by friction only. Based on consideration of prior DEM studies including Thornton [50], samples of different initial densities were prepared by using a coefficient of interparticle friction, $\mu =0.01$ for dense samples and $\mu =0.3$ for loose samples during isotropic compression. It is worth noting that the samples were not generated to a particular target relative density as is often the case in experimental soil mechanics studies. Rather here the sample preparation approach adopted follows standard practice adopted in DEM simulations. While the void ratio of the dense samples is denoted ${e}_{\mathrm{min}}$, the void ratio of the loose samples is denoted ${e}_{\mathrm{max}}$; these ${e}_{\mathrm{min}}$ and ${e}_{\mathrm{max}}$ values cannot be mapped directly to ${e}_{\mathrm{min}}$ and ${e}_{\mathrm{max}}$ values obtained using standard experimental procedures used in soil mechanics. The void ratio, $e$, in 2-D is the ratio of the total area of the void space in the periodic cell to the total area of the solid particles. To determine the area of the particles, the overlap area of the discs in a clump was subtracted from the total area of the discs in the clump. The distribution of contact normal orientations was checked to confirm the samples were isotropic prior to shearing. Isotropic compression simulations were not terminated until the target stress was reached and until any variation in the average number of contacts per particle (i.e., the coordination number) was less than 0.0001 for at least 500,000 simulation timesteps. This was to ensure the samples reached static equilibrium before shearing.

After isotropic compression, the samples were subjected to strain controlled biaxial compression under a constant confining pressure of ${\sigma }_{x}=$ 100 kPa in the $x$-direction and a deformation rate in the $y$-direction of 0.0005 m/s (Fig. 3). With this applied deformation rate, the maximum inertial number for the simulations is 1.76 × 10^–6, hence, the simulations here were quasi-static, following the inertial number limit for quasi-static simulations specified by Da Cruz et al. [51]. Use of different $\mu$ values during compression enabled the packing density to be varied, however all samples were sheared at $\mu$=0.3. Prior to shearing, the dense samples generated with $\mu$=0.01 during isotropic compression were equilibrated using $\mu$=0.3 for 10,000,000 timesteps. Using this approach 18 samples were isotropically compressed and subject to biaxial shear compression. The effects of different confining pressure is beyond the scope of this study.

Fig. 3

Illustration of a sample under biaxial shear

4 Macroscopic behaviour

4.1 Initial void ratio

Figure 4 shows the effects of convexity, sphericity and AR on the initial void ratios, ${e}_{\mathrm{min}}$ and ${e}_{\mathrm{max}}$ for the samples studied here in comparison to the experimental data by Altuhafi et al., [2], the 2-D DEM data by Azéma and Radjaï [4] and the 3-D DEM data by Nie et al. [18]. The ${e}_{\mathrm{min}}$ and ${e}_{\mathrm{max}}$ generally increased as both convexity and sphericity increased for the range of particle shapes considered here. As the particle AR increased, the ${e}_{\mathrm{min}}$ and ${e}_{\mathrm{max}}$ generally decreased. This trend agrees with both the 2-D DEM data by Azéma and Radjaï [4] and the 3-D DEM data by Nie et al. [18], over a similar range of particle shapes. However, the opposite trend was observed in prior studies where the sphericity and convexity values were lower than those studied here (i.e., C < 0.98 and S < 0.95); in the experimental data by Yasin [23] and Altuhafi et al., [2] and the DEM data by Nie et al. [18], ${e}_{\mathrm{min}}$ and ${e}_{\mathrm{max}}$ generally decreased as both convexity and sphericity increased. These observations indicate that the effect of particle shape on initial sample packing depends on the range of particle shapes considered; the initial void ratio first decreases with an increase in convexity and sphericity but increases as the particles become more spherical. Also, it is important to appreciate that the fabric in experimental samples may be different from that for DEM samples owing to the difference in the sample preparation methods.

Fig. 4

Effect of particle shape on packing: a convexity b sphericity c aspect ratio (“2D-DEM” denotes the current study.)

4.2 Shear strength and dilatancy

Figure 5 shows the stress–strain responses and the volumetric strain responses for selected samples. These samples were sheared until an axial strain, ${\epsilon }_{a}\approx$ 50%, so that the conditions at the critical state could be examined. The critical state is the state at which there is no change in the volume or the void ratio of a sample during shearing. As shown in Fig. 4a, in all cases, the stress ratio $(q/p)$ for the initially dense samples (${\mu }_{\mathrm{prep}}$ $= 0.01$) increased with axial strain, ${\varepsilon }_{a}$, to a peak and then declined until a plateau was reached, where $q={\sigma }_{y}-{\sigma }_{x}$ and $p=\left({\sigma }_{y}+{\sigma }_{x}\right)/2$. The inset of Fig. 5a indicates the true stiffnesses of the assemblies and shows a smooth response at ${\varepsilon }_{a}$ < 1% in comparison to the fluctuations observed at large strains, in agreement with the data presented by Athanassiadis et al., [52]. As expected, after an initial slight contraction (inset of Fig. 5c), these initially dense samples exhibited a dilative volumetric response (negative ${\varepsilon }_{\mathrm{vol}}$) (Fig. 5c), where the volumetric strain is denoted as ${\varepsilon }_{\mathrm{vol}}$. The trends obtained here for the samples are similar to those obtained from experimental studies involving drained triaxial compression of real sands [19, 53]. We are therefore reasonably confident that the observations made on the particle-scale mechanics in Sect. 5 are relevant to sand behaviour. At large strains (${\varepsilon }_{a}$ > 40%), the volumetric strain tended to increase with increasing particle AR. For the initially loose samples (${\mu }_{\mathrm{prep}}$ $= 0.3$), again for all particle shapes considered, $q/p$ increased monotonically until a plateau was reached (Fig. 5b). The volumetric strain response for these initially loose samples also depended on particle shape. While the samples with particle AR ≤ 1.2; S ≥ 0.991 exhibited a contractive response before reaching the critical state (Fig. 5d), the samples with particle AR = 1.5; S ≤ 0.966 exhibited both a contractive response (at ${\varepsilon }_{a}$ < 6%) and a dilative response before eventually reaching the critical state at large strains (${\varepsilon }_{a}$ > 40%).

Fig. 5

Effect of particle shape on the a stress–strain responses for selected dense samples (${\mu }_{prep}$ $= 0.01$) and b loose samples (${\mu }_{prep}$ $= 0.3$) c volumetric strain responses for the dense samples and d the loose samples of clusters

The effects of convexity, sphericity and AR on the peak $q/p$ for the initially dense samples (${\mu }_{prep}$ $= 0.01$) and the critical state $q/p$ for all tested samples are presented in Fig. 6. The peak $q/p$ was determined as the maximum $q/p$ attained between the start of a simulation and the critical state; the critical state $q/p$ was determined by taking the mean of the $q/p$ from the start of the critical state to the end of the simulations. The start of the critical state was determined as the axial strain beyond which there is no noticeable change in volumetric strain or void ratio of the samples during shearing, this was observed at ${\varepsilon }_{a}$ values between 35 and 50%, especially for the assemblies of elongated particles studied here. The critical state was reached earlier in assemblies of circular particles in comparison to non-spherical particles [21]. The peak $q/p$ values generally decreased as both the sphericity and the convexity of the particles increased (Fig. 6a and b), while the peak $q/p$ increased as AR increased (Fig. 6c). The peak $q/p$ data correlated best with AR (R² = 0.60), followed by sphericity (R² = 0.40), the weakest correlation was found for convexity (R² = 0.26). Similar trends were observed at the critical state (Fig. 6d-f), although a higher correlation was observed for the critical state $q/p$ (R² = 0.49 for convexity, R² = 0.83 for sphericity and R² = 0.94 for AR) in comparison to the peak. The observations made here generally agree with the experimental data presented by Yasin [23] and Altuhafi et al. [2] in their studies on the effect of particle shape on the behaviour of natural sands. Jiang et al. [40] in their 3-D DEM study on the effect of particle size distribution on the critical state behaviour of spherical and non-spherical particle assemblies stated that shear strength decreased with an increase in the sphericity of the particles.

Fig. 6

Effect of: a convexity on peak $q/p$ b sphericity on peak $q/p$ c AR on peak $q/p$. d convexity on critical state $q/p$ e sphericity on critical state $q/p$ f AR on critical state $q/p$. Data points of the same value will appear as one data point. R² in bracket excludes the data for the sample of circular particles (AR = 1.0). (“2D-DEM” denotes the current study.)

The relationship between the strength and dilatancy for the samples were also examined as shown in Fig. 7. There is no clear link between particle shape and the peak dilatancy factor for the samples studied here; a linear correlation (R² = 0.78) was observed between the peak $q/p$ and the peak dilatancy factor, ${-d\varepsilon }_{v}/{d\varepsilon }_{q}$ (Fig. 7a) considering all of the data. The data in Fig. 7a are replotted in Fig. 7b again to establish the relationship between strength and dilatancy as proposed by Bolton [54] for the samples of different particle shapes studied here. The linear relationship found between strength and dilatancy for the samples can be generally fitted as follows:

$$ \phi_{p} = \phi_{c} + 0.649\psi_{{{\text{max}}}} $$

(3)

where $\phi_{p}$ is the peak friction angle, ${\phi }_{c}$ is the critical state friction angle, ${\psi }_{\mathrm{max}}$ is the maximum dilatancy angle.

Fig. 7

a Relationship between peak $q/p$ and maximum dilatancy factor $({{-d\varepsilon }_{v}/{d\varepsilon }_{q})}_{\mathrm{max}}$ b Excess friction angle versus peak dilatancy angle for samples with various particle shapes

5 Microscopic behaviour

5.1 Coordination number

Figure 8 shows the evolution of coordination number, $Z$, during shearing for selected samples; $Z$ is the average number of contacts made by particles in a sample. $Z$ for the initially dense samples (${\mu }_{prep}$ $= 0.01$) drastically decreased after a slight increase at the beginning of the shearing (as shown in the inset of Fig. 8a) and reached a stable value at ${\varepsilon }_{a}$ > 10%. Dense packings respond to shearing through contact slippage at the microscale and packing dilation which results in the reduction of $Z$ [22]. $Z$ for the initially loose samples (${\mu }_{prep}$ $= 0.3$) ultimately increased marginally after a slight initial decrease ${\varepsilon }_{a}$ < 0.5% (inset of Fig. 8b). In all cases the coordination number reached a steady value while the overall sample volume continued to evolve (refer to Figs. 5c and d).

Fig. 8

Effect of particle shape on coordination number ($Z$) evolution during shearing for selected: a dense samples b loose samples of clusters

The effects of convexity, sphericity and AR on $Z$ at the critical state are shown in Fig. 9. The critical state $Z$ values are the mean values of $Z$ from the start of the critical state (explained above) to the end of the simulations. The critical state $Z$ generally decreased as both convexity and sphericity increased (Figs. 9a and b); as particle AR increased, critical state $Z$ generally increased (Fig. 9c). In the earlier 3-D DEM study by Jiang et al. [40], it was established that the critical state $Z$ tends to decrease as the particles sphericity increases. Gong and Liu [6] also showed in their 3-D DEM study that the critical state $Z$ increased as the $AR$ of the particles in their granular assemblies increased (Fig. 9c). The initial increase observed here in the critical state $Z$ as both sphericity and convexity increased indicates the differences in the $Z$ between the sample of 2-disc clusters (first data points in Figs. 9a and b) and the sample of 3-disc clusters (second data points in Figs. 9a and b). Both samples contained particles at AR = 1.5 but different sphericity values. A better correlation was obtained between critical state $Z$ and sphericity (R² = 0.72) in comparison to convexity (R² = 0.36), the highest correlation was obtained for AR (R² = 0.89) for the range of particle shapes studied here.

Fig. 9

Effect of: a convexity on critical state $Z$ b sphericity on critical state $Z$ c AR on critical state $Z$ R² in bracket excludes the data for the sample of circular particles (AR = 1.0)

5.2 Relationship between coordination number, void ratio and strength

To understand the factors that determine how particle shape influences the behaviour of granular materials at the macroscopic scale, the relationship between coordination number, void ratio and strength were determined at both the peak and the critical state. Figure 10 shows the relationship between $Z$ after isotropic compression (initial $Z$), $e$ after isotropic compression (initial $e$) and the peak $q/p$. The peak $q/p$ values for the samples prepared with ${\mu }_{prep}$ $= 0.01$ are represented by closed circles (these are the maximum values on Fig. 4a). The peak $q/p$ values for the samples prepared with ${\mu }_{prep}$ $= 0.3$ are represented by open circles (these are the mean $q/p$ values within 5 ≤ ${\varepsilon }_{a}$≤ 25 presented in Fig. 4b). A linear correlation (R² = 0.95) was obtained between the peak $q/p$ and the initial $Z$; the peak $q/p$ generally increased as initial $Z$ increased. A slightly higher correlation (R² = 0.98) was obtained between peak $q/p$ and initial $e$, peak $q/p$ generally decreased as the initial $e$ values of the samples increased. These observations indicate that the initial state of the samples influences their behaviour before the critical state just as is the case in experiments on sand. Rothenburg and Bathurst [11] also reported a good correlation between the initial $Z$ and the peak friction angle for granular assemblies of planar elliptical particles with varying particle AR. Generally, for the samples studied here, the higher the AR of the particles (i.e., the lower the sphericity of the particles), the higher the initial $Z$ and the lower the initial $e$ value, hence the increase in peak $q/p$. Matsushima [16] and Salot et al., [12] also noted that the multiple contacts made by non-spherical particles is responsible for the high strength they exhibit.

Fig. 10

a Peak $q/p$ plotted against $Z$ after isotropic compression (initial $Z$). b Peak $q/p$ plotted against $e$ after isotropic compression (initial $e$). Open circles represent loose samples (${\mu }_{prep}$ $= 0.3$) while closed circles represent dense samples (${\mu }_{prep}$ $= 0.01$)

Figure 11 shows the relationships between the critical state $Z$, the critical state $e$, the critical state mechanical $e$ and the critical state $q/p$ of the samples. The mechanical $e$ is the $e$ value of the samples when rattlers, i.e. particles with contacts ≤ 1, are considered to be part of the void space. These particles do not contribute to stable stress state within the sample [50]. While the critical state $q/p$ increased linearly with critical state $Z$ (Fig. 11a), the critical state $q/p$ generally decreased as critical state $e$ increased (Fig. 11b). A lower correlation (R² = 0.81) was obtained for the relationship between critical state $q/p$ and critical state $e$ in comparison to the relationship between critical state $q/p$ and critical state $Z$ (R² = 0.94). A slightly higher correlation (R² = 0.94) was obtained for critical state mechanical $e$ (Fig. 11c). These trends indicate that to predict strength at the critical state, both $Z$ and mechanical $e$ are more important parameters than $e$, for the samples studied here. It has been previously reported that higher $Z$ implies more stable contact network structure [21], hence the correlation between critical state $q/p$ and critical state $Z$. Although $Z$ generally decreased as $e$ increased at the critical state as expected (Fig. 11d), the relationship is not perfectly linear (R² = 0.90).

Fig. 11

a Critical state $q/p$ plotted against critical state $Z$ b Critical state $q/p$ plotted against critical state $e$ c Critical state $q/p$ plotted against critical state mechanical $e$ d Critical state $Z$ plotted against critical state $e$

5.3 Fabric anisotropy

The effect of particle shape on the fabric of the samples was investigated by determining the particle orientation anisotropy, ${a}_{\mathrm{p}}$ and the contact normal orientation anisotropy, ${a}_{\mathrm{cn}}$. A second-order particle orientation fabric tensor ($\Phi_{{{\text{pij}}}} )$ defined by Satake [55] was calculated as:

$$ \Phi_{{{\text{pij}}}} = \frac{1}{{N_{p} }}\mathop \sum \limits_{p = 1}^{{N_{p} }} n_{pi}^{p} n_{pj}^{p} $$

(4)

where $N_{p}$ is the total number of particles in the system and the vector ${\varvec{n}}_{{\varvec{p}}}$ gives the particle orientation.

Similarly, the second-order contact normal orientation tensor was calculated as:

$$ \Phi_{{{\text{cij}}}} = \frac{1}{{N_{c} }}\mathop \sum \limits_{c = 1}^{{N_{c} }} n_{ci}^{c} n_{cj}^{c} $$

(5)

where $N_{c}$ is the total number of contacts in the system and the vector ${\varvec{n}}_{{\varvec{c}}}$ gives the contact normal orientation. The fabric anisotropy is quantified as the difference between the major and minor eigenvalues, i.e., $a_{{{\text{cn}}}} = \Phi_{c1} - \Phi_{c2}$ or $a_{p} = \Phi_{p1} - \Phi_{p2}$. The closer the fabric anisotropy is to zero, the closer a sample is to an isotropic packing. The evolution of ${a}_{p}$ during shearing for selected samples are shown in Fig. 12. A nonlinear increase in ${a}_{p}$ was observed as shearing progressed, the particle orientations generally became more anisotropic as ${\varepsilon }_{a}$ increased. The same trend was found for the initially dense samples (Fig. 12a) and the initially loose samples (Fig. 12b), although the evolution of ${a}_{p}$ for the initially loose samples appeared smoother.

Fig. 12

Effect of particle shape on the evolution of particle orientation anisotropy during shearing for selected: a dense samples (${\mu }_{\mathrm{prep}}$ $= 0.01$) b loose samples (${\mu }_{\mathrm{prep}}$ $= 0.3$)

The effects of convexity, sphericity and AR on ${a}_{p}$ at ${\varepsilon }_{a}$ = 40% for the initially dense samples (${\mu }_{\mathrm{prep}}$ $= 0.01$) are presented Fig. 13. An initial increase in ${a}_{p}$ was observed as both convexity and sphericity increased, but ${a}_{p}$ declined with further increase in convexity and sphericity (Fig. 13a and b). A general increase in ${a}_{p}$ was observed as particle AR increased (Fig. 13c).

Fig. 13

Effect of: a convexity on particle orientation anisotropy at ${\varepsilon }_{a}$ = 40% b sphericity on particle orientation anisotropy at ${\varepsilon }_{a}$ = 40% c $AR$ on particle orientation anisotropy for dense samples at ${\varepsilon }_{a}$ = 40%

Figure 14 shows the evolution of ${a}_{cn}$ during shearing for selected samples. As shown in Fig. 14a, ${a}_{cn}$ for the initially dense samples (${\mu }_{\mathrm{prep}}$ $= 0.01$) increased to a peak before decreasing to a plateau in a manner similar to the evolution of shear strength (Fig. 4a). On the other hand, ${a}_{cn}$ for the initially loose samples (${\mu }_{\mathrm{prep}}$ $= 0.3$) increased monolithically until it reached a plateau (Fig. 14b). Similar trends were shown in Maeda et al. [15] and Rothenburg and Bathurst [11] in their DEM study on the micromechanics of sand with particle shape effects.

Fig. 14

Effect of particle shape on the evolution of contact normal anisotropy during shearing for selected: a dense samples b loose samples of clusters

Figure 15 shows the effects of convexity, sphericity and AR on ${a}_{\mathrm{cn}}$ at the critical state. The critical state ${a}_{cn}$ was determined by taking the mean of ${a}_{\mathrm{cn}}$ from the critical state to the end of the simulations. While a general decrease in the critical state ${a}_{\mathrm{cn}}$ was observed as both the convexity and the sphericity of the particles increased (Fig. 15a and b), the critical state ${a}_{cn}$ generally increased as particle AR increased (Fig. 15c). A better correlation was obtained between critical state ${a}_{cn}$ and sphericity (R² = 0.71) in comparison to convexity (R² = 0.33), the highest correlation was obtained for AR (R² = 0.86). A linear correlation (R² = 0.92) was found between the ${a}_{p}$ and the percentage of rattlers, %Rattlers, within the samples (Fig. 16a); the higher the %Rattlers within the samples, the lower the ${a}_{p}$ (i.e., the less anisotropic the samples). The critical state ${a}_{cn}$ increased linearly (R² = 0.93) with critical state $Z$ (Fig. 16b); the more the average number of contacts per particle within the samples, the higher the anisotropy of contact normal orientations for the samples. A linear correlation (R² = 0.96) was also found between critical state $q/p$ and critical state ${a}_{cn}$; the higher the critical state ${a}_{cn}$, the higher the critical state $q/p$ (Fig. 16c).

Fig. 15

Effect of: a convexity on contact normal anisotropy b sphericity on contact normal anisotropy c AR on contact normal anisotropy at the critical state

Fig. 16

a Particle orientation anisotropy plotted against %Rattlers at ${\varepsilon }_{a}$ = 40% b Critical state contact normal anisotropy plotted against critical state $Z$ c Critical state $q/p$ plotted against critical state contact normal anisotropy

Figure 17 shows the effect of particle sphericity on the preferred orientations for the particles, the branch vectors and the contact normals within the initially dense samples (${\mu }_{\mathrm{prep}}$ $= 0.01$), at ${\varepsilon }_{a}$ = 40%. The vector joining the centers of two particles in contact is referred to here as the particle branch vector. The preferred orientation is the major principal direction of the fabric. As shown in Fig. 12, the particles within the samples started from an isotropic state (i.e., from ${a}_{p}$ ≈ 0) and continuously became anisotropic with preferential orientations within 0–2.0° at ${\varepsilon }_{a}$ = 40% (Fig. 17a). It was observed that as the samples were sheared, the particles gradually took a preferential orientation towards the horizontal (i.e., perpendicular to the loading direction or the major principal stress). Preferential orientations for the particles initially decreased as sphericity increased and later increased with further increase in sphericity. Figure 18 indicates that the particles with S = 0.9662 (Fig. 18a) are more preferentially oriented towards the horizontal (i.e. within the + 20° and − 20°) than the particles with S = 0.9839 (Fig. 18b). Although the preferred orientations for the branch vectors within the samples varied with particle sphericity (Fig. 17b), sphericity did not systematically influence how the branch vectors were preferentially oriented. There was no significant change in the preferred orientations for the contact normals as sphericity increased, until S > 0.9924 when the preferential orientations for contact normals generally decreased as sphericity increased (Fig. 17c). At S < 0.9924, the contact normals with the samples were preferentially oriented at ≈ 90° to the horizontal (i.e., parallel to the direction of loading or the major principal stress). Peña et al., [10] showed the major principal direction of the fabric for a sample of highly elongated particle (i.e., lower sphericity) is higher in comparison to that for a sample of particles with AR = 1.0.

Fig. 17

Preferred orientation for: a particles b branch vectors c contact normals for the initially dense samples at ${\varepsilon }_{a}$ = 40%

Fig. 18

Colour map of particle orientations for dense samples of 3-disc clusters with $AR$=1.5 at ${\varepsilon }_{a}$ = 40%: a $S$= 0.9662 b $S$= 0.9839

5.4 Particle moment and friction mobilisation

The low shear strength associated with samples of spherical and circular particles is typically related to the ability of circular/spherical particles to rotate, in other words the lack of rotational resistance provided by the inter-particle contacts. Referring to Fig. 19a, here the particle moment acting on a specified particle is calculated as:

$$ \vec{M}_{cl,k} = \mathop \sum \limits_{j = 1}^{{N_{d} }} \mathop \sum \limits_{i = 1}^{{N_{c} }} \left( {\vec{L}_{ci,j,k} \times \vec{T}_{ci,j,k} + \vec{L}_{ci,j,k} \times \vec{N}_{ci,j,k} } \right) $$

(6)

where $N_{d}$ is the number of discs in a clump, $N_{c}$ is the number of contacts for a disc belonging to the $k$ clump, $\vec{L}_{ci,j,k}$ is the contact-centroid branch vector, ${\overrightarrow{T}}_{ci,j,k}$ is the contact tangential force vector and ${\overrightarrow{N}}_{ci,j,k}$ is the contact normal force vector. As the simulations are quasi-static, the net moment acting on every particle, ${\overrightarrow{M}}_{cl,k}\approx 0$. In other words, the summation of the contact shear force induced moment on each particle is balanced by that of the contact normal force induced moment. Therefore, to quantify the mobilised moment resistance across the sample, the contribution from the contact shear (normal) force induced moments is isolated, which is indicated by the mean value of the contact shear (normal) force induced moment magnitude:

$$ \left\langle {\left| {\vec{M}_{T} } \right|} \right\rangle = \frac{{\mathop \sum \nolimits_{k = 1}^{{N_{p} }} \left| {\vec{M}_{T,k} } \right|}}{{N_{p} }} = \frac{{\mathop \sum \nolimits_{k = 1}^{{N_{p} }} \left| {\mathop \sum \nolimits_{j = 1}^{{N_{d} }} \mathop \sum \nolimits_{i = 1}^{{N_{c} }} \left( {\vec{L}_{ci,j,k} \times \vec{T}_{ci,j,k} } \right)} \right|}}{{N_{p} }} $$

(7)

$$ \left\langle {\left| {\vec{M}_{N} } \right|} \right\rangle = \frac{{\mathop \sum \nolimits_{k = 1}^{{N_{p} }} \left| {\vec{M}_{N,k} } \right|}}{{N_{p} }} = \frac{{\mathop \sum \nolimits_{k = 1}^{{N_{p} }} \left| {\mathop \sum \nolimits_{j = 1}^{{N_{d} }} \mathop \sum \nolimits_{i = 1}^{{N_{c} }} \left( {\vec{L}_{ci,j,k} \times \vec{N}_{ci,j,k} } \right)} \right|}}{{N_{p} }} $$

(8)

where $\left\langle {\left| {\vec{M}_{T} } \right|} \right\rangle$ and $\left\langle {\left| {\vec{M}_{N} } \right|} \right\rangle$ are the isolated mean values of the magnitude of the contact shear and normal force induced moment, respectively. ${N}_{p}$ is the total number of particles in the assembly and $\left|{\overrightarrow{M}}_{T,k}\right|$ and $\left|{\overrightarrow{M}}_{N,k}\right|$ are, respectively, the magnitude of the contact shear and normal force induced moment computed on $k$ clump. Basically, when subjected to shearing, particles with low sphericity and large aspect ratio are capable of providing more moment resistance therefore a larger stress ratio at critical state is expected as a consequence of the increased stability of the force chains.

Fig. 19

Calculation of $|\sum {\overrightarrow{M}}_{T}|$ and $|\sum {\overrightarrow{M}}_{N}|$: a Illustration of moments at contacts b Illustrative sample with three clumps for which the detailed values are provided in Table 2

To illustrate these metrics, i.e. $|\sum {\overrightarrow{M}}_{T}|$ and $|\sum {\overrightarrow{M}}_{N}|$, Fig. 19b shows an example in which three clumps are created in the simulation cell. The triangles denote the centroid of the clumps whereas the solid dots represent the center of the discs. Particle contacts are illustrated with red crosses. The shear force induced moment and the normal force induced moment at each contact belonging to clump A are computed and tabulated in Table 2. Clearly the magnitude of the sum of the contact shear force induced moment, $|\sum {\overrightarrow{M}}_{T}|$, is very close to that of the contact normal force induced moment, $|\sum {\overrightarrow{M}}_{N}|$, indicating the simulations are quasi-static (this can also be proved by the force equilibrium in $x$ and $y$ directions, i.e., $\sum {\overrightarrow{T}}_{x}+\sum {\overrightarrow{N}}_{x}\approx 0$; $\sum {\overrightarrow{T}}_{y}+\sum {\overrightarrow{N}}_{y}\approx 0$).

Table 2 Clump A: moment transmitted at each contact illustrated in Fig. 17b

Figures 20a-b show the effects of sphericity and AR on $<\left|{\overrightarrow{M}}_{T}\right|>$, at the critical state. (Data for $<\left|{\overrightarrow{M}}_{N}\right|>$ are equivalent and so not presented), While $<\left|{\overrightarrow{M}}_{T}\right|>$ generally decreased with an increase in particle sphericity (Fig. 20a), $<\left|{\overrightarrow{M}}_{T}\right|>$ increased linearly as particle AR increased (Fig. 20b). As expected, similar trends were obtained for $<\left|{\overrightarrow{M}}_{N}\right|>$. A correlation was found between the $q/p$ and the mean values of the moment components transmitted by the particles at the critical state (Fig. 20c). Generally, the higher the moment transmitted within the samples, the higher the critical state $q/p$. In order to understand the effects of sphericity and AR on friction mobilisation within the samples and the role played by friction on strength, the friction mobilisation index, ${I}_{m}$ presented in Majmudar and Behringer [56] and Azéma and Radjaï [25] as $\left|{F}_{t}\right|/{\mu F}_{n}$ was calculated. ${I}_{m}$ indicates the extent to which a contact is far from the Coulomb failure criterion, ${I}_{m}=1$ shows that a contact is at the Coulomb failure criterion [56]. Its average, ${I}_{\mathrm{mf}}$ = $\langle \left|{F}_{t}\right|/{\mu F}_{n}\rangle$ at the critical state decreased as sphericity increased with a correlation of (R² = 0.81) as shown in Fig. 20d. ${I}_{\mathrm{mf}}$ increased linearly (R² = 0.90) with an increase in the particle AR; in agreement with the 3-D DEM study on the influence of particle shape on limiting bulk friction in direct shear tests conducted by H $\ddot{a}$ rtl and Ooi [7] (Fig. 20e). Azéma and Radjaï [25] in their 2-D DEM study also showed that ${I}_{\mathrm{mf}}$ at the critical state increased as particle AR increased. A significantly high correlation (R² = 0.98) was found between ${I}_{\mathrm{mf}}$ at the critical state and critical state $q/p$ (Fig. 20f). This indicates that the higher the friction mobilised within the samples, the higher the strength. From another perspective as the aspect ratio increases the increase in $<\left|{\overrightarrow{M}}_{T}\right|>$ provides stability to the force chains so that more force chains can remain in a stable state even though a substantial amount of friction is mobilised at the contacts making up these force chains.

Fig. 20

Effect of: a Sphericity on critical state $<\left|{\overrightarrow{M}}_{T}\right|>$ b AR on critical state $<\left|{\overrightarrow{M}}_{T}\right|>$ c Critical state $q/p$ plotted against critical state $<\left|{\overrightarrow{M}}_{T}\right|>$ d Sphericity on critical state ${I}_{\mathrm{mf}}$ e AR on critical state ${I}_{\mathrm{mf}}$ (Note: while the µ value used during shearing in both H $\ddot{a}$ rtl and Ooi [7] and Azema and Radjai [25] is 0.5, µ = 0.3 in the current study) f Critical state $q/p$ plotted against critical state ${I}_{\mathrm{mf}}$

5.5 Particle rotation

The response of the samples to shearing at the microscale was explored by quantifying the rotation of particles within the samples during shearing. Here, particle rotation at a particular ${\varepsilon }_{a}$ is calculated as the change in particle orientation from the beginning of the simulation (i.e., at ${\varepsilon }_{a}$ = 0%) to the ${\varepsilon }_{a}$ of concern. Figure 21 shows the effect of particle sphericity on the mean particle rotation at different stages during shearing. In agreement with results from the experimental study on shape effects in 2D by Marteau and Andrade [21], it was observed here that particles continued to rotate from the beginning to the end of shearing, regardless of the initial state of the samples. At ${\varepsilon }_{a}$ = 1%, particle rotation generally increased as particle sphericity increased (Fig. 21a). At this stage, particle rotation within the initially dense samples (${\mu }_{prep}$ $= 0.01$) is distinctively lower in comparison to particle rotation within the initially loose samples (${\mu }_{prep}$ $= 0.3$). It is reasonable to expect the initially loose samples to have more room for particle rearrangement close to the beginning of shearing, hence, the higher particle rotation exhibited. At ${\varepsilon }_{a}$ = 10%, less distinction was observed in the particle rotation within the initially loose samples in comparison to the initially dense samples at lower sphericity values (Fig. 21b), although, particle rotation still generally increased with particle sphericity. At ${\varepsilon }_{a}$ = 40% where the samples are expected to be closer to or at their critical states (i.e., where $e$ is expected to be similar for the same sample with different initial states), the least distinction in particle rotation between initially dense and initially loose samples was obtained (Fig. 21c). However, in agreement with the 3-D DEM study conducted by Nie et al. [18], particle rotation was observed to generally increase as the particle sphericity increased even at the critical state (i.e. at ${\varepsilon }_{a}$ = 40%). It has been previously reported in experiemtal studies on shape effect that the susceptibility to rotation increases as particle sphericity increases [21, 22]. The observations made here provide evidence for the shear strength mobilised by the samples. The higher the sphericity of the particles within a sample, the higher the particle rotation (i.e. the lower the resistance to rotation), hence, the lower strength exhibited by the samples containing the more spherical particles. Salot et al. [12] also suggested that the unrealistically low shear strength exhibited by samples with spherical particles during shearing is because the convexity of such particles allows for rolling at the contacts. While non-spherical particles are less susceptible to excessive rolling, spherical particles are able to resist rolling only through frictional contacts as they interact through single points [57]. The higher the coordination number of a granular assembly, the higher the resistance to particle rotation, rotation is frustrated in dense packings while particles in loose packings are more free to rotate [22].

Fig. 21

Effect of sphericty on mean particle rotation at: a ${\varepsilon }_{a}$ = 1% b ${\varepsilon }_{a}$ = 10% c Critical state, ${\varepsilon }_{a}$ = 40%. Open circles represent loose samples (${\mu }_{\mathrm{prep}}$ $= 0.3$) while closed circles represent dense samples (${\mu }_{\mathrm{prep}}$ $= 0.01$)

Figures 22a and b show the evolution of the rate of particle rotation within the initially dense (${\mu }_{\mathrm{prep}}$ $= 0.01$) and the initially loose samples (${\mu }_{\mathrm{prep}}$ $= 0.3$), respectively. The rate of particle rotation, $\frac{|d\left(\mathrm{mean} \mathrm{rotation}\right)|}{{d\varepsilon }_{a}}$, for the initially dense samples increased to a peak before declining to a plateau at large strains (Fig. 22a). In contrast, the rate of particle rotation in the initially loose samples monotonically decreased until it reached a plateau at large strains. The peak observed in the rate of particle rotation within the initially dense samples is similar to the peak dilatancy shown in Fig. 22c. The peak particle rotation values for the initially dense samples tend to decrease as the particle sphericity decreased. It was generally observed as shown in Fig. 22d that during shearing, peak $q/p$ occurred first, followed by peak rotation rate, peak dilatancy occurred last in most of the samples studied here. Although the ${\varepsilon }_{a}$ at which the three peaks occurred were similar at lower sphericity values and became wider with further increase in sphericity. Although Rothenburg and Bathurst [11] did not consider particle rotation, they also showed that maximum shear strength occurred well before peak dilatancy was achieved for samples of planar elliptical particles with different eccentricities (elongations).

Fig. 22

Effect of particle shape on: a the evolution of mean particle rotation rate during shearing for dense samples and b loose samples c the evolution of dilatancy factor for dense samples d ${\varepsilon }_{a}$ for maximum dilatancy, peak $q/p$ and peak mean particle rotation

6 Conclusions

The effects of particle shape characteristics such as convexity, sphericity and aspect ratio on the behaviour of sheared granular materials were studied using DEM simulations. Granular assemblies of systematically varied particle shapes and different initial densities were prepared and subjected to drained biaxial shearing. The microscopic characteristics of the granular assemblies such as the coordination number, the percentage of non-contacting particles, fabric anisotropy, particle moment, friction mobilisation at contacts and particle rotation were analysed to understand the factors underlying the macroscopic behaviour of the granular assemblies. The main conclusions from the study are summarised below:

• (1) Key macro-scale data such as the strength and dilatancy confirm that the DEM simulations here are effective analogues of sand. The macro-scale responses align with key expected behaviour characteristics of sand as observed in experiments. This means we can be reasonably confident that the observations made on the particle-scale mechanics are relevant to sand behaviour.

• (2) The critical state strength correlates highly with the critical state coordination number, the void ratio and the mechanical void ratio. However, in comparison with the void ratio, the coordination number and the mechanical void ratio gave higher correlations and are considered more important parameters to predict critical state strength. The initial state (i.e., the initial coordination number and void ratio after isotropic compression) of the samples determine the behaviour at the peak.

• (3) While contact normal anisotropy correlates with strength and evolved in a manner similar to the shear strength of the samples, the anisotropy of particle orientation continued to increase during shearing. Particle orientation anisotropy at large strains decreased as the percentage of the rattlers within the samples increased. Particle orientation anisotropy gave a lower correlation with strength when compared with the contact normal anisotropy.

• (4) The magnitude of moment transmitted by particles and the friction mobilised at contacts correlate with strength. At the critical state, the mean contact shear (normal) force induced moments ($<\left|{\overrightarrow{M}}_{T}\right|> \mathrm{and } <\left|{\overrightarrow{M}}_{N}\right|>)$ and the mean friction mobilised at contacts decreased with particle sphericity and increased with aspect ratio. The critical state strength increased as both the mean particle moment and the mean mobilised friction increased.

• (5) Particle rotation is important in understanding the response of granular materials to shearing. While particles progressively rotate within samples during shearing, the rate of particle rotation depends on sample density. The rate of particle rotation in the initially dense samples increased to a peak before declining to a plateau at large strains. For the initially loose samples, the rate of particle rotation decreased monotonically until it reached a plateau without the peak that characterised the dense samples. Particle rotation clearly increased with particle sphericity especially at lower axial strains, this increase became less pronounced at large strains. Essentially, the higher the sphericity of the particles, the lower the resistance to particle rotation, hence, the lower the strength exhibited as particles become more spherical.

• (6) By comparing the data from our 2-D DEM study with the data from both experimental and 3-D DEM studies, we demonstrate that the trends observed in our study agree well with the observations made in 3-D systems of granular assemblies subjected to drained shearing. Our analysis of the rate of particle rotation within the dense and loose granular systems studied here provides a new micro-mechanical data which supports the evolution of shear strength observed during shearing. While previous studies on the effect of particle shape determined the magnitude of particle rotation and related this to shear strength, the relationship between particle shape and the rate of particle rotation shown here is not evident in earlier studies.

• (7) Despite the agreement between our 2-D DEM data and the earlier works done in 3-D, our data show that the magnitudes of parameters such as the shear strength, coordination number, mobilised friction and particle rotation are lower in 2-D in comparison to 3-D data. This observation which can be attributed to the out-of-plane interactions ignored in 2-D indicates that 2-D quantities may not directly match 3-D quantities. We acknowledge that the out-of-plane system response ignored in 2-D simulations may not accurately represent the contact evolution and the volumetric response of 3-D granular samples, hence the disparity in the magnitude of quantified parameters. However we must also acknowledge that we can more easily visualise 2-D systems and so the inherent value of 2-D simulations remains, despite the advances in computational resources and improvements in DEM software. In the future, three-dimensional representations of real sand particles should be studied to confirm the relevance of the observations made here to the behaviour of sand.