On interface property characterization and performance of fiber-reinforced cementitious composites

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1 Concrete Science an Engineering, Vol. 1, September 1999, pp On interace property characterization an perormance o iber-reinorce cementitious composites Z. Lin 1, T. Kana an V. C. Li 1 (1 Avance Civil Engineering Materials Research Laboratory, Department o Civil an Environmental Engineering, University o Michigan, Ann Arbor, MI , USA ( Kajima Technical Research Institute, Tobitakyu, Chou-shi, Tokyo 18, Japan RESEARCH PAPERS ABSTRACT It has been well recognize that iber/matrix interace properties have signiicant eects on the perormance o a iber-reinorce cement composite, incluing its racture toughness, tensile an lexural strength an uctility. Proper characterization o the interace properties via micromechanics moels can lea to eective tools or esigning high perormance an cost-eective cement-base materials. In this paper, a micromechanics moel is evelope to characterize the interace properties at single iber pullout level. In the moel, interacial racture toughness, rictional bon strength an post-eboning slip-harening coeicient are explicitly accounte or. Fiber rupture an iber strength reuction ue to incline iber pullout are also consiere. The complete composite briging stress versus crack opening curve ( B - δ relation is erive analytically. Implications o the present moel on various composite properties, incluing composite tensile strength, racture energy an complementary energy (a measure o uctility, are iscusse along with an example o PVA iber reinorce cement composites. 1. INTRODUCTION Introuction o ibers in a cement-base brittle matrix can signiicantly increase the ailure strain an racture toughness o the composite by orers o magnitue [1-4]. In orer to achieve high perormance an cost-eectiveness o such composites, quantitative materials esign tools are neee. Micromechanics moels have been proven to be very eective in high perormance an costeective cement-base composites evelopment [5-7]. A generation o super-uctile short-iber cement composites so-calle Engineere Cementitious Composites (ECC have been evelope an start to gain momentum in real-worl applications [8-1]. Presente in this paper is a generic micromechanics moel that can be use or almost all the iber types, incluing steel ibers an synthetic (polymeric ibers with both hyrophobic an hyrophilic nature. The current moel extens the work by Maalej et al. [3] to inclue chemical boning an slipepenent interacial properties. It provies not only a more realistic representation o iber-matrix interacial behaviors in many cases but also signiicantly broaens the range o viable iber types in achieving both high perormance an cost-eectiveness, especially or those ibers with strong chemical bon to cement matrix such as Poly-Vinyl-Alchohol (PVA iber, which will be investigate in etail. Fig. 1 shows the scope o the present work within the perormance riven esign approach (PDDA [11]. The remainer o this paper is organize as ollows. A theoretical single iber eboning an pullout moel is irst presente with three most important, physically meaningul parameters: (1 Chemical bon strength quantiie by interacial racture toughness, ( constant rictional bon strength or small sliing an (3 slipharening coeicient that characterizes the increasing eective rictional bon uring large sliing (pullout stage [7, 1]. Then, a composite briging stress versus crack opening ( B - δ relation is erive in close-orm base on the single iber pullout moel an ranom iber istribution assumption. As a unamental material Fig. 1 Scope o the present research within the Perormance Driven Design Approach. ISSN en cours/99 RILEM Publications S.A.R.L. 173

2 Concrete Science an Engineering, Vol. 1, September 1999 property, the B - δ relation has been well recognize as a crucial link between composite constituents an overall composite properties. Most irect an important implications o this moel incluing tensile strength an uctility are presente in the ollowing section along with an example o PVA iber-cement composite. Conclusions rawn rom the current research are mae in the en.. INTERFACE CHARACTERIZATION A commonly-use technique to investigate ibermatrix interacial behaviors is single iber pullout. Fig. shows schematically the set-up or a typical single iber pullout test. The specimens are abricate accoring to the technique or microibers escribe in [13] in orer to ensure accurate alignment. The tensile loa on the iber is measure with a miniature loa cell o 1-Newton capacity. Fig. 3 is a typical pullout curve or a PVA iber o 14.8 µm iameter. The iber embement length L e is chosen in such a way that iber rupture is avoie uring the whole process an both eboning an rictional pullout behaviors can be completely capture. As can be seen rom Fig. 3, there are three stages associate with the loa-isplacement curve: initial elastic stretching o the iber ree length (the portion not embee, ollowe by eboning stage, which is simulate in the present moel by a moe-ii tunneling crack avance with non-zero crack-tip racture toughness. The eboning stage continues until reaching the maximum loa an a istinct loa rop occurs. This loa rop is an inication o chemical boning because it woul not appear i the interace is rictionally bone only. Physically, the loa rop represents the transition rom both chemical bon an rictional bon controlle eboning stage to the pullout stage with rictional bon only. Ater complete eboning o the interace, chemical bon oes not exist but rictional bon coul eectively increase ue to abrillation o iber surace sliing against surrouning matrix. A concave upwar portion o the curve inicates this so-calle slip-harening behavior, which has been investigate elsewhere [7, 1]. A theoretical single iber eboning an pullout moel is erive, base on simple stress analysis an energy balance principle (see Appenix -A. The main assumptions mae in this moel are: (1 Fibers are o high aspect ratio (> 1 so that the en eect on the total eboning loa is negligible. This assumption is generally satisie or most available ibers, an it greatly simpliies the analysis without losing accuracy. Fig. Schematic o a single iber pullout test set-up. Fig. 3 Single iber pullout curve o a PVA iber: test result vs. moel preiction ( During the eboning stage, the slip-epenent eect is negligible since relative slippage between the iber an the matrix in the ebone portion is small. Hence, the rictional stress within the ebone zone remains at a constant τ. (3 Poisson s eect is negligible. For lexible iber/cement systems, Poisson s eect is usually iminishe ue to inevitable slight misalignment an surace roughness o the iber (no alignment at all in a ranom, short iber composite [1]. (4 Elastic stretch o the iber ater complete eboning is negligible, compare with slip magnitue. Following the etaile erivation in Appenix A, we have the theoretical pullout loa P versus pullout isplacement δ (P-δ relation: 3 3 πτe 1 η π GE P = δ, δ δ or the eboning stage, an: ( ( < < P = π τ 1 β δ δ / L δ δ, δ δ L e e (1 ( or the pullout stage, where L e is the iber embement length an δ correspons to the isplacement at which ull-eboning is complete. It is given by: 174

3 Lin, Kana, Li τ Le 1 η 8GL e 1 η δ (3 = E E where η = V E /V m E m, V an V m are the volume ractions o iber an matrix respectively. In terms o iber eboning length L, the eboning loa P can be also expresse as: 3 / P = π τ L π G E (4 At ull-eboning, L = L e ; hence, the maximum eboning loa is given by: Pa = Pb π GE 3 / (5 where P b = π τ L e is the initial rictional pullout loa. Equation (5 can be conveniently use to calibrate the chemical bon strength G an rictional bon strength τ rom the maximum eboning loa P a an the initial rictional pullout loa P b. The slip-harening coeicient β is obtaine rom ( by best-itting the rictional pullout portion o the P-δ curve. The ashe curve in Fig. 3 is the moel result with three interacial parameters: G = 6. J/m, τ = 3. MPa an β =.5. To urther veriy the moel, the maximum eboning loa P a as a unction o iber embement length L e or the same PVA iber is preicte using (5 with the same set o parameters. As can be seen rom Fig. 4, goo agreement between moel preiction an experimental result is oun. The simple P-δ relations in equations (1 to (3 are the builing blocks in constructing the composite briging law. It is also important to stuy the iber alignment eect on the pullout loa an iber in-situ strength because in an actual short iber composite ibers are ranomly oriente. The eect o iber alignment on the pullout loa, so-calle snubbing eect, was investigate in [14, 15]. An empirical relation is given by: = P P e φ φ (6 where is the snubbing coeicient. Kana an Li [16] investigate the eect o iber alignment on iber in-situ strength. Base on the experimental results or PVA ibers, a similar exponential expression was oun to characterize this eect well: = φ u φ u e (7 where u is the in-situ iber strength an is the iber strength reuction coeicient. Fig. 5 shows such a iber strength reuction eect on PVA ibers. The parameters introuce in this section, such as the slip-harening coeicient β, snubbing eect coeicient, an iber strength reuction coeicient, are empirical, curve-itting parameters. They are etermine by single Fig. 4 Maximum iber eboning stress as a unction o iber embement length or PVA (RMU iber: experiment vs. moel preiction. Fig. 5 Eect o iber inclination on in-situ iber strength o PVA (RMU ibers (ater Kana an Li [16]. iber pullout (straight or incline tests as iscusse herein an in [7, 15 an 16]. The range o β or polymeric ibers such as Polypropylene, Polyethylene an PVA ibers is between.5 to.5. Snubbing coeicient ranges rom.5 to.9. The iber strength reuction coeicient is about.3 or the PVA ibers, an it has not been teste or other iber types. 3. COMPOSITE BRIDGING STRESS- CRACK OPENING RELATION Crack briging law is a unamental material property o iber reinorce composites. It is cast in terms o the crack briging stress vs. the crack opening relation ( B -δ relation. Base on the P-δ relation o a single iber pullout, the composite B -δ curve can be obtaine by averaging over the contributions o only those iniviual ibers that 175

4 Concrete Science an Engineering, Vol. 1, September 1999 cross the matrix crack plane as the crack opens up [17]: B 4V π / L / cosφ φ ( δ = P( δ e p( φ p( z zφ π φ= z = (8 where p(φ an p(z are the probability ensity unctions o the orientation angle φ an the centroial istance o a iber rom the crack plane z, respectively. The relation between ecreasing iber iameter an increasing number o ibers has been consiere in the above equation. For a 3-D ranom istribution, p(φ = sinφ an p(z = /L. It shoul be pointe out that in the crack briging case, the crack opening δ results rom two-sie iber eboning an one-sie pullout (shorter embement sie. Thereore, P(δ in equation (8 shoul be moiie rom the one-sie eboning result (equation (1 as ollows: 3 3 P = π Eτ δ 1 η / 4 π EG / (9 By changing variable, equation (8 can be re-cast into the ollowing orm[3]: V π / 1 φ B( δ = P( δφ e ( φ φ (1 π φ=,, l sin l l = where is normalize iniviual iber embement length (= /(L /. In orer to consier the inluence o iber rupture, when carrying out the integration (1, we nee to keep iscounting the contribution o broken ibers as the crack opening δ increases. To illustrate, a typical integration omain at an arbitrary crack opening δ is plotte in Fig. 6. There are several important non-imensional variables associate with this integral omain, whose physical meanings are explaine in the ollowing: Critical iber embement length (normalize by L / or iniviual ibers: ( φ l c = L c e / (11 u where L c = is the normalize critical embement τl length o an aligne iber (φ = in absence o chemical τl bon [3] an = ( τ δ 8G /, with δ = is E ( 1 η a relative measure o chemical bon strength to the rictional bon strength. Equation (11 is obtaine rom equations (4, (6 an (7 with P set to be the critical value P = π u (φ / 4. It eines the potential iber rupture subomain: ibers that locate above this curve will eventually rupture as a matrix crack opens up. From (11, we can eine the critical iber embement length or aligne iber (φ = with both chemical bon an rictional bon: L c = Lc / (1 Fibers at all orientation angles have the potential to rupture i its hal-length L / > L c (i.e. Lc < 1. Clearly, chemical bon tens to promote iber rupture. Also, by setting φ = π/ in (11, another limiting iber length can be eine: ( π / Lr = Lce (13 I L r > L (i.e. L r >, no iber rupture will occur regarless o orientation angle. Also in Fig. 6, the variable (δ = c (φ = φ c eines the critical embement length at which eboning o the iber has complete at crack opening δ an just survive rupture. For iber length between L r an L c, some ibers will rupture, epening on their orientation angles. The limiting iber rupture angle φ b : φ b = 1 ln 1 L L c c, φb π/ (14 eines the potential iber rupture space in terms o orientation angle: only those ibers orient at an angle higher than φ b will eventually rupture. Current iber rupture angle φ c (δ: Fig. 6 Integration omain at an arbitrary crack opening δ or normal chemical bon strength case: L c > 1. φ δ δ c = 1 ln (15 ( L c δ 4 L c It eines the iber orientation angle above which ibers must have rupture at a given crack opening (the riving orce or iber rupture. This quantity is boune between φ b an π/. As crack opening δ increases, φ c starts rom π/ an approaches φ b. Note that within the potential iber rupture space, whether a speciic iber will rupture or not still epens on its embement length. 176

5 Lin, Kana, Li Equation (15 is generally not applicable to uctile metal ibers ue to the elasticity assumption in its erivation. For uctile ibers, i not too ar beyon yieling is expecte, (15 coul be a goo estimation. Base on Fig. 6, it is not iicult to evaluate the lethan sie o (1 analytically by taking only the shae omains marke as eboning an pullout ibers in the igure. Fibers that have broken or have been pulle-out (embement length less that crack opening are eliminate. Close-orm expressions or the crack briging are inclue in Appenix B, which orm the theoretical basis or iscussions in the ollowing sections. For more etaile erivation, one may reer to the previous works [3] an [16]. It is worth consiering a special case o extremely strong chemical boning ( >>1. In this case, part or all o ibers will rupture either uring the eboning process or even beore eboning (recall that there is a threshol or initiation o eboning governe by chemical bon strength (re. equation (1. Fig. 7 shows such a case. Fibers belonging to φ r < φ < π/ an < < 1 will rupture uring the eboning initiation process, where φ r is given by: Fig. 7 Integration omain at an arbitrary crack opening δ or extreme chemical bon strength case: L c < 1, L r <. φr = 1 ln (16 L c The maximum briging stress is attaine at the initial stage (δ =, an it can be estimate by summing the contributions rom eboning an non-eboning (rupture beore eboning ibers: V φr 1 φ B = e sinφl φ (17 V π / 1 φ φ φ φ ue sin l r where = an = 1 / Vτ ( L /. It results in: V Fig. 8 Pre-ully-eboning briging stress - crack opening curves or a system with low tenency o iber rupture. ( r π φ V / r u e e sinφr cosφ B = 4 φr e ( sinφr cosφr 4 (18 Figs. 8 to 1 show graphically the non-imensional crack briging stress - crack opening curves erive analytically. Figs. 8 an 9 are the pre-ully-eboning curves or iber/matrix systems with low tenency o iber rupture (L c =. an high tenency o iber rupture (L c =.5. In both Figs. 8 an 9, the parameter / L c (rom. to.9 inicates the relative chemical bon strength, Fig. 9 Pre-ully-eboning briging stress - crack opening curves or a system with high tenency o iber rupture. given other iber an interace characteristics such as iber length, iameter, strength an rictional bon strength. It can be seen rom these two igures, chemical 177

6 Concrete Science an Engineering, Vol. 1, September 1999 tenency is low, chemical bon increases the maximum briging stress much more ramatically than the case with high iber rupture tenency (L c =.5. Also, at the en o eboning stage, because o less severe iber rupture, the briging stress remains higher at a larger corresponing crack opening in L c =. case. This also contributes to larger post-eboning racture energy - area uner the escening part o the B -δ curve (see Figs. 1 an 11. In the case o slip-harening (Fig. 1, both briging stress an racture energy can be urther increase, epening on the slip-harening coeicient. Fig. 1 Post-ully-eboning briging stress - crack opening curves or a system with low tenency o iber rupture (continuation o the curves in Fig MODEL IMPLICATIONS ON MATERIALS DESIGN FOR DESIRED PROPERTIES With the analytical tools evelope, it is convenient to preict the composite overall properties rom its constituents - iber, matrix an interace. In this section, some irect implications rom the current micromechanics moel are presente, incluing tensile strength, racture energy an complementary energy (a measure o material uctility. Tensile Strength, Fracture Energy, Ductility Fig. 11 Post-ully-eboning briging stress - crack opening curves or a system with high tenency o iber rupture (continuation o the curves in Fig. 9. The tensile strength preicte by the moel is essentially the maximum briging stress or a iber/cement composite which shows multiple cracking (re. [18]. Fig. 13 shows the epenency o tensile strength on iber length in a nonimensional orm. Given iber volume raction V, iameter, iber strength u an rictional bon strength τ, the composite tensile strength cu monotonically increases with iber length in zero chemical boning case (the lowest curve in Fig. 13. This is consistent with the result erive by Fig. 1 Eect o slip-harening interace on the post-ullyeboning B -δ relation. bon tens to increase initial briging stress an reuce the complementary area to the let o the B -δ curve up to the maximum briging stress point (this area is reerre as the complementary energy [4]. When iber rupture Fig. 13 Normalize composite tensile strength as a unction o normalize iber length an relative chemical bon strength. 178

7 Lin, Kana, Li Fig. 14 Normalize composite racture energy as a unction o normalize iber length an relative chemical bon strength. Maalej et al. [3]. As chemical bon increases, tensile strength or short iber length can be ramatically increase. The plateaus at small iber lengths inicate that composite tensile strength is controlle by the initial eboning stress which is in turn governe by chemical bon. As iber becomes longer, subsequent eboning ater initial eboning results in higher briging stress an composite tensile strength. From this igure, it is clear that when iber is too long or chemical bon is too strong, composite tensile strength becomes almost inepenent o iber length. Physically, those two limiting cases lea to severe iber rupture an composite tensile strength shoul be mainly controlle by iber volume raction an iber strength. Shown in Fig. 14 is composite racture energy as a unction o iber length an the relative chemical bon strength. This racture energy is the energy absorbe by briging ibers in the crack plane an it is evaluate by integrating the area uner the post-ully-eboning portion o a briging stress - crack opening curve (Figs The peak shown in each curve in Fig. 14 inicates that there exists an optimal iber length to maximize the racture energy o a iber composite. It is also clear that chemical bon tens to reuce racture energy because it promotes more iber rupture. The concept o multiple cracking in iber reinorce brittle matrix composites has been extensively stuie. Marshall an Cox [18] propose a simple means o etermining the conition or steay state cracking, require or multiple cracking. Base on a J-integral analysis, the conition can be written in terms o the complementary energy G c an the crack tip toughness J : Gc J (19 where G c is eine in terms o iber briging property via the B -δ curve: G δ c cu cu B = cu δ δ δ ( an cu an δ cu are the maximum briging stress an the corresponing crack opening. Graphically, G c is simply the complementary area to the let o the B -δ curve up to peak stress. The above iscussion suggests that high uctility o a short iber composite requires large complementary energy. Equation (19 becomes a very important esign criterion or uctile iber composites. Using the generic micromechanics moel evelope in this research, one can easily compute this quantity with ierent combinations o iber, matrix an interace properties an choose the one that satisies conition (19 to guarantee the uctility o such engineere iber composites. Shown in Fig. 15 are the normalize complementary energy as a unction o iber length an relative chemical bon strength. Fig. 15 Normalize composite complementary energy as a unction o normalize iber length an relative chemical bon strength. An Example - PVA iber/cement composites Recently, Polyvinyl Alcohol (PVA iber has been recognize or its high potential as reinorcement in high perormance cementitious composites ue to its high strength. This type o iber can also orm strong chemical bon to the cement matrix. In orer to maximize the potential o PVA ibers as viable reinorcement or cement base composites which have both high strength an high uctility, analytical material esign tools are certainly neee. To emonstrate the useulness o the present micromechanics moel, two types o cement composites mae rom two types o PVA ibers (RMU an RK with 14 an 4 µm iameter respectively are investigate. Table 1 lists the iber an interace parameters or both PVA ibers. The interacial properties were erive rom the simple iber pullout tests iscusse previously. PVA-RMU composite contains 1.5% volume raction o RMU ibers while PVA-RK composite has % RK ibers. 179

8 Concrete Science an Engineering, Vol. 1, September 1999 Table 1 Fiber an interacial parameters or PVA-RMU an RK ibers Fiber E L τ G u, Type (GPa (mm (µm (MPa (Nm (MPa RMU ,.3 RK ,.3 Table Matrix mix proportion Cement San Water Super Viscous Plasticizer Agent The matrix mix proportion or the two composites are liste in Table. They were teste ater seven ays water curing an one ay air rying. Stanar og-bone shape specimens were use. Fig. 16 shows the uniaxial tension test results or both composites. As can be seen rom this igure, PVA-RMU has higher tensile strength but much lower uctility than the PVA-RK composite. This is because RMU iber is much thinner in iameter. Although it is a higher iber strength than the RK iber, uner similar interacial conitions, RMU ibers suere more severe iber rupture, which leas to lower uctility. The contrast in uctility between these two composites can be explaine more clearly using the complementary concept later on. Fig. 17 shows the comparison between moel preictions an tensile test results in terms o tensile strength. The moel preicts higher tensile strength or PVA-RMU than or PVA-RK, consistent with experimental observations. Note however, that the PVA-RMU oes not satisy the multiple cracking conition (equation (19 (see iscussion to ollow an the measure tensile strength cannot be expecte to be the same as the calculate maximum briging stress. Also, the current tensile strengths or both composites are very close to their limit values an reuction in iber length will not cause much loss in tensile strength. From the racture energy preictions (Fig. 18, the current iber lengths (1 mm or RK, 6 mm or RMU are well over the optimal values or maximum racture energy. The ierence in tensile strength as L approaches zero is ue to the iber strength ierence. Fig. 19 illustrates the complementary energy as a unction o iber length or both composites. Uner current composite esign, PVA-RK has a G c value o 3.4 J/m while PVA-RMU has only.8 J/m. Compare to the racture toughness o cement paste, which is about J/m, the PVA-RK composite satisies the multiple cracking conition (19 an this is the reason behin the contrast in tensile uctility o the two composites. Fig. 16 Uniaxial tensile test results or PVA-RMU an PVA-RK composites. Fig. 17 Composite tensile strength as a unction o iber length or both PVA-RMU an RK composites: experiments vs moel preictions. Fig. 18 Preicte composite racture energy as a unction o iber length or both PVA-RMU an RK composites. Fig. shows the eect o chemical bon strength G on both racture energy an tensile strength o the PVA- RMU composite. Clearly, chemical bon ecreases the racture energy ramatically while increases the tensile 18

9 Lin, Kana, Li Fig. 19 Preicte composite complementary energy as a unction o iber length or both PVA-RMU an RK composites. (3 Via parametric stuy presente in this paper, it is oun that in iber rupture cases, interacial racture toughness (chemical bon ten to increase composite tensile strength but ecreases composite racture energy an complementary energy as well as uctility. Single iber pullout problem is a complicate soli mechanics problem. The erivation inclue in the present stuy is only an approximate solution to this complicate problem, base on simpliie orce an energy balance. The attempt has been mae to capture the physical essence o the interacial parameters in a simple, easy-to-use ormulation without getting into complex mathematics. In the current micromechanics moel, the iber ispersion eect an the statistical istribution o iber strength are not inclue. The comparison between the two iber/cement composites PVA-RMU an PVA-RK is meant to emonstrate the useulness o the moel in iber section or esire properties (say, uctility rather than just a comparision, which has been the central theme o this work. ACKNOWLEDGMENTS Support o this work has been provie by the National Science Founation (CMS-9616 to the University o Michigan. Fig. Preicte composite racture energy an tensile strength as a unction o chemical bon strength or the PVA-RMU composite. strength to the limit set by iber strength an volume raction. The kink in the tensile strength curve is ue to the transition rom normal chemical boning to extreme chemical boning case as iscusse previously (Re. equation (18. It is important to tailor the interacial properties incluing chemical bon to achieve both high strength an high uctility. 4. CONCLUSIONS AND DISCUSSIONS (1 Several important interacial behaviors characterize: chemical bon, rictional bon, an slip-harening. These interacial parameters can be etermine rom a single iber pullout test. ( An analytical briging stress-crack opening relation is erive, which can be use as a materials esign tool or esire properties an cost-eectiveness via intelligent iber selection (type, strength, length, content,, matrix moiication an iber/matrix interace tailoring. REFERENCES [1]Krenchel, H. an Hansen, S., New recipes an new prouction techniques or high perormance FRC-materials, in High Perormance Fiber Reinorce Cement Composites, Reinhart H. W., Naaman, A. E., Es., E& FN Spon, Lonon, 199, pp [] Naaman, A. E., SIFCON: Tailore properties or structural perormance, in High Perormance Fiber Reinorce Cement Composites, Reinhart H. W, Naaman, A. E., Es., E& FN Spon, Lonon, 199; pp [3] Maalej M., Li, V. C. an Hashia, T., Design an Structural Applications o Stress-Crack With Relations in Fiber Reinorce Concrete, J. o Engineering Mechanics 8 ( [4] Li, V. C., Kana, T. an Lin, Z., The Inluence o Fiber/Matrix Interace Properties on Complementary Energy an Composite Damage Tolerance, in Key Engineering Materials: Proc. 3r Con. on Frac. & Strength o Solis, Hong Kong, 1997, pp [5] Li, V. C. an Leung, C. K. Y., Steay state an multiple cracking o short ranom iber composites, ASCE J. Engng Mech. 118 ( [6] Li, V. C. an Wu, H. C., Conitions or pseuo strain-harening in Fiber Reinorce Brittle Matrix Composites, J. Appl. Mech. Rev. 45 (8 ( [7] Lin, Z. an Li, V. C., Crack briging in iber reinorce cementitious composites with slip-harening interaces, J. 181

10 Concrete Science an Engineering, Vol. 1, September 1999 Mech. Phys. Solis 45 (5 ( [8] Li, V. C. an Kana, T., Engineere cementitious composites or structural applications, ASCE J. Mater. Civil Eng. 1 ( ( [1] Kabele, P., Li, V. C., Horii, H., Kana, T. an Takeuchi, S., Use o BMC or Ductile Structural Members, in Proc. o 5th Int. Sym. on Brittle Matrix Composites, Warsaw, Polan, 1997; pp [1] Kana, T., Watanabe, S. an Li, V. C., Application o Pseuo Strain Harening Cementitious Composites to Shear Resistant Structural Elements, in Proc. Fracture Mechanics o Concrete Structures-3, AEDIFICATIO Publishers, Freiburg, Germany, Oct., 1998, pp [11] Li, V. C., From micromechanics to structural engineering - the esign o cementitious composites or civil engineering applications, JSCE J. Struct. Mech. Earthquake Eng. 1 ( ( [1] Wang, Y., Li, V. C. an Backer, S., Moeling o iber pull-out rom a cement matrix, Int l. J. Cement Comp. Lightweight Concrete 1 (3 ( [13] Katz, A., Li, V. C., A Special technique or etermining the bon strength o carbon ibers in cement matrix by pullout test, J. Materials Science Letters 15 ( [14] Morton, J. an Groves, G. W., The eect o metal wires on the racture o a brittle matrix composite, J. Mater. Sci. 11 ( [15] Li, V. C., Post-crack scaling relations or iber reinorce cementitious composites, ASCE J. Mater. Civil Engng. 4 (1 ( [16] Kana, T. an Li, V. C., Interace property an apparent strength o a high strength hyrophilic iber in cement matrix, ASCE J. Mater. Civil Engng. 1 (1 ( [17] Li, V. C., Wang, Y. an Backer, S., A micromechanical moel o tension-sotening an briging toughening o short ranom iber reinorce brittle matrix composites, J. Mech. Phys. Solis 39 (5 ( [18] Marshall, D. B. an Cox, B. N., A J-integral metho or calculating steay-state matrix cracking stress in composites, Mech. Mater. 7 ( [19] Gao, Y. C., Mai, Y.-W. an Cotterell, B., Fracture o iberreinorce materials, J. Appl. Math. Phys. 39 ( Appenix A APPENDIX Derivation o P-δ relation with combine chemical bon an rictional bon Consier a iber with length L e embee in a matrix (Fig. A1. A portion o the iber/matrix interace is ebone. The ebone zone length is L, within which a constant rictional stress τ is assume to exist. Chemical bon strength or the bone region is G. The task here is to erive the relation between iber pullout loa P or stress an iber pullout isplacement δ. From equilibrium requirements, the one-imensional Fig. A1. stress istribution in the ebone region is given by: z ( z = L (A-1 m z 1 z/l m (A- where an m are the normel stresses in the iber an the matrix respectively at z =, an they are given by: 4τL = (A-3 m (A-4 where V an V m are the volume ractions o iber an matrix respectively. Deine the relative isplacement between iber an matrix in the ebone region as: = m z u z u z Then, = ( 4τLV = V m = = (A-5 z u z um z z m z (A-6 z z z E Em Substituting (A-1 through (A-4 an applying the bounary conition (z = =, we have: ( z z Lz z = 4τ τ 1 η 1 η E (A-7 E E where η= VE. The iber pullout istance is then VmEm given by δ τ η = ( L = L L 1 (A-8 E E Now, we nee to in = (L, G, τ,. An energy base eboning criterion is employe in establishing such a relation [19]. For any ininitesimal avance o the ebone zone, A, the energy conservation requires 18

11 Lin, Kana, Li Pu = U W GA (A-9 It ollows that where u is the axial isplacement at the pulle iber en (work conjugate o applie loa P = π τ η η / 4, U is the = L ( GE ( change in strain energy in the system, W is the change in (A-16 energy issipate by riction at the interace, an G A is or: the energy consume in the avance o ebone zone. On 3 the other han, i the ebone zone is consiere as a π GE ( 1 η P= π/ 4= πτl( 1 η stress bounary o the system, it can be shown that [19]: (A-17 U = 1 In a single iber pullout specimen, η can be neglecte ( Pu W (A-1 without losing accuracy. Thereore, equation (5 in the From (A-9 an (A-1, we have: main text results rom (A-17. Also, when L = L e, maximum eboning loa is reache. The corresponing is- GA = 1 ( Pu W (A-11 placement is given by (A-16 an (A-8: where u an W can be obtaine rom: τl e( 1 η 8GL e 1 η L ( z u (A-1 E z V = δ = ( (A-18 ( E L e L E E c Combining (A-17 an (A-8, we have: an 3 3 πτ L E W = π ( 1 η π GE ( z z (A-13 P = δ (A-19 In equation (A-1, I η =, (A-19 reuces to equation ( in the main Ec = VE VmEm (A-14 text. By using relevant equations alreay erive, (A-11 becomes: Ater eboning, the interace is controlle by rictional bon only. By taking S = δ-δ in equation (1 in the τ η τ η η main text an rom orce balance, it is reaily shown that 8 L( 1 16 L GE( = (A-15 or iber pullout stage, P= π τ 1 β δ δ / L δ δ (A- ( ( e Appenix B Analytical crack briging relation For L r < L < L c B δ δ φ π g 41 ( ( 1 u = G φ ( c, 41 ( ( 1 Lc φc δ δ A(, L c A φ φ δ φ π c, A ( 1 ( c, u = 4 ( φ, ( L ( φ, δ G b 1 c A b Lc A( φb, δ Lr A φ ( b, ( 1 c δ δ δ G( b, ( 1 L c B( b, a, L c φ δ φ φ δ B( φb, φa, δ B φ ( b, φa, L ( 1 cδ r 1 183

12 Concrete Science an Engineering, Vol. 1, September 1999 B where For L > L c δ δ φ π g 41 ( ( 1 u = G φ ( c, 41 ( ( 1 LcA φc δ δ (, L ca φ φ δ φ π ( c, 1 A c, u = u φ = 4 π L G L δ G π c, c, G π, ( 1 c ( = L r δ δu φ δ L cg( a, L φ c δ G( φa, G δ φa, ( 1 c Lr L c 1 βl Vτ L /, c = = τ L = 8G / τδ, δ, δ = δ 1, δ / L / E 1 η L u c =, = Lc/ ( L / = Lc Lc, L Lr / L / L ce τ L r φ φ φ u c b a = Lc e ( 1 ( φ δ φ L c = 1 δ ln ( c Lc c 4L, = / 1 c 1 1 = ln ( L c L c = 1 ln L L = c c = π / = = For 3-D istribution: [ ] = φ G φ, 1 e sinφ cosφ 4 g G π /, = = [ ] φ π A φ, 1 / e cosφ sinφ e 4 φ e sin φ cos φ B( φ, φ, = φ1 e sinφ cosφ ( 1 1 For -D istribution: = φ G φ, 4 e cosφ sinφ π 1 g G π /, = = [ ] [ ] π φ A φ, 4 / e e cosφ sinφ π( 1 φ e ( cosφ sinφ B( φ, φ, = 4 1 φ π( 1 1 e ( cosφ1 sinφ 1 184