# Packing and Strength Properties of Agglomerates of Platelet Particles

Trieu-Duy Tran, Farhang Radjai, Jean Philippe Bayle, Lhassan Amarsid, Saeid Nezamabadi

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Trieu-Duy Tran, Farhang Radjai, Jean Philippe Bayle, Lhassan Amarsid, Saeid Nezamabadi. Packing and Strength Properties of Agglomerates of Platelet Particles. IOP Conference Series: Earth and Environmental Science, 2025, 1480, pp.012084. 10.1088/1755-1315/1480/1/012084. hal-05493490

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# Packing and Strength Properties of Agglomerates of Platelet Particles

To cite this article: T. D. Tran et al 2025 IOP Conf. Ser.: Earth Environ. Sci. 1480 012084

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PACKING AND STRENGTH PROPERTIES OF AGGLOMERATES OF PLATELET PARTICLES

T. D. Tran¹,²,⁵, F. Radjai¹, J.-P. Bayle², L. Amarsid³ and S. Nezamabadi¹,⁴

¹ LMGC, University of Montpellier, CNRS, Montpellier, France

² CEA, DES, ISEC, DMRC, University of Montpellier, Marcoule, France

³ CEA, DES, IRESNE, DEC, Cadarache F-13108 Saint-Paul-lez-Durance, France

⁴ IATE, CIRAD, INRAE, SupAgro, University of Montpellier, F-34060, Montpellier, France

⁵ Center for Environmental Intelligence, VinUniversity, 100000 Hanoi, Vietnam

✉tranduypfiev@gmail.com

**Abstract:** We use DEM simulations to analyze the microstructure and bulk modulus of dense frictionless agglomerates composed of platelets of increasing platyness. We show that the solid fraction increases from that of a packing of spherical particles as the platyness increases, but declines again beyond a peak value. The coordination number follows the same trend whereas the contact coordination number remains nearly constant and equal to its isostatic value of 12 when three single contacts are attributed to each face-face contact and two single contacts to each face-edge contact. We show that the evolution of the microstructure is echoed by the bulk elastic modulus, which considerably increases with platyness.

## 1. Introduction

There is growing interest in a quantitative understanding of the influence of particle shape on the microstructure and mechanical behavior of granular materials. Particle shape involves many different features such as angularity, elongation, platyness, and nonconvexity. In all cases, the effects of particle shape can be quantitatively characterized by using spheres as a reference shape or by defining a shape parameter that interpolates between spherical shape and the particle shape of interest [1, 2, 3, 4].

In this paper, we use this methodology to investigate the microstructure of granular materials composed of platelets of pentagonal cross section. In this case, the shape parameter varies between sphere and infinitely thin platelets. The granular samples are agglomerates prepared by radial compaction without friction and simulated by means of the discrete element method (DEM). Plate particles are encountered, for instance, in clayey soils, where the observed local ordering and clustering of the particles is often attributed to the cohesive interactions between particles. But, it is well known that platy particles in dry cohesionless particles tend also to form locally-ordered structures [5]. We analyze the packing fraction, coordination numbers, and the bulk elastic modulus of the agglomerates as a function of shape parameter.

## 2. Methodology

The particle shape can be described as thick pentagonal plates with rounded edges. In other words, they are Minkowski sums of a sphere of diameter $h$ with a regular pentagon of circumdiameter $D$. Each plate consists of five vertices, five edges, and one face. We define a platyness parameter $η$ as

$$ \eta = \frac{D}{D + h} \qquad (1) $$

As $η$ is increased from zero, the particle shape evolves from a sphere to a thin platelet; see Fig. 1(a).

The agglomerates are built by placing the particles without overlap inside a spherical ball of variable radius and applying a constant pressure on the ball as described in [4]. The ball radius decreases according to the equation of dynamics and the particles get jammed in a balanced configuration in finite time. The friction coefficient between particles during compaction is set to zero to obtain

<figure><img src="image_2.png" /></figure>

Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
the densest states corresponding to the random close packing for each value of η. Figure 1(b) displays two examples of the agglomerates for η = 0.5 and η = 0.91. The particles are jammed in a disordered configuration, but close to the boundary they tend to align with the internal surface of the ball all the more that η is high.

<figure><img src="image_3.png" /><figcaption>Figure 1. (a) Particles of increasing platyness η from left to right. (b) Agglomerates composed of platelets for shape parameter η = 0.5 (left) and η = 0.91 (right), built by radial compaction.</figcaption></figure>

### 3. Microstructure

Let us first consider the packing fraction $\Phi$ obtained by dividing the total volume $V_p$ of the particles by the ball volume $V$. Figure 2(a) shows $\Phi$ as a function of $\eta$. An interesting observation is the non-monotonic variation of $\Phi$ with $\eta$. In the limit $\eta = 0$ (representing spherical particles), we obtain $\Phi = \Phi_0 \approx 0.62$. This value is slightly below that ($\approx 0.635$) under periodic boundary conditions [6]. This slightly lower value of $\Phi$ is a consequence of the higher porosity close to the boundary as a result of wall-induced correlations in particle positions. As $\eta$ increases from zero, $\Phi$ increases above $\Phi_0$ and achieves values as large as 0.72 for $\eta = 0.5$. However, for even higher values of $\eta$, $\Phi$ declines again, reaching the value 0.69 for $\eta = 0.91$. Note that for all values of $\eta > 0$, the packing fraction $\Phi$ is consistently higher than that of spherical particles. The non-monotonic variation of $\Phi$ with $\eta$ has been observed for other shapes in 2D and 3D [7, 1, 4].

<figure><img src="image_4.png" /><figcaption>Figure 2. Packing fraction $\Phi$ (a) and coordination number $Z$ and contact coordination number $Z_c$ (b) as a function of platyness $\eta$.</figcaption></figure>

The lowest-order parameter that quantifies particle connectivity is the coordination number. However, when considering two platelets, different types of contact must be distinguished: face-face, face-edge, face-vertex, edge-edge, edge-vertex, and vertex-vertex. Face-face contacts represent three geometrical constraints corresponding to the hindrance of relative rotations between two particles. A
face-face contact can therefore be modeled as a set of three simple independent contacts and we refer to them as ‘triple’ contacts. In the same way, face-edge contacts represent two geometrical constraints and may be described as ‘double’ contacts. All other contacts are single contacts. Therefore, we distinguish the usual coordination number Z, denoting the average number of neighboring particles in contact with a given particle, from the contact coordination number Zc, representing the average number of contacts per particle. For platelets, Zc is therefore defined as

$$Z_c = \frac{2N_c}{N_p} = \frac{2(N_s + 2N_d + 3N_t)}{N_p} \quad (2)$$

where Ns, Nd and Nt denote the numbers of simple, double and triple contacts, and Nc and Np are the total numbers of contacts and particles, respectively [8].

Figure 2(b) displays Z and Zc as a function of η. In the absence of inter-particle friction, the isostatic state of packings implies Zc = 6 for spherical particles and Zc = 12 for platelets when particles are perfectly rigid. We see that the values of Zc align closely with their isostatic values, the small offsets reflecting the finite rigidity of the particles [9]. Interestingly, the coordination number Z increases with η until η = 0.5 (thick platelets), but then declines as η further increases (thin platelets) due to the increase of the proportion of face-face interactions.

## 4. Bulk modulus

To illustrate the influence of particle shape on the mechanical behavior, we consider here only the bulk elastic modulus K of the agglomerates as a function of η. We evaluate K by applying a small radial compressive volume change ΔV and measuring the resulting pressure change Δp. The bulk modulus is given by $K = -VΔp/ΔV$. The volumetric strain ΔV/V is sufficiently small to avoid frictional slip and particle rearrangements during the application of the probe, so that the response is purely elastic. We used strain values below ΔV/V = 10⁻¹¹ and obtained a linear response and thus well-defined values of K.

<figure><img src="image_5.png" /><figcaption>Figure 3. Bulk modulus K as a function of shape parameter η as measured from simulations (red continuous line) and as predicted by the Effective Medium Theory (black dashed line).</figcaption></figure>

Figure 3 shows K as a function of η, as determined through simulations and predicted by the Effective Medium Theory (EMT). For thick platelets (η < 0.55) we observe a gradual increase of K. Subsequently, the rate of increase of K becomes significantly larger. As the η varies from a nearly spherical shape to η = 0.91, the bulk modulus increases by a factor of 20. To understand this relationship between K and η, we may use the EMT, which provides an upper bound for the
elastic moduli and offers an accurate estimation for the bulk modulus [8]. The EMT is built upon the assumption of uniform strain fields, implying relative particle displacements determined by the strain tensor. It leads to the following expression for the bulk modulus $K_t$:

$$K_t = \frac{n_c k_n \langle \ell^2 \rangle}{9}, \qquad (3)$$

where $n_c$ denotes the number density of contacts and $\ell$ is the distance between particle centers. Theoretical bulk modulus $K_t$ is shown in Fig.3 for all values of $\eta$, based on the values of $n_c$ and $\langle \ell^2 \rangle$ calculated from the simulations. We see that $K_t$ consistently remains slightly higher, yet in close proximity, to the directly measured values of $K$ obtained from the numerical simulations. Equation (3) indicates that the bulk modulus $K_t$ is controlled by two structural characteristics: (i) the number density of contacts $n_c$, depending on both the packing fraction $\Phi$ and the contact coordination number $Z_c$, and (ii) $\langle \ell^2 \rangle$. As $\eta$ increases, $n_c$ exhibits a rapid and notable raise, while $\langle \ell^2 \rangle$ slightly decreases due to the increase of the number of face-face contacts. Consequently, the increase of $K$ with $\eta$ can be attributed to the faster growth of $n_c$ compared to the decrease rate of $\langle \ell^2 \rangle$.

## 5. Conclusion

We used DEM simulations to investigate the microstructure and bulk modulus of random close packings of platelets of increasing platyness built by a radial compaction method. We showed that the agglomerates have contact coordination numbers that are consistent with their isostatic state in the absence of friction between particles. We also evidenced a non-monotonous variation of the packing fraction as a function of the platyness parameter. Another interesting finding was the strong amplifying effect of platyness on the bulk modulus, which reveals also a transition from thick platelets to thin platelets, resulting from the variation of the contact density and average distribution of particle centers in the agglomerate. Many other features of the microstructure and the effect of particle shape on the tensile strength will be presented in a forth-coming paper.

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