# Modeling Soft Granular Materials Under Compaction: Strain Rate Effect

Saeid Nezamabadi, Farhang Radjai, Kamilia Ayed, Serge Mora, Jean-Yves Delenne

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Saeid Nezamabadi, Farhang Radjai, Kamilia Ayed, Serge Mora, Jean-Yves Delenne. Modeling Soft Granular Materials Under Compaction: Strain Rate Effect. IOP Conference Series: Earth and Environmental Science, 2025, 1480, pp.012049. 10.1088/1755-1315/1480/1/012049. [hal-05493399](https://hal.inrae.fr/hal-05493399v1)

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# Modeling Soft Granular Materials Under Compaction: Strain Rate Effect

To cite this article: S. Nezamabadi et al 2025 IOP Conf. Ser.: Earth Environ. Sci. 1480 012049

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MODELING SOFT GRANULAR MATERIALS UNDER COMPACTION:
STRAIN RATE EFFECT

S. Nezamabadi¹,², F. Radjai¹, K. Ayed¹, S. Mora¹ and J.-Y. Delenne²

¹ LMGC, Univ Montpellier, CNRS, Montpellier, France

² IATE, INRAE, Montpellier SupAgro, Univ Montpellier, Montpellier, France

✉ saeid.nezamabadi@gmail.com

**Abstract:** Soft granular materials are assemblies of particles that undergo large deformations without breakage under load. Clay aggregates and wet agglomerates are examples of soft particles. In this paper, we study the compaction of such materials for a better understanding of the effect of softness and we focus more specifically on the strain rate effect. We use a numerical approach that is based on a combination of an explicit total Lagrangian formulation of the Material Point Method with the Contact Dynamics method. For different values of the strain rate, we study the compaction of an assembly of soft grains up to very high levels of packing fraction in terms of the global stress, grain shape and coordination number. We show three regimes of compaction: rearrangements without significant shape change, rearrangements induced by shape change, and shape change without significant rearrangements. The transition points depend on the strain rate.

**1. Introduction**

Many materials are composed of a disordered arrangement of soft grains, a characteristic prevalent in various substances such as food products, metal powders, colloidal suspensions, and clays. Unlike hard-grain models, the interaction between disorder and grain deformations in these materials results in unique physical and mechanical properties. The notable deformability of soft grains under low confining pressure enables packing fractions to exceed the random close packing of hard grains [1, 2]. Consequently, rheological properties like compressibility and shear strength, as well as microstructural characteristics, are governed by a combination of grain rearrangements, grain shape change and particle volume variation. The diverse forms of the microstructure depend on grain properties (compressibility, plasticity...) and interaction forces (friction, adhesion...) between grains.

In this work, to model the rheology and microstructure of soft granular systems, we employ an explicit total Lagrangian formulation of the Material Point Method (TLMPM) to compute grain deformations, coupled with the Contact Dynamics Method (CDM) for treating inter-grain interactions [3]. This novel approach marks a departure from our previous works [4, 5, 6, 7], where an implicit updated Lagrangian formulation of MPM had been used with the CD method. Furthermore, in our new TLMPM-CD method, the contact algorithm, based on the CD method, manages contacts between material points, differing from our previous methodology where contacts were addressed using background mesh nodes. We apply this approach to investigate the compaction of an assembly of elastic grains, with a particular interest in exploring the behavior at high packing fraction values as influenced by material parameters and strain rate loading.

**2. Compaction of granular assemblies**

We investigate an assembly of 500 elastic disks (grains) enclosed within a rectangular box formed by four rigid walls, as depicted in Figure 1a. By imposing a downward movement on the top wall at a constant velocity of $v_i$, the assembly undergoes uniaxial compression. The grain diameters are uniformly distributed within the range $D \in [3:4]$ mm. The properties of the elastic grains are as follows: Young's modulus $E = 10$ MPa, Poisson's ratio $\nu = 0.45$, and density $\rho = 1000$ kg/m$^3$. The initial setup is prepared using CD simulations. There is no friction between grains and, between

<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.
grains and walls. Gravitational acceleration is disregarded to prevent stress gradients. To investigate the effects of strain rate, we employ four values for the imposed velocity on the top wall: $v_i = 0.2, 1.0, 2.0,$ and $10.0$ m/s. The simulation time steps are defined by keeping $v_i \Delta t$ constant and equal to $2 \times 10^{-5}$ mm.

The simulations are performed using the TLMPM-CD approach proposed in [3]. In this study, two-dimensional simulations are conducted under plane strain conditions, with the third dimension represented by assuming unit length for the grains in that direction. The computational domain is meshed using four-node elements, and cubic spline interpolation is employed.

<figure><img src="image_3.png" /><figcaption>Figure 1. A snapshot of the initial configuration of an assembly of 500 grains (a); Snapshots of the compacted assemblies for the packing fraction $\Phi \simeq 0.95$ and for the imposed velocities on the top wall $v_i = 0.2$ (b), $1.0$ (c) and $10.0$ (d) m/s.</figcaption></figure>

Figure 1 shows both the initial configuration and snapshots depicting the compacted state for different values of $v_i$ at a packing fraction $\Phi \simeq 0.95$ ($\Phi = V_s/V$, where $V_s$ represents the total volume of grains and $V$ is the volume of the simulation box). The packing fraction could achieve high values essentially due to a change in grain shape from circular to nearly polygonal, facilitating efficient space filling. It is worth noting that while there are small differences in grain rearrangements, the compacted assemblies exhibit similar features for all values of $v_i$.

Figure 2a displays the mean assembly vertical stress $\sigma_{yy}$ as a function of $\Phi$ for different values of $v_i$. Initially, for small values of $v_i$, $\sigma_{yy}$ exhibits fluctuations near zero, contrary to the behavior observed at $v_i = 10.0$ m/s. However, $\sigma_{yy}$ then shows a substantial increase for all values of $v_i$. As noted previously, at high packing fractions, any remaining small pores can only be filled through grain shape change and variations of grain volume under high-stress conditions. Additionally, it is observed that the same value of $\Phi$ is reached for a value of $\sigma_{yy}$ that increases with $v_i$. The evolution of $\sigma_{yy}$ especially for $v_i = 10.0$ m/s reveals three regimes.

To track the evolution of the assembly, we monitor the mean incremental grain displacement, $d\bar{Y}_g$, in the $y$-direction. Given that velocity is imposed on the top wall, we anticipate significant variations in grain displacements in the $y$-direction as the compaction process transitions from grain displacements and rearrangements to elastic grain deformations. Figure 2b shows $d\bar{Y}_g$ normalized by the imposed incremental displacement of the top wall, $dY_{top wall}$, plotted against the packing fraction $\Phi$ for all values of $v_i$. Positive displacements are oriented downward. Initially, grains are carried along by the motion of the top wall, resulting in a mean grain displacement slightly less than but close to that of the top wall. As more grains come under the influence of the top wall, the mean grain displacement increases. At a critical value of $\Phi$, the ratio $d\bar{Y}_g/dY_{top wall}$ suddenly drops below 0.5, followed by oscillations around 0.5. This ratio reflects the normalized incremental displacement of the system's
mass center due to the vertical displacement gradient from the top wall to the bottom wall. Notably, the critical value of $\Phi$ observed increases with $v_i$. The observed oscillations indicate grain rearrangements, with their amplitude slightly decreasing as $\Phi$ increases. These oscillations demonstrate that grains continue to move and rearrange until reaching high levels of $\Phi$. Additionally, a well-defined frequency is apparent, suggesting that the rearrangements occur on a system-wide scale, with a global accommodation of grain displacements. Indeed, individual grain displacements would average out and fail to induce global oscillations of the system's center of mass. It is also noteworthy that the frequency of these oscillations increases proportionally to $v_i$.

To quantify shape change, we introduce a shape parameter $S$ defined as:

$$S = 1 - \left\langle \frac{4\pi A}{\rho^2} \right\rangle, \qquad (1)$$

where $P$ and $A$ denote the perimeter and surface area of the grains, respectively. When a grain is perfectly circular, $S$ equals zero, and it increases towards 1 as the shape deviates from circularity. Figure 2c shows $S$ as a function of $\Phi$. The grain shape begins to change significantly at a critical value of $\Phi$. The minor variations with $v_i$ are attributable to grain relaxation time depending on $v_i$. In the simulations, the maximum level of shape change observed is $S \simeq 0.1$ for $\Phi \simeq 0.98$. At such high $\Phi$ levels, grains tend to exhibit nearly regular hexagonal shapes, as depicted in the snapshots of Figure 1. Considering the definition of $S$ in equation (1), the shape parameter for a regular hexagon is $S = 1 - \sqrt{3\pi}/6 \simeq 0.1$.

Figure 2d displays the variation of the coordination number $Z$ with $\Phi$. Initially, $Z$ exhibits slight increase from its initial value. This initial value tends to be higher for larger values of $v_i$, which can be attributed to greater grain deformations. Subsequently, a plateau is observed where $Z$ remains nearly constant. Then, beyond a critical value of $\Phi$, $Z$ linearly increases with $\Phi$. $Z$, approaching 6 as $\Phi$ tends towards 1. This specific value denotes the formation of a hexagonal network among grains, each with an average of 6 sides; see Fig. 1. This evolution of $Z$ with $\Phi$ contrasts with the expected power-law behavior reported in previous studies [8, 1, 7, 3].

The observed trend of $Z$ together with those of $S$ and $\sigma_{yy}$ suggest that the compaction process undergoes three regimes. In the first regime, as in the case of rigid grains, the compaction proceeds through grain rearrangements. These rearrangements are chaotic and occur through grain displacements that do not exactly follow the motion of the top wall. In the second regime, despite the jammed configuration of the grains, grain shape change begins to occur and induces further global and consistent rearrangements that lead to higher packing fraction. In this regime, $Z$ remains constant. In the third regime, grain shape change prevails and rearrangements are rare. As a result, $Z$ and $S$ increase with $\Phi$ at a constant rate. The transitions between these regimes depend on the strain rate.

### 3. Conclusions

The TLMPM-CD approach is employed to simulate the uniaxial compression of an elastic soft granular assembly. This approach couples an explicit total Lagrangian formulation of the Material Point Method for bulk grain behavior with the implicit Contact Dynamics method for the treatment of frictional contacts between soft grains. We show that, in contrast to rigid grains, soft grains continue to undergo significant rearrangements even after jamming. The evolution of the packing is analyzed in terms of the global stress, grain shape and the grain connectivity as a function of the packing fraction. We have shown three regimes: rearrangements without shape change, rearrangements induced by shape change, and shape change without significant rearrangements. Higher compaction rates lead to higher values of the packing fraction for the transitions between these regimes.

### Acknowledgements

The authors acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-20-CE08-0011 (PaMaCo project).
<figure><img src="image_4.png" /><figcaption>Figure 2. Mean assembly vertical stress, $\sigma_{yy}$, normalized by the grain Young's modulus $E$ (a), ratio of the mean incremental grain displacement to the top wall displacement in $y$-direction $d\bar{Y}_g/dY_{top wall}$ (b), coordination number $Z$ (c), and grain shape parameter $S$ (d) as a function of packing fraction $\Phi$.</figcaption></figure>

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