Mechanical Properties of Cement Mortar: Development of Structure-Property Relationships

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1 International Journal of Concrete Structures and Materials Vol.5, No.1, pp.3~10, June 011 DOI /IJCSM Mecanical Properties of Cement Mortar: Development of Structure-Property Relationsips Tewodros Tekeste Gebrab 1 and Parviz Sorousian (Received August 10, 010, Revised November 5, 010, Accepted January 4, 011 Abstract: Teoretical models for prediction of te mecanical properties of cement mortar are developed based on te morpology and interactions of cement ydration products, capillary pores and microcracks. Te models account for intermolecular interactions involving te nano-scale calcium silicate ydrate (C-S-H constituents of ydration products, and consider te effects of capillary pores as well as te microcracks witin te ydrated cement paste and at te interfacial transition zone (ITZ. Cement mortar was modeled as a tree-pase material composed of ydrated cement paste, fine aggregates and ITZ. Te Hasin s bound model was used to predict te elastic modulus of mortar as a tree-pase composite. Teoretical evaluation of fracture tougness indicated tat te frictional pullout of fine aggregates makes major contribution to te fracture energy of cement mortar. Linear fracture mecanics principles were used to model te tensile strengt of mortar. Te predictions of teoretical models compared reasonably wit empirical values. Keyword: structure-property relationsips, mecanical properties, cement mortar, ITZ, fine aggregates. 1. Introduction Teoretical models were developed for prediction of te mecanical properties of cement mortar using background models developed for ydrated cement paste. 1 Te paste models account for te intermolecular interactions between nano-scale calcium silicate ydrate (C-S-H structures, pullout of micro-scale calcium ydroxide (CH crystals, and te effects of capillary pores and microcracks on lowering te mecanical properties of ydrated cement paste (HCP. 1 Te work reported erein employed tese background teoretical models of ydrated cement paste to model te mecanical performance of cement mortar accounting for te effects of fine aggregates, te ITZ, and te srinkage microcracks developing predominantly at te ITZ. Te ITZ is usually expressed as a region aving a tickness of 30~50 µm, in wic te ydrated cement paste composition differs significantly from tat of te bulk cement paste. Te ITZ as muc iger porosity and CH volume fraction, and lower contents of C-S-H gel and unydrated cement paste wen compared wit te bulk cement paste. Tis is attributed, to some extent, to te low packing of C-S-H gels near te surface of aggregate particle.,3 Te ITZ constitutes te weakest pase of mortar. 4 1 Dept. of Construction Engineering, Texas Tec University, Box 4310, Lubbock, TX , USA. tewodros.gebrab@ttu.edu. Dept. Civil and Environmental Engineering, Micigan State University, Room 3546, Engineering Building, East Lansing, MI , USA. Copyrigt 011, Korea Concrete Institute. All rigts reserved, including te making of copies witout te written permission of te copyrigt proprietors.. Modulus of elasticity of cement mortar Cracks propagate in cement mortar mostly troug te ydrated cement paste and along te ITZ. Tis indicates tat te ITZ influences te mecanical properties of cement mortar. In modeling te elastic modulus of cement mortar, te effect of ITZ was considered by evaluating mortar as a tree-pase composite material. In order to develop te elastic modulus model of cement mortar, te modulus of elasticity of every constituent pase of te composite material was determined. For ydrated cement paste, te background elastic modulus model (Eq. (1 was used. Te modulus of elasticity of te ITZ in cement mortar (E im is a modified version of te elastic modulus model of ydrated cement paste. Te capillary porosity of te ITZ was considered to be about twice tat of te bulk ydrated cement paste. Te basis for te assumption is tat te maximum porosity in ITZ is tree times tat of cement matrix. 5 Since te minimum possible porosity in ITZ is te same as tat of cement matrix, we ave taken te average porosity for ITZ, wic is twice tat of cement matrix. Tis variation in porosity was accounted for in developing te elastic modulus of te ITZ (E im based on tat of te ydrated cement paste (. Te modulus of elasticity of fine aggregates (E s ranges from 70 GPa to 90 GPa. 6 A sensitivity analysis was conducted to evaluate te significance of tis range in determining te modulus of elasticity of cement mortar. ( 1 p 0.77E ( π 4p 0.37 o a b p were elastic modulus of ydrated cement paste, E o intrinsic modulus of elasticity of ydrated cement paste, a & b are alf (1 3

2 of te major and minor axes lengts of te ellipsoidal capillary pores, respectively, and p capillary porosity. Te upper and lower bounds of te modulus of elasticity of cement mortar were calculated from an equation relating te modulus of elasticity of a composite material to its sear modulus (G m and bulk modulus (K m (Eq. (. 7 Te upper and lower bounds of te sear modulus and bulk modulus were calculated using te Hasin s modulus of elasticity model for tree-pase composite materials (Eqs. (3~(7. 7 Te steps followed for te computation of te upper and lower bounds for te modulus of elasticity of cement mortar are presented below. Te upper and lower bounds of te elastic modulus of mortar, E m, were approximated as follows: 7 were E and ν are te modulus of elasticity and Poisson s ratio of a material, respectively. From te above relationsips, te bulk modulus and sear modulus of ydrated cement paste and fine aggregates were calculated as follows (considering ν 0. for bot te HCP 8 and fine aggregates 9 : Te bulk modulus of HCP, K, K ( ν ( 0. ( 0.56 Te bulk modulus of sand (fine aggregate, K s, ( K m- G m- 3K m- + G m- E K G m+ m+ m 3K m+ + G m+ were K m+ and K m- are te upper and lower bounds of te bulk modulus of cement mortar, respectively; and G m+ and G m- are te upper and lower bounds of te sear modulus of cement mortar, respectively. Tese parameters were calculated as follows: Lower bound of te bulk modulus of cement mortar, K m-, ( E s K s ( ν ( 0. ( 0.56 E s Te sear modulus of HCP, G, Te sear modulus of sand (fine aggregate, G s, E s G ( + ν ( (10 (11 1 V 3 t r K m- K K s K i G i (3 E s E s G s ( + ν ( E s (1 were is volume fraction of sand, V is volume fraction of HCP, K s is bulk modulus of sand, K is bulk modulus of HCP, K i is bulk modulus of ITZ, G i is sear modulus of ITZ, and t r is te ratio of te tickness of te ITZ to te radius of an equivalent sperical fine aggregate. All tese parameters are defined below. Upper bound of te bulk modulus of cement mortar, K m+, K s K m+ V K K s t r K i G i Lower bound of te sear modulus of cement mortar, G m-, 1 V G m- G G s 0.4V t s r K i G i G i were G is sear modulus of HCP. Upper bound of te sear modulus of cement mortar, G m+, G s G m+ V G G s t r K i G i Te next step was to determine all te parameters of te above equations. Te bulk modulus and sear modulus of any material are related to its modulus of elasticity troug Eqs. (7 and (8: 7 E K ( ν E G ( + ν (4 (5 (6 (7 (8 Based on te above relationsips, te bulk modulus and te sear modulus of te ITZ of cement mortar could also be determined as follows: K i 0.56E i G i 0.4E i (13 (14 wit te elastic modulus of sand (E s ranging from 70 to 90 GPa, te corresponding values of K s and G s range from 39. to 50.4 and 9.4 to 37.8, respectively. A sensitivity analysis conducted to assess te significance of tese ranges in determining te mecanical properties of mortar is presented below. Te elastic modulus of te ITZ (E i is a function of te elastic modulus ( and te capillary porosity (P of te ydrated cement paste in cement mortar. Te ratio of te tickness of te ITZ to te radius of an equivalent sperical fine aggregate, t r,can be expressed as follows: 7 t r t i --- r s (15 were t i is tickness of te ITZ wic is a function of aggregate size; and r s is te radius of an equivalent sperical aggregate. Tickness of ITZ (t i is proportional to te size of te aggregate it envelopes. 7 It can tus be approximated using te reported values of te maximum ITZ tickness in concrete. A linear relationsip was assumed between te aggregate size and te tickness of te ITZ using a size range of 0.1 to 4 mm and 4 to 5 mm for fine and coarse aggregates, respectively. Te maximum tickness of te ITZ in concrete is about 50 µm.,10 Tis tickness was considered to correspond to a 5 mm aggregate size. No ITZ was assumed to occur for an aggregate size lower tan 0.1 mm. Te value of t i was 4 International Journal of Concrete Structures and Materials (Vol.5 No.1, June 011

3 tus considered to ranged from 0 to 3.9 µm for aggregate size ranging from 0.1 to 4 mm. A sensitivity analysis was conducted to investigate te significance of tis range in determining te mecanical properties of mortar, and te results are presented in te next section. Te significance of te tree pases in determining te mecanical properties of cement mortar also depends upon teir respective volume proportions in te mortar. Te calculation of te volume fractions of ydrated cement paste ( V, fine aggregate ( and ITZ ( V i in cement mortar is presented below. Since te maximum porosity of ITZ (P i is tree times 11 and te minimum is equal to tat of cement matrix, an average porosity of ITZ of about twice tat of te HCP was assumed. Wen very large volume fractions of sand are used in cement mortar, te workability of te fres mix tends to deteriorate. Tis damage to workability results in te formation of large voids and micro-defects, wic would undermine te performance of mortar. A reasonable range for te volume fraction of sand considered in tis investigation is from 0.5 to 0.6. A mortar mix proportion, were sand and bulk ydrated cement paste constitute 55% and 45%, respectively, of te mortar volume was considered to represent conventional mortar mixtures. Te volume of te ITZ surrounding a single sand particle in cement mortar (V 1i was calculated using Eq. (16 based on te relationsip between aggregate size and te ITZ discussed above. A circular cylinder wit aspect (eigt-to-diameter ratio equal to 1 was considered as a geometric model of sand (Fig. 1. Tis geometric model implies tat: V 1i π ( r s + t i 3 3 ( r s (16 Assuming a uniform aggregate size, te number of sand particles per unit volume of cement mortar (N s can be calculated as follows: N s V 3 1s π r s (17 were V 1s is te volume of a single cylindrical sand particle of radius r s and eigt r s, as sown in Fig. 1. Te total volume of te ITZ (V i can ten be calculated as te product of N s and V 1i, as follows: V i V 1i N s (18 Since t i is proportional to r s, for a constant, V i and V remain constant wit a cange in aggregate size, as sown in Table Sensitivity analysis of te effect of te fine aggregate elastic modulus on te modulus of elasticity of cement mortar Since te elastic modulus of te fine aggregates used in cement mortar ranges from 70 MPa to 90 MPa, a sensitivity analysis was conducted to assess te significance of tis range of aggregate elastic modulus in determining te modulus of elasticity of cement mortar. In tis sensitivity analysis, viable ranges of ydrated cement paste properties were considered (for example, E o equal to 38 GPa, and capillary pore aspect ratio, a/b, equal to. Based on te results presented in Table 1, te values of V 0.443, V i and 0.55 were considered for evaluation of te elastic modulus of cement mortar. Te elastic modulus of ydrated cement paste was calculated by substituting te above values into Eq. (1, and simplifying it to obtain: ( 5.85( 1 p π 4p p 0.37 (19 were and p are te modulus of elasticity and capillary porosity of HCP, respectively. Te modulus of elasticity of te ITZ was calculated by modifying Eq. (19, assuming tat P i is twice tat of te HCP. 5 Te value of te elastic modulus of te ITZ (E i was multiplied by a factor of 0.75 to account for te 5% volume fraction of te directionally oriented CH crystals in te transition zone. ( E i 3.4( 1 p π 8p p 0.37 (0 Fig. 1 Te geometric model of a sand particle. and E i were calculated using Eqs. (19 and (0 for a reasonable range of capillary porosity in cement mortar. Tree aggregate sizes (0.6,.3 and 4 mm were considered to investigate te influence of te variation in aggregate size, and te results suggested tat te upper and lower bounds of sear modulus and bulk modulus of mortar did not cange considerably wit aggregate size. Te reason for tis is tat te bulk modulus and te sear modulus of a material depend upon te volume fractions and elastic moduli of te individual pases in te composite material and not on te Table 1 Volume fractions of te ITZ and te ydrated cement paste for different aggregate sizes (at constant aggregate volume in cement mortar. r s (m t i (m t r V 1i (m 3 N s V i V 0.30E E E-03.04E-1 3.4E E E E E E E E E E E International Journal of Concrete Structures and Materials (Vol.5 No.1, June 011 5

4 aggregate size. Wile te aggregate surface area decreases wit increasing aggregate size, te volume fraction of te ITZ remains constant as te aggregate size canges because its tickness increases wit aggregate size. Te upper and lower bounds of te elastic modulus of cement mortar (E m+ and E m- for different elastic moduli of fine aggregates were calculated using Eq. (. Te results are sown in Fig. in order to assess te influence of te variation in te fine aggregate elastic modulus (E s on te modulus of elasticity of mortar (E m. It can be observed in Fig. tat te variations in te upper bound of E m tend to be more pronounced wen compared wit tose in te lower bound. However, it could be generally noted tat te effect of te variation in te fine aggregate elastic modulus, witin te given range, on te elastic modulus of cement mortar is not of great significance. Based on tis observation, an average value for te elastic modulus of fine aggregate, 80 GPa, was considered as a representative value in development of te model. Once te upper and lower bounds of te elastic modulus of cement mortar were establised, te next step was to develop a reasonable model for te modulus of elasticity of cement mortar, witin tese bounds. For tis purpose, te predictions of an empirical model of elastic modulus, 1 wic was based on substantial experimental data, are compared wit te teoretical predictions of lower and upper bounds in Fig.. Tis comparison suggests tat te empirical values (reflecting experimental results fall witin te upper and lower bounds of te teoretical model, leaning more towards te lower bound. Based on tis observation, te lower-bound model of elastic modulus was considered to reasonably represent te elastic modulus of cement mortar. Te modulus of elasticity of cement mortar was tus represented as follows: were is te modulus of elasticity of ydrated cement paste, given by Eq. (1. 4. Mecanistic model for te fracture tougness of cement mortar It was noted earlier tat major contributions are made to te fracture energy of HCP by te pull-out of CH crystals. In te case of cement mortar, one sould add te contributions of te sand (fine aggregate pull-out to fracture energy. Te pull-out of fine aggregates is expected to make major contributions to fracture tougness by increasing te surface area involved in frictional pull-out. Sand particles are assumed to ave roug surfaces, wit HCP interacting wit sand particles as sown in Fig. 3. Fine aggregates also offer a crack-sielding effect wic enances te process zones in front of te crack tip, and tus contributes to fracture tougness. Pull-out of sand particles would involves pononic friction as te dendrites of HCP bonding to te sand surface sear off (Fig. 3. It is assumed tat eac dendrite of HCP is subjected to a searing force by te dendrites on te sand surface, acting at te median contact point. Tis causes a diagonal tensile stress on te ydrated cement paste dendrite, generating a 45-degree diagonal crack sown in Fig. 4. Te surface area of te ydrated cement paste subjected to tis stress condition was approximated to be alf te pulled-out surface area of sand. Te surface area of sand subjected to pull-out (after debonding was computed by assuming a cylindrical model of sand sown in Fig. 5. Te saded region in Fig. 5 is te mean surface area of sand subjected to frictional pull-out. Tis region is assumed to occur, on te average, at te lower quarter of te cylinder eigt. 9K E m- G m- m K m- + G m- (1 5. Energy released during debonding of sand particles from ydrated cement paste Wit an aggregate elastic modulus (E s of 80 GPa, Eq. (1 yields te following expression for te modulus of elasticity of cement mortar in terms of te elastic modulus of ydrated cement paste. E m E ( Te interfacial surface area of sand tat is subjected to debonding (followed by pull-out, A sd, can be expressed as follows: A sd πr s (3 were A sd is interfacial surface are of sand, and r s is te radius of sand particle. In order to account for te porosity of te ITZ (assumed to be Fig. Comparison of te predictions of teoretical model for te elastic modulus of cement mortar wit tose of an empirical model based on substantial test data. Fig. 3 Surface rougness of fine aggregate, and interaction of te ydrated cement paste (HCP at te interface. 6 International Journal of Concrete Structures and Materials (Vol.5 No.1, June 011

5 6. Energy released during pull-out of sand particles from ydrated cement paste Te surface area of sand subjected to frictional pull-out ( A sp is te saded area sown in Fig. 5 (excluding te bottom surface: A sp πr s P (9 Fig. 4 Te crack pattern of te HCP dendrite under te force P generated during sand pull-out. Wen te dendrites of ydrated cement paste sear off due to sand pull-out, te diameter of fractured dendrites is about 1.414d'(due to te diagonal sear effect, as sown in Fig. 4. Due to te assumption tat te dendrites of bot te ydrated cement paste and te sand are interlocked (Fig. 3, te total surface area of HCP subjected to sear stress ( A sp is about times A sp. Terefore, A sp 1.414πr s P (30 Te energy (U csp1 released due to CH pull-out upon interfacial fracture over A sp can be calculated as te product of G p and A sp (Eq. (31, were G p is te energy released per unit fractured surface area (due to CH pull-out in ydrated cement past, and is considered to be equal to 7.88 J/m. Fig. 5 Surface of sand particle (saded area subjected to frictional pull-out. U csp1 7.88A sp U csp1 39.4πr s P (31 (3 twice tat of HCP, P, Eq. (3 was furter modified as follows: A sd πr s P (4 Debonding of sand occurs essentially troug cracking of te interface zone, and involves debonding of C-S-H globules. Te energy lost due to debonding of C-S-H globules is about 1.56 J/ m. Terefore, te energy released (U sd, Joules due to te debonding of sand from HCP (i.e., due to debonding of C-S-H globules in te ITZ can be calculated as follows: U sd 1.56A sd (5 were A sp is te total surface area of HCP subjected to sear stress; and U csp1 is te energy released due to CH pull-out upon interfacial fracture. Let U csp represent te energy released due to CH pull-out at te bottom surface of sand particle. It can be calculated as te product of G p and te base area of te model sand particle sown in Fig. 5: U csp 7.88πr s P (33 Te total energy released due to CH pull-out during te pull-out of sand particles from HCP (U csp is, terefore, te sum of U csp1 and U csp : U sd 3.1πr s 1 P ( (6 U csp U csp1 + U csp (34 Te fracture energy released per unit area of mortar (G sd, J/m, due to C-S-H/C-S-H debonding, can be calculated as te total debonding energy released due to one sand particle divided by te projected area of sand on a plane perpendicular to te pull-out direction (see Eqs. (7 and (8. G sd 3.1πr s P πr s (7 U csp 67.3πr s P (35 were U csp is te total energy released due to CH pull-out during te pull-out of sand particles from HCP. Te energy released per unit fractured area of ITZ (G csp, in J/ m, due to CH pull-out, can be calculated as te total energy released as one sand particle pulls out divided by te projected area of te sand on a plane perpendicular to te pull-out direction: G sd P ( (8 were, G sd is fracture energy (J/m released per unit area of mortar due to C-S-H/C-S-H debonding. G csp 67.3πr s P πr s G csp P ( (36 (37 International Journal of Concrete Structures and Materials (Vol.5 No.1, June 011 7

6 were G csp is te total energy per unit fractured area released during CH pull-out from HCP. Te total energy release rate per unit fractured area of te ITZ in mortar (G im can now be calculated as te sum of G csp and G sd : G im G csp + G sd G im P ( (38 (39 Te oter factor wic contributes to te fracture tougness of cement mortar is te sielding effect of sand particles against crack growt. Tis crack sielding effect results from bridging of te two crack surfaces by sand particles near te crack tip. Tis penomenon promotes multiple microcracking aead of te crack tip in ydrated cement paste. Te region were tese microcracks form is called te crack process zone. For typical volume fractions of sand used in mortar, formation of te process zone furter increases te fracture tougness of ydrated cement paste by about 65%. Terefore, wit a total fracture tougness of ydrated cement paste (G o of 9.44 J/m, te modified fracture tougness of ydrated cement paste in mortar will be about J/m. Fracture tougness of mortar (G m can now be calculated as te sum of te fracture tougness of te ITZ and tat of HCP, proportional to teir respective volume fractions in cement mortar: G m G im + G V (40 were is te sand (and te small ITZ volume fraction in mortar; and V is te ydrated cement paste volume fraction in mortar. As noted in previous sections, and V are used as reasonable examples. It can also be sown tat G G o (1- P and G o J/m. Substituting tese values into Eq. (40 yields: G m 70.4( 1 P ( 1 P G m P ( (41 (4 te fracture tougness of cement mortar wic range from 55 to 75 J/m. 13 A sensitivity analysis was conducted using te teoretical model in order to assess te significance of sand volume fraction in determining te fracture tougness of cement mortar. Te relationsips between te fracture tougness of mortar (G m and te volume fraction of sand ( for capillary porosities (P of 0.3 is plotted in Fig. 7 (for te typical conditions introduced earlier. Te increase in sand volume fraction witin practical ranges is observed to produce a minor rise in te fracture tougness of cement mortar. Similar trends were observed at oter levels of capillary porosity. It sould be noted tat te addition of sand markedly increases te fracture tougness of cement mortar over tat of neat cement paste. 7. Tensile strengt model of cement mortar Te tensile strengt model of cement mortar was developed by applying te Griffit equation: σ mt E m G m πl m (43 were E m is te modulus of elasticity of cement mortar; G m is te fracture tougness of cement mortar; and l m is alf te critical crack lengt at wic unstable crack propagation occurs in mortar. Many microcracks form in cement mortar prior to any loading due to te restrained srinkage effect. Te restraint of drying srinkage partly results from te presence of te dimensionally stable and ig-modulus aggregates witin te cement paste. Microcracks tend to initiate at te ITZ, and ten propagate into te ydrated cement paste. Tese cracks may extend over and bridge between two adjacent sand particles. 3,14 In tis researc, te critical crack lengt in cement mortar is considered to extend between two sand particles, as sown in Fig. 8. Te initial crack lengt in mortar ( l m can be estimated using Fig. 8 as te sum of te center-to-center distance between two sand particles (l and te size of sand (r s : Te fracture tougness calculated using tis model wit reasonable capillary porosity (P levels ranges from 44 to 76 J/m (Fig. 6, wic is in conformance wit te experimental values of l m ( l + r s (44 Fig. 6 Fracture tougness of mortar as a function of te capillary porosity of HCP. Fig. 7 Fracture tougness of mortar as a function of te volume fraction of sand for a capillary porosity of International Journal of Concrete Structures and Materials (Vol.5 No.1, June 011

7 Fig. 8 Te critical srinkage crack in cement mortar bridging between two adjacent sand particles. Srinkage cracks are more likely to occur around larger aggregates because teir interfaces are weaker (due to attraction of greater quantities of bleeding water and also because te restrained srinkage stresses tend to be greater in teir vicinity. For tis reason, te aggregate size in mortar were crack is likely to initiate was considered to be 4 mm (i.e., close to te upper bound of fine aggregate particle size. If we assume a uniform sand particle size in cement mortar, te center-to-center spacing of sand particles (l can be calculated for a sand volume fraction of 0.55 as: Fig. 9 Tensile strengt of cement mortar as a function of te capillary porosity of HCP. 3 πr s 0.55l 3 (45 For a 4 mm diameter sand, 1 -- π( 3 3 l mm 0.55 (46 Substituting te result of Eq. (46 into Eq. (44 yields l m 4.4 mm. Te tensile strengt model of cement mortar was ten obtained by substituting Eqs. (, (4 and (44, for E m, G m and l m, respectively, into Eq. (43. Te resulting model is sown below. ( σ mt P [ 1.β β] β β 9.3 (47 were P is te capillary porosity of ydrated cement paste; and β is defined as: β α( 1 P and α is defined as: π 4P α P (48 (49 Te relationsip between te tensile strengt of cement mortar ( σ mt and te porosity of HCP (P (Eq. (47 is plotted in Fig. 9. Te prediction of te model is compared wit experimental results. A sensitivity analysis was conducted to investigate te effect of sand volume fraction on te tensile strengt of cement mortar. Fig. 10 sows te relationsips between te tensile strengt of cement mortar ( and te volume fraction of sand ( for a typical σ mt 1/ Fig. 10 Tensile strengt of mortar as a function of te volume fraction of sand for a capillary porosity of 0.3. capillary porosity (P of 0.3. Teoretical predications indicate a sligt increase in tensile strengt wit increasing sand volume fraction witin te range considered ere. It sould be noted tat excess quantities of sand can compromise workability (at constant water/cement ratios, tereby complicating te production of cement mortar and tus increasing te potential for generation of large defects wic are damaging to tensile strengt. 8. Conclusions Te following conclusions can be drawn from te teoretical work presented in tis paper: 1 Te elastic modulus model of cement mortar was developed by considering te contributions and interactions of te ydrated cement paste, sand particles, and te ITZ. Te model was furter verified and refined using te experimental trends reported in te literature. Te fracture tougness model of cement mortar igligted te significance of te contributions made by fine aggregates troug frictional pull-out and arrest/diversion of cracks. Te predictions of te fracture tougness model compared favorably wit reported experimental results. 3 Te tensile strengt model of cement mortar was based on te corresponding elastic modulus and fracture tougness models, considering te effects of aggregates on restrained srinkage International Journal of Concrete Structures and Materials (Vol.5 No.1, June 011 9

8 microcracking of mortar. Predictions of te tensile strengt model compared favorably wit experimental results. References 1. Gebrab, T. T. and Sorousian, P., Mecanical Properties of Hydrated Cement Paste: Development of Structure-Property Relationsips, International Journal of Concrete Structures and Materials, Vol. 4, No. 1, 010, pp. 37~43.. Dir, R. K., Key Features in View of Modeling te Permeability of Concrete, Cement Combinations for Durable Concrete Proceedings of te International Conference Held at te University of Dundee, Scotland UK, 005, pp. 591~ Bisscop, J. and Van Mier, J. G. M., Quantification of Srinkage Microcracking in Young Mortar wit Fluorescence Ligt Microscopy and ESEM, Heron, Vol. 44, No.4, 1999, pp. 45~ Scrivener, K. L., Materials Science of Concrete I, Skalny, J. P., Ed., American Ceramic Society, 1989, pp. 17~ ttp://ciks.cbt.nist.gov/garbocz/104itz/node1.tm, Hsu, T. T. C., Slate, F. O., Sturman, G. M., and Winter, G., Microcracking of Plain Concrete and te Sape of te Stress- Strain Curve, American Concrete Institute Journal Proceedings, Vol. 60, No., 1963, pp. 09~4. 7. Mindess, S., Young, J. F., and Darwin, D., Concrete, nd Ed., Prentice Hall, Pearson Education, Inc., NJ, 003, pp. 308~ Poon, C. S., Lam, L., and Wong, Y. L., Effects of Fly As and Silica Fume on Interfacial Porosity of Concrete, Journal of Materials in Civil Engineering, Vol. 11, Issue 3, 1999, pp. 197~ Meta, P. K. and Monteiro, P. J. M., Concrete: Microstructure, Properties, and Materials, nd Ed., McGraw-Hill Com. Inc., Scrivener, K. L., Crumbie, A. K., and Laugesen, P., Te Interfacial Transition Zone between Cement Paste and Aggregate in Concrete, Interface Science, Vol. 1, 004, pp. 411~ Kliszczewicz, A. and Ajdukiewicz, A., Differences in Instantaneous Deformability of s/pc According to te Kind of Coarse Aggregate, Cement and Concrete Researc, Vol. 4, No., 00, pp. 63~ Ramakrisnan, N. and Arunacalam, V. S., Effective Elastic Modulus of Porous Solids, Journal of Material Science, Vol. 5, 1990, pp. 3930~ Giaccio, G. and Zerbino, R., Failure Mecanism of Concrete: Combined Effect of Coarse Aggregates and Strengt Level, Advanced Cement Based Materials, Vol. 31, 1998, pp 41~ Liu J., Mukopadyay, A. K., and Zollinger, D. G., Contribution of Aggregates to te Bonding Performance of Concrete, Paper Submitted to te Transportation Researc Board for Presentation and Publication, Wasington D.C., International Journal of Concrete Structures and Materials (Vol.5 No.1, June 011