Bonding Effect in Sands by DEM

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2007; 69:1158–1193Published online 31 July 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1804

A simple and efficient approach to capturing bonding effect innaturally microstructured sands by discrete element method

Mingjing Jiang1,∗,†, Hai-Sui Yu2 and Serge Leroueil3

1Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China2School of Civil Engineering, The University of Nottingham, Nottingham, NG7 2RD, U.K.

3Department of Civil Engineering, Universite Laval, Ste-Foy, Que., Canada G1K 7P4

SUMMARY

A discrete element modelling of naturally microstructured sands is very important to geomechanics.This paper presents a simple discrete element model for naturally microstructured sands with the aimto efficiently capture the effect of cementation between particles (bonds). First, a simple bond contactmodel was proposed by introducing a rigid-plastic bond element into the conventional contact model fordry granular material. Second, efficient numerical techniques were investigated to implement this contactmodel into the distinct element method (DEM). Then, a two-dimensional DEM code was developedto simulate a series of isotropic compression tests on the materials of different densities and bondingstrengths. Finally, the DEM results were examined in comparison with the experimental data on artificiallybonded sands obtained by Rotta et al. (Geotechnique 2003; 53(5):493–502). In addition, we discussed theyielding mechanism, the Coop and Willson criteria on weak/strong bonding (J. Geotech. Eng. (ASCE)2003; 129(11):1010–1019) and the strong bonding phenomenon observed by Rotta et al. based on the DEMdata. The study shows that the DEM model is able to capture the main features of naturally microstructuredsands, such as variations of yielding and bulk modulus against bonding strength or material density. Inaddition, it is shown that the gross yielding (the yielding defined in terms of strains) is largely relatedto bond breakage; Coop and Willson criteria are generally reasonable; and the strong bonding in theexperimental data obtained by Rotta et al. comes from that their bonded materials start at different pointson the same compression line. Copyright q 2006 John Wiley & Sons, Ltd.

Received 12 January 2006; Revised 19 May 2006; Accepted 20 May 2006

KEY WORDS: naturally microstructured sands; simple bond contact model; discrete element method;yielding; bulk modulus; bonding effect

∗Correspondence to: Mingjing Jiang, Department of Geotechnical Engineering, Tongji University, Shanghai 200092,China.

†E-mail: [email protected], [email protected]

Contract/grant sponsor: EPSRC; contract/grant number: GR/R85792/01

Copyright q 2006 John Wiley & Sons, Ltd.

CAPTURING BONDING EFFECT IN NATURALLY MICROSTRUCTURED SANDS 1159

1. INTRODUCTION

Natural sands are sometimes named as ‘problematic soil’ in the geotechnical community andhave recently attracted great attention of many geo-researchers, since their mechanical behaviourare evidently distinct from the clean sands usually employed in laboratory. It is now recognizedthat their peculiar behaviour results from their microstructure which is mainly characterized withbonding materials between particles or aggregates (bonds) and fabric. Bonds can be produced innatural sands through various processes, such as carbonate sands with calcium carbonate depositedfrom supersaturated pore fluid to form calcarenites [1–4] or the quartzitic sandstone depositedfrom water flowing through the soil to form iron oxide [5]. Figure 1 provides a scanning electronmicrograph (SEM) of a natural sand, Lower Greensand obtained by Cuccovillo and Coop [6], inwhich white material is the bond. Such cementation plays an important role in their stress–strainand strength behaviour, which leads to their mechanical behaviour mainly controlled not onlyby stress history and density as described by classical geomechanics, but also by the strength ofthe bonds [7–12]. Hence, it is very important to consider the effect of bonds in establishing themechanics on naturally microstructured sands, which has been considered as one of key futuretasks of modern geomechanics [9, 10].

In the light of specimen type used in laboratory, two main approaches have been used to studymicrostructured soils and have their own features:

(i) Natural soil specimens retrieved from the field [2–6, 8, 9, 13, 14]. In this approach, thebehaviour is usually investigated in comparison with that of their reconstituted counterpart.In addition to a high spatial variability of the specimens, the main difficulty lies in thedisturbance of the microstructure that can occur during the sampling process which couldlead to breakage of the bonds as a result of unloading [15, 16]. Moreover, recent evidenceshows that the sample cutting process in laboratory, in which specimens of a targeted sizeare generated from in situ samples, may give rise to additional disturbance [11].

(ii) Artificially cemented specimens made up by adding a bonding agent to soil materials,such as Portland cement, gypsum, lime, etc. [1, 17–24]. They can also be generated bymore complicated processes in which several steps are needed in sequence to form bondingmaterials [25–28]. In this approach, the behaviour is usually examined in comparison totheir clean and uncemented counterpart.

The aforementioned two approaches have led to a framework of the macroscopic mechanics onnaturally microstructured sands [5, 8–12, 17]. However, microscopic information on natural sandsunder loading is still scare. For example, it is believed that the gross yielding (the yielding definedin terms of strains) on macroscopic scale may be associated with bond breakage on microscopicscale. But, there has been few microscopic data to verify this statement. The main reason may liein extreme difficulty in monitoring sufficient microscopic data, such as on bonds, in the specimensin geo-laboratory even with advanced technologies such as X-ray [29], the stereophoto-grammetrictechnique [30], or particle image velocimetry [31].

A tool that appears to be promising for the macroscopic and microscopic investigations onmicrostructured sands is the distinct element method (DEM), which is a numerical simulationtechnique originally developed for dry granular materials by Cundall and Strack [32, 33]. Un-like finite element method, this technique treats soils as an assembly of discrete elements, startswith basic constitutive laws at interparticle contacts and can provide macroscopic and micro-scopic responses of the particle assembly under different loading conditions. Compared with the

Copyright q 2006 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 69:1158–1193DOI: 10.1002/nme

1160 M. JIANG, H.-S. YU AND S. LEROUEIL

Interparticle bonding

Void

Particles

0.5 mm

Figure 1. Interparticle bonding of natural sands: SEM image of LowerGreensand (after Cuccovillo and Coop [6]).

aforementioned experimental approaches, DEM is able to facilitate sample reproducibility, andmonitor macroscopic/microscopic data simultaneously and continuously in a non-destructive man-ner. DEM has been used to examine several aspects of soil behaviour, such as granular mechanics[34–36], creep theory [37] and anisotropy [38] of soils, mechanics of crushable soils [39–42],constitutive models for granular flow [43], the effective stress and shear strength functions ofunsaturated granulates [44, 45]. It has also been employed to investigate microstructured soilsunder one-dimensional compression and biaxial shear tests [46, 47]. It is mostly useful to under-stand the relation between bond properties at the level of contacts between particles and observedmacroscopic behaviour, and can help better understand the behaviour of naturally microstructuredsands. This constitutes one of strong motivations for this study. We believe that a discrete elementmodelling of naturally microstructured sands may be as important to geomechanics as the originalDEM to granular materials [32, 33].

The main objective of the paper is to provide a simple discrete element model for naturallymicrostructured sands with the aim to efficiently capture the effect of bonds. First, a simplybond contact model is proposed, which was preliminarily introduced in References [46, 47], byintroducing a rigid bond element into the conventional contact model [32, 33]. Second, efficientnumerical techniques are exploited in order to implement this contact model into DEM. Then, atwo-dimensional (2-D) DEM code incorporating the model, which is based on References [46–48],is used to simulate a series of isotropic compression tests on the samples of different bondingstrengths or void ratios. Finally, the main macroscopic DEM results are examined in comparisonwith the experimental data on artificially bonded sands obtained by Rotta et al. [22]. The yielding



mechanism, the Coop and Willson criteria on weak/strong bonding [15] which will be brieflydescribed in Section 6.2, and the strong bonding phenomenon in Reference [22] are also discussed.Several other features of naturally microstructured sands, such as particle crushing [39, 49], fabric[5], anisotropy [23, 24], particle shape or grade [1] and strain localization are not discussed.

2. SIMPLE CONTACT MODEL FOR NATURALLYMICROSTRUCTURED SANDS

In this section, after briefly reviewing existing DEM contact models for soil mechanics, we shallintroduce a bond contact model used in this study for naturally microstructured sands.

2.1. Current contact models

Since the pioneering works by Cundall and Strack [32, 33], DEM has been used with much successin studying a variety of aspects of soil behaviour with various contact laws. These contact lawsinclude:

(i) the linear-elastic normal and tangential contact laws, in addition to tangential failure con-trolled by the Mohr–Coulomb criterion [32–35, 43, 48]. These contact laws are widely usedin DEM analyses, usually constituting bases for other contact laws as standard contact laws;

(ii) the non-linear-elastic normal contact law applied to 2-D problems with disks [50], three-dimensional (3-D) problems with spheres [51, 52] or ellipsoids [53], which are establishedon the Hertz theory of contact mechanics. These contact laws are more rigorous, but costmore CPU time in numerical simulations, than the standard contact laws;

(iii) the contact laws for clays with adhesion force [54] or with repulsive force [38, 55],possibly including creep [37]. These contact laws are employed to investigate some specificmechanism of reconstituted soils in soil mechanics, such as anisotropy mechanism [35, 55]and creep theory [37];

(iv) the contact laws considering rolling resistance, by introducing three model parameters andone variable related to particle overlaps [56], or by introducing only one additional modelparameter [57], into the standard contact laws. In addition to capturing the reasonablestructure within shear band [56], such laws lead the coefficient of internal friction to areasonable value for granular materials [57];

(v) the contact laws for crushing soils [40–42], which is in essence a kind of cementation modelbeing inexplicitly incorporated into the commercial software PFC3D [58]; the contact lawsfor unsaturated granular [44, 45] by introducing both capillary water bond and slider intothe standard contact model;

(vi) the contact laws for microstructured geomaterial, with two rigid-plastic bond elementsemployed into the standard contact laws [46, 47] or with another rigid-plastic bond elementfurther introduced into rolling contact model [59, 60]. Similar contact laws incorporatingbond rolling resistance can also be found in References [61–63].

2.2. Simple bond contact model

Recent experimental evidence clearly suggests that the bonds produce both traction and shearresistance, and can be broken in an irreversible manner due to excessive traction or shear forces,



Bond

Tangentialcontact model

Spring Bond

Bond

Slider

contact modelNormal

DashpotSpring

Divider

Dashpot

Figure 2. Simple contact model proposed for bonds in the DEM.

hence playing an important role in its stress–strain and strength behaviour. In addition, bonds maylead to the resistance against relative particle rotation at a contact (bond rolling resistance) andcould also be broken due to excessive particle rotation [59–64]. Several complete bond modelsincorporating bond rolling resistance can be found in literature, e.g. References [59–64]. It isthe authors’ opinion that such mechanism has to be included if bond size is relatively large incomparison to grain size, but may be neglected if not. In this study, for simplicity, the contactmodel in References [46, 47] was chosen which captures only the tension and shear resistances ofbonds while neglecting the rolling resistance. Such a choice has a potential advantage: unlike thematerial of rolling resistance which in general is macroscopically associated with a couple-stresscontinuum, a material with no bond rolling resistance may be macroscopically described withinclassical continuum mechanics; hence, the information on such a material may be directly used toestablish practical constitutive models for natural soils.

Since our simple bond contact model was only introduced preliminarily in References [46, 47],we shall give more information on this model here. In comparison to the standard contact model forgranular material, the key innovation in the model is that a rigid-plastic bond element is introduced,respectively, into the standard normal and tangential contact model [32, 33]. The bond element isillustrated as a ‘bond’ in this model shown in Figure 2 [46, 47]. This bond element has its peakstrength R, with its behaviour defined in terms of external force F and displacement u as follows:

u = 0, if F�R−; the bond is intact (1a)

u = ∞, if F�R+; the bond is broken and F = 0 (1b)

where superscripts ‘+’ and ‘−’ represent the limit value approaching R from upper and lowerrange, respectively, and F = R is a singular point.

We introduce the rigid-plastic bond element in proper positions, as illustrated in Figure 2, leadingto a physical model for bonds. This simple bond contact model consists of a normal (tangential)



component to resist traction (shear) force, but neglects bond rolling resistance. As shown inFigure 2, the normal and tangential components of the model are similar in their principle. Likethe standard contact model for granular media [32, 33], they both include a spring reflecting anelastic behaviour of the bond before failure and a dashpot that allows energy dissipation andquasi-static deformations in DEM analyses. Unlike the standard contact model [32, 33], both thenormal and tangential contact models include a rigid bond element controlled by Equations (1).The rigid bond element is set to be in parallel with divider in the normal contact model or sliderin the tangential contact model to produce tension or shear resistance, respectively, in order torepresent the main action of bonding materials. The normal model includes a divider to simulatethe fact that no traction force is transmitted through the contact when the bond is broken and theparticles are separated. The tangential model includes a slider that provides the contact a shearresistance controlled by the Mohr–Coulomb criterion. Note that the physical model for bonds inFigure 2 will reduce to the standard contact model [32, 33], when the bond elements are brokenor excluded. The introduction of such elements will give a contact main mechanical responsesignificantly different from that in References [32, 33], which will be introduced below.

Figure 3 provides the main mechanical performance of the normal (tangential) bond contactmodel in terms of normal (tangential) relative displacement un (us) and contact force Fn (Fs).The normal (tangential) model is mainly characterized by a constant stiffness parameter Kn (Ks)

(in N/m in a 2-D system) and a normal (tangential) bonding strength Rnb (Rtb) (in MN/m). Thislinearity between un (us) and Fn (Fs) through Kn (Ks) comes from the fact that the model isbased on the standard contact model [32, 33] in which the linear-elastic normal (tangential) contactlaws are used. It is largely in agreement with the most recent observation in laboratory by Mabilleet al. [64] and Delenne et al. [62] on bonds between rods. However, it would be possible in themodel to consider non-linear Kn (Ks).

In Figure 3(a), the performance of the normal bond contact model is different in tension andcompression. In tension, the normal model provides an attractive contact force which firstly is aproduct of Kn and tensional un , and which then abruptly reduces from Rnb to zero once the bondis broken. In compression, the model presents a repulsive contact force always as a product of Knand compressive un . Note that the model can be easily extended to include different stiffness intension and compression, respectively.

In Figure 3(b), the tangential bond contact model performs similarly in negative to positive us .The shear force increases linearly with the tangential displacement until its peak value. After thebond is broken, the strength of the contact abruptly drops to the shear strength associated withnormal force and defined by the Mohr–Coulomb criterion, with its net reduction as Rtb. Notethat a bond is broken when the bonding strength is exceeded either in tension or in shear. SinceEquation (3) below is adopted in this study, the bond is broken when the particles are separated. Thisensures that interparticle friction coefficient always has a physical interpretation in the tangentialcontact model.

Note that, in contrast to recoverable capillary water bond for unsaturated soils [45], the bondin naturally microstructured sands breaks irrecoverably, i.e. a pair of separated particles does notobtain bonding strength any more even if they are in contact again. In general, two additionalparameters, i.e. Rnb and Rtb, are needed in comparison to the standard contact model [32, 33, 65].These two parameters can be measured in laboratory, such by the method proposed in References[62, 64] and will not be discussed in this paper. Instead, we shall focus on the discrete modellingof naturally microstructured sands in this study. In the next section, we shall introduce how toefficiently implement this bond contact model in DEM codes.



(a)

Normal tension displacement

Normal bond

strength nbR

(MN) kn

Nor

mal

tens

ion

forc

e

nF

nuNormal compressive displacement

Nor

mal

com

pres

sive

forc

e

Tangential displacement

Tangential bond

strength tbR

(MN)

k

Tang

entia

lfor

ce

Strength due to friction

sF

su

Yield point

(MN)

Strength due to friction

Tangential bond

strength tbR

(MN)

(b)

Figure 3. Mechanical performance of the simple contact model proposed for bonds: (a) normal contactmodel; and (b) tangential contact model.

3. DEM INCORPORATING SIMPLE CONTACT MODEL

3.1. NS2D DEM code

Based on previous work by the authors, a 2-D DEM code has been improved for naturallymicrostructured soils (NS) incorporating the proposed simple model, which has techniques similar



to those proposed by Cundall and Strack [32, 33]. The code was firstly developed at UniversiteLaval [45–48] and then improved further [43, 44, 57, 59] by the authors. It is named NS2D hereafter.Each particle of the soil mass is a rigid disk that is identified independently with its own mass, m,moment of inertia, I0 and contact properties as illustrated in Figures 2 and 3. The total unbalancedforce for motion F (m)

i (i represents x or y direction) and moment M0 acting on each particle arecomputed and then used to estimate the instantaneous acceleration of each particle, a, based onNewton’s second Law. The acceleration a is used to calculate velocities and then displacementsin the x and y directions. This is repeated at each time increment until the simulation is stopped.For each particle, the normal and tangential contact forces for motion, denoted by F (m)

n and F (m)s ,

respectively, are summed up over the p neighbours, giving

�2xi�t2

= 1

m

p∑r=1

F (m)ri ,

�2��t2

= 1

I0

p∑r=1

Mr0 (2)

In the NS2D code, F (m)n is calculated by

F (m)n = Fn + Dn − Rnb (3)

where Fn is the normal contact force calculated from the overlaps of particles and is positive at acompressive contact in the study. Dn is the normal damping force and Rnb is the normal bondingstrength that becomes zero when bond is broken due to excessive tangential or normal tensionalforce. Hence, a positive (Fn − Rnb) is the normal contact force due to load. Once the particlesseparate, we have Fn = 0, Dn = 0 and Rnb = 0.

F (m)s is determined by

F (m)s = Fs + Ds (4)

where Fs is the tangential contact force and Ds is the tangential damping force. Ds = 0 stands inthe case that Fs exceeds the peak shear strength Fpeak

s for contacts with intact bonds or residualshear strength F resid

s for contacts without intact bonds.

In this study, Fpeaks and F resid

s are suggested, respectively, to be

Fpeaks = Fn tan�� + Rtb, F resid

s = Fn tan�� (5)

where tan�� is called the interparticle friction coefficient, which is somewhere denoted by � in this

paper. If two particles are separated due to normal tension force in excess of Rnb, then F (m)n = 0,

F (m)s = 0. Damping, as well as frictional sliding, is used in DEM analyses to dissipate energy due

to the dynamic formulation of the model.The vertical and horizontal strains (the principal values of the strain tensor in our DEM

simulations) are obtained by the position change of rigid boundaries as follows:

�v =�hv/hv, �h = �hh/hh (6)

where �hv and �hh is the vertical and horizontal deformation of the specimen, respectively, whilehv and hh are the specimen length in the vertical and horizontal direction, respectively.



The horizontal and vertical normal stress is defined as follows:

�h =( ∑right wall

fh

)/hv, �v =

( ∑top wall

fv

)/hh (7)

where fh is the horizontal component of the contact force acting on the right vertical wall, andfv is the vertical component of the contact force acting on the top horizontal wall. Note that, in theanalyses, fluid pore pressure is not modelled, i.e. the soil is dry or saturated but under fully drainedcondition. Hence, effective stresses are thus equal to total stresses calculated by Equation (7).In addition, a unit length (m) is implicitly included in Equation (7) in the direction vertical tothe plane.

Because NS2D is a 2-D DEM code, the density of specimen is described in term of planar voidratio ep by

ep = (A − Ag)

Ag(8)

where A and Ag are the total area of the specimen and the sum of the area of all grains (disks),respectively. Note that Equation (8) does not consider overlaps between particles for simplicity.The reason for such choice and its effect will be discussed later.

3.2. Translational-axis method

Equations (3)–(5) are the numerical approaches we used in the NS2D code to implement theproposed contact model. Equations (4)–(5) represent the tangential contact model shown inFigure 3(b) by following the Mohr–Coulomb criterion. They are very easy to understand, andhence will not be illustrated here. Equation (3) corresponds to the normal contact model proposedand may be uneasy to understand. We shall give a further explanation through the followingsimple example, in which a pair of particles in bonded contact moves quasi-statically (Dn = 0)on a horizontally positioned frictionless wall, as shown in Figure 4. The process is described infour steps.

(i) In Figure 4(a), the pair of particles stays stationary on the wall. By Equation (3), the bondcontact undergoes a normal contact force Fn0 = Rnb (>0). The particles still remain incontact since F (m)

n = Fn0 − Rnb = 0 in this case, and Rnb produces an overlap (normaldisplacement) un0 = Rnb/Kn (>0) between particles.

(ii) Then, if any other forces applied tend to separate the particles gradually, in which Dn = 0still holds, as shown in Figure 4(b). Consequently, both Fn and un will decrease gradually atthe contact. In addition, F (m)

n = (Fn − Rnb) becomes negative with its magnitude increasinggradually, i.e. F (m)

n becomes a kind of attractive force resisting the separation of the particlesas a bond.

(iii) Once the particles start to separate, i.e. un = 0 and Fn = 0, as shown in Figure 4(c),Equation (3) leads to (Fn0 − Fn) = Rnb>0. On the other hand, Rnb is set to zero inEquation (3) at this time as the result of bond breakage. That is to say, the separation ofthe particles is a singular or bifurcation point for (Fn0 − Fn): it can be either Rnb or zero.

(iv) Finally, after the separation of the particles, the overlap becomes negative and the bondremains broken, as shown in Figure 4(d). In this stage, Fn ≡ 0, and Rnb ≡ 0 and F (m)

n ≡ 0.



Overlap

nu >0

Contact force

nF = nbR - F

ForceF < nbR

ForceF < nbR

Overlap

nu =0

Contact force

nF = nbR - F

ForceF = nbR

ForceF = nbR

Overlap

nu <0

Contact force

nF =0

ForceF > nbR

ForceF > nbR

Frictionlesswall

Overlap 0nu

= nnb KR >0

Contact force

0nF = nbR

Particle 1 Particle 2Particle 1 Particle 2

Particle 1 Particle 2 Particle 1 Particle 2

(a) (b)

(d)(c)

Figure 4. Description of the four stages of a pair of bonded particles in DEM: (a) initial positionwith intact bond at contact, without external force F; (b) transition position with the bond response aselastic, F<Rnb the normal bond strength; (c) singular position with bond at its peak strength or broken,

F = Rnb; and (d) post-failure position with the bond broken and the particles separated, F>Rnb.

Now, let us use (Fn0 − Fn) and (un0 − un), namely translated variables here, to depict thisprocess, which is shown in Figure 5. Figure 5 shows that this process is characterized firstly by alinearly elastic relationship between (Fn0 − Fn) and (un0 − un), both larger than zero with theirratio equal to Kn . Then, after (Fn0−Fn) reaches Rnb, its value abruptly reduces to zero followed byinfinite (un0−un). By comparing Figure 5 with the normal bond model in tension in Figure 3(a), itis easy to find that both the figures demonstrate the same elastic-brittle-plastic performance. Hence,Equation (3) in essence provides a contact the exact performance as required by the normal model,from the viewpoint of ‘translational axis’. This method is named as ‘translational-axis method’.An obvious advantage of this method is that the normal bond contact model can be efficientlyimplemented in DEM codes. In addition, the condition for Equation (3) at the separation of theparticles leads to an implicit assumption that the bond ruptures after the separation of particles.Thus, another advantage of the ‘translational-axis method’ is that interparticle friction coefficientalways has a physical interpretation in the tangential contact model.

Equations (3)–(5) have been used to implement the simple bond contact model into the 2-DDEM code by the authors. Several debugging tests have been carried out to verify the revisedDEM code, NS2D, and we shall not present them here.



Normal bond

strength nbR

(MN)

kn

( 0nF - nF )

( 0nu - nu )

Tra

nsla

ted

norm

al te

nsio

n fo

rce

Translated normal displacement

Figure 5. Normal contact response described in terms of translated variables:‘translational-axis method’ (Fn0 = Rnb, un0 = Rnb/Kn).

The tests considered in this study are isotropic compression DEM tests on materials of differentbonding strengths and densities, which programme will be introduced in Section 4.

4. DEM TEST PROGRAMME

In this section, we shall introduce DEM test programme in detail. Since one of our main aims isto compare the DEM results with the experimental data in Reference [22] (see Section 5 below),we shall also briefly present the programme in Reference [22] somewhere for clearness. First, weshall introduce grain size distribution of DEM material and specimen size considered in our study.Then, we shall describe how to generate DEM bonded specimen. Finally, we shall present testprocedure and the examined aspects on the material. Note that current geo-lab techniques are notable to measure all particle properties at micro-level in a specimen, such as particle shapes, particleroughness and interparticle bonding strengths, and that these properties may change during loading.Hence, it is impossible to obtain such experimental data for DEM models to quantitatively simulatenaturally microstructured sands. However, such an exact micro-level agreement is not required inusual problems. What is required, on the other hand, is a quantitative agreement between themacro-level behaviour of the model and reality.

4.1. Grain size distribution and specimen size

The real material in Reference [22] is represented here by an idealized DEM material in our study.Figure 6(a) provides the grain size distribution of unbonded material and Portland cement usedby Rotta et al. [22]. The grain size distribution is 27.8% medium sand (0.2mm<d<0.6mm),33.4% fine sand (0.06mm<d<0.2mm), 31.3% silt (0.002mm<d<0.06mm), and 7.5% clay



100

80

60

40

20

05 6 7 8 9 10

Particle diameter (mm)

Perc

enta

ge s

mal

ler

(%)

(b)

(a) Grain size: mm

Per

cent

fine

r by

wei

ght

Figure 6. Particle size distributions in: (a) the experiment [22]; and (b) the DEM study.

(d<0.002mm). Figure 6(b) presents a grain size distribution of the particles used in this study.The DEM material is of a maximum diameter of 9.0mm, a minimum diameter of 6.0mm, anaverage grain diameter d50 = 7.6mm and uniformity coefficient Cu = d60/d10 = 1.3. Note that thegrain size distribution of DEM material will remain the same before and after bonding, whilethat of real material will probably change slightly in laboratory. However, Reference [22] has notprovided the grain size distribution of the bonded material. Moreover, our main aim is to capturethe bonding effect by a DEM model. Hence, the real grain size distribution is not used in theDEM tests.

The number of particles of each specific radius shown in Figure 6(b) was calculated in DEM by

N(i) = P(i)

rs(i)PN (9)



where N(i) is the total number of particle i with specific radius r(i) and P(i) is the weight percentageof particle i . N is the total number of particles of different radii used in the DEM analyses. Parameters is 2 for the disks used in this study. P is a variable obtained by

P =n p∑i=1

P(i)

rs(i)(10)

where n p is the number of types for particles, i.e. 10 used in the DEM samples.The DEM samples have a width of 200mm and a height that depends on the selected planar

void ratio ep. For ep = 0.30 described here as medium-dense, the height is about 178mm. Somecomparative tests were performed on the bonded material in different sizes to evaluate if thegeometric difference in samples could influence their behaviour. However, no significant differencehas been found in the preliminary study. In all cases, the total number of particles in one DEMsample is 620. This small number of disks was chosen in light of two factors: (a) The numericalexperiment should represent an element test like that in geotechnical laboratory; (b) The observedbehaviour of a discrete assembly should represent a continuum material. This requires that thenumber of particles should be large enough to enable the specimen to demonstrate a rather ‘stable’and reasonable response under loading. Previous DEM analyses in literature show that 500 particlesappear to be the bottom line satisfying this latter requirement, although fewer particles have alsobeen used to examine some specific issues. In order to reduce possible effects resulting from thedifferences in particle sizes and constitutive models of contacts in the study, frictionless boundarieswere used, see Section 2.2. Hence, it is believed that the number of particles chosen for the studyis reasonable.

In addition, the cylindrical specimens in Reference [22], which size is 50mm diameter× 100mmhigh, are represented here by 2-D DEM samples. This simplification is also acceptable. This isbecause both 2-D and 3-D assemblies are a kind of mechanical system, they must obey and sharebasic laws. It is these laws that would enhance understanding the behaviour of natural soils andsubsequently establishing their practical macro-constitutive models. In addition, we shall investigatethe behaviour of bonded materials under isotropic compression tests (explained later), in whichthe principal stresses satisfy �1 = �2 = �3 as in Reference [22].

4.2. Generation of DEM bonded specimen

Large voids and bonds are usually regarded as two main features of naturally microstructured soilsin comparison to reconstituted soils [8–12]. In Reference [22], a mixture of dry soil and Portlandcement was compacted at a large void ratio, isotropically compressed at specific stresses and filledwith water to form bonds. Similar to Reference [22], the DEM bonded samples were prepared intwo steps: (a) generation of unbonded samples at different densities using the undercompactiontechnique [48] for DEMs; (b) creation of bonds at all contacts in the samples. The main reasonsfor using undercompaction technique come from: (a) the possibility of generating loose specimens,which is difficult for usual sample preparation technique [32, 33]; (b) the difficulty of producinghomogeneous samples by isotropic compression, which leads to the central part being looserthan the outer part; (c) the fact that most natural soils were formed in essentially 1-D condition;(d) the observation that when a soil sample is compressed one-dimensionally, the bottom partof the specimen generally has a void ratio differing from the upper part. The undercompaction



technique [48] overcomes these shortcomings. It can be briefly described as follows.

(i) The sample is constituted of several horizontal layers, four here. Consequently, thetotal number of particles is divided by four and each layer has the same number ofparticles.

(ii) The particles of each layer are randomly distributed in an area corresponding to a planarvoid ratio of about 0.65, i.e. about twice the planar void ratio targeted after compaction.They are then statically compacted, with the lateral and bottom walls being fixed and thetop wall moving downward at a constant speed till that the targeted planar void ratio isreached.

(iii) In order to ensure that all the layers undergo a final average planar void ratio ep f , the bottomlayer, 1, must be compacted at a planar void ratio ep(1) larger than ep f . Similarly, the secondlayer, 2, must follow such a way that the layers 1 and 2 have an average planar void ratioep(1+2) larger than ep f , etc. The final layer is directly compacted at ep(1+2+···+n) = ep f .Hence,

ep(1)>ep(1+2)> · · · >ep(1+2+···+n) = ep f (11)

The distribution of ep(1), ep(1+2), . . . , ep(1+2+···+n) = ep f has to be determined by trial anderror. However, an equation suggested in Reference [48] was used in this study. For example, toobtain a ep f of 0.30, the initial void ratios are ep(1) = 0.360, ep(1+2) = 0.353, ep(1+2+3) = 0.333,ep(1+2+3+4) = 0.30.

After the unbonded specimens were prepared at the required density, all particle contacts werebonded with a set of bond model parameters for further testing. The whole procedure for thesample preparation can be described as follows.

(i) Unbonded state. The soil is firstly loaded one-dimensionally by using the undercompactiontechnique up to a targeted planar void ratio. During this phase, an interparticle frictioncoefficient � is selected to be 1.0 for obtaining loose samples, and the wall–particle frictionis set to be zero in order to improve the homogeneity. Four different values of planar voidratios were considered: 0.34 (described here as very loose), 0.32 (loose), 0.30 (medium-dense) and 0.28 (dense). Such choices are to investigate the effect of densities on themechanical behaviour of bonded material like [22].

(ii) Equilibrium state. The friction coefficient � is decreased to a value of 0.5, which is a typicalvalue used in DEM analyses. The sample is then unloaded to a vertical stress of 12.5 kPa.During this stage, a slight change in specimen height may occur in the sample before itarrives at equilibrium state.

(iii) Bonded state. All the contacts between particles are then bonded with a set of bond modelparameters. Because the effect of cements may be dependent on sedimentation process,which is different at different sites in the world and which is one of our future works, ithas been considered in the present study that Rnb = Rtb = R for simplicity, with a widerange of values as 0, 1, 5, 10, 20 and 30MN/m. Hence, only one additional parameter, R,was used here in comparison to the standard contact model [32, 33]. The different R isused to simulate different cement contents in Reference [22]. A reasonable range of contactparameters may be our future task. After bonding, � is maintained at a value of 0.5, exceptfor two special cases (shown in Figure 18) in which � = 0.2 was used.



Particle

200 mm

18

3m

m

Figure 7. A typical bonded sample used for the DEM tests.

Figure 7 illustrates a typical bonded sample of planar void ratio as 0.334 generated for the DEMtests in this study, in which bonding material are marked as black dot lines at particle contacts.Several large voids and particles are also noted. Figure 7 shows that both large voids and bonds,the two main features of naturally microstructured sands, have been captured in the sample.

Other material parameters are the normal and tangential spring stiffnesses, which were taken,respectively, as 1.5× 1010 and 1.0× 1010 N/m. These values were chosen by the first author’sresearch experience and by trials and errors to produce reasonable numerical results similar toexperimental data.

4.3. Test procedure and examined aspects

The boundary conditions before and during testing are illustrated in Figure 8. The DEM bondedsamples are prepared with the condition shown in Figure 8(a), in which frictionless sidewalls arefixed and in which a vertical pressure of 12.5 kPa is applied through movable frictionless top/bottomwalls. During testing, all four walls are movable and a set of pressures is applied identically onthe samples through these walls, as shown in Figure 8(b). The initial pressure applied is 25 kPa,with incremental ratio as 1.5 for the subsequent loadings, i.e. the pressure applied is 25.0, 37.5,56.26 kPa, respectively, in a similar manner to [22]. Note that the wall–particle friction/bondingis always set as zero in order to eliminate any boundary effects. A total of 24 numerical testshave been performed on the materials of �= 0.5 but having different bonding strengths and initialvoid ratios in a similar manner as in Reference [22]. In addition, two isotropic compression testshave been carried out on the very-loose unbonded material and the medium-dense material ofRnb = Rtb = R = 1.5MN, but with �= 0.2 for the both. The results of the tests are described onep–log �′

v scale, with the former 24 tests mainly used to examine yielding, post-yield compressionlines, bulk modulus, etc., which are then compared with experimental data in Reference [22] toshow the capability of the model. The latter two special tests were designed to verify the Coopand Willson criteria [15] on weak bonding and strong bonding (explained later in Section 6).



Vertical pressure

v = 12.5 kPa Movable frictionless rigid boundary

Fixed frictionless rigid boundaries

Particle assembly

Movable frictionless rigid boundary

B

H

Movable frictionless rigid boundaries

Particle assembly

v ,

Vertical pressure

v , applied incrementally



vh

hh

Horizontal pressure

h = v h

σ

σ σ

σ

σ

(a) (b)

σ

Figure 8. Boundary conditions in the DEM tests: (a) sample preparation; and(b) isotropic compression tests.

Table I provides a summary of test programme for experiments in Reference [22] and DEM testsin this study. Table I shows that the DEM tests in this study follow the procedures of specimengeneration, bonding process, formal tests and examination manner employed in Reference [22],except for six items:

(i) 2-D specimens are used here while cylindrical specimens were used in Reference [22]. Thisleads to the fact that the planar void ratio 0.28, 0.30, 0.32 and 0.34 correspond, respectively,to void ratio 0.46, 0.54, 0.57 and 0.62 in examining the effect of densities, which are termedas dense, medium-dense, loose and very loose materials, respectively;

(ii) The bonding strength is controlled here by different value of Rnb = Rtb = R, while cementpercentage was employed in Reference [22]. This allows that R = 10, 20, 30MN correspondapproximately and, respectively, to 1, 2 and 3% in Reference [22] in investigating the effectof bonding strength;

(iii) Initial curing (bonding) void ratios are directly achieved by undercompaction techniquehere for simplicity while they were controlled by isotropic compression in Reference [22].Hence, there seems to be difference in their initial state before isotropic compression.However, our preliminary study, in which the curing process in Reference [22] was strictlysimulated, showed no significant difference on the main part of compression curves of thebonded material between current process and that in Reference [22];

The reason for choosing the current process in Item (iii) will be explained in Section 6. We shallintroduce the main comparison between the DEM numerical and experimental data in the nextsection.



Table I. Summary of test programme for experiments in Reference [22] and DEM tests in this study.

Rotta et al. [22] DEM in this study

Name of method Undercompaction Undercompactionmethod (Ladd [66]) method [48]

Number of 3 4compaction layers

Specimen Compaction criteria Ladd [66] Jiang et al. [48]generation

Grain size distribution Figure 6(a) Figure 6(b)

Sample size Cylindrical specimens 2-D specimens; about200× 180 (mm×mm)

Method to control Isotropic compression Undercompactioninitial void ratio method [48]

Bonding (Planar) Void ratio 0.46, 0.54, 0.57, 0.62 0.28, 0.30, 0.32, 0.34(curing) at bonding (Planar void ratio)process

Bonding index Cement percentage: Bonding strength:0, 1, 2, 3 0, 1, 5, 10, 20, 30 (MN)

Test condition Isotropic compression Isotropic compression

Total number of tests 18 25

Compression curve Compression curve

Initial bulk modulus Initial bulk modulus

Formal tests Primary yield loci, post- Gross-yield loci, post-and yield compression lines yield compression lines

examination Examined aspects Incremental yield stress Incremental yield stressmanner against curing void ratio against curing void ratio

Yield stress against Yield stress againstcement content bonding strength

No Mechanism of gross-yielding,bond breakage

No Coop and Willson criteriaon weak and strongbonding [15]

No Strong bonding behaviourin Reference [22]

5. COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL DATA

Of the 24 DEM numerical tests performed, only some will be presented in this section to comparewith the experimental data in Reference [22]. We shall first examine their isotropic compressioncurves. Then we shall compare their bulk modulus. Finally, we shall investigate their yielding onmacroscopic scale. Focus will be put on the effect of different bonding strengths and densities.



5.1. Isotropic compression curves

Figure 9 presents the variation of planar void ratio ep with applied mean stress �′m in the isotropic

compression tests observed on the specimens of different cement contents in laboratory [22]and of different bonding strengths in the DEM tests. The DEM material of bonding strengthRnb = Rtb = R = 10, 20, 30MN is compared, respectively, to the experimental material of cementcontent 1, 2 and 3%. They are shown in pairs in Figures 9(I–III), respectively, in which theexperimental data is shown in (a) and the DEM data in (b).

Like the uncemented sand shown in the experimental data, the numerical curve obtained forthe very-loose unbonded material in Figures 9(I(b)–III(b)) provides a reference curve for theother DEM tests. This curve represents the intrinsic soil response in its destructured (uncemented)state, and is sometimes named intrinsic compression line (ICL) [22] or normal compression line(NCL) [67] in the geotechnical community. This numerical reference curve shows three features:there is a significant reduction in void ratio (ep) against �′

m when the pressure is relatively small(<0.4MPa); once �′

m exceeds this latter value, the rate of void ratio reduction becomes smaller;this is true until the pressure becomes close to 10MPa when the void ratio reduction rate increasesagain. The first feature can be observed in the experimental data, while the second and third onesare not. The second feature in the DEM data may come from that the particle crushing is notconsidered in the DEM study while it often occurs in naturally microstructured sands under highstress level [39–42, 67]. Hence, the small void ratio reduction indicates that there is a little spacefor the assembly to be compacted further as it arrives at a relatively dense state. This featureis in agreement with the observation on 1-D compression experiment on Alumina powder [68],where crushability of particles is very small. However, in comparison with the experimental data inFigures 9(I(a)–III(a)), the third feature of the numerical reference curve for �′

m�10MPa appears tobe unusual. This peculiar feature is largely due to the fact that a linear-elastic normal contact modelwas chosen in the DEM studies and overlap between particles is excluded from calculating ep byEquation (8). With the linear contact law, the overlap � between two particles of radius r1 and r2,increases linearly with normal contact force, and hence with �′

m , where � = r1 + r2 − D with Dequal to the distance between the particle centres. However, the overlap area A0 between the twoparticles increases non-linearly and evidently with the increasing of �. Theoretically, omitted here,the ratio of A0/� increases with �. This is to say, A0 increases rapidly with � and subsequentlywith �′

m in the DEM simulations. As can be seen easily from Equation (8), the neglect of A0leads to a smaller ep than a realistic planar void ratio. Hence, the higher �′

m is, the larger isthe difference between the calculated ep and its actual counterpart. This gives rise to the secondsignificant reduction of ep as shown in Figures 9(I(b)–III(b)). Large normal contact stiffness or anon-linear normal contact law may reduce the effect of overlaps on ep. In addition, overlaps canbe all or partly included in calculating ep in order to improve the performance of the numericalsimulations at high stress level. Nevertheless, these methods are not adopted in this study forsimplicity. This is because: (a) particles in DEM are idealized as rigid disks and a small amountof overlaps may reasonably represent elastic deformation of particles. It is very complicated todecide how much overlaps should be included in calculating ep; (b) the ignorance of A0 shouldaffect ep for both bonded and unbonded materials identically at high �v . Such ignorance may notaffect the difference between bonded and unbonded materials; and (c) the yield stress discussedbelow never falls within the range of this second significant reduction of ep. The ignorance thusdoes not change our conclusions.



0.01 0.1 1 10 1000.24

0.26

0.28

0.30

0.32

0.34

R= 0 MNVery-looseLooseMedium-denseDense

0.01 0.1 1 10 1000.24

0.26

0.28

0.30

0.32

0.34

0.01 0.1 1 10 1000.24

0.26

0.28

0.30

0.32

0.34

R= 0 MN

I(b)

II(b)

III(b)

Applied mean stress(MPa)



Plan

ar v

oid

ratio

Plan

ar v

oid

ratio

Plan

ar v

oid

ratio

Medium-denseLooseVery-looseR= 0 MN

Dense

Medium-denseLooseVery-loose

Dense

Yield point

III(a)

I(a)

II(a)

Figure 9. Isotropic compression responses observed on specimens of different cement contentin laboratory [22] and of different bonding strengths in the DEM tests: (I) experimental dataof cement content 1% (a); DEM data of R = 10MN (b); (II) experimental data of cementcontent 2% (a); DEM data of R = 20MN (b); and (III) experimental data of cement content

3% (a); DEM data of R = 30MN (b).



As for bonded materials, Figure 9 shows that there is slight difference between the numericaland experimental data: the gross-yielding stress in light of volumetric strain (also known aspreconsolidated pressure in the geotechnical community) is well defined for the DEM samples(except for dense specimens); however, it is more progressive for the artificially cemented sands.For example, the very-loose bonded DEMmaterial of R = 10MN obviously defines a yielding pointwith its stress value about 0.64MPa in Figure 9(I(b)), while its counterpart of cement content 1%shows a progressive yielding in Figure 9(I(a)). Such difference may attribute to material difference.In the DEM materials, bonding exists only at particle contacts. However, in the artificially bondedsands, bonding materials may exist at particle contacts, voids and particle surfaces. A clear definitionof the yielding has often been reported in the literature (e.g. References [6, 9, 11]). Nevertheless,in comparison with the each corresponding reference curve, Figure 9 shows that the bondedDEM samples capture several salient features of the artificially cemented sands in Reference [22]as follows.

(i) They are able to go in part of the ep–�′m space that is not accessible to the unbonded

material; at a given consolidation stress, their void ratio is generally larger than theunbonded counterpart. This feature has been often reported in the literature for structuredand naturally/artificially cemented soils (e.g. Reference [9]).

(ii) The pre-yield deformation is small; deformation increases abruptly if consolidation pressureis larger than the yield stress. A typical example is the compression curve of the very-loosebonded material of R = 10MN in Figure 9(I(b)).

(iii) Given a bonding strength (cement content for the experimental data), such as R = 30MN,the post-yield compression curves of the material of different densities tend to coincidewith each other (the dot line in Figure 9(III(b))).

The observation in Figure 9 shows that the DEM model can give predictions similar to the exper-imental data in Reference [22]. We shall further examine them later. Because of the slight differenceon the performance at the yielding, we shall adapt the primary yield for the experimental data [22]but use the gross yielding for the DEM tests for the further analyses below. The definitions ofprimary yield and gross yield are illustrated in Figures 10(a) and (b), respectively. In contrast to thegross yielding that corresponds to the most-gradient point on a compression curve (about 4.5MPain Figure 10(b)), the primary yield in Figure 10(a) represents a state where the stress–strain curvedeviates from the initial linear behaviour (about 1700KPa in the figure). The primary yield inFigure 10(a) is believed to represent the point at which breakage of the cement bonds com-mences [22]. The gross yielding is also associated with bond breakage, which will be discussed laterin Section 6. These two yielding may give rise to an identical value, such as in the DEM simulationof the very-loose specimen of R = 10MN in Figure 9(I(b)). They may lead to different values, whichdepends on the microstructure of soils. However, even in this latter case, it is believed that these twoyielding should be strongly linked to each other through bond breakage. Hence, it is acceptable touse the primary yield for the experimental data [22] and the gross yielding for the DEM tests in com-parison. We choose the gross yielding for the DEM data because: (a) unlike the experimental data inReference [22], the gross yielding is well defined in most cases; (b) the gross yielding has beenwidely used as a design index for practical engineering in the geotechnical community and as an im-portant parameter in constitutive modelling of natural soils in geomechanics. We shall compare thevariations of bulk modulus and yielding observed in Reference [22] and in the DEM tests in the nextsubsections.



0 .01 0. 1 1 10 1000. 24

0. 26

0. 28

0. 30

0. 32

0. 34

DEM te sts

Applied mean stress (MPa)

Plan

ar v

oid

ratio

(a) (b)

Gross-yield point

Figure 10. Definitions of: (a) primary yield in the experimental data [22];and (b) gross-yield in the DEM tests.

0 500 1000 1500 20000.28

0.30

0.32

0.34 R= 10 MN R= 20 MN R= 30 MN

Cur

ing

void

rat

io

Bulk modulus (MPa)(a) (b)

Figure 11. Variations of initial bulk modulus with curing (bonding) void ratio observedon specimens: (a) of different cement content in laboratory [22]; and (b) of different

bonding strengths in the DEM tests.

5.2. Bulk modulus

Figure 11 presents variations of bulk modulus with curing (bonding) void ratio observed onspecimens of different cement content in laboratory [22] (Figure 11(a)), and of different bondingstrengths in the DEM tests (Figure 11(b)). The modulus is calculated by the incremental meanstress over the incremental volumetric strain between the initial and yield points. Figure 11 shows



that the DEM numerical data is very similar to the experimental counterpart:

(i) Given a bonding strength (cement content for the experimental data), bulk modulus reduceswith the increasing of curing void ratio ep (the void ratio of the sample being bonded).For example, for a bonding strength of 20MN, bulk modulus reduces from 1072.7MPa atep = 0.2965 to 293.8MPa at ep = 0.334 in the DEM tests.

(ii) For a given curing void ratio, bulk modulus increases with the increasing of bonding strength(cement content). For example, for ep = 0.2965, bulk modulus increases from 705.6MPaat R = 10MN to 1617.4MPa at R = 30MN in the DEM tests.

However, Figure 11 also shows slight difference between the experimental and DEM data, whichmay give some useful indications. The experimental data in Figure 11(a) shows that the increasein the bulk modulus caused by an increase in cement content is much more pronounced at lowervoid ratios, and that the bulk moduli tend towards a unique low value as the curing void ratioincreases. Such coupled effect of density and cementation on the mechanical behaviour has beenalso reported elsewhere on artificially bonded soils, e.g. Reference [18]. Such tendency is not sowell observed in the DEM data. For example, the DEM tests predict a bulk modulus still varyingfrom 200MPa at R = 10MN to 536.7MPa at R = 30MN for the very-loose materials, instead of aunique value similar to experimental data. It is the authors’ opinion that, in laboratory, the bondingmaterials may be artificially formed not only at interparticle contacts, but also on the particlesurfaces. Some bonding materials may also fill in some voids of the soil. Hence, the observedmechanical behaviour may come from the combination of several factors in laboratory, instead ofinterparticle bonding alone, which leads to the coupled effect. Such effect may be simulated byDEM but is not included in the present study. In the DEM tests, the effects of bonding strengthand density are investigated separately, which can be regarded as another advantage of DEM testsover experiments.

5.3. Yielding

Figure 12 provides yield loci and post-yield compression lines (PYCL) observed on specimensof different cement contents in laboratory [22] (Figure 12(a)), and of different bonding strengthsin DEM tests (Figure 12(b)). Like the experimental data in Reference [22], the yield loci andthe PYCL in Figure 12(b) can be represented simply as straight lines. Note that they cannot bestraight over an extended pressure range, since the lines cannot cross the ICL. Figure 12 showsthat the numerical simulations capture the main feature of the experimental data: both yield lociand PYCL expand and steepen as the bonding strength (cement content for experimental data)increases. For example, yield loci and PYCL of R = 30MN are above those of R = 10MN IN theDEM tests, with the former slopes larger than the later. By extrapolating the PYCLs and the ICLshown in Figure 12, it can be seen that slow convergence happens for numerical data, regardlessof the bonding strength, which is similar to the experimental data for different cement content.Such a slow convergence seems not unusual for artificially cemented sands, and is regarded asa function of the base material used in geotechnical engineering, e.g. Reference [22]. However,Figure 12 also shows slight difference between the DEM and experimental data. The yield loci bythe numerical data in Figure 12(b) come from the gross-yielding points; thus, they are observedto be very close to PYCL. In contrast, the yield loci deduced from the experimental data inFigure 12(a) are deduced from primary yielding points; hence, they lie between the ICL andPYCL. The coincidence of the yield loci and the PYCL in the DEM data appears to be consistent



0.01 0.1 1 100.24

0.26

0.28

0.30

0.32

0.34

Plan

ar v

oid

rati

o


Intrinsic compression line Yield loci: R=10 MN Yield loci: R=20 MN Yield loci: R=30 MN Post-yield compression line: R= 10 MN Post-yield compression line: R= 20 MN Post-yield compression line: R= 30 MN

(a) (b)

Figure 12. Yield loci and post-yield compression lines observed on specimens: (a) of different cementcontent in laboratory [22]; and (b) of different bonding strengths in the DEM tests.

0 0.28

0.29

0.30

0.31

0.32

0.33

0.34 R= 10 MN R= 20 MN R= 30 MN

Plan

ar v

oid

ratio

at b

ondi

ng

Incremental yield stress (MPa)(b)(a)1 2 3 4 5 6 7 8

Figure 13. Variations of incremental yield stress with curing void ratio observed on specimens: (a) ofdifferent cement content in laboratory [22]; and (b) of different bonding strengths in DEM tests.

with the observation on the natural calcarenite tested by Cuccovillo and Coop [5, 67], in which theyield locus practically coincided with the PYCL. Hence, we may conclude that the DEM modelcan capture the features of the yield loci and PYCL of bonded materials.

Figure 13 presents variations of incremental yield stress (structural increase in yield stress��y) with curing void ratio observed on specimens of different cement content in Reference [22](Figure 13(a)), and of different bonding strengths in the DEM tests (Figure 13(b)). The incrementalyield stress is defined as the difference between the yield stress on the isotropic compression curveof bonded specimens and the stress on the ICL at the corresponding void ratio. Figure 13 shows



0 5 10 15 20 25 30 351E-3

0.01

0.1

1

10

100

Bonding void ratio: 0.34Bonding void ratio: 0.32

Bonding void ratio: 0.30Bonding void ratio: 0.28

Gro

ss-y

ield

ing

stre

ss (

MPa

)

Bonding strength (MN)(a) (b)

Figure 14. Variations of yield stress with cement content observed on specimens: (a) of different cementcontent in laboratory [22]; and (b) of different bonding strengths in DEM tests.

that the DEM simulations completely reflect the two features of the experimental data:

(i) for specimens cured at the same void ratio, ��y increases with the increasing of bondingstrength (cement content for the experimental data); For example, the loose material inthe DEM simulations show its yield stress as 1.19MPa at R = 10MN and as 5.01MPa atR = 30MN;

(ii) for specimens with the same bonding strength, ��y increases with the reduction of curingvoid ratio. For example, for a bonding strength of 10MN. ��y increases from 0.63MPa atep = 0.334 to 3.03MPa at ep = 0.2965.

The first feature clearly results from the bonding strength of bonds. The second arises fromthe fact that an increase in density gives rise to an increase in the number of contact pointsbetween the soil particles where bonds can be formed. This could be confirmed by the DEMstudy.

Figure 14 provides variations of yield stress against cement content observed in laboratory [22](Figure 14(a)), and against bonding strength measured in the DEM tests (Figure 14(b)). The DEMmaterials are very-loose, loose, medium-dense and dense, having R from 0.0 to 30.0MN. Theartificially cemented soil in Reference [22] is characterized with cement content changing from 0.0to 3.0% and with initial void ratios as 0.47, 0.54, 0.57 and 0.62, respectively. The full comparisonbetween Figures 14(a) and (b) shows that the DEM materials demonstrate the variation of yieldstress in a very similar way to the artificially cemented soil:

(i) the yield stress increases with bonding strength (cement content for the experimental data)for a given density;

(ii) for a given bonding strength (cement content), the yield stress reduces with the increasingof void ratio; moreover,

(iii) the increase in the yield stress caused by an increase in bonding strength (cement content)is much more pronounced at lower void ratios.



The first two items are consistent with the observation in Figure 13 and can been easily found inFigure 14. Item (iii) can be explained by the DEM curves of ep = 0.34 and 0.28 as examples. Forep = 0.34, the gross-yielding stress increases significantly from 0.25MPa at R = 5MN to 2.3MPaat R = 30MN. In contrast, the boned specimen of ep = 0.28 shows little change of the yield stressfrom R = 5 to 30MN. Note that Item (iii) indicates that the relative contribution of cementationto the soil behaviour in isotropic compression reduces with decreasing curing void ratio.

The comparison between the DEM and experimental data [22] in this section indicates thatthe DEM model proposed can capture the main mechanical features of bonded sands. Conse-quently, in the next section, we shall carry out further investigation on some important issues inReference [22].

6. DISCUSSION

In this section, we shall discuss two important issues in Reference [22]: the link between yieldingand bond breakage; the strong bonding phenomenon (explained later). The first issue will bediscussed by examining the relation between gross yielding and bond breakage, based on DEMdata obtained from the material tested at different initial densities and different bonding strengths.The second issue will be discussed with the help of the DEM tests on the medium-dense bondedmaterial, after examining Coop and Willson criteria on weak bonding and strong bonding [15, 67]by the DEM tests.

6.1. Link between yielding and bond breakage

It is believed in Reference [22], as well as in References [5, 8–12, 17], that yielding of naturallymicrostructured sands is largely related to bond breakage. However, there is little verification ofthis statement in the literature. We shall examine it below based on the DEM numerical data onmaterials of different initial densities and different bonding strengths.

Figure 15(a) presents the variation of planar void ratio ep with applied mean stress �′m , whereas

Figure 15(b) provides the broken contact ratio as a function of �′m , for the very-loose, loose

and medium-dense DEM materials of the bonding strength Rnb = Rtb = R = 10MN. The brokencontact ratio is the proportion of the initially bonded contacts that have been broken and can beregarded as a damage index of the bonds. Like the gross yielding in Figure 15(a), Figure 15(b)shows that the yielding can be well defined for the samples in terms of bond breakage (the largestgradient point on the curve) for all specimens, which namely the micro-yielding hereafter. Thereis no or few bond breakage when �′

m is smaller than the micro-yielding stress. Once �′m exceeds

the micro-yielding stress, a large amount of bonds break. By comparing Figure 15(a) with (b),with the help of the dotted lines 1–3, it is observed that the micro-yielding stress appears tobe equal to or slightly smaller than the respective gross-yielding stress. For example, the very-loose bonded material gives the two yielding the same values as 0.64MPa, while the medium-dense bonded material leads to the gross-yielding (micro-yielding) stress as 3.2MPa (3.0MPa).Hence, the DEM results here indicate that the gross yielding appears to be in essence associatedwith breakage of bonds, which supports the conceptual statement in geomechanics by severalresearchers [5, 8–12, 17] that the gross yielding of naturally microstructured sands must be relatedto bond breakage. For further confirmation, we shall next examine the numerical data deduced forthe materials of different bonding strengths.



1 2 3

(a)

(b)

0 .01 0. 1 1 1 0 1 0 00.2 4

0.2 6

0.2 8

0.3 0

0.3 2

0.3 4

R= 0 MN

Loose

0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

Bro

ken

cont

act r

atio

Loose

Plan

ar v

oid

ratio



Very-loose

Medium-dense

Very-loose

Medium-dense

Figure 15. Yielding observed in DEM tests on materials of R = 10MN but of different densities from:(a) compression curves; and (b) broken contact ratio.

Figure 16(a) presents the influence of bonding strength on isotropic compression curves forthe very-loose DEM materials of R = 0, 1, 5, 10, 20, 30MN, whereas Figure 16(b) provides thecorresponding variations of broken contact ratio. Again, like the gross-yielding in Figure 16(a),Figure 16(b) demonstrates that the micro-yielding can be well defined for the very-loose samplesof different bonding strengths. No or few bonds break when �′

m is smaller than the micro-yielding value; however, once �′

m exceeds the micro-yielding, a large amount of bonds break.In addition, comparison between Figures 16(a) and (b) also shows that the micro-yielding stressseems to be equal to or slightly smaller than the respective gross-yielding stress. For instance,



0.01 0.1 1 10 1000.24

0.26

0.28

0.30

0.32

0.34

0.01 0.1 1 10 1000.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4 5

Applied mean stress (MPa)(b)

(a)

Plan

ar v

oid

ratio

Bro

ken

cont

act r

atio

R=0 MN

R= 1 MN

R= 5 MN

R=10 MN

R=20 MN

R=30 MN

R= 1 MN R= 5 MN R= 10 MN R= 20 MN R= 30 MN


Figure 16. Yielding observed in DEM tests on very-loose materials of different bonding strengths from:(a) compression curves; and (b) broken contact ratio.

the material of R = 10MN produces the two yielding the same values as 0.641MPa, while thematerial of R = 20MN has the gross-yielding stress as 1.44MPa and the micro-yielding stress as1.40MPa. Hence, we may draw a conclusion from Figures 15 and 16 that the yielding of naturallymicrostructured sands is largely related to bond breakage as stated in References [5, 8–12, 17, 22].This is in agreement with our previous study on microstructured soils under 1-D compressiontests [46, 47]. It also shows that the two yielding definitions in Figure 10 are inter-associated, eventhough bonding exists only at particle contacts in the DEM materials while bonding materials mayexist at particle contacts, voids and particle surfaces in the artificially bonded sands.



v

w

s

v

w

s

vw

s

(a)

(b)

(c)

w weak bondings strong bonding

intrinsic NCL

true yield

gross yield

ln p’

ln p’

ln p’

Figure 17. Weak bonding (w) and strong bonding (s) criteria used by Coop and Willson [15] in investigatingthe factors that might influence the effect of bonds on the compression behaviour of a sand: (a) strength

of bonds; (b) position of normal compression line (NCL); and (c) initial volume.

6.2. Strong bonding phenomenon in Reference [22]We shall first verify Coop and Willson criteria on weak bonding and strong bonding [15, 67] withour DEM model. Then, we shall discuss the reason why Rotta et al. [22] have only observedone strong bonding phenomenon in their experiments, with the help of the DEM tests on themedium-dense bonded material.

Figure 17 provides schematic representation of the factors that might influence the effect ofinter-particle cementing on the compression behaviour of a natural sand, proposed by Coop and



Willson [15, 67]. A salient phenomenon in Figure 17 is that yielding in compression may occurabove or below ICL, which is named NCL by Coop and Willson [15, 67] in the figure. The formerone is termed ‘strong bonding (s)’, while the latter ‘weak bonding (w)’ in References [15, 67]in their interpretation of the behaviour observed on two natural sands. They believe that whetherthe bonding is weak or strong depends on three factors [15, 67] which, regardless of particlebreakage/crushing, are:

(i) Firstly and primarily, the amount and strength of the cement deposited, with a smalleramount clearly being present for the weak bonding while a larger amount for the strongbonding, as indicated in Figure 17(a);

(ii) Secondly, the position of the intrinsic NCL. The same yield stress may be regarded asstrong bonding if NCL is below conditions at yielding, or weak bonding if NCL is above,as shown in Figure 17(b);

(iii) Thirdly and finally, the initial density. A denser bonded material, even though it has thehigher yield stress in compression, may show the weak mode of behaviour, whereas a looserbonded material of the lower yield stress may show the strong mode, as demonstrated inFigure 17(c).

The criteria aforementioned were deduced from the experimental data. We shall here verifythe Coop and Willson criteria with our DEM model so that we can then understand why Rottaet al. [22] have only observed one strong bonding phenomenon in their experiments. Figure 18presents the variation of planar void ratio ep with applied mean stress �′

m in the isotropic DEMcompression tests, with the focus on the three factors: strength of bonds (Figure 18(a)); positionof NCL (Figure 18(b)); and initial density (Figure 18(c)). It can be seen:

(i) in Figure 18(a) that the DEM medium-dense material gives a weak bonding behaviour atbonding strength R = 1MN but a strong bonding behaviour at R = 10MN. This confirmsthe first Coop and Willson criterion that the amount and strength of the cement is a factorinfluencing the effect of inter-particle cementing on the compression behaviour of a naturalsand, with a smaller amount clearly for the weak bonding while a larger amount for thestrong bonding.

(ii) in Figure 18(b) that the response of the two unbonded very loose materials shows thatNCL of �= 0.5 (NCL1 in the figure) is well above that of � = 0.2 (NCL2 in the figure),which is reasonable since larger � produces larger resistance against compaction under agive pressure. The two bonded DEM medium-dense materials, which have R = 1MN for�= 0.5 (Material 1) and R = 1.5MN for � = 0.2 (Material 2), respectively, predict almostthe same gross-yielding stress, with the compression curves approaching their own NCLafter yielding. In addition, the gross-yielding stress of the two bonded materials lies betweenNCL1 and NCL2, indicating that Material 1 should be regarded as a weak bonding sinceNCL1 is above it, while Material 2 a strong bonding since NCL2 is below it. This confirmsthe second Coop and Willson criterion that the position of the intrinsic NCL is anotherfactor influencing the effect of inter-particle cementing on the compression behaviour of anatural sand.

(iii) in Figure 18(c) that the very-loose bonded DEM material, even though it has the loweryield stress in compression, shows the strong mode of behaviour, whereas the medium-dense bonded material of the higher yield stress shows the weak mode. Note that thesetwo bonded materials both have R = 1MN. Hence, Figure 18(c) confirms the third Coop



0.01 0.1 1 100.26

0.28

0.30

0.32

0.34

NCL

0.01 0.1 1 100.26

0.28

0.30

0.32

0.34

Very-loose, unbonded (ICL)

NCL

0.01 0.1 1 100.24

0.26

0.28

0.30

0.32

0.34

Plan

ar v

oid

ratio

Very-loose,unbonded, � = 0.5Medium-dense, R=1.0 MN, � = 0.5Very-loose,unbonded, � = 0.2

NCL1

O

NCL2




Plan

ar v

oid

ratio

Plan

ar v

oid

ratio

Weak bonding

Strong bonding

Very-loose, unbonded (ICL) Medium-dense,R=1 MN Medium-dense,R=10 MN

Weak bonding

Strong bonding

yield point Same

Strong bonding

Weak bonding

Very-loose,R=1 MNMedium-dense,R=1 MN

(a)

(b)

(c)

Medium-dense, R=1.5 MN, � = 0.2

Figure 18. The factors influencing the effect of bonds on the compression behaviour observed in the DEMtests: (a) strength of bonds; (b) position of normal compression line (NCL); and (c) initial density.



0.01 0.1 1 10

Plan

ar v

oid

ratio

NCL


Strong bonding

Start pointin [22]

0.34

0.32

0.30

0.28

0.26

Weak bonding

Medium-dense,unbonded (ICL2)

Medium-dense,R=1 MN

Very-loose, unbonded (ICL1)

Figure 19. DEM simulation of strong bonding phenomenon in the experiment [22].

and Willson criterion that initial density is an important factor influencing the effect ofinter-particle cementing on the compression behaviour of a natural sand.

The agreement between Figures 17 and 18 shows that our DEM model is promising in the studyof natural sands and confirms the Coop and Willson criteria. However, Rotta et al. [22] did notobserve the weak bonding phenomenon in their experiments on the artificially bonded sand, andthey simply attributed the weak bonding phenomenon to the cementation happening after a stronggeological over-consolidation. This aspect will be discussed hereunder.

Figure 19 provides the variation of ep with �′m in the isotropic DEM compression tests observed

on the unbonded very-loose (NCL), unbonded medium-dense specimens and the medium-densespecimen of R = 1MN. In comparison to NCL (or ICL1 in the figure), the medium-dense materialis initially at a state of over-consolidation which has resulted from the compaction during specimengeneration. In addition, the medium-dense specimen of R = 1MN almost starts at the same initialpoint as its unbonded counterpart; hence, it starts at the compression curve of its unbondedcounterpart (ICL2 in the figure), which is the procedure adopted by Rotta et al. [22] in theexperiments. Figure 19 shows that ICL2 deviates from ICL1 initially, due to their difference ininitial density; but they coincide with each other with the increasing of �′

m . This behaviour isin agreement with the observation on reconstituted/unbonded sands in laboratory that isotropiccompression samples of all initial densities tend to a unique NCL [67]. More importantly, Figure 19shows that, although the DEM bonded medium-dense material predicts a weak bonding behaviourin comparison to NCL, it does give a prediction of strong bonding behaviour in comparison to ICL2at which the bonded material starts. Note that this latter phenomenon has also observed on ourDEM loose materials, which is omitted here. Recall that in Figures 9(I(b)–III(b)), the dense bondedmaterials show no gross-yielding stress over the NCL. Other DEM results on the dense materials,omitted here, also show little difference in compression curves between unbonded and bondedstates. Hence, Figure 19 can mainly explain why in Reference [22] all their bonded materials



show one strong bonding phenomenon in their experiments. This is probably because their bondedmaterials start at different ‘loose’ (or relatively loose) points on the same compression curve ofan unbonded material, which accidentally is NCL or ICL; hence, the effect of bonds naturallymakes their compression curves over this NCL, which are regarded as strong bonding accordingto the Coop and Willson criteria. Note that another possibility may be that bond agent may changeparticle strength or interparticle friction of the material, which then also naturally leads to strongbonding phenomenon when the material starts even at dense points on the same compression curveof an unbonded material. More works are needed in the future to examine this latter possibility.Nevertheless, we may conclude that the strong bonding phenomenon in Reference [22] comesfrom the fact that their bonded materials start at different points on the same compression curveof an unbonded material.

In addition, Figure 19 appears to confirm the statement in Reference [22] that weak bonding phe-nomenon has resulted from ‘the cementation happening after a strong geological over-consolidation’.In comparison to NCL, the medium-dense material is initially at a state of over-consolidation.Hence, its subsequent bonding can be regarded as ‘the cementation happening after a stronggeological over-consolidation’. As a result, the material predicts a weak bonding phenomenonin comparison to NCL, as shown in Figure 19. However, the material, as shown by the case ofR = 10MN in Figure 18(a), can also predict a strong bonding. In fact, the loose and medium-densematerials in this study all start at ‘a strong over-consolidation’. But, most of their bonded materialsshow a strong bonding. Hence, we believe that the statement in Reference [22] on weak bondingis incomplete, and better to be replaced by Coop and Willson criteria. This is why initial curing(bonding) void ratios are directly achieved by undercompaction technique in our study, instead ofthe procedure in Reference [22].

7. CONCLUDING REMARKS

This study presented a simple discrete element modelling of naturally microstructured sands byefficiently capturing the effect of cementing material between particles (bonds). This is particularlyimportant for understanding the behaviour of natural soils and subsequently establishing theirpractical macro-constitutive models. Hence, the model here may be as important to geomechanicsas the original DEM for dry granular materials [32, 33]. The main features of the study lie in:(a) a simple bond contact model [46, 47], in which bond rolling resistance is not accounted for,was used and hence the information on such material can be directly used to establish practicalconstitutive models for natural soils within classical continuum mechanics; (b) efficient numericaltechniques were proposed to implement this simple contact model into DEM; (c) the DEM codeNS2D, developed by the authors, were used to carry out numerical tests and the observationswere compared with recent experimental data on artificially bonded sands in Reference [22];(d) the DEM data were further used to discuss two important issues in Reference [22]. The mainconclusions from this study are:

(i) The simple bond model has been successfully implemented into a 2-D DEM code NS2Dwith the proposed efficient techniques such as ‘translational-axis’ method. Using NS2Dand the undercompaction technique, the DEM samples generated can capture two of mainfeatures of natural soils, i.e. looseness and bonds, for numerical experiments.

(ii) The proposed simple DEM model can efficiently capture the main mechanical behaviourof bonded sands. The bonded DEM materials are able to go in part of the ep–�′

m space



that is not accessible to the unbonded material; at a given consolidation pressure, their voidratio is generally larger than unbonded counterpart. The pre-yield deformation is small;deformation increases abruptly if consolidation pressure is larger than the yield stress. Givena bonding strength, the post-yield compression curves (PYCL) of the material of differentdensities tend to coincide with each other. For a given bonding strength, an increase invoid ratio reduces the yield stress and the bulk modulus, whereas an increase in bondingstrength increases the yield stress and the bulk modulus at a given void ratio. Both yield lociand PYCL expand and steepen as the bonding strength increases. The relative contributionof bonds to the soil behaviour in isotropic compression reduces with decreasing curingvoid ratio.

(iii) The proposed simple DEM model can investigate some important phenomena in bondedsands. The gross-yield curve of bonded sands is largely associated with bond breakage,confirming the deduction for naturally microstructured sands in the geotechnical community.In general, it is reasonable to use the Coop and Willson criteria in evaluating weak bondingand strong bonding [15, 67], which include the three factors influencing the effect of bonds:strength of bonds, position of normal compression line and initial density. In addition, thesolely observed strong bonding in the Rotta et al. experiments [22] lies in that their bondedmaterials start at different points on the same compression line. But, their interpretation forthe weak bonding is not as complete as Coop and Willson criteria.

The DEM model in this study can be extended into 3-D conditions, as well as in other areassuch as strain localization analysis of naturally microstructured sands and its compaction bandanalysis [69]. The results can be useful to further understand constitutive models for bondedmaterial, e.g. Reference [70]. In addition, one of our future works is to carry out physical testswhich should be linked to naturally microstructured sands and which can be exactly simulated byusing DEM.

ACKNOWLEDGEMENTS

The DEM code NS2D used in this study was first developed in 2000–2002 at Laval University, Canada,during the first author’s previous post-doctoral fellowship financially supported by NSERC, Canada. Thenumerical investigation was carried out during the first author’s current post-doctoral fellowship fundedby an EPSRC grant with number GR/R85792/01. The authors would like to thank Prof. F. Schnaid,Federal University of Rio Grande do Sul, Brazil; Dr Matthew Coop, Imperial College London, U.K. fortheir discussions on some work in the paper.

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Bonding Effect in Sands by DEM

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