PROCEEDINGS OF THE 8TH INTERNATIONAL CONFERENCE ON DISCRETE ELEMENT METHODS (DEM8)

CRITICAL STATE BEHAVIOUR OF GRANULAR MATERIALS

Md. Mizanur Rahman1, Hoang B. K. Nguyen2, Hung-Chun Wang3 and Md. Arif Shahriar4

1Associate Professor

School of Natural and Built Environment, University of South Australia Email: Mizanur.Rahman@unisa.edu.au

2Research Associate School of Natural and Built Environment, University of South Australia

Email: Khoi.Nguyen@unisa.edu.au 3Masters student

School of Natural and Built Environment, University of South Australia Email: wanhy118@mymail.unisa.edu.au

4Department of Civil Engineering, Bangladesh University of Engineering & Technology Email: shahriar.buet.civi14@gmail.co

Keywords: DEM; consolidation; shearing behaviour; cyclic loading.

Abstract: The behaviour of granular materials has been widely examined under laboratory conditions; however, the real micro-mechanics behind the observed behaviour has not been fully understood. Hence, discrete element method (DEM) was adopted for investigating the behaviour of granular materials under the critical state soil mechanics (CSSM) framework. It was found that DEM is able to simulate stress and strain path tests (constant volume, cyclic loading, anisotropic consolidation, etc.), which are difficult to conduct in laboratory condition, and produces the qualitative response of granular materials i.e. captures important characteristic features of shearing response. The critical state (CS) data from a series of triaxial simulations formed a unique critical state line (CSL) in the classical e-log(p′) space, regardless of consolidation and drainage conditions. The CSSM framework was also found suitable for synthesizing both monotonic and cyclic behaviour in DEM. The uniqueness of CSL was also observed for micro-mechanical quantities such as number of contacts and fabric anisotropy. The qualitative correlations between macro- and micro-parameters, which are anchor concepts for soil modelling, were established.

1 INTRODUCTION The behaviour of granular materials has been investigated in laboratory triaxial testing

program, which assesses the contractive/dilative tendency or liquefaction susceptibility. The drained response of granular materials has been commonly classified as contractant (C) or dilatant (D) behaviour. A loose specimen often exhibits C behaviour, whereas D behaviour is commonly observed in a dense specimen. When shearing loose specimen, volumetric contraction with deviatoric stress (q) hardening occurs throughout this stage until reaching an equilibrium state i.e. dεv=dq=0, where εv is volumetric strain, q=(σʹ1-σʹ3), σʹ1 and σʹ3 are major and minor effective principal stresses respectively. For D behaviour, q hardening takes place until reaching initial peak strength, and then q softens toward an equilibrium. In this case, the dense specimen initially exhibits volumetric contraction and then phase-transforms

to volumetric dilation. However, under constant volume (undrained condition) a loose specimen (contractive) associated with strain softening after initial strain hardening, whereas dense specimen (dilative) only exhibited strain hardening. Note, the start of strain softening under undrained condition is also known as the triggering of liquefaction.

An equilibrium state of effective stress or volumetric strain at later stage of shearing is often called critical state (CS), which is an anchor concept of critical state soil mechanics (CSSM) framework [1-4]. The CS line (CSL), as defined by a set of CS data points in e-log(pʹ) space, is also a reference state for modelling contractive and dilative behaviour. Contractive behaviour is observed for a soil state above the CSL and dilative behaviour are observed for a state below the CSL; where e is void ratio and pʹ is mean effective stress. The state parameter (ψ), which is the difference between e at current state and e on the CSL at the same pʹ [5], is a mathematical expression of such a state index and can be easily implemented in a constitutive model formulation. It has been assumed that the CSL is unique irrespective of drainage/shearing conditions. While many experimental studies suggest that the CSL is independent of drainage conditions [4, 6, 7], some studies found a difference in CSLs from drained and undrained triaxial compression tests [8]. In addition, some studies have shown that the CSL may be affected by consolidation condition [9-11], while CSL determined in other studies showed no difference [12-14]. It would be desirable to develop a large dataset to enhance our qualitative understanding of the link between these conditions.

Discrete element method (DEM) has recently become a common approach for qualitative understanding of soil behaviour under a CSSM framework [15-18], as it does not suffer from experimental limitation of non-uniformity and allows the understanding of internal structure of soil. There have been previous studies on CS behaviour of the assembly of idealized particles e.g. discs or balls [16, 17, 19]; however, natural soils often have some degree of angularity. Therefore, this study adopted the assemblies of 3D ellipsoid particles to observe the response of granular material under different conditions and also capture some angularity features of real soils. Furthermore, considering the advantage of DEM, a set of cyclic loading simulations were also performed to provide more understanding about the link between monotonic and cyclic loading behaviour under CSSM framework.

2 METHODOLOGY This study conducted various simulations with different shearing and consolidation

conditions. This DEM study does not associate with fluid mechanics. Therefore, the typical undrained condition in the laboratory were replicated by maintaining constant volume. Note, the simulations in this study were named in the following manner: ‘IC’= Isotropic consolidation; ‘K0C’= K0 consolidation; ‘CV’= Constant volume condition; ‘CD’= Conventional drained condition; ‘CCV’= Cyclic constant volume condition. The specimen of ellipsoid particles was generated within periodic boundary (see Fig. 1). All these particles were assigned a normal contact stiffness of 108 N/m, a ratio of tangential to normal contact stiffness of 1.0, a coefficient of friction at particle contacts of 0.50 and the coefficient of (rotational and translational) body damping of 0.05. A linear contact model and periodic boundary condition were used in this study.

Fig. 1. Specimen of around 5000 ellipsoid particles.

3 NUMERICAL STABILITY OF SIMULATION

To achieve a quasi-static condition in DEM simulations, micro-mechanical measures such as inertial number (I), pseudo-static measures and specimen size, have been commonly investigated in the literature [20, 21]. The inertial number (I) can be defined by the equation,

𝐼 = 𝜀𝑑! 𝜌!/𝑝′; where 𝜀 is shear strain rate (1/s), d is the particle diameter (m), pʹ is the

mean stress (kPa), and ρg is the particle density (kg/m3). The calculated I values should be smaller than the recommended value of 0.003 for quasi-static conditions [21].

The study adopted OVAL [22] for all DEM simulations. OVAL does not directly consider the conventional I. However, two variables (χ1 and χ2) were checked to determine the numerical stability of the simulation in OVAL; where χ1 is the average force imbalance on a particle divided by the average magnitude of a contact force; χ2 is the average moment imbalance on a particle divided by both the average magnitude of a contact force and the average particle radius. For the applied strain rate, the highest value χ1 and χ2 were below the recommended value of 0.005 [22] and considered that simulations were quasi-static.

The time step of this study was 1; and the particle density (ρg) was 9.766x108 (kg/m3), which is approximately 3.69x105 times larger than 2650 kg/m3 for real geo-material as reported in Perez, Kwok [20]. To optimize the simulation time, the particle density was scaled up in the literature, notably Thornton [23] used a scale-up factor of 1012. A similar scale-up approach was also observed in Ng [24]. The strain rate of 0.00001% per time step was found as suitable for all DEM simulations [15, 25].

4 EFFECT OF CONSOLIDATION CONDITION

Two simulations with similar initial states (e0=0.511-0.514, pʹ0=480kPa) but different consolidation condition (i.e. isotropic and K0) were performed to assess the effect of consolidation condition on the CS behaviour. The initial state parameter (ψ0) of two simulations are quite close i.e. -0.103 to -0.106.

Fig. 2. Effect of consolidation condition on soil behaviour in a) q-εq, b) q-pʹ, c) CN-εq, d)

CN-pʹ, e) FvM-εq and f) FvM- pʹ spaces.

There are some discrepancies between the two simulations in the earlier stage. However, the stresses (q and pʹ) in Figs. 2a and b coincides when they were approaching CS. Similarly, the micro-mechanical quantities such as coordination number (CN) and von Mises fabric (FvM) reached similar values at CS in Figs. 2c to 2f. Note, the micro-mechanical quantities came to the same CSL for such specimen of ellipsoids, which was reported in Nguyen, Rahman [26]. This finding confirmed that CSL is unique regardless of consolidation history. It should be noted that the discrepancies in earlier strains may contribute to different characteristic features of soil behaviour (e.g. phase transformation, instability). But the CS is still unique.

5 EFFECT OF SHEARING CONDITION

a) b)

CS points

Start of shearing

c) d)

𝐶𝑁 = 3.91 + 0.36 ×𝑝ζ𝑝-

..//CSL

CS points

Start of shearing

e) f)

CS zone(Nguyen et al. 2018)

Start of shearing

CS points

Two simulations with similar initial states (e0=0.538, pʹ0=50kPa) but different shearing condition (i.e. constant volume and conventional drained) were performed to assess the effect of shearing condition on the CS behaviour. The initial state parameter (ψ0) of two simulations is both -0.130. IC-CV-01 exhibited strain hardening during constant volume shearing, whereas IC-CD-01 showed strain softening after initial strain hardening during conventional drained shearing. Eventually, the two simulations reached different CS values (see Fig. 3a).

Fig. 3. Effect of shearing/drainage condition on the soil behaviour in a) q-εq, b) q-pʹ, c) CN-

εq, d) CN-pʹ, e) FvM-εq and f) FvM- pʹ spaces. Due to the nature of the shearing condition, the observed behaviours were different.

However, both simulations were approaching the same CSL at the end in Fig. 3b. Similarly, the CS values for CN and FvM were also different due to the nature of shearing condition. However, these micro-mechanical quantities at CS lied on the same CSL in Figs. 3d and f. This also confirmed the uniqueness of CSL for a granular material.

a) b)

CS points

c) d)

CS points

𝐶𝑁 = 3.91 + 0.36 ×𝑝ζ𝑝-

..//CSL

Start of shearing

e) f)

CS points

CS zone(Nguyen et al. 2018)

6 CSSM FRAMEWORK

Fig. 4 shows unique CSLs in different spaces i.e. e-log(pʹ), q-pʹ, CN-pʹ and FvM-pʹ. Note, the data for FvM at CS did not form a unique line with pʹ. However, the results lied in a narrow zone.

Fig. 4. CSL for different consolidation and shearing conditions under monotonic triaxial

loading in the (a) e-log(pʹ), (b) q-pʹ, (c) CN-pʹ and (d) FvM-pʹ spaces.

7 CYCLIC LOADING BEHAVIOUR The triaxial cyclic loading simulations with cyclic deviatoric stress (qamp) symmetrical

with q=0kPa were simulated to develop relation between cyclic stress ratio (CSR=qamp/2pʹ0) and the number of cycles required for initial liquefaction (NL). Fig. 5 showed the simulations, IC-CCV-01 and IC-CCV-02, for the same specimens but with different qamp.

Fig. 6a showed the effect of initial states i.e. e0 on the CSR. One set of simulations with the same initial states formed one unique line in the CSR-NL space. Higher e0 gave lower CSR. However, the relationship between CSR and NL is still highly dependent on the state of soil. Therefore, a value of CRR20, CSR at 20 cycles, was defined for each set with the same initial state. CRR20 formed a unique exponential relationship with ψ0.

𝑒 = 0.68 − 0.02 ×𝑝ζ𝑝+

,.-.

a) b)

CSL (M=0.90)

𝐶𝑁 = 3.91 + 0.36 ×𝑝ζ𝑝-

..//CSL

c)

CS zone

d)

Fig. 5. Cyclic loading behaviour in a) q-εq, b) q-pʹ, c) CN-pʹ and d) FvM- pʹ spaces.

Fig. 6. CSSM framework for cyclic liquefaction assessment: (a) CSR- NL and (b) CRR-ψ0.

8 CONCLUSIONS This study adopted DEM to investigate the behaviour of granular material under different conditions and to validate the existing CS behaviour under the CSSM framework. The findings in this study were listed as the followings:

• The CSL was confirmed to be unique regardless of consolidation (isotropic or K0) or shearing (drained or undrained) conditions. The micro-mechanical quantities also formed a unique relationship with the macro-mechanical parameter i.e. pʹ. This also validated the link between micro- and macro- mechanical parameters.

a)

17th cycle

35th cycle

b)

Liquefaction initiation(pʹ <10% of pʹ0)

Start of butterfly shape(mobility) – IC-CCV-01

Start of butterfly shape(mobility) – IC-CCV-02

CSL

c)

Liquefaction initiationCN≈4

d)

a) b)

𝐶𝑅𝑅#$ = 0.106 exp (−17Φ$) + 0.143

• The cyclic resistance i.e. CRR of granular material can be predicted by using the state parameter, which was defined by the CSL from monotonic loading simulations.

REFERENCES: [1]. Lashkari, A., et al. (2019), Instability of Particulate Assemblies under Constant Shear

Drained Stress Path: DEM Approach. International Journal of Geomechanics, 19(6). [2]. Rabbi, A.T.M.Z., M.M. Rahman, and D.A. Cameron (2019), Critical State Study of

Natural Silty Sand Instability under Undrained and Constant Shear Drained Path. International journal for geomechanics, 19(8): p. 04019083.

[3]. Zhang, J., et al. (2018), Characterizing Monotonic Behavior of Pond Ash within Critical State Approach. Journal of Geotechnical and Geoenvironmental Engineering, 144(1): p. 04017100.

[4]. Been, K., M. Jefferies, and J. Hachey (1991), The critical state of sands. Geotechnique, 41(3): p. 365-381.

[5]. Been, K. and M.G. Jefferies (1985), A state parameter for sands. Géotechnique, 35(2): p. 99-112.

[6]. Sladen, J., R. D'hollander, and J. Krahn (1985), The liquefaction of sands, a collapse surface approach. Canadian Geotechnical Journal, 22(4): p. 564-578.

[7]. Bobei, D.C., et al. (2009), A modified state parameter for characterizing static liquefaction of sand with fines. Canadian Geotechnical Journal, 46(3): p. 281-295.

[8]. Yamamuro, J.A. and P.V. Lade (1998), Steady-State Concepts and Static Liquefaction of Silty Sands. Journal of Geotechnical and Geoenvironmental Engineering, 124(9): p. 868-877.

[9]. Finno, R.J. and A.L. Rechenmacher (2003), Effects of Consolidation History on Critical State of Sand. Journal of Geotechnical and Geoenvironmental Engineering, 129(4): p. 350-360.

[10]. Fourie, A.B. and L. Tshabalala (2005), Initiation of static liquefaction and the role of K0 consolidation. Canadian Geotechnical Journal, 42(3): p. 892-906.

[11]. Rabbi, A.T.M.Z., M.M. Rahman, and D.A. Cameron (2018), Undrained behavior of silty sand and the role of isotropic and K0 consolidation. Journal of Geotechnical and Geoenvironmental Engineering, 144(4): p. 04018014.

[12]. Chu, J. and D. Wanatowski (2008), Instability Conditions of Loose Sand in Plane Strain. Journal of Geotechnical and Geoenvironmental Engineering, 134(1): p. 136-142.

[13]. Rahman, M.M., H.B.K. Nguyen, and A.T.M.Z. Rabbi (2018), The effect of consolidation on undrained behaviour of granular materials: A comparative study between experiment and DEM simulation. Geotechnical Research, 5(4): p. 199-217.

[14]. Nguyen, H.B.K., M.M. Rahman, and A.B. Fourie (2017), Undrained behaviour of granular material and the role of fabric in isotropic and K0 consolidations: DEM approach. Géotechnique, 67(2): p. 153-167.

[15]. Nguyen, H.B.K., et al., The effect of consolidation path on undrained behaviour of sand - a DEM approach, in Computer Methods and Recent Advances in Geomechanics. 2015, CRC Press. p. 175-180.

[16]. Huang, X., et al. (2014), Discrete-element method analysis of the state parameter. Geotechnique, 64(12): p. 954-965.

[17]. Zhao, J. and N. Guo (2013), Unique critical state characteristics in granular media considering fabric anisotropy. Géotechnique, 63(8): p. 695-704.

[18]. Kuhn, M.R. (2016), The critical state of granular media: convergence, stationarity and

disorder. Géotechnique, 66(11): p. 902-909. [19]. Sitharam, T. and J.S. Vinod (2009), Critical state behaviour of granular materials from

isotropic and rebounded paths: DEM simulations. Granular matter, 11(1): p. 33-42. [20]. Perez, J.C.L., et al. (2016), Exploring the micro-mechanics of triaxial instability in

granular materials. Géotechnique, 66(9): p. 725-740. [21]. da Cruz, F., et al. (2005), Rheophysics of dense granular materials: Discrete simulation

of plane shear flows. Physical Review E, 72(2): p. 021309. [22]. Kuhn, M.R., OVAL and OVALPLOT: Programs for analyzing dense particle

assemblies with the discrete element method. 2006, 98p. [23]. Thornton, C. (2000), Numerical simulations of deviatoric shear deformation of

granular media. Géotechnique, 50(1): p. 43-53. [24]. Ng, T. (2006), Input Parameters of Discrete Element Methods. Journal of Engineering

Mechanics, 132(7): p. 723-729. [25]. Nguyen, H.B.K. and M.M. Rahman (2017), The role of micro-mechanics on the

consolidation history of granular materials. Australian Geomechanics Journal, 52(3): p. 27-35.

[26]. Nguyen, H.B.K., M.M. Rahman, and A.B. Fourie (2018), Characteristic behaviour of drained and undrained triaxial tests: A DEM study. Journal of Geotechnical and Geoenvironmental Engineering, 144(9): p. 04018060.