Original papers Granular Matter 6, 75–86 c© Springer-Verlag 2004DOI 10.1007/s10035-004-0167-9

Fundamentals of cohesive powder consolidation and flowJurgen Tomas

Abstract The fundamentals of cohesive powder consoli-dation and flow behaviour are explained to combine rea-sonably particle and continuum mechanics. The influenceof elastic-plastic repulsion and, consequently, stressingpre-history dependent adhesion is demonstrated by thenew model “stiff particles with soft contacts” and the con-tact force equilibrium. With this as the physical basis,incipient powder consolidation, yield and cohesive steady-state flow are explained. These models are used to eval-uate shear cell test results as constitutive functions forcomputer aided apparatus design for reliable flow.

Keywords Particle contact mechanics, Adhesion, Vander Waals force, Constitutive models, Powder flow prop-erties

Symbolsa separation, nmA area, particle contact area, m2

CH Hamaker constant, JD particle size, µmE modulus of elasticity, kN/mm2

F force, nNFH0 adhesion force without contact deformation, nNP pressure, kPapf micro-yield strength of contact, MPar radius, nmε porosity

Received: 10 July 2003

J. TomasMechanical Process Engineering,The Otto-von-Guericke-University,Universitatsplatz 2,D – 39 106 Magdeburg, Germanye-mail: juergen.tomas@vst.uni-magdeburg.deTel.: ++391 67 18783Fax: ++391 67 11160

The author would like to acknowledge his previous co-work-ers Dr. T. Groger, Dr. Th. Kollmann, Dr. B. Reichmann andhis present co-workers Dr. W. Hintz, Dr. S. Aman, and Dr.W. Schubert for providing relevant information and theoreticaltips. The advises from H.-J. Butt [22] and S. Luding [23] withrespect to the fundamentals of particle and powder mechan-ics were especially appreciated during the colaboration of theproject “shear dynamics of cohesive, fine-disperse particle sys-tems” of the joint research program “Behaviour of GranularMedia” of German Research Association (DFG).

κ elastic-plastic contact consolidation coeff.κA elastic-plastic contact area coefficientκp plastic contact repulsion coefficientϕi angle of internal friction, degϕst stationary angle of internal friction, degρ density, kg/m3

σ normal stress, kPaσ1 major principal stress, kPaσ2 minor principal stress, kPaσ0 isostatic tensile strength of unconsolidated

powder, kPaτ shear stress, kPa

Indicesb bulkc compressiveK total contacte effectiveel elasticH adhesioni internaliso isostaticl liquidM centre of Mohr circleN normalpl plasticpre preshearR radius of Mohr circles solidS shearst stationaryV pre-consolidationVdW van der WaalsZ tensile0 initial, zero point

1IntroductionThe well-known flow problems of cohesive particulate sol-ids in storage and transportation containers, conveyors orprocess apparatuses – mainly mentioned by Jenike [1] –lead to bridging, channelling, oscillating mass flow ratesand particle characteristics associated with feeding anddosing problems [2]. Taking into account this list of techni-cal problems, it is essential to deal with the fundamentalsof cohesive powder consolidation and flow behaviour. Inde-pendently of development of the discrete element method

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in rock mechanics, first comprehensive physical models forfine particles were published 1976/78 by Molerus [3,4] andcontinued 1983/91 by Tomas [5,6].

By the new model “stiff particles with soft contacts”,which is used for fine to ultra-fine particles, and the forceequilibrium of a representative or characteristic contact,universal equations are presented which include theelastic-plastic particle contact behaviours with adhesion,load-unload hysteresis, and thus, energy dissipation anda history dependent adhesion force function. With this asthe physical basis, incipient powder consolidation, yieldand cohesive steady-state flow are explained. This paper isfocused on the method, i.e., how to develop an innovativecombination of particle and continuum mechanics as com-fortable as possible to obtain analytically usable results.This derivation has not been published up to now.

2Particle contact constitutive modelThe well-known failure hypotheses of Tresca, Coulomb-Mohr, the yield locus concept of Jenike [1] and Schwedes[2] were supplemented from Molerus [3,4] by the cohesivesteady-state flow criterion. The consolidation and non-rapid, frictional flow of cohesive powders was explained bythe adhesion forces at particle contacts [4]. His advancedtheory is the suitable basis of extended and generalisedmodels in particle and powder mechanics [7–12] which isshown now.

In principle, there are four essential deformation effectsin particle-surface contacts and their force-displacementbehaviours can be explained as follows, see in detail thecomprehensive literature review of Tomas [11] with 151references:

• Elastic contact deformation which is reversible, inde-pendent of deformation rate and consolidation timeeffects and valid for all particulate solids,

• Plastic contact deformation with adhesion which isirreversible, deformation rate and consolidation timeinvariant, e.g. mineral powders,

• Viscoelastic contact deformation which is reversibleand dependent on deformation rate and consolidationtime, e.g. bio-particles,

• Viscoplastic contact deformation which is irreversibleand dependent on deformation rate and consolidationtime, e.g. nanoparticles fusion.

2.1Normal force – displacement model of a characteristicparticle contactThis paper is intended to focus on the model of isotropic,stiff, linear elastic, mono-disperse spherical particles thatare approaching to soft contacts by attractive adhesionforces of smooth particle surfaces, Fig. 1. Thus, this softor compliant contact displacement is assumed to be smallhK/d << 1 compared to the size (diameter) of the stiffparticle. The particles may have a certain material stiff-ness so that the volume deformation is negligible. Duringsurface stressing the rigid particle is not so much deformedthat it undergoes a certain change of the particle shape.

This is equivalent to a model of “healing contacts” afterstressing and deformation. In contrast, soft particle mattersuch as biological cells or macromolecular organic materialdo not behave this way.

These particle contact mechanics should be explainedby a typical force-displacement diagram of Fig. 1[7,9–11]. The zero-point of this diagram hK = 0 is equiv-alent to the characteristic adhesion separation of a di-rect contact aF=0. After approaching FN ∝ a−2 (witha = aF=0 − hK) from an infinite distance −∞ to a mini-mum separation a = aF=0 the sphere-sphere contact with-out any contact deformation is formed by the long-rangeattractive adhesion force FH0 (the so-called jump in).Then the contact may be loaded from point FH0 to Y and,as the response, is elastically deformed with an approx-imated circular contact area, Fig. 1b). With increasingexternal normal load this soft contact starts at a pres-sure pf with plastic yielding at the point Y, Fig. 1c). Thisyield point Y is located below the abscissa, i.e. the con-tact force equilibrium FN = 0 includes a certain elastic-plastic deformation as response of adhesion force (1+κ).

FH0. The micro-yield surface is reached and this maxi-mum pressure pf can not be exceeded and results in acombined elastic-plastic yield limit of the nanoplate-platecontact with an annular elastic zone and a circular centre.

The total force can be obtained by the particle con-tact force equilibrium between attraction (−) and elasticas well as soft plastic repulsion (+) or force response (rKcontact radius, r∗

K coordinate of annular elastic contactarea):∑

F = 0 = −FH0 − pV dW · π · r2K − FN + pf · π · r2

K,pl

+2 · π ·rK∫

rK,pl

pel(r∗K) · r∗

Kdr∗K (1)

A very comfortable, linear force-displacement model withthe particle centre approach (height of flattening or over-lap) of both particles hK is obtained [7,11]:

FN + FH0 = π · r1,2 · pf · (κA − κp) · hK (2)

This line is shown in Fig. 1 as elastic-plastic yield limitwith the averaged or reduced particle radius

r1,2 = (1/r1 + 1/r2)−1 (3)

at contact of two smooth spheres 1 and 2. Thus predomi-nant plastic yielding behaviour provided, the contact stiff-ness decreases with smaller particle size d = 4 · r1,2 (ormicro-roughness radius of non-deformed contact) espe-cially of cohesive nanopowders:

kN,pl =dFN

dhK= π · r1,2 · pf · (κA − κp) (4)

A confined plastic field is formed inside of the contactcircle. The elastic-plastic contact area coefficient κA rep-resents the dimensionless ratio of plastic particle contactdeformation area Apl to total contact deformation areaApl +Ael. This includes a certain elastic displacementmainly at the perimeter of the contact circle:

κA =23

+13

· Apl

Apl + Ael(5)

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Fig. 1. Force-displacement diagram of recalculated charac-teristic contact deformation of cohesive limestone particlesas spheres, median diameter d50 = 1.2 µm. Pressure andcompression are defined as positive but tension and extensionare negative, above panel. After approaching from an infinitedistance −∞ to this minimum separation aF=0 the sphere-sphere-contact without any contact deformation is formed bythe attractive adhesion force FH0 (the so-called “jump in”).As the response, from FH0 – Y the contact is elastically com-pacted, forms an approximated circular contact area, panel (b)and starts at the yield point Y at pmax = pf with plastic yield-ing, panel (c). This displacement is expressed by annular elasticAel and circular plastic Apl contact area. Between the elastic-

plastic yield limit and the adhesion limit the elastic domainis located. If the contact would be unloaded, e.g. between thepoints U – A, the contact recovers elastically according to Eq.(7) to a displacement hK,A. If one applies a certain pull-offforce FN,Z =−FH,A as given in Eq. (11) but here negative, theadhesion limit at failure point A is reached and the contactplates fail and detach with the increasing distance, panel (d).The material data are: modulus of elasticity E = 150 kN/mm2,plastic micro-yield strength pf = 350 N/mm2, Poisson ratioν = 0.28, Hamaker constant CH,sls = 3.8 10−20 J, equilibriumseparation for dipole interaction aF=0 = 0.336 nm, elastic-plas-tic contact area coefficient κA = 5/6 and the plastic repulsioncoefficient κp = 0.15

The plastic repulsion coefficient κp describes a dimen-sionless ratio of attractive van der Waals pressure pV dW

(adhesion force per unit planar surface area) to repulsiveparticle micro-hardness pf for a plate-plate model (e.g.pV dW ≈ 3–600 MPa):

κp =pV dW

pf=

CH,sls

6 · π · a3F=0 · pf

(6)

The characteristic adhesion distance lies in a molecularscale aF=0 ≈ 0.3–0.4 nm and depends mainly on the prop-erties of liquid-equivalent packed adsorbed layers. Thisseparation can be estimated for a molecular interaction

potential minimum−dU/da = F = 0 = Fat + Frep orforce equilibrium. Provided that these molecular contactsare stiff enough compared with the soft particle contactbehaviour influenced by these mobile adsorption layersdue to molecular rearrangement, this separation aF=0is assumed to be constant. The Hamaker constant (in-dex solid-liquid-solid) CH,sls using the Lifshitz theory isrelated to continuous media and depends on their dielec-tric constants and refractive indices [13].

Constant mechanical properties provided, the finer theparticles the smaller is the yield point Y in Fig. 1 [9].Thus, an initial and exclusive elastic contact deformation

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at loading has no relevance for the irreversible consoli-dation of fine to ultra-fine powders and can be excludedhere.

Thus, one has to consider an elastic-plastic contactbehaviour which is characterized by the elastic-plasticyield limit, Eq. (2). This line can not be crossed. Belowthe yield limit the elastic domain begins. The model isequivalent to the yield surface in continuum mechanics.Any load FN yields an increasing displacement hK . But,if one would unload, beginning at arbitrary point U, theelastically deformed, annular contact zone recovers along aparabolic curve U – A and the plate-plate contact remainswith a “frozen” radius rK,A or reduced plastic displace-ment hK,A

FN,unload =23

· E ∗ ·√

r1,2 · (hK − hK,A)3 + FN,Z (7)

with the averaged modulus of elasticity E* of both parti-cles 1 and 2 (ν Poisson ratio):

E∗ = 2 ·(

1 − ν21

E1+

1 − ν22

E2

)−1

(8)

This model was obtained by combining the Hertz law withEq. (2) [11]. Along the symmetric curve A – U the contactmay be reloaded:

FN,reload = −23

· E ∗ ·√

r1,2 · (hK,U − hK)3 + FN,U (9)

If one replaces the contact radius r2K,A = r1,2 · hK,A after

contact flattening in the sum of attractive particle contactforces as given from Eq. (1)∑

Fat = 0 = −FH0 − pV dW · π · r2K,A − FN,Z (10)

a plausible adhesion force-displacement relation isobtained which shows the increased pull-off force levelFN,Z , here negative:

FN,Z = −FH0 − π · r1,2 · pV dW · hK,A (11)

The adhesion or contact failure limit at point A in thediagram of Fig. 1 is reached and the contact plates detachwith the increasing distance a = aF=0 + hK,A−hK [7,11].This actual particle separation a can be considered for thecalculation by means of a long-range hyperbolic adhesionforce curve FN,Z ∝ pV dW (a) ∝ a−3 [13] from the plate-plate model as given from Eq. (6).

These theoretical predictions of normal force-displace-ment behaviours are very useful to describe the compres-sion of primary elastic, elastic-plastic particle compoundsor elastic-plastic granules at conveying and handling oper-ations. Elastic-plastic load and Hertzian unload curves,Eq. (2) and Eq. (7), and the unload/reload hysteresis wereexperimentally confirmed for coarse granules [14].

2.2Linear adhesion force – normal force modelThe slopes of elastic-plastic yield and adhesion limits inFig. 1 are characteristics of irreversible particle contactstiffness or compliance. Consequently, if one eliminatesthe center approach hK of the loading and unloadingfunctions, Eqs. (2) and (7), an implied positive function

between the contact pull-off force FH,A =− FN,Z at thedetachment point A is obtained for the normal force atthe unloading point FN = FN,U :

FH,A,(1) = FH0 + κ · (FN + FH0) − π · r21,2 · pV dW

×[

3 · (FN + FH0)2 · r2

1,2 · E∗ ·(

1 +FH,A,(0) − FH0

FN + FH0

)]2/3

(12)

The unloading point U is stored in the memory of the con-tact as pre-consolidation history. This general non-linearadhesion force model, dashed curve in Fig. 2, implies thedimensionless, elastic-plastic contact consolidation coeffi-cient κ, Eq. (14), the adhesion force FH0( r1,2) without anycontact flattening and, additionally, the influence of adhe-sion, stiffness, average particle radius r1,2 and the averagemodulus of elasticity E* in the last term of the equation.The slope of the adhesion force is reduced with increasingradius of surface curvature r1,2 which is in agreement withthe practical experiences in powder handling [1,2]. Thisnon-linear adhesion-normal force function FH = f(FN ) canbe linearised as shown in Fig. 2:

FH =κA

κA − κp· FH0 +

κp

κA − κp· FN

= (1 + κ) · FH0 + κ · FN (13)The dimensionless elastic-plastic contact consolidationcoefficient κ is given by the slope of adhesion force FH

influenced by predominant plastic contact failure:

κ =κp

κA − κp(14)

This displacement coefficient κ characterises the irrevers-ible particle contact stiffness or softness as well. A shal-low slope implies low adhesion level FH ≈ FH0 becauseof stiff particle contacts, but a large slope means softcontacts, or i.e., a cohesive powder flow behaviour. Thismodel considers, additionally, the flattening of soft parti-cle contacts caused by the adhesion force κ·FH0. Thus, thetotal adhesion force FH consists of a stiff contribution FH0and a displacement influenced component κ. (FH0 + FN ),Fig. 2.

All the essential material parameters are collected inTable 1. Thus, Eq. (13) can be interpreted as a general lin-ear particle contact constitutive model, i.e. linear in forces,but non-linear concerning material characteristics [7].

3Cohesive powder flow criteriaThe micro/macrotransition of normal and tangential par-ticle contact forces to stresses at a representative volumeelement of the powder should be designed as comfortableas possible to obtain analytically usable results.

3.1Micro/macrotransition, orientation of the shear zoneand the yield locusTo formulate the failure conditions at particle contacts ofthe shear zone, one can use the Molerus theory [3]. Butinstead of a perfect plastic model [3],

FH = FH0 +pV dW

pf· FN = FH0 + κp · FN (15)

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Fig. 2. Adhesion force – normal force diagram of recalculatedparticle contact forces of limestone according to Eqs. (12) and(13), median diameter d50 = 1.2 µm, surface moisture XW

= 0.5 %. Adhesion force parameters are the elastic-plasticcontact consolidation coefficient κ = 0.224, recalculated fromEq. (30), and the adhesion force without any contact flattening

FH0 = 2.64 nN. For comparison, the total adhesion force FH,tot

after a time consolidation of 24 h is also shown in Fig. 2 by afull line. This time variable slope is given by the elastic-plasticand viscoplastic contact consolidation coefficient κvis = 1.014which is obtained by the sum of rheological models [9]

Table 1. Material parameters of adhesion force functions FH(FN ) in Fig. 2 and contact friction

Instantaneous contact Eq.consolidation

plastic repulsion coefficient κp = pV dWpf

= CH,sls

6·π·a3F=0·pf

(6)

elastic-plastic contact area coefficient κA = 23 + Apl

3·(Apl+Ael)(5)

elastic-plastic contact consolidationcoefficient κ = κp

κA−κp(14)

averaged particle radius r1,2 = (1/r1 + 1/r2)−1 (3)

averaged modulus of elasticity E∗ = 2 ·(

1−ν21

E1+ 1−ν2

2E2

)−1(8)

Poisson ratio νmicro-yield strength of contact pf

Adhesion force of sphere-sphere contact without FH0

any contact deformation, see e.g. [3,13]contact friction coefficient

∑FS∑FN

= tan ϕi (20), (22)

the more complex, elastic-plastic particle contact modelEq. (13) is used to describe the failure conditions of thecharacteristic particle contact:

(1) The contact areas are small in comparison to the char-acteristic particle size d, so that these may be consid-ered as contact points.

(2) As a result of spherical symmetry, the contact pointsare uniformly distributed over the particle surfacewith the same probability.

(3) The packing structure is assumed to be isotropic andrandom with uniform porosity ε in any cross-sectionalareas as well as in the powder bulk.

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(4) Due to these assumptions an isotropic normal FN andshear or tangential force FS distribution in the uni-form distributed contacts is at least plausible.

(5) There is a direct correlation between isostatic stressesand contact forces as a superposition of three uniaxialnormal stresses which are oriented orthogonal to eachother.

(6) Thus, the relation between stresses and contact forcesfor characteristic monodisperse particles amounts to:

σ, τ =1 − ε

ε· FN , FS

d2 (16)

σM , σR =1 − ε

ε· FM , FR

d2 (17)

(7) With regard to the spherical symmetry and the uni-form probability of contact normal forces a uniformangle between radius and centre contact force compo-nents FR·cosα, FM ·cosα and normal vectors is desig-nated by α.

(8) Therefore, instead of radius σR and centre contactstresses σM , the normal and shear force magnitudesorientated with this angle α against direction of majorprincipal stress may be generally expressed by:

FN = FM + FR · cos 2α (18)

FS ≤ FR · sin 2α (19)

(9) If the shear force should be completely activated inthe shear zone, then the ‘=’ is valid.

(10) With regard to the uniform probability of contactforces and stresses, uniform angles of friction ϕi atcontacts and in the bulk are assumed [3].

The ratio of shear force/normal force is used to describesimultaneous sliding or steady-state flow at particle con-tacts (tanϕi coefficient of internal friction):

FR sin 2α

(1 + κ) · FH0 + (1 + κ) · (FM + FR cos 2α)≤ tanϕi (20)

To eliminate the failure angle α from Eq. (20), equality isonly achieved for a value of α where the left-hand side inbrackets is a maximum [3]. This is valid for the commonrelation of the characteristic orientation of normal vec-tor of the shear zone against direction of major principalstress:

α =π

4+

ϕst

2(21)

The condition for incipient or instantaneous contact fail-ure is consequently ( FHM and FHR are the adhesion forcecomponents of contact pre-consolidation)FR · [sin 2α − (1 + κ · FHR/FR) · tan ϕi · cos 2α]

(1 + κ) · FH0 + FM + κ · FHM≤ tan ϕi (22)

but now with an auxiliary angle β:

tanβ = (1 + κ · FHR/FR) · tanϕi (23)

Equality in Eq. (22) is only achieved for a value of α wherethe left-hand side in brackets is a maximum. This is validfor the failure angle α of the characteristic orientation ofthe shear zone [3]:

α =π

4+

β

2(24)

The larger the term of Eq. (23) in brackets κ · σV R/σR ≥0, expressed here as the ratio of radius stresses for con-solidation pre-history σV R and yielding σR, the more isthe difference from the normally expected relation of α =π/4 + ϕi/2. This latter relation is predicted to be validonly for stiff particle contacts without any contact consol-idation κ = 0.

Next, the auxiliary angle β has to be eliminated fromEq. (22) and a square function is obtained with the posi-tive root:

FR = sin ϕi

×[√

((1 + κ) · FH0 + κ · FHM + FM )2 − κ2 · F 2HR · cos2 ϕi

−κ · FHR · sin ϕi

](25)

This model is expressed in stresses, i.e. from Eq. (17) thepre-consolidation forces FHR, FHM are transferred intoradius σV Rand centre stresses σV M . By Eqs. (25) and (30)the general non-linear equation of the yield locus of a cohe-sive powder is obtained:

σR = sin ϕi · σV R

×[√√√√

( tan ϕsttan ϕi

· (σV M +σ0)−σV M +σM

σV R

)2

− sin2 (ϕst − ϕi)cos2 ϕst · tan2 ϕi

− sin (ϕst − ϕi)cos ϕst

](26)

This constitutive equation (26) combines the radius σR

and centre σM of Mohr circles of incipient yield withradius σV R and centre σV M of Mohr circles of arbitrarypre-consolidation state as envelope.

3.2Consolidation, Incipient yield and steady-state flowIt is worth to note here that the stressing pre-historyof cohesive powder flow is stationary (steady-state) and,from Eqs. (17) and (20), results significantly in a cohesivestationary yield locus (critical state line). This stationaryyield locus is expressed in radius stress-centre stress coor-dinates (radius σV R = σR,st and centre σV M = σM,st ofMohr circle for steady-state flow)

σR,st = sinϕst · (σM,st + σ0) (27)

or can be shown in an equivalent shear stress-normal stressdiagram of Fig. 3:

τ = tanϕst · (σ + σ0) (28)

This shear zone is characterised by a dynamic equilibriumof simultaneous contact shearing, unloading and failing,creating new contacts, loading, reloading, unloading andshearing again with a certain negative intersection of theabscissa.

σ0 =1 − ε0

ε0· FH0

d2 (29)

This isostatic tensile strength σ0 of an unconsolidatedpowder without any particle contact deformation isobtained from the adhesion force FH0 with the initialporosity ε0 of very loose packing [4].

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Fig. 3. Shear stress – normal stress diagram of yield char-acteristics of a cohesive powder. Only positive shear stressesτ(σ) are taken into consideration, the negatives -τ(σ) meanshear in counter-direction. Mainly, compressive stresses(pressures) σ occur in powder mechanics and are definedas positive. Tensile stresses are negative. In general, thesteady-state flow of a cohesive powder is cohesive. Hence,the total normal stress consists of an external contributionσ, e.g. by weight of powder layers, plus (by absolute value)an internal contribution by pre-consolidation dependentadhesion (tensile stress σZ). The consolidation and yield lociand the stationary yield locus are completely described onlywith three independent flow parameters, i.e. angle of inter-nal friction ϕi, stationary angle of internal friction ϕst, isostatic

tensile strength of an unconsolidated powder σ0 plus the influ-ence of characteristic pre-consolidation stress or average pres-sure at steady-state flow σM,st = (σ1+σ2)/2, shown in boxesin panel (a). This pre-consolidation state can also be expressedby the major principal stress σ1 (consolidation stress). σ2 is theminor principal stress at shear stress τ = 0 and σR,st = (σ1-σ2)/2 is the radius stress at steady-state flow. Thus, the flowparameters cohesion τc, uniaxial tensile strength σZ1 (tension),uniaxial compressive strength σc (pressure), isostatic tensilestrength σZ (tension) and isostatic pressure σiso depend di-rectly on the centre stress of steady-state flow (responsible forpre-consolidation) σM,st. σV M and σV R are centre and radiusstresses of an arbitrary pre-consolidation state, panel (b)

For the combination of angle of internal friction ϕi forincipient contact failure (slope of yield locus) with thestationary angle of internal friction ϕst the relation wasderived as follows [4,8]:tanϕst = (1 + κ) · tanϕi (30)The softer the particle contacts, the larger are the differ-ences between these friction angles and consequently, themore cohesive is the powder response.

The shear tests gives us parallel yield loci of constantdistance, i.e. as linear function of pre-consolidation stressσM,st which can be approximated as straight lines. Toobtain this comfortable linear function for incipient yield,

the square root function Eq. (26) is linearised by Taylorseries expansion at the end point of incipient yield andtransition to steady-state flow, Fig. 3:σR = σR (σM = σM,st)

+dσR

dσM

∣∣∣∣σM=σM,st

· (σM − σM,st) (31)

This linear yield locus is simply to use:

σR = sin ϕi ·[σM +

(sin ϕst

sin ϕi− 1)

· σM,st +sin ϕst

sin ϕi· σ0

](32)

τ = tanϕi · (σ + σZ) = tanϕi ·(

σ +σR,st

sin ϕi− σM,st

)(33)

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Fig. 4. Yield loci (YL) and stationary yield locus (SYL) oflimestone powder, milling product with median particle size d50

= 1.2 µm, solid density ρs = 2740 kg/m3, surface moisture XW

= 0.5% accurately analysed by Karl Fischer titration; shearrate vS = 2 mm/min. The approximately parallel arrangementof yield loci with constant distance is good to see. Thus thephysical model, Eqs. (28) and (33), is applied by straight lineregression with very good fit ≥ 0.98. The three flow param-eters are obtained as: angle of internal friction ϕi = 33–43◦,

stationary angle of internal friction ϕst = 38◦, extrapolated iso-static tensile strength of the unconsolidated powder σ0 = 1.34kPa. The particle shape of the milling product differs signifi-cantly from smooth spheres used as model in Fig. 1 and Fig.2. Thus, these models may be extended by approximations ofthe real shape, like spheres with hemispherical micro-rough-ness or clumps of spheres. To describe the force-displacementbehaviour of such rough contacts is not solved yet

The total normal stress consists of an external contribu-tion σ, e.g. by weight of powder layers, plus (by absolutevalue) an internal contribution by the pre-consolidationdependent adhesion, the isostatic tensile stress σZ :

σZ =σR,st

sin ϕi− σM,st =

(sin ϕst

sin ϕi− 1

)· σM,st

+sin ϕst

sin ϕi· σ0 (34)

It is worth to note here that σZ and the cohesion τc =σZ tanϕi depend on the consolidation pre-history whichis given by the influence of centre stress σM,st or averagepressure of steady-state flow.

These linear yield loci are found to be in accordancewith experimental experience, e.g. for limestone powder,see Fig. 4.

The consolidation locus represents the envelope of allMohr circles for incipient consolidation, i.e. the radiusand centre stresses between steady-state flow and isostaticpressure σiso, Fig. 3. Provided that the particle contactfailure is equivalent to that between incipient powder flowand consolidation, one can write for a linear consolidationlocus with negative slope −sinϕi which is symmetricallywith the linear yield locus:

σV R = sinϕi · (−σV M + σiso) (35)

Due to this symmetry between yield and consolidationlocus, one can directly estimate the isostatic powder com-pression σV 1 = σV 2 = σV M = σiso from Fig. 3 for theradius stress σV R = 0:

σiso = 2 · σM,st + σZ =σR,st

sin ϕi+ σM,st

=(

sin ϕst

sin ϕi+ 1

)· σM,st +

sin ϕst

sin ϕi· σ0 (36)

Both isostatic stresses σZ and σiso limit on left and rightthe yield surface and determine the consolidation pre-his-tory of a cohesive powder.

3.3Linear constitutive equations and the three flowparametersUsing these radius and centre stresses, the essential flowparameters are compiled as one set of linear constitu-tive equations, i.e. for instantaneous consolidation, theso-called consolidation locus (CL),

σR = sinϕi · (−σM + σM,st) + σR,st (37)

for incipient yield, the yield locus (YL),

σR = sinϕi · (σM − σM,st) + σR,st (38)

and for steady-state flow, the stationary yield locus (SYL):

σR,st = sinϕst · (σM,st + σ0) (39)

The consolidation locus and the instantaneous yieldlocus describe the limits of incipient plastic deformationduring consolidation and flow. The stationary yield locusis the envelope of all Mohr circles for steady-state flowwith a certain intersection of the abscissa σ0. These yield

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Fig. 5. Powder strength - consolidation stress diagram ofconstitutive consolidation functions of limestone powderfor instantaneous flow (t = 0) and time consolidation t =24 h. The shear test is equivalent to a pre-consolidationby the consolidation stress σ1 in a cylindrical mould (wallshear stress τ = 0 provided) and the measurement of theuniaxial compressive strength (unconfined yield strength)σc or stressing in counter-direction the uniaxial tensilestrength σZ1. The classification of the powder flow behaviours

according to Jenike’s flow function ffc = σ1/σc is also shown[1,7]. The physical model, Eqs. (40) and (41), is applied bystraight line regression with sufficiently good fit ≥ 0.95. Butthe flow parameters differ a bit from the data evaluation ofyield loci. With an averaged angle of internal friction ϕi = 37◦,there are the stationary angle of internal friction ϕst = 43◦ andthe isostatic tensile strength of the unconsolidated powder σ0

= 0.65 kPa

Table 2. Flowability assessment and elastic-plastic contact consolidation coefficient κ(ϕi = 30◦) calculated by Eq. (42)

flow function ffc κ-values ϕst in deg evaluation examples

100–10 0.01–0.107 30.3–33 freeflowing dry fine sand4–10 0.107–0.3 33–37 easy flowing moist fine sand2–4 0.3–0.77 37–46 cohesive dry powder1–2 0.77–∞ 46–90 very cohesive moist powder< 1 ∞ – non flowing moist powder

functions are completely described only with three flow ormaterial parameters plus the characteristic pre-consolida-tion stress σM,st or average pressure influence:

(1) ϕi – incipient particle friction of failing contacts, i.e.Coulomb friction;

(2) ϕst – steady-state particle friction of failing contacts,increasing adhesion by means of flattening of contactexpressed with the contact consolidation coefficientκ, or by friction angles (ϕst − ϕi) as shown in Eq.(26). The softer the particle contacts, the larger arethe difference between these friction angles the morecohesive is the powder;

(3) σ0 – extrapolated isostatic tensile strength of uncon-solidated particle contacts without any contact defor-mation, equals a characteristic cohesion force in anunconsolidated powder;

(4) σM,st – previous consolidation influence of an addi-tional normal force at particle contact, characteristiccentre stress of Mohr circle of pre-consolidation statedirectly related to powder bulk density. This aver-age pressure influences the increasing isostatic ten-sile strength of yield loci via the cohesive steady-stateflow.

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Fig. 6. Characteristic constitutive functions of stiff and com-pliant particle contact behaviours, free flowing and cohesivepowder behaviours, and finally, stiff incompressible and softcompressible powders [9]:a Force – displacement diagram of characteristic contact defor-mation according to Fig. 1,b Adhesion force – normal force diagram of particle contactforces according to Fig. 2,c Shear stress – normal stress diagram of yield and consolida-tion loci (YL, CL) and stationary yield locus (SYL) accordingto Fig. 3 and Fig. 4,

d Powder strength – consolidation stress diagram of consoli-dation functions acc. to Fig. 5,e Radius stress – centre stress diagram of yield and consolida-tion loci (YL, CL) and stationary yield locus (SYL) accordingto Eqs. (28), (37) and (38),f Bulk density – consolidation stress diagram of compressionfunction [8–10,12,20]

This pre-consolidation stress σM,st is also the key param-eter of the so-called compression functions, i.e. thecompression rate dρb/dσM,st = f(σM,st), bulk den-sity ρb = f(σM,st) and the mass-related compressionwork Wm,b = f(σM,st), the specific power consumptionPm,b,pre = f(σM,st) of steady-state flow and the mass-related preshear work Wm,b,pre = f(σM,st) of the cohesivepowder which are explained in a parallel paper [20].

3.4Consolidation functions and flow functionWith the derivation of the linear yield locus, Eq. (32), theuniaxial compressive strength or unconfined yield stressσc is simply found for σc = 2.σR (σ2 = 0 and σR =σM ),Fig. 5:

85

σc =2 · (sin ϕst − sin ϕi)

(1 + sinϕst) · (1 − sin ϕi)· σ1

+2 · sin ϕst · (1 + sinϕi)

(1 + sinϕst) · (1 − sin ϕi)· σ0 (40)

This typical correlation σc(σ1) for a cohesive powder isalso known as “flow function” [2] – do not confuse it withthe shear stress-shear rate τ (γ) flow function in rheology.Equivalent to this linear function of the major principalstress σ1 and using again Eq. (32), the absolute value ofthe uniaxial tensile strength σZ,1 is also found for σZ,1 =2.σR (σ1 = 0 and σR = −σM ):

σZ,1 =2 · (sin ϕst − sin ϕi)

(1 + sinϕst) · (1 + sinϕi)· σ1

+2 · sin ϕst

1 + sin ϕst· σ0 (41)

Again it is worth to notice here, that both σc and σZ,1depend on the pre-consolidation level of the shear zonewhich is expressed here by the applied consolidation stressfor steady-state flow σ1. A considerable time consolidationunder this major principal stress σ1 after one day storageat rest is also shown in Fig. 5. Equivalent linear functionson physical basis are also used to describe these time con-solidation effects [8].

Obviously, the flow behaviour is mainly influenced bythe difference between the friction angles (sinϕst − sin ϕi)in Eqs. (40) and (41) which is as a measure of the adhe-sion force slope κ in the general linear particle contactconstitutive model, Eq. (13). The softer the particle con-tacts, the larger are the differences between these frictionangles (ϕst − ϕi) and consequently, the more cohesive isthe (fine to ultra-fine) powder response. Due to these con-solidation functions, a small slope designates a free flow-ing (coarse) particulate solid with very low adhesion levelbecause of stiff particle contacts but a large slope impliesa very cohesive powder flow behaviour because of soft par-ticle contacts, Fig. 5. Thus, one can find directly the cor-relation between the elastic-plastic contact consolidationcoefficient κ and the powder flowability expressed by theflow function ffc according to Jenike [8]:

κ =1 + (2 · ffc − 1) · sin ϕi

tanϕi · (2 · ffc − 1 + sin ϕi)

×√√√√

1

1 −(

1+(2·ffc−1)·sin ϕi

2·ffc−1+sin ϕi

)2 − 1 (42)

Table 2 shows directly the correlation between flowabilityor powder strength and the increasing of adhesion forcewith pre-consolidation.

Thus, the influence of micro-properties as particlecontact stiffness on the macro-behaviour as powder flowproperties can be qualitatively demonstrated in Fig. 6.Increasing contact compliance determine decreasing slopeof the elastic-plastic yield limit and increasing inclinationof the adhesion limit. As the result, the slope of the normalforce-adhesion force function increases. Next, the differ-ence between the stationary angle ϕst and angle of internalfriction ϕi of the powder becomes larger. Consequently,the slope of the powder consolidation function increasesand the powder is more compressible.

4Summary and ConclusionsBy the model “stiff particles with soft contacts” andthe contact force equilibrium, universal equations werederived and explained which include the elastic-plasticparticle contact behaviours with adhesion, load-unloadhysteresis and a history dependent adhesion force func-tion. With this as the physical basis, a complete setof models for incipient powder consolidation, yield andsteady-state flow has been shown. The yield surfaces dueto theory of plasticity can be expressed by simple linearequations in centre and radius stresses σM ,σR:

�Y L,SY L,CL = 0

=

σR−sin ϕi · (σM − σM,st)−σR,st yield locus

σR,st − sin ϕst · (σM,st+σ0) stationary yield locus

σR−sin ϕi · (−σM + σM,st)−σR,st consolidation locus

(43)

The consolidation and yield loci and the stationary yieldlocus are completely described only with three flow param-eters, i.e. angle of internal friction ϕi, stationary angleof internal friction ϕst, isostatic tensile strength of anunconsolidated powder σ0 plus the influence of character-istic pre-consolidation stress or average pressure at steady-state flow σM,st.

This physical approach can be used for modern dataevaluation of various of powder flow properties concern-ing particle size distribution (nanoparticles to granules),moisture content (dry, moist and wet), material proper-ties (minerals, chemicals, pigments, waste, plastics, foodetc.) which have been tested and evaluated, e.g. [5,6,10,15]. Thus, these models are also applied to evaluate thetest data of a new oscillating shear cell [16,17] and a press-shear-cell in the medium pressure range from 50 to 2000kPa of liquid saturated, compressible filter cakes [18,19]and for dry powders [20]. These contact models are alsoneeded to simulate the shear dynamics of cohesive pow-ders using the discrete element method (DEM) and to cal-ibrate these simulations by shear cell measurements [21].

Finally, these models are used as constitutive func-tions for computer aided data evaluation of shear cell testresults and process apparatus design for reliable flow.

The influence of particle surface properties, e.g. as con-tact stiffness, on the powder flow properties can be directlyinterpreted, Fig. 6, and practically used to design partic-ulate products in process industries.

References1. A. W. Jenike, Storage and flow of solids, Engng. Exp. Stat.

Bull. No. 123, (Univ. Utah, 1964)2. J. Schwedes & D. Schulze, Lagern von Schuttgutern,

In: H. Schubert (Ed.), Handbuch der Mechanischen Ver-fahrenstechnik, (Whiley-VCH, Weinheim, 2003), p. 1137–1253

3. O. Molerus, Theory of yield of cohesive powders, PowderTechnol. 12 (1975), p. 259–275

4. O. Molerus, Effect of interparticle cohesive forces on theflow behaviour of powders, Powder Technol. 20 (1978),p. 161–175

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5. J. Tomas, Untersuchungen zum Fließverhalten von feuch-ten und leichtloslichen Schuttgutern (investigation of theflow behaviour of moist and soluble bulk solids, Ph.D. The-ses), Freiberger Forschungshefte A 677 (1983), p. 1–133

6. J. Tomas, Modellierung des Fließverhaltens von Schutt-gutern auf der Grundlage der Wechselwirkungskrafte zwis-chen den Partikeln und Anwendung bei der Auslegung vonBunkeranlagen (modelling of the flow behaviour of bulksolids on basis of the particle interaction forces and appli-cation at bunker plant design, Sc.D. Theses), (Habilitation,Bergakademie Freiberg, 1991)

7. J. Tomas, Assessment of mechanical properties of cohe-sive particulate solids – part 1: particle contact con-stitutive model, Particulate Sci. & Technol. 19 (2001),p. 95–110

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9. J. Tomas, Zur Mechanik trockener kohasiver Schuttguter,Schuttgut 8 (2002), p. 522–537

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12. J. Tomas, Product Design of Cohesive Powders – Mechan-ical Properties, Compression and Flow Behaviours, Chem.Engng & Technol. 27 (2004), p. 607–618

13. J. N. Israelachvili, Intermolecular and surface forces, (Aca-demic Press, London, 1992)

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15. M. Medhe, B. Pitchumani & J. Tomas, Flow characterisa-tion of fine powders using material characteristic parame-ters, J. Advanced Powder Technol. (2004) in press

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18. B. Reichmann & J. Tomas, Expression behaviour of fineparticle suspensions and the consolidated cake strength,Powder Technol. 121 (2001), p. 182–189

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21. R. Tykhoniuk, S. Luding & J. Tomas, Simulation derScherdynamik kohasiver Pulver, Chem.- Ing.- Technik 76(2004), p. 59–62

22. S. Ecke & H.-J. Butt, friction between individual micro-contacts, J. Colloid Interface Sci. 244 (2001), p. 432–435

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