Abstract An adhesive elasto-plastic contact model forthe discrete element method with three dimensional non-spherical particles is proposed and investigated to achievequantitative prediction of cohesive powder flowability. Sim-ulations have been performed for uniaxial consolidationfollowed by unconfined compression to failure using thismodel. The model has been shown to be capable of predict-ing the experimental flow function (unconfined compressivestrength vs. the prior consolidation stress) for a limestonepowder which has been selected as a reference solid in theEurope wide PARDEM research network. Contact plasticityin the model is shown to affect the flowability significantlyand is thus essential for producing satisfactory computationsof the behaviour of a cohesive granular material. The modelpredicts a linear relationship between a normalized uncon-fined compressive strength and the product of coordinationnumber and solid fraction. This linear relationship is in linewith the Rumpf model for the tensile strength of particu-late agglomerate. Even when the contact adhesion is forcedto remain constant, the increasing unconfined strength aris-ing from stress consolidation is still predicted, which hasits origin in the contact plasticity leading to microstructuralevolution of the coordination number. The filled porosity ispredicted to increase as the contact adhesion increases. Underconfined compression, the porosity reduces more graduallyfor the load-dependent adhesion compared to constant adhe-sion. It was found that the contribution of adhesive force
S. C. Thakur (B) · J. P. Morrissey · J. Sun · J. Y. OoiSchool of Engineering, The University of Edinburgh,King’s Buildings, Edinburgh EH9 3JL, Scotland, UKe-mail: [email protected]
J. F. ChenSchool of Planning, Architecture and Civil Engineering, Queen’sUniversity Belfast, Belfast BT9 5AG, Northern Ireland, UK
to the limiting friction has a significant effect on the bulkunconfined strength. The results provide new insights andpropose a micromechanical based measure for characterisingthe strength and flowability of cohesive granular materials.
Keywords Discrete element method · Contact model ·Cohesive material · Uniaxial test · Plasticity · Unconfinedstrength · Stress history dependence · Flow function ·PARDEM
List of symbols
d Particle diameter (m)davg Average particle diameter (m)e Co-efficient of restitutiong Gravitational constant (m/s2)
m∗ Equivalent mass of the particles (kg)N Number of particlesn Non-linear index parameterP Pressure (kPa)u Unit normal vectorZ Coordination number at peakZi Instantaneous coordination numberFat Average adhesive strength at contact (N)f0 Constant adhesive strength at first contact (N)ft Contact tangential force (N)fct Coulomb limiting tangential force (N)fnd Normal damping force (N)f ci Contact force (N)
fatp Average tensile force at the peak (N)fts Tangential spring force (N)ftd Tangential damping force (N)fhys Hysteretic spring force (N)
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384 S. C. Thakur et al.
fts(n−1) Tangential spring force at previous time step(N)
ε Total bulk deformationεa Bulk axial strainεp Total plastic deformationβn Normal dashpot co-efficientβt Tangential dashpot coefficientφ Angle of friction (◦)� fts Incremental tangential force (N)δ Total normal overlap (m)δmax Maximum normal overlap (m)δp Plastic overlap (m)η Sample bulk porosityηc Consolidated bulk porosityηf Fill porosityμ Co-efficient of frictionμr Coefficient of rolling frictionτi Total applied torque (N m)ωi Unit angular velocity vector (radian/s)λp Contact plasticityλb Bulk plasticityγ̇ Shear rate
1 Introduction
Many powders and particulate solids are stored and han-dled in large quantities in various industries such as phar-maceutical, food, and chemical industries: they constituteover 75 % of material feedstock in industry [1]. One ofthe common issues experienced by many of these mate-rials is the flow difficulty arising from material cohesion.Poor flow affects storage, transfer, production, packing, com-paction, fluidisation, distribution, and end use of the productin a negative manner, resulting in, for example, arching andratholing in silo storage, segregation in blending, under- andover-dosage in filling, which ultimately lead to economiclosses.
The flowability of cohesive solids is often measured usingthe flow function introduced by Jenike [2], which describesthe unconfined strength as a function of the consolidationstress. The flow function of a cohesive solid is an impor-tant material property for appropriate, efficient, and eco-nomic design of bulk handling equipment. Apart from themore traditional direct shear tests such as the Jenike cir-cular cell [2] and the Carr–Walker [3] or Schulze annularring cell [4], indirect uniaxial shear tests are often used toevaluate the flowability for a cohesive material [5–11]. Theflow function of a cohesive solid shows the manifestationof the unconfined yield strength that arises from the histor-ical consolidation stress and therefore the cohesive strengthis stress-history dependent. Such stress-history dependentcohesive behaviour must be captured if a numerical modelis to successfully simulate the cohesive powder flow. Thispaper describes the development of a phenomenological dis-crete element method (DEM) model coupled with a calibra-tion methodology for quantitative prediction of such powderflow behaviour.
1.1 DEM contact models for adhesion
A number of contact models [12–21] have been proposed formodelling cohesive powders. However, the common adhe-sion models including JKR [14], DMT [12], and Matuttisand Schinner [21], which are elastic contact models, maynot be able to capture the stress history dependent behaviourshown in experiments of cohesive powders [5–11].
Since the contact area between two fine particles is verysmall, even moderate forces (in the order of 1 nN) can causeplastic deformation at the contact [22] which can give riseto a stress history dependent behaviour. Therefore, it is pro-posed that contact plasticity has an important role in cor-rectly simulating the stress history dependency. Walton andBraun [23] were probably the first to introduce a normal forcedisplacement (NFD) elasto-plastic bilinear spring model,based on an approximation of finite element analysis (FEA)results, to account for the plasticity at the contact for cohe-sionless solids. Ning and Thornton [24] also presented anNFD contact model for elasto-plastic spheres, in which theyused a normal force displacement relationship that followsthe non-linear Hertzian relationship [25] until the contactyields and thereafter assumed a linear model. For unloadingafter plastic deformation the NFD contact model again fol-lows the normal force-displacement relationship proposedby Hertz [22], but uses a larger effective radius allowingfor contact enlargement resulting from irreversible plasticdeformation.
Based on a FEA study on spherical particles, Vu-Quocand Zhang [26] found that after the initial plastic yielding,the NFD contact model is neither linear as assumed by Thorn-
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Micromechanical analysis of cohesive granular materials 385
ton [27] nor following the Hertzian relationship. They arguedthat Thornton’s NFD contact model produces a softer force-displacement relationship and a smaller coefficient of restitu-tion for the same maximum normal displacement level com-pared to the FEA results. Vu-Quoc and Zhang [26] proposedan NFD contact model with nonlinear unloading stiffnesswhich is independent of the applied load.
The first NFD contact model with elasto-plastic defor-mation and adhesion was introduced by Thornton and Ning[16]. For cohesive solids, the plastic deformation in the con-tact region causes a larger effective radius of the deformedcontact region, and upon unloading a larger resultant pull-off force is observed. They modelled such behaviour usinga modified JKR curve with a larger contact radius. Wal-ton and Johnson [28] argued that the modified JKR modelhas a complicated formulation and may have difficulty tomeld with friction and rolling resistance. A more elabo-rate and detailed contact model accounting for load, timeand rate dependent visco-elastic, plastic, visco-plastic, adhe-sion and dissipative behaviour was proposed by Tomas [29].In the contact model initial loading response is Hertzianwhich switches to linear elasto-plastic loading after yielding.Contact unloading is elastic and follows Hertzian contact.The contact model also takes account of hysteretic behav-iour during the unloading/reloading cycle. A linear adhesionlimit accounts for increasing adhesion with increasing plasticdeformation. However this model is computationally inten-sive. By ignoring the initial non-linear portion before yield-ing on the basis that fine particles show a negligible range ofHertz-like behaviour [30], Tykhoniuk et al. [31] showed thatsimpler piece-wise linear models such as the ones proposed in[32] produced similar results at a smaller computational cost.Luding et al. [15,33] further improved the piece-wise linearmodel by recognizing the loading history dependent natureof the unloading/reloading stiffness by making the unload-ing/reloading stiffness a function of the previously experi-enced maximum overlap. This has introduced an element ofnonlinearity in unloading/reloading path. Walton and John-son [28] also proposed a contact model similar to Tomas andLuding’s model but separated the rate of increase of the pull-off force from the slope of the tensile force-displacementunloading curve which requires an additional parameter todescribe the adhesive behaviour.
1.2 DEM studies of uniaxial test
A number of DEM studies have been conducted to investi-gate the behaviour of cohesive solids using the above men-tioned contact models. Moreno-Atanasio et al. [34] con-ducted DEM simulations of a uniaxial test using JKR elas-tic adhesive contact model to investigate the effect of cohe-sion on the flowability of a polydisperse particulate system.
The mechanical properties of glass beads were used as DEMinput parameters. They argued that for consolidation stressesin the range of 0–10 kPa the model produced unconfinedyield strengths that could be classified as highly cohesiveaccording to Jenike’s classification. The unconfined strengthincreased with increasing consolidation stress showing stresshistory dependence. It may be argued that the stress historydependence is because the cohesive powder forms a looseinitial structure which collapses on the application of loadresulting in particle re-arrangement into a denser packing sothat the deformation does not recover significantly on theremoval of load, especially at low stress levels. The use of anelastic contact model may be suitable for simulating somematerials such as elastic glass beads at low stresses wheredeformation is mainly due to particle re-arrangement. How-ever, for many industrial processes, the stress can be muchhigher and the materials are evidently not elastic. In the caseof cohesive solids, an adhesive elastic contact model may notbe able to represent the realistic behaviour, where the perma-nent plastic deformation gives rise to stress history dependentbehaviour.
Hassanpour and Ghadiri [35] conducted both experimen-tal and DEM studies on the flowability of powders usingindentation and uniaxial compression tests on a small assem-bly of powders compacted at low pressures. They adopted acontact model based on the Hertzian analysis for the elasticregime, Thornton and Ning’s [16] model for the plastic defor-mation and the JKR theory [14] for the adhesion force. DEMsimulations based on single particle mechanical propertiesshowed that the unconfined yield strength varies linearly withthe consolidation stress, which is similar to the experiment,but there is a poor quantitative agreement between experi-ment and simulation, with a discrepancy up to 160 %. Thediscrepancy was attributed to rough estimation of the adhe-sion parameter and the adoption of spherical particle shapein the DEM simulation.
The great majority of DEM studies in the literature wereconducted using either 2D disks or 3D spheres. A number ofnumerical and experimental studies [36–39] have shown thatparticle shape affects the flowability of the powder. Whilespherical particles with higher rolling friction are introducedto simulate particle shape by many researchers (e.g. [40]),it has been reported that spherical particles cannot repre-sent the “real” solids regardless of the angle of inter-particlefriction [41]. The spherical particles fails to capture inter-locking related dilation, voidage distribution, and mater-ial shear strength arising from interlocking [41]. Therefore,it is important to introduce an appropriate degree of non-sphericity in particle shape to capture the behaviour of realsolids which are very rarely spherical. This study deploys3D non-spherical particle using multi-sphere technique asdescribed in [42] which is also being used by many others[43–47].
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1.3 Overview and focus of study
In this paper, the development of an adhesive elasto-plasticDEM contact model for cohesive powders is presented. Themodel comprises a nonlinear hysteretic spring model toaccount for the elastic–plastic contact deformation and anadhesive force component that is a function of the plasticcontact deformation. The proposed contact model is concep-tually similar to existing models [15,17,18] but has the addi-tional capability of non-linearity for the loading and unload-ing paths. Atomic Force Microscopy (AFM) measurementsof the contact force displacement curve of very fine particlesof fumed silica and titania have shown smooth non-linearbehaviour during both loading and unloading [48]. The non-linear behaviour after plastic yielding has also been reportedby Vu-Quoc and Zhang [26]. In order to model both linearand nonlinear NFD behaviour of real solids, a power law isproposed for both loading and unloading paths in the contactmodel.
The predictive capability of the proposed DEM contactmodel is evaluated by attempting to simulate the behaviourof a limestone powder under quasi-static and slow shearingregimes. The study explores the full spectrum of cohesivebehaviour from filling of a space (fill porosity) to loadingunder confined compression, and finally unconfined loadingto failure. A key focus is to produce a quantitative modelfor the unconfined strength as a function of the consolida-tion stress. The calibration of the model parameters employsexperimental data of bulk solid characteristics, recognisingthe differences between the model and the real solids at theparticle level. The details of the model and computationalmethodology are described in the following section, followedby modelling strategy and the DEM implementation to simu-late confined and unconfined uniaxial loading to failure. TheDEM predictions are compared with the experimental resultsbefore the effects of various parameters on filled porosity,compressibility, and unconfined strength are explored. Thecauses of the stress history dependency of the unconfinedstrength of cohesive solids are then investigated. Finally, theeffect of limiting frictional criteria on the unconfined strengthis explored.
2 Computational methodology
2.1 Equations of motion
The discrete element method (DEM) was first developed byCundall and Strack [49] as a tool for analysing quasi-staticproblems related to densely packed granular materials. Inter-action is modelled using the soft contact approach whererigid particles are allowed to overlap each other at the con-tact point with small overlaps, typically less than 1 % of the
particle diameter at each time step. The contact force is cal-culated according to the contact model as a function of theoverlap.
The changes in positions and velocities of the particles dueto the contact and gravitational forces are calculated from theintegration of Newton’s motion equations. For particle i thetranslational motion equation is
mid2
dt2 xi = f i + mi g (1)
where mi is the mass of the particle, t is time, xi is its position,f i is the summation of all forces acting on the particle ( f i =∑
c f ci ), and g is acceleration due to gravity. The rotational
motion equation for particle i is given as:
Iid
dtωi = Ti (2)
where Ii is the moment of inertia for particle i,ωi is its angu-lar velocity and Ti is the total torque acting on it, which isdefined by Eq. 3, where lc
i is the vector from the centre ofparticle i to the contact point and f c
i is the contact force.
Ti =Nc∑
i=1
lci × f c
i (3)
A sufficiently small integration time step is required to ensurethe stability of simulation by having a sufficient number oftime steps within each collision. The time step is usuallycalculated as a percentage of the Rayleigh time step [50–52],typically less than 10 %.
2.2 Visco-elasto-plastic adhesive frictional contact model
The DEM contact model proposed here is based on the physi-cal phenomena observed in adhesive contact between micronsized particles or small agglomerates [53]. When two parti-cles or agglomerates are pressed together, they undergo elas-tic and plastic deformations. It is assumed that the pull-off(adhesive) strength increases with an increase of the plasticcontact area. A non-linear contact model that accounts forboth the elastic–plastic contact deformation and the contact-area dependent adhesion is proposed. The schematic dia-grams of particle contact and normal force-overlap ( fn − δ)
curve for this model are shown in Fig. 1a.The loading, unloading, re-loading, and adhesive branches
are characterised by five parameters: the virgin loading stiff-ness parameter k1, the unloading and reloading stiffness para-meter k2, the constant adhesive strength fo, the adhesivestiffness parameter kadh and the stiffness exponent n. Dur-ing the initial loading, the contact model follows the virginloading path, k1, upon unloading of the contact; the con-tact will switch from the virgin loading path to the unload-ing/reloading path, k2. During reloading, the contact forceinitially follows along the reloading k2 path but switches to
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Micromechanical analysis of cohesive granular materials 387
Fig. 1 Normal contact force-displacement function for the contact model a non-linear b linear
the virgin loading k1 path when the previous maximum load-ing force is reached. Unloading along the k2 path below theplastic overlap δp results in the development of an adhe-sive force until the maximum adhesive force is reached at−kadhδn
min + f0. Further unloading past this point resultsin a reduction in both the normal overlap and the adhesiveforce until separation occurs (δ = 0). If reloading of thecontact occurs while on the adhesion branch, the contactwill reload along a k2 path (there is an infinite number ofk2 paths depending on the point of first unloading), until thevirgin loading k1 path is reached, and will continue load-ing along k1 path on further increase of load. The contactinformation is lost when the particles are separated. Thismodel does not consider hysteretic behaviour during reload-ing/unloading of the contact below δmin as in Tomas’s model.If k1 is set equal to k2 the model is reduced to an elastic contactmodel.
The unloading/reloading stiffness k2 is not load depen-dant as in Luding’s model [15]. The model has been imple-mented with a power law exponent parameter n to describethe shape of all the three loading-unloading branches—theyall become linear when n = 1 (see Fig. 1b). Both linear andnonlinear trends for normal force displacement relationshiphave been reported in experiments (e.g. [48]) and numeri-cal studies (e.g. [26]). In this first study of the model, wefocus on studying the influence of the contact plasticity andadhesion parameters (k2, kadh and fo) without invoking thenonlinearity by setting n = 1 which reverts the contact modelto a piece-wise linear version that is conceptually similar toseveral existing models [15,28]. The simulations conductedusing the non-linear model can be found in [54].
The contact model as described above was implementedthrough the API in EDEM� v2.4 (and subsequent versions),a commercial DEM code developed by DEM Solutions Ltd[55]. The total contact normal force, f n, is the sum of thehysteretic spring force, fhys , and the normal damping force,fnd :
f n = ( fhys + fnd)u, (4)
where, u is the unit normal vector pointing from the contactpoint to the particle centre. The force-overlap relationship fornormal contact, fhys, is mathematically expressed by Eq. 5.
fhys =⎧⎨
⎩
f0 + k1δn i f k2(δ
n − δnp) ≥ k1δ
n
f0 + k2(δn − δn
p) i f k1δn>k2(δ
n − δnp)> − kadhδn
f0 − kadhδn i f − kadhδn ≥ k2(δn − δn
p)
(5)
The normal damping force, fnd, is given by:
fnd = βnνn (6)
where νn is the magnitude of the relative normal velocity,and βn is the normal dashpot co-efficient expressed as:
βn =√√√√
4m∗k1
1 +(
π
In e
)2 (7)
with the equivalent mass of the particles m∗ = (mi m j/mi +m j ), where m is the mass of the respective particle, and thecoefficient of restitution e given to the simulation code as aninput parameter.
The contact tangential force, f t , is given by the sum oftangential spring force, f ts, and tangential damping force,f td., as given by:
f t = (f ts + f td
). (8)
The tangential spring force is expressed in incremental terms:
f ts = f ts(n-1) + �f ts, (9)
where f ts(n-1) is the tangential spring force at the previoustime step, and �f ts is the increment of the tangential forceand is given by:
�f ts = −ktδt, (10)
where kt is the tangential stiffness, and δt is the incrementof the tangential displacement. While varying values for thetangential stiffness have been used in the literature, in this
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388 S. C. Thakur et al.
study it is set as 2/7k1 [23].The tangential damping force isproduct of tangential dashpot coefficient, βt , and the relativetangential velocity, vt , as given by Eq. (11):
f td = −βtνt. (11)
The dashpot coefficient βt is given by:
βt =√√√√
4m∗kt
1 +(
π
In e
)2 (12)
The limiting tangential friction force is calculated using theCoulombic friction criterion with the normal force modifiedby the adhesives forces given by:
fct ≤ μ(| fhys + kadhδn − fo|
)(13)
where fct is the limiting tangential force, and μ is the frictioncoefficient.
The default EDEM rolling friction model is adopted inthis study. The total applied torque, τ i, is given by:
τ i = −μr | fhys |Riωi, (14)
where μr is the coefficient of rolling friction, Ri is the dis-tance from the contact point to the particle centre of massand ωi is the unit angular velocity of the object at the contactpoint.
There is some limited evidence in support of Eq. 13 in theexperimental results from Skinner and Gane [56], who con-ducted micro-friction experiments between soft metal stylusand a hard smooth surface of graphite or diamond in a scan-ning electron microscope and found that the attractive forcecan be considered as an additive term to the normal forcein calculation of limiting friction. Savkoor and Briggs [57],and Thornton and Yin [58] derived similar equation basedon contact mechanics theory. The limiting friction criteriais consistent with the criteria set in other existing tangentialforce models [15,17,28,58].
The literature is rather vague about the tangential forcemodels. One of the questions to be addressed in the areaof tribology is whether the relationship between frictionand normal force is linear. A number of microscopic inter-particle friction experiments have reported the relationshipbetween friction and normal force [56,59–64]. Both non-linear (Hertzian and JKR) and linear (Coulombic) relation-ships were found between the frictional and normal forces.Briscoe and Kremnitzer [59] found non-linear relationshipat tensile loads and linear relationship at compressive loads.They assumed a single asperity contact. In contrast, Ruthset al. [61] found a linear dependence in a study conductedon boundary friction of two different atomic silane mono-layer with low adhesion in a single asperity contact with sur-face forces apparatus (SFA) and frictional force microscopy(FFM). Schwarz et al. [63] conducted a study using frictionalforce spectroscopy on nano-size carbon compounds with a
single asperity and found that the frictional force was pro-portional to the 2/3 power of load similar to the theoreticalmodel based on a Hertzian-type tip–sample contact. Frictionforce experiments by Jones et al. [60] found non-linear rela-tionship between the normal and friction forces for ballotiniindicating single asperity contacts and linear relationship formaterials including alumina, limestone powder, zeolite, andtitania supporting multi-asperity contact.
Although both linear and nonlinear relationships havebeen found between normal load and friction in microscopicfriction experiments, we have assumed a linear relationshipin this study for two main reasons. Firstly, real materials gen-erally have multi asperity contacts and the literature suggestslinear relationship for multi asperity or plastic contacts [60].Secondly, elasto-plastic behaviour for fine adhesive parti-cles can be expected in most cases of industrial process-ing and handling where averaged macroscopic stresses areabove 1 kPa [22], and the nonlinear behaviour observed atlow stresses in FFM can be ignored.
3 Modelling strategy and experimental characterisation
The above contact model in its full generic form captures thekey elements of the frictional-adhesive contact mechanics inthat: f0 provides the van der Waals type pull-off strength; k1
and k2 provide the elastic–plastic contact; kadh provides theadhesion unloading stiffness; the exponent n provides non-linearity and the resulting contact plasticity defines the totalcontact adhesion (see Fig. 1a). The model is thus expected tobe capable of modelling truly micron sized particles to studyphenomena such as fine agglomeration, attrition and flow.
To apply the model at the single particle level requiresthe model parameters to be determined at the true particle-particle interaction level. This would require one to considerthe enormous complexity of interfacial interaction includ-ing the influence of surface topology and chemistry andproperties of the interstitial media etc. Whilst many studieshave been reported on these measurements using microscopicmeasurement techniques such as AFM, Nano-indentationetc., the measurements tend to be on either highly idealisedparticle (such as specially manufactured perfect sphere) orsuffer from enormous scatter and uncertainty with regard tothe accuracy of the measurement [31,65]. Additionally, it isprohibitive to model each and every individual particle andcohesion arising from several different phenomena includ-ing van der Waals, capillary bridge and electrostatic forcesseparately, even in a very small system of fine powders. Forexample, a uniaxial test simulation of a cylindrical sampleof 40 mm diameter and 80 mm in height with 4.7 μm sizedlimestone powder containing >1012 particles may take in theorder of 60 years if this was to be simulated with a 4 core,64-bit computer.
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Micromechanical analysis of cohesive granular materials 389
This study focuses on an intermediate scale between themicro- and macro-scales, aiming to produce a phenomeno-logical contact model that can reproduce the bulk cohesivestrength, stress history dependency, and other behaviour evi-denced in experiments. This study deploys 3D non-sphericalparticles in conjunction with a calibration strategy recognis-ing the differences between the model and the real solids atthe particle level to reproduce the bulk granular friction ofa particulate system. The calibration of a mesoscopic DEMcontact model requires experimental measurements charac-terizing the mechanical behaviour of powders. The mechani-cal behaviour of cohesive powders can be carefully measuredusing element tests such as biaxial test, true triaxial and hol-low cylinder tests. However in practice these tests are expen-sive and slow to conduct and are almost never performed formany industrial applications requiring material characteri-sation. Here we have investigated a simpler technique thatcould be used for filling this important gap with the focusof providing test data for model calibration and simulationvalidation.
In this study, the Edinburgh Powder Tester (EPT) [5] isemployed. The EPT is a semi-automated uniaxial tester,providing rapid measurements of various bulk mechani-cal properties of powders, inc- the stress–strain and thestress–porosity response during confined compression aswell as the stress–strain response during unconfined com-pression including the peak unconfined strength. The repro-ducibility of results has been assessed for various mate-rials with a coefficient of variation of typically less than7 % [45,54].
In an EPT test, the sample is poured into the consolida-tion cylinder of diameter 40 mm and height 80 mm. Thesample is loaded by a weight and the force is recorded by aload cell attached to the consolidation plunger. To minimisethe effect of the friction between the particles and bound-aries [6], the sample is allowed to compress from both thetop and bottom in the EPT. After the sample is loaded fora selected consolidation time, the consolidation plunger isautomatically retracted and the mould is manually slid downthe pedestal, exposing a free standing column of consoli-dated powder sample. The unconfined sample is loaded tofailure by a motor driven test piston at a speed of 0.4 mm/s,which was so chosen to conduct the test rapidly withoutaffecting the measured unconfined yield strength. Watanabeand Groves [66] found that the unconfined strength of deter-gent samples was unaffected when the piston speed variedfrom 0.084 to 0.43 mm/s. The unconfined strength is auto-matically recorded, as well as the whole load-displacementcurve. Consolidation stresses in the range of 16–96 kPa wereapplied in this study with 1 min consolidation time. ESKAL500� (KSL Staubtechnik GmbH), a limestone powder withparticles of 4.7 μm mean diameter was tested in this study(Fig. 2).
Fig. 2 Scanning electron microscopic (SEM) image of ESKAL 500
Fig. 3 Paired particle of aspect ratio = 1.5
4 Numerical simulation set-up
The DEM was used to simulate a series of uniaxial testexperiments in a cylindrical mould. The proposed con-tact model was only applied to particle-particle interac-tions. Particle-geometry interactions were modelled usingthe Hertz-Mindlin (no-slip) contact model and hence noparticle-geometry adhesion was allowed. Non-spherical par-ticles were used, each consisting of two overlapping (paired)spheres of 1 mm diameter (d) giving a particle aspect ratioof 1.5 (Fig. 3). This relatively simple two-sphere shape waschosen based on the findings from two studies. The first study[67] showed that accurate representation is not necessary toproduce satisfactory predictions as long as there is sufficientshape interlocking to generate the bulk friction. The secondstudy showed that for purely spherical system, the maximumbulk friction under direct shear testing saturates at ∼0.8 evenfor sliding friction of up to 2.0, whereas for two overlappingsphere with aspect ratio of 1.5, a full spectrum of bulk fric-
Adhesive strength at first contact orconstant adhesive strength, f0, (N )
−0.002 to − 0.05
Adhesive parameter stiffness, kadh (kN/m) 0.1–100
Particle static friction, μsf 0.5
Particle rolling friction, μrf 0.001
Wall friction, μwf 0
Top and bottom platen friction, μPf 0.1
Wall shear modulus, N/m2 1010
Poisson’s ratio for particle to particleinteraction
0.3
Poisson’s ratio for particle to wallinteraction
0.25
Simulation time-step (s) 8 × 10−7 to 2 × 10−6
tion up to ∼1.9 can be predicted [68]. Thus, by choosing thetwo-sphere model, we can expect the DEM model to be ableto capture a wide range of bulk frictional characteristics forany real complex shape particles.
The parameters used in the simulations are listed inTable 1. The loading stiffness k1 was chosen so that it issufficiently stiff to avoid excessive particle overlap but nottoo stiff to avoid significant computational cost, and yet pro-vide a close match to the experimental loading response. Inorder to reduce the computational time further, density scal-ing [52] was used as simulations are in quasi-static regime.The significance of dynamic effects in granular material isoften evaluated by inertial number. The inertial number isused to describe the relative importance of inertia and con-fining stresses [69]:
I = γ̇ davg√P/ρ
(15)
where γ̇ = shear rate, davg = average particle diameter, P =pressure, and ρ = particle density.
An inertial number of less than 1×10−4 was observed forall simulation parameters and confirms that the simulationsare in the quasi-static regime [69]. The simulation time stepwas chosen to be equal to 0.1
√m/k2: no noticeable differ-
ence in results was found between simulations with time stepof 0.3
√m/k2 and 0.1
√m/k2. The ratio of k2/k1 was varied
from 1 to 100 by increasing k2. Since the constant adhesivestrength f0 and the load-dependent adhesion represent dif-ferent origins leading to cohesion, they are studied separatelyin Sect. 6 for the former and 7 for the latter.
DEM simulations using the proposed contact model wereconducted for a series of uniaxial compression tests in a cylin-
Fig. 4 DEM simulation of uniaxial test: a loading, b unloading, and cunconfined compression
drical mould of 15 mm diameter with top and bottom platens.This represents a scale of approximately 1/3 of the originalexperimental set-up, to reduce the computational cost. Theinitial filling height varied with DEM input parameters. How-ever, the consolidated aspect ratio of the sample for the DEMsimulations was kept in a narrow range of 1.2–1.4 which wasused in the experiment. Each simulation consists of threestages—filling the cylindrical mould to form the initial pack-ing used for all stress levels; confined consolidation to therequired stress level and subsequent unloading; and finallyunconfined compression of the sample to failure after theremoval of the mould. The process is visualised in Fig. 4.
The random rainfall method was adopted to form a ran-dom packing. To ensure that the system reached a quasi-staticstate, loading only commenced when the kinetic to potentialenergy ratio was less than 10−5 with a constant coordina-tion number. The potential energy in the system is calculatedbased on a datum level of zd = 0mm, which in this studyrelates the bottom of the mould. The confined consolidationprocess was conducted by moving the top platen downwardsat a constant speed of 5 mm/s (strain rate ≈ 0.2s−1) to applya vertical compression. After consolidating the sample to thedesired stress, the load on the assembly was released by mov-ing the top platen upwards at the same constant speed. Thelateral confining walls were then removed and the unconfinedsample was allowed to relax for 0.1 seconds. This allowedthe kinetic energy generated from the removal of the con-fining wall and upward retreat of the top platen to dissipate.The sample was then crushed to failure by moving the topplaten downwards again, at a constant rate of 2 mm/s (strainrate ≈ 0.1 s−1). To investigate the effect of applied strainrate, simulations were conducted with varying strain rate ina range of 0.02–60 s−1
, for elastic (k2 = k1) and elasto-plastic(k2 = 100k1) case. For both cases, it was found that confinedand unconfined stress-strain behaviour does not change sig-nificantly with strain rate below 0.5s−1. Therefore, strain rate
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Micromechanical analysis of cohesive granular materials 391
smaller than 0.5s−1 was chosen in this study. The failure ofthe sample was characterised by a drop in stress accompa-nied by a drop in the coordination number (Z ) (see Fig. 15).The bottom platen remained stationary in all stages.
5 Numerical repeatability and prediction of flowability
Three numerical samples (samples 1, 2 and 3 below) withthe same model parameters were created using the randomparticle generator implemented within the particle factory inthe commercial code. These samples were then consolidatedto 100 kPa prior to an unconfined compression test to failureto assess the variability in the results related to the gener-ation of different particle packings in the assemblies. Thescatter of these numerical samples was evaluated. Figures 5
Fig. 5 Confined compression-axial stress (σa) versus axial strain (εa)
for three random simulations and one large size simulation
Fig. 6 Unconfined compression-axial stress (σa) versus axial strain(εa) for three random simulations and one large size simulation. Contactmodel parameters used are: f0 = −0.002 N, k2 = 100 and kadh =0 kN/m (Note: N = number of particles, ηf = porosity corresponding to0.5 kPa stress)
and 6 present the axial stress-strain response during con-fined and unconfined compression, respectively. For confinedcompression, some variations in the stress-strain behaviourare noted. For unconfined compression, the average uncon-fined strength was found to be 6.6 kPa with a coefficient ofvariation (COV) = 3.4 %. The small COV indicates that ran-domly generated particle assembly has minor effect on thebulk response but is not independent of random assemblycreation. While three data points would not usually be con-sidered sufficient for rigorous statistical analysis, the verylow COV indicates that the numerical scatter introduced bythe random initial packing is small. The presence of numer-ical scatter should always be checked and used in the inter-pretation of the numerical results.
To investigate the influence of numerical sample size, anadditional simulation (sample 4) with 10,000 paired-sphereparticles with same particle aspect ratio (1.5) and otherDEM input parameters as the simulation with 2,200 parti-cles was performed. The number of particles were increasedby increasing the diameter of the sample but keeping the sam-ple aspect ratio same as the simulation with 2,200 particles.The results are included in Figs. 5 and 6. The comparisonshows that the sample size does not have a significant effecton the prediction. For confined compression, the 10,000 par-ticle system predicted ∼1.5 % smaller compression at thepeak compared to the average peak strain of the sampleswith 2,200 particles. This can be attributed to a lower filledporosity for the 10,000 particle system because of a smallerboundary effect. For unconfined compression to failure, theunconfined peak strength is also within the scatter of the mea-surement for 2,200 particles. A more rigorous assessment ofsample size would require more repeat random simulationsto establish the statistical scatter which is beyond the scopeof this paper.
The capability of the model is explored for predict-ing the unconfined yield strength of ESKAL 500 (a PAR-DEM reference solid) under different consolidation stresses.Figure 7 shows the predicted axial stress-strain responsesduring unconfined uniaxial compression of samples whichhave been consolidated at five stress levels: 16, 36, 56, 76,and 96 kPa. The initial loading stiffness increases as the con-solidation stress increases. This has arisen from the changein the packing structure with decreasing porosity as consol-idation stress increases. The maximum stress during uncon-fined compression (i.e. the unconfined strength σu) is plottedagainst the consolidation stress (see Fig. 8). For the parame-ters chosen, the model predicted a flow function that is ingood quantitative agreement (within ∼12 %) with the exper-imental results. The simulation results show a strong depen-dence on the consolidation stress history as measured in theexperiments. The reason for this stress-history dependency isrelated to the contact plasticity: the micromechanical aspectswill be elucidated in the following section. Simulations were
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392 S. C. Thakur et al.
Fig. 7 Predicted unconfined axial stress (σa)-strain (εa) relationship.Contact model parameters used are: f0 = −0.003 N, k2 = 3.5 andkadh = 0 kN/m
Fig. 8 Predicted versus test flow function for a limestone powder.Contact model parameters used are: f0 = −0.003 N, k2 = 3.5 andkadh = 0 kN/m
also conducted using the modified JKR contact model withan elastic Hertzian contact in the EDEM code version 2.4[70]. The surface energy of 1 J/m2 and particle shear mod-ulus of 107N/m2 were used with other parameters kept thesame as in Table 1. Figure 8 also shows the simulation resultsusing the modified JKR cohesive model. The results showonly a slight increase in unconfined strength as consolida-tion stress increases, giving an increasingly large discrepancywith the experimental observations with increasing consoli-dation stress when the JKR model is used. The results shownin Figs. 7 and 8 indicate that the implemented model is supe-rior at capturing the salient features of a real cohesive pow-der.
Fig. 9 a Linear elasto-plastic contact model with constant contactadhesion b typical bulk stress strain response during confined com-pression
6 Micromechanical analyses of porosity, plasticity andcohesion
The origin of the stress history dependent cohesion strengthpredicted by the model is explored here from a micromechan-ical point of view. The relationships between bulk materialproperties; namely the unconfined strength, the bulk plastic-ity and the porosity; and the microstructural properties areexplored. Simulations of confined followed by unconfinedcompression, as described in Sect. 3, were performed forparticle stiffness k2 varying from 1 to 100 kN/m as listed inTable 1, and consolidation stress levels from 16 to 96 kPa.The level of contact plasticity was changed prior to the gen-eration of the particles and was maintained for both the con-fined consolidation and unconfined compression to failure.The adhesion stiffness parameter kadh was set to zero in thefirst instance so that the influence of the constant pull-offforce f0 can be explored (see Fig. 9a).
In an elasto-plastic contact, the contact plasticity λp maybe defined as the ratio of the maximum plastic deformation δp
to the total deformation δ at the contact. For the linear versionof the contact model (n = 1, see Fig. 9a), λp becomes a simplefunction of the loading k1 and unloading k2 stiffnesses:
λp = δp
δ= 1 − k1
k2(16)
Similarly the bulk plasticity (λb) can be defined as the ratioof the bulk plastic deformation εp to the total deformation ε,as shown in Fig. 9b.
λb = εp
ε(17)
Figure 10 shows the predicted flow functions for a rangeof contact plasticity (induced by varying the stiffness ratiok1/k2). The slope of the flow function is an indication ofthe level of cohesion for a given set of parameters. The levelof cohesion can increase from two sources: increasing coor-dination number and flattening of the contact under load-ing. It is evident that the flow function is strongly depen-
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Micromechanical analysis of cohesive granular materials 393
Fig. 10 Effect of particle contact plasticity on flow function( f0 = −0.002 N, Kadh = 0 kN/m)
dent on the contact plasticity: a cohesive material with aconstant particle-level adhesion force f0 can change frombeing only slightly cohesive to moderately cohesive whenthe contact plasticity λp increases from 0 (elastic-no residualinter-particle contact deformation after unloading) to nearly1 (no recovery of elastic deformation after unloading). Whenthe contacts are elastic, the stress-history dependency largelydisappears. Increasing the level of contact plasticity, whichreduces elastic rebound, allows the consolidated assembly tomaintain a lower consolidated porosity following unloadingof the sample. The lower consolidated porosity in-turn leadsto a higher number of inter-particle contacts, which gener-ates a higher unconfined strength at which the assembly fails.Since load dependent adhesion was intentionally set as zero(kadh = 0: see Fig. 9a), the increasing level of cohesion withincreasing contact plasticity relates solely to the increasingcontacts between particles.
The loss of stress-history dependent unconfined strengthwhen contacts are elastic explains why adhesive models withelastic contact such as JKR and DMT models have difficultiesin adequately capturing the stress history effect of cohesivesolids. Most fine cohesive powders exhibit plasticity even atrelatively low consolidation stress where a modest amountof force may lead to plastic yielding and irreversible defor-mation at the tiny particle-particle contact [71]. As shownby Hietsland [72], a modest amount of plastic deformation atparticle contact may cause a dramatic increase in the strengthof consolidated powders, which is in line with the simulationresults in this study. From the mesoscopic perspective wherea DEM particle represents a local assembly of primary par-ticles, it is easy to see the presence of contact plasticity. Anappropriate level of plasticity in the contact model should beincluded when modelling cohesive powders.
Fig. 11 Bulk plasticity (εp / ε) as a function of particle plasticity
Fig. 12 Confined compression-axial stress (σa) versus axial strain (εa)
with λp = 0
Figure 11 shows the effect of particle contact plastic-ity λp on bulk plasticity λb. It is seen that the bulk plas-ticity increases with increasing λp. When λp is small, thebulk plasticity arises predominantly from particle rearrange-ment which is greater during initial compression from theloose filled state; this give rise to a larger bulk plasticityfor a smaller consolidation stress. As λp increases and con-tributes to overall deformation, the bulk plasticity increasesas expected, but this must approach unity as the contact plas-ticity λp approaches unity. On its own, bulk plasticity cannotfully explain the stress history effect of a material becauseparticles with elastic contacts can have very different bulkplasticity (see Fig. 11) but very little stress history effect(see Fig. 10), whilst particles with elastic–plastic contactscan have similar bulk plasticity but very strong stress historydependence.
Figure 12 shows the effect of contact plasticity λp onthe loading-unloading response under confined compression.Three cases are shown: elastic contact (λp = 0), almost rigid
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Fig. 13 Confined compression-porosity (η) versus axial strain (εa)
with λp = 0.99
plastic contact (λp = 0.99) and an intermediate value ofplasticity (λp = 0.5). Even for elastic contact, significantbulk plastic deformation can arise from particle rearrange-ment. As contact plasticity increases, plastic contact defor-mation increases under loading resulting in a stiffer responseon unloading. The softer loading responses with increasingλp appears surprising at first since the loading parameter k1
was set to be constant for all three cases. The answer lies inthe larger initial sample porosity when λp is larger. A closerlook at the DEM results show that a larger contact plasticityλp gives rise to a greater degree of clustering during the fillingprocess which resulted in larger voids between the clusters:this gives rise to a greater initial porosity and hence a softerresponse during compression. This is further highlighted inFig. 13, where the variation in sample porosity with consol-idation stress is plotted. At low consolidation stresses thereis a significant difference in the observed porosities whichconverge onto a single loading curve at greater consolidationstresses (say 5 kPa). For the unloading path, particle contactplasticity significantly affects the bulk unloading stiffness asexpected. As the contact plasticity decreases the assemblyrebounds to a higher consolidated porosity.
Next the relationship between porosity and unconfinedyield strength is explored. Figure 14 plots the unconfinedyield strength (σu) against the consolidated porosity afterunloading (ηc) for different contact plasticity λp. For eachλp, σ1 varies from 20 kPa to 100 kPa, with the porosity at 20kPa being the highest in all cases. As the consolidation stressincreases, the unconfined yield strength increases whilstthe porosity decreases, similar to the findings from con-ventional soil/powder consolidation experiments [73]. Thedependence on contact plasticity is evident where increas-ing λp has resulted in an increasing range of consolidatedporosity ηc which gives rise to an increasing range of uncon-fined strength. However the lines are unique below a λp of0.8, so the bulk porosity alone is not sufficient to account for
Fig. 14 Unconfined strength (σu) as a function of consolidated poros-ity (ηc) at different contact plasticity
Fig. 15 Normalized unconfined strength (σa d2/ f0, σu d2/ f0) vs. coor-dination number (Zi , Z) : f0 = −0.002 N unless stated explicitly
the history-dependent strength across materials with differ-ent contact plasticity. Above λp of 0.8 the results seem toconverge. This is because as the level of contact plasticitytends above 0.8, the coordination number reaches a limitingvalue for a certain consolidated porosity (only true when fo
= constant and kadh = 0).Figure 10 above has shown how the unconfined strength
increases with increasing consolidation stresses at differentlevels of contact plasticity. To explore the mechanism for thisstress-history dependence, the stress-strain loading paths tofailure for several simulations (see Fig. 7) are re-plotted interms of the normalised axial stress and the instantaneouscoordination number (Zi ) in Fig. 15. The axial stress (σa)
is normalized with f0/d2 which relates to particle adhesivestrength. The arrows indicate the direction of loading, fromthe start of unconfined loading to post-peak. As loading pro-gresses, the axial stress increases strongly with only a verysmall decrease in the coordination number before reachingthe peak (unconfined strength). Further loading of the sample
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Micromechanical analysis of cohesive granular materials 395
following the peak leads to a significant drop in the observedcoordination number which is associated with increasingdilation (increasing volume) [2,73] and decreasing post peakstrength. This trend is consistent for all simulations in thisstudy.
Superimposed on Fig. 15 are the normalized unconfinedstrength σu d2/ f0 and the coordination number Z at the peakfor all simulations with kadh = 0 (denoted by �). All datapoints collapse into a single ‘critical curve’ for the full rangeof contact plasticity and consolidation stresses in Fig. 10.This critical line is analogous to the concept of critical state insoil mechanics [74]. It indicates that with a constant contactadhesion fo (kadh = 0), the microscopic mechanism forthe increasing bulk cohesion under increasing stress (stress-history effect) is due to the increasing number of contacts asa result of both the consolidation stress and contact plasticity.
7 Interaction between adhesion parameters
Cohesion in bulk materials may arise from different sourcesof adhesion at particle level and these are represented by twoparameters in the contact model: f0 and kadh . In the sectionabove, we explored the bulk cohesion arising only from a con-stant adhesive strength f0 coupled with contact plasticity. Inthis section we study the interaction between the two. Simula-tions with constant adhesion only, load-dependent adhesion( f0 = 0, kadh = 0) only, and with both f0 and kadh not equalto zero were performed. The effect of these parameters on fillporosity, compressibility, and flow function are investigatedbelow.
We first explore the effect of contact adhesion on fill poros-ity (ηf) which is defined as the sample porosity at a very lownominal consolidation stress of 0.3 kPa after filling. The fillporosity depends on the method of filling which is described
Fig. 16 Variation of porosity (ηf ) at end of filling with kadh and f0 :k1 = 1 and k2 = 10 kN/m
in Sect. 4. While comparing the effect of kadh and fo onthe filled porosity it is important to note that fo is an adhe-sive strength (N) whilst kadh is adhesive stiffness (N/m). It istherefore difficult to compare the effect of these parameterson the filled porosity directly.
Figure 16 shows the fill porosity arising from two dif-ferent scenarios: varying f0 with kadh = 0 and varyingkadh with f0 = 0. For the system with zero adhesion(kadh = 0, f0 = 0), the porosity is 41 % (as shown by dashedline in the figure) which is consistent with the findings forcohesionless paired non-spherical particle with aspect ratioof 1.5 [43]. This can be compared with the porosity of 36 %for random packed mono-disperse frictionless spheres, show-ing the effect of non-sphericity and friction on porosity. Asthe contact adhesion increases either by increasing kadh orf0, the filled porosity also increases. Contact adhesion causesthe particles to stick together during the filling process andform local clusters leading to chain like structures and thushigher porosity. The adhesive forces provide a higher resis-tance to counteract the effect of gravity force and providemechanical stability. These forces restrict the relative move-ment between particles and significantly reduce the densifi-cation due to rolling and sliding between particles [75].
The porosity initially increases slowly with increasingadhesion parameters, before it increases rapidly after reach-ing the inflection point and finally reaches a plateau. It shouldbe noted that the adhesion strength at contact is not fullymobilized during the filling process. For a fair comparisonof the effect of adhesion parameters on the porosity wouldrequire calculation of average mobilized adhesive (tensile)force in each system. The average tensile force ( fat ) isdefined as the ratio of total tensile force to the number oftensile contacts in the system. The Fig. 17 shows the result inFig. 16 re-plotted in terms of average tensile force ( fat ) nor-malized by gravitational force ( fg). It is notable to find thatfor both adhesion parameters, the interparticle force ratio
Fig. 17 Interparticle force ratio versus fill porosity
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396 S. C. Thakur et al.
Fig. 18 Consolidated porosity (ηc) as a function of consolidationstress (σ1) : k1 = 1 and k2 = 10 kN/m
( fat/ fg) versus porosity relationship converges to a singleline. This suggests that porosity relates strongly to mobilisedadhesive (tensile) force regardless of constant adhesion orload dependent adhesion.
Next we explore the effect of adhesion on bulk compress-ibility. The consolidated bulk porosity of the sample wascalculated from the height of the consolidated sample afterunloading. Figure 18 shows the consolidated porosity as afunction of the consolidation stress for different adhesionparameters. As the consolidation stress increases, the con-solidated bulk porosity reduces for all adhesion parametersinvestigated. The porosity of the sample increases as the levelof adhesion (coming from fo and kadh) increases for thesame consolidation stress. It is noted that the rate of decreasein porosity for kadh is noticeably slower than those with anonzero f0 because the adhesive force is proportional to kadh
and is thus higher at a higher stress level, making the samplemore difficult to compact. Additionally, for constant adhe-sion, porosity decreases more rapidly on the first applicationof stress and then slows down as the stress increases fur-ther. The rapid decrease in porosity at lower stress can beattributed to higher particle rearrangement resulting from thelooser packing formed.
Finally, we investigate the effect of adhesion on the com-puted flow function. Figure 19 shows that the unconfinedcompressive strength increases with the consolidation stressfor all the adhesion parameters, showing the stress historydependency phenomenon. Within the range of the consoli-dation stress studied, the predicted unconfined strength forcases with both non-zero kadh and f0 is found to be approx-imately the sum of the contributions from kadh and f0 sep-arately when all the other DEM input parameters are keptconstant. In this example, the slope of the flow function forthe case with kadh = 0.1 kN/m and f0 = 0 N is greater com-pared to that with kadh = 0 kN/m and f0 = −0.002 N. In the
Fig. 19 Predicted flow function for different kadh and f0 with k1 = 1and k2 = 10 kN/m
former case, the unconfined strength increases because boththe coordination number Z and the load-dependent adhesionincreases with loading whilst in the latter case, the strengthonly increased as a function of increasing Z .
It is evident from the above discussion that the consoli-dated porosity (ηc) and thus the coordination number (Z)
play a pivotal role in characterising the bulk cohesion. Herewe explore how the mesoscopic contact parameters relate tomicroscopic cohesion. Rumpf [76] was the first to propose asimple model relating tensile strength (σt) to average adhe-sive strength (Fat ) at the particle level for a system of hardmono-disperse spheres with a random isotropic packing asfollows:
σt = Fat (1 − η)Z
πd2 (18)
where d is the diameter of particle, η is the porosity, and Z isthe coordination number. It should be noted that this equationwas developed for an ideal packing of hard spheres withisotropic and homogenous distribution of stresses. However,our experimental setup of uniaxial compaction is anisotropicin nature. The effect of anisotropy has been investigated byQuintanilla et al. [77] and they reported that the anisotropydoes not affect the relationship significantly.
Rumpf derived the equation for tensile strength of agglom-erates, whilst in this study, our focus is on the bulk compres-sive strength. The relationship between tensile strength and
Fig. 20 Mohr circle for uniaxial tension and compression
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Micromechanical analysis of cohesive granular materials 397
Fig. 21 Normalized unconfined compressive force (σu d2/ fatp) as afunction of coordination number (Z) and consolidated porosity (ηc)
unconfined compressive strength can be derived from theconstruction of Mohr Circle (see Fig. 20) by assuming a lin-ear cohesive-frictional material. The unconfined compressivestrength σu is thus related to the unconfined tensile strengthσt by the simple expression:
σu = 1 + sin φ
1 − sin φσt (19)
where, φ is angle of internal friction.It is thus proposed that the compressive strength would
have the same micromechanical origins as the tensilestrength. From Eqs. 18 and 19, the normalized unconfinedstrength can be defined as:
σu
( fatp/d2)∝ (1 − ηc)Z (20)
where fatp is the average tensile contact force evaluated forthe whole particle system at the unconfined strength (σu)
state. The relationship between the normalized unconfinedstrength and the product of Z and solid volume fraction(1−ηc) for the full range of adhesion parameters is shown inFig. 21. A unique linear relationship exists between the twoquantities, which is analogous to the finding from Rumpf[76]. This proposes that the bulk unconfined strength for thesame size particle is governed by the contact tensile force atfailure, the coordination number and the solid fraction. Thisrelationship reveals an important microstructural mechanismfor bulk cohesion and can be used to facilitate unifying thecharacterisation and modelling of cohesive granular materi-als with different adhesion parameters.
It may first seem counter-intuitive why the unconfinedcompressive strength should relate to the tensile contactforce. The effect of frictional force can be elucidated bythe example simulations shown in Fig. 22. The figure showsthe stress strain result during unconfined compression. Twopairs of simulations with the same simulation parameters
Fig. 22 Unconfined stress strain behaviour as a function of limitingfriction (case I: f0 = −0.002 N, k2 = 10 and Kadh = 0.1 kN/m, caseII: f0 = −0.007 N, k2 = 10 and kadh = 0 kN/m)
but different limiting friction criteria fct ≤ μFn and fct ≤μ|( fn + kadhδn − fo)| were conducted: the former has beenexplored elsewhere [13] and the latter was adopted in thepresent model (see Eq. 8). It can be seen that there is a hugereduction in the computed unconfined strength as well as ini-tial stiffness if the limiting tangential force does not includethe tensile strength component. This can be seen for bothcases: combination of constant adhesion and load depen-dent adhesion ( fo & kadh—case I) and constant adhesion( f0—case II). When friction limit does not include tensile(adhesive) component, the tensile force remains similar butthe unconfined strength reduces significantly due to reducedshearing resistance. This confirms that the contribution ofadhesive force to the limiting friction has a significant effecton the bulk unconfined strength. This is also in line with theobservation that under uniaxial unconfined compression, themode of failure is predominantly shear failure rather thantensile failure [73,74].
8 Conclusions
DEM simulations and micromechanical analysis of cohe-sive powders using an adhesive elasto-plastic contact modelhave been presented. The model is capable of capturing theimportant stress-history dependency of powders’ unconfinedcohesive strength, as observed in experiments. This suggeststhat the elasto-plastic adhesive model may be used to sim-ulate cohesive solids subjected to different flow and stressregimes.
The particle contact plasticity has been found to be essen-tial for capturing the stress history dependence and to producea realistic flow function. Micromechanical analysis revealedthat increased particle contact plasticity increases the bulk
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398 S. C. Thakur et al.
plasticity. The contact plasticity prevents excessive elasticrebound at the contact level which leads to a lower porosityon the application of stress. For constant adhesive strength,the unconfined strength has been shown to correlate withthe instantaneous coordination number at the peak state.This provides a microscopic explanation for the unconfinedstrength variation and also indicates that the coordinationnumber can be used as a state variable to avoid invokingthe history effect. In some sense, this correlation is a newmicroscopic (or meso-scale) flow function.
When the contact adhesive strength increases, the filledporosity also increases. Higher adhesive forces allow theparticles to stick together during the filling process, lead-ing to stronger chain like structure and ultimately higherfilled porosity. The adhesive forces provide a high resistanceto counteract the effect of gravity force and provide somemechanical stability which restricts the relative movementbetween particles. For both load dependent kadh and constantadhesive strength f0, a unique relationship between porosityand mobilized adhesive force was found. On the applicationof confined compression, the load-dependent adhesion pro-vides a greater resistance to volumetric compression becausethe adhesive strength is proportional to kadh and is thus higherat higher stress levels, making the sample more difficult tocompact.
While comparing the effect of adhesion parameters onunconfined strength within a range of consolidation stresses,it has been found that the predicted unconfined strength forcases with both non-zero kadh and f0 is approximately thesum of the contributions from kadh and f0 separately; whenall the other DEM input parameters are kept constant. A linearrelationship has been established between the normalizedunconfined strength and the product of coordination numberand solid volume fraction. This gives a general microscopic(or meso-scale) flow function for different materials withdistinct cohesion strength origins.
It has been found that contribution of adhesive force to thelimiting friction has a significant effect on bulk unconfinedstrength. Failure to include the adhesive contribution in thecalculation of the frictional resistance may lead to under-prediction of unconfined strength and incorrect failure mode.
Acknowledgments The support of the European Commission underthe Marie Curie Initial Training Network for the PARDEM Project isgratefully acknowledged. The authors would also like to thank Prof.Stefan Luding and Dr. Hossein Ahmadian for useful discussions.
References
1. Nedderman, R.M.: Statics and Kinematics of Granular Materials.Cambridge University Press, Cambridge (2005)
2. Jenike, A.W.: Storage and flow of solids, bulletin no. 123. Bull.Univ. Utah 53(26) (1964)
3. Carr, J.F., Walker, D.M.: An annular shear cell for granular mate-rials. Powder Technol. 1(6), 369–373 (1968)
4. Schulze, D.: A new ring shear tester for flowability and time con-solidation measurements. In: 1st International Particle TechnologyForum, pp. 11–16, Denver, CO, USA (1994)
5. Bell, T.A., Catalano, E.J., Zhong, Z., Ooi, J.Y., Rotter, J.M.: Eval-uation of the Edinburgh powder tester. In: Proceedings of thePARTEC, pp. 2–6 (2007)
6. Enstad, G., Ose, S.: Uniaxial testing and the performance of a palletpress. Chem. Eng. Technol. 26, 171–176 (2003)
7. Freeman, R., Fu, X.: The development of a compact uniaxial tester.In: Part. Sci. Anal. Edinburgh, UK, pp. 1–6 (2011)
8. Parrella, L., Barletta, D., Boerefijn, R., Poletto, M.: Comparisonbetween a uniaxial compaction tester and a shear tester for thecharacterization of powder flowability. KONA Powder Particle J.26, 178–189 (2008)
9. Röck, M., Ostendorf, M., Schwedes, J.: Development of an uniaxialcaking tester. Chem. Eng. Technol. 29, 679–685 (2006)
10. Williams, J.C., Birks, A.H., Bhattacharya, D.: The direct measure-ment of the failure of a cohesive powder. Powder Tech. 4, 328–337(1971)
11. Zhong, Z., Ooi, J.Y., Rotter, J.M.: Predicting the handlability of acoal blend from measurements on the source coals. Fuel 84, 2267–2274 (2005)
12. Derjaguin, B., Muller, V.M., Toporov, Y.P.: Effect of contact defor-mations on the adhesion of particles. J. Colloid Interface Sci. 53,314–326 (1975)
13. Gilabert, F.A., Roux, J.N., Castellanos, A.: Computer simulationof model cohesive powders: influence of assembling procedure andcontact laws on low consolidation states. Phys. Rev. E. 75, 011303(2007)
14. Johnson, K.L., Kendall, K., Roberts, A.D.: Surface energy and thecontact of elastic solids. Proc. R. Soc. Lond. A. Math. Phys. Sci.324, 301 (1971)
18. Walton, O.R., Johnson, S.M.: DEM Simulations of the effects ofparticleshape, interparticle cohesion, and gravity on rotating drumflows of lunar regolith. In: Earth and Space, Honolulu, Hawaii, pp.1–6 (2010)
19. Molerus, O.: Theory of yield of cohesive powders. Powder Technol.12, 259–275 (1975)
20. Brilliantov, N., Albers, N., Spahn, F., Pöschel, T.: Collision dynam-ics of granular particles with adhesion. Phys. Rev. E. 76, 051302(2007)
21. Matuttis, H., Schinner, A.: Particle simulation of cohesive granularmaterials. Int. J. Mod. Phys. C. 12, 1011–1021 (2001)
23. Walton, O.R., Braun, R.L.: Viscosity, granular-temperature, andstress calculations for shearing assemblies of inelastic, frictionaldisks. J. Rheol. 30, 949–980 (1986)
24. Ning, Z., Thornton, C.: Elastic–plastic impact of fine particles witha surface. In: Powders and Grains, pp. 33–38 (1993)
25. Hertz, H.: On the contact of elastic solids. J. Reine Angew. Math.92, 110 (1881)
26. Vu-Quoc, L., Zhang, X.: An elastoplastic contact force-displacement model in the normal direction: displacement-drivenversion, Proc. R. Soc. London. Ser. A Math. Phys. Eng. Sci. 455,4013–4044 (1999)
123
Micromechanical analysis of cohesive granular materials 399
27. Thornton, C.: Coefficient of restitution for collinear collisionsof elastic-perfectly plastic spheres. J. Appl. Mech. 64, 383–386(1997)
28. Walton, O.R., Johnson, S.M.: Simulating the effects of interpar-ticle cohesion in micron-scale powders. In: AIP Conference onProceedings, Golden, CO, pp. 897–900 (2009)
29. Tomas, J.: Assessment of mechanical properties of cohesive par-ticulate solids. Part 1: particle contact constitutive model. Part. Sci.Technol. 19, 95–110 (2001)
30. Tomas, J.: Particle Adhesion Fundamentals and Bulk Powder Con-solidation, Kona (2000)
31. Tykhoniuk, R., Tomas, J., Luding, S., Kappl, M., Heim, L., Butt,H.-J.: Ultrafine cohesive powders: from interparticle contacts tocontinuum behaviour. Chem. Eng. Sci. 62, 2843–2864 (2007)
32. Luding, S., Tykhoniuk, R., Tomas, J.: Anisotropic material behav-ior in dense, cohesive-frictional powders. Chem. Eng. Technol. 26,1229–1232 (2003)
33. Luding, S., Manetsberger, K., Mullers, J.: A discrete model for longtime sintering. J. Mech. Phys. Solids. 53, 455–491 (2005)
34. Moreno-atanasio, R., Antony, S.J., Ghadiri, M.: Analysis of flowa-bility of cohesive powders using distinct element method. PowderTechnol. 158, 51–57 (2005)
35. Hassanpour, A., Ghadiri, M.: Characterisation of flowability ofloosely compacted cohesive powders by indentation. Part. Part.Syst. Charact. 24, 117–123 (2007)
36. Aoki, R., Suzuki, M.: Effect of particle shape on the flow and pack-ing properties of non-cohesive granular materials. Powder Technol.4(2), 102–104 (1971)
37. Li, J., Langston, P.A., Webb, C., Dyakowski, T.: Flow of sphero-disc particles in rectangular hoppers—a DEM and experimentalcomparison in 3D. Chem. Eng. Sci. 59, 5917–5929 (2004)
38. Roberts, T.A., Beddow, J.K.: Some effects of particle shape andsize upon blinding during sieving. Powder Technol. 2, 121–124(1968)
39. Chong, Y.S., Ratkowsky, D.A., Epstein, N.: Effect of particle shapeon hindered settling in creeping flow. Powder Technol. 23, 55–66(1979)
40. Wensrich, C.M., Katterfeld, A.: Rolling friction as a technique formodelling particle shape in DEM. Powder Technol. 217, 409–417(2012)
41. Cleary, P.W.: DEM prediction of industrial and geophysical particleflows. Particuology 8, 106–118 (2010)
42. Favier, J.F., Abbaspour-Fard, M.H., Kremmer, M., Raji, A.O.:Shape representation of axi-symmetrical, non-spherical particlesin discrete element simulation using multi-element model parti-cles. Eng. Comput. 16, 467–480 (1999)
43. Härtl, J., Ooi, J.Y.: Experiments and simulations of direct sheartests: porosity, contact friction and bulk friction. Granul. Matter.10, 263–271 (2008)
44. Chung, Y.C., Ooi, J.Y.: Confined compression and rod penetra-tion of a dense granular medium? Discrete. In: Modern Trends inGeomechanics, pp. 223–239 (2006)
45. Thakur, S.C., Ahmadian, H., Sun, J., Ooi, J.Y.: An experimental andnumerical study of packing, compression, and caking behaviour ofdetergent powders. Particuology 12, 2–12 (2014)
46. Kodam, M., Bharadwaj, R., Curtis, J., Hancock, B., Wassgren,C.: Force model considerations for glued-sphere discrete elementmethod simulations. Chem. Eng. Sci. 64, 3466–3475 (2009)
47. Kruggel-Emden, H., Rickelt, S., Wirtz, S., Scherer, V.: A study onthe validity of the multi-sphere discrete element method. PowderTechnol. 188, 153–165 (2008)
48. Jones, R.: Inter-particle forces in cohesive powders studied byAFM: effects of relative humidity, particle size and wall adhesion.Powder Technol. 132, 196–210 (2003)
49. Cundall, P.A., Strack, D.L.: A discrete numerical model for gran-ular assemblies. Geotechnique 1, 47–65 (1979)
50. Rayleigh, L.: On waves propagated along the plane surface of anelastic solid. Proc. Lond. Math. Soc. 4,4–11 (1885)
52. Sheng, Y., Lawrence, C.J., Briscoe, B.J., Thornton, C.: Numericalstudies of uniaxial powder compaction process by 3D DEM. Eng.Comput. 21, 304–317 (2004)
53. Jones, R.: From single particle AFM studies of adhesion and fric-tion to bulk flow: forging the links. Granul. Matter. 4, 191–204(2003)
54. Morrissey, J.P.: Discrete Element Modelling of Iron Ore Pellets toInclude the Effects of Moisture and Fines, Doctoral dissertation,University of Edinburgh (2013)
55. DEM Solutions: EDEM 2.3 User Guide, Edinburgh, Scotland, UK(2010)
56. Skinner, J., Gane, N.: Sliding friction under a negative load. J. Phys.D. Appl. Phys. 5, 2087 (1972)
57. Savkoor, A., Briggs, G.: The effect of tangential force on the contactof elastic solids in adhesion. Proc. R. Soc. Lond. A. Math. Phys.Sci. 356, 103–114 (1977)
58. Thornton, C., Yin, K.: Impact of elastic spheres with and withoutadhesion. Powder Technol. 65, 153–166 (1991)
59. Briscoe, B., Kremnitzer, S.L.: A study of the friction and adhe-sion of polyethylene-terephthalate monofilaments. J. Phys. D Appl.12(4),505 (1979)
60. Jones, R., Pollock, H.M., Geldart, D., Verlinden-Luts, A.: Fric-tional forces between cohesive powder particles studied by AFM.Ultramicroscopy 100, 59–78 (2004)
61. Ruths, M., Alcantar, N.A., Israelachvili, J.N.: Boundary Friction ofAromatic Silane Self-Assembled Monolayers Measured with theSurface Forces Apparatus and Friction Force Microscopy, Society,pp. 11149–11157 (2003)
62. Berman, A., Drummond, C., Israelachvili, J.: Amontons’ law at themolecular level. Tribol. Lett. 4, 95–101 (1998)
63. Schwarz, U.D., Zwörner, O., Köster, P., Wiesendanger, R.: Quanti-tative analysis of the frictional properties of solid materials at lowloads. I. Carbon compounds. Phys. Rev. B. 56, 6987 (1997)
64. Ecke, S.: Friction between individual microcontacts. J. ColloidInterface Sci. 244, 432–435 (2001)
65. Heim, L., Farschi, M., Morgeneyer, M., Schwedes, J., Butt, H.J.,Kappl, M.: Adhesion of carbonyl iron powder particles studiedby atomic force microscopy. J. Adhes. Sci. Technol. 19, 199–213(2005)
66. Watanabe, H., Groves, W.L.: Caking test for dried detergents. J.Am. Oil Chem. Soc. 41, 311–315 (1964)
67. Chung, Y.: Discrete Element Modelling and Experimental Valida-tion of a Granular Solid Subject to Different Loading Conditions.The University of Edinburgh. PhD dissertation (2006)
68. Härtl, J., Ooi, J.Y.: Numerical investigation of particle shape andparticle friction on limiting bulk friction in direct shear tests andcomparison with experiments. Powder Technol. 212, 231–239(2011)
69. Midi, G.D.R.: On dense granular flows. Eur. Phys. J. E. Soft Matter.14, 341–65 (2004)
70. DEM Solutions Ltd.: EDEM 2.4 Theory Reference Guide, Edin-burgh, Scotland, UK (2011)
71. Luding, S., Alonso-Marroquin, F.: The critical-state yield stress(termination locus) of adhesive powders from a single numericalexperiment. Granul. Matter. 13, 109–119 (2011)
72. Hiestand, E.N.: Principles, tenets and notions of tablet bonding andmeasurements of strength. Eur. J. Pharm. 44 (1997)
73. Schofield, A., Wroth, P.: Critical State Soil Mechanics. McGraw-Hill, New York (1968)
74. Wood, D.M.: Soil Behaviour and Critical State Soil Mechanics.Cambridge University Press, Cambridge (1990)
123
400 S. C. Thakur et al.
75. Yang, R.Y., Zou, R.P., Yu, A.B.: Computer simulation of the pack-ing of fine particles. Phys. Rev. E. 62, 3900–3908 (2000)
76. Rumpf, H.: The strength of granules and agglomerate. In: Knepper,W. (ed.) Agglomeration. Wiley, New York (1962)
77. Quintanilla, M.A.S., Castellanos, A., Valverde, J.: Correlationbetween bulk stresses and interparticle contact forces in fine pow-ders. Phys. Rev. E. 64, 1–9 (2001)
