# Evolving pore orientation, shape and size in sheared granular assemblies

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## Abstract

This paper presents new insights into the deformation response of sheared granular assemblies by characterising pore space properties from discrete element simulations of monodisperse particle assemblies in two-way cyclic shearing. Individual pores are characterized by a modified Delaunay tessellation, where tetrahedral Delaunay cells can be merged to form polyhedral cells. This leads to a natural partition of the pore space between individual pores with tetrahedral and polyhedral geometry. These are representative of small compact pores and larger well-connected pores, respectively. A scalar measure of pore orientation anisotropy during shearing is introduced. For triaxial shearing, larger pores align in the loading direction, while small pores are aligned perpendicular to the larger pores. Pore anisotropy mobilises at a slower rate than contact anisotropy or macroscopic stress state, and hence, is an important element to characterise in granular assemblies. Further, the distribution of pore volume remains isotropic. Pore shape was found to be a good micro-scale indicator of macroscopic density, with a strong relationship between averaged shape factor and macroscopic void ratio. Combining results for pore shape and orientation reveals an interesting interplay, where large elongated pores were aligned with the loading direction. These results highlight the importance of considering pore space characteristics in understanding the behaviour of granular materials.

## Keywords

Pore characterisation Pore geometry Orientation tensor Anisotropy Shape factor## 1 Introduction

The macroscopic response of granular materials is influenced by microstructural characteristics of particles, their contacts and interstitial pores (or voids) [22]. The shape and size distribution of discrete particles forming the granular assembly can affect packing structure and strength properties [34, 35]. Interparticle contacts form a complex network and the associated anisotropy of this contact network provides insights into the nature of stress transmission and the development of shear strength [7, 26, 32]. Particles and their associated contact networks have been extensively studied and this is aided by the fact that both can be regarded as discrete objects.

In contrast, the effect of pores (or more generally, pore space characteristics) on mechanical response of the medium has received relatively limited attention. This may be attributed to the complexity associated with the description of continuous, entirely interconnected pore space in granular materials. Prior studies have focused on how pore space characteristics affect fluid flow and transport properties, such as conductivity [3, 33], water retention [4, 8] and filtration [24, 30].

However, features of the pore space can also provide insights into deformation characteristics in sheared granular assemblies. Oda et al. [23] observed the formation of elongated pores in bi-axial compression tests on photoelastic disks and quantified this pore anisotropy using a scan-line technique. Ghedia and O’Sullivan [6] extended this method to analyse digital images, while Muhunthan et al. [21] and Muhunthan and Chameau [20] employed a similar technique in exploring the concept of yielding and an ultimate state boundary surface.

An alternate approach is to tessellate the pore space, or identify *individual* pores forming the granular assembly. Common approaches include the classical Delaunay and Voronoi tessellations, which typically rely on the centroidal coordinates of particles. Other tessellation methods such as the approach proposed in Li and Li [16] explicitly consider the interparticle contact points. Space tessellation techniques have also been considered in the formulation of a micromechanical strain tensor for a granular assembly [2, 14, 15, 27]. Li and Dafalias [18] explicitly considered a pore fabric measure in their model of anisotropic critical state based on the length of void vectors obtained from a slight modification of the space tessellation proposed by Li and Li [16]. They proposed that limiting states of pore space fabric would be attained at critical state conditions (in addition to the usual constraints of critical state soil mechanics). These examples suggests that the use of space tessellation is a useful means to gain fundamental understanding of deformation characteristics.

This study investigates how pore space characteristics affect the deformation response of sheared granular assemblies by employing the space tessellation method. Pore-scale data for sheared granular assemblies are extracted from numerical simulations employing the discrete element method, and individual pores are identified using the modified Delaunay tessellation [1]. While it is acknowledged that there is no unique way to tessellate the pore space, several studies [10, 25, 31] have suggested that the modified Delaunay tessellation provides a physically representative delineation of the pore space. Two particular properties of the pore space are investigated in detail: (i) pore orientation and the associated anisotropy of pores, and (ii) pore shape as an indicator for macroscopic density. The influence of pore size is also considered indirectly, while some elements of the contact networks are briefly discussed.

## 2 Pore characterisation from numerical simulations of sheared granular assemblies

### 2.1 Discrete element simulations

- 1.
CP-L: axisymmetric test with constant radial pressure (mimicking a drained triaxial test) for a

*loose*sample \(\left( e_{0}=0.68\right) \). - 2.
CP-D: axisymmetric test with constant radial pressure for a

*dense*sample \(\left( e_{0}=0.56\right) \). - 3.
CV-D: axisymmetric test with constant volume condition (mimicking an undrained triaxial test) for a

*dense*sample \(\left( e_{0}=0.59\right) \). - 4.
SS-D: simple shear test with constant volume condition for a

*dense*sample \(\left( e_{0}=0.59\right) \).

### 2.2 Pore space tessellation

The pore space is tessellated using the modified Delaunay tessellation proposed by Al-Raoush et al. [1]. This method initially considers the classical Delaunay tessellation, which produces a space-filling set of tetrahedral cells, where the vertices of the tetrahedron are the centres of the four particles forming the tetrahedron. The classical Delaunay tessellation over-segments the pore space, and thereby does not capture the wide variation in pore sizes (that is, the existence of small and large pores) and the variable connectivity of the pores (as tetrahedrons have exactly four four faces). Merging of these tetrahedral Delaunay cells using the modified Delaunay tessellation addresses both these concerns to produce a more physically representative tessellation of the pore space.

Each Delaunay cell has an inscribed sphere contained entirely within the pore space. Part of the inscribed sphere may be located outside the Delaunay tetrahedron and intersect neighbouring inscribed spheres. Individual Delaunay cells are then merged with adjacent cells if their respective inscribed spheres overlap. This results in a space-filling set of polyhedral *pore units* which are representative of individual pores. Figure 3 provides several examples of pores identified using the modified Delaunay tessellation, and further details can be found in Sufian et al. [31].

### 2.3 Properties of individual pores

For each individual pore, there are several geometric properties of interest in this study: (i) pore volume, (ii) pore orientation, and (iii) pore shape.

Pore volume, \(V_{p}\), is calculated by subtracting the solid volumes from the total volume of the unit cells, and can be determined analytically with the assumption of non-overlapping particles (which is reasonable for the numerical simulations considered in this study).

## 3 Pore orientation and induced anisotropy of the pore space

In Eq. 3, the superscript *k* refers to a particular subset of pores. This concept of partitioned subsets was previously applied to the interparticle contact network [32], where dominant contributors to shear strength were identified. Sufian et al. [32] found a unique linear relation between the mobilised internal friction and the anisotropy of a partitioned subnetwork. The current paper considers whether a partitioned subset of pores can be identified as the dominant contributors to pore space anisotropy. In addition to considering the complete collection of pore units, this study will also investigate subsets of unmerged and merged pores, as delineated by the modified Delaunay tessellation and denoted \(k\in \left\{ u,\,m\right\} \), respectively. An absence of any value for *k* implies a quantity calculated for the complete collection of pores. Note that unmerged pores are representative of smaller pores in the assembly, while merged pores represent larger pores. Hence, the \(\left\{ u,\,m\right\} \) partition also explores the influence of pore size on the anisotropy of the pore space.

*pore space fabric tensor*. \(N_{p}^{k}\) is the number of pores within the \(k^{\text {th}}\) subset. A scalar measure of the degree of anisotropy can be defined by the second invariant of \(D_{ij}^{k}\):

### 3.1 Orthogonal preferential alignment of unmerged and merged pores

*x*–

*z*plane).

When considering the complete collection of pores, there is a preferential alignment in the direction of expansion. This implies that in the axisymmetric simulations, pores are orientated horizontally in the loading phase, before becoming vertical in the unloading phase and returning to a horizontal preference in the reloading phase. This is consistent with the observations in the simple shear simulation, where pores align to the right in the loading phase (which is the direction of tilt and the macroscopic direction of expansion), while they are orientated to the left in the unloading phase and align to the right in the reloading phase.

A more detailed perspective is provided by considering the anisotropy in the \(\left\{ u,\,m\right\} \) partition. Unmerged pores are the dominant contributor to pore space anisotropy, with \(a_{p}^{u}>a_{p}^{m}\) and this is particularly noticeable in the CP-L (Fig. 4a) and CP-D (Fig. 4b) simulations. In contrast, the magnitude of pore anisotropy is significantly smaller in the constant volume shear simulations (CV-D in Fig. 4c and SS-D in Fig. 4d). These cases show a near linear trend in pore anisotropy due to the (relatively) small strain range considered in these simulations. Similar linear response in pore anisotropy was found in the CP-L and CP-D samples for shearing to \(\left| \varepsilon _{a}\right| =2\%\) (not presented here).

In all simulations, unmerged pores exhibit strong preferential orientation in the expansion direction (perpendicular to the compression loading direction), while merged pores have a slight orientational preference parallel to the compression direction. This is schematically shown in Fig. 5 and was also observed in numerical simulations of bi-axial compression tests and direct shear tests [10, 11].

This orthogonal preferential orientation of merged and unmerged pores can be explained by considering the interparticle contacts along the edges of a pore unit. Recall that a pore unit formed by the modified Delaunay tessellation has polyhedral geometry (Fig. 3). Some edges of the polyhedral unit cell may contain interparticle contacts, while other edges will have a separation distance between non-interacting particles. Suppose \(\kappa _{c}=\frac{N_{c}}{N_{e}}\) is the ratio of active contacts, where \(N_{c}\) is the number of contacting edges and \(N_{e}\) is the number of edges of the pore unit. Also of importance is the ratio of sliding contacts, \(\kappa _{s}=\frac{N_{s}}{N_{c}}\), where \(N_{s}\) is the number of contacts which are sliding (i.e., at the Coulomb friction limit). Consider also the magnitude of the interparticle normal force, \(f_{n}\) of the active contacts. The average force magnitude of contacts forming a pore unit is defined as \(\kappa _{f}=\left\langle f_{n}/\overline{f_{n}}\right\rangle \), where the force is normalised by the mean normal force, \(\overline{f_{n}}\), and the average is taken only over active contacts. The parameters \(\kappa _{c}\), \(\kappa _{s}\) and \(\kappa _{f}\) effectively capture the influence of the contact network on individual pores.

### 3.2 Steady state conditions and lag in the response of pore anisotropy

In addition to this asymmetry, comparison of the pore and contact anisotropy indicates that pore space anisotropy is substantially smaller than observed for the contact network with \(\left| a_{p}\right| <0.5\), while \(\left| a_{c}\right| <1.0\) (similar trends are noted for the partitioned subsets of pores and contacts). This was also noted in Oda et al. [23] (albeit with different measures of contact and pore space anisotropy) and reflects the fact that a highly anisotropic pore space (e.g. all pores aligned in a particular direction) would not form a mechanically stable assembly when subject to shear.

Figure 8 also shows that the stress ratio response has a slight asymmetry (\(\frac{q}{p}=0.71,\;0.59\) for critical state compression and extension conditions, respectively). However, a direct correlation with the stress ratio is hindered by the fact that pore space anisotropy \(\left( a_{p}\right) \) evolves at a much slower rate than the contact network anisotropy \(\left( a_{c}\right) \) and the macroscopic stress response. This is most evident by considering the unloading and reloading phases in Fig. 8. In the unloading phase, the strain required to change loading direction (or anisotropy direction) is \(\left| \delta \varepsilon _{a}\right| =0.8\%\) for the stress ratio, while it is \(\left| \delta \varepsilon _{a}\right| =2.5\%\) for contact anisotropy and \(\left| \delta \varepsilon _{a}\right| =4.1\%\) for pore anisotropy. In the reloading phase, sign reversal requires \(\left| \delta \varepsilon _{a}\right| =2.0,\;7.1,\;14.3\%\) for stress ratio, contact anisotropy and pore anisotropy, respectively. This apparent lag in the response of the pore space reflects the fact that re-alignment of pores requires a substantial displacement of particles, which in a “dense” or slow-flowing granular assembly (with low inertia number) takes substantial amount of time. In contrast, the stress state can switch directions quickly by adjusting the magnitude of contact forces initially and then allowing the contacts to re-align as observed by Sufian et al. [32].

### 3.3 Isotropic directional distribution of pore volume in sheared assemblies

*k*denoting partitioned subsets is omitted. \(T_{ij}\) is given by:

Note that \(M_{ij}\) incorporates both competing effects of distribution by number and distribution by volume, and hence, differs from the representative value analysis presented in Li and Yu [17] which is typically used to explore directional distribution of normal and tangential force magnitude independent of any contact normal anisotropy [32]. Within the context of this paper, it is desirable to explore pore distribution by number and volume together.

Figure 9a shows that the anisotropy parameter \(a_{t}\approx 0\) in sheared granular assemblies, with some slight deviations away from perfect isotropy (as shown in the inset to Fig. 9a), but this deviation is two orders of magnitude less than the observed values for \(a_{p}\). Fig. 9b shows that the global orientation tensor is approximately isotropic with diagonal elements defined by the mean free pore volume. Note that this is not observed when considering the unmerged or merged pores separately. Therefore, deforming granular assemblies maintain isotropic pore volume distribution for all the simulations considered in this study. This suggest that when subject to external mechanical loading, particles adjust in a manner to maintain isotropic volume distribution, despite the induced anisotropy in pore orientation.

## 4 Pore shape as an indicator for the macroscopic density

Further insights into deformation characteristics can be gained by exploring pore shape properties. Figure 4a, b show that CP-L and CP-D specimens show similar pore anisotropy in loading, although the loose specimen undergoes contraction during shearing and the dense dilates (Fig. 1b). This difference in their volumetric behaviour can be captured by exploring pore shape.

*free*volume and hence, represents a loose assembly. In the \(\left\{ u,\,m\right\} \) partition, Fig. 10b shows that smaller unmerged pores have more regular geometry, while larger merged pores are elongated. The observation of large elongated pores is also in good agreement with experimental observations by Oda et al. [23] on photoelastic disks.

The average shape factor \(\left\langle \beta \right\rangle \) can be viewed as a measure of shape anisotropy, while \(a_{p}\) provided a measure of orientational anisotropy of the granular assembly. Note that these two forms of anisotropy have opposing trends. The loose assembly (CP-L) exhibits a lower degree of orientational anisotropy (Fig. 4) but a higher degree of shape anisotropy compared to the dense assembly (CP-D). This suggests that deformation characteristics may be represented as a competition between shape and orientational anisotropy.

There is an interesting interplay between shape and orientation of pores when combining the results of the orientational analysis in Sect. 3 and the above shape analysis. Smaller unmerged pores are more rounded and align in the expansion direction, while larger merged pores are more elongated and align in the compression direction (schematically shown in Fig. 5). This is reflected in the directional averaged \(\beta \) in Fig. 13.

In the initial loading phase for the CP-D simulation (Fig. 13a), sub-horizontal pores (\(\theta =0^{\circ }-30^{\circ }\), perpendicular to the loading direction) are more rounded, while sub-vertical pores (\(\theta =60^{\circ }-90^{\circ }\)) are more elongated, and pores with \(\theta =30^{\circ }-60^{\circ }\) fall in between. As the loading direction is changed in the unloading phase, sub-horizontal pores become more elongated (as they are now aligned parallel to the loading direction), while sub-vertical pores become more rounded (and so on for the reloading phase). This demonstrates that pore shape properties are closely tied to the orientational anisotropy of the pore space, which is confirmed in the CV-D simulation (Fig. 13b). Although the average shape factor was constant in the CV-D simulation, there is a clear directional dependence, and in fact the trends noted for the CP-D simulation are equally applicable for the CV-D simulation.

## 5 Conclusion

- 1.
Average shape factor was a micro-scale indicator of macroscopic void ratio. The loose assembly comprised more elongated pores, while the dense assembly consisted of more rounded pores.

- 2.
Local shear induced deformation anisotropy, where large pores are aligned in the direction of compression, while smaller pores are aligned perpendicular to the larger pores and in the direction of extension. This orthogonal preferential orientation of small and large pores can be related to the interparticle contacts along the edges of a pore.

- 3.
There exists an interplay between pore shape, size and orientation, with smaller rounded pores aligning in the expansion direction, while larger elongated pores aligned in the compression direction. This suggests that the deformation response in sheared granular assemblies may be represented by a competition between shape and orientational anisotropy.

- 4.
Macroscopic deformation response has a time lag, as demonstrated by larger levels of macroscopic shear strain required to mobilise pore anisotropy. This lag in the anisotropy response was also observed in DEM simulations reported by Zhao and Guo [36], where anisotropy in particle orientation (for non-spherical particles) evolved at a slower rate than the contact anisotropy. This highlights the importance of considering pore space anisotropy, in conjunction with contact network and particle orientation anisotropy, when describing the macroscopic response of granular materials.

- 5.
Conventional definitions of critical state conditions (based on void ratio and stress state) can be complemented with pore space fabric measures. The pore anisotropy parameter in this study showed evidence of critical state characteristics and can be implemented into continuum-based models, such as Li and Dafalias [18]. However, further studies (including different particle size distribution and stress states) are required to confirm the uniqueness of this critical state.

- 6.
The directional distribution of pore volume remained isotropic. This implied that in the compression direction, there were fewer larger pores, while in the expansion direction there were many smaller pores, and this effectively balanced to provide a net isotropic distribution of pore volume. This suggests that there is a constraint on the overall deformation, that is, particles rearrange to ensure isotropic volume distribution, but further studies are required to confirm this hypothesis.

## Notes

### Funding

This study was funded by the Australian Research Council (Grant No. DP150104123)

### Compliance with ethical standards

### Conflict of interest

None of the authors have a conflict of interest.

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