A Theoretical Frame to Study Stability of Fresh Concrete

8/6/2019 A Theoretical Frame to Study Stability of Fresh Concrete

http://slidepdf.com/reader/full/a-theoretical-frame-to-study-stability-of-fresh-concrete 1/11

Materials and Structures (2006) 39:81–91

DOI 10.1617/s11527-005-9036-1

A theoretical frame to study stability of fresh concrete

N. Roussel

Received: 4 November 2004 / Accepted: 24 February 2005C RILEM 2006

Abstract A theoretical frame able to describe the seg-

regation under gravity of a mono-sized granular mate-

rial in a yield stress suspending fluid is presented. Two

stability criteria are proposed. The first one deals with

the stability of a single sphere in a yield stress fluid

while the second one takes into account the mechani-

cal interactions between the particles that could lower

the risk of instability. The two criteria are then experi-

mentally validated in the case of polystyrene spheres in

a cement paste. Image analysis techniques are used to

measure the final solid fraction in the segregated zone.The comparison of the experimental and theoretical re-

sults is followed by an analysis of the possible use of

such criteria in a mix proportioning method and a short

discussion of the influence of the change of rheological

properties with time on the segregation phenomenon.

Resume Un cadre th´ eorique permettant de d ecrire le

ph´ enomene de s´ egr egation sous gravit e de spheres

mono-disperses dans un fluide a seuil est pr esent e.

Deux crit eres de stabilit e du m´ elange sont propos´ es.

Le premier concerne la stabilit e d’une sphere uniquedans un fluide a seuil alors que le deuxieme prend

en compte les interactions m´ ecaniques entre spheres

qui peuvent r eduire le risque d’instabilit e du m´ elange.

Les deux crit eres sont valid es exp´ erimentalement dans

le cas de billes de polystyr ene m´ elang´ ees a une

pˆ ate de ciment. Une technique d’analyse d’images

N. Roussel

LCPC, Paris, France

est utilis´ ee pour mesurer la fraction volumique solide

finale atteinte dans les zones s egr eg´ ees. Cette val-

idation exp´ erimentale est suivie d’une courte anal-

yse de l’utilisation possible de tels crit eres dans une

m´ ethode de formulation de b´ etons et de l’influence de

la thixotropie et du vieillissement sur la stabilit e d’un

m´ elange.

1. Introduction

“More fluid”. This is one of the big trends of the last

twenty years in the field of modern concretes. As we

do not have any tool to predict the needed workabil-

ity in terms of the shape of the element to cast, the

spacing of steel bars and the chosen casting process,

the only way to ensure that the casting will fill a com-

plicated formwork is to use the most fluid concretes.

This finally leads to the elaboration of Self Compact-

ing Concretes (SCC) [1]. Quickly, however, engineers

realized that very high concrete fluidity may have two

main drawbacks: the pressure on the formwork was al-most hydrostatic [2] and theconcrete stability wasquite

difficult to ensure [3].

Concrete is a multiphasic material. The densities

of the numerous components entering traditional con-

crete mix fitting vary between 1000 kg/m3 (water) and

3200 kg/m3 (cement). Even lighter material may be

used in the case of lightweight concrete. With such

a mixture, gravity quickly becomes the enemy of

homogeneity. In the field of cementitious materials,



82 Materials and Structures (2006) 39:81–91

heterogeneities induced by gravity are divided in two

categories depending on the phase that is migrating:

bleeding and segregation. Both are induced by the den-

sity difference between the components but bleeding

phenomenon is concerned with the water migration

whereas segregation is concerned with the movement

of the coarser particles.The work presented here focuses on segregation in

the case of an homogeneous (i.e. non bleeding) non-

Newtonian fluid containing solid inclusions. This can

be applied to various length scale of observation. For

example segregation in mortars would be predicted

from paste rheology and the volume fraction, shape

and size distribution of the sand particles. This type of

multi-scale physical approach was not adapted to tra-

ditional concretes, the behaviour of which was domi-

nated by grain to grain contacts. Any attempt to predict

a physical phenomena had to take into account all theinteractions between the particles, even the smallest

[4]. The behavior of modern concretes, without even

speaking of SCC, is getting closer to the behavior of a

fluid suspension. The physical concept of inclusions in

a suspending fluid becomes adapted to the description

of each of the constitutive phase of the concrete and was

already used in a paper by Saak and co-workers [5] to

study the stability of SCC (the reading of which, by the

way, initiate most of the experimental work presented

here and should be advised to anybody interested in

segregation).Attempts to correlate the rheology of the fresh con-

crete to its stability can be found in the literature [6,7].

Figures linking either empirical test results or even

quantitative measurement of segregation to the slump

or the slump flow of a given concrete have often been

plotted. Although a correlation may, in certain cases, be

obtained, no information about the underlying physics

will ever be obtained this way. Segregation is a multi-

phasic separation phenomenon (the minimum number

of phases is two: a suspending fluid and some solid

inclusions). As such, the only relevant approach is amultiphasic one. The rheological behavior of concrete

has no role to play; only the rheological behavior of the

suspending fluid(s) does matter. When correlations be-

tween concrete behavior and its stability are obtained,

it is only because concrete behavior is strongly linked

to the behavior of its suspending fluid(s).

Segregation (or stability property) is often associ-

ated to static sedimentation. The particles “fall” to-

wards the bottom of a given sample or of a formwork

as their density is higher than the density of suspend-

ing fluid. However, it must not be forgotten that, if the

inclusions density is lower than the suspending fluid

density, the situation may be the opposite. This is true

for lightweight concrete and this is the case in the ex-

perimental results presented in this paper. The physical

phenomena are of course the same no matter in whichdirection the particles are moving. Moreover, it must

not be forgotten that some other reasons than gravity

may induce segregation. Indeed, obstacles or confined

flows may generate flow conditions in which the sus-

pending fluid is not able to “carry” its particles. In this

paper, we will not deal with this type of dynamic seg-

regation.

The aim of this paper is to present two criteria for

the stability of identical spheres in a yield stress fluid.

They are not of course directly suitable to the com-

plete prediction of segregation in real concretes butthey provide a basic theoretical frame that allowed us

to understand many of our experimental data. The pa-

per is divided into three parts. In a first theoretical part,

the two stability criteria are demonstrated. Using these

results, several possible segregation cases are isolated

and analysed. In a second part, experimental results

are presented. Using image analysis, the final pack-

ing fraction of polystyrene spheres in various cement

pastes is measured. Finally, in a third part, experimen-

tal results and theoretical predictions are analysed and

successfully compared. This confirms the validity of the proposed approach.

2. Theoretical analysis of the problem

2.1. Sphere in a yield stress fluid: movement

criterion and settling velocity

The following piece of work is a generalization to yield

stress fluids of the work done by He and co-workers [8]

in the case of the Bingham model.

We assume in this study that the behavior of thecement paste may be approximated by a yield stress

model of the following general form:

γ = 0 → τ < τ 0 (1a)

γ = 0 → τ = τ 0 + f(γ ) (1b)

where tau 0 is the yield stress, γ the shear rate and f is a

positive continuously increasing function of the shear



Materials and Structures (2006) 39:81–91 83

rate with f(0) = 0. In the particular case of a Bingham

fluid with a plastic viscosity μp, f(γ ) = μpγ

In laminar regime, the drag force Fs on a sphere of

diameter d with a moving velocity Vs in a Newtonian

fluid with viscosity μ writes:

Fs = 3π dμVs (2)

The idea is to substitute in Eq. (2) the apparent vis-

cosity to the Newtonian viscosity to reflect the non-

Newtonian nature of the fluid surrounding the particle.

The apparent viscosity μa of a general yield stress fluid

writes if the shear stress overcomes the yield stress:

μa =τ

γ =

τ 0 + f (γ )

γ (3)

In order to calculate the apparent viscosity, oneneeds to determine the shear rate around the particle.

On a dimensional point of view, Vs /d is equivalent to

a shear rate induced by particle to fluid relative move-

ment. It can then be equated to an average shear rate

by introducing a constant k:

γ = k Vs

d(4)

The calculation of the average shear rate constant

k is beyond the scope of this paper. However, it can

be noted that it depends on the shape of the particlesand on the thickness of the unyielded layer around the

particle shown on Fig. 1. Ansley and Smith [9] obtained

for a Bingham fluid k = 24/7π ∼= 1.09. For practical

purposes, as shown by He and co-workers [8], k may

be treated as constant and equal to 1.

Fig. 1 Yielded and unyielded regions around a sphere moving

in a non-Newtonian fluid.

When studying segregation in concrete, the speed at

which thephysical phenomena take place is low enough

to consider a succession of quasi static states. Thus, at

equilibrium, the gravitational and buoyant force combi-

nation Fm on a sphere of density ρs in a fluid of density

ρf is equal to the drag force Fs:

Fm =πd3

6(ρs − ρf )g = Fs (5)

Introducing Eq. (4) in Eq. (5),

kd

18(ρs − ρf )g = τ 0 + f

kVs

d

(6)

The absolute value of the moving velocity of the

sphere becomes:

Vs =d

k f −1

kd

18|ρs − ρf |g− τ 0

(7)

If there is no segregation, there is no particle to fluid

relative movement. Vs is then equal to zero and Eq. (6)

reduces to:

d

18|ρs − ρf |g =

τ 0

k (8)

Eq. (8) constitutes a stability criterion.

An alternative way to write this criterion is to statethat, for segregation to occur, the sphere diameter has to

be larger than a critical diameter dc for k = 1. Spheres

smaller than dc are stable.

dc =18τ 0

|ρs − ρf |g(9)

This means that, in a given granular repartition, if

we neglect any interactions between particles or any

lattice effects [3], some small particles may be stable

while larger ones may be segregating. Moreover, themoving speed of these larger particles increases with

their diameters.

Whereas there exists a complete analytical solution

for a sphere moving through a Newtonian fluid [10],

in the case of a yield stress fluid, many solutions (but

all approximated) can be found in the literature and

compared with Eq. (9).

Bethmond and co-workers [11] have shown that a

general form of the previous criterion may be written




as:

dc =Kτ 0

|ρs − ρf |g(10)

Let us focus on this aspect. The stress tensor de-

creases from the surface of the sphere to zero as thedistance from the sphere tends to infinity. It results that,

beyond some critical distance, the yielding criterion is

not satisfied, so that the fluid should remain rigid de-

spite sphere motion. This rigid region is bounded by

a surface Sc surrounding the sphere beyond which the

yielding criterion is not fulfilled. At the approach of Sc,

the velocity tends towards zero so that the amplitude

of the shear rate tends towards zero which implies that

the shear stress tends towards τ 0. When the sphere ve-

locity decreases to zero (incipient motion or stoppage),

Sc should approach S but the exact value of Sc staysunknown.

The simplest approach [12] consists in assuming

that, just when flow starts or stops, the force exerted

by the fluid corresponds to a shear stress vector at the

surface of the object (S= Sc) of amplitude equal to the

yield stress and directed tangentially to the sphere sur-

face. The k value is then obtained by integration and is

equal to 3π /2 ∼= 4.71. This value is the lowest that can

be admitted on a theoretical point of view (a lower one

would mean that Sc < S, which would be inconsistent).

However, using plasticity theory, Hill [13] demon-strated that Sc does not tend towards S when the sphere

velocity decreases. Some rigid zones are located at the

top and the bottom of the sphere (Cf. Fig. 1) increasing

the value of Sc and thus the theoretical value of K. The

theoretical expression of Sc obtained in a similar man-

ner by Ansley and Smith [9] leads to a K value equal

to 21π /4 = 16.5.

It has to be noted here that Saak and co-workers [5]

used a relation obtained by [14] with K=1.5. This rela-

tion can not be correct as K is lower than the minimum

consistent value (4.71). This relation was however ex-perimentally validated in [14] but it was followed by

an erratum.

In the present analytical work, the approximation

used to obtain a solution lies in the k coefficient used

in Eq. (4) that averages the shear rate field inside Sc.

From numerical simulations, Beris [15] obtained K

= 20.97 and unyielded zones around the spheres very

similar to the ones obtained analytically by Ansley and

Smith [9]. This K value was used by Petrou and co-

workers in their very interesting experimental works

[16, 17] on aggregate settlement in concrete.

There is an even larger discrepancy on the experi-

mental values of K that can be found in the literature.

From an average experimental value on several vari-

ous materials, Jossic and Magnin [18] obtained K =

17.24. But, in general, the existing data give signifi-cantly dispersed values for K (between 11 and 25 [19]).

This might be due either to the general uncertainty con-

cerning the experimental determination of the incipient

motion or of the yield stress from rheometrical tests.

As it seems difficult to find a definitive answer about

the value of K, in the following, the value K = 18

obtained in Eq. (9) will be used.

2.2. Converging spheres in a Bingham fluid and

prediction of the final compaction state after

segregation

What will happen to the particles that are moving if

we neglect as a first approximation the hydrodynamic

interactions between particles (It will be demonstrated

further in this paper that this assumption is of course

not licit in the case of a concentrated suspension of

spheres in a yield stress fluid)?

Let us consider a granular skeleton consisting of

identical spheres (i.e. same diameter, same density).

We assume in this part that the diameter of the spheres

is higher than the critical diameter. Thus spheres start tomove in the suspending fluid and segregation occur. For

identical spheres in a Bingham fluid (plastic viscosity

μp, yield stress τ 0), the moving speed given by Eq. (7)

is constant and is equal to

Vs =d

μp

d

18(ρs − ρf )g− τ 0

(11)

This means that, in an infinite domain, as all the

spheres move at the same speed, the average distancebetween spheres stay constant and is equal to the initial

average distance that only depends on the initial solid

fraction. However, during casting, the flowing domain

is not infinite and some moving spheres reach a bound-

ary (free surface, bottom of the formwork according

to the sign of (ρs − ρf ) and stop moving. As soon as

some spheres stop moving, the spheres still moving are

converging towards the fixed ones. One could then ex-

pect the packing fraction in this zone to converge to




the maximum random packing fraction of a group of

spheres of identical diameters (φmax = 0.74).

In the case of fine particles, it has been demonstrated

that after an initial sedimentation phase, a slow consol-

idation phenomenon takes place [20]. This is true with

cement grouts, with clay suspensions and in general

with any very small particles suspension, the perme-ability of which may reach low values when the solid

concentration increases. The consolidation phase is in

fact a sedimentation phenomenon slowed down by the

difficulty of the liquid to be drained out of the skele-

ton [21]. In the case of small particles with high sur-

face forces involved, the sedimentation may be slowed

down but the final solid fraction will be more or less

the maximum random packing of the particles.

In the case of coarse particles (sand, gravel), on a

theoretical point of view, the same Newtonian fluid will

always be able to instantaneouslycross theskeleton. Noconsolidation phase should appear and the final solid

fraction after sedimentation should be the maximum

random packing of a group of spheres.

However, as it will be demonstrated here, when the

particles are getting closer to each other, the non New-

tonian nature of the suspending fluid limits the solid

fraction reached after segregation. The following phe-

nomenon has to be kept in mind because it limits the

segregation tendency of a given mixture : two con-

verging spheres stop moving towards each other when

the driving force (gravitational and buoyant force) be-comessmaller than a critical value Fc. Infact, each cou-

ple of spheres generates in the separating gap a small

squeezing flow. The needed driving force to bring the

spheres closer to each other increases as the gap be-

tween the sphere decreases. In the case of two identical

spheresmoving towards eachotherin a Newtonian fluid

at a speed Vs, the force acting on one sphere is [22],

F =3π μd2Vs

4δ(12)

with δ the shortest gap distance between the two

spheres. This means that, for a given driving gravi-

tational and buoyant force Fm = F, as δ is decreas-

ing (spheres getting closer to each other), the speed

at which the spheres are getting closer is decreasing.

Thus, in the case of a Newtonian fluid, it maytake some

time (without even taking into account the permeabil-

ity of the whole skeleton) but the final packing fraction

will converge towards the maximum random packing

fraction. This is not the case for a Bingham fluid.

By substituting the apparent viscosity of a Bingham

fluid in Eq. (12) and assuming that Eq. (4) still holds,

one obtains:

F = 3π d2

Vs(μpγ + τ 0)4δγ

= 3π d2

Vsμp

4δ+ 3π d

3

τ 04δ

(13)

At sphere stoppage, Vs tends towards zero and Eq.

(13) becomes:

Fc =3π τ 0d3

4δ(14)

In the case of non identical spheres, a coefficient

M may be introduced taking into account an eventualradius difference between the two spheres. In this case,

d is the diameter of the smallest particle (read [23] for

the general formulation of the problem).

Fc = Mτ 0d3

δ(15)

M = 3π /4 when the spheres are identical.

As the driving force Fm is given and known for a type

of spheres and a given Bingham fluid, it can be com-

pared to Fc, which depends on the packing state. As

the spheres are getting closer, Fc increases and, when

Fm = Fc, the spheres stop converging. Thus, the pack-

ing fraction may be lower than the maximum random

packing fraction and segregation may stop before this

expected packing state.

At stoppage, the gravitational and buoyant force

combination Fm is equal to Fc. The shortest gap dis-

tance in this particular configuration writes:

δ =9τ 0

2|ρs − ρf |g(16)

In the general case of non-identical spheres, Eq. (16)

becomes:

δ =6Mτ 0

π |ρs − ρf |g(17)

The shortest gap distance between two spheres was

linked to the sphere solid fraction by De Larrard [4] in

terms of the particle diameter d and the maximum solid




fraction o∗

.

δ = d

φ

φ∗

− 13

− 1

(18)

Knowing the yield stress of a given cement paste,

the lowest δ value (i.e. the lowest gap distance betweenthe spheres) that can be reached before spheres can not

get closer can be calculated using Eq. (17). Using Eq.

(18), the solid fraction φ associated to the obtained δ

value may be calculated. This solid fraction is the one

reached after segregation has occurred. It is the final

packing state.

What happens if the calculated final solid fraction is

lower than the initial solid fraction? This means that the

initial gap distance between the spheres is lower than

the one that could be reached after segregation. This

state can not of course be reached naturally but it can bereached during a mixing process, the energy of which

reduces the δ value. Such a mix is very stable as, if one

sphere moves, then it has to converge towards an other

sphere. But, as the force needed is higher than gravity

and buoyant forces, this can not happen and there can

not be any sphere to sphere relative displacement. This

means that the mixed solid fraction fulfils the following

criterion:

φ ≥ φ∗ 9τ 0

2|ρs − ρf |dg

+ 1−3

(19)

Using the theory presented above and applied to

identical spheres in a cement paste, several cases may

be obtained and dealt with:

Case 1: the diameter of the spheres is lower than the

critical diameter given by Eq. (8)

The spheres are stable. They do not have the ability

to move as the stress generated by gravity and buoyant

forces is not sufficient to overcome the cement paste

yield stress.

Case 2: the diameter of the spheres is higher than thecritical diameter given by Eq. (8). The final predicted

solid fraction in the segregated zone using Eq. (17) and

Eq. (18) is lower than the solid fraction initially mixed,

which then fulfils the inequality (19).

The yield stress of the suspending fluid is not suf-

ficient to stabilize the spheres. However, they can not

move as the gravity and buoyant forces combination

is not sufficient to overcome the force needed to bring

closer two spheres.

Case 3: the diameter of the spheres is higher than the

critical diameter given by Eq. (8). The final predicted

solid fraction in the segregated zone using Eq. (17) and

Eq. (18) is higher than the solid fraction initially mixed,

which then does not fulfil the inequality (19).

The yield stress of the suspending fluid is not suf-

ficient to stabilize the spheres. They will move and asegregated zone will appear. The final sphere volume

fraction reached in this zone is the one predicted by

Eq. (17) and Eq. (18) but is not equal to the maximum

packing fraction of a group of spheres (except in the

case of a purely Newtonian suspending fluid).

Finally, it should be kept in mind that, according to

the two above criteria, segregation tendency in a given

mixture depends on the yield stress of the suspend-

ing Bingham fluid but not on its plastic viscosity. This

affirmation is confirmed by Petrou and co-workers ex-

perimental results [16,17]. The plastic viscosity has of course an influence on the separation velocity if the

particle is not stable but does not affect whether or not

the particle is stable. One could object that, if the plas-

tic viscosity is high enough, then, even if the particle

is not stable, the separation velocity will be so low that

the displacement of the particles before setting will be

negligible. To have an idea of the order of magnitude

of the plastic viscosity that would be needed to prevent

a sand (Dmax 4 mm, density 2700 kg/m3) from settling

in a cement paste (density 1900 kg/m3), let us do the

following calculation:We assume that segregation is negligible if the sand

particles displacement before setting is less than 1 mm

and that setting occurs after 4 hours. The settling veloc-

ity of the 4 mm sand particle should then be lower than

7.10−8 m/s. The viscosity needed to reach this veloc-

ity may be calculated using Eq. (11) for a yield stress

equal to zero and is higher than 9000 Pa.s, which is at

least 9000 times higher than a standard cement paste.

Of course, one could try to find an equilibrium between

yield stress and plastic viscosity, to get a quasi-stable

mixture but, most of the time, the chosen yield stresswill have to be very close to the critical yield stress

that fully prevents a sphere from moving under its own

weight.

3. Experimental results

In this part, experimental results are presented. They

consist in the observation of the segregation of different




volume fractions of polystyrene aggregates in cement

pastes displaying various yield stresses. The cement

paste yield stress was measured using a Vane test pro-

cedure. After setting, each sample wascut in two halves

and numerical pictures of the segregation state were

taken. These images were analyzed and the final solid

fraction after segregation was calculated in each sam-ple and compared to the solid fraction that was initially

mixed.

3.1. Materials, preparation and rheological

measurements

3.1.1. Materials

The prepared suspending fluids were cement pastes.

Class A HTS cement was used. The superplasticizer(Chrysofluid OPTIMA 100) amount was varied be-

tween 0.7 and 1% of the cement weight and the water to

cement weight ratio W/C was varied between 0.26 and

0.30. The studied inclusions were polystyrene spheres

(POLYSBETO c, BS Technologies S.A.). Their aver-

age diameter was 3.0 mm. The dispersion around this

average valuewas about 0.5 mm andtheirrandom pack-

ing fraction was measured and equal to 0.70 ± 0.01,

which is higher than 0.63 for identical spheres. This

results might come from the small size deviation of the

polystyrene spheres used here.

3.1.2. Preparation procedure

Thesuperplasticiser was introduced in thewaterand the

obtained solution was mixed at 260 rpm for 60 s (mixer:

Rayneri Turbotest). The cement was then added. The

obtained suspension was mixed at 700 rpm for 15 min

and then at 260 rpm for 15 min. This (long) mixing

aims at giving the superplasticizer enough time to

play its full role in the mixture and to limit any fur-ther irreversible evolution of the rheological behavior

[24, 25]. One small sample was then used to mea-

sure the yield stress (see next section). The rest of

the prepared cement pastes was carefully mixed by

hand with the polystyrene spheres and poured in a

cylindrical mould. It has to be noted that it seemed

not technically possible in our experiments to incorpo-

rate a volume fraction of polystyrene spheres higher

than 0.45.

Fig. 2 The vane test. Two

different geometrics were

used to cover the studied

yield stress range. FL 10

geometry: height = 60 mm:

radius= 20mm

FL 100 geometry: height =

16 mm: radius: 11 mm

3.1.3. Rheological measurements

The yield stress of the cement paste was measured us-

ing a HAAKE ViscoTester R VT550 with a Vane tool

similar to the one shown in Fig. 2. The yield stress of

the various cement pastes was measured using the Vane

procedure described by N’Guyen and Boger [26]. The

rotation speed was equal to 0.4 min−1. It can be noted

that two different Vane test geometries were used to

measure the yield stresses of the tested mixtures. In-

deed, as the yield stresses of the studied mixtures var-

ied from 0.1 Pa to several hundreds Pa, it was necessary

to change the geometry of the rotating tool in order to

measure an acceptable torque with a sufficient preci-

sion in the entire studied range of yield stresses.

3.2. Image analysis method

The freeware ImageJ c[27] was used inthis work. Each

sample was carefully cut in two halves and numeri-cal pictures of the cut section were taken. Because of

the obvious colour difference between cement paste

and polystyrene spheres, no further treatment of the

obtained grey levels pictures was needed (Fig. 3(a)).

When segregation occurred, a zoom on the converging

zone was done (Figure 3(b)) in order to get rid of any

edge effects. A threshold treatment was then applied on

the picture (Fig. 3(c)). A granular analysis was carried

out by the software (Fig. 3(d)) and the volume fraction




of the spheres in the studied zone was then calculated.

In this paper, only the average volume fraction in the

converging zone is studied and a simplified procedure

could be used. If the initial sphere volume fraction is

noted f 0, the height of the mould is h0 and the height

of the converging zone is noted h1 (see Fig. 3(a)), the

final average volume fraction in the converging zoneshould be f 0h0/h1. Both methods give about the same

results. This fact is important as it means that the sphere

volume fraction is not far from being homogeneous in

the converging zone. However, it should not be for-

gotten that this may not be the case for larger scale

testing. The initial (mixed) and final volume fractions

of the tested samples are plotted on Fig. 4 in terms of

the yield stress of the suspending cement paste. The

squares represent the theoretical initial state of he sam-

ple and is in fact the spheres volume fraction initially

mixed in the sample whereas the crosses represent themeasured spheres volume fraction in the converging

zones. If no segregation occurred, the plotted sphere

and square representing a given mix can not be disso-

ciated.

Fig. 3 Image analysis of the sample (a) Numerical picture of the

segregated sample and studied zone. (b) Zoom on the converg-

ing zone. (c) Numerical picture after a threshold treatment. (d)

Numerical picture after analysis. The total surface of the sample

of the surface of the inclusions is calculated. (Continued )

Fig. 3 (Continued )

4. Results analysis and discussion

Depending on the yield stress of the cement paste, some

of the prepared mixtures were stable while some others




Fig. 4 Initial and final packing state in terms of cement paste

yield stress. The predicted sample final solid fraction in the con-

verging zone is plotted in dashed lines.

were not. In the particular case of the mixture with the

cement paste displaying a 4.0 Pa yield stress, it was ob-served that most of the spheres were stable whereas the

largest ones did move. This is a meaningful illustra-

tion of the critical diameter defined in Eq. (9). In this

case, this critical diameter was equal to 3.5 mm, which

is about the size of the largest spheres.

For the studied spheres (average diameter =

3.0 mm), the predicted critical yield stress obtained

with K = 18 in Eq. (10), under which the spheres are

not stable, is 3.8 Pa. This limit is plotted as a straight

line in Fig. 4. This theoretical boundary between sta-

bility and instability is in agreement with the obtainedexperimental results. For all the samples prepared with

cement pastes with a yield stress lower than this value,

the final packing state was higher than the initial one.

Moreover, from these experimental results, the K value

in Eq. 10 can also be recomputed in the two cases be-

low and above the critical yield stress. It is comprised

between 17 and 29. This is in agreement with K = 18

obtained in the theoretical approach presented in this

paper but it is not sufficient to conclude on the exact

value of this parameter. However, K = 18 seems like a

safe and reasonable value.

When segregation occurred, the final volume frac-

tion in the converging zone is of course higher than the

initial one. But, as predicted by the theory proposed

here, it is lower than the maximum packing fraction of

a group of spheres (0.74). Apart from the valueobtainedwith the cement pastes with the lowest yield stress (0.71

for 0.1 Pa), it is also lower than the random packing

fraction measured for theses spheres (0.70). Moreover,

it increases when the yield stress of the cement pastes

(i.e. the suspending fluid) decreases. Finally, the agree-

ment between the final predicted state and the measured

one is good. It can be noted that the final packing frac-

tion in the converging zone does not depend on the

initial volume fraction but just on the yield stress of

the cement paste. It has however to be noted that the

model predicts that the packing fraction tends towardsthe maximum packing fraction (0.74) when the yield

stress tends towards zero. On a practical point of view,

this could be reached but probably only by applying a

strong vibration to the samples.

As already suggested in the paper by Saak and co-

workers [5], such stability criteria may be a basis for a

mix design method. The first criterion (Eq. (9)) allows

the choice of the rheological behaviour of the suspend-

ing fluid (cement paste or mortar) for a given granular

materials (sand or gravel) whereas thesecond (Eq. (19))

allows the choice of the amount of granular materials.Of course, work is still needed to take into account

the dispersion of the granular sizes, the non spherical

shape of most granular materials and the possible over-

lap between particle species distribution. The figures

given in the analysis below are certainly not correct

but the underlying physics should stay the same. As a

first step, the use of the first criterion (Eq. (9)) leads

to the following yield stresses: a cement paste (density

1900 kg/m3) should display a yield stress of the order

of magnitude of 2 Pa to prevent a sand (Dmax 4mm;

density 2700 kg/m3

) from settling. Similarly, a mortar(density 2200 kg/m3) should display a yield stress of

the order of magnitude of 10 Pa to prevent a gravel

(Dmax 20 mm; density 2700 kg/m3) from settling. The

second way to ensure stability of the mix is based on

the second criterion. It consists in introducing enough

granular materials in the mix for the mixed solid frac-

tion to fulfill the inequality (19) knowing that each par-

ticle on its own is unstable. The interactions between

the particles should stabilize the mixture. In the case




of a cement paste displaying a 2 Pa yield stress and

the above sand, applying Eq. (19), one finds that, by

introducing 900 kg/m3, a stable mortar should be ob-

tained. Similarly, in the case of a mortar displaying a

10 Pa yield stress and the above gravel, one finds that,

by introducing 700 kg/m3 of gravel, a stable concrete

should be obtained.In the approach proposed here, the strong evolution

in time of the behaviour of cement pastes [28, 29] has

not been taken into account. To study the consequences

of this evolution on the segregation phenomenon, we

have to separate two aspects: the thixotropic behaviour

and the aging phenomenon. Thixotropy is by definition

a reversible phenomenon. If a cement paste is left at rest,

its flocculation will increase. This flocculation is how-

ever reversible by mixing the sample strongly enough.

This is not the case with aging phenomenon, which af-

fects in an irreversible way the rheological behaviour.In the case of a thixotropic material, a simplified way to

describe what happens is the following: the yield stress

is very low while the material is flowing (low floccula-

tion of the suspension) but, as soon as the material stops

(i.e. end of casting), the yield stress strongly increases

(fast increase of the flocculation state) and then keeps

on increasing with time. This means that the 10 Pa yield

stress needed to prevent the gravel from settling in the

above example is the yield stress just after the end of

casting. The fact that the yield stress keeps on increas-

ing gives a further security against segregation. In thecase of an aging material, an irreversible evolution of

the rheological behaviour occurs. The yield stress may

either increase (influence of the hydration process for

example) or decrease (delayed action of the superplas-

ticizer for example), which can create strong instabili-

ties in the material. The only way to take this risk into

account in the above calculation is to carefully study

the evolution in time of the rheological behaviour and

introduce in the two above criteria the minimum value

reached by the yield stress before setting.

5. Conclusion

A theoretical frame able to describe the segregation un-

der gravity of a mono-sized granular material in a yield

stress suspending fluid has been proposed. Of course,

the case of real concrete is far more complex but the

phenomena that have been isolated and studied in this

work are similar to what happens in real concretes. As

demonstrated in the last part of this paper, this theoret-

ical frame may be applied to study the stability of sand

in a cement paste or gravels in a mortar.

Two stability criteria have been written. The first

one deals with the stability of a single sphere in a yield

stress fluid while the second one takes into account

the mechanical interactions between the particles thatcould lower the risk of instability.

The two criteria have been experimentally validated

in the case of polystyrene spheres in a cement paste.

Image analysis techniques have been used to measure

the final solid fraction in the segregated zone. The com-

parison of the experimental and theoretical results was

followed by an analysis of the possible use of such cri-

teria in a mix fitting method and a discussion of the

influence of the evolution of the rheological behaviour

in time on the segregation phenomenon.

Acknowledgements The author would like to thank P. Coussot

(LMSGC, France) for the always useful and pleasant scientific

discussions.

References

1. Okamura H, Ozawa K (1995) Mix design for self-

compacting concrete. Proc. of JSCE Concrete Library In-

ternational 25:107–120.

2. Billberg P (2003) Form pressure generated by self-

compacting concrete. Proceedings of the 3rd internationalRILEM Symposium on Self-Compacting Concrete, Reyk-

javik, Iceland, (RILEM PRO33), pp. 271–280.

3. Wallevik O (2003) Rheology—a scientific approach to de-

velop self-compactingconcrete. Proceedings of the 3rd inter-

national RILEM Symposium on Self-Compacting Concrete,

Reykjavik, Iceland, (RILEM PRO33), pp. 23–31.

4. De Larrard F (1999) Concrete mixture proportioning. E &

FN Spon, London.

5. Saak AW, Jenning H, Shah S (2001) New methodology for

designing self-compacting concrete. ACI materials Journal

98(6):429–439.

6. Cussigh F, Sonebi M, De Schutter G (2003) Project testing

SCC-segregation test methods. Proceedings of the 3rd inter-

national RILEM Symposium on Self-Compacting Concrete,Reykjavik, Iceland, (RILEM PRO33), pp. 311–322.

7. Daczko JA (2003)A comparison of passing ability test meth-

ods for self consolidating concrete. Proceedings of the 3rd

international RILEM Symposium on Self-Compacting Con-

crete, Reykjavik, Iceland, (RILEM PRO33), pp. 335–344.

8. He YB, Laskowski JS, Klein B (2001) Particle movement

in non-Newtonian slurries: the effect of yield stress on

dense medium separation. Chemical Engineering Science

56:2991–2998.

9. Ansley RW, Smith TN (1967) Motion of spherical particles

in a Bingham plastic. A.I.Ch.E. Journal 13:1193–1196.




10. Batchelor GK (1967) An introduction to fluid mechanics.

Cambridge University Press, Cambridge.

11. Bethmond S, d’Aloıa Schwarzentruber L, Stefani C, Le Roy

R (2003) Defining the stability criterion of a sphere sus-

pended in a cement paste: a way to study the segregation

risk in self compacting concrete (SCC). Proceedings of the

3rd international RILEM Symposium on Self-Compacting

Concrete, Reykjavik, Iceland, (RILEMPRO33), pp. 94–105.

12. Du Plessis MP, Ansley RW (1967) J. Pipeline Division

(ASCE) 2(1).

13. Hill R (1950) The mathematical theory of plasticity. Oxford

University Press, Oxford.

14. Dontula P, Macosko CW (1999) Yield stress of Orbitz. Rhe-

ology Bulletin 68(1):5–6.

15. Beris AN, Tsamopoulos JA, Armstrong RC, Brown RA

(1985)Creepingmotion of a spherethrough a Bingham plas-

tic. Journal of Fluid Mechanics 158:219–244.

16. Petrou MF, Wan B, Galada-Maria F, Kolli VG, Harries KA

(2000) Influence of mortar rheology on aggregate settlement.

ACI Materials Journal 97(4):479–485.

17. Petrou MF, Harries KA, Galada-Maria F, Kolli VG (2000) A

unique experimental method for monitoring aggregate set-tlement in concrete. Cem. Conc. Res. 30:809–816.

18. Jossic L, Magnin A (2001) Traınee et stabilite d’objet en

fluide a seuil. les cahiers de rheologie 18(1). (Only available

in French).

19. Chhabra RP (1993) Bubbles, drops, and particles in non-

Newtonian flows. CRC Press, Boca Raton, Florida, USA.

20. Burger R (2000) Phenomenological foundation and math-

ematical theory of sedimentation–consolidation processes.

Chemical Engineering Journal 80:177–188.

21. Roussel N, Lanos C (2004) Particle fluid separation in shear

flow of dense suspensions: experimental measurements on

squeezed clay pastes. Appl. Rheol. 14:256–265.

22. Yang SM, Leal LG, Kim YS (2002) Hydrodynamic inter-

action between spheres coated with deformable thin liquid

films. Journal of Colloid and Interface Science 250:457–

465.

23. Rodin GJ (1996) Squeeze film between two spheres in a

power-law fluid. J. Non-Newtonian Fluid Mech. 63:141–

152.

24. Banfill PFG, Saunders DC (1981) On the viscosimetric ex-

amination of cement pastes. Cem. Conc. Res. 11:363–370.

25. Caufin B, Papo A (1983) Rheological testing methods for

gypsum plaster pastes. RILEM materials and structures

95:315–321.

26. Nguyen QD, Boger DV (1985) Direct yield stress measure-

ment with the vane method. J. Rheol. 29:335–347.

27. Rasband W, ImageJ, Image processing and analy-

sis in Java. downloadable on the ImageJ homepage(http://rsb.info.nih.gov/ij/).

28. Lei WG, Struble LJ (1997) Microstructure and flow be-

haviour of fresh cement paste. J. Am. Ceram. Society

80(8):2021–2028.

29. Roussel N (2005) Fresh cement paste steady and transient

flow behaviour. Cem. Conc. Res. (Accepted for publication).

Documents

A Theoretical Frame to Study Stability of Fresh Concrete