Effect of particle type on the shear behaviour of granular materials

Highlights

• The effect of convex true and multi-sphere ellipsoids on shear strength was studied.

• The effects of the two particle types on fabric anisotropy were compared.

• The different shear strength was explained in terms of micromechanics.

Abstract

In discrete element method (DEM) simulations, multi-sphere (MS) clumped and convex particles are two main particle models that are used to study the mechanical behaviours of granular materials. Of interest is the evaluation of the effect of multiple contacts between clumped particles or single contacts between convex particles on the mechanical behaviours of granular materials. In this context, a series of drained triaxial compression tests were conducted on convex true (CT) ellipsoids and MS ellipsoids with aspect ratios (ARs) ranging from 1.0–2.0. The microscale results indicate that at a given AR, the critical friction angle φ_c changes with the particle type, whereas the peak friction angle φ_p is nearly independent of the particle type. The anisotropic analysis provides underlying mechanisms of the shear strength evolution from two perspectives. First, the anisotropies of granular materials are essential to shear strength as the deviatoric (q)-to-effective mean (p′) stress ratio can be expressed as the sum of the anisotropies, i.e., $q/p^{\prime }\approx 0\text{.4}{a}_{\text{c}}+0\text{.4}{a}_{\text{n}}+0\text{.6}{a}_{\text{t}}$, where a_c, a_n and a_t are the normal contact anisotropy, normal contact force anisotropy and tangential contact force anisotropy, respectively. For all samples, a_c and a_n underpin the shear strength and are influenced by the particle type. The similar φ_p displayed by the CT and MS ellipsoids does not translate to similar a_n and a_c but similar a_c+a_n for the two particle types. In addition, owing to their larger a_c+a_n, the CT ellipsoids have a higher φ_c than the MS ellipsoids. Second, there is a satisfactory linear relationship between $q/p^{\prime }$ and a_c within strong and non-sliding (sn) contacts $a_{\text{c}}^{\text{sn}}$ (i.e., q/p′ = $k{a}_{\text{c}}^{\text{sn}}$), where k is the fitting parameter. Accordingly, in the peak state, the subtle difference in shear strength is attributed to the greater $a_{\text{c}}^{\text{sn}}$ in the CT ellipsoids than in the MS ellipsoids that is counteracted by the smaller k. However, in the critical state, the greater difference in $a_{\text{c}}^{\text{sn}}$ between the CT and MS ellipsoids is partially offset by the smaller difference in $$ k, causing a higher φ_c in the CT ellipsoids than in the MS ellipsoids.

Introduction

Particle shapes, as an important factor that influences the macroscopic properties of granular materials, have been widely studied in numerical simulations with the development of the discrete element method (DEM) (Alizadeh Behjani, Hassanpour, Ghadiri, & Bayly, 2017; Alonso-Marroquín, 2008; An & Li, 2013; Asachi, Behjani, Nourafkan, & Hassanpour, 2020; Behjani, Rahmanian, Fardina bt Abdul Ghani, & Hassanpour, 2017; Galindo-Torres & Pedroso, 2010; Gong, Liu, & Cui, 2019; Härtl & Ooi, 2011; Mack, Langston, Webb, & York, 2011; Maeda et al., 2010; Pournin et al., 2005; Qian, Yao, Li, Xiao, & Luo, 2020; Saint-Cyr, Delenne, Voivret, Radjai, & Sornay, 2011; Wellmann, Lillie, & Wriggers, 2008; Zhou, Zou, Pinson, & Yu, 2011). In these studies, the aspect ratio (AR) is commonly used to quantitatively describe particle shapes and represents the ratio of the long and short axes of an ellipse or ellipsoid. Zheng and Hryciw (Zheng & Hryciw, 2015) reported that AR reflects the sphericity of granular materials, which is often used in the Krumbein–Sloss chart (Krumbein & Sloss, 1951). However, previous studies have been inconclusive on the influence of AR on the shear strength of materials. For example, Ng (Ng, 2009, Ng, 2001) reported that the stress–strain behaviour, which represents the shear strength, was independent of AR, but subsequent research showed that the shear strength decreased slightly as AR increased. However, Gong and Liu (Gong & Liu, 2017), Zhao et al. (Zhao, Evans, & Zhou, 2018), Rothenburg (Rothenburg & Bathurst, 1992) and Azéma and Radjai (Azema & Radjai, 2010) have indicated that the critical shear strength is an increasing nonlinear function of AR. Gong and Liu (Gong & Liu, 2017) and Rothenburg (Rothenburg & Bathurst, 1992) found that the peak shear strength displayed a unimodal peak as AR increased, whereas Zhao et al. (Zhao et al., 2018) observed that the peak shear strength was unaffected. The different observations may be attributed to the different particle models used in the studies. For example, convex true (CT) ellipsoids were used in the simulations performed by Ng (Ng, 2009) and Zhao et al. (Zhao et al., 2018), whereas multi-sphere (MS) ellipsoids were used in the studies performed by Azéma and Radjai (Azema & Radjai, 2010) and Gong and Liu (Gong & Liu, 2017). Clumped and convex particles are two main particle models used in DEM studies to simulate the mechanical behaviours of granular materials. In the former particle model, the clumped particles are composed of MSs, possibly with multiple contacts among the particles, whereas in the latter particle model, there is only one contact between any two given convex particles. For some research questions (e.g., how does AR influence the shear behaviour of granular materials?), both clumped and convex particle models can be adopted. However, it is unclear whether the adoption of two different particle models will lead to different macroscale and microscale responses. In this context, herein, we examine the influence of multiple contacts between clumps and a single contact between convex particles on the mechanical behaviours of granular materials.

In terms of micromechanics, the anisotropy of force and fabric is the source of the shear resistance of granular materials (Guo and Zhao, 2013, Ouadfel, 2001, Zhao et al., 2018). To establish a relationship between shear strength and stress-induced anisotropy, for cohesionless granular materials, the deviatoric (q)-to-effective mean (p′) stress ratio, $q/p^{\prime }$, can be expressed as the sum of the anisotropies under triaxial conditions (Ouadfel, 2001; Zhao et al., 2018), as follows:

$$q/p^{\prime }=0\text{.4}(a_{\text{c}}+a_{\text{n}}+1\text{.5}{a}_{\text{t}}+a_{\text{dn}}+1\text{.5}{a}_{\text{dt}})$$

where a_c, a_n, a_t, a_dn and a_dt are the normal contact anisotropy, normal contact force anisotropy, tangential contact force anisotropy, normal branch vector anisotropy and tangential branch vector anisotropy, respectively. The anisotropic coefficients can be calculated from generalised fabric and force tensors, as confirmed by previous studies (Guo & Zhao, 2013; Ouadfel, 2001; Kanatani, 1984; Markauskas, Kačianauskas, Džiugys, & Navakas, 2009; Sufian, Russell, & Whittle, 2017). By comparing the values between the normal contact force f_n, tangential contact force f_t and the mean normal contact force $\overline{f_{\text{n}}}$, a quad-partition is considered comprising four subnetworks: (1) strong and sliding contact network ($fn>\overline{f_{\text{n}}}$ and $ft>{\mu }_{\text{b}}\overline{f_{\text{n}}}$); (2) strong and non-sliding contact network ($fn>\overline{f_{\text{n}}}$ and $ft<{\mu }_{\text{b}}\overline{f_{\text{n}}}$); (3) weak and sliding contact network ($fn<\overline{f_{\text{n}}}$ and $ft>{\mu }_{\text{b}}\overline{f_{\text{n}}}$); and (4) weak and non-sliding contact network ($fn<\overline{f_{\text{n}}}$ and $ft<{\mu }_{\text{b}}\overline{f_{\text{n}}}$). These subnetworks are denoted as ss, sn, ws, and wn, respectively (Alonso-Marroquin, Luding, Herrmann, & Vardoulakis, 2005; Thornton & Antony, 1998; Radjai, Wolf, Jean, & Moreau, 1998; Sufian et al., 2017) and ${\mu }_{\text{b}}$ is the interparticle friction coefficient. As reported in the literature, the anisotropic coefficient of contacts in the strong and non-sliding contact network $a_{\text{c}}^{\text{sn}}$ (note that the value of $a_{\text{c}}^{\text{sn}}$ is obtained based on the equations of ${\text{a}}_{\text{c}}$ but restricting to the strong and non-sliding contacts, i.e., the subnetwork sn.) is the dominant contributor to the shear strength (Nie, Zhu, Wang, & Gong, 2019) and exhibits a linear relationship with the stress ratio for spherical particle systems (Sufian et al., 2017; Zhou, Wu, Ma, Ng, & Chang, 2018) as follows:

$$q/p^{\prime }=k{a}_{\text{c}}^{\text{sn}}$$

where k is a fitting parameter (≈0.4 for spherical particle systems). As the particle type varies, the anisotropies may change, which can be reflected by changes in the microscopic behaviours of granular materials. To obtain a better understanding of the effect of particle type on the shear strength of granular materials, the microscopic behaviours and stress-induced anisotropies must be explored, wherein grain-scale modelling using DEM is conducted. This approach has been demonstrated to reproduce some features (e.g., strength, deformation, porosity, etc.) of granular materials (Azéma, Radjaï, Peyroux, & Saussine, 2007; Brown et al., 2011; Gong & Liu, 2017; Gong, Liu et al., 2019; Qu, Feng, Wang, & Wang, 2019).

In the present study, the influence of particle type on the shear behaviours of particle packing is studied through a series of drained triaxial compression tests using DEM modelling. Two forms of ellipsoid-shaped particles (i.e., CT and MS ellipsoids) are generated and examined. A macroscale and microscale analysis of the simulation results is discussed along with proposed underlying mechanisms related to the influence of particle type on the shear strength via the anisotropic coefficient.