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Constitutive flow models for characterizing therheology of fresh mortar and concrete1
S.E. Chidiac and F. Mahmoodzadeh
Abstract: Constitutive equations for fresh mortar and fresh concrete provide the characterization of the mixture’s flow andthe quantification of the rheological properties. This paper presents a constitutive material model for mortar and concretethat builds on previous research. It postulates that (a) the shear stress is the sum of three components: static interaction be-tween particles, dynamic interaction between particles, and collision of particles; and (b) the cell is a representative volumeof the mixture. For fresh concrete, the effects of particle collisions are negligible due to high concentration, and the equationreduces to the Bingham model. Experimental data reported in the literature was employed to evaluate the predictive capabil-ities of the constitutive equations. The model results are found to compare very well with the measured experimental dataand the difference is within the measurement errors.
Key words: mortar, concrete, constitutive equations, rheology, workability, cell method.
Résumé : Les équations de comportement pour le mortier frais et le béton frais permettent de caractériser l’écoulement dumélange et de quantifier les propriétés rhéologiques. Cet article présente un modèle de comportement des matériaux, le mor-tier et le béton basé sur le travail des recherches antérieures. Il présume que (a) la contrainte en cisaillement est la sommede trois composantes : l’interaction statique entre les particules, l’interaction dynamique entre les particules et la collisiondes particules, et (b) la cellule constitue un volume représentatif du mélange. Pour le béton frais, les effets des collisions en-tre les particules sont négligeables en raison de la forte concentration, et l’équation se résume au modèle Bingham. Les don-nées expérimentales rapportées dans la littérature ont été utilisées pour évaluer les capacités prédictives des équations decomportement. Les résultats du modèle se comparent très bien aux données expérimentales mesurées et la différence estdans la plage des erreurs de mesure.
Mots‐clés : mortier, béton, équations de comportement, rhéologie, maniabilité, méthode des cellules.
[Traduit par la Rédaction]
1. Introduction
Characterizing the rheological properties and behaviour offresh concrete is an important step toward controlling thequality of concrete (Chidiac et al. 2000, 2003; Ferraris et al.2001). This stems from the fact that placement of fresh con-crete, namely transportation, pumping, casting, and consoli-dation, depends on the rheological properties of freshconcrete. Moreover, the significance of characterizing theflowability of fresh concrete is becoming crucial especiallyfor new categories of concretes such as self-compacting con-crete (SCC) where more stringent requirements are needed(Ferraris and deLarrard 1998; Ferraris et al. 2001; ACI Com-mittee 238 2008).At present, there are no standard test methods for charac-
terizing the rheological properties of fresh concrete, namelyyield stress and plastic viscosity. Researchers have, however,developed different types of concrete rheometers and corre-sponding models for estimating the rheological properties offresh concrete (Ferraris et al. 2001). A review of these testmethods has shown that the proposed methods yield proper-
ties that are statistically comparable but the values for theproperties are different (Ferraris and Brower 2003). A stand-ardized test method, once developed, will be an excellent toolto control the properties of fresh concrete but not to designthe concrete mixture. Researchers have developed models toquantify the rheological properties of concrete on the basisof the composition, specifically the work of Ferraris and de-Larrard (1998), Roshavelov (2005), and Mahmoodzadeh andChidiac (2011). A critical review of these rheological modelshas shown that the models put forward by Ferraris and de-Larrard and Mahmoodzadeh and Chidiac do provide a repre-sentative description for concrete rheological properties,namely yield stress and plastic viscosity (Chidiac and Mah-moodzadeh 2009).With the exception of the work published by Lu et al.
(2008), there are no mathematical models based on funda-mental principles that have been proposed in the literaturethat describe the constitutive behaviour of fresh mortar. Forconcrete, plasticity and visco-plasticity based models havebeen proposed in the literature to model the flow behaviour(Chidiac and Habibbeigi 2005). However, these models as-
Received 26 April 2011. Revision accepted 21 February 2012. Published at www.nrcresearchpress.com/cjce on April 2012.
S.E. Chidiac and F. Mahmoodzadeh. Department of Civil Engineering, McMaster University, 1280 Main Street West, Hamilton, ONL8S 4L7, Canada.
Corresponding author: S.E. Chidiac (e-mail: [email protected]).1This paper is one of a selection of papers in this Special Issue on Innovations and IT.
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40 –482 (2013) doi:10.1139/L2012-025 Published by NRC Research Press
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sume as a priori that the material will obey the Bingham ma-terial model and that the corresponding rheological propertiesare known. Moreover, these models fail to discriminate be-tween mixtures on the basis of the composition.This paper presents a mathematical description of the pro-
posed constitutive equations for characterizing the flow offresh mortar and concrete. A brief review of rheology as itpertains to concrete and mortar is first presented. This is fol-lowed by the description of the constitutive models for mor-tar and fresh concrete that build on the work of Lu et al.(2008) and Mahmoodzadeh and Chidiac (2011). Evaluationof the models is carried out by comparing the model resultswith experimental data reported in the literature.
2. Rheology — a theoretical background
Fresh concrete and mortar are composed of cement par-ticles, aggregates, and water. They can be characterized assuspended solid particles in viscous medium (Ferraris anddeLarrard 1998; Ferraris et al. 2001). The constitutive equa-tions required for simulating the flow of fresh concrete andmortar are difficult to develop because the mixture possessesparticles that have varying gradation, shape, surface, texture,and angularity. Moreover, the model needs to account forparticle interaction. This section provides a brief review ofthe rheological models for both mortar and concrete thathave been proposed in the literature.
2.1. ConcreteRheology, which is defined as “the study of deformation
and flow” (Harrison 1940), is a measure that relates shearforce applied to a material to the rate of deformation orchange of shape experienced by the material. This relationbetween shear stress and strain rate is referred to as a constit-utive equation. This section provides the form of the steadystate non-Newtonian constitutive equations proposed in theliterature to model the flow of fresh concrete.Bingham model, represented by eq. [1], relates the shear
stress (t) and shear strain rate ( _g) with a first order polyno-mial. The corresponding parameters t0 and h0 are referredto, respectively, as yield stress and plastic viscosity.
½1� t ¼ t0 þ h0 _g
Herschel and Bulkley (H-B), which is defined by eq. [2],is a combination of three parameters, yield stress, plastic vis-cosity and power index, n. Accordingly, H-B model is ex-pected to provide better predictions over a wider range ofshear rates, specifically for the case of strain softening andstrain hardening, in comparison to the Bingham model (Fer-raris and deLarrard 1998; Ferraris et al. 2001).
½2� t ¼ t0 þ h0 _gn
Both Bingham model and H-B model have been used pri-marily to estimate the rheological properties of concrete us-ing experimental measurements. There are other models thathave been proposed to provide an estimate of the rheologicalproperties based on the composition of the mixture. Reviewof these models has revealed that only the models proposedby Mahmoodzadeh and Chidiac (2011) and Ferraris and de-Larrard (1998) provide good estimates, and that the former
model yields consistent values for the rheological properties(Chidiac and Mahmoodzadeh 2009).
2.2. MortarModels proposed in the literature characterizing the flow
of fresh mortar are cited in Lu et al. (2008), Epsing (2004),and Hu (2005). However the majority of these models arephenomenological. In this research it is postulated that theflow can be represented by three interactions: static interac-tion between particles, dynamic interaction between particles,and collision between particles and that these three interac-tions are independent. Accordingly, the model can be repre-sented by
½3� t ¼ t0 þ tDI þ tcollisions
where t0 is the shear stress due to static interaction betweenthe particles, tDI is due to dynamic interaction between theparticles, and tcollisions is due to collisions of the particles. To-ward the development of a fundamental constitutive rheologi-cal model for mortar, Lu et al. (2008) assumed that theparticles were rigid, non-cohesive, and well distributed andthat the amount of air was negligible. They have, also, ac-counted for the high concentration of suspended particles,the different size and shape of the particles, and the interac-tion and collision of the particles during flow which are ne-cessary requirements to afford a representative description ofthe flow of fresh mortar (Lu et al. 2008).
3. Constitutive equations for fresh mortar
The proposed model for characterizing the flow of freshmortar postulates that the shear stress arises due to differentcauses and the components are additive (Lu et al. 2008). Ac-cordingly, shear stress is the sum of stresses due to static in-teraction, dynamic interaction, and particle collisions.
3.1. Yield stressYield stress, which is one of Bingham rheological proper-
ties, is the term that accounts for the static interaction be-tween the particles. Yield stress proposed by Mahmoodzadehand Chidiac (2011) is adopted for this study and is given by
½4� t0 ¼ tiyð4Þ3
�4�1� yð4Þ7
�4�1þ yð4Þ10
�� 25yð4Þ3
�1þ yð4Þ4
�þ 42yð4Þ5
24
35
and
½5� yð4Þ ¼ ð4=4max Þ1=3 1� CY
mG
mW
þ rwVair
� �� �
where tI is the “intrinsic” yield stress and is a function of theshape of the particles; y(4) is the ratio of particle size to cellsize; 4 is the volumetric fraction of solid material refers topacking density; 4max is the maximum packing density of thewhole mixture; mG and mW are the mass of gravel and waterof the mixture, respectively; rw is the density of water; Vair isthe volume of air; and CY is a fitting parameter. It should benoted that the proposed model for yield stress differs from
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that of Lu et al. (2008), where the latter assumed that theyield stress for cement paste is equal to that of mortar.
3.2. Particle interactionsLu et al. (2008) postulated that the interaction between two
adjacent particles can be mathematically represented by themodel shown in Fig. 1. Accordingly, the effect of a two par-ticle interaction takes the following form:
½6� tDI ¼ hp 1þ y
1� y
� �_g and y ¼ 34A
1þ 1:654A
where hp is the plastic viscosity of cement paste and 4A is thepacking density of aggregate. However, for mortar there aremore than two particles that are interacting at one time. Toaccount for multi-particle interaction, the cell method devel-oped by Mahmoodzadeh and Chidiac (2011) is employed.The concept is schematically represented in Fig. 2, and itconsists of a rigid particle surrounded by fluid.By accepting the cell as a representative volume, it implies
that the particles, which are located at the centre, do notcome in contact with each other and that the particle interac-tion is limited to the interaction of the cells. Accordingly,Mahmoodzadeh and Chidiac (2011) have developed anequivalent model that accounts for cell interaction, and isgiven by
½7� tDI ¼ hwhiyð4Þ3
�4�1� yð4Þ7
�4�1þ yð4Þ10
�� 25yð4Þ3� �
1þ yð4Þ4�þ 42yð4Þ5
24
35 _g
and
½8� yð4Þ ¼ ð4=4max Þ1=3 1� Cp
mC
mW
� �
where hw is the viscosity of water, hi is the intrinsic viscosityand is a function of the particle shape, mC is the mass of ce-ment in the mixture, and Cp is a fitting parameter.
3.3. Particle collisionThe effect of particle collision is included by modifying
the particle collision model proposed by Lu et al. (2008).These effects are calculated based on the energy dissipationdue to collision of particles moving in parallel horizontalplanes. The velocity of the particle is divided into two parts,
mean flow velocity (yM) and fluctuation velocity (yF), andthat collision occurs due to fluctuation velocity, where
½9� y ¼ yM þ yF
In this formulation, the fluctuation velocity is assumed tobe constant for all the particles and is found by solving thepartial differential equation, PDE (Lu et al. 2008),
½10� NcollisionDP@u
@Y¼ NcollisionFAPSþ NcollisionDE
where
½11� DP ¼ mmð1þ 3Þ 0:083D@u
@Yþ 0:25yF
� �
and
½12� DE ¼ m
4ð1� 32Þ 0:212D
@u
@Y
� �2
þ ð0:6365yFÞ2( )
þ m
40:4244D
@u
@Y
� �2
� 0:4244D@u
@Y� m 0:212D
@u
@Yð1þ 3Þ þ 0:6365yFð1þ 3Þ
� � �2 !
where Ncollision is the number of collisions, DP is the averagemomentum change of the two particle collision in the meanflow direction, @u=@Y is the velocity gradient of mortar, FAP
is the force acting on cement paste by a single aggregate par-ticle (equal to drag force), S is the average distance between
the two particles, DE is the energy loss due to the two parti-cle collision, m is the friction coefficient of particles, m is theaverage mass of aggregate particles, 3 is the coefficient ofelastic restitution, and D is the average diameter of particlesin a horizontal plane.
Fig. 1. The effect of a two particle interaction (Lu et al. 2008).
Fig. 2. Schematic description of the cell method.
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In the Lu et al. (2008) model, the Reynolds number, Re,given by
½13� Re ¼ rPðD=2Þ2 _ghP
is assumed to have a high value and consequently the dragcoefficient (CD) is assumed constant. Accordingly, the dragforce is represented by
½14� FAP ¼ CDrPy2F2
pD2
4
� �
where rP is the density of cement paste. Recognizing that therange of mortar strain rate can vary from 0.1 s–1 for gravitylevelling to 100 s–1 during pumping (Saak et al. 2001), andthat the particle size can vary from 0.5 mm to 2 mm, onecan deduce that Re can have a high value of one or greaterbut it can also have a value as low as 0.001. For the lattercase, CD can no longer be considered constant and the corre-sponding FAP can be calculated from (Happel and Brenner1983)
½15� FAP ¼ 6phpD
2
� �yF
By adopting eq. [15], the drag force captures the specific-ity of the mixture and its placement, namely shear strain rate,particle size, density, and viscosity of cement paste. By ap-plying momentum conservation principles, the shear stressdue to the collision of the particles becomes
½16� tcollisions ¼ NcollisionðkpFAP þDPÞwhere kp is the normal stress coefficient. A new approach isproposed below to quantify Ncollision and D.
3.3.1. Average particle diameterThe average diameter of the particles, D0, shown in Fig. 3,
assuming that they are spherical in shape, can be obtainedfrom
½17� D0 ¼ log�1
Xki¼1
fi log ðDiÞ
Xki¼1
fi
0BBBB@
1CCCCA
where Di is the diameter of the particles in class i, and fi thecorresponding frequency. The average area, AAVE, is then de-termined by assuming certain geometric distribution. For ex-ample, Lu et al. (2008) assumed that the angle a has auniform distribution as shown in Fig. 3. Accordingly, theaverage area can be obtained from
½18� AAVE ¼
Rp=20
pD0
2cos ðaÞ
� �2
da
Rp=20
da
¼ p
8D2
0
and by recalling that the average area is a function of theaverage particle diameter of the horizontal plane, D, that is,
½19� AAVE ¼ p
4D2
Lu et al. (2008) were then able to develop an expressionfor calculating the average particle diameter of the horizontalplane,
½20� D ¼ffiffiffiffiffiffi0:5
pD0 ¼ 0:707D0
However, after analyzing the cutting plane as it moves ver-tically, the authors discovered that it is best to assign the ver-tical parameter, Y, to have a uniform distribution, asillustrated in Fig. 3c. Accordingly, the average area becomes
½21� AAVE ¼
RD0=2
0
p½ðD0=2Þ2 � Y2�dYRD0=2
0
dY
¼ p
6D2
0
Substituting eq. [19] into eq. [21], a revised relationship forthe average diameter is obtained and is given by
½22� D ¼ffiffiffi2
3
rD0 ¼ 0:816D0
3.3.2 Number of collisionsTo calculate the number of collisions, one needs to first
calculate the number of particles in a unit material volume,N. From Fig. 3, the following formulation can be used (Luet al. 2008),
½23� N ¼ 4A
A�AVE
Lu et al. (2008) ignored the distance between the particlesand substituted eq. [19] into eq. [23] to obtain the number ofparticles. To overcome this simplification, the following aver-age area is proposed for the calculation of N,
½24� A�AVE ¼
RD0=2
0
p ðD0=2Þ2 � Y2 �
dY
Rb0
dY
¼ pD30
12b¼ pD2
0
6
� �y
where b is the cell size as shown in Fig. 2. Subsequently, onecan compute the number of particles,
½25� N ¼ 4A
ðpD20=6Þy
The number of collisions is obtained using the followingexpression (Lu et al. 2008):
½26� Ncollision ¼ NSp
Sp þ D
� �f
where f is the frequency of collision and Sp is the distancebetween particles in the assumed horizontal plane.
3.3.3. Frequency of collisionsDifferent formulations were proposed in the literature for
calculating the frequency of collisions. Marrucci and Denn
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(1985) assumed that the frequency is equal to shear strainrate but Probstein et al. (1994) argued that it is much higher.In this study, a similar argument is presented to find the fre-quency of collisions.By using the relative velocity of the particles with respect
to each other, the only velocity that leads to collisions is thefluctuation velocity (yf). For a mixture with high particleconcentration, a representative particle is always surroundedin all directions by other particles. The particle moves withthe velocity yf towards one of the adjacent particles. Recog-nizing that yf has an arbitrary direction, the average of fluc-tuation velocity (yf) of the adjacent particle can beconsidered equal to zero. Accordingly, the frequency of colli-sions can be estimated from
½27� f ¼ yf
S
Substituting eqs. [25] and [27] into eq. [26] yields
½28� Ncollision ¼ 4A
pD20=6
� �1
y
� �Sp
Sp þ D
� �yf
S
� �
By examining Fig. 4, which provides an illustration of twoadjacent particles, the following formulation can be extracted,
½29� Sp þ D ¼ ðSþ D0Þcos ðqÞ ¼ D0
ycos ðqÞ
Recalling the volume of a spherical particle, Vparticle, where
½30� VParticle ¼ p
6D3
0
By substituting eq. [29] and [30] into eq. [28], and definingthe number of particles per unit volume, N, as
½31� N ¼ 4A
VParticle
And considering that the average of cos(q) is,
½32�
Rp=20
cos ðqÞdqRp=20
dq
¼ 2
p
the number of particle collisions can be calculated from,
½33� Ncollision ¼�cos ðqÞ=y
�� ðD=D0Þ�
cos ðqÞ=y�� cos ðqÞ
24
35Nyf ffi Nyf
In summary, the proposed constitutive equations for mortarconsist of eqs. [3], [4], [7], and [16] along with the accompa-nying equations.
4. Constitutive equation for fresh concrete
The proposed flow model for mortar was extended tomodel the flow of fresh concrete. For concrete, the concen-tration of suspended particles in the mixture is high in com-
Fig. 3. (a) Average particle diameter in a horizontal plane; (b) schematic horizontal plane before simplification; and (c) schematic horizontalplane after simplification (Lu et al. 2008).
Fig. 4. Relationship between Sp + D and S + D0.
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parison to mortar. As a consequence, the effect of particlecollision becomes negligible. By setting the shear stress termdue to particle collision equal to zero in eq. [3], it becomesthe Bingham model, given in eq. [1]. Using experimentaldata reported in the literature, Mahmoodzadeh and Chidiac(2011) and Chidiac and Mahmoodzadeh (2009) have shownthat the proposed models are adequate and consistent in pre-dicting the rheological properties of fresh concrete namelyyield stress, given in eq. [4], and plastic viscosity, given ineq. [7].
5. Evaluation of the mortar constitutiveequationsEvaluation of the model consists of two parts: the first part
is to validate the rheological properties including viscosityand yield stress based on experimental measurements carriedout by Ferraris and deLarrard (1998), and the second part isto evaluate the ability of the constitutive equation to charac-terize the mortar flow using experimental work reported byHu (2005).
5.1. Rheological propertiesFerraris and deLarrard (1998) carried out an extensive test-
ing program to measure the rheological properties of mortarand concrete. The data corresponding to normal slump con-crete were used to demonstrate the adequacy of the proposedmodel to predict the rheological properties of concrete (Chi-diac and Mahmoodzadeh 2009). In this study, the data corre-sponding to normal mortar are used to evaluate rheologicalproperties of mortar. Details of the experimental program arereported in Ferraris and deLarrard (1998) and the experimen-tal results are given in Table 1. The model predictions of thefour mixtures are also given in Table 1 and shown in Figs. 5and 6. The results demonstrate that the proposed model pre-dictions are very good.
5.2. Constitutive flow of mortarAn experimental study was conducted by Hu (2005) to in-
vestigate the flow characteristics of fresh mortar. The mortarmixture was composed of type I Portland cement, river-sandfine aggregate, and water. The mixture proportions were 0.4water to cement ratio, and 2.0 sand to cement ratio. The par-ticle size ranged between 0.6 mm and 1.18 mm. A Brook-field rheometer was used to measure shear stress versusshear strain rate. The loading and unloading sequences,shown in Fig. 7, indicate an initial pre-shear cycle prior tothe commencement of the test. The test includes a loadingcycle, referred to as up-curve, and an unloading cycle, re-ferred to as down-curve. The rate is constant for both the upand down curve. It should be noted that the loading and un-
loading sequences shown in Fig. 7 do not permit the determi-nation of shear stress growth which is the maximum stressfrom rest and is equal to the static yield stress. The equiva-lent term in the proposed model is t0, the shear stress due tostatic interaction between the particles. Experimental resultsfrom Hu (2005) are shown in Fig. 8. An understanding ofthese results is merited prior to the application of the pro-posed constitutive equations. The results show a jump in theshear stress at the onset of the up-curve before quickly de-creasing to shear stress values in the ramp of 500 Pa. Subse-quently, the experimental results indicate a small increase inshear stress as the strain rate went from 20 s–1 to 100 s–1.Although the pre-shear cycle was intended to break-downthe structure, the recorded response indicates a significant re-sistance to aggregate movement as they move through the ce-ment paste at low speed. The scientific interpretation of theseresults suggests that the Reynolds number is low at the onsetof the test, and thus the drag force is proportional to the ve-
Table 1. Mortar mixtures proportion and properties (Ferraris and deLarrard 1998) and corresponding model predictions.
MixtureSand(kg/m3)
Finesand(kg/m3)
Cement(kg/m3)
Water(kg/m3)
Packingdensity
Maximumpackingdensity
Yield stress (Pa) Plastic viscosity (Pa·s)
Exp. ModelError(%) Exp. Model
Error(%)
1 1082 336 475 298 0.699 0.799 1061 829 22 38 35 72 984 305 635 296 0.701 0.785 1263 1378 9 34 33 43 973 302 628 304 0.693 0.785 1001 1023 2 31 29 64 961 298 621 312 0.685 0.785 764 778 2 19 26 38
Fig. 5. Predicted versus measured rheological properties accordingto Mahmoodzadeh and Chidiac (2011): (a) yield stress and (b) plas-tic viscosity.
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locity of the aggregates and plastic viscosity of the cementpaste. Accordingly, the drag coefficient is not constant, andthe drag force to be determined according to eq. [15].As the shear strain continues to increase, the aggregates
move at high speed through the cement paste. Accordingly,the drag force becomes proportional to the square of the ve-locity, and the drag coefficient can be considered constant.Equation [14] is then used to calculate the correspondingdrag force.For the unloading portion of the test, the same argument
can be presented for the high shear strain rate portion of thetest. As the test continues, the same trend is observed evenwhen the strain rate drops below 20 s–1. The results are at-tributed to the breakdown of the cement paste structure. Ac-cordingly, drag force is only affected by Re and not theviscosity of the cement paste. Therefore, eq. [14] is applica-ble for the full unloading cycle.The model results are shown in Fig. 8. Comparing with
the experimental data, it can be concluded that the interpreta-tion of the experimental results is sound and that the pro-posed model is capable of characterizing the behaviour ofmortar mixture during the loading and unloading cycle. The
model results also show a linear trend once the structure ofthe paste is broken for both the up and down-curve. More-over, if one accepts Hu’s (2005) argument, then the down-curve can be used to characterize the steady state flow ofmortar that is a linear relation between shear stress and strainrate. However, to capture the flow of mortar as a continuumusing a finite element framework, the model needs to accountfor both the loading and unloading especially given the areabetween the up and down-curve provides a measure of thixo-tropy — reversible, time-dependent reduction in viscositysubject to shear. This flow behaviour is captured in the pro-posed constitutive equation.
6. Conclusions
This paper presents a constitutive model for fresh mortarand concrete consisting of three components: static interac-tion, dynamic interaction, and collision. The first and secondcomponents are adopted from the work of Mahmoodzadehand Chidiac (2011) and, respectively, represent the yieldstress and plastic viscosity. The last component is a modifiedversion of the Lu et al. (2008) model by introducing the con-cept of cell method. The evaluation of the model has re-vealed that
1. The proposed model is capable of predicting the rheologicalproperties of mortar and fresh concrete.
2. Experimental results support the postulation that the shearstress is the sum of static interaction, dynamic interaction,and particle collisions.
3. For different ranges of shear strain rate, the governing con-stitutive equations of mortar are different and the proposedmodel can depict that behaviour.
4. For normal slump concrete, it is proven fundamentally thatthe flow of fresh concrete obeys the Bingham model.
5. Revised relationship is proposed for determining averageparticle diameter.
6. Shear stress due to collisions of particles is formulated toaccount for low and high Reynolds number.
7. Proposed model has the ability to provide a measure ofthixotropy.
AcknowledgmentsThis research was partially funded through grants from the
Fig. 8. Predicted versus Hu (2005) measured shear stress and shearstrain rate.
Fig. 6. Yield stress and plastic viscosity versus the ratio of the par-ticle radius to the cell radius.
Fig. 7. Loading pattern for measuring shear stress versus shear strainrate (Hu 2005).
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Natural Sciences and Engineering Research Council of Can-ada (NSERC) and McMaster University's Centre for EffectiveDesign of Structures.
ReferencesACI Committee 238. 2008. 238.1R-08 Report on measurements of
workability and rheology of fresh concrete, American ConcreteInstitute, Committee 238. Farmington Hills, Michigan USA. 70 pages.
Chidiac, S.E., and Habibbeigi, F. 2005. Modeling the rheologicalbehaviour of fresh concrete: an elasto-viscoplastic finite elementapproach. Computers and Concrete, 2(2): 97–110.
Chidiac, S.E., and Mahmoodzadeh, F. 2009. Plastic viscosity of freshconcrete – a critical review of predictions methods. Cement andConcrete Composites, 31(8): 535–544. doi:10.1016/j.cemconcomp.2009.02.004.
Chidiac, S.E., Maadani, O., Razaqpur, A.G., and Mailvaganam, N.P.2000. Controlling the quality of fresh concrete – a new approach.Magazine of Concrete Research, 52(5): 353–363. doi:10.1680/macr.2000.52.5.353.
Chidiac, S.E., Maadani, O., Razaqpur, A.G., and Mailvaganam, N.P.2003. Correlation of rheological properties to durability andstrength of hardened concrete. Journal of Materials in CivilEngineering, 15(4): 391–399. doi:10.1061/(ASCE)0899-1561(2003)15:4(391).
Epsing, O. 2004. Rheology of cementitious materials, effect ofgeometrical properties of filler and fine aggregate. Thesis for theDegree of Licentiate of Engineering. Department of BuildingTechnology, Building Materials, Chalmers University Of Technol-ogy, Goteborg, Sweden.
Ferraris, C.F., and Brower, L.E. 2003. Comparison of concreterheometers. Concrete International, 25(8): 41–47.
Ferraris, C.F., and deLarrard, F. 1998. Testing and modeling of fresh
concrete rheology. Building and fire research laboratory NationalInstitute of Standards and Technology Gaithersburg, MD 20899.
Ferraris, C.F., deLarrard, F., and Martys, N. 2001. Fresh concreterheology – recent developments. Materials Science of ConcreteVI. Edited by S. Mindess and J. Skalny. The American CeramicSociety, 735 Ceramic Place, Westerville, OH 43081. pp. 215–241.
Happel, J., and Brenner, H. 1983. Low Reynolds number hydro-dynamics: with special applications to particulate media, Springer.
Harrison, V.G.W. 1940. The science of rheology. Nature, 146(3705):580–582. doi:10.1038/146580a0.
Hu, J. 2005. A study of effects of aggregate on concrete rheology,Ph.D. thesis, Iowa State University.
Lu, G., Wanga, K., and Rudolphi, T.J. 2008. Modeling rheologicalbehaviour of highly flowable mortar using concepts of particle andfluid mechanics. Cement and Concrete Composites, 30(1): 1–12.doi:10.1016/j.cemconcomp.2007.06.002.
Mahmoodzadeh, F., and Chidiac, S.E. 2011. Rheological models forpredicting plastic viscosity and yield stress of fresh concrete.Cement and Concrete Research, [In review]
Marrucci, G., and Denn, M.M. 1985. On the viscosity of concentratedsuspension of solid spheres. Rheologica Acta, 24(3): 317–320.doi:10.1007/BF01332611.
Probstein, R.F., Sengun, M.Z., and Tseng, T.C. 1994. Bimodal modelof concentrated suspension viscosity for distributed particle sizes.Journal of Rheology, 38(4): 811–829. doi:10.1122/1.550594.
Roshavelov, T. 2005. Prediction of fresh concrete flow behaviourbased on analytical model for mixture proportioning. Cement andConcrete Research, 35(5): 831–835. doi:10.1016/j.cemconres.2004.09.019.
Saak, A.W., Jennings, H.M., and Shah, S.P. 2001. New methodologyfor designing self-compacting concrete. ACI Materials Journal,98(6): 429–439.
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