Construction and Building Materials 40 (2013) 869–874

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Construction and Building Materials

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Influence of porosity on compressive and tensile strength of cement mortar

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." Strength decreases with increasing porosity." Suitability of existing expressions relating strength and porosity is assessed." Extended Zheng model is good representation of experimental data." Compressive/tensile strength ratio decreases with increase porosity.

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:StrengthPorosityCement mortar

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

⇑ Corresponding author. Tel.: +86 25 83786551; faxE-mail address: cxdong1985@hotmail.com (X. Che

a b s t r a c t

The compressive, flexural and splitting tensile strength of cement mortar has been measured and inter-preted in terms of its porosity. The authors first reviewed the existing porosity–strength 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 porosity–strength 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 compressivestrength and indirect tensile (splitting tensile and flexural) strength of cement mortar is not constant,but is porosity dependent. The ratio decreases with increase porosity values of cement mortar.

� 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 [1–3]. It has also been dis-covered that porosity has an important role in the frost resistanceof concrete [4–6]. 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 materialscreated such an upsurge in research activities that our knowledgeconcerning the relationship between pore structure and frostresistance of concrete is much more complete than the strength–porosity 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 [8–10] and the results have less than satisfactory.

ll rights reserved.

: +86 26 83786986.n).

In the field of more basic research, the pore structure of cement-based materials has been a dominant topic [2,11–14]. But experi-mentally measurement of a relevant porosity parameter hasproved to be extremely difficult 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 influence of porosity on the strength of cement-basedmaterial 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 fit for compressive and tensile strengths. Someexcellent reviews [18–20] 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

870 X. Chen et al. / Construction and Building Materials 40 (2013) 869–874

most suitable for one material a quite different equation is mostsuitable for a second material. Clearly some simplification isdesirable. Despite the relatively large number of experimentalinvestigations that have been conducted to characterize the linkbetween strength and porosity, few systematic evaluations havebeen extended beyond simple expressions for tensile or compres-sive strength of a specific 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 andflexural 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 field applications.

The objective of this paper is to determine the compressivestrength, splitting tensile, and flexural strength of cement mortar,and to study how porosity influences the magnitude of and therelationship between these mechanical properties. In addition,the existing strength–porosity 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 strength–porosity relationship. Cement mortar samples were preparedfrom ordinary Portland cement 42.5. The fine aggregate used for mortar specimenswas river quartzite sand. The sand was passed through a No. 4 sieve before use. Fourwater–cement ratio (w/c), 03, 0.5, 0.6 and 0.7, were used for cement mortar. Thecorresponding sand–cement 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 flexural strengthof the mix proportions were carried out on the long surface of prism specimensusing a bend tester (ASTM C 348 [24]). Similar to the compressive tests, flexuraltests were carried out on triplicate specimens and average flexural strength valueswere 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 flexural 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 ¼ ðWssd �WdÞðWssd �WwÞ

� 100% ð1Þ

Fig. 1. Experimental data on compressive strength–porosity dependence. Graphs ofthe best fit obtained for existing models tested are shown.

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), and Ww

is the weight of saturated specimen (g).This method has been used to measure the porosity of the cement-based mate-

rials successfully [15,28–30].

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 ¼ r0ð1� pÞb ð2Þ

where 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 ¼ r0e�kp ð3Þ

where k is the empirical constant.Schiller [17], on the basis of the study of set sulfate plasters,

proposed the relation (Eq. (4)):

r ¼ n lnp0

p

� �ð4Þ

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 ¼ r0 � cP ð5Þ

where c is the empirical constant.Results of fitting previously mentioned models of strength–

porosity relations are given in Figs. 1–3. Values of parameters r0

in 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 flawsremaining 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 fit-ting parameter is used in practical applications. For cement-basedmaterials, the constant r0 contains microstructure factors in-volved, like density of cement particle and C–S–H, particle size dis-tribution and size, and density of flaws [34–36]. The model of

Fig. 2. Experimental data on flexural strength–porosity dependence. Graphs of thebest fit obtained for existing models tested are shown.

Fig. 3. Experimental data on splitting tensile strength–porosity dependence.Graphs of the best fit obtained for existing models tested are shown.

X. Chen et al. / Construction and Building Materials 40 (2013) 869–874 871

Schiller 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 fit. The values of those param-eters are approximately the same all mixes studied. Simple linearrelationship of Hasselman model shows artificial 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. 1–3,the model of Hasselman overestimates the observed strength dropwith increasing porosity. Thus, although Hasselman’s model ap-pears to embody a rational concept, it is quantitatively subject toquestion. Recently, Hyun et al. [40] suggested that the empiricalconstant b in Balshin’s 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-shin’s model is different from Hasselman’s model, the basic con-cept in these two models is similar, since load bearing area and

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]. Ryshkewitch’s 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 Ryshkewith’s 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) Poisson’s ratio can either increase and decrease with increasingporosity, with it converging to a fixed value, are open to question[41]. For the model of Balshin, the value of b is merely a way ofobtaining the best fit and have no physical significance, 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 fit is obtained if both p0 and n are fitted freely. It is alsoshown in Figs. 1–3 that Ryshkewithch’s exponential and Schiller’slogarithmic 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 fitting strength datausing the models of Ryshkewithch and Schiller.

It is necessary to point out that the models summarized above,which were based on specific 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 ðpc1

Þ, 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).

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

r ¼ffiffiffiffiffiffiffiffi2Ecpa

rð6Þ

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 ¼ pc � ppc

� ��m

ð7Þ

where �r is the average value of pore size; m is the ratio of calculatedaverage distance to the nearest pore, m reflects 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, thepore distribution is classified as clustered, for cement-based mate-rials, m = 0.85 [50]; pc is the percolation porosity at failurethreshold.

Fig. 5. Comparison of predicted and observed flexural 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 Building Materials 40 (2013) 869–874

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

r ¼ a � pc � ppc

� �m=2

� KIc ð8Þ

KIc ¼ffiffiffiffiffiffiffiffi2cE

pð9Þ

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

KIc ¼ KIcopc � p

pc

� �� ð1� p2=3Þ

� �1=2

ð10Þ

where KIco is the fracture toughness of pore-free material.An important feature that differentiates Eq. (10) from other

expressions [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 fieldinteraction effects are so large that they cannot be neglected.

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

r ¼ a � KIcopc � p

pc

� �1þm

� ð1� p2=3Þ" #1=2

ð11Þ

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

r ¼ r0pc � p

pc

� �1:85

� ð1� p2=3Þ" #1=2

ð12Þ

The theoretical curves for strength against porosity are shownin Figs. 4–6. 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 Zheng’s 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 Zheng’s model reflects the randomnature of microstructure in cement-based materials. This modelrequires two parameters to define the strength characteristics ofcement mortar and the parameter r0 and pc can account thechanges in loading regime (splitting tension, flexure orcompression).

Fig. 4. Comparison of predicted and observed compressive strength.

4. Relation between compressive and indirect tensile strengthof cement mortar

The flexural 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 influencesthe correlation between indirect tensile and compressive strength.From a number of other investigators [7,21,55–57], a simple powerlaw model has become one of the most widely used analyticalmodels for describing the relationship between the indirect tensile(splitting tensile/flexural) strength and compressive strength ofconcrete. 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:

Fig. 7. Effect of porosity on the ratio between compressive strength and splittingtensile strength of cement mortar.

X. Chen et al. / Construction and Building Materials 40 (2013) 869–874 873

rC

rF¼ 4:12 � p�0:236 ð13Þ

rC

rS¼ 7:45 � p�0:221 ð14Þ

where rC is the compressive strength of cement mortar (MPa); rS isthe splitting tensile strength of cement mortar (MPa); and rF is theflexural 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 coefficient (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 forhydrated cement paste. They found a linear relation betweencompressive/splitting tensile strength ratio and porosity. The ratiodecrease linearly with increase porosity values. That the trends

Fig. 8. Effect of porosity on the ratio between compressive strength and flexuralstrength of cement mortar.

indicated by Eqs. (13) and (14) are in conformity with the findingsof Odler and Robler [58].

5. Conclusions

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

(1) Ryshkewithch’s exponential and Schiller’s logarithmic for-mulae for the porosity–strength 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 artificialintercept 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 Zheng’smodel are good representations of the experimental dataon the strength of cement mortar. This model requires twoparameters to define the strength characteristics of cementmortar and the parameters can account the changes in load-ing regime (splitting tension, flexure or compression). Basedon the generality of the assumptions used in the derivationof the extended Zheng’s 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 andflexural) 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 financial support.

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