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InfluenceofporosityoncompressiveandtensilestrengthofcementmortarDATASETinCONSTRUCTIONANDBUILDINGMATERIALSMARCH2013ImpactFactor:2.3DOI:10.1016/j.conbuildmat.2012.11.072

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t" Extended Zheng model is good representation of experimental data.creases

ile (splitting tensile and exural) strength of cement mortar is not constant,The ratio decreases with increase porosity values of cement mortar.

created such an upsurge in research activities that our knowledgeconcerning the relationship between pore structure and frostresistance of concrete is much more complete than the strengthporosity relationship. This does not mean that no efforts have beenmade for the development of quantitative relationships betweenstrength and porosity but rather that these efforts have been spo-radic [810] and the results have less than satisfactory.

material has already been investigated. Taking an empirical ap-proach, Powers [11] was able to deduce an equation which relatesthe compressive strength of mortar cubes to a function of the gel-space ratio. Schiller [17] using a theoretical approach deduced anequation relation the strength of material to the porosity. He ap-plied this equation to experimental data on gypsum plasters andobtained a good t for compressive and tensile strengths. Someexcellent reviews [1820] of the effect of porosity on the strengthof concrete presented some of the more important empirical andtheoretical equation for relating strength to porosity. The profusionof the possible equation is enormous and whilst one equation is

Corresponding author. Tel.: +86 25 83786551; fax: +86 26 83786986.

Construction and Building Materials 40 (2013) 869874

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evE-mail address: cxdong1985@hotmail.com (X. Chen). 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The fact that a reduction of porosity in a solid material increasesits strength in general, and the strength of cement-based materialsin particular, was recognized long ago [13]. It has also been dis-covered that porosity has an important role in the frost resistanceof concrete [46]. Furthermore, porosity has a role in the relation-ship between mechanical properties of concrete, such as thecompressive strength-modulus of elasticity relationship [7]. Thepractical importance of durability of cement-based materials

In the eld of more basic research, the pore structure of cement-based materials has been a dominant topic [2,1114]. But experi-mentally measurement of a relevant porosity parameter hasproved to be extremely difcult in cement-based materials,because of the special character of the hydration products formed[15]. Hence the results obtained will depend not only on the mea-suring principle but also on the drying method used prior to theporosity measurements [16]. But even with these problems solved,a connection between the porosity and strength has to be estab-lished. The inuence of porosity on the strength of cement-basedCement mortarStrengthPorosity strength and indirect tens

but is porosity dependent." Compressive/tensile strength ratio de

a r t i c l e i n f o

Article history:Received 5 July 2012Received in revised form 26 September2012Accepted 21 November 2012

Keywords:0950-0618/$ - see front matter 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.conbuildmat.2012.11.072with increase porosity.

a b s t r a c t

The compressive, exural and splitting tensile strength of cement mortar has been measured and inter-preted in terms of its porosity. The authors rst reviewed the existing porositystrength relationships(Ryshkewithch, Schiller, Balshin and Hasselman model) and assessed the suitability of existing relation-ships. The Zheng model for porous materials has been used to evaluate the porositystrength relationshipof cement mortar. Over the porosity ranges examined, the extended Zheng model is good representationof the experimental data on the strength of cement mortar. Based on the generality of the assumptionsused in the derivation of the extended Zheng model, this model for cement mortar can be applied forother cement-based materials. The experimental data also show that the ratio between compressive" Strength decreases with increasing porosity." Suitability of existing expressions relating strength and porosity is assessed.Inuence of porosity on compressive and

Xudong Chen , Shengxing Wu, Jikai ZhouCollege of Civil and Transportation Engineering, Hohai University, Nanjing, China

h i g h l i g h t s

" Strength and porosity of cement mortar has been measured.

Construction and

journal homepage: www.elsll rights reserved.ensile strength of cement mortar

SciVerse ScienceDirect

uilding Materials

ier .com/locate /conbui ldmat

suitable for a second material. Clearly some simplication isdesirable. Despite the relatively large number of experimental

been extended beyond simple expressions for tensile or compres-

relationship between these mechanical properties. In addition,

from ordinary Portland cement 42.5. The ne aggregate used for mortar specimenswas river quartzite sand. The sand was passed through a No. 4 sieve before use. Four

using a bend tester (ASTM C 348 [24]). Similar to the compressive tests, exuraltests were carried out on triplicate specimens and average exural strength values

3. Test results and discussion

Quite a few relationships involving strength and porosity ofengineering materials have been reported in the literature [20].Historically, several general types of model have been developedfor cement-based materials.

Balshin [31], from his study of the tensile strength of metalceramics, suggested the relation (Eq. (2)):

r r01 pb 2where r is the strength, r0 is the strength at zero porosity, b is theempirical constant.

Ryshkewitch [32], from a study of the compressive strength ofAl2O3 and ZrO2, obtained the relation (Eq. (3)):

r r0ekp 3where k is the empirical constant.

uilding Materials 40 (2013) 869874were obtained. Splitting tensile tests were run on cubical specimens(70.7 70.7 70.7 mm) according to BS 1881-117 [25].

2.3. Determination of porosity

After the exural tests, three pieces from each specimen were weighed underwater and in the saturated surface-dry (SSD) [26] condition, thus enabling the bulkvolume to be calculated. It was assumed that any volume change during drying orre-saturation was negligible; this volume was used to calculate the bulk density ofeach sample after drying (in the worst case, the bulk volume change due to dryingwould be approximately 1.5% [26,27]). Each specimen was then dried in a carbon-dioxide free oven at 105 C until it reached constant weight. The difference inweight between in the water-saturated and oven-dry conditions was used to calcu-late the porosity expressed as a percentage of the bulk specimen volume. The datawhich are presented are the average of three replicates. The porosity was calculatedusing the following equation:

p Wssd WdWssd Ww 100% 1

where p is the porosity (100%),Wssd is the specimen weight in the saturated surface-dry (SSD) condition (g),Wd is the specimen dry weight after 24 h in oven (g), andWwwatercement ratio (w/c), 03, 0.5, 0.6 and 0.7, were used for cement mortar. Thecorresponding sandcement ratio (s/c) for all cement mortars is 1.2. Mixing wasdone in a small mixer. Casting was completed in two layers which were compactedon a vibrating table. The cast specimens were covered with polyurethane sheet anddamped cloth in a 20 2 C chamber and were demoulded at the age of 1 day. Forstrength and porosity tests, the specimens were cured in saturated limewater at20 2 C until the test age 7 and 28 days.

2.2. Strength measurements

Compressive tests were run on specimens according to ASTM C 349 [23]. Thespecimens (40 40 160 mm) were prepared according to ASTM C 348 [24]. Threespecimens were tested for each mix proportions. Flexural tests for exural strengthof the mix proportions were carried out on the long surface of prism specimensthe existing strengthporosity relationship have been reviewedand compared with experimental results.

2. Experimental details

2.1. Materials and mix compositions

An adequate number of series of cement mortar compositions were prepared tostudy the strengthporosity relationship. Cement mortar samples were preparedsive strength of a specic material. None of these encompassesboth compressive and tensile strength for cement-based materials.

The compressive and tensile strength of concrete are importantdesign parameters in civil engineering. The splitting tensile andexural test has been reported as two indirect measure of the ten-sile strength of cement-based materials [21,22]. It has been usedwidely in practice due to its testing ease, simplicity of specimenpreparation, and possible eld applications.

The objective of this paper is to determine the compressivestrength, splitting tensile, and exural strength of cement mortar,and to study how porosity inuences the magnitude of and theinvestigations that have been conducted to characterize the linkbetween strength and porosity, few systematic evaluations havemost suitable for one material a quite different equation is most

870 X. Chen et al. / Construction and Bis the weight of saturated specimen (g).This method has been used to measure the porosity of the cement-based mate-

rials successfully [15,2830].Schiller [17], on the basis of the study of set sulfate plasters,proposed the relation (Eq. (4)):

r n ln p0p

4

where n is the empirical constant, p0 is the porosity at zero strength.Hasselman [33] suggested the equation of a linear relationship

between strength and porosity for different refractory materials(Eq. (5)):

r r0 cP 5where c is the empirical constant.

Results of tting previously mentioned models of strengthporosity relations are given in Figs. 13. Values of parameters r0in models of Hasselman, Balshin, and Ryshkewithch correspondto the strength of nonporous material or equivalently to theextrapolated strength of specimens to the zero porosity. It shouldalso be mentioned that the estimated value of the parameter r0(strength at zero porosity) may not always provide a reliable esti-mate of the material nonporous response. Other microscopic awsremaining in the material under these conditions can control itsstrength, and this aspect is not explicitly taken into account inthe above models. Hence, one should be careful with how this t-ting parameter is used in practical applications. For cement-basedmaterials, the constant r0 contains microstructure factors in-volved, like density of cement particle and CSH, particle size dis-tribution and size, and density of aws [3436]. The model ofFig. 1. Experimental data on compressive strengthporosity dependence. Graphs ofthe best t obtained for existing models tested are shown.

uildiFig. 2. Experimental data on exural strengthporosity dependence. Graphs of thebest t obtained for existing models tested are shown.

X. Chen et al. / Construction and BSchiller has a vertical asymptote at zero porosity, and the value ofparameter k depends on the base of the logarithm so its value ismerely a way of obtaining the best t. The values of those param-eters are approximately the same all mixes studied. Simple linearrelationship of Hasselman model shows articial intercept withthe abscissa at porosity less than the initial porosity and predictsnegative strength at higher porosities. A pore-initiated-failuremodel for glass at low values of strength at higher porosity was of-fered by Hasslman [37] in the explanation of the load-bearingareas. In treating failure initiation from this complex, Hasselmanand Fulrath [38] used the cylindrical model solved by Bowie [39]and assumed that crack extension parallel to the surface of thespecimen triggered catastrophic failure. As shown in Figs. 13,the model of Hasselman overestimates the observed strength dropwith increasing porosity. Thus, although Hasselmans model ap-pears to embody a rational concept, it is quantitatively subject toquestion. Recently, Hyun et al. [40] suggested that the empiricalconstant b in Balshins model is related with the stress concentra-tion around pores in the porous materials. The stress concentrationfactor of the pores depends on the pore geometry and orientationwith the direction of applied stress. Although the equation of Bal-shins model is different from Hasselmans model, the basic con-cept in these two models is similar, since load bearing area and

Fig. 3. Experimental data on splitting tensile strengthporosity dependence.Graphs of the best t obtained for existing models tested are shown.stress concentration around the pores are closely related to eachother. For example, the loading bearing area is reduced withincreasing the porosity, which causes stress concentration aroundthe pores [41,42]. Ryshkewitchs model is based on the assumptionthat the relative strength of porous material is equal to the ratio ofthe minimum solid area to the cell area normal to the referencestress [43]. Rice [44] suggested that the Hasselman model haveshown to less accurate than the minimum solid area approach.However, it is generally found that the minimum solid area canbe related to the porosity of relatively low volume fraction ofporosity [45,46] (p 6 0.4 pc, where pc is the critical porosity thatcorresponds to the percolation limit of the solid phase). Also, theassumption of the Ryshkewiths model, namely, that (a) the appli-cation of a hydrostatic pressure to the composite sphere assem-blage can adequately represent the stress and strain response toother stresses and that the pressure is uniformly experienced byall of the various hollow spheres comprising the model body, and(b) Poissons ratio can either increase and decrease with increasingporosity, with it converging to a xed value, are open to question[41]. For the model of Balshin, the value of b is merely a way ofobtaining the best t and have no physical signicance, thus leav-ing us with no respective to predict this value. Although the initialporosity of the material enters in the model of Schiller, the pre-dicted strength increase with the decrease in porosity is too highand better t is obtained if both p0 and n are tted freely. It is alsoshown in Figs. 13 that Ryshkewithchs exponential and Schillerslogarithmic formulae for the strength of cement mortar are numer-ically indistinguishable except in the neighborhood of the ex-tremes of 0% and 100% porosity. In general the overestimatedzero-porosity strength is a consequence of tting strength datausing the models of Ryshkewithch and Schiller.

It is necessary to point out that the models summarized above,which were based on specic structures. The microstructural evo-lution of a material with increasing porosity is a 3D connectivityproblem. According to the percolation theory, there exist two crit-ical porosity levels [46,47]. When the porosity reaches the criticalporosity value pc1 , a microstructural transition occurs from fullyisolated and closed pores with nearly spherical or ellipsoidalshapes to open and interconnected with complex shapes. Finally,the effective strength or elastic modulus vanishes when the poros-ity reaches the second critical value (pc).

Grifths model of fracture [48] is usually taken as a classic the-ory to explain how the mechanical performance is related to poros-ity. Grifth found that the critical stress incurs crack propagationwithin a brittle material and can be expressed by:

r 2Ecpa

r6

where E is the modulus of elasticity, c is the fracture surface energyand a is the half length of an internal crack.

Ficker [49] suggested that the average value of pore size in por-ous materials can be written as,

r pc ppc

m7

where r is the average value of pore size;m is the ratio of calculatedaverage distance to the nearest pore, m reects the randomness ofpore distribution, the degree of randomness can be sued to classifythe distribution of porosity in each location, if m is close to 1, thepores are considered randomly distributed, for m less than 1, the

ng Materials 40 (2013) 869874 871pore distribution is classied as clustered, for cement-based mate-rials, m = 0.85 [50]; pc is the percolation porosity at failurethreshold.

Therefore, according to the brittle fracture theory proposed byGrifth [48] early in 1920, Zheng et al. [50] suggested that thestrength of porous materials with porosity p can be written as:

r a pc ppc

m=2 KIc 8

KIc 2cE

p9

where KIc is the fracture toughness of porous material; a is a coef-cient concerning stress state.Wagh et al. [51] given the porositydependence of the fracture toughness as:

KIc KIco pc ppc

1 p2=3

1=210

where KIco is the fracture toughness of pore-free material.

4. Relation between compressive and indirect tensile strengthof cement mortar

The exural and splitting tensile tests are much cheaper, sim-pler and quicker to carry out because the samples are smaller,and the set up time for the tests is much less. All quantitative datareported so far referred exclusively to compressive strength [7]. Inthis section, we explore the role of porosity and how it inuencesthe correlation between indirect tensile and compressive strength.From a number of other investigators [7,21,5557], a simple powerlaw model has become one of the most widely used analyticalmodels for describing the relationship between the indirect tensile(splitting tensile/exural) strength and compressive strength of

Fig. 5. Comparison of predicted and observed exural strength.

Fig. 6. Comparison of predicted and observed splitting tensile strength.

Table 1Estimated values for r0 and pc.

Loading regime pc r0 Corr. coeff. (R)

Compression 0.562 69.4 0.989Splitting tension 0.768 9.74 0.996Flexure 0.783 5.56 0.993

872 X. Chen et al. / Construction and BuildiAn important feature that differentiates Eq. (10) from otherexpressions [44,52] relating the fracture toughness to porosity isthat it takes into account the effect of stress concentration inducedby the presence of pore. It has been demonstrated experimentally[53] and theoretically [18,39,54] that the stress concentration dueto the presence of pores and the annular crack pore stress eldinteraction effects are so large that they cannot be neglected.

Substituting Eq. (10) into (8), one obtains:

r a KIco pc ppc

1m 1 p2=3

" #1=211

Assuming that r0 = aKIco is the strength of pore-free materials, thenthe following equation can be easily obtained:

r r0 pc ppc

1:85 1 p2=3

" #1=212

The theoretical curves for strength against porosity are shownin Figs. 46. The experimental results are generally in good agree-ment with the theoretical curves. The application of the theoreticalequation to the experimental data leads to the constants given inTable 1. The extended Zhengs model is a rigorous mathematicalformula that of a simple symmetry. It postulates no assumptionson either physical properties or processes or microstructures. Thus,it is believe that the extended Zhengs model reects the randomnature of microstructure in cement-based materials. This modelrequires two parameters to dene the strength characteristics ofcement mortar and the parameter r0 and pc can account thechanges in loading regime (splitting tension, exure orcompression).Fig. 4. Comparison of predicted and observed compressive strength.ng Materials 40 (2013) 869874concrete. From the experimental results, we can write a newexpression for the ratio between indirect tensile strength and com-pressive strength, as a function of porosity:

X. Chen et al. / Construction and BuildirCrF

4:12 p0:236 13

rCrS

7:45 p0:221 14

where rC is the compressive strength of cement mortar (MPa); rS isthe splitting tensile strength of cement mortar (MPa); and rF is theexural strength of cement mortar (MPa).

The empirical relationship suggested in Eqs. (13) and (14) areplotted in Figs. 7 and 8. It can be seen that the predicted resultsfrom Eqs. (13) and (14) showed a relative good relationship be-tween porosity and compressive-indirect tensile strength ratio ofcement mortar. The correlation coefcient (R), which indicateshow much of the total variation in the dependent variable can beaccounted for by the regression equation, was obtained as 0.959and 0.973 for Eqs. (13) and (14) in this study, respectively. Further-more, it may be inferred from Figs. 7 and 8 that weaker (higherporosity) cement mortar has a lower compressive strength-indi-rect tensile strength ratio, whereas stronger cement mortar (lowerporosity) has higher compressive-indirect tensile strength ratio.Odler and Robler [58] also suggested that the ratio of compressivestrength and split tensile strength is porosity dependent for

Fig. 7. Effect of porosity on the ratio between compressive strength and splittingtensile strength of cement mortar.hydrated cement paste. They found a linear relation betweencompressive/splitting tensile strength ratio and porosity. The ratiodecrease linearly with increase porosity values. That the trends

[10] Lian C, Zhuge Y, Beecham S. The relationship between porosity and strength

Fig. 8. Effect of porosity on the ratio between compressive strength and exuralstrength of cement mortar.for porous concrete. Constr Build Mater 2011;25:42948.[11] Powers TC. Structure and physical properties of hardened Portland cement

paste. J Am Ceram Soc 1958;41(1):16.[12] Diamond S, Dolch WL. Generalized log-normal distribution of pore sizes inindicated by Eqs. (13) and (14) are in conformity with the ndingsof Odler and Robler [58].

5. Conclusions

The dependence of compressive, splitting tensile and exuralstrength on porosity for cement mortar was analysed empiricallyand theoretically in this paper. The following conclusions can bedrawn:

(1) Ryshkewithchs exponential and Schillers logarithmic for-mulae for the porositystrength relationship of cement mor-tar are numerically indistinguishable except in theneighborhood of the extremes of 0% and 100% porosity. Sim-ple linear relationship of Hasselman model shows articialintercept with the abscissa at porosity less than the initialporosity and predicts negative strength at higher porosities.Although the initial porosity of the material enters in themodel of Schiller, the predicted strength increase with thedecrease in porosity is too high.

(2) Over the porosity ranges examined, the extend Zhengsmodel are good representations of the experimental dataon the strength of cement mortar. This model requires twoparameters to dene the strength characteristics of cementmortar and the parameters can account the changes in load-ing regime (splitting tension, exure or compression). Basedon the generality of the assumptions used in the derivationof the extended Zhengs model, this model for cement mor-tar can be applied for other cement-based materials.

(3) The experimental data also show that the ratio betweencompressive strength and indirect tensile (split-tensile andexural) strength of cement mortar is not constant, but isporosity dependent. The ratio decreases with increase poros-ity values of cement mortar.

Acknowledgement

The authors are grateful to the National Natural Science Foun-dation (Nos. 50979032 and 51178162) for the nancial support.

References

[1] Yudenfreund M, Hanna KM, Skalny J, Odler I, Brunauer S. Hardened Portlandcement paste of low porosity, V: compressive strength. Cem Concr Res1972;2:73143.

[2] Auskern A, Horn W. Capillary porosity in hardened cement paste. J Test Eval1973;1:749.

[3] Pantazopoulou SJ, Mills RH. Microstructural aspects of the mechanicalresponse of plain concrete. ACI Mater J 1995;92(6):60516.

[4] Hasan M, Ueda T, Sato Y. Stressstrain relationship of frost-damaged concretesubjected to fatigue loading. J Mater Civil Eng 2008;20(1):3745.

[5] Zuber B, Marchand J. Modeling the deterioration of hydrated cement systemsexposed to frost action, Part 1: description of the mathematical model. CemConcr Res 2000;30:192939.

[6] Hain M, Wriggers P. Computational homogenization of micro-structuraldamage due to frost in hardened cement paste. Finite Elem Anal Des2008;44:23344.

[7] Popovics S. Method for developing relationships between mechanicalproperties of hardened concrete. J Am Concr Inst 1973;70:7958.

[8] Popovics S. New formulas for the relationships for the prediction of the effectof porosity on concrete strength. J Am Concr Inst 1985;82:13646.

[9] Pann KS, Yen T, Tang CW, Lin LD. New strength model based on watercementratio and capillary porosity. ACI Mater J 2003;100(4):3118.

ng Materials 40 (2013) 869874 873hydrated cement paste. J Colloid Interf Sci 1972;38(1):23444.[13] Zhang M, Li H. Pore structure and chloride permeability of concrete containing

nano-particles for pavement. Constr Build Mater 2011;25(2):60816.

[14] Das BB, Kondraivendhan B. Implication of pore size distribution parameters oncompressive strength, permeability and hydraulic diffusivity of concrete.Constr Build Mater 2012;28(1):3826.

[15] Day RL, Marsh BK. Measurement of porosity in blended cement pastes. CemConcr Res 1988;18:6373.

[16] Vodak F, Trtik K, Kapickova O, Hoskova S, Demo P. The effect of temperature onstrengthporosity relationship for concrete. Cem Concr Res 2004;14:52934.

[17] Schiller KK. Strength of porous materials. Cem Concr Res 1971;1:41922.[18] Kendall K, Howard AJ, Birchall JD, Parrt PL, Proctor BA, Jefferis SA. The relation

between porosity, microstructure and strength, and the approach to advancedcement-based materials. Philo Trans Roy Soc Lond 1983;SA310(1511):13953.

[19] Kumar R, Bhattacharjee B. Porosity, pore size distribution and in situ strengthof concrete. Cem Concr Res 2003;33:15564.

[20] Li L, Aubertin M. A general relationship between porosity and uniaxial strengthof engineering materials. Can J Civil Eng 2003;30:64458.

[21] Raphel JM. Tensile strength of concrete. J Am Concr Inst 1984;81(2):15865.[22] Torrent RJ. A general relation between tensile strength and specimen

geometry for concrete-like materials. Mater Struct 1977;10(4):18796.[23] ASTM C349. Standard test method for compressive strength of hydraulic-

cement mortars (using portions of prisms broken in exure). ASTM C349-08,Annu book ASTM Stand 04.01.

[24] ASTM C348. Standard test method for exural strength of hydraulic-cementmortars. ASTM C348-02, Annu book ASTM Stand 04.01.

[25] BS 1881: Part 117. Testing concrete method for the determination of tensilesplitting strength. British Standard Institute.

[26] Galle C. Effect of drying on cement-based materials pore structure as identiedby mercury intrusion porosimetry, a comparative study between oven-,vacuum-, and freeze-drying. Cem Concr Res 2001;31(10):146777.

[27] Brue F, Davy CA, Skoczylas F, Burlion N, Bournon X. Effect of temperature onthe water retention properties of two high performance concretes. Cem ConcrRes 2012;42(2):38496.

compressive strength of mortar based on this paste. Cem Concr Res2006;36:17403.

[37] Hasselman DPH. On the porosity dependence of the elastic moduli ofpolycrystalline refractory materials. J Am Ceram Soc 1962;45:4523.

[38] Hasselman DPH, Fulrath RM. Effect of small fraction of spherical porosity onelastic moduli of glass. J Am Ceram Soc 1964;47:523.

[39] Bowie OL. Analysis of an innite plate containing radial cracks originating atthe boundary of an internal circular hole. J Math Phys 1956;35(1):6071.

[40] Hyun SK, Murakami K, Nakajima H. Anisotropic mechanical properties ofporous copper fabricated by unidirectional solidication. Mater Sci Eng A2001;299:2418.

[41] Rice RW. Comparison of stress concentration versus minimum solid area basedmechanical propertyporosity relations. J Mater Sci 1993;28:218790.

[42] Rice RW. Comparison of physical propertyporosity behavior with minimumsolid area models. J Mater Sci 1996;31:150928.

[43] Carniglia SC. Working model for porosity effects on the uniaxial strength ofceramics. J Am Ceram Soc 1972;55(12):6108.

[44] Rice RW. Relation of tensile strengthporosity effects in ceramics to porositydependence of Youngs modulus and fracture energy, porosity character andgrain size. Mater Sci Eng A 1989;112:21524.

[45] Krstic VD. Porosity dependence of strength in brittle solids. Theor Appl FractMec 1988;10:2417.

[46] Winslow DN, Cohen MD. Percolation and pore structure in mortars andconcrete. Cem Concr Res 1994;24:2537.

[47] Ye G. Percolation of capillary pores in hardening cement pastes. Cem Concr Res2005;35:16776.

[48] Grifth AA. The theory of rupture and ow in solids. Trans Roy Soc Lond A1920;221:63.

[49] Ficker T. Fractal strength of cement gels and universal dimension of fracturesurfaces. Theor Appl Fract Mec 2008;50:16771.

874 X. Chen et al. / Construction and Building Materials 40 (2013) 869874[28] Matusinovic T, Sipusic J, Vrbos N. Porositystrength relation in calciumaluminate cement pastes. Cem Concr Res 2003;33:18016.

[29] Papayianni I, Stefanidou M. Strengthporosity relationships in lime-pozzolanmortars. Constr Build Mater 2006;20:7005.

[30] Chindaprasirt P, Rukzon S. Strength, porosity and corrosion resistance ofternary blend Portland cement, rice husk ash and y ash mortar. Constr BuildMater 2008;22:16016.

[31] Balshin MY. Relation of mechanical properties of powder metals and theirporosity and the ultimate properties of porous-metal ceramic materials. DoklAskd SSSR 1949;67(5):8314.

[32] Ryshkevitch R. Compression strength of porous sintered alumina and zirconia.J Am Ceram Soc 1953;36(2):658.

[33] Hasselman DPH. Grifth aws and the effect of porosity on tensile strength ofbrittle ceramics. J Am Ceram Soc 1969;52:457.

[34] Roy DM, Gouda GR. Porositystrength relation in cementitious materials withvery high strengths. J Am Ceram Soc 1973;56:54950.

[35] Relis M, Soroka I. Compressive strength of low-porosity hydrated Portlandcement. J Am Ceram Soc 1980;63:6904.

[36] Li YX, Chen YM, Wei JX, He XY, Zhang HT, Zhang WS. A study on therelationship between porosity of the cement paste with mineral additives and[50] Zheng M, Zheng X, Luo ZJ. Fracture strength of brittle porous materials. Int JFract 1992;58:515.

[51] Wagh AS, Singh JP, Poeppel RB. Dependence of ceramic fracture properties onporosity. J Mater Sci 1993;28:358993.

[52] Roberts JTA, Ueda Y. Inuence of porosity on deformation and fracture of UO2. JAm Ceram Soc 1972;55(2):11724.

[53] Palchik V, Hatzor YH. Crack damage stress as a composite function of porosityand elastic matrix stiffness in dolomites and limestones. Eng Geol2002;63:23345.

[54] Lawrence P, Majumdar AJ, Nurse RW. The application of the Mackenzie modelto the mechanical properties of cements. Cem Concr Res 1971;1:7599.

[55] Nielsen LF. Strength development in hardened cement paste: examination ofsome empirical equations. Mater Struct 1993;26:25560.

[56] Rashid MA, Mansur MA, Paramasivam P. Correlations between mechanicalproperties of high-strength concrete. J Mater Civil Eng 2002;14(3):2308.

[57] Choi Y, Yuan RL. Experimental relationship between splitting tensile strengthand compressive strength of GFRC and PFRC. Cem Concr Res2005;35:158791.

[58] Older I, Robler M. Investigations on the relationship between porosity,structure and strength of hydrated Portland cement pastes. II. Effect of porestructure and degree of hydration. Cem Concr Res 1985;15:40110.