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1 Introduction

Granular materials such as sand and gravels are ubiquitous in our daily life, and the understanding of such materials are of utmost interest to a number of practical issues ranging from landslides to industrial processes. One of the simplest presence form of dry granular materials is the sandpile which is constructed by pouring sand particles from a point source onto a horizontal plane. The slope of the sandpile is termed as “angle of repose” . In terms of this conceptually simple granular system, one may intuitively think that the maximum pressure at the bottom would occur right under the apex of the sandpile. However, a variety of experimental studies reveal that a local pressure dip exists under the apex of the sandpile [1–6]. To understand this interesting phenomenon, some researchers developed theoretical models to explore the origin of sandpiles [7–9]. Such theoretical studies, which were conducted in the framework of continuum mechanics, have shed some light on the interesting phenomena observed in sandpiles such as arching in the force network and pressure dip at the bottom.

But, it is worth noting that the macroscopic angle of repose and its associated phenomena in fact originate from the discrete nature of granular media [10–12]. Thus the grain-scale modeling such as molecular dynamics (MD) and discrete element method (DEM) are preferred by researchers in the investigation of sandpiles. Lee and Herrmann [13], Luding [14], Zhou et al. [15] and Goldenberg and Goldhirsch [16] performed the MD or DEM simulations to examine the effect of inter-particle friction on the angle of repose. Liffman et al. [17] and Li et al. [18] used idealized spherical particles to mimic the formation of sandpile and examine the force distribution in or underneath the pile. Alternatively using irregular-shaped particles to build up numerical models, Matuttis [19], Matuttis et al. [20], Zhou and Ooi [21] and Zhou et al. [22] studied the effect of particle shape on the formation of the sandpile and scrutinized its related phenomena such as arching and pressure dip .

The MD and DEM simulations have to some extent provided a fundamental understanding into the origin of sandpile , whereas most of these numerical studies tended to reproduce the characteristic responses observed in the experiments or predicted by the theoretical models, and make a qualitative and phenomenological interpretation. Few studies have been carried out to probe an effective quantitative relationship between the macroscopic angle of repose and micromechanical indices characterizing the microstructures and inter-particle force network. Thus no satisfactory explanation has been achieved to account for the formation of sandpile and the origin of angle of repose from a micromechanical perspective.

This paper describes an DEM study of the formation of sandpile and angle of repose by focusing on the effect of particle shape. The sandpiles are created with clumped particles of different aspect ratios or circularities. The effect of particle shape on the angle of repose is discussed. Particular attention has been given to a complete quantitative analysis of the microstructures in the sandpile by taking into consideration the effect of particle shape. An attempt has also been made to explore a quantitative link between the macroscopic angle of repose and the micromechanical parameters, such that the angle of repose is clarified from a microscopic perspective.

2 Numerical Implementation

The DEM program PFC2D [23] was used in current study to mimic the formation of sandpiles comprising particles of different shapes. In total six particle shapes, as described in Fig. 1, were considered, including the disk-shaped particle (Fig. 1a), elongated-shaped particle (Figs. 1b–e), and triangular-shaped particle (Fig. 1f). Except for the disk-shaped particle, the particles of other shapes were generated by clumping two or three disks to form a rigid integrated entity which cannot be broken apart. The particle size of clumped particles was measured by the diameter of circular particles which have the same area as those clumped ones, with the cumulative size distribution curve given in Fig. 2. The indices of aspect ratio (AR) and circularity (CR) were employed to characterize the shape of particles. The aspect ratio of a particle is here defined to be the ratio of the length of its minor axis over that of its major axis [24–27], and the circularity is expressed to be [28–30]:

Fig. 1
figure 1

Description of particle shape

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Fig. 2
figure 2

Particle size distribution curve

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$$\text{CR} = \frac{{(2\pi {\text{R}})^{2} }}{{{\text{P}}^{2} }} = \frac{{4\pi {\text{A}}_{\text{p}} }}{{{\text{P}}^{2} }}$$
(1)

where R is the radius of an equivalent particle having the same area; Ap and P are the particle area and perimeter, respectively. The values of the two shape parameters for the concerned particle shapes are given in Table 1.

Table 1 Shape parameter and angle of repose
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The particle density is 2.65 g/cm3. The contact behavior between particles is governed by the linear elastic contact model built-up in PFC2D, with the normal and tangential stiffness of particles specified to be 1.0 × 109 N/m. The stiffness of bottom wall in both normal and tangential directions is also given to be 1.0 × 109 N/m. The inter-particle and particle-wall friction coefficients are both set to be 0.5.

In this numerical study, the sandpiles are constructed by pouring the particles which are initially generated within a narrow domain, onto a bottom wall, as schematically described in Fig. 3. The narrow rectangular domain in fact serves as a point source, with the dimension being 0.14 × 1.6 m2. After the particles have fallen and rested on the bottom wall, the narrow domain is moved upward, with a clearance of about 50 dm left between this domain and the apex of sandpile, where dm is the mean particle size. Then a new batch of particles are generated within this updated domain, and the pouring of particles is once again executed to create another layer of particles in the sandpile. This process is repeated in total thirteen times, such that each sandpile consists of thirteen layers, and each layer has 200 particles. With the omission of the particles finally settling at a certain position off the main part of sandpile, each sandpile contains about (slightly less than) 2600 particles.

Fig. 3
figure 3

Schematic illustration of creation of sandpile

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The description of fabric anisotropy relating respectively to the contact and particle orientations follows that of Dai et al. [31, 32].

3 Results and Discussions

3.1 Effect of Particle Shape on the Angle of Repose

The angles of repose for both left and right parts of sandpiles are plotted respectively against the aspect ratio AR and circularity CR, as given in Fig. 4. It is shown that the angle of repose decreases with the increase of both AR and CR, implying that the more irregular particle shape is, the larger the angle of repose is. As indicated in Table 1, the angle of repose of shape A (disk shaped) is on the average the smallest and that of shape E (three-particle clumped) is the largest. It is also noted that the angle of repose is correlated better with the circularity CR, as compared with the aspect ratio AR, which has in fact been verified by the root mean square deviations shown in Fig. 4.

Fig. 4
figure 4

The relations of angle of repose with shape parameters: a α vs. AR; b α vs. CR

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3.2 Micromechanical Analysis

  1. (a)

    Contact orientations

Figure 5 describes the relations between the shape parameters and the fabric anisotropy of contact orientations for both left and right parts of sandpiles. It is seen that the anisotropy magnitude an shows a negative correlation with AR and CR, and an is the lowest for the disk-shaped case (shape A). Figure 6 plots the deviation angle Δϕn of the principal anisotropy direction relative to the vertical direction, against the angle of repose α. It is seen that Δϕn decreases with the angle of repose α. It is also interesting to find that there seems to exist a linear relationship between Δϕn and α, in which the summation of Δϕn and α approximates to a constant 47º.

Fig. 5
figure 5

The relations of fabric anisotropy with the shape parameters: a an vs. AR; b an vs. CR

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Fig. 6
figure 6

The relationship between the deviation angle Δϕn and the angle of repose α

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  1. (b)

    Particle orientations

Figure 7 depicts the relations of the fabric anisotropy of particle orientations with the aspect ratio. Figure 7a shows that the anisotropy magnitude a0 decreases with the increase of AR. That is, the more irregular particle shape is, the more intense the fabric anisotropy of particle orientations tends to be. In the meantime the principal anisotropy direction ϕ0 for the left part of the case shape B, as can be seen in Fig. 7b, is a negative value, referring to a direction below the horizontal plane (in the fourth quadrant of Cartesian coordinate system). As particle shape becomes more irregular, ϕ0 increases with the decrease of AR, and ϕ0 for the left part of the case Shape E has even become a positive value denoting a direction above the horizontal plane (in the first quadrant). On the contrary the principal anisotropy direction for the right part of sandpiles transits from a direction above the horizontal plane (the shape B) to a direction beneath it (the shape E) as the aspect ratio AR decreases.

Fig. 7
figure 7

The relations of fabric anisotropy with the aspect ratio: a a0 vs. AR; b ϕ0 vs. AR

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Of great interest is the observation that the two best fit lines cross at the point (0.6, 0). It can be thus inferred that with the aspect ratio AR varying, the two principle anisotropy directions respectively for the left and right parts of sandpiles are expected to reach a compromise state where both left and right parts share the same principal anisotropy direction, and that this characteristic state takes place at AR = 0.6, with the common principal anisotropy direction being ϕ0 = 0º ̶ the horizontal direction.

  1. (c)

    Bottom response

The average normal and friction forces are plotted against the relative positions, as given in Fig. 8. Obviously, a pressure dip is observed at the middle of sandpile for the distribution of normal forces. Correspondingly, the friction force at the middle is nearly zero.

Fig. 8
figure 8

Average normal and friction force at the bottom

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4 Conclusions

  1. (a)

    The angle of repose decreases as the aspect ratio AR (or circularity CR) increases, and its correlation with CR seems to be better than that with AR.

  2. (b)

    The fabric anisotropy magnitude an of contact orientations in both left and right parts of sandpiles shows a negative correlation with AR (or CR), with the disk-shaped case being the smallest and the case shape E being the largest. The angle of repose is found to be dependent on the fabric anisotropy, with its summation with the deviation angle Δϕn being a constant 47º in this study.

  3. (c)

    It is revealed that the more irregular particle shape is, the more intense the fabric anisotropy of particle orientations tends to be. The principal anisotropy direction for both left and right parts of sandpiles will experience rotations as the shape parameter AR varies. A characteristic aspect ratio AR = 0.6 has thus been identified, at which the left and right parts of sandpiles share the same principal anisotropy direction ̶ the horizontal direction.

  4. (d)

    A pressure dip and the zero friction response are observed at the middle of sandpiles, which agrees well with previous laboratory observations.