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An analytical study on the complete strain hardening process of ultra high toughness cementitious composites under direct tension load
Engineering Fracture Mechanics 102 (2013) 249–256
Contents lists available at SciVerse ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
An analytical study on the complete strain hardening process of ultra high toughness cementitious composites under direct tension load Xiang-Rong Cai a,⇑, Bai-Quan Fu b a b
Building Science Research Institute of Liaoning Province, Shenyang 110005, China College of Urban Construction, Shenyang Jianzhu University, Shenyang 110167, China
a r t i c l e
i n f o
Article history: Received 30 May 2012 Received in revised form 3 December 2012 Accepted 10 February 2013
Keywords: Fiber reinforced materials Strain hardening Multiple cracking Analytical models
a b s t r a c t The ideal tensile stress–strain curve of ultra high toughness cementitious composites (UHTCC) shows trilinear stress–strain relationship. And its strain hardening process can be divided into multiple cracking zone and post multiple cracking zone. In this paper, the general expression of the fiber bridging stress laws in the crack plane is presented, and the stress–strain models are established for both zones based on the law. Furthermore, the critical fiber length between fiber pull-out and fiber rupture in the post multiple cracking zone are deduced. And the precondition to guarantee the post multiple cracking zone is deduced. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The brittleness and insufficient durability of concrete have been identified as the bottlenecks which hinder the development of concrete structures. In order to improve the toughness of concrete and to enhance its damage tolerance, deformability capacity and anti-cracking ability under tensile load, intensified research has been conducted on fiber reinforced cementitious composites. As a result, a new family of high performance fiber reinforced cementitious composites named as ultra high toughness cementitious composites (UHTCC) was developed. This class of composites has many excellent performance to solve the brittleness and insufficient durability of concrete with fiber content typically 2% by volume or less. For example, its tensile strain capacity is more than 3%, above 300 times of that of ordinary concrete [1–3]. The compressive strength of UHTCC is in the range of high strength, and the compressive strain capacity is approximately double that of ordinary fiber reinforced concrete [4–6]. The shear strength is about three times higher than that of plain concrete and its peak shear strain is over 20 times the failure strain of plain concrete [7]. Its magnitude of fracture energy is the highest ever reported for a fiber reinforced cementitious composite. The area of the damage zone has been observed to be on the order of 1000 cm2 in double cantilever beam specimens with dimensions of 490 585 35 mm, and the energy absorption of the off-crack plane inelastic deformation process has been measured to be more than 50% of the total fracture energy of up to 34 kJ/m2 [8] The crack width under peak load can be limited to less than 0.1 mm, even 0.06 mm [1,2]. The anti-cracking performance of UHTCC is obviously superior to concrete. The crack width of UHTCC under restrained drying shrinkage is about 0.03–0.05 mm at 50% relative humidity [9]. Furthermore, a study on the bending behavior of functionally graded composite beam crack-controlled by UHTCC demonstrated that, in addition to improving bearing capacity, the corrosion-induced ⇑ Corresponding author. Tel.: +86 024 233 88143. E-mail address: [email protected] (X.-R. Cai). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.02.006
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Nomenclature a df Ec E Em f g Gc Gd K Lc Le Lf P p p(/) p(z) q Vf x z
emc emu d d0 /
rfu rmc rmu s s0 ss g c
crack spacing fiber diameter elastic modulus of composite elastic modulus of fiber elastic modulus of matrix snubbing coefficient snubbing factor shear modulus of composite chemical debonding energy slip-hardening coefficient critical fiber length the embedded length of fibers fiber length the force loaded at the end of a fiber increment of crack width probability density function of the orientation angle probability density function of the centroidal distance z the number of separated pieces of matrix counted from the fiber center fiber volume fraction fiber end slippage distance of centroid of fiber from crack plane strain corresponding to the first cracking strength of matrix strain corresponding to the stress at the end of the multiple cracking process the displacement of the fiber-loaded end the corresponding displacement at the maximum stress orientation of fiber with respect to tensile loading direction ultimate tensile strength of fiber first cracking strength of matrix stress at the end of the multiple cracking process frictional sliding shear stress frictional sliding shear stress at the tip of debonding zone where no slip occurs interfacial chemical bonding coefficient of effective fiber length strain increment caused by the fiber slipping in the post multiple cracking process
damage can be effectively prevented [10]. Therefore, it is potential to solve the security risk and insufficient durability inherent with today’s concrete structures by using this class of composites. One of the most distinct features of UHTCC is the inelastic tensile behavior. Based on the shape of the stress–strain curve, UHTCC can be divided into two categories, one with bilinear tensile stress–strain curve and the other one with trilinear tensile stress–strain curve. For the one with bilinear stress–strain curve, the stress–strain curve can be divided into linear elastic zone and multiple cracking zone. And for the other one with trilinear stress–strain curve, the stress–strain curve can be divided into linear elastic zone, multiple cracking zone and post multiple cracking zone. The excellent mechanical performance of UHTCC is due to the tensile strain hardening process which is strongly related to the multiple cracking zone and post multiple cracking zone. Compared with the composites with bilinear tensile stress–strain curves, those composites with trilinear tensile stress–strain curves possess higher tensile strength, tensile strain and toughness, and can be regarded as ideal fiber reinforced cementitious composites [11]. However, it is more difficult to achieve post multiple cracking behaviors in random discontinuous fiber reinforced composites compared to continuous aligned ones. But, it is by no means impossible. The challenge is to determine the proper conditions under which post multiple cracking will occur, despite the reduced efficiency of stress transfer in the random and discontinuous bridging fibers and the separate matrix after multiple cracking process. As will be clear from the discussion to follow, the precondition for post multiple cracking is proposed, and the post multiple cracking models for the cases of fiber length less than critical length and more than critical length are established respectively. 2. Typical tensile stress–strain curves of UHTCC The typical tensile stress–strain curves obtained from uniaxial tension tests are given in Fig. 1. After the first tensile crack formed in the matrix, the bridging fibers across the crack will control the extension of crack and bear the load released by the cracked matrix. If the fibers have sufficient strength to sustain the additional load, the composite will continue to bear load
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6
8
(a)
6
stress
stress
4 multiple cracking zone
2
0
(b)
0
1
2
3
4
strain
4 multiple cracking zone post multiple cracking zone
2 0
0
1
2
strain
3
4
5
Fig. 1. Typical tensile stress–strain curves of UHTCC: (a) bilinear model and (b) trilinear model.
without failure. And the fibers will transfer the load back to the adjacent uncracked matrix through interfacial shear. When the transferred load reaches the matrix cracking strength, another crack will form in the matrix. This process then repeats itself until a set of periodic subparallel cracks are formed. This process is corresponding to multiple cracking zone in the tensile stress strain curve. In the multiple cracking zone, the stress keeps constant or increases slightly with the strain increasing, and the stress–strain curve is almost linear as shown in Fig. 1. When the crack spacing is so small that the transferred load cannot induce new cracks in the matrix, the multiple cracking process ended. After this, the composite will come into the failure phase if the bridging fibers are not able to bear additional load, and the tensile stress strain curve presents clearly bilinear tensile model as shown in Fig. 1a. Otherwise, the composite will come into the post multiple cracking phase if the bridging fibers are able to bear additional load. In this phase, the cement matrix lost its load-carrying capacity, and all the force is applied to the fibers. With a further increase in deformation, the bridging fibers will be pulled out gradually form the separated matrix. Because of the slip hardening effect, the composite stress increases with the strain increasing until the ultimate tensile strength is reached. The tensile stress–strain curve of this type of composite presents clearly trilinear tensile model as shown in Fig. 1b. 3. The strain hardening analysis in the multiple cracking zone The fiber bridging law is one of the main control parameters for establishing the stress–strain relationship of UHTCC. In this paper, a general expression of the fiber bridging stress laws in the crack plane was presented by referring to the existing laws mentioned above and considering the fiber/matrix debonding criteria of both strength and energy criterion respectively. For a composite with fiber volume fraction Vf, the composite stress can be obtained by integrating over the contributions of those individual fibers that cross the matrix crack plane
rB ¼
4gV f
pd2f
Z
p 2
/¼0
Z
L
ð 2f Þ cos /
PðdÞef / pð/ÞpðzÞdzd/
ð1Þ
z¼0
where P(d) is the relationship between the force P loaded at the end of a fiber and the displacement d of the fiber-loaded end, p(/) is the probability density function of the orientation angle / and p(z) is the probability density function of the centroidal distance z of fibers from the crack plane. For a uniform 3-D random distributions, p(z) = 2/Lf and p(/) = sin /. g is the coefficient of effective fiber length. In this paper, a brief introduction on the tensile stress–strain relationship during the multiple cracking processes is presented. The detailed derivation and discussion of the strain hardening analysis in multiple cracking zone sees our previous research achievements in the literature of Cai et al. [12]. For strength debonding criterion, the bridging stress-crack opening relation can be presented as
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cdf 2 ~d ~d þ ~d ~d qLf
r~ B ð~dÞ ¼ 2
ð2Þ
L 2 pf =2 Þ. ~ B ¼ rB =r0 ; ~ where C ¼ sssi ; r d ¼ d=ðLf =2Þ; ~ d ¼ s0 Lf =Ef df ; r0 ¼ 12 gg s0 V f ðdf Þ; g ¼ 4þf 2 ð1 þ e f For energy debonding criterion, the bridging stress-crack opening relation can be presented as
r~ B ð~dÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ Þ ~d 4ð~d þ a ~d d~
~ ¼ 2Gd =s0 Lf where a Comparing Eq. (2) with Eq. (3), the general expression of the fiber bridging law can be presented as
ð3Þ
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r~ B ð~dÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~d ~d n2 þ 4 ~d ~d
2
~d 6 ~d 1 n 4
! ð4Þ
qffiffiffiffi 2 Cd For strength debonding criterion, n ¼ 2 qL f ; and for energy debonding criterion, n ¼ 2 ~da~ . When ~ d¼~ d ð1 n4 Þ; the maximum f 2 n2 of fiber bridging stress can be reached: rB;max ¼ r0 ð1 þ 4 Þ: Generally, n 1, thus rBmax r0 . To see whether debonding is governed by strength or energy criterion for a certain composite system, one can carry out pull-out tests with different fiber radii or different fiber volume fractions and obtained the result form the trend of experimental data using the following equation [13]: 1
Ppeak =Ppostpeak ¼ 1 þ ½ðY 2 YÞ1=2 cos h ðYÞ1=2 =ð2qL=df Þ
ð5Þ
where Ppeak and Ppostpeak are the peak load and post peak load at the load–displacement curve obtained from the pull-out test. L is the embedded length of fibers, q is a parameter depending on moduli of fiber and matrix as well as fiber volume fraction in the pull out specimen, the expression of is given above. If Y is a constant with respect to df or q, debonding is governed by strength creterion, if Y varies with df and q, debongding is governed by energy creterion. The second parameter to control the performance of UHTCC is the cracking strength of matrix. In this paper, only the flaw size distribution of matrix is considered to determine the cracking strength of matrix. Referring to the literatures [14,15], the flaw size distribution is supposed to obey Weibull distribution. And the mortar matrix is treated as nonlinear brittle material and cohesive crack model is adopted. Referring to the literature [16], the finite element method is used to calculate the cracking strength of the matrix. When the cracking strength at flaw size of 2ci is equal to the fiber bridging stress, matrix cracking begins. i.e.
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ~d ~d 2 @ r0 c þ 4 ~ ~ A ¼ rmc ðci Þ d d
ð6Þ
Half of the crack width can be obtained as:
dðci Þ ¼ ki d
ð7Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ki ¼ ð1 1 rmcr0ðci Þ þ 14 c2 Þ2 14 c2 The maximum debonding length of single fiber is given by
Li ¼
Lf 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 þ 4ki c
ð8Þ
The new crack occurs at the position with distance of Li away form the existing crack. The tensile strain of the composite here can be calculated by
eðci Þ ¼ e0 þ
Z
cmax
ci
2dðci Þ f ðcÞdc 100% Li
ð9Þ
where e0 is the elastic tensile strain at the first cracking stress, and cmax is the maximum flaw size radium in the matrix. Thus the tensile stress–strain during the multiple cracking processes is given by
(
rðci Þ ¼ rmc ðci Þ eðci Þ ¼ e0 þ
R cmax ci
2dðci Þ f ðcÞdc Li
100%
ð10Þ
The accuracy of the model proposed above was verified in the literature of Cai et al. [12] by experiment results obtained form literature [16]. 4. The strain hardening analysis in the post multiple cracking zone 4.1. The fiber pull-out process analysis In this zone, the matrix has been separated into pieces with length of a by the multiple cracks, and the fibers are pulled out from the separated matrix, as shown in Fig. 3. In order to simplify the analysis, the crack width at the end of the multiple cracking zone is not consider temporarily, that is to say only the increment of the crack width is considered in the analysis of this part. The fiber elastic elongation is neglected when it is in the pull-out process, and only the fiber slippage is considered. Furthermore, the crack width increment caused by the fiber slippage is recorded as p, i.e. p = d dc. Supposing that there n pieces in the fiber length Lf, i.e. Lf = na. When the composite elongates under tensile load, the pieces separate from each other. In this process, the fiber retracts and slips in the matrix by overcoming the friction in the interface. The friction load in one direction is balanced by the load in the opposite direction.
X.-R. Cai, B.-Q. Fu / Engineering Fracture Mechanics 102 (2013) 249–256
253
4.2. The critical fiber length Analyzing a single fiber pulled out from the separated matrix in the direction of / = 0, as shown in Fig. 2. In order to calculate the fiber slippage, the fiber pull-out process is regarded as the movements of matrix pieces relative to fiber, i.e. the matrix pieces move form the fiber center to the fiber end. When the increment of the crack width is p, the moving distance of piece q counted from the fiber center is calculated as
1 x¼p q 2
ð11Þ
The interface shear stress caused by matrix pieces’ slipping in the piece q is calculated as follows:
sq ¼ s0 þ Kp q
1 2
ð12Þ
Thus the stress transferred between fiber and matrix in the piece q is given by:
rq ¼
4a df
s0 þ Kp q
1 2
ð13Þ
Supposing that there are still n0 pieces which contacted with the fiber when the crack width increment is p, as shown in L þnp Fig. 3, the total stress transferred in half of the unit (in the range of f 2 ) is calculated as follows: n0
rmax ¼
q¼ 2 X
rq ¼
q¼1
an0 Kpn0 2s0 þ df 2
ð14Þ
L
f where n0 ¼ aþp Eq. (14) is the maximum fiber stress at the crack width increment p, the calculated result is given as follows:
rmax ¼
a Lf Kp Lf 2s0 þ df a þ p 2 aþp
ð15Þ
When fiber is pulled out from the direction of /, the maximum fiber stress is given as follows by considering the snubbing effect:
rmax;/ ¼
a Lf Kp Lf ef / 2s0 þ df a þ p 2 aþp
ð16Þ dr
max;/ When rmax,/ < rfu, all fibers will be pulled out rather than rupture.It is known that d/ > 0 is always satisfied, thus the maximum fiber stress increases with the angle / increase. Therefore, the fiber ruptures gradually from / ¼ p2 to / ¼ 0. max The peak point of rmax p can be obtained from drdp ¼ 0:
8 > <
L K4s
ppeak ¼ Lf Kþ4s00 a f
ð17Þ
2 > : rmax;peak ¼ ðLf Kþ4s0 Þ 8d K f
When rmax,peak < rfu, all fibers will be pulled out, i.e.
Fig. 2. Sketch map of single fiber pulled out from separated matrix.
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Fig. 3. Stress redistribution at fiber rupture.
ðLf K þ 4s0 Þ2 < rfu 8df K
ð18Þ
The calculated result is given as follows
Lf < Lc ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2df K rfu 4s0 K
ð19Þ
where Lc is the critical fiber length. 4.3. The case of fiber pull-out When Lf 6 Lc , all fibers will be pulled out. The average stress in the piece q counting from the fiber center is given by:
r q ¼ rmax
i¼q X
1 2
ri þ rq ¼
i¼1
2a df
(
s0 ðn0 þ 1Þ þ
" # ) Kp ðn0 Þ2 1 þ qðKp 2s0 Þ Kpq2 2 2
ð20Þ
Thus the average stress in n0 /2 pieces can be obtained: 0
q¼n
2 X2 an0 r n20 ¼ 0 r q ¼ n q¼1 df
s0 þ
Kpðn0 1Þðn0 þ 1Þ 3n0
ð21Þ
This is also the average stress along half of the fiber length counting from the center of the fiber (including the crack opening part between two pieces, seen Fig. 2). Therefore, the average fiber stress is calculated as follows:
r Lf ¼ r n20 ¼
Lf df
1 K c Lf K ca2 ð1 þ cÞ s0 þ 1þc 3Lf 3 1þc
ð22Þ
where c is strain increment caused by the fiber slipping in the post multiple cracking process, and c ¼ np ¼ pa. Lf The average fiber stress along the length of Lf + np, i.e. the average stress of all fibers with the same direction but different embedded length in the crack plane, is given by:
r fur ¼
Lf df ð1 þ cÞ2
s0 þ
K cLf K ca2 ð1 þ cÞ 3Lf 3ð1 þ cÞ
ð23Þ
Thus the composite stress is given by:
r ¼ Vf
Z
p 2
r fur cos/ sin /d/ ¼
0
V f Lf 2df ð1 þ cÞ2
s0 þ
K cLf K ca2 ð1 þ cÞ 3Lf 3ð1 þ cÞ
ð24Þ
When a is very small relative to Lf, Eq. (24) can be simplified as
r¼
V f Lf 2df ð1 þ cÞ2
s0 þ
K cLf 3ð1 þ cÞ
ð25Þ
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255
Considering the snubbing effect g and the coefficient of effective fiber length g, the average fiber stress is given by:
r ¼ r0
1 ð1 þ cÞ2
K cLf 3s0 ð1 þ cÞ
1þ
ð26Þ
4.4. The case of fiber rupture When Lf > Lc, the fibers rupture gradually in the post multiple cracking zone. If rmax,/ = rfu, the critical limit of fiber rupture is given by
n0c ¼
Lf A þ 4s0 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðLf A þ 4s0 Þ2 8Adf rfu ef / 2aA
ð27Þ
When n0 6 n0c , fiber ruptures. Stress redistributed after fiber rupture, as shown in Fig. 3. The fiber stress distribution changes from polygonal line 1 to polygonal line 2. Supposing that the stress rebalances between the piece x and piece x + 1: n0c
x X
2 X
q¼1
q¼xþ1
rq ¼
rq ¼
rfu
ð28Þ
2
The calculation results is given by
x¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4s20 þ df Acrfu 2s0
ð29Þ
2Aac
Thus the average stress of the fiber in n0 /2 pieces can be obtained:
x 2 X 2aq 2a 1 8a 1 1 ¼ 0 ð2s0 þ ApqÞ s0 þ Ap q s0 x2 þ Ap x2 þ x n 1 df df 2 n df 3 6
r n20 ¼ 0 2
ð30Þ
Thus the average stress in n/2 pieces can be obtained:
n0 n
r fr ð/Þ ¼ r n20 ¼
8a ndf
1 2 1 x þ x 3 6
s0 x2 þ Ap
ð31Þ
When rmax,/ > rfu, the angle range of fiber rupture at the crack width increment p is given as follows:
! ! df rfu ð1 þ cÞ2 1 p / ¼ ln ; f 2s0 Lf ð1 þ cÞ þ AcLf 2
ð32Þ
The average bridging stress at the crack plane is given by:
rf ¼
Z
/
r fur ð/Þcos/ sin /d/ þ
Z
0
p 2
r fur ð/Þcos/ sin /d/
ð33Þ
/
Thus the composite stress is given by
r ¼ V f rf
ð34Þ
Considering the snubbing effect g and the coefficient of effective fiber length g, composite stress is given by
r ¼ ggV f rf
ð35Þ
4.5. The critical condition for the existence of post multiple cracking zone In order to guarantee the existence of post multiple cracking zone, i.e. after the multiple cracking phase, the stress increases with the strain increasing, it requires that
dr=dc > 0
ð36Þ
That is to say when the Eq. (36) is satisfied, the strain hardening behavior in the post multiple cracking process can be obtained and the stress–strain curve presents trilinear model. Otherwise, the stress–strain curve presents bilinear model without post multiple cracking zone. For the case of fiber pull-out, the following expression must be satisfied to guarantee the strain hardening behavior in the post multiple cracking zone.
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K > K crit ¼
12rmc df ð1 þ cÞ
ggV f L2f ð1 2cÞ
ð37Þ
i.e. the minimum threshold of the slip hardening coefficient to guarantee the post multiple cracking zone is given by
K 0crit ¼
12rmc df
ggV f L2f
ð38Þ
5. Conclusions Ultra high toughness cementitious composites (UHTCC) possess clear bilinear or trilinear tensile stress–strain curves and exhibit pseudo strain hardening and multiple cracking under tensile load. This paper established the relationship between material parameters and macroscopic tensile strain hardening performance by considering the random distribution properties of fiber orientation and fiber location. For the multiple cracking zone, the effects of micro-parameters on the strain and crack width are also analyzed. For the post multiple cracking zone, the critical fiber length between fiber pull-out and fiber rupture are deduced. The precondition for post multiple cracking zone is proposed. The theoretical models proposed in this paper can be used either for the optimization of composite properties or for the prediction of composite macromechanical performance. References [1] Li HD. Experimental research on ultra high toughness cementitious composites. PhD thesis. Dalian: Dalian University of Technology; 2008 [in Chinese]. [2] Xu SL, Cai XR. Experimental study on mechanical properties of ultra high toughness fiber reinforced cementitious composite. J Hydraul Engng 2009;40(9):1055–63 [in Chinese]. [3] Li HD, Xu SL. Tensile and flexural properties of ultra high toughness cementitious composite. J Wuhan Univ Technol [Mater Sci Ed] 2009;24(4):677–83. [4] Xu SL, Cai XR, Zhang YH. Experimental measurement and analysis on axial compressive stress–strain curve of ultra high toughness cementitious composites. China Civil Engng J 2009;42(11):79–85 [in Chinese]. [5] Xu SL, Cai XR. Experimental study and theoretical models on compressive properties of ultrahigh toughness cementitious composites. ASCE J Mater Civil Engng 2010;22(10):1067–77. [6] Li VC, Kanda T. Engineered cementitious composites for structural applications. Innov Forum ASCE J Mater Civil Engng 1998;10(2):66–9. [7] Li VC, Mishra DK, Naaman AE, et al. On the shear behavior of engineered cementitious composites. J Adv Cem Mater 1994;1(3):142–9. [8] Maalej M, Li VC, Hashida T. Effect of fiber rupture on tensile properties of short fiber composites. ASCE J Engng Mech 1995;121(8):903–13. [9] Lim YM, Wu HC, Li VC. Development of flexural composite properties and dry shrinkage behavior of high performance fiber reinforced cementitious composites at early age. J Mater 1999;96(1):20–6. [10] Xu SL, Li QH. Theoretical analysis on flexural performance of functionally graded composite beam crack-controlled by ultrahigh toughness cementitious composites. Sci China Series E: Technol Sci 2009;39(6):1081–94. [11] Shen RX, Cui Q, Li QH. New fiber reinforced cementitous composite. Beijing: China Construction Material Industry Press; 2004 [in Chinese]. [12] Cai XR, Xu SL, Fu BQ. A statistical micromechanical model of multiple cracking for ultra high toughness cementitious composites. Engng Fract Mech 2011;78:1091–100. [13] Leung CK, Li VC. Strength-based and fracture-based approaches in the analysis of fibre debonding. J Mater Sci Lett 1990;9:1140–2. [14] Li VC, Stang H. Interface property characterization and strengthening mechanisms in fiber reinforced cement based composites. Adv Cem Based Mater 1997;6(99):1–20. [15] Wu HC, Li VC. Stochastic process of multiple cracking in discontinuous random fiber reinforced brittle matrix composites. Int J Damage Mech 1995;4(1):83–102. [16] Wang S. Micromechanics based matrix design for engineered cementitious composites. PhD thesis. The University of Michigan; 2005.
Contents lists available at SciVerse ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
An analytical study on the complete strain hardening process of ultra high toughness cementitious composites under direct tension load Xiang-Rong Cai a,⇑, Bai-Quan Fu b a b
Building Science Research Institute of Liaoning Province, Shenyang 110005, China College of Urban Construction, Shenyang Jianzhu University, Shenyang 110167, China
a r t i c l e
i n f o
Article history: Received 30 May 2012 Received in revised form 3 December 2012 Accepted 10 February 2013
Keywords: Fiber reinforced materials Strain hardening Multiple cracking Analytical models
a b s t r a c t The ideal tensile stress–strain curve of ultra high toughness cementitious composites (UHTCC) shows trilinear stress–strain relationship. And its strain hardening process can be divided into multiple cracking zone and post multiple cracking zone. In this paper, the general expression of the fiber bridging stress laws in the crack plane is presented, and the stress–strain models are established for both zones based on the law. Furthermore, the critical fiber length between fiber pull-out and fiber rupture in the post multiple cracking zone are deduced. And the precondition to guarantee the post multiple cracking zone is deduced. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The brittleness and insufficient durability of concrete have been identified as the bottlenecks which hinder the development of concrete structures. In order to improve the toughness of concrete and to enhance its damage tolerance, deformability capacity and anti-cracking ability under tensile load, intensified research has been conducted on fiber reinforced cementitious composites. As a result, a new family of high performance fiber reinforced cementitious composites named as ultra high toughness cementitious composites (UHTCC) was developed. This class of composites has many excellent performance to solve the brittleness and insufficient durability of concrete with fiber content typically 2% by volume or less. For example, its tensile strain capacity is more than 3%, above 300 times of that of ordinary concrete [1–3]. The compressive strength of UHTCC is in the range of high strength, and the compressive strain capacity is approximately double that of ordinary fiber reinforced concrete [4–6]. The shear strength is about three times higher than that of plain concrete and its peak shear strain is over 20 times the failure strain of plain concrete [7]. Its magnitude of fracture energy is the highest ever reported for a fiber reinforced cementitious composite. The area of the damage zone has been observed to be on the order of 1000 cm2 in double cantilever beam specimens with dimensions of 490 585 35 mm, and the energy absorption of the off-crack plane inelastic deformation process has been measured to be more than 50% of the total fracture energy of up to 34 kJ/m2 [8] The crack width under peak load can be limited to less than 0.1 mm, even 0.06 mm [1,2]. The anti-cracking performance of UHTCC is obviously superior to concrete. The crack width of UHTCC under restrained drying shrinkage is about 0.03–0.05 mm at 50% relative humidity [9]. Furthermore, a study on the bending behavior of functionally graded composite beam crack-controlled by UHTCC demonstrated that, in addition to improving bearing capacity, the corrosion-induced ⇑ Corresponding author. Tel.: +86 024 233 88143. E-mail address: [email protected] (X.-R. Cai). 0013-7944/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2013.02.006
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X.-R. Cai, B.-Q. Fu / Engineering Fracture Mechanics 102 (2013) 249–256
Nomenclature a df Ec E Em f g Gc Gd K Lc Le Lf P p p(/) p(z) q Vf x z
emc emu d d0 /
rfu rmc rmu s s0 ss g c
crack spacing fiber diameter elastic modulus of composite elastic modulus of fiber elastic modulus of matrix snubbing coefficient snubbing factor shear modulus of composite chemical debonding energy slip-hardening coefficient critical fiber length the embedded length of fibers fiber length the force loaded at the end of a fiber increment of crack width probability density function of the orientation angle probability density function of the centroidal distance z the number of separated pieces of matrix counted from the fiber center fiber volume fraction fiber end slippage distance of centroid of fiber from crack plane strain corresponding to the first cracking strength of matrix strain corresponding to the stress at the end of the multiple cracking process the displacement of the fiber-loaded end the corresponding displacement at the maximum stress orientation of fiber with respect to tensile loading direction ultimate tensile strength of fiber first cracking strength of matrix stress at the end of the multiple cracking process frictional sliding shear stress frictional sliding shear stress at the tip of debonding zone where no slip occurs interfacial chemical bonding coefficient of effective fiber length strain increment caused by the fiber slipping in the post multiple cracking process
damage can be effectively prevented [10]. Therefore, it is potential to solve the security risk and insufficient durability inherent with today’s concrete structures by using this class of composites. One of the most distinct features of UHTCC is the inelastic tensile behavior. Based on the shape of the stress–strain curve, UHTCC can be divided into two categories, one with bilinear tensile stress–strain curve and the other one with trilinear tensile stress–strain curve. For the one with bilinear stress–strain curve, the stress–strain curve can be divided into linear elastic zone and multiple cracking zone. And for the other one with trilinear stress–strain curve, the stress–strain curve can be divided into linear elastic zone, multiple cracking zone and post multiple cracking zone. The excellent mechanical performance of UHTCC is due to the tensile strain hardening process which is strongly related to the multiple cracking zone and post multiple cracking zone. Compared with the composites with bilinear tensile stress–strain curves, those composites with trilinear tensile stress–strain curves possess higher tensile strength, tensile strain and toughness, and can be regarded as ideal fiber reinforced cementitious composites [11]. However, it is more difficult to achieve post multiple cracking behaviors in random discontinuous fiber reinforced composites compared to continuous aligned ones. But, it is by no means impossible. The challenge is to determine the proper conditions under which post multiple cracking will occur, despite the reduced efficiency of stress transfer in the random and discontinuous bridging fibers and the separate matrix after multiple cracking process. As will be clear from the discussion to follow, the precondition for post multiple cracking is proposed, and the post multiple cracking models for the cases of fiber length less than critical length and more than critical length are established respectively. 2. Typical tensile stress–strain curves of UHTCC The typical tensile stress–strain curves obtained from uniaxial tension tests are given in Fig. 1. After the first tensile crack formed in the matrix, the bridging fibers across the crack will control the extension of crack and bear the load released by the cracked matrix. If the fibers have sufficient strength to sustain the additional load, the composite will continue to bear load
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6
8
(a)
6
stress
stress
4 multiple cracking zone
2
0
(b)
0
1
2
3
4
strain
4 multiple cracking zone post multiple cracking zone
2 0
0
1
2
strain
3
4
5
Fig. 1. Typical tensile stress–strain curves of UHTCC: (a) bilinear model and (b) trilinear model.
without failure. And the fibers will transfer the load back to the adjacent uncracked matrix through interfacial shear. When the transferred load reaches the matrix cracking strength, another crack will form in the matrix. This process then repeats itself until a set of periodic subparallel cracks are formed. This process is corresponding to multiple cracking zone in the tensile stress strain curve. In the multiple cracking zone, the stress keeps constant or increases slightly with the strain increasing, and the stress–strain curve is almost linear as shown in Fig. 1. When the crack spacing is so small that the transferred load cannot induce new cracks in the matrix, the multiple cracking process ended. After this, the composite will come into the failure phase if the bridging fibers are not able to bear additional load, and the tensile stress strain curve presents clearly bilinear tensile model as shown in Fig. 1a. Otherwise, the composite will come into the post multiple cracking phase if the bridging fibers are able to bear additional load. In this phase, the cement matrix lost its load-carrying capacity, and all the force is applied to the fibers. With a further increase in deformation, the bridging fibers will be pulled out gradually form the separated matrix. Because of the slip hardening effect, the composite stress increases with the strain increasing until the ultimate tensile strength is reached. The tensile stress–strain curve of this type of composite presents clearly trilinear tensile model as shown in Fig. 1b. 3. The strain hardening analysis in the multiple cracking zone The fiber bridging law is one of the main control parameters for establishing the stress–strain relationship of UHTCC. In this paper, a general expression of the fiber bridging stress laws in the crack plane was presented by referring to the existing laws mentioned above and considering the fiber/matrix debonding criteria of both strength and energy criterion respectively. For a composite with fiber volume fraction Vf, the composite stress can be obtained by integrating over the contributions of those individual fibers that cross the matrix crack plane
rB ¼
4gV f
pd2f
Z
p 2
/¼0
Z
L
ð 2f Þ cos /
PðdÞef / pð/ÞpðzÞdzd/
ð1Þ
z¼0
where P(d) is the relationship between the force P loaded at the end of a fiber and the displacement d of the fiber-loaded end, p(/) is the probability density function of the orientation angle / and p(z) is the probability density function of the centroidal distance z of fibers from the crack plane. For a uniform 3-D random distributions, p(z) = 2/Lf and p(/) = sin /. g is the coefficient of effective fiber length. In this paper, a brief introduction on the tensile stress–strain relationship during the multiple cracking processes is presented. The detailed derivation and discussion of the strain hardening analysis in multiple cracking zone sees our previous research achievements in the literature of Cai et al. [12]. For strength debonding criterion, the bridging stress-crack opening relation can be presented as
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cdf 2 ~d ~d þ ~d ~d qLf
r~ B ð~dÞ ¼ 2
ð2Þ
L 2 pf =2 Þ. ~ B ¼ rB =r0 ; ~ where C ¼ sssi ; r d ¼ d=ðLf =2Þ; ~ d ¼ s0 Lf =Ef df ; r0 ¼ 12 gg s0 V f ðdf Þ; g ¼ 4þf 2 ð1 þ e f For energy debonding criterion, the bridging stress-crack opening relation can be presented as
r~ B ð~dÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ Þ ~d 4ð~d þ a ~d d~
~ ¼ 2Gd =s0 Lf where a Comparing Eq. (2) with Eq. (3), the general expression of the fiber bridging law can be presented as
ð3Þ
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r~ B ð~dÞ ¼
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~d ~d n2 þ 4 ~d ~d
2
~d 6 ~d 1 n 4
! ð4Þ
qffiffiffiffi 2 Cd For strength debonding criterion, n ¼ 2 qL f ; and for energy debonding criterion, n ¼ 2 ~da~ . When ~ d¼~ d ð1 n4 Þ; the maximum f 2 n2 of fiber bridging stress can be reached: rB;max ¼ r0 ð1 þ 4 Þ: Generally, n 1, thus rBmax r0 . To see whether debonding is governed by strength or energy criterion for a certain composite system, one can carry out pull-out tests with different fiber radii or different fiber volume fractions and obtained the result form the trend of experimental data using the following equation [13]: 1
Ppeak =Ppostpeak ¼ 1 þ ½ðY 2 YÞ1=2 cos h ðYÞ1=2 =ð2qL=df Þ
ð5Þ
where Ppeak and Ppostpeak are the peak load and post peak load at the load–displacement curve obtained from the pull-out test. L is the embedded length of fibers, q is a parameter depending on moduli of fiber and matrix as well as fiber volume fraction in the pull out specimen, the expression of is given above. If Y is a constant with respect to df or q, debonding is governed by strength creterion, if Y varies with df and q, debongding is governed by energy creterion. The second parameter to control the performance of UHTCC is the cracking strength of matrix. In this paper, only the flaw size distribution of matrix is considered to determine the cracking strength of matrix. Referring to the literatures [14,15], the flaw size distribution is supposed to obey Weibull distribution. And the mortar matrix is treated as nonlinear brittle material and cohesive crack model is adopted. Referring to the literature [16], the finite element method is used to calculate the cracking strength of the matrix. When the cracking strength at flaw size of 2ci is equal to the fiber bridging stress, matrix cracking begins. i.e.
0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ~d ~d 2 @ r0 c þ 4 ~ ~ A ¼ rmc ðci Þ d d
ð6Þ
Half of the crack width can be obtained as:
dðci Þ ¼ ki d
ð7Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ki ¼ ð1 1 rmcr0ðci Þ þ 14 c2 Þ2 14 c2 The maximum debonding length of single fiber is given by
Li ¼
Lf 4
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c2 þ 4ki c
ð8Þ
The new crack occurs at the position with distance of Li away form the existing crack. The tensile strain of the composite here can be calculated by
eðci Þ ¼ e0 þ
Z
cmax
ci
2dðci Þ f ðcÞdc 100% Li
ð9Þ
where e0 is the elastic tensile strain at the first cracking stress, and cmax is the maximum flaw size radium in the matrix. Thus the tensile stress–strain during the multiple cracking processes is given by
(
rðci Þ ¼ rmc ðci Þ eðci Þ ¼ e0 þ
R cmax ci
2dðci Þ f ðcÞdc Li
100%
ð10Þ
The accuracy of the model proposed above was verified in the literature of Cai et al. [12] by experiment results obtained form literature [16]. 4. The strain hardening analysis in the post multiple cracking zone 4.1. The fiber pull-out process analysis In this zone, the matrix has been separated into pieces with length of a by the multiple cracks, and the fibers are pulled out from the separated matrix, as shown in Fig. 3. In order to simplify the analysis, the crack width at the end of the multiple cracking zone is not consider temporarily, that is to say only the increment of the crack width is considered in the analysis of this part. The fiber elastic elongation is neglected when it is in the pull-out process, and only the fiber slippage is considered. Furthermore, the crack width increment caused by the fiber slippage is recorded as p, i.e. p = d dc. Supposing that there n pieces in the fiber length Lf, i.e. Lf = na. When the composite elongates under tensile load, the pieces separate from each other. In this process, the fiber retracts and slips in the matrix by overcoming the friction in the interface. The friction load in one direction is balanced by the load in the opposite direction.
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4.2. The critical fiber length Analyzing a single fiber pulled out from the separated matrix in the direction of / = 0, as shown in Fig. 2. In order to calculate the fiber slippage, the fiber pull-out process is regarded as the movements of matrix pieces relative to fiber, i.e. the matrix pieces move form the fiber center to the fiber end. When the increment of the crack width is p, the moving distance of piece q counted from the fiber center is calculated as
1 x¼p q 2
ð11Þ
The interface shear stress caused by matrix pieces’ slipping in the piece q is calculated as follows:
sq ¼ s0 þ Kp q
1 2
ð12Þ
Thus the stress transferred between fiber and matrix in the piece q is given by:
rq ¼
4a df
s0 þ Kp q
1 2
ð13Þ
Supposing that there are still n0 pieces which contacted with the fiber when the crack width increment is p, as shown in L þnp Fig. 3, the total stress transferred in half of the unit (in the range of f 2 ) is calculated as follows: n0
rmax ¼
q¼ 2 X
rq ¼
q¼1
an0 Kpn0 2s0 þ df 2
ð14Þ
L
f where n0 ¼ aþp Eq. (14) is the maximum fiber stress at the crack width increment p, the calculated result is given as follows:
rmax ¼
a Lf Kp Lf 2s0 þ df a þ p 2 aþp
ð15Þ
When fiber is pulled out from the direction of /, the maximum fiber stress is given as follows by considering the snubbing effect:
rmax;/ ¼
a Lf Kp Lf ef / 2s0 þ df a þ p 2 aþp
ð16Þ dr
max;/ When rmax,/ < rfu, all fibers will be pulled out rather than rupture.It is known that d/ > 0 is always satisfied, thus the maximum fiber stress increases with the angle / increase. Therefore, the fiber ruptures gradually from / ¼ p2 to / ¼ 0. max The peak point of rmax p can be obtained from drdp ¼ 0:
8 > <
L K4s
ppeak ¼ Lf Kþ4s00 a f
ð17Þ
2 > : rmax;peak ¼ ðLf Kþ4s0 Þ 8d K f
When rmax,peak < rfu, all fibers will be pulled out, i.e.
Fig. 2. Sketch map of single fiber pulled out from separated matrix.
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Fig. 3. Stress redistribution at fiber rupture.
ðLf K þ 4s0 Þ2 < rfu 8df K
ð18Þ
The calculated result is given as follows
Lf < Lc ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2df K rfu 4s0 K
ð19Þ
where Lc is the critical fiber length. 4.3. The case of fiber pull-out When Lf 6 Lc , all fibers will be pulled out. The average stress in the piece q counting from the fiber center is given by:
r q ¼ rmax
i¼q X
1 2
ri þ rq ¼
i¼1
2a df
(
s0 ðn0 þ 1Þ þ
" # ) Kp ðn0 Þ2 1 þ qðKp 2s0 Þ Kpq2 2 2
ð20Þ
Thus the average stress in n0 /2 pieces can be obtained: 0
q¼n
2 X2 an0 r n20 ¼ 0 r q ¼ n q¼1 df
s0 þ
Kpðn0 1Þðn0 þ 1Þ 3n0
ð21Þ
This is also the average stress along half of the fiber length counting from the center of the fiber (including the crack opening part between two pieces, seen Fig. 2). Therefore, the average fiber stress is calculated as follows:
r Lf ¼ r n20 ¼
Lf df
1 K c Lf K ca2 ð1 þ cÞ s0 þ 1þc 3Lf 3 1þc
ð22Þ
where c is strain increment caused by the fiber slipping in the post multiple cracking process, and c ¼ np ¼ pa. Lf The average fiber stress along the length of Lf + np, i.e. the average stress of all fibers with the same direction but different embedded length in the crack plane, is given by:
r fur ¼
Lf df ð1 þ cÞ2
s0 þ
K cLf K ca2 ð1 þ cÞ 3Lf 3ð1 þ cÞ
ð23Þ
Thus the composite stress is given by:
r ¼ Vf
Z
p 2
r fur cos/ sin /d/ ¼
0
V f Lf 2df ð1 þ cÞ2
s0 þ
K cLf K ca2 ð1 þ cÞ 3Lf 3ð1 þ cÞ
ð24Þ
When a is very small relative to Lf, Eq. (24) can be simplified as
r¼
V f Lf 2df ð1 þ cÞ2
s0 þ
K cLf 3ð1 þ cÞ
ð25Þ
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255
Considering the snubbing effect g and the coefficient of effective fiber length g, the average fiber stress is given by:
r ¼ r0
1 ð1 þ cÞ2
K cLf 3s0 ð1 þ cÞ
1þ
ð26Þ
4.4. The case of fiber rupture When Lf > Lc, the fibers rupture gradually in the post multiple cracking zone. If rmax,/ = rfu, the critical limit of fiber rupture is given by
n0c ¼
Lf A þ 4s0 þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðLf A þ 4s0 Þ2 8Adf rfu ef / 2aA
ð27Þ
When n0 6 n0c , fiber ruptures. Stress redistributed after fiber rupture, as shown in Fig. 3. The fiber stress distribution changes from polygonal line 1 to polygonal line 2. Supposing that the stress rebalances between the piece x and piece x + 1: n0c
x X
2 X
q¼1
q¼xþ1
rq ¼
rq ¼
rfu
ð28Þ
2
The calculation results is given by
x¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4s20 þ df Acrfu 2s0
ð29Þ
2Aac
Thus the average stress of the fiber in n0 /2 pieces can be obtained:
x 2 X 2aq 2a 1 8a 1 1 ¼ 0 ð2s0 þ ApqÞ s0 þ Ap q s0 x2 þ Ap x2 þ x n 1 df df 2 n df 3 6
r n20 ¼ 0 2
ð30Þ
Thus the average stress in n/2 pieces can be obtained:
n0 n
r fr ð/Þ ¼ r n20 ¼
8a ndf
1 2 1 x þ x 3 6
s0 x2 þ Ap
ð31Þ
When rmax,/ > rfu, the angle range of fiber rupture at the crack width increment p is given as follows:
! ! df rfu ð1 þ cÞ2 1 p / ¼ ln ; f 2s0 Lf ð1 þ cÞ þ AcLf 2
ð32Þ
The average bridging stress at the crack plane is given by:
rf ¼
Z
/
r fur ð/Þcos/ sin /d/ þ
Z
0
p 2
r fur ð/Þcos/ sin /d/
ð33Þ
/
Thus the composite stress is given by
r ¼ V f rf
ð34Þ
Considering the snubbing effect g and the coefficient of effective fiber length g, composite stress is given by
r ¼ ggV f rf
ð35Þ
4.5. The critical condition for the existence of post multiple cracking zone In order to guarantee the existence of post multiple cracking zone, i.e. after the multiple cracking phase, the stress increases with the strain increasing, it requires that
dr=dc > 0
ð36Þ
That is to say when the Eq. (36) is satisfied, the strain hardening behavior in the post multiple cracking process can be obtained and the stress–strain curve presents trilinear model. Otherwise, the stress–strain curve presents bilinear model without post multiple cracking zone. For the case of fiber pull-out, the following expression must be satisfied to guarantee the strain hardening behavior in the post multiple cracking zone.
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K > K crit ¼
12rmc df ð1 þ cÞ
ggV f L2f ð1 2cÞ
ð37Þ
i.e. the minimum threshold of the slip hardening coefficient to guarantee the post multiple cracking zone is given by
K 0crit ¼
12rmc df
ggV f L2f
ð38Þ
5. Conclusions Ultra high toughness cementitious composites (UHTCC) possess clear bilinear or trilinear tensile stress–strain curves and exhibit pseudo strain hardening and multiple cracking under tensile load. This paper established the relationship between material parameters and macroscopic tensile strain hardening performance by considering the random distribution properties of fiber orientation and fiber location. For the multiple cracking zone, the effects of micro-parameters on the strain and crack width are also analyzed. For the post multiple cracking zone, the critical fiber length between fiber pull-out and fiber rupture are deduced. The precondition for post multiple cracking zone is proposed. The theoretical models proposed in this paper can be used either for the optimization of composite properties or for the prediction of composite macromechanical performance. References [1] Li HD. Experimental research on ultra high toughness cementitious composites. PhD thesis. Dalian: Dalian University of Technology; 2008 [in Chinese]. [2] Xu SL, Cai XR. Experimental study on mechanical properties of ultra high toughness fiber reinforced cementitious composite. J Hydraul Engng 2009;40(9):1055–63 [in Chinese]. [3] Li HD, Xu SL. Tensile and flexural properties of ultra high toughness cementitious composite. J Wuhan Univ Technol [Mater Sci Ed] 2009;24(4):677–83. [4] Xu SL, Cai XR, Zhang YH. Experimental measurement and analysis on axial compressive stress–strain curve of ultra high toughness cementitious composites. China Civil Engng J 2009;42(11):79–85 [in Chinese]. [5] Xu SL, Cai XR. Experimental study and theoretical models on compressive properties of ultrahigh toughness cementitious composites. ASCE J Mater Civil Engng 2010;22(10):1067–77. [6] Li VC, Kanda T. Engineered cementitious composites for structural applications. Innov Forum ASCE J Mater Civil Engng 1998;10(2):66–9. [7] Li VC, Mishra DK, Naaman AE, et al. On the shear behavior of engineered cementitious composites. J Adv Cem Mater 1994;1(3):142–9. [8] Maalej M, Li VC, Hashida T. Effect of fiber rupture on tensile properties of short fiber composites. ASCE J Engng Mech 1995;121(8):903–13. [9] Lim YM, Wu HC, Li VC. Development of flexural composite properties and dry shrinkage behavior of high performance fiber reinforced cementitious composites at early age. J Mater 1999;96(1):20–6. [10] Xu SL, Li QH. Theoretical analysis on flexural performance of functionally graded composite beam crack-controlled by ultrahigh toughness cementitious composites. Sci China Series E: Technol Sci 2009;39(6):1081–94. [11] Shen RX, Cui Q, Li QH. New fiber reinforced cementitous composite. Beijing: China Construction Material Industry Press; 2004 [in Chinese]. [12] Cai XR, Xu SL, Fu BQ. A statistical micromechanical model of multiple cracking for ultra high toughness cementitious composites. Engng Fract Mech 2011;78:1091–100. [13] Leung CK, Li VC. Strength-based and fracture-based approaches in the analysis of fibre debonding. J Mater Sci Lett 1990;9:1140–2. [14] Li VC, Stang H. Interface property characterization and strengthening mechanisms in fiber reinforced cement based composites. Adv Cem Based Mater 1997;6(99):1–20. [15] Wu HC, Li VC. Stochastic process of multiple cracking in discontinuous random fiber reinforced brittle matrix composites. Int J Damage Mech 1995;4(1):83–102. [16] Wang S. Micromechanics based matrix design for engineered cementitious composites. PhD thesis. The University of Michigan; 2005.