Evaluating the particle rolling effect on the characteristic features of granular material under the critical state soil mechanics framework

Abstract

The discrete element method (DEM) has been extensively used to capture the macroscopic and particulate response of granular materials. Although particle rolling (i.e. controlled by rolling resistance) has been acknowledged as a major contributing factor towards micro-mechanical behaviour of idealized spherical granular material, its influence on characteristic behaviour has not been thoroughly investigated within critical state soil mechanics (CSSM) framework. For instance, the influence of particle rolling on characteristic features of undrained and drained behaviour (e.g. phase transformation, characteristic state, instability, dilatancy, critical state) and the state parameter, (ψ) has not been captured. In this study, a series of constant volume (CV) and drained triaxial compression simulations were undertaken using a rolling resistance linear contact model, deployed within a DEM software. The CSSM framework was centrally used to assess the influence of particle rolling tendencies/resistance on CV and drained behaviours from both a macro- and micro-mechanical standpoint. The study advanced the current understanding of the influence of rolling resistance on CS-related behaviour.

Graphic abstract

1 Introduction

The critical state line (CSL) in e-log p′ space has been used as a reference line to predict granular material behaviour; where e is the void ratio and p′ is the mean effective stress. A granular material with a state (i.e. e and p′) above the CSL exhibits contractive behaviour, whilst a state below the CSL exhibits dilative behaviour. Been and Jefferies [4] defined such a state with the state parameter (ψ), as the difference between current e and the e on the CSL (measured at same p′) as presented in Fig. 1. In general, a positive ψ (above CSL) indicates contractive behaviour, whilst negative ψ (below CSL) indicates dilative behaviour. ψ has also been correlated with many characteristic behaviour of soil such as instability [26, 28, 59, 67, 70, 73, 74, 87], generation of excess pore water pressure (Δu) [69, 92], phase transformation (PT) and characteristic states (ChS) [55, 60, 92] and cyclic shear behaviour [69, 73, 74]. The CSL in q − p′ space defines intrinsic material properties such as the constant volume friction angle (ϕ_cv) and critical state stress ratio M = (q/p′)_cs. The concept of state dependency and intrinsic material properties at CS have been used in many state-dependent constitutive models [16, 34, 42–44, 48]. Popular state-dependent constitutive models such as an earlier version of the SANISAND models described in Li and Dafalias [43] assume particle arrangement in specimens are the same i.e. formed via the same deposition/reconstitution method. However, specimens prepared by different deposition methods may have different stress–strain behaviour with same or different CSL [85] and has to be considered as different/partially different soil for constitutive modelling. The initial particle arrangement and the evolution of their interlocking, which is often indirectly captured by the mobilised inter-particle friction angle [20, 79], affects the observed stress–strain behaviour of granular materials. Therefore, understanding the evolution of particle arrangement, along with CS, is needed to develop further understanding of soil behaviour. However, capturing the micro-mechanical response of sand is challenging in an experimental setting. Experimental techniques such as Scanning Electron Microscopy (SEM) and Computed Tomography (CT) scanning have been used to capture different aspects of the micromechanical behaviour of granular materials with limited success [18, 25, 30, 89]. Although some micro-CT scanning techniques exist, which allow for the soil fabric behaviour to be captured in triaxial compression [18, 19], capturing the continuous fabric evolution behaviour using such techniques may be deemed expensive and time-consuming.

Fig. 1

Conceptualisation of the state parameter, ψ

The discrete element method (DEM), proposed by Cundall and Strack [14], has become popular as an alternative approach to develop a qualitative understanding of soil behaviour due to its capability of generating a numerical assembly of particles with a desired grain size distribution, friction coefficient and shape, and allowing an in-depth understanding of micromechanics [23, 26, 28, 59, 63]. Thus, DEM has proven to be a useful tool in capturing the micromechanical interactions of granular materials and is capable of providing a physical explanation of the overall behaviour of such materials. The role of fabric and contact density on the characteristic behaviour of granular materials have been explored using DEM. However, in many of these recent studies, rolling characteristics of particles have been neglected within the DEM contact model [22, 23, 93]. This is a crude assumption as many DEM studies employ spherical particles, which characteristically exhibit large particle rolling tendencies. Through X-ray and photo-elastic techniques, Oda et al. [66] and Oda and Kazama [65] suggested particle rolling is a dominant micro-deformation mechanism which controls the dilatancy and peak strength behaviours of granular materials. Their experimental study reported that rolling resistance can somewhat indirectly capture particle shape and surface texture features of granular material. Zhao et al. [95] studied the shear behaviour of variable particle shapes within a rolling resistance model in DEM and observed that particle shape could somewhat be captured by rolling resistance. However, the model was only effective in doing so for particle shapes which have slight distortion from the spherical particle. Assemblies mobilised with rolling resistance usually show higher shear strength and volumetric dilation [32, 54, 65, 94]. Liu et al. [46] reported similar observations but only until a threshold rolling resistance value of 0.3. However, Plassiard et al. [68] showed there was no clear link between rolling resistance with peak dilatancy or drained CS strength. Although, other DEM studies have observed that the addition of rolling resistance enhances the peak and residual friction angle [1]. Iwashita and Oda [32] observed shear band formation and strain localization behaviour to intensify with the inclusion of rolling resistance, which led to higher dilative tendencies. Zhao and Guo [94] and Liu et al. [46] observed fabric anisotropy to increase with the inclusion of rolling resistance. Some studies have observed the CS to strengthen with rolling resistance, although the application of CSSM to evaluate the influence of rolling resistance on constitutional (characteristic) behaviour has not been performed [17, 100]. For example, Zhao and Guo [94] & Chang et al. [10] investigated the influence of rolling resistance within CSSM framework and observed an upwards shift of the CSL in e-log p′ space with the inclusion of rolling resistance. Although, CSSM (i.e. evaluating the performance of ψ) was not used to capture the influence of particle rolling on characteristic features and constitutional shearing behaviour (e.g. phase transformation, characteristic state, undrained instability state, maximum dilatancy).

In this study, a rolling resistance linear contact model (RRLCM) is deployed in DEM to advance the understanding of the influence of particle rolling on shearing behaviour. The influence of particle rolling (or rolling resistance) on macro-mechanical features (e.g. instability, phase transformation (PT), characteristic states (ChS), peak dilatancy, CS etc.) was evaluated, advancing the understanding of the influence of particle rolling on the characteristic response of granular material. The influence of particle rolling on characteristic features was also captured via ψ (i.e. CSSM framework was utilised) and their behaviours could be bridged with the observed micro-mechanical response in the RRLCM (i.e. contact density, fabric, contact network). Currently, the ability for the CSSM (i.e. ψ) to capture the influence of particle rolling on soil behaviour is not understood. Consequently, existing state-dependent framework (e.g. the flow-rule equation in [43]) and state-dependent constitutive models and their ability to capture the influence of particle rolling on soil behaviour is not known. Through assessing the response between particle rolling resistance and the aforementioned characteristic features of triaxial compression behaviour, the ability for a state-dependent constitutive framework to capture the particle rolling effect may be analysed.

Additionally, through a focus on the DEM-observed characteristic behaviour of granular material, the understanding of the CSSM in DEM may be built upon. For instance, the recently debated [60, 92] CSSM-related phenomenon—the equivalency of the PT and ChS in undrained and drained triaxial shearing, could be analysed. Although the happening of the PT and ChS corresponds to the transition between contractive and dilative behaviours in undrained and drained triaxial shearing, the micro-mechanical influence on the happening of this transition has not been investigated. The micro-mechanical behaviour is captured at characteristic states, in order to assess the micro-mechanical influence on the happening of the PT and ChS states.

2 DEM model details

DEM software, PFC^3D (Itasca [31] was used to simulate constant volume (CV or undrained) and drained triaxial compression tests. Inter-particle forces and displacements were resolved using the rolling resistance linear contact model (RRLCM). The RRLCM is identical in nature to the conventional linear contact model (see Supplemental Data), however with a rolling spring and dashpot installed at the contact (Fig. 2). The rolling spring simulates the linear elastic (non-tension) frictional behaviour caused by particle rolling, whilst the dashpot accounts for any viscous behaviour due to particle rolling. It should be noted, whilst twisting may occur at the contact, the behaviour resulting from particle twisting is not incorporated in the model. Many DEM studies which attempted to analyse the influence of particle rolling behaviour have assumed constant values for either normal stiffness (k_n), tangential stiffness (k_s) and/or rolling stiffness (k_r) over an entire particle size distribution [32, 46, 54, 84]. In reality, particle stiffness has been observed to be particle size-dependent [86]. In this study, k_n, k_s and k_r of individual particles are computed based on their radius and potential deformation of particles at the contact. The deformability of granular material which share contact can be described by the elastic constants of the material, i.e. Young’s modulus (E) and Poisson’s ratio (ν). In this study, k_n, k_s and k_r are a direct function of an effective modulus (E*) and k^* = k_n/k_s, which are emergent properties of E and ν. The method of deformability installed in the contact model and the mathematical formulations of k_n, k_s and k_r are further described in Appendix A. An average contact overlap ratio representative of sand particles was maintained. That is, an average overlap ratio less than 6% was achieved [51]. An E* = 50 MPa was selected and critical state stress ranges (i.e. p′_cs) observed in this study aligned with those observed in experimental studies of identical graded granular material [75]. Unlike other studies, this study therefore evaluates the rolling behaviour of particles where contact stiffness varied with radius and potential deformation. The full mathematical description of the inherited RRLCM is presented in “Appendix B”.

Fig. 2

Rheology of rolling contact mechanism used in study

Various approaches have been used to mimic the rolling resistance (or potential) created at a contact. Two of the more common methods of modelling rolling resistance in DEM is through either- 1) an eccentricity parameter [98, 99] with a unit length which is related to a rolling moment arm between two contacting particles or 2) a dimensionless shape parameter, δ [35] which apparently reflected the contact width between contacting entities and their relative particle shapes. A more commonly used approach is adopting a dimensionless rolling resistance parameter (μ_r) due to its established physical basis:

$$\mu_{\text{r}} = \tan \beta$$

(1)

where β corresponds to the maximum angle of a plane at which a particle sitting on it possesses a resisting moment equal to its natural rolling torque. The mathematical definition of μ_r is conveniently in close association with the definition of angle of repose and therefore sliding friction (μ). Further, Ai et al. [2] indicated that the μ_r definition is preferable in measuring rolling resistance due to its clear physical basis. μ and μ_r were used simultaneously in all simulations. In this study, various μ_r values were used to investigate the effect of particle rolling on overall behaviour.

3 Triaxial simulations

In this study, 90 triaxial tests with different consolidation stresses, p′₀ (48–500 kPa), different initial void ratio, e₀ (0.594–0.889) and different μ_r (0–1) were simulated. A summary of DEM input parameters and test conditions are presented in Tables 1 and 2 respectively. The range of 0 ≤ μ_r ≤ 1 was selected as at μ_r ≈ 1 a threshold rolling value is approached, and minimal change in soil and CS behaviour were observed.

Table 1 Summary of the specimen testing in DEM

Table 2 Summary of DEM simulations

Spherical particles were used to represent clean sand with median particle size, D₅₀ of 0.279 mm. The particle size distribution of the assembly is presented in Fig. 3. Particles were initially generated with no initial contact within a cubic domain bounded by six rigid and frictionless walls. A staged isotropic compression technique was used to achieve the desired void ratio before consolidation, e_i. During this process, a smaller μ (≈ 0–0.4) was used to form dense assemblies, whilst a large μ (≈ 0.6–1) was used to form loose assemblies. After the desired e_i was achieved, μ was set to a representative value for granular material (μ = 0.5) before the commencement of consolidation in all triaxial simulations. All specimens were isotropically compressed to different p′₀.

Fig. 3

DEM assembly and its particle size distribution

In undrained (constant volume) shearing constant volume was ensured through a servo-control mechanism applied to the vertical walls while shearing strains were applied at the top and bottom walls. Drained shearing was achieved through strain control of the top and bottom walls, whilst the stresses on the vertical walls were controlled simulating constant cell pressure.

4 Numerical stability of DEM simulations

The deterministic nature of elastoplastic theories suggests that the mechanical behaviour of granular material in DEM assemblies should be independent of size i.e. number of particles [36]. Figure 4 presents the shearing responses for three specimens with different particle number (N) and strain deformation rates. TRR26 and TRR31 were conducted with the same strain rate of strain of 0.001 s⁻¹ but with N = 10,506 and N = 17,921 respectively. TRR33 was conducted with N = 10,506 and a strain rate five times smaller than TRR26 (0.0002 s⁻¹). Negligible discrepancies in their q–ε_q and q–p′ paths in Fig. 4a, b suggest that a strain rate of 0.001 s⁻¹ with N = 10,506 achieves numerical stability. This specimen size and strain rate was used in all further simulations in this study.

Fig. 4

Effect of specimen size and strain rate on DEM simulation in: a q − ε_q, b q − p′ and c CN − ε_q spaces

The coordination number, CN = 2N_c/N [78], is a micromechanical entity which quantifies the contact density of a granular assembly; where N_c is the number of particle contacts. Figure 4c shows that specimen size and strain rate had a negligible influence in their CN-ε_q paths. The inertial number (I) proposed by da Cruz et al. [15] has been used as a performance parameter in DEM which assesses whether a quasi-static deformation of particles in the assembly throughout shearing is upheld. I is defined as,

$$I = \dot{\varepsilon }d\left( {\rho_{g} /p} \right)^{0.5}$$

(2)

where $\dot{\varepsilon }$ is the shear strain rate, $d$ is the mean particle diameter, and $\rho_{g}$ is the particle density. All simulations satisfied the condition of I < 10⁻³ i.e. the influence of particle kinematics on the observed shear behaviour corresponded to a quasi-static condition [47, 50].

5 Results

The major and minor principal stresses (σ′₁₁ and σ′₃₃) were computed from the stress tensor as suggested by Christoffersen et al. [11]:

$$\sigma_{ij}^{\prime } = \frac{1}{V}\mathop \sum \limits_{{c\epsilon N_{c} }} f_{i}^{c} l_{j}^{c}$$

(3)

where $V$ is the total assembly volume, $N_{c}$ is the total number of contacts, $f_{i}^{c}$ is the ith component of contact force at contact c and $l_{j}^{c}$ represents the jth branch vector connecting the centres of two contacting particles, at contact c. Only particle to particle contacts are considered in Eq. 3 so that the effect of the rigid boundary end-restraint is reduced [21, 49]. p′ and q (in triaxial space) were derived from the stress tensor as below:

$$p^{\prime} = \frac{{\sigma '_{11} + \sigma '_{33} }}{3}$$

(4)

$$q = \sigma '_{11} - \sigma '_{33}$$

(5)

5.1 Constant volume tests

Figure 5 shows the response during CV shearing for six samples with different μ_r (0 ≤ μ_r ≤ 1). All tests were conducted with p′₀ of 100 kPa and similar e₀ values (between 0.594 and 0.601). Each simulation was continued up to ε_q of 50%. Due to the very dense nature of the presented tests, a strict condition of CS was sometimes not reached, as the condition dΔu = 0 was not satisfied. Note, Δu was indirectly derived from the difference between p′ in CV and drained paths (Δu = p′_D − p′_CV) as described by Sitharam et al. [81]. In cases where the CS condition of dΔu = 0 was not reached, an extrapolation approach [8, 55, 75] was used. All specimens underwent strain hardening until a near CS was reached, and therefore exhibited non-flow (NF) behaviour. Specimens with large μ_r values manifested the strongest behaviour characterised by prolonged strain hardening legs, resulting in the manifestation of larger CS strengths. An identical trend reoccurred in effective stress path space (Fig. 5b), suggesting the increase in strength with μ_r is largely due to the enhancement of frictional properties. In Fig. 5a, b, M = q/p′ or CS strength increased with μ_r, but became less sensitive to changes as μ_r → 1 indicating a theoretical state of maximum rolling resistance is approached. The DEM study of Huang et al. [26, 28] detected similar behaviour in the case of μ.

Fig. 5

Influence of μ_r on CV behaviour in a q − ε_q, b q − p′ and c Δu − ε_q spaces

The influence of μ_r on Δu is evaluated in Fig. 5c. When ε_q < 25%, assemblies mobilised with low μ_r were observed to exhibit the largest negative pore water pressure changes. Although, assemblies with larger μ_r, generate negative Δu at larger rates past intermediate (≈ 25%,) strains. A clear trend between Δu and μ_r at CS is hard to observe as most tests are yet to reach CS and are changing at different rates.

Earlier studies suggested that a large CN is often associated with a large CS strength or dense assembly [22, 59]. To evaluate the effect of μ_r on CN, the isotropic compression paths are presented first in Fig. 6a. A negligible influence of μ_r on CN was observed, i.e. these paths were almost identical. The evolution of CN during shearing was then presented in Fig. 6b. Although CN at the start of shearing is identical for all simulations, during shearing, a specimen with larger μ_r value achieved a lower CN at the critical state and vice versa. For instance, assemblies with large rolling tendencies (lower μ_r) achieved a critical state strength via a larger number of weaker contacts. On the other hand, the efficiency of strong contact formation was superior for assemblies mobilised with a larger μ_r. The strong formation efficiency or interlocking behaviour may therefore be tuned via μ_r. Although, these findings are important in understanding the qualitative influence of particle rolling on contact density, the RRLCM may have the tendency to generate magnitudes of CN which may not be realistic [95].

Fig. 6

Influence of μ_r on a isotropic consolidation behaviour in CN − p′ space, b CV shear behaviour in CN − ε_q space

The contact force networks at CS for four undrained specimens with similar initial states but varying μ_r illustrate this effect (Fig. 7). A profound influence of particle rolling on the contact force network and the contact anisotropy throughout shearing can be observed. In a free-rolling environment (i.e. μ_r = 0, Fig. 7a) a dense contact network was formed (i.e. large CN). The network was rife with contacts largely disorientated to the direction of loading, i.e. weak contact forces were predominate [52]. Following Radjai et al. [72], an assembly which possesses a contact network of this type may behave as an interstitial fluid. In agreement, Zhou et al. [97] indicated the reduction in μ_r conforms to a lubrication or softening effect of particles in contact. When μ_r was large, strong contacts with larger magnitude (i.e. a hardening effect) dominated the force chain network, ultimately with a reduction of CN at CS. The amount of sliding or rolling contacts are therefore limited [72] and ultimately interlocking capabilities are enhanced. Guo and Su [24] in their laboratory experiments and Huang et al. [26, 28] in their DEM simulations also observed interlocking behaviour to be closely related with particle angularity—a parallel that could be drawn between particle rolling resistance and angularity. Although an interlocking effect can be simulated via rolling resistance, through capturing the twisting resistance at the contact, the interlocking could be better approximated.

Fig. 7

Influence of rolling resistance on observed contact network at CS: a TRR22 (μ_r = 0), b TRR25 (μ_r = 0.2), c TRR28 (μ_r = 0.6), d TRR30 (μ_r = 1.0)

The change in force chain networks could also be explained by a transitional behaviour from rolling-dominanting contact networks to sliding-dominating contacts networks, as μ_r is varied. Reportedlty, for the granular mixtures mobilised with small μ_r values, the displacement of contacts are dominated by particle rolling. As μ_r increased, the participation of sliding at the contacts was increased, in lieu of rolling behaviour [27]. Nevertheless, a clear strengthening effect is observed with the addition of μ_r.

Many studies have reported the internal structure or spatial arrangement (i.e. soil fabric) of granular material to be highly anisotropic which influences the soil behaviour [56, 64, 91]. Similar to CN, observing the anisotropic fabric and its evolution throughout shearing may assist in explaining the macro-response of a granular material. To quantify anisotropic fabric, Satake [80] and Rothenburg and Bathurst [78] proposed similar second-order fabric tensors. Via the former, the fabric tensor, F may be expressed as:

$${\mathbf{F}} = F_{ij} = \frac{1}{{N_{c} }}\mathop \sum \limits_{k = 1}^{{N_{c} }} n_{i}^{k} n_{j}^{k} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {F_{11} } & {F_{12} } & {F_{13} } \\ \end{array} } \\ {\begin{array}{*{20}c} {F_{21} } & {F_{22} } & {F_{23} } \\ \end{array} } \\ {\begin{array}{*{20}c} {F_{31} } & {F_{32} } & {F_{33} } \\ \end{array} } \\ \end{array} } \right)$$

(6)

where n^k is the directional unit vector for the kth contact, F_ij is scalar counterpart of F. Rahman et al. [76] showed that the deviatoric fabric and its evolution throughout shearing could be represented by the von Mises fabric, F_vM= 3(F_J2D)^0.5[77], where F_J2D represents the second invariant of F:

$$F_{J2D} = \frac{1}{6}\left[ {\left( {F_{11} - F_{22} } \right)^{2} + \left( {F_{11} - F_{33} } \right)^{2} + \left( {F_{22} - F_{33} } \right)^{2} } \right] + F_{12}^{2} + F_{13}^{2} + F_{23}^{2}$$

(7)

Figure 8 illustrates the evolution of deviatoric fabric for each of the CV simulations. The deviatoric fabric intensified to a peak F_vM and then reduced until CS was reached where the changes of von Mises fabric dF_vM approached zero. In agreement with Li and Dafalias [41], dF_vM= 0 may be considered as a supplementary condition of CS. Specimens mobilised with large μ_r values displayed a large peak and CS F_vM. Assemblies with large strong contact networks typically exhibit large anisotropy [26, 28], which potentially explains why assemblies with large μ_r exhibit large peak F_vM or deviatoric anisotropies. The observations also highlight that the deviatoric fabric is largest for assemblies mobilised with large μ_r [96].

Fig. 8

Influence of μ_r on CV behaviour in F_vM − ε_q space

The phase transformation (PT) state corresponds to the transition from contractive to dilative behaviour, which may be observed in the undrained shearing of dense and medium-dense specimens. Mathematically the PT can be defined as when dp′ = 0 [38] and corresponds to the ‘knee’ in the effective stress path as shown in Fig. 9a. Alternatively, dΔu = 0 has been used to define the PT state as shown in Fig. 9b [83]. The later description (dΔu/dε_q = 0) was used to define PT in this study. The rolling tendencies manifested within an assembly was observed to influence the PT. Specifically, the PT was observed at slightly larger ε_q, Δu and q values for assemblies with larger μ_r. In Fig. 9c the PT point observed at dΔu = 0 was projected on the CN evolution path for each simulation. Interestingly, the PT point does not correspond to a change in contact behaviour, i.e. a change from decreasing contact density to increasing contact density. Literature has linked the local minimum of the CN path to the quasi-steady state [88], although it seems there is no relation with the classical PT point. The PT point was also projected on the fabric evolution path (Fig. 9d). It seems, the occurrence of PT cannot be explained by a sudden change (local maximum) in the granular material deviatoric fabric or spatial arrangement. The CN and F_vM behaviour has been useful in explaining CS behaviour from a micromechanical standpoint [3, 26, 28, 57], although explaining the occurrence of PT from these two micromechanical entities is challenging.

Fig. 9

Influence of μ_r on PT state: a at dp′ = 0, and b dΔu = 0, c PT projected on CN evolution path, dPT projected on F_vM evolution path

At the happening of PT a sharp change from contractive to dilative behaviour is observed. Although, such behaviour is not believed to arise as a result of DEM imposed testing conditions. For instance in Ishihara [29] study of Toyoura sand, a sharp change from contractive to dilative behaviour was also observed. In the experimental study of Murthy et al. [55] a clear ‘knee’ in the effective stress path could also be observed—both studies yielded similar ‘knee’ or PT behaviour as this manuscript.

5.2 Drained shearing tests

Figure 10 shows the drained shearing responses for samples with p′₀ of 100 kPa, e₀ within a narrow range of 0.594–0.601 and varying of μ_r values. In Fig. 10a, all specimens exhibited dilative behaviour to a peak deviatoric stress (q_peak) followed by strain-softening to CS. Similar to the observations made for constant volume shearing, an increase in CS strength is associated with a rise in μ_r (see Fig. 10a). For large μ_r, hardening occurred over a larger range of ε_q, thus a higher q_peak was reached at larger deviatoric strains. Specimens with large μ_r manifested steep strain softening (i.e. brittle) behaviour.

Fig. 10

Influence of μ_r on drained response of clean sand in a q − ε_q, b ε_v − ε_q, c d_peak − μ_r, d CN − ε_q and e F_vM − ε_q spaces

The highly dilative nature of assemblies mobilised with large μ_r is illustrated in ε_v − ε_q space; where ε_v represents the volumetric strain (Fig. 10b). Volume dilation shows to increase with μ_r. Although, when μ_r > 0.3, insignificant increase in volume dilation occurs. Each of the drained tests peak dilatancy, i.e. $d_{peak} = \left( {\frac{{{\text{d}}\varepsilon_{q}^{p} }}{{{\text{d}}\varepsilon_{v}^{p} }}} \right)_{peak} ,$ was captured (Fig. 10c); where superscript “^p” refers to plastic. Note, following many researchers ${\text{d}}\varepsilon_{q}^{p} \approx {\text{d}}\varepsilon_{q}$ and ${\text{d}}\varepsilon_{v}^{p} \approx {\text{d}}\varepsilon_{v}$ were assumed for large strain behaviour of granular materials [43, 73, 74]. Figure 10c supports the supposition that there is insignificant change in dilatancy behaviour at ranges of μ_r > 0.3 is further supported. Although in q − ε_q space at μ_r > 0.3, the change in peak strength is more significant. Thus, at this range of μ_r it seems an enhancement of frictional properties (i.e. M or ϕ_cv) rather than dilatancy properties is responsible for strengthening the materials response via strong contacts. When μ_r < 0.3, the response is governed more equally by both dilatancy as well as frictional behaviour. In Fig. 10c, the peak dilatancy, was observed to increase initially with μ_r although Plassiard et al. [68] observed peak dilatancy to be independent of rolling resistance. In Fig. 10d, e the evolution of CN and F_vM during drained shearing is presented. The behaviour is identical to what is observed throughout CV simulations.

Similar to the PT state in undrained shearing, a characteristic state (ChS) exists in drained shearing which signifies the transition between contractive and dilative behaviour [38]. The ChS in ε_v − ε_q space is presented in Fig. 11a. The ChS can be pinpointed as the state of dε_v = 0, i.e. dilatancy, d = dε_v/|dε_q| = 0. Similar to the PT, the ChS is influenced by μ_r. The ChS occurred at larger ε_v and ε_q for specimens mobilised with large μ_r for specimens sheared with similar initial states. Thus, in both CV and drained shearing of medium dense/dense specimens, initial contractive tendencies were prolonged for assemblies mobilised with larger rolling resistance. It is likely, this is due to the unease of displacing strong interlocking contacts, and thus dense assemblies stay in a contractive state for longer, until contacts may be displaced at larger shear stresses. The ChS point was projected on the CN and F_vM evolution paths (Fig. 11b, c). Similar to observations in CV shearing (Fig. 9), no change in soil micromechanics was observed at the ChS states in drained shearing. This supports the notion of continuum mechanics that the happening of PT and ChS states are related to the responses of dilative tendencies, not due to a change of particle contact density or particle spatial arrangement.

Fig. 11

Influence of rolling resistance on ChS

6 Critical state behaviour

6.1 Influence of rolling resistance on CSL

The CS data points from both CV and drained simulations with five different μ_r of 0, 0.1, 0.3, 0.5 and 0.7 are plotted in the e-log p′ space as shown in Fig. 12a. The CSLs can be represented by the following power function as proposed by Li et al. [45]:

$$e_{cs} = e_{ \lim } - \lambda \left( {\frac{{p^{\prime}}}{{p_{a} }}} \right)^{\xi }$$

(8)

where p_a is the atmospheric reference stress i.e. p_a = 100 kPa and e_lim, λ and ξ are dimensionless curve fitting parameters. The CSLs were independent of drainage condition aligning with early experimental studies [5, 9, 82, 90, 6, 71] and DEM studies [60]. These CSLs are curved and shifted upwards with increasing μ_r. Previous literature had suggested the curvature of the CSL to only be attributed to particle breakage (Konrad 1998). However, the DEM particles were unbreakable and therefore suggest otherwise—which is consistent with other DEM studies inheriting different contact models and particle shapes [59, 61, 62]. The pronunciation of the CSL curvature became more prominent at large μ_r values which led to merging CSLs at higher p′ (> 1000 kPa) irrespective of μ_r. A conversing trend for CS data points at higher p′ was also found with particle angularity in DEM [61]. However, Huang et al. [26, 28] using a simplified Hertz–Mindlin contact model where particle rotations were allowed (following Calvetti and Emeriault [7]) in DEM software LAMMPS found almost parallel CSLs with very mild curvature up to p′ of 10,000 kPa for two frictional constraints. They also observed an unusual increase of critical state void ratio with increasing p′ for μ ≥ 0.5. μ = 0.5 was used in this study and the unusual behaviour was not observed. Such difference with this study may be attributed to the difference in contact models, implementation platform i.e. PFC & LAMMPS and limited CS data in Huang et al. [26, 28]. The CS data points for other μ_r values are also projected with these CSLs in Fig. 12b. The data suggests there is negligible influence of μ_r on CSL in the e-log p′ space when μ_r ≥ 0.4 and one CSL may be assumed in this range of μ_r.

Fig. 12

Influence of μ_r on CS response of clean sand a influence on q_cs, b influence on CN_cs, c influence on F_vM(cs)

The CSLs in the q − p′ space were also μ_r-dependant (Fig. 12c). Within the range of 0 ≤ μ_r ≤ 1, M increased from 0.73 to 1.38 with a noticeable change from 0.73 to 1.15 for μ_r only increased from 0.0 to 0.3. The increasing trend of M with μ_r is very similar to increasing M with increasing particle angularity in DEM [61]. Therefore, rolling resistance between particles contribute to the CSLs and its curvature in the e-log p′ space in a similar manner to particle shape and thus, μ_r may be inferred as contributor to frictional properties of particle shape.

6.2 Change in CS behaviour with μ_r

A series of CV and drained simulations, where shearing commenced from similar initial states (p′₀ = 100 kPa, e₀ = 0.5975 ± 0.0035) for both drained and CV simulations were performed and the influence of rolling resistance on q, CN and F_vM at CS was captured (Fig. 13). The change in μ_r with the three quantities are similar between drainage conditions. In Fig. 13a, q at CS (q_cs) increases gradually with μ_r before plateauing at approximately μ_r = 0.4. Thus, at larger μ_r ranges the influence of particle rolling resistance on q_cs is negligible. Similar tendencies were also observed for CN and F_vM in Fig. 13b, c respectively. Although strictly not a threshold μ_r value, the change in CS behaviour becomes relatively unnoticeable as μ_r → 1. Liu et al. [46] reported a threshold μ_r value of 0.3 where at μ_r ≥ 0.3 the strength of granular material shows negligible change. Similar is observed here, although micromechanical entities CN and F_vM are observed to change differently at μ_r ≥ 0.3. Therefore, a threshold rolling value may only be applicable for selected entities/performance parameters rather than applying for the entire material response. Further, it should be acknowledged that the change in contact model (e.g. contact stiffness algorithms) and model parameters (e.g. μ) which may influence particle rolling behaviour may lead to different observations relating to a threshold μ_r value.

Fig. 13

Effect of μ_r on CSL in a e-log(p′) space, b projected data in e-log(p′) space and c q − p′ space

6.3 Rolling resistance and associated micro-mechanics for CSL

Both linear [22] and power [58] functions have been used to express the relationship between CN and p′ at CS. Providing a better fit for the data in this study, a power function was used:

$$CN_{cs} = A + B \times \left( {\frac{{p^{\prime}}}{{p_{a} }}} \right)^{C }$$

(9)

where CN_cs is the CN at CS and A, B and C are fitting constants. A strong relation between p′ and CN_cs for each μ_r value, irrespective of drainage condition was observed (see Fig. 14a). For any given p′_cs value the CN_cs decreased with increasing μ_r. In Fig. 14b, the contact networks of two assemblies, TRR10 and T14, which possess μ_r values of 0 and 0.3, but similar p′_cs values reflect this change in contact density. The contact network for TRR10 relies on a larger number of weak contacts to achieve a similar CS strength when compared to T14. For specimen TRR10, p′_cs = 363 kPa is achieved via widespread weak contact forces which may be easily displaced or rearranged, whilst T14 achieves p′_cs = 354 kPa through the efficient formation of strong contact forces which possess larger shear resistance. Ultimately, a change in rolling tendencies substantially impacts fabric related entities such as contact orientation and contact intensity at CS.

Fig. 14

a Influence of μ_r on CN_CS in CN-log (p′) space, b Specimen TRR10 (µ_r = 0) contact force network, c Specimen T14 (µ_r = 0.3) contact force network

For all CV and drained simulations, the F_vM(cs) behaviour was captured by CSLs in F_vM − log p′ space (Fig. 15). No influence of drainage condition on observed CS fabric behaviour. Firstly, it is encouraging that CS fabric behaviour is analogous with overall observed CS behaviour, e.g. CSL in e − log p′ space—further showing fabric is highly influential on soil behaviour and advancing the theory that fabric should be encapsulated under CSSM framework [41]. The CSL in F_vM − log p′ space shifted upwards with μ_r. Huang et al. [26, 28] observed identical behaviour, with the case of μ. The increasing anisotropy with μ_r may be explained by the intensification of strong contact formation (observed in Fig. 7), which primarily contributes to the deviatoric stress (and therefore likely deviatoric fabric) component of the assembly [52, 72].

Fig. 15

Influence of μ_r on CS behaviour in F_vM − log(p′) space

7 Particle rotation and angular velocity

In PFC^3D, the angular velocity, which represents rotational tendency of individual particles, can be captured throughout shearing. At prescribed ε_q values, the angular velocity for each particle was recorded and the average angular velocity, ω_avg of the entire assembly was calculated. In Fig. 16, the variation of ω_avg and average angular velocity at critical state, ω_cs for a series of drained shearing tests (with similar initial states, i.e. p′₀ = 100 kPa and e₀ = 0.5975 ± 0.0035) are plotted against rolling resistance. A very strong relationship between μ_r and ω_avg was observed. Low rolling resistance corresponded to high particle rotations and vice versa. The observations indicate ω_avg can be used within DEM to quantify rolling behaviour. From herein, ω_avg is used as an indirect measure to capture particle rolling tendencies.

Fig. 16

Particle angular velocity as a function of μ_r

In Fig. 17a, the influence of e₀ and p′₀ on the rotational tendencies of granular material is assessed. All specimens had μ_r = 0.1 and exhibited NF behaviour (Fig. 17b). Although unpronounced, a trend in Fig. 17 is that during strain hardening, ω_avg increases. Further, particle rolling was observed to increase with e₀ (on comparison between TRR34 and TRR35), i.e. rolling is more prone to occur in loose specimens. The influence of p′₀ on rolling tendencies (on comparison of TRR35 and TRR38) is negligible.

Fig. 17

a Influence of ω_avg on e₀ and p′₀ b strain hardening behaviour of TRR34, TRR35 & TRR38

In Fig. 18, the variation of ω_avg is plotted against the post-compression state parameter, ψ₀ for μ_r = 0.1 and μ_r = 0.3. Notably, contractive specimens with higher ψ₀ corresponded to large rotational/rolling behaviour. In dilative specimens rolling/rotational behaviour still existed although tendencies were minimised. The trend line shifts downwards in the ω_avg − ψ₀ space as μ_r increased. Thus, depending on μ_r, each ω_avg − ψ₀ possesses its own characteristic rolling tendency.

Fig. 18

Influence of rotational tendencies on material state

8 Influence of rolling resistance on instability state

Instability triggering is a state which corresponds to the peak q in undrained shearing before strain softening occurs and is also known to be the onset of static liquefaction. The instability state is a characteristic feature of undrained behaviour of loose sand. It has been characterised using approaches, such as the collapse line (CL) [82] or the instability line (IL) [13, 37]. Perhaps the instability stress ratio, η_IS = (q/p′)_peak is better suited to quantify instability behaviour as it can be linked with mechanical performance parameters such as ψ₀ [53, 87, 39]. Yang [87] developed an exponential relationship and Rahman and Lo [53] modified the relationship between η_IS and ψ in the form of:

$$\eta_{IS} = A + \exp \left( { - B\psi } \right)$$

(10)

where A and B are curve fitting parameters. Equation (10) captures the instability behaviour as a function of soil state. In Fig. 19a the relationship between η_IS and ψ₀ for varying μ_r values is captured. The instability data points in the η_IS − ψ₀ space showed a clear upward shift with increasing μ_r. However. in the free-rolling case, a steeper reduction in η_IS occurs than those assemblies mobilised with a finite μ_r. Through a reanalysis of DEM data, Nguyen et al. [59] asserted a steep slope in η_IS − ψ₀ space to be attributed to low particle angularity and interlocking capabilities. Aligning with the force chain networks in Fig. 7 it can be suspected that the steep behaviour shown for μ_r = 0 is also due to poor interlocking and highly lubricative behaviour. Therefore, a link between particle angularity and rolling resistance share a relation may be made between the two studies. That is, it is possible that μ_r can be used as a tuneable parameter in DEM to indirectly control particle angularity.

Fig. 19

Influence of particle rolling on instability behaviour a η_IS − ψ₀, b F_vM(IS) − ψ₀ spaces

When the F_vM at η_IS, F_vM(IS) is plotted against ψ₀ (Fig. 19b), two separate relationships are yielded. Following on from Fig. 19a, when μ_r = 0, a steeper decrease of F_vM(IS) occurs with increasing ψ_0, in comparison to those assemblies mobilised with a finite μ_r value. Thus, for μ_r = 0 the behaviour observed in η_IS − ψ₀ is reflected in F_vM(IS) − ψ₀ space. Again, this behaviour could be attributed to the associated low interlocking capabilities. Secondly, when μ_r > 0 the influence of μ_r on F_vM(IS) appears to be nullified. In fact, a single trend line can be approximated to represent the trend in F_vM(IS) − ψ₀.

9 Influence of rolling resistance on PT and ChS

Both PT and ChS states correspond to the change over from contractive to dilative tendency and, therefore, they can be assumed as a state of zero dilatancy and equivalent. Thus, the dilatancy equation of SANISAND model [43] can be manipulated to derive the following relation between stress ratio at PT (η_PT) and ChS (η_ChS) with ψ –

$$\eta_{PT/ChS} /M = \exp \left( { - B\psi } \right)$$

(11a)

However, some experimental studies suggested the equivalence between η_PT and η_ChS [12, 40], whilst others observed different behaviour [92]. Therefore, the equivalence of these states and the functional relation has significant importance in granular material modelling.

It is found in Fig. 20a that assemblies with large rolling resistance possesses larger η_PT or η_ChS at the same ψ₀, however both η_PT and η_ChS possess single relation with ψ₀ for each μ_r i.e. η_PT and η_ChS are equivalent. When η_PT and η_ChS were normalised by M (Fig. 20b), a single trend line is developed, further indicating the equivalence of η_PT and η_ChS, and support normalisation in constitutive relation. However, the best fit function is slightly different than Eq. (11) and presented as below:

Fig. 20

Influence of particle rolling in a η_PT/ChS − ψ₀, b η_PT/ChS/M − ψ₀ and c F_vM(PT/ChS)/M − ψ₀ spaces

$$\eta_{PT/ChS} /M = A\exp \left( { - B\psi } \right)$$

(11b)

While this observation confirms the relationship between characteristic features with ψ₀ in the state-dependent constitutive modelling [43], it also indicates possible adjustment. The deviatoric fabric at PT and ChS states, F_vM(PT/ChS) (normalised with M) also shared a unique relation with ψ₀, irrespective of μ_r (Fig. 20c).

10 Conclusions

A series of CV and drained triaxial simulations were undertaken using the RRLCM within PFC^3D. The influence of particle rolling on clean sand behaviour was captured through the CSSM framework. The key conclusions drawn from the study are summarised below:

• i. Rolling resistance has significant influence on the observed behaviour of granular material. Assemblies mobilised with high rolling resistance possessed highly frictional behaviour with dilative tendencies. Assemblies with low rolling resistance manifested less dilative tendencies. Thus, the dilative tendency shared a dependency with μ_r. Although, at approximately μ_r > 0.3, the influence of μ_r on volume dilation and peak dilatancy was almost unnoticeable.

• ii. Strong contact formation thrived for assemblies mobilised with large rolling resistance inferring that interlocking behaviour is influenced by rolling tendencies/resistance. A potential by-product of the strong contact formation is strain softening or brittle failure after peak deviatoric stress, q. Assemblies mobilised with low rolling resistance were incapable of forming strong interlocking contacts and instead yielded contact networks that were rife with weak contact forces. μ_r could therefore be used as a parameter to indirectly control the interlocking capabilities of a granular assembly.

• iii. The influence of μ_r on characteristic features of undrained shearing (i.e. PT and η_IS) and drained shearing (i.e. ChS) were captured. For dense/medium-dense specimens, initial contractive tendencies were prolonged in terms of deviatoric strain (ε_q) for assemblies mobilised with large rolling resistance. The stress ratio (q/p′) at PT, ChS and instability states, i.e. η_PT, η_Ch and η_IS respectively, were all shown to increase with μ_r. η_PT, η_ChS and η_IS also showed correlations with ψ₀ for different values of μ_r. The study observed an equivalence between η_PT and η_ChS.

• iv. CS parameters were significantly influenced by particle rolling. The CSL in e-log p′ space shifted upwards when rolling resistance increased. When μ_r was large, the curvature of the CSL became increasingly pronounced. Thus, the geometry of the CSL was influenced by rolling behaviour rather than particle breakage, which was suggested in the literature. An increase of M was observed with μ_r. Although at large ranges of μ_r, even though M continued to increase, changes in other mechanical entities such as F_vM, CN, q and the CSL in e-log p′ space were mostly unnoticeable.

• v. A larger deviatoric fabric was observed with increasing μ_r. F_vM(IS) and F_vM(PT) showed relationships with initial state parameter (ψ₀) with respect to μ_r.

Although the study presents valuable DEM findings relating to the influence of particle rolling on the characteristic (undrained and drained) behaviour of granular material, it should be noted that several rolling resistance models/frameworks exist, as noted in Ai et al. [2]. Utilising a rolling resistance model/algorithm which differs from that used in this study may lead to observed behaviour which diverges from the observations in this study. The influence of particle rolling on soil behaviour has been isolated in the manuscript, although sliding and twisting behaviour which occurs at the contact may also influence granular material behaviour.

Appendices

Appendix A: Method of deformability

1.1 Computation of stiffness parameters

Through the method of deformability:

$$k_{n} = \frac{{AE^{*} }}{L}$$

(12)

$$k_{s} = \frac{{k_{n} }}{{k^{*} }}$$

(13)

$$k_{r} = k_{s} R_{r}$$

(14)

here A = πr² represents the contact area. Depending on the type of entities in contact r is computed:

$$r = \left\{ {\begin{array}{*{20}c} {\hbox{min} \left( {R^{\left( 1 \right)} , R^{\left( 2 \right)} } \right), \,for\, \space particle - particle \space \,contacts } \\ {R^{\left( 1 \right)} , \,for \space\, particle - wall \space\, contacts} \\ \end{array} } \right.$$

(15)

where $R^{\left( 1 \right)}$ denotes the radius of contact entity 1. In similar nature the contact length (L) is computed based on contact type:

$$L = \left\{ {\begin{array}{*{20}c} {R^{\left( 1 \right)} + R^{\left( 2 \right)} , \,for \space\, particle - particle \space\, contacts} \\ {R^{\left( 1 \right)} , \,for \space\, particle - wall \space\, contacts} \\ \end{array} } \right.$$

(16)

E^* and k^* = k_n/k_s are input parameters in the study and are discussed and presented in Table 1 in the manuscript. The rolling radius, R_r and its formulation is also presented in the manuscript.

Appendix B: Rolling resistance linear contact model

The rolling resistance linear contact model (RRLCM) utilised in this study adds to the conventional linear contact model commonly used in DEM through the installation of a rolling spring and dashpot at the contact (Fig. 21).

Fig. 21

Rolling resistance linear contact model. After Iwashita and Oda [32]

In a rheological sense, the rolling spring signifies the presence of an elastic resisting moment between contacting pieces ($M_{r}^{k} )$, whilst the dashpot signifies the presence of a viscous moment at the contact $(M_{r}^{d} )$. The overall rolling resistance moment (M_r) may be mathematically defined as:

$$M_{r} = M_{r}^{k} + M_{r}^{d}$$

(17)

to effectively utilize the RRLCM in a numerical modelling environment, M_r must be updated incrementally with respect to the time step. At time t + Δt, $M_{r}^{k}$ may be computed via:

$$M_{r, \Delta t + t}^{k} = \left\{ {\begin{array}{*{20}c} {M_{r, t}^{k} + \Delta M_{r}^{k} } \\ { \left| {M_{r, t + \Delta t}^{k} } \right| \le M^{*} } \\ \end{array} } \right.$$

(18)

here M* represents the maximum (limiting) resisting torque ($M^{ *} = \mu_{r} \bar{R}F_{n}$) and $\bar{R} = \left( {r_{i} r_{j} } \right)/\left( {r_{i} + r_{j} } \right)$; where F_n is the normal contact force, $\bar{R}$ represents the effective radius of the contact, $r_{i}$ and $r_{j}$ are the radii of contacting entities. When a boundary or wall element is the contacting entity, r → ∞. Notice, when $\mu_{r} = 0$, $M^{ *} = 0$ and therefore a free-rolling environment is created. $M_{r}^{k}$ is the incremental rolling resistance torque observed at time, t + Δt and is computed through consideration of the rolling stiffness, k_r and the relative rotation between two contacting particles, θ_r,

$$\Delta M_{r}^{k} = - k_{r} \Delta \theta_{r}$$

(19)

where k_r = k_sR_r. A limitation of the model is defining a k_r which has strong physical basis. k_r is related to k_s based on an idealized consideration that the moment generated at a contact due to shear displacement is equivalent to the moment generated due to rolling displacement. $M_{r}^{d}$ in Equation 17 is also updated with respect to the time step, t + Δt:

$$M_{r, t + \Delta t}^{d} = \left\{ {\begin{array}{*{20}c} { - C_{rr}\, if \,\left| {M_{r, t + \Delta t}^{k} } \right| < M^{ *} } \\ { 0 \,if \,\left| {M_{r, t + \Delta t}^{k} } \right| = M^{ *} } \\ \end{array} } \right.$$

(20)

where $C_{r} = 2_{r} \sqrt {I_{r} k_{r} }$ is the viscous damping rolling coefficient, η_r is the critical viscous damping ratio and I_r is the equivalent moment of inertia about the contact point between two contacting entities. Instead of an oscillating resisting moment applied at the contact, as applied in some rolling resistance contact models [2], the applied resisting moment at the contact within a quasi-static system is stable and therefore the packing behaviour of the assembly is stabilized. For such reasons, this model is often used in DEM simulation of quasi-static systems. Using various modified versions of this model in DEM along with the tuning of μ_r, some have captured the influence of particle rolling on the behaviour of granular material in triaxial compression. In particular some observed that with the addition of rolling resistance, shear strength and dilatancy increases [32, 46, 54, 94], shear banding and strain localization intensifies [32, 33, 54, 84] and fabric anisotropy intensifies [46].