Particle size eect on EPS lightweight concrete

porosities ranging from 10% to 50% have been investigated. Compressive tests results have conrmed the presence of a

1. Introduction

It had been observed that the compressivestrength of expanded polystyrene (EPS) lightweightconcrete increases with a decrease in EPS bead size,

erved.

* Corresponding author. Tel.: +33 164 153749; fax: +33 164153741.

E-mail addresses: miled@lcpc.fr (K. Miled), sab@lami.enpc.fr(K. Sab), Leroy@lami.enpc.fr (R. Le Roy).

Mechanics of Materials 39 (0167-6636/$ - see front matter 2006 Elsevier Ltd. All rights resparticle size eect on the EPS concrete compressive strength. Further, it is observed that this size eect is very pronouncedfor low porosity concretes and becomes negligible for very high porosity concretes. Based on EPS concrete failure modesanalysis, a phenomenological model has been proposed with a view to explaining the EPS concrete particle size eect andpredicting its normalized compressive strength according to the concrete (macro) porosity (p) and to the ratio /lc

, where

(lc) is the width of the EPS concrete matrix fracture process zone (FPZ). The model predictions have been then comparedwith experimental results, showing a good agreement. 2006 Elsevier Ltd. All rights reserved.

Keywords: Lightweight concrete; Particle size eect; Volume size eect; Characteristic material length; Macro porositycompressive strength: Experimental investigation and modelling

K. Miled a,b, K. Sab a,*, R. Le Roy a

a Ecole Nationale des Ponts et Chaussees, LAMI, Institut Navier, 6 et 8, Avenue Blaise Pascal, Cite Descartes,

Champs-sur-Marne, 77455 Marne-La-Vallee, Franceb Laboratoire Central des Ponts et Chaussees, Division BCC, 58, BD Lefebvre, 75732 Paris cedex 15, France

Received 5 July 2005; received in revised form 20 April 2006

Abstract

It had been observed [Parant, E., Le Roy, R., 1999. Optimisation des betons de densite inferieure a` 1. Tech. rep., Lab-oratoire Central des Ponts et Chaussees, Paris, France; Le Roy, R., Parant, E., Boulay, C., 2005. Taking into account theinclusions size in lightweight concrete compressive strength prediction. Cem. Concr. Res. 35 (4), 770775; Ganesh Babu,K., Saradhi Babu, D., 2002. Behaviour of lightweight expanded polystyrene concrete containing silica fume. Cem. Concr.Res. 2249, 18; Laukaitis, A., Zurauskas, R., Keriene, J., 2005. The eect of foam polystyrene granules on cement com-posite properties. Cem. Concr. Compos. 27 (1), 4147] that the compressive strength of expanded polystyrene (EPS) light-weight concrete increases signicantly with a decrease in EPS bead size (/), for the same concrete (macro) porosity (p)(EPS volume fraction). To conrm that this scaling phenomenon is an intrinsic particle size eect which is related tothe EPS bead size (/) and not aected by a volume size eect related to the specimen size (D), compressive tests have beencarried out on homothetic EPS concrete specimens containing homothetic EPS beads. Moreover, ve concrete (macro)doi:10.1016/j.mechmat.2006.05.0082007) 222240

www.elsevier.com/locate/mechmat

(/). Finally, the major conclusions are summarisedin Section 5.

2. Experimental investigation

This experimental investigation is the continua-tion of the one made by Parant and Le Roy in1999 (Parant and Le Roy, 1999; Le Roy et al.,2005). That is why their ultra high strength mortar

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 223for the same concrete density. This scaling phenom-enon was observed rst in 1999 by Parant and LeRoy on the basis of an experimental investiga-tion aiming to formulate and optimize an EPSconcrete of a density ranging from 600 kg/m3 to1400 kg/m3 and possessing structural strength qual-ity. Three sizes of polystyrene beads were used inthis investigation (1, 3 and 6 mm) with a very highstrength mortar matrix. Results of compressivetests, performed on EPS concrete cylindrical speci-mens of 110 mm diameter and 220 mm height andof 1400 kg/m3 density and containing 1 mm EPSbeads, had showed an increase of 50% in the EPSconcrete compressive strength in comparison withstrengths given by specimens of the same dimen-sions and the same density but containing 6 mmEPS beads. However, for lower EPS concrete densi-ties (or higher EPS volume fractions), Parant and LeRoy observed that EPS bead size inuence on thelightweight concrete compressive strength becomesnegligible.

This phenomenon was then conrmed by GaneshBabu and Saradhi Babu (2002) with structural EPSconcretes of higher densities (ranging from 1440 kg/m3 to 1850 kg/m3) and with two EPS bead sizes:6.3 mm and 4.75 mm. Further, by Laukaitis et al.(2005) who observed the same scaling phenomenonon ultra lightweight EPS concrete (of density rang-ing from 150 kg/m3 to 300 kg/m3) made of a foammatrix (mortar with embedded polystyrene) andEPS beads of diameters ranging from 2.5 mm to10 mm. In fact, for the same concrete density, com-pressive tests results showed an increase of 40% inthe compressive strength given by 2.55 mm EPSbeads concrete in comparison with the one givenby 510 mm EPS beads concrete.

The main purpose of this paper is to conrm rstthat this scaling phenomenon is an intrinsic particlesize eect which is related to the EPS bead size (/)and not aected by a volume size eect related tothe specimen size (D). Thereafter, to identify thephysical origin of this size eect and to determinethe law governing it. The paper is organised as fol-lows. In Section 2, the results of an experimentalinvestigation performed on 15 EPS concrete mixesare presented. Section 3 is devoted to the explana-tion of the physical origin of this size eect on thebasis of the analysis of EPS concrete failure modes.In Section 4, a phenomenological model is proposedto predict the EPS concrete normalized compressivestrength according to the concrete (macro) porosity

(p) (EPS volume fraction) and to the EPS bead sizematrix was used to made the EPS concretes. Thismatrix is made with CEM I 52.5 cement, class Asilica fume (according to NF P 18 502), a roundedquartz sand of a maximum diameter estimated a pri-ori at 300 lm and a polycarboxylate-based super-plasticizer. It has a water to cement ratio of 0.26,a silica fume to cement ratio of 0.3 and a superplast-icizer dosage of 0.9%. Mix proportions of the EPSconcrete matrix are summarised in Table 1.

Moreover, to highlight the EPS bead size inu-ence on EPS concrete compressive strength, threesizes of EPS beads were used: /1 = 1 mm,/2 = 2.5 mm and /3 = 6.3 mm (Fig. 1). These sizespresent a ratio of homothety of 2.5.

Furthermore, since Parant and Le Roy (1999)had already explored a domain of EPS concretedensities ranging from 600 kg/m3 to 1400 kg/m3

and had observed that for very low density con-cretes the size eect becomes negligible, it was cho-sen to investigate the ve following EPS concretedensities: 1200, 1400, 1600, 1800 and 2000 kg/m3.In fact, with these densities it was hoped to observean important particle size eect on the EPS concretecompressive strength. To obtain these densities, thepolystyrene amount was varied in the EPS concretemixes. The EPS volume fraction (p) considered hereas the concrete (macro) porosity was determined bythe following formula:

p qmatrix qconcreteqmatrix qEPS

;

where qmatrix, qconcrete and qEPS are respectively thedensities of matrix, EPS concrete and EPS beads.

Table 1Mix composition of the EPS concrete mortar matrix

Mix component Proportion (kg/m3)

Cement CEM I 52.5 HTS du Teil 961.9Rounded quartz ne sand Semanez 786.00Silica fume 288.60Superplasticizer Optima 175 8.30Water 244.30

Matrix density (kg/m3) 2289.1

0ining

224 K. Miled et al. / Mechanics of Materials 39 (2007) 222240Table 2Mix designs (kg/m3) of 1 mm EPS beads concretes

Mix M1 10 M1 2Cement 839.35 754.56Sand 685.86 616.58Silica fume 251.83 226.39Superplasticizer 7.24 6.51Water 213.17 191.64EPS beads (/1 = 1 mm) 2.55 4.31p (%) 12.8 21.7qconcrete (kg/m

3) 2000 1800

Fig. 1. (110 220 mm) EPS concrete cylinders contaThe latter decreases when the EPS bead size in-creases. In fact, q1 = 33 kg/m

3 for /1 = 1 mm,q2 = 19 kg/m

3 for /2 = 2.5 mm and q3 = 17 kg/m3

for /3 = 6.3 mm (Le Roy et al., 2005). The 15 mixdesigns are summarised in Tables 24.1 These mixeswere prepared in a standard concrete mixer; water,superplasticizer, silica fume, sand and cement aresuccessively introduced in the mixer. After 5 minof mixing, when the mortar became homogeneous,EPS beads were introduced and the mixing wasmaintained two more minutes at low speed. Fur-ther, the EPS concrete was poured in the mouldswithout vibration to avoid segregation. Specimenswere demoulded after 48 h and protected from dess-ication with an aluminium paper, and then stored inlaboratory conditions (22 3 C) for 28 days. Foreach EPS concrete (macro) porosity, each EPS beadsize and each specimen size, four samples were cast.Moreover, for each concrete porosity, it wasdecided to report each value and not the averagevalue because porosity of the samples varies slightly

1 M1,M2 andM3 represent mixes containing respectively 1 mm,2.5 mm and 6.3 mm EPS beads. The following number indicatesthe (macro) porosity (p) ranging from 10% to 50%.M1 30 M1 40 M1 50669.78 585.00 500.22547.30 478.02 408.74200.96 175.52 150.085.78 5.05 4.32170.11 148.58 127.046.07 7.84 9.6030.5 39.4 48.31600 1400 1200

respectively 1 mm, 2.5 mm and 6.3 mm EPS beads.(notably for low porosity EPS concretes). Finally, tomake the comparison between the compressivestrengths of the three EPS bead sizes concretes eas-ier for the same concrete porosity, it was decided tocompare the normalized compressive strengthsgiven by the ratio rp;/rmatrix

, where r(p,/) is the EPS

concrete compressive strength for a given (macro)porosity (p) and EPS bead size (/) and rmatrix isthe matrix average compressive strength. In fact,the mixes were prepared separately for the threeEPS bead sizes concretes and it was found that theirmatrixes average compressive strengths were slightlydierent; r1matrix 161 MPa for /1 = 1 mm, r2matrix 149 MPa for /2 = 2.5 mm and r3matrix 154:5 MPafor /3 = 6.3 mm.

2.1. Results of modulus of elasticity tests

Static modulus of elasticity tests were carried outon the (110 220 mm) EPS concrete cylinders foreach EPS bead size. The results of these testsshowed that there is no particle size eect on theelastic modulus of EPS concrete. In fact, thismechanical material property does not depend on

20

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 225Table 3Mix designs (kg/m3) of 2.5 mm EPS beads concretes

Mix M2 10 M2 the inclusion size (/), but depends only on the con-crete (macro) porosity (p) and decreases withincreasing (p) (which is expected). Moreover, its var-iation with (p) seems to be linear (Fig. 2).

Fig. 2. The variation of EPS concrete modulus of elasticity wi

Cement 839.40 754.66Sand 685.90 616.65Silica fume 251.85 226.42Superplasticizer 7.24 6.51Eau 213.19 191.66EPS beads (/2 = 2.5 mm) 2.42 4.09p (%) 12.7 21.5qconcrete (kg/m

3) 2000 1800

Table 4Mix designs (kg/m3) of 6.3 mm EPS beads concretes

Mix M3 10 M3 20Cement 839.40 754.66Sand 685.99 616.80Silica fume 251.88 226.48Superplasticizer 7.24 6.51Water 213.22 191.71EPS beads (/3 = 6.3 mm) 2.16 3.66p (%) 12.7 21.5qconcrete (kg/m

3) 2000 1800M2 30 M2 40 M2 502.2. RVE for the EPS concrete compressive strength

To ensure that the representative volume element(RVE) for the EPS concrete compressive strength is

th concrete (macro) porosity (p), for each EPS bead size.

669.91 585.17 500.42547.41 478.16 408.91200.99 175.57 150.145.78 5.05 4.32170.14 148.62 127.095.77 7.44 9.1230.4 39.2 48.01600 1400 1200

M3 30 M3 40 M3 50669.91 585.17 500.42547.62 478.43 409.24201.07 175.67 150.265.78 5.05 4.32170.21 148.70 127.205.16 6.65 8.1530.3 39.1 47.91600 1400 1200

already reached, it is necessary to verify if compres-sive tests results are aected or not by a volumesize eect, which is related to the specimen size

(D) and more precisely to the ratio D/

. In fact, it

had been observed that the nominal compressivestrength (rN) (as well as the nominal tensilestrength) of quasi-brittle materials like concrete

decreases when the ratio D/

increases (Kadlecek

and Spetla, 1967; Wu, 1991), and it seems that

(rN) follows a power law rN / D/ a

; a > 0

.

However, it had been proved numerically (Saband Laalai, 1993; Laalai and Sab, 1994) that a crit-

ical specimen size (Dc) (i.e. a critical ratio Dc/ ) existsbeyond which the volume size eect vanishes andthe RVE will be reached (Fig. 3). That is why to sep-arate the particle size eect problem of the specimensize (D), compressive tests were carried out onhomothetic EPS concrete cylinders containinghomothetic EPS beads. Since the three EPS beadsizes present a ratio of homothety of 2.5, only twocylindrical specimen sizes having also a ratio ofhomothety of 2.5 were considered: (44 88 mm)(D1 = 44 mm) and (110 220 mm) (D2 = 110 mm)(Fig. 4). Thus, it follows that: D1/1

D2/2

44and D1/2

D2/3

17:6.Compressive tests on the homothetic EPS con-

crete specimens containing 1 mm EPS beads havegiven very similar results using cylinders of size

(44 88 mm) D/ 44

and of size (110 220 mm)D/ 110

(Fig. 5). Thus, for 1 mm EPS beads con-crete compressive strength, the RVE is alreadyreached with a specimen size of (44 88 mm) corre-sponding to a ratio D/

equal to 44. Therefore, it is

concluded that Dc/

6 44.

Fig. 3. Volume size eect.

226 K. Miled et al. / Mechanics of Materials 39 (2007) 222240Fig. 4. (110 220 mm) and (44 88 mm)However, with 2.5 mm EPS beads concretes, itwas observed a slight decrease (notably for highporosity concretes) in compressive strengthsEPS concrete homothetic cylinders.

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 227obtained with specimens of size (110 220 mm)D/ 44

, in comparison with those determined

from specimens of size (44 88 mm) D/ 17:6

(Fig. 6). Therefore, it is concluded that specimens

Fig. 5. Normalized compressive strengths obtained with 1 mm EPS

Fig. 6. Normalized compressive strengths obtained with 2.5 mm EPof size (110 220 mm) are representative for2.5 mm EPS beads concrete compressive strength

since D/

44, whereas specimens of size

(44 88 mm) are not representative for 2.5 mm

beads concrete homothetic cylinders D/ 110 and D/ 44

.

S beads concrete homothetic cylinders D/ 44 and D/ 17:6

.

EPS beads concrete, and thus compressive strengthsdetermined from these specimens are aected by aslight volume size eect. As a consequence, it ismost likely that the RVE is not yet reached with(110 220 mm) specimens containing 6.3 mm EPSbeads since D/

17:6, and thus compressive tests

results slightly overestimate the 6.3 mm EPS beadsconcrete compressive strengths (notably those ofhigh porosity concretes).

2.3. Particle size eect

Results of compressive tests made on (110 220 mm) concrete cylinders containing 2.5 mm

EPS beads D/ 44

and on (44 88 mm) concretecylinders containing 1 mm EPS beads D/ 44

showed an increase in the normalized compressivestrength given by 1 mm EPS beads concrete in com-parison with the one given by 2.5 mm EPS beadsconcrete, for the same concrete (macro) porosity

D/ 17:6

conrmed the presence of a particle size

eect on the EPS concrete strength. In fact, for aconcrete porosity of 0.24 for example, there is anincrease of 30% in the normalized compressivestrength given by 2.5 mm EPS beads concrete incomparison with the one given by 6.3 mm EPSbeads concrete (Fig. 8). However, recall that theseresults are most likely aected by a slight volume

size eect since D/

17:6, and thus they slightly

overestimate the compressive strengths of 2.5 mmand 6.3 mm EPS beads concretes (notably those ofhigh porosity concretes).

Furthermore, since the RVE for the EPS con-crete compressive strength is already reached withspecimens of size (110 220 mm) for 1 mm and2.5 mm EPS beads concretes, and it is also nearlyreached for 6.3 mm EPS beads concrete (notablyfor low porosity concretes where the particle sizeeect is very important), compressive tests were car-ried out on (110 220 mm) EPS concrete cylindersfor each EPS bead size and each concrete porosity.

228 K. Miled et al. / Mechanics of Materials 39 (2007) 222240(Fig. 7). For a concrete porosity of 0.23 for exam-ple, an increase of 29% was observed with 1 mmEPS beads. However, this tendency becomes negligi-ble for high porosity concretes.

Results of compressive tests made on (110 220 mm) concrete cylinders containing 6.3 mm

EPS beads D/ 17:6

and on (44 88 mm) con-crete cylinders containing 2.5 mm EPS beadsFig. 7. Normalized compressive strengths obtained with 2.5 mm anResults of these tests conrmed that the lower theinclusion size (/), the greater the concrete normal-ized compressive strength, for the same (macro)porosity (p) (Fig. 9). In fact, for a (macro) porosityof 0.24, there is an increase of 60% in the normalizedcompressive strength with 1 mm EPS beads concretein comparison with the one given by 6.3 mm EPSbeads concrete. For the same concrete porosity,

D d 1 mm EPS beads concretes homothetic cylinders / 44 .

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 229there is also an increase of 18% with 1 mm EPSbeads concrete in comparison with 2.5 mm EPSbeads concrete and an increase of 35% with2.5 mm EPS beads concrete in comparison with6.3 mm EPS beads concrete. However, this tendencydecreases with increasing EPS concrete (macro)

Fig. 8. Normalized compressive strengths obtained with 6.3 mm and

Fig. 9. EPS concrete normalized compressive strengths obtained with(macro) porosity (p).porosity (p) and vanishes for very high porosities(pP 0.5).

On the other hand, it was observed that the EPSconcrete failure mode depends greatly on its(macro) porosity (p). In fact, for low porosity con-cretes exhibiting an important particle size eect, a

2.5 mm EPS beads concretes homothetic cylinders D/ 17:6

.

(110 220 mm) cylinders for each EPS bead size, versus concrete

Fig. 10. Longitudinal sections of two broken (110 220 mm) EPS concrete specimens characterized by low porosities, and containingrespectively 6.3 mm and 2.5 mm EPS beads: a quasi-brittle and localised failure.

230 K. Miled et al. / Mechanics of Materials 39 (2007) 222240quasi-brittle failure mode was observed which ischaracterized by a few localised longitudinal split-ting macro-cracks (Fig. 10). The latter are generallyinitiated around EPS beads in zones where the max-imum extension strain for the matrix is reached.

However, for very high porosity concretes exhib-iting a negligible size eect, a ductile failure modewas observed which is characterized by diusemicro-cracks distributed in the whole mortar matrixaround EPS beads (Fig. 11). Moreover, specimenswere capable to retain load after failure without fulldisintegration. Thus, it is concluded that the EPSFig. 11. Longitudinal sections of two broken (110 220 mm) EPS conrespectively 6.3 mm and 2.5 mm EPS beads: a ductile failure mode.concrete particle size eect depends greatly on itsfailure mode.

Based on the results of this experimental investi-gation, the presence of a particle size eect on theEPS concrete compressive strength, which hadalready been mentioned in literature, is conrmed.Moreover, it is concluded that this scaling phenom-enon related to the EPS bead size (/) depends alsoon the EPS concrete (macro) porosity (p), since itis very pronounced for low porosity concretesand becomes negligible for very high porosityconcretes.crete specimens characterized by a high porosity, and containing

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 2313. Particle size eect explanation

3.1. A brief review of deterministic size eects in

materials and structures

Particle size eect had been observed rst inmetallic materials. In fact, Hall (1951) and Petch(1953) seem to have been the rst to discover thisphenomenon when they had observed accidentallythat the yield stress (r) of a mono-crystalline mildsteel decreases when increasing its grains size (d).Moreover, they had found that this tendencyfollows a power law (r / da) with (a = 1/2).Thereafter, it was observed that the strength ofpolycrystalline metals and (particle/ber) reinforcedmetal matrix composites also decreases with increas-ing grain or (particle/ber) size (Friedel, 1964; Kellyand Lilholt, 1969; Fleck et al., 1994; Smyshlyaevand Fleck, 1994; Lloyd, 1994; Stolken and Evans,1998; Kouzeli and Mortensen, 2002; Haque andSaif, 2003; Despois et al., 2004; Fivel, 2004). As itwas well explained by Arzt (1998) in his compara-tive review of size eects in materials, particle sizeeects on metallic material properties emanate gen-erally from the competition or coupling betweentwo dierent size dependencies through the interac-tion of two length scales; one is the dimension char-acteristic of the physical phenomenon involved,denoted generally as the characteristic materiallength. The other is some micro-structural dimen-sion, denoted as the size parameter or the mate-rial length parameter.

Particle size eects had also been observed inconcrete-like materials. In fact, it was observed thatthe compressive strength of normal weight concretedecreases with increasing its aggregates maximumsize (Walker and Bloem, 1960; Cordon and Gilles-pie, 1963; Hobbs, 1972; Stock et al., 1979). How-ever, this particle size eect is less pronouncedthan the one observed on the EPS lightweight con-crete compressive strength, which explains probablythe fact that it had aroused little interest in theresearch community (de Larrard and Tondat,1993), and that its physical origin is remainedunexplained.

In quasi-brittle materials like concrete, a largefracture process zone (FPZ) exists in which stressesare redistributed and energy dissipated. Thus, largecracks can grow before reaching the maximum load.As a consequence, structures made from these mate-rials exhibit a deterministic size eect on their nom-

inal strength which was brought to light byHillerborg et al. (1976). To account for this scalingphenomenon, Bazant had derived from energy con-siderations and asymptotic matching techniques alaw which is applicable for geometrically similarpre-cracked structures (Bazant, 1984; Bazant andPfeier, 1987):

rN Bft1 bp with b

LL0

; 1

where (rN) is the nominal stress at failure of a struc-ture of size L, B is a parameter depending on thestructure geometry, ft is the material tensile strengthand L0 is a characteristic material length related tothe width lc of the FPZ. The parameter b corre-

sponding to the ratio LL0

is denoted: brittleness

number. From formula (1), it is clear that thegreater b, the more brittle the structure failure.Moreover, formula (1) reduces to the linear fracturemechanics as b!1. Thereafter, this law had beenimproved by Bazant to account for the size eect oninitially non-cracked structures (Bazant, 1996).

Karihaloo et al. (2006) have recently proposedthe following deterministic size eect law fornotched quasi-brittle structures (which generalizesand improves the one proposed by the rst authorin 1999 (Karihaloo, 1999), which is based on the c-tious crack model (FCM) originated by Hillerborget al., 1976):

rNft D1 1 L=lcD2

1=2 D12D2

Llc

1 L=lcD3

1;

2where D1, D2 and D3 are coecients to be deter-mined experimentally. From formula (2), it is clear

that (rN! D1) as (L! 0) and rN ! D1D32D2

as

(L!1). In other words, the structure nominalstrength (rN) attains nite asymptotic values at bothsize extremes. Moreover, Karihaloo et al. (2006)have proved experimentally and theoretically thatthe deterministic structural size eect becomesstronger as the size of the crack increases relativeto the size of the structure but weakens as the sizeof the crack reduces.

Finally, other structural size eect laws for con-crete-like materials have been proposed by Carpin-teri and his co-workers in the last decade based onmulti-scale fractal analysis (Carpinteri, 1994; Carp-interi and Chiaia, 1997). Moreover, Carpinteriand Pugno (2005) have recently proposed simple

analytical laws to predict the strength of structures

to the strength of its matrix. That is why this ratio

232 K. Miled et al. / Mechanics of Materials 39 (2007) 222240is denoted brittleness mesoscopic number andwritten bm analogous to the brittleness numbercontaining re-entrant corners, with generalizedstress-singularity powers dierent from (1/2).

3.2. EPS concrete compressive failure analysis

On the basis of the failure modes observed exper-imentally, it is stipulated that the EPS concretefailure occurs in two phases: a rst phase ofmicro-crack initiation, which can be seen as theend of the elastic linear response of EPS concreteunder a compressive loading. This phase is assumedto be governed only by the concrete (macro) poros-ity (p), since it was observed experimentally thatthere is no particle size eect on the EPS concretemodulus of elasticity. Then, a second phase ofmicro-cracking (and/or) macro-crack propagationis entered, where one of these two failure modesor both modes can occur. In fact, during this secondphase, micro-cracks appear rst in rather diuseway within the EPS concrete matrix fracture processzone (FPZ) and then coalescence of these micro-cracks occurs to form macro-cracks, which cangrow in a stable way before reaching the maximumload (as it is the case in quasi-brittle materials likeconcrete). However, for very high porosity EPSconcretes, coalescence of micro-cracks seems tonot take place since it was observed that failureoccurs by progressive diuse micro-cracking withinthe whole matrix around EPS beads. Based onthese assumptions, it is concluded that the particlesize eect observed on the EPS concrete compres-sive strength emanates from the second phase ofmicro-cracking (and/or) macro-crack propagationthrough the competition between these two failuremodes. Thus, it is assumed that the size eect is gov-erned by the competition between two length scales.The rst is a characteristic material length lc govern-ing the rst failure mode of micro-cracking, whereasthe second is a geometric length lg(p,/) governingthe second failure mode of macro-crack propaga-tion and depending on the concrete (macro) poros-ity (p) and on the EPS bead size (/). Consequently,the EPS concrete second failure phase will be gov-erned by the ratio

lgp;/lc

. Furthermore, it will be

assumed that the greaterlgp;/

lc

, the more brittle

the concrete failure mode and thus the greater thedecrease in its compressive strength with referenceb proposed by Bazant (Bazant, 1984; Bazant andPfeier, 1987) in his structural size eect analysis.Thus, the EPS concrete normalized compressivestrength, for a given (p) and (/), will be expressedas:

rp;/rmatrix

rinitN p radN p; bm; 3

where rinitN p represents the normalized stress re-quired to initiate the rst micro-crack for a givenconcrete (macro) porosity (p), whereas radN p; bmrepresents the normalized additional stress requiredfor the second phase of micro-cracking (and/or)macro-crack propagation.

3.3. The mesoscopic brittleness number bm

To determine bm, it is necessary to identify thegeometric length lg(p,/) and the characteristic mate-rial length lc. The latter is naturally identied as thewidth of the EPS concrete matrix FPZ, which isapparently related to the maximum size lm of thematrix heterogeneities. Moreover, since for concretestructures, the FPZ width is often equal to threetimes the maximum size of material inhomogeneities(Pijaudier-Cabot and Bazant, 1987), the EPS con-crete characteristic material length lc, at the meso-scopic scale, has also been xed at three times themaximum size lm of the mortar matrix heterogene-ities. The latter corresponds here to the maximumsize of sand grains which has been identied owingto SEM observations and xed at 0.25 mm. Thus, lcis xed at 0.75 mm.

The geometric length lg is a dimension character-izing the EPS concrete micro-structure. It dependson the EPS bead size (/) and varies also with theconcrete (macro) porosity (p). Since lg(p,/) governsthe second failure phase of micro-cracking (and/or)macro-crack propagation, it is assumed that lg(p,/)controls the size of these micro-cracks and macro-cracks. For very low porosities (p! 0), the EPSconcrete micro-structure is characterized by a fewvery largely spaced EPS beads embedded in thematrix (Fig. 12). In this case, it was observed thatfailure occurs by propagation of a few localised lon-gitudinal splitting macro-cracks initiating from EPSbeads. Therefore, it is stipulated that the size ofthese macro-cracks is governed by the EPS bead size(/), and thus lg(0

+,/) = limp! 0lg(p,/) = /.However, for very high porosity concretes which

failure occurs by progressive diuse micro-crackingwithin the whole matrix around EPS beads, it is

stipulated that the size of micro-cracks is controlled

by the average EPS beads spacing e(p,/) whichdepends on (/) and (p). Thus, the geometric lengthlg(p,/) for very high porosity concretes has beenidentied as this average beads spacing which hadbeen estimated by de Larrard and Tondat (1993)as:

pmax 1=3 !

cubic (FCC) arrangement of mono-sized spheres,that is: pmax = 0.74.

Finally, for intermediate concrete porosities(0 < p 6 pmax), it is assumed that lg(p,/) is rangingbetween (/) and e(p,/). That is why it has beenapproached for all porosities as:

lgp;/ / ~lgp 0 < p 6 pmax 5with ~lgp ppmax 1

53

ppmax

2 ppmax 1

.

In fact, the non-dimensional function ~lgp hasbeen approached by a third-degree polynomial(Fig. 13) with respect to the following conditions:

limp!0 ~lgp 1, ~lgpmax 0, limp!0 d~lgpdp 0 and

d ~lgdp

pmax dedp

pmax 13pmax. Thus, the brittle-

ness mesoscopic number bm will be given by the fol-lowing formula:

bm /lc

~lgp: 6

3.4. Asymptotic analysis of EPS concrete particle size

eect

Fig. 12. The geometric length lg(p,/) for very low and very highporosity EPS concretes.

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 233ep;/ /p

1 ; 4

where pmax is the maximum packing density of anordered mixture with mono-sized spheres, whichcorresponds to the packing density of a face-centredFig. 13. The variation of the non-dimensional geometric lTo determine the law governing the EPS concreteparticle size eect, an asymptotic analysis corre-sponding to the cases (bm 6 1) and (bm!1) hasbeen conducted.ength ~lgp with EPS concrete (macro) porosity (p).

4. Particle size eect model

Based on the asymptotic analysis, it is concludedthat the EPS concrete normalized strength rp;/rmatrix

,

234 K. Miled et al. / Mechanics of Materials 39 (2007) 222240When (bm 6 1), it is concluded that the EPS con-crete particle size eect vanishes. In fact, for veryhigh porosity concretes (p! pmax) and for a giveninclusion size (/), there is a porosity (p*) for whichthe geometric length lg(p*,/) (corresponding in thiscase to the average EPS beads spacing e(p*,/))becomes smaller than the characteristic materiallength lc (corresponding to the matrix FPZ width).Consequently, micro-cracks appear progressivelyin a diuse way within the matrix FPZ which widthwill be forced to not exceed the EPS beads spacing.Thus, the FPZ at failure will cover the whole mortarmatrix which explains the ductile failure modeobserved experimentally for very high porosity con-cretes and the absence of particle size eect in thiscase. Further, for very low porosity concretes(p! 0) when the geometric length lg(0+,/) (corre-sponding in this case to the EPS bead size /) willbe in the same range of the matrix heterogeneitiesmaximum size lm, the material can be consideredas size-homogeneous at the microscopic scale, andthus it does not exhibit a particle size eect on itscompressive strength. That is why it is concludedthat when (lg(p,/) 6 lc), there is no particle sizeeect on the EPS concrete compressive strength.In this case, the normalized additional stressradN p; bm, required for the EPS concrete secondfailure phase of micro-cracking (and/or) macro-crack propagation, attains its maximum value andthus the EPS concrete normalized compressive

strength rp;/rmatrix

will be given by an upper bound

denoted g0(p), which depends only on the concrete(macro) porosity (p).

When (bm!1), it is assumed that the EPS con-crete failure mode becomes brittle. In fact, when thegeometric length lg(p,/) becomes very large in com-parison with the characteristic material length lc, theenergy dissipated in the FPZ by micro-crackingbecomes negligible in comparison with the materialfracture energy needed for crack propagation, andthus the EPS concrete failure occurs as soon asthe rst initiated micro-crack becomes a macro-crack propagating in an unstable way. In this case,the normalized additional stress radN p; bm requiredhere mainly for the macro-crack propagation tendsasymptotically to zero as (bm!1). Consequently,the particle size eect vanishes and the normalized

EPS concrete compressive strength rp;/rmatrix

will be

given by a lower bound denoted g1(p), which isidentied to the normalized stress rinitN p required

to initiate the rst micro-crack.for a given (macro) porosity (p), is ranging betweenthe lower bound g1(p) (bm!1) and the upperbound g0(p) (bm 6 1). Thus, the following phenom-enological size eect model is proposed to predictthe EPS concrete normalized strength according tothe EPS bead size (/) and to the concrete (macro)porosity (p):

rp;/rmatrix

g0p if bm6 1;g1p|{z}rinitN

fbmg0pg1p|{z}radN

if bmP 1;

8>:

7where f(bm) is the EPS concrete particle size eectlaw. This law is ranging between 1 when (bm 6 1)and zero when (bm!1). The expressions ofg0(p), g1(p) and f(bm) have been determined basedon the experimental results and on theoretical andnumerical calculations.

4.1. The upper bound g0(p)

This function must tend to 1 close to zero poros-ity, that is: limp!0g0(p) = 1. In other words, the EPSconcrete strength must tend to its mortar matrixstrength. However, this function is minimal for(p = pmax). In this case, it is assumed that there ispercolation of EPS beads and thus the material issupposed to loose completely its strength, that is:g0(pmax) = 0. An hyperbolic function has been con-sidered to approach g0(p), with respect to the twoprevious conditions. This form had already beenused by Le Roy et al. (2005) in their EPS concretecompressive strength modelling:

g0p c0 1 ppmax c0 ppmax 0 6 p 6 pmax; 8

where c0 is a tting coecient. The latter has beenidentied based on the experimental results ob-tained with (110 220 mm) EPS concretes speci-mens for very high porosities (0.5 6 p 6 0.66),2

2 Normalized compressive strengths, obtained with EPS con-cretes having (macro) porosities ranging from 0.4 to 0.66 andrepresented with full forms in Fig. 14, result from the experi-mental investigation made by Parant and Le Roy (Parant and Le

Roy, 1999; Le Roy et al., 2005).

us EPS concrete (macro) porosity (p).

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 235concrete normalized strength will be given by g0(p)(Fig. 14). Therefore, c0 has been xed at 0.45.

4.2. The lower bound g (p)where the particle size eect vanishes and the EPS

Fig. 14. The upper bound g0 vers1

This function gives for a given concrete (macro)porosity (p) the EPS concrete normalized compres-sive strength when its failure is brittle, i.e. at thebeginning of the rst failure phase of micro-crackinitiation. It is assumed to be equal to zero for(p = pmax). An hyperbolic function has also beenconsidered (of the same form as g0(p)) to approachg1(p):

g1p c1 1 ppmax d1 ppmax 0 < p 6 pmax; 9

where c1 and d1 are two tting coecients. Toidentify these coecients, the behaviour of g1(p)close to zero porosity has been investigated. In fact,the theoretical case of a single spherical void (a cav-ity) embedded in an innite 3D elastic-brittle med-ium subjected to a uniform compressive stress R1at innity has been considered (Fig. 15). Moreover,since the failure mechanism observed experimen-tally is a tensile failure mode characterized by longi-tudinal splitting cracks starting from EPS beads, itis concluded that the initiation of these cracks isdue mainly to a maximum extension strain reachedaround EPS beads (Miled et al., 2004). Therefore,the brittle failure criterion is expressed as:

Fig. 15. A spherical void embedded in an innite (3D) elastic-brittle matrix subjected to a uniform compressive stress R1 atinnity.

236 K. Miled et al. / Mechanics of Materials 39 (2007) 222240maxM crMmatrix; 10where crMmatrix is the critical extension strain sup-ported by the 3D innite homogeneous matrix.Thus, to determine the EPS concrete normalizedstrength, it is necessary to compute for the samecompressive loading R1 the maximum elastic exten-sion strain in the matrix Mmatrix and also in the in-nite medium around the spherical void maxM . In fact,for a unit compressive loading (R1 = 1) for exam-ple, it follows that:

limp!0

g1p MmatrixmaxM

: 11

To compute the elastic extension strain M, theequivalent elastic extension deformation proposedby Mazars (1984) has been considered, which is ex-pressed as:

M X3i1max0; i2

vuut ; 12where i, i = 1 3 are the principal strains. It fol-lows from (12) that:

Mmatrix 2

pmm

R1Em

; 13

where Em is the matrix modulus of elasticity and mmis the matrix Poisson coecient.

The maximum elastic extension strain in the 3Dinnite medium around the void maxM is given by:

maxM xA2 zA2

q; 14

where x(A) and z(A) are extension deformations atpoint A around the hole (which are respectivelyequal to x(B) and z(B) at point B (Fig. 15)). Thesedeformations had already been derived analyticallyby Southwell and Gough (Southwell and Gough,1926; Wang and Shrive, 1999):

xA zA 31 4mm 5m2m

14 10mmR1Em

: 15

Thus,

maxM 32

p 1 4mm 5m2m14 10mm

R1Em

: 16

The normalized compressive strength of the 3D

innite brittle medium containing a spherical voiddepends so only on the matrix Poisson coecientmm and it is expressed analytically as:

limp!0

g1p mm14 10mm

31 4mm 5m2m: 17

Thus, for mm = 0.2 it follows that:

limp!0

g1p 0:5: 18

There is so a great drop in the EPS concrete strengthwith reference to the matrix strength close to zeroporosity.

Thereafter, to approach the slope value of g1(p)when (p! 0), compressive tests have been simu-lated on periodic body-centred cubic (BCC) latticescontaining spherical holes (of a maximum packingdensity pmax of p

3

p8, that is: pmax 0:68). Two

porosities nearby 0 have been considered:p1 = 5 1004 and p2 = 1003. 3D nite element(FE) calculations using the FE code CASTEM2000have been carried out on (1/8) of BCC lattices unitcells (making use of the cell symmetries). Moreover,an elastic-brittle model has been considered for thematrix with the same failure criterion used for the3D innite medium containing the spherical void.Uniaxial compressive tests have been simulatedwhere periodic conditions have been applied onthe mesh boundary faces (Miled, 2005). For a unitcompressive stress loading (R = 1) for example,it follows that:

limp!0

g1p MmatrixmaxM p

19

with Mmatrix 2

pmm REm

2

pmm

Em. For these calcula-

tions, Em can take any arbitrary value since the EPSconcrete normalized strength depends only on mm(equal here to 0.2), when the concrete failure is brit-tle. Moreover, to compute maxM in the 3D BCC lat-tice unit cell, a mesh renement around the holehas been performed with a view to obtaining accu-rate results and ensuring their mesh independency(Fig. 16). The following results have been obtained:g1(5 1004) = 0.477 and g1(1003) = 0.460. Thus,the slope value of g1(p) close to zero porosity hasbeen approached by a linear regression of these re-sults and also of the analytical result obtained for(p! 0). That is:

limp!0

dg1pdp

40: 20

From (18) and (20), it follows that c1 = 8.59 03 0210 and d1 = 1.72 10 = 2c1.

4.3. The particle size eect law f(bm)

When (bm > 1), a power law of bm has been pro-posed to predict the particle size eect on the EPS

4.4. An overview of the proposed model

The phenomenological particle size eect modelproposed to predict the EPS concrete normalized

Fig. 16. Distribution of the elastic extension strain M in the (1/8) of a BCC lattice unit cell having a porosity p1 = 5 1004: the maximumis reached around the hole.

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 237concrete compressive strength:

fbm bma; 21where the power a is xed at (1/3) by tting theexperimental normalized compressive strengths ob-tained with (110 220 mm) specimens for the threeEPS bead sizes concretes.Fig. 17. The model predictions for the three EPS bead sizesstrength according to its (macro) porosity (p) andto the EPS bead size (/) will be expressed as:

rp;/rmatrix

g0p if bm 6 1;g1p bm1=3g0p g1p if bm P 1;

(

withconcretes considered in the experimental investigation.

he ex

238 K. Miled et al. / Mechanics of Mg0p 0:45 1 ppmax

;

Fig. 18. Comparison between the model predictions and t(110 220 mm) specimens for the three EPS bead sizes concretes.0:45 ppmax

g1p 1 ppmax

2 1 58:14 ppmax and

bm /lc

p

pmax 1

5

3

ppmax

2 p

pmax 1

!;

pmax 0:74

Fig. 17 shows the model predictions for 1 mm,2.5 mm and 6.3 mm EPS beads concretes. It is soclear that for a given (p) and (/), the size eectmodel is governed by the matrix FPZ width lc. Thischaracteristic material length is assumed here to beequal to three times the matrix heterogeneities max-imum size lm. Moreover, to optimize the EPS con-crete compressive strength (i.e. to attain the upperbound g0), it will be recommended based on thismodel to use an heterogeneous mortar matrix ofheterogeneities maximum size which must be inthe same range of that of EPS beads.

Finally, the model predictions have been com-pared with the normalized compressive strengthsobtained with (110 220 mm) specimens for thethree EPS bead sizes concretes considered in theexperimental investigation. The particle size eect

perimental normalized compressive strengths obtained with

aterials 39 (2007) 222240model reproduces in a very satisfactory manner theexperimental results given by 1 mm and 2.5 mmEPS beads concretes which are not aected by a vol-ume size eect. Furthermore, the model predictionstake into account the slight volume size eectobserved on the normalized strengths obtained with(110 220 mm) specimens and 6.3 mm EPS beadsize, since they slightly underestimate these experi-mental results (Fig. 18).

5. Conclusions

An experimental investigation was conducted onthree EPS lightweight concretes having (macro)porosities ranging from 10% to 50% and containingthree EPS bead sizes, with a view to conrming thepresence of an intrinsic particle size eect on theEPS concrete compressive strength which hadalready been mentioned in literature. Moreover, toseparate the size eect problem of the specimen size(D), compressive tests were carried out on homo-thetic EPS concrete specimens containing homo-thetic EPS beads. Results of these tests showedthat strengths obtained with homothetic specimens

having a ratio D/P 44

are not aected by a

eect vanishes.

K. Miled et al. / Mechanics of Materials 39 (2007) 222240 239volume size eect. Therefore, it is concluded that for1 mm and 2.5 mm EPS beads concretes, the RVEfor the EPS concrete compressive strength is alreadyreached with a specimen size of (110 220 mm).However, it is concluded that results of compressivetests obtained with 6.3 mm EPS beads concretespecimens of size (110 220 mm) slightly overesti-mate the 6.3 mm EPS beads concrete compressive

strengths, since the RVE is not yet reached forD/ 17:6

.On the other hand, compressive tests results con-

rmed the presence of a particle size eect on theEPS concrete compressive strength since it wasobserved that the lower the EPS bead size (/), thegreater the concrete compressive strength, for thesame concrete (macro) porosity (p). Moreover, itwas observed that this particle size eect is very pro-nounced for low porosity concretes which failuremode is quasi-brittle, whereas it becomes negligiblefor very high porosity concretes which failure modeis more ductile.

On the basis of the analysis of EPS concrete fail-ure modes observed experimentally, it is assumedthat the EPS concrete failure occurs in two phases:a rst phase of micro-crack initiation governed bythe concrete (macro) porosity (p), followed by a sec-ond phase of micro-cracking (and/or) macro-crackpropagation where one of these two failure modesor both modes can occur. As a consequence, it isconcluded that the particle size eect on the EPSconcrete compressive strength emanates from thesecond phase of micro-cracking (and/or) macro-crack propagation through the competition betweenthese two failure modes. Thus, it is stipulated thatthe size eect is governed by the competitionbetween two length scales: a characteristic materiallength lc which governs the rst failure mode anda geometric length lg(p,/) which governs the secondfailure mode and depends on the concrete (macro)porosity (p) and on the EPS bead size (/). There-fore, the EPS concrete particle size eect will be

governed by the ratiolgp;/

lc

denoted brittleness

mesoscopic number bm (analogous to the brittle-ness number b proposed by Bazant in his struc-tural size eect analysis), and thus it is assumedthat the greater bm, the more brittle the concretefailure mode. The brittleness mesoscopic numberbm has been then identied. The characteristic mate-rial length lc is identied as the width of the EPSconcrete matrix fracture process zone (FPZ) which

has been xed at three times the mortar matrix het-A power size eect law of bm has been proposedto ensure the transition from g0(p) to g1(p) when(bm > 1). The power of this scaling law is xed at(1/3) by tting the experimental results.

Eventually, the model predictions have beencompared with experimental results, showing agood agreement.

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Particle size effect on EPS lightweight concrete compressive strength: Experimental investigation and modellingIntroductionExperimental investigationResults of modulus of elasticity testsRVE for the EPS concrete compressive strengthParticle size effect

Particle size effect explanationA brief review of deterministic size effects in materials and structuresEPS concrete compressive failure analysisThe mesoscopic brittleness number beta mAsymptotic analysis of EPS concrete particle size effect

Particle size effect modelThe upper bound g0(p)The lower bound g infin (p)The particle size effect law f( beta m)An overview of the proposed model

ConclusionsReferences