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ARTICLE IN PRESSG ModelARTIC-803; No. of Pages 13Particuology xxx (2015) xxx–xxx

Contents lists available at ScienceDirect

Particuology

j our na l ho me page: www.elsev ier .com/ locate /par t ic

uantified evaluation of particle shape effects from micro-to-macrocales for non-convex grains

. Yanga, J.F. Wangb, Y.M. Chenga,∗

Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, ChinaDepartment of Civil and Architectural Engineering, City University of Hong Kong, Hong Kong, China

r t i c l e i n f o

rticle history:eceived 28 July 2014eceived in revised form0 November 2014ccepted 30 January 2015

eywords:uantitative analysis

a b s t r a c t

Particle shape plays an important role in both the micro and macro scales responses of a granular assem-bly. This paper presents a systematic way to interpret the shape effects of granular material duringquasi-static shearing. A more suitable shape descriptor is suggested for the quantitative analysis ofthe macroscale strength indexes and contact parameters for non-convex grains, with special consid-eration given to the peak state and critical state. Through a series of numerical simulations and relatedpost-processing analysis, particle shape is found to directly influence the strain localisation patterns,microscale fabric distributions, microscale mobilisation indexes, and probability distribution of the nor-

hape factortatistical analysisicro–macro indexes

malised contact normal force. Additionally, the accuracy of the stress–force–fabric relationship can beinfluenced by the average normal force and the distribution of contact vectors. Moreover, particle shapeplays a more important role than do the confining pressures in determining the friction angle. Strong forcechains and the dilation effect are also found to be strongly influenced by the high confining pressure.

© 2015 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy ofSciences. Published by Elsevier B.V. All rights reserved.

ntroduction

Particle shape can directly influence the structural featuresf granular assemblies, which ultimately control the mechanicalroperties of the granular material. Generally speaking, the peakriction angle and the peak dilation angle of circular disks (2D)r spheres (3D) are significantly lower than those of natural sand,hich has an irregular particle shape. Many previous studies have

imed to determine the shape effect of granular materials. Twoain approaches exist for investigating the shape effect. The first

pproach is to establish the rolling resistance model (Iwashita &da, 1998; Jiang, Leroueil, Zhu, Yu, & Konrad, 2009; Jiang, Yu, &arris, 2005). However, the real rotation mechanism (instead ofn artificial mechanism) is constrained by the particle’s geometry,hich may induce an asymmetric stress tensor for an individualarticle. The second approach generates an appropriate model ofhe grain shape, in which the modelling geometry of the irregulararticles can be divided into two main groups by contact: (1)

Please cite this article in press as: Yang, Y., et al. Quantified evaluation ofgrains. Particuology (2015), http://dx.doi.org/10.1016/j.partic.2015.01

smooth-convex shape and (2) a non-convex shape. Smooth-onvex particles can be generated using arbitrary functions oruperquadric formulations based on previous work. The simplest

∗ Corresponding author. Tel.: +852 27666042.E-mail address: [email protected] (Y.M. Cheng).

ttp://dx.doi.org/10.1016/j.partic.2015.01.008674-2001/© 2015 Chinese Society of Particuology and Institute of Process Engineering, C

smooth-convex shape is an ellipse. An ellipse shape has been usedin many studies (Ng, 1994; Rothenburg & Bathurst, 1992). Thenon-convex particles can be formed by polygonal (Mirghasemi,Rothenburg, & Matyas, 2002; Seyedi Hosseininia, 2012) or by com-bining clusters (Abedi & Mirghasemi, 2011; Lu & McDowell, 2007;Jensen, Bosscher, Plesha, & Edil, 1999). Some advanced engineeringtechniques (Digitial, SEM, and X-ray) and robust algorithms havealso been applied to establish realistic microscale particle geome-try for the three-dimensional (3D) condition (Fu et al., 2006, 2012;Wang, Park, & Fu, 2007a; Alonso-Marroquín & Wang, 2009; Ferellec& McDowell, 2010; Liu et al., 2013; Williams, Chen, Weeger, &Donohue, 2014). There is no doubt that real physical grains are 3D ingeometry; however, more artificial assumptions are required in therealistic geometry algorithms. Additionally, 3D simulations requiresignificantly higher-performance devices, or even parallel analysis.Many previous studies have demonstrated that a 2D discrete ele-ment model can adequately capture various complex mechanicalfeatures of granular materials (Rothenburg & Bathurst, 1989, 1992;Ng, 1994; Luding, 2005; Jiang, Yu, & Harris, 2006; Wang, Dove, &Gutierrez, 2007b; Abedi & Mirghasemi, 2011; Seyedi Hosseininia,2012, 2013; Zhou, Huang, Wang, & Wang, 2013; Jiang, Chen, Tapias,

particle shape effects from micro-to-macro scales for non-convex.008

Arroyo, & Fang, 2014). Moreover, the visual deformation patternsand force chains are easily captured by 2D analysis; therefore, thisstudy uses 2D numerical simulations, which are sufficient for thefundamental study of the physical and mechanical properties of

hinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

dx.doi.org/10.1016/j.partic.2015.01.008dx.doi.org/10.1016/j.partic.2015.01.008http://www.sciencedirect.com/science/journal/16742001http://www.elsevier.com/locate/particmailto:[email protected]/10.1016/j.partic.2015.01.008

IN PRESSG ModelP2 uology xxx (2015) xxx–xxx

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etowftimtfi

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fi

ri

Fig. 1. (a) Schematic of the particle shapes; (b) illustration of different particle shapeindexes.

Table 1Quantitative value of the shape indexes.

Category Aspect ratio Circularity SF (%) AF (%)

Circular 1.0 1.0 0 0Elongated1 0.952 0.952 4 0.1Elongated2 0.909 0.909 8 0.4Elongated3 0.80 0.80 19.8 1.9Elongated4 0.667 0.667 33.3 3.88

ARTICLEARTIC-803; No. of Pages 13 Y. Yang et al. / Partic

he granular assemblages. The authors here choose combining clus-ers to generate non-convex particles for the shape effect analysis,hich can lead to a better understanding of the problem.

Although non-convex particles are well recognised in the realranular world, they are still not fully understood either on aaboratory experimental scale or through simulated numerical

ethods, particularly for quantitative analysis. A quantitative anal-sis study is thus important to understand the more detailedehaviour of granular materials. Many different quantitative shapeescriptors have been proposed for this purpose in previous stud-

es (Cho, Dodds, & Santamarina, 2006; Sukumaran & Ashmawy,001; Ehrlich & Weinberg, 1970; Powers, 1953; Krumbein, 1941;adell, 1932), but very few comparisons of mechanical proper-

ies are performed on these common particle shape definitions,articularly for non-convex shapes. In the present study, eight dif-erent particle shapes are evaluated using four quantitative shapeescriptors to address this insufficiency. This methodology pro-ides an additional approach for a suitable and adequate shapendex. Next, quantitative relationships between particle shape andtrength indexes are investigated. In addition, the patterns of strainocalisation for different shapes are captured. The relationshipsetween particle shape and microscale fabric parameters are also

valuated at the peak state (peak stress ratio(�f�n

)max

and crit-

cal state (stress ratio(�f�n

)= constant). Moreover, the accuracy

f stress–force–fabric (SFF) (Christoffersen, Mehrabadi, & Nemat-asser, 1981; Rothenburg & Bathurst, 1989; Guo & Zhao, 2013; Li

Yu, 2013; Seyedi Hosseininia, 2013) is investigated, with spe-ial consideration of the accuracy of average contact normal forcesnd the distribution of contact vectors. In this paper, an evaluationf particle interlocking effects from peak state to critical state islso presented to address the particle shape effect using a prob-bilistic approach (Zhou et al., 2013). The distributions of forcehains for different shapes are also compared quantitatively by thehape descriptor. The inter-particle force network is a striking fea-ure that determines the mechanical properties of granular massRadjai, Jean, Moreau, & Roux, 1996; Sun, Jin, Liu, & Zhang, 2010). Itan be used to describe the strength variation at critical state in aicroscale model. Additionally, the authors find that the relation-

hip between the strong force network and the confining pressuresan explain the decrease in the strength indexes with the increasedonfining intensities.

This study presents a comprehensive analysis of particle shapeffects using the discrete element method (DEM), which captureshe micro mechanical behaviour of the granular assembly. Section 2f this paper describes the definitions for the particle shape indexes,hich are analysed and compared in a later section. The general

ormulation of the SFF relationship is also briefly introduced inhis section. Next, the DEM simulations and results for the biax-al drained tests are given in Section 3. Quantified macroscopic and

icroscopic responses are also interpreted by the shape index inhis section. The main conclusions of this study are presented in thenal section.

article shape description and SFF relationship

Using the DEM method, rigid particles with soft contacts can besed to reflect the contact geometry and evaluate the fundamen-al features of a cohesionless material. Seven non-convex particlesre employed in the present DEM analysis, where each irregulararticle is expanded using a standard element to prevent incorrectoments of inertia. Hence, the combined clump density is modi-

Please cite this article in press as: Yang, Y., et al. Quantified evaluation of particle shape effects from micro-to-macro scales for non-convexgrains. Particuology (2015), http://dx.doi.org/10.1016/j.partic.2015.01.008

ed and reassigned(�clump =

Vclump�diskVoverlap+Vclump

). Next, the numerical

esults are compared. The particle shape categories are shownn Fig. 1(a) and Table 1. Four simple quantitative shape indexes

Elongated5 0.50 0.5 45.8 13.2Triangular 0.957 0.722 34.5 10.4Rhombus 0.804 0.710 35.1 (N = 40) 12.9

33.6 (N = 80) 12.96

dx.doi.org/10.1016/j.partic.2015.01.008

ARTICLE IN PRESSG ModelPARTIC-803; No. of Pages 13Y. Yang et al. / Particuology xxx (2015) xxx–xxx 3

rmal v

(es2

A

C

S

A

wD

Fig. 2. Illustration of the associated contacts, contact no

Fig. 1(b)) are considered: elongation (AR), circularity, shape param-ters SF and AF. The definitions of these indexes are presented byeveral investigators (Cho et al., 2006; Sukumaran & Ashmawy,001) as follows:

R = ELminELmax

, (1a)

ircularity = Dmax-inDmin-out

, (1b)

F =∑

i∈N∣∣˛i grain∣∣N45o

100%, (1c)

F =∑

i∈N(ˇi grain − 180o

)2 − [((360o)2/N][ ] , (1d)


3(

180o)2 − ((360o)2/N

here ELmin and ELmax are the smallest axis and largest axis,max-in and Dmin-out are diameters of the largest inscribed circle and

Fig. 3. Schematic of the numerical speci

ector (n), contact force vector (f), and contact vector (l).

smallest circumscribed circle, respectively, ˛i grain is the differencein orientation between the particle chord vector and the relatedcircle, ˇi grain is the difference between 180◦ and the internal angleof the particle, and N is the number of interval points.

The detailed values for the above shapes are shown in Table 1,and the sampling interval is set at 9◦ (N = 40) to represent thedegree of angularity and the surface roughness for each shape. Asmall interval (4.5◦) is also chosen to capture the quantities of therhombus, where the difference between 4.5◦ and 4.5◦ is very small.Hence, the interval number (N = 40) is adequate for the quantitativeanalysis of the particle shape, as was suggested by the initial sup-porters of this idea (Sukumaran & Ashmawy, 2001). In this paper,different definitions for the particle shape are compared to obtainan improved curve-fitting analysis and understanding.


The definition of a micro-structural stress tensor is presentedby previous researchers (Christoffersen et al., 1981; Rothenburg &Bathurst, 1989) and is applied in this study, where the stresses atthe boundary and contact forces within the assembly are in static

men and particle size distribution.


ARTICLE IN PRESSG ModelPARTIC-803; No. of Pages 134 Y. Yang et al. / Particuology xxx (2015) xxx–xxx

F e princa

et

�

wwtntb

tcr

�

ig. 4. Macroscopic behaviour of the numerical results: (a) relationship between thnd the axial strain; (c) error bars of the stress ratio in the critical state.

quilibrium. The macroscale stress tensor for the microscale quan-ities of the contact forces and contact vectors is as follows:

ij =1V

∑c∈Nc

f ci lcj , (2)

here f ci

is the ith component of the contact force at contact cithin the assembly; lc

jis the jth component of the contact vec-

or at contact c; V is the volume of the assembly; and Nc is the totalumber of contacts. Fig. 2 illustrates these variables. Because mul-iple disks are combined to generate a clump, several contacts maye present in a collision with the studied clump.

For the orientation of the contact normal(

n = (cos �, sin �))

and

he contact tangential(

t =(−sin �, cos �

)), the equations above

an be replaced by an integral expression for the whole polaregion:


ij =NcV

2�∫0

fi(�)lj(�)E(�)

d� = NcV

2�∫0

(fn

(�)ni + ft

(�)ti)(ln

(�)nj +

ipal stress ratio and the axial strain; (b) relationship between the volumetric strain

where E(�)

is the normalised contact orientation distribution for

any angle �, and f̄i(�)

and l̄j(�)

are the average force and theaverage contact vectors associated with the same contact normal.

The second-order Fourier series expressions describe the polardistribution of the contact normal, the contact forces, and thecontact vectors (Eq. (4b)) within a loaded granular assembly. Thedensity functions are as follows (Rothenburg & Bathurst, 1989; Li& Yu, 2013; Seyedi Hosseininia, 2013):

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

E(�)

= 12�

[1 + a cos 2

(� − �a

)]f n

(�)

= f 0[1 + an cos 2

(� − �f

)]ft

(�)

= −f0at sin 2(� − �t

), (4a)


lt(�)tj)E(�)

d�, (3)


Please cite this article in press as: Yang, Y., et al. Quantified evaluation of particle shape effects from micro-to-macro scales for non-convexgrains. Particuology (2015), http://dx.doi.org/10.1016/j.partic.2015.01.008


Fig. 5. Peak friction angle (a), critical friction angle (b), and peak dilation angle (c) versus different shape quantitative indexes.



F at thes ntact

{

wmfw

ig. 6. Macroscale granular deformation observations and microscale rose diagramstrain distribution, and (c) statistical particle contact orientation and the related co

ln(�)

= l0[1 + bn cos 2

(� − �bn

)]lt

(�)

= −l0bt sin 2(� − �bt

) , (4b)


here a represents the anisotropy coefficients of the contact nor-al and �a describes the principal direction of the contact normal.

0̄ is the average normal contact force for different � with the sameeight an and at are the magnitude coefficients of the contact force

critical state (15% axial strain): (a) painted grid distribution, (b) accumulated shearnormal force.

anisotropy. �f and �t are the principal directions of the contact nor-mal and tangential forces. l̄0 is the average normal contact vectorfor different � with the same weight bn and bt are the magnitudesof anisotropy for the average contact normal and tangential vectors


�bn and �bt describe the principal directions of the average contactnormal vector and tangential vector.

The values of the anisotropy magnitudes and the principal direc-tions can be obtained from the fabric tensors and can be easily



F differa ential

cYGc&u

f

l

b2n + �f

) �bn

− �b

ig. 7. Illustration of the accuracy of the SFF relationship with collected data for nisotropic coefficients of contact normal (b), contact normal force (c), contact tang

aptured from the discrete data (Wang, Dove, & Gutierrez, 2007c;in, Chang, & Hicher, 2010; Li & Yu, 2013; Seyedi Hosseininia, 2013;uo & Zhao, 2013). f̄0 and l̄0 are different from the average normalontact vector and the average normal force over all contacts (Guo

Zhao, 2013; Li & Yu, 2013; Rothenburg & Bathurst, 1989), partic-larly for the anisotropic behaviour, and are presented as follows:

0 = Fnii =1

2�

2�∫0

f n(�)

d� = 1Nc

∑c∈Nc

f cn nini1 + acklnknl

/= 1Nc

∑c∈Nc

f cn , (5a)

0 = Vnii =1

2�

2�∫0

ln(�)

d� = 1Nc

∑c∈Nc

lcnnini1 + acklnknl

/= 1Nc

∑c∈Nc

lcn. (5b)

√√√√√√√√√√√√√√

a2 + a2n + a2t + 2aan cos 2

(�a −

2abn cos 2(�a −

2anbn cos 2(�f(


�f�n

=

√ 2atbn cos 2 �t − �bn2anat cos 2

(�f − �t

)2 + aan cos 2

(�a − �f

)+ abn cos 2

(�a − �b

ent average contact normal forces (Eq. (11)) (a); relationship between respective force (d) and SF.

The anisotropy coefficients can be captured through the eigen-values of these fabric tensors. Additionally, the cosine and sine ofthese principal directions can be obtained from the eigenvectors ofthese tensors.

Based on the Mohr criterion, the invariants of the average stresstensor are as follows:

�n = �11 + �222 ,

�f =√(

�11 − �222

)2+ �212.

(6)

Substituting Eqs. (4) and (5) into Eq. (3) and eliminating thepolynomials of third order and greater, the general form of the two-stress invariants ratio can be obtained:

b2t ++ 2aat cos 2

(�a − �t

)+)

+ 2abt cos 2(�a − �bt

)+

n

)+ 2anbt cos 2

(�f − �bt

)+) ( )


+ 2atbn cos 2 �t − �bt ++ 2bnbt cos 2

(�bn − �bt

)n

)+ anbn cos 2

(�f − �bn

)+ atbt cos 2

(�t − �bt

) . (7)



F r aniss

ts

2ana

f

N

D

p1(m(

ig. 8. Contact normal vector anisotropy coefficient (a) and contact tangential vectotate).

For disk or sphere particles, the magnitudes of the average con-act normal vector and the tangential vector anisotropy are verymall, and the equation above can be reduced to

�f�n

=

√a2 + a2n + a2t + 2aan cos 2

(�a − �f

)+ 2aat cos 2

(�a − �t

)+

2 + aan cos 2(�a − �f

)With the assumption of coincident �a = �f = �t, Eq. (8) can be

urther decomposed to

�f�n

=√a2 + a2n + a2t + 2aan + 2aat + 2anat

2 + aan =a + an + at

2 + aan . (9)

umerical simulation results and discussion

EM sample preparation

In the present biaxial numerical simulation, the numerical sam-le initially contains 9506 circular particles with dimensions of


00 mm (W) × 200 mm (H). The numerical specimen illustrationFig. 3), which can easily be used to capture the internal defor-

ation of a granular sample, was first proposed by Jiang et al.2006) for effectively evaluating the deep penetration mechanisms

otropy coefficient (b) versus axial strain (polar distribution of Elongated5 at critical

t cos 2(�f − �t

). (8)

in granular materials. The particle size distribution (PSD) is alsoshown in Fig. 3. The mean particle diameter (d50) is 1.62 mm with

a uniformity coefficient of Cu = 1.47. After an initial porosity (0.16)is obtained, seven irregular particles will replace the initial diskwith an equivalent area and the same centre location. The ori-entation of the irregular particles is arbitrarily distributed in therange of 0◦ to 360◦. A comparison of the mechanical properties isperformed, using a biaxial loading test on single shapes. Duringthe servo-control mechanism, the sample is subjected to a targetconsolidated pressure (100, 300, and 500 kPa). In this study, theloading rate is a constant strain rate of 5% per minute, and theconfining stress on the lateral walls is constant. All simulationsin this paper are maintained in a plane strain condition. The con-tact model selection also requires careful consideration. For the


collision and small-strain problems, the difference between the lin-ear contact and non-linear contact model is obvious (O’Sullivan,2011; Kumar, Lmole, Magnanimo, & Luding, 2014). However, thenumerical specimens in this study are all within the process of



Fu

qt1e

Table 2Input microscale parameters in DEM simulations.

Sand particles Density (kg/m3) 2600Normal/shear contact stiffness (N/m) 1 × 108Inter-particle frictional coefficient 0.5Local non-viscous damping 0.45

Fw

ig. 9. Comparison of the relationship between the stress force and the fabric eval-ation from the general SFF relationship with the simplified forms.


uasi-static shearing, where the small-strain response is ignored inhe element tests. Additionally, previous study (Mirghasemi et al.,997) has found that choosing a suitable magnitude for the lin-ar contact stiffness will produce macroscopic behaviour similar to

ig. 10. The probabilistic distribution of the friction mobilisation: (a) the mobilisation inith SF at the peak state and critical state.

Confiningboundaries

Normal contact stiffness (N/m) 1 × 107Frictional coefficient 0.0

Hertzian behaviour, especially for the large-strain response. More-over, many researchers adopted linear contact models and wereable to adequately represent the mechanical responses of granularmedia (Jensen et al., 1999; Jiang et al., 2009; Abedi & Mirghasemi,2011; Seyedi Hosseininia, 2012, 2013; Kumar et al., 2014). There-fore, for simplicity, the authors used the linear contact model inthe numerical simulations. It is well known that using only a slid-ing mechanism may not be efficient to control the granular packingwithin a quasi-static system. Therefore, local non-viscous dampingis artificially introduced in this study, to avoid the non-physicalvibrations that develop at the contacts. This damping model only


damps the acceleration motion. The non-dimensional dampingconstant is also frequency-independent. The damping model hasbeen discussed in detail by O’Sullivan (2011). The values of thenumerical parameters are provided in Table 2. The internal-particle

dex Im at the peak state and critical state and (b) the average value of Im correlated


ING ModelP1 uology

frYauip

S

bqte

t

(

(

(tbmttmgapr(crtfU1

i(2sfcGtdlfcpitbsiditd

M

F

explain the variation of macroscale strength indexes in Fig. 5.The evolution of the mean contact normal vector anisotropy

(parameter bn) and the contact tangential vector anisotropy(parameter bt) is presented in Fig. 8(a) and (b) and is determined


riction coefficient follows previous studies of the authors and otheresearchers (Wang et al., 2007a; Abedi & Mirghasemi, 2011; Jiang,an, Zhu, & Utilis, 2011; Seyedi Hosseininia, 2012, 2013). Addition-lly, the particle density, damping coefficient and contact stiffnesssed for the present study are very close to those in the above stud-

es. For a better comparison of particle shape effects, equal inputarameters are used for the mimic samples.

trength and deformation for the prior shape descriptor

The particle shape can directly influence granular stress–strainehaviour, especially for the critical state, as shown in Fig. 4(c). Theuantitative analysis for the particle shape effect is shown, withhe strength and dilation characteristics (Figs. 4 and 5). The differ-nt particle shape descriptors (Eq. (1)) are considered to evaluate

he trend of the peak friction angle(

sin (∅mobilized)max =(�f�n

)max

)Fig. 5(a)), critical friction angle

(sin∅mobilized =

�f�n

= constant)

Fig. 5(b)), and peak dilation angle(

sin max =(

− dε1+dε2dε1−dε2)

max

)Fig. 5(c)) through the fitting comparisons of the exponential func-ion. The comparative results show that the SF index provides theest fit, where the associated adjusted residual square is approxi-ately 1.0. The aspect ratio and AR indexes produce poor results in

he discrete regression analysis. Hence, the SF index is suggestedo evaluate the granular mechanical behaviour from the micro-to-

acroscale in the following sections. The peak friction in theseranular assemblies is dramatically increased for a low SF value,nd slightly decreased for a large SF coefficient. The shape of thearticle, particularly for a higher SF value, will play an importantole in the global mechanical behaviour. The critical friction anglemean value in the critical state) is directly increased by the SFoefficients. Particles with higher SF values are more difficult toealign at the large deformation state, which may directly increasehe critical friction angle. The peak dilation angle then decreasesaster than the peak friction angle associated with a larger SF value.nless otherwise specified, the following section is studied under00 kPa.

Previous studies have proved that shear can induce strain local-sation within granular material under rigid or flexible boundariesDesrues & Viggiani, 2004; Gao & Zhao, 2013; Jiang et al., 2011,014). It is clear in Fig. 6 that the ultimate shear bands (i.e., 15%train) for the irregular particles are much more obvious than thoseor the disk sample. Accumulated shear strain in Fig. 6(b) was cal-ulated using the mesh-free strain approach proposed by Wang,utierrez, and Dove (2007d). Nevertheless, the localisation pat-

erns exhibit disparities for different samples. Particle shape canirectly influence the evaluation of strain localisation in the granu-

ar media. Additionally, the rose distributions of the contact normalorce magnitudes and associated numbers are investigated at theritical state (15% strain) for typical shapes (Fig. 6(c)) to evaluate therogressive behaviour of the granular media using advanced visual-

sation techniques to display two micro-anisotropy parameters inhe same figure. Fig. 6(c) can also be used to explain the differencesetween the macro-mobilised strength and shear bands for thesehapes. A more intense degree for the contact normal force cannduce a higher critical strength. Additionally, the low anisotropicistribution of the contact number for circular particles cannot eas-

ly generate an obvious strain concentration. The development ofhese micro-statistical variables can help explain the anisotropyevelopment in later parts of the study.


icro-anisotropic parameters and SFF

To quantify the arrangement of contacts, the second-orderourier series (Eqs. (4) and (5)) represent the shape of the contact

PRESS xxx (2015) xxx–xxx

orientation distribution, the mean contact force distribution, andthe mean contact vector distribution.

Fig. 7(a) shows the accuracy of the SFF relationship for differ-ent average contact normal forces versus the axial strain for two

cases. f0 = 1Nc∑c∈Nc

f cn will induce a significantly higher stress ratio

than f0 = 1Nc∑c∈Nc

f cn nini1+ac

klnknl

, which is nearly the same as the mon-

itored data. Hence, the average normal force that is selected isvery important. The figure is useful for determining the effect ofthe induced anisotropies in Fig. 7(b)–(d), where the critical stateparameters are the mean values within the critical regimes. Theparticle SF coefficients can directly influence fabric developmentin the granular assembly at the peak/critical state. With the excep-tion of the contact normal anisotropy coefficient, the magnitudesof these fabric parameters are larger at the peak state. These fig-ures show that the anisotropy coefficients at the critical state areincreasing nearly linearly with the associated particle SF coefficient.Additionally, the contact normal, which can describe the geometrycontact, shows the highest increasing gradient. However, particleshape displays a threshold effect for these fabric parameters, exceptfor the contact tangential anisotropic coefficient, which has only aminor contribution to the friction and dilation angle at the peakstate. This microscale statistical information can also be used to


Fig. 11. (a) Probability distribution function of normalised normal force at the crit-ical state (15% axial strain); (b) relationship between the strong force exponent andSF.



Fig. 12. Strength indexes and PDF of normalised normal force under different confining pressures: (a) peak friction angle, (b) critical state friction angle, (c) peak dilationa ial str

utctadhtv(

ngle, (d) macro-responses of Circle, and (e) Elongated5 at the critical state (15% ax

sing the discrete data. The selected rose distributions of the con-act vectors at the critical state are also shown for the Elongated5ategory, where the Fourier series fit with the collected data fromhe numerical results. These results are interesting and useful, butre seldom considered in studies. The magnitudes of bn and btepend on the contact normal distribution along the vertical and


orizontal directions through the whole deviatoric loading. Andheir related values have a smaller magnitude than the minimumalues of the other anisotropy coefficients except for Elongated5Fig. 8(a) and (b)), where the values of bn and bt for Elongated5 at

ain).

the critical state are close to the minimum value of at in Fig. 7(d).Hence the anisotropic coefficients of the contact vectors and therelated principal angles can also be influenced by the particle’sshape, especially for a high SF value.

The general SFF formulation (Eq. (7)) may be more accurate thanthe simplified expression (Eqs. (8) and (9)) in Fig. 9. Hence, the


general SFF relationship is suggested as a means of verifying themonitored data in the granular assemblages and the microscalefabrics evaluation. The magnitude of the contact vector cannot beignored for the high SF grains.


ING ModelP1 uology

F

id

I

picsubls

tipt

P

woetmvp

C

uccdtfaifTtcpm

C

ttadihaAcmp


riction mobilisation and force chain distribution

The probability distribution of the sliding friction mobilisationndex Im can determine the shape effect (Zhou et al., 2013). Theefinition of Im is:

m =∣∣f ct ∣∣�f cn

. (10)

The probability value of Im at the plastic portion (Im = 1.0) in theeak state is significantly higher than in the critical state, which

s always associated with a larger particle interlocking effect, andan explain the strength variation between the peak and criticaltates. The particle shape may have a threshold effect, based on theniform gradient of the sliding friction from the centre of the shearand (Fig. 10(a)). Fig. 10(b) shows that the average Im increases

inearly with SF, and the incremental gradient and magnitude areignificantly lower at the critical state than at the peak state.

The semilogarithmic plot of the probability distribution of con-act normal force normalised by the mean contact force f cn is givenn Fig. 11 at the critical state. For all of the shapes in this paper, therobability density of the normalised contact normal force essen-ially follows a power-law decay:

(f cn

)∝

{(f cn /

〈f cn

〉)˛f cn <

〈f cn

〉eˇ(1−f

cn /〈f cn 〉) f cn >

〈f cn

〉 , (11)here ̨ and ̌ represent the variation indexes. Here we only focus

n the distribution of the strong force, which resists the majorxternal load transfer. ̌ decreases with increasing SF, which showshe intensity of inhomogenous contact normal force shift. This

icroscale force chain information can also be used to explain theariation in an in Fig. 7(c). Additionally, the higher strong forceercentage can increase the strength indexes.

onfining pressure effect

Fig. 12 shows the strength indexes of biaxial compression testsnder different confining pressures. The peak friction angle andritical state friction angle are slightly decreased with the increasedonfining pressures. Whereas, the peak dilation angle is highlyecreased. Particle shape still plays an important role in the fric-ion angles, even under higher confining pressures, and especiallyor the critical state. Additionally, the strain value at peak frictionngle and peak dilation angle is enlarged by the increased confin-ng stress in Fig. 12(d). There is also a small saturation SF valueor the peak dilation angle under the higher confining stress level.he variation in these strength indexes can also be explained byhe strong force distribution in Fig. 12(e). The green arrow indi-ates that the coefficient of ̌ decreases with increasing confiningressure. This microscale information is useful for interpreting theacroscale properties.

onclusions

This paper quantitatively analyses the particle shape factorhrough a series of numerical studies of non-convex irregular par-icles, using a significant amount of computer analysis. SF provides

better evaluation of the granular mechanical behaviour thano other particle shape indexes. The critical state friction angle

ncreases linearly with the SF value. The strong force chain in theigher SF granular assemblies will occupy a larger portion of thessemblage contacts, which will increase the packing resistance.


dditionally, the peak friction/dilation angle shows a nonlinearorrelation with SF where there exists a threshold value. Further-ore, all of the strength indexes decrease with increasing confining

ressure, especially for the dilation angle. It is remarkable that

PRESS xxx (2015) xxx–xxx

the coefficient of ̌ for the contact force decreases with increas-ing confining pressure. This study also found that shear localisationpatterns are sensitive to the particle shape. It is difficult to generatea shear band for circular particles with a low anisotropic level of thecontact normal.

The average contact normal force is also found to influence theaccuracy of the SFF formulation. The magnitudes of fabric param-eters are higher at the peak state, except for the contact normalanisotropic coefficient with a larger SF. The contact tangential forceanisotropic coefficient shows a linear, increasing relationship withSF at both the peak and steady states, but its contribution to themacroscale strength is smaller than that of the other fabric param-eters. The anisotropic coefficients of the contact vectors and therelated preferred angles can also influence the SFF relationship.Hence, the general formulation of the SFF relationship is suggestedfor representing the mechanical behaviour of irregular granularmedia.

The probability distributions of friction mobilisation and contactforce display an inhomogeneous distribution within the granularpacking. The microscale average Im increases with SF magnitude,especially at the peak state. The higher strong force percentagealso increases with SF. The end of contact force distribution is usedto explain how macroscale strength varies with the particle shapeindex and confining pressure level.

Acknowledgements

The authors would like to thank the Hong Kong Polytechnic Uni-versity for the Ph.D. studentship through account RT1c. We alsowish to thank Dr. Zhihong Zhao at the Tsinghua University for thepaper review and comments. We also benefitted from discussionwith Prof. Zhenyu Yin, at the Shanghai Jiao Tong University, andfrom technical discussion with Dr. Ning Guo, at the Hong KongUniversity of Science and Technology.

References

Abedi, S., & Mirghasemi, A. A. (2011). Particle shape consideration in numerical sim-ulation of assemblies of irregularly shaped particles. Particuology, 9, 387–397.

Alonso-Marroquín, F., & Wang, Y. (2009). An efficient algorithm for granular dynam-ics simulations with complex-shaped objects. Granular Matter, 11(5), 317–329.

Cho, G.-C., Dodds, J., & Santamarina, J. C. (2006). Particle shape effects on packing den-sity, stiffness, and strength: Natural and crushed sands. Journal of Geotechnicaland Geoenvironmental Engineering, 132, 591–602.

Christoffersen, J., Mehrabadi, M., & Nemat-Nasser, S. (1981). A micromechanicaldescription of granular material behaviour. Journal of Applied Mechanics, 48,339–344.

Desrues, J., & Viggiani, G. (2004). Strain localization in sand: An overview of theexperimental results obtained in Grenoble using stereo photogrammetry. Inter-national Journal for Numerical and Analytical Methods in Geomechanics, 28(4),279–321.

Ehrlich, R., & Weinberg, B. (1970). An exact method for characterization of grainshape. Journal of Sedimentary Research, 40(1), 205–212.

Ferellec, J. F., & McDowell, G. R. (2010). A method to model realistic particle shapeand inertia in DEM. Granular Matter, 12(5), 459–467.

Fu, X., Dutt, M., Bentham, A. C., Hancock, B. C., Cameron, R. E., & Elliott, J. A. (2006).Investigation of particle packing in model pharmaceutical powders using X-ray microtomography and discrete element method. Powder Technology, 167(3),134–140.

Fu, X., Huck, D., Makein, L., Armstrong, B., Willen, U., & Freeman, T. (2012). Effectof particle shape and size on flow properties of lactose powders. Particuology,10(2), 203–208.

Gao, Z., & Zhao, J. (2013). Strain localization and fabric evolution in sand. InternationalJournal of Solids and Structures, 50(22), 3634–3648.

Guo, N., & Zhao, J. (2013). The signature of shear-induced anisotropy in granularmedia. Computers and Geotechnics, 47, 1–15.

Iwashita, K., & Oda, M. (1998). Rolling resistance at contacts in simulation of shearband development by DEM. Journal of Engineering Mechanics, 124, 285–292.

Jensen, R. P., Bosscher, P. J., Plesha, M. E., & Edil, T. B. (1999). DEM simulation of gran-


ular media—Structure interface: Effects of surface roughness and particle shape.International Journal for Numerical and Analytical Methods in Geomechanics, 23,531–547.

Jiang, M., Yu, H.-S., & Harris, D. (2005). A novel discrete model for granular materialincorporating rolling resistance. Computers and Geotechnics, 32, 340–357.

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ARTICLEARTIC-803; No. of Pages 13Y. Yang et al. / Partic

iang, M., Yu, H.-S., & Harris, D. (2006). Discrete element modelling of deep penetra-tion in granular soils. International Journal for Numerical and Analytical Methodsin Geomechanics, 30(4), 335–361.

iang, M., Leroueil, S., Zhu, H., Yu, H.-S., & Konrad, J.-M. (2009). Two-dimensional dis-crete element theory for rough particles. International Journal of Geomechanics,9, 20–33.

iang, M., Yan, H. B., Zhu, H. H., & Utili, S. (2011). Modeling shear behavior and strainlocalization in cemented sands by two-dimensional distinct element methodanalyses. Computers and Geotechnics, 38(1), 14–29.

iang, M., Chen, H., Tapias, M., Arroyo, M., & Fang, R. (2014). Study of mechanicalbehavior and strain localization of methane hydrate bearing sediments with dif-ferent saturations by a new DEM model. Computers and Geotechnics, 57, 122–138.

rumbein, W. C. (1941). Measurement and geological significance of shape androundness of sedimentary particles. Journal of Sedimentary Research, 11(2),64–72.

umar, N., Imole, O. I., Magnanimo, V., & Luding, S. (2014). Effects of polydispersity onthe micro–macro behavior of granular assemblies under different deformationpaths. Particuology, 12, 64–79.

iu, R., Yin, X., Li, H., Shao, Q., York, P., He, Y., et al. (2013). Visualization and quantita-tive profiling of mixing and segregation of granules using synchrotron radiationX-ray microtomography and three dimensional reconstruction. Internationaljournal of pharmaceutics, 445(1), 125–133.

i, X., & Yu, H.-S. (2013). On the stress–force–fabric relationship for granular mate-rials. International Journal of Solids and Structures, 50, 1285–1302.

u, M., & McDowell, G. R. (2007). The importance of modelling ballast particle shapein the discrete element method. Granular Matter, 9(1–2), 69–80.

uding, S. (2005). Anisotropy in cohesive, frictional granular media. Journal of Physics:Condensed Matter, 17(24), S2623–S2640.

irghasemi, A. A., Rothenburg, L., & Matyas, E. L. (1997). Numerical simulations ofassemblies of two-dimensional polygon-shaped particles and effects of confin-ing pressure on shear strength. Soils and Foundations, 37(3), 43–52.

irghasemi, A., Rothenburg, L., & Matyas, E. (2002). Influence of particle shapeon engineering properties of assemblies of two-dimensional polygon-shapedparticles. Geotechnique, 52, 209–217.


g, T.-T. (1994). Numerical simulations of granular soil using elliptical particles.Computers and Geotechnics, 16, 153–169.

’Sullivan, C. (2011). Particulate discrete element modelling. New York, NY: CRC Press.owers, M. C. (1953). A new roundness scale for sedimentary particles. Journal of

Sedimentary Research, 23(2), 117–119.

PRESS xxx (2015) xxx–xxx 13

Radjai, F., Jean, M., Moreau, J. J., & Roux, S. (1996). Force distributions indense two-dimensional granular systems. Physical Review Letters, 77(2),274–277.

Rothenburg, L., & Bathurst, R. (1989). Analytical study of induced anisotropy inidealized granular materials. Geotechnique, 39, 601–614.

Rothenburg, L., & Bathurst, R. (1992). Micromechanical features of granular assem-blies with planar elliptical particles. Geotechnique, 42, 79–95.

Seyedi Hosseininia, S. E. (2012). Discrete element modeling of inherentlyanisotropic granular assemblies with polygonal particles. Particuology, 10,542–552.

Seyedi Hosseininia, S. E. (2013). Stress–force–fabric relationship for planar granularmaterials. Geotechnique, 63, 830–841.

Sukumaran, B., & Ashmawy, A. K. (2001). Quantitative characterisation of the geom-etry of discretparticles. Geotechnique, 51, 619–627.

Sun, Q., Jin, F., Liu, J., & Zhang, G. (2010). Understanding force chains in dense granularmaterials. International Journal of Modern Physics B, 24(29), 5743–5759.

Wadell, H. (1932). Volume, shape, and roundness of rock particles. The Journal ofGeology, 40(5), 443–451.

Wang, L., Park, J. Y., & Fu, Y. (2007). Representation of real particles for DEMsimulation using X-ray tomography. Construction and Building Materials, 21(2),338–346.

Wang, J., Dove, J., & Gutierrez, M. S. (2007b). Discrete-continuum analysis of shearbanding in the direct shear test. Geotechnique, 57, 513–526.

Wang, J., Dove, J., & Gutierrez, M. S. (2007c). Determining particulate–solidinterphase strength using shear-induced anisotropy. Granular Matter, 9,231–240.

Wang, J., Gutierrez, M. S., & Dove, J. E. (2007). Numerical studies of shearbanding in interface shear tests using a new strain calculation method. Inter-national Journal for Numerical and Analytical Methods in Geomechanics, 31(12),1349–1366.

Williams, K. C., Chen, W., Weeger, S., & Donohue, T. J. (2014). Particle shape char-acterisation and its application to discrete element modelling. Particuology, 12,80–89.

Yin, Z. Y., Chang, C. S., & Hicher, P. Y. (2010). Micromechanical modelling for effect


of inherent anisotropy on cyclic behaviour of sand. International Journal of Solidsand Structures, 47(14), 1933–1951.

Zhou, B., Huang, R., Wang, H., & Wang, J. (2013). DEM investigation of particle anti-rotation effects on the micromechanical response of granular materials. GranularMatter, 15, 315–326.

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