Experimental study on fracture performance of ultra-high toughness cementitious composites with J-integral

Engineering Fracture Mechanics 96 (2012) 656–666

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Experimental study on fracture performance of ultra-high toughness cementitious composites with J-integral Wen Liu a,c, Shilang Xu b,c,⇑, Qinghua Li b a

Key Laboratory of Soil & Water Conservation and Desertification Combating, Ministry of Education, Beijing Forestry University, Beijing 100083, China Institute of Advanced Engineering Structures and Materials, Zhejiang University, Hangzhou 310058, China c Institute of Structural Engineering, Dalian University of Technology, Dalian 116024, China b

a r t i c l e

i n f o

Article history: Received 2 February 2012 Received in revised form 3 August 2012 Accepted 5 September 2012

Keywords: Ultra-high toughness cementitious composites Ductile fracture Double J-integral criterion JR curve

a b s t r a c t Fracture test of three-point flexure was taken on ultra-high toughness cementitious composites (UHTCCs), which is a strain-hardening material, to investigate its ductile fracture performance. On the basis of nonlinear fracture mechanics, a double J-integral criterion was introduced to evaluate its fracture property. Moreover, its JR curve was studied and simplified. The results showed that the JR curve had three stages, with the two separations as the initial cracking point and the crack localizing point. In the stable stage, equal energy was required for equal quantity of crack growth. Moreover, the crack covering area was applicable to describe the crack growth of UHTCC. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Concrete is the most widely used engineering material all over the world. However, due to its brittle character of low tensile strength and low strain capacity, this material has weak cracking resistance. A lot of money and labor have to be input to repair concrete structures [1]. In order to improve such poor properties of concrete, fiber reinforced concrete (FRC) has been developed in recent several decades. And one is ultra-high toughness cementitious composites (UHTCCs), which exhibits the characteristics of multiple cracking and pseudo strain hardening under uniaxial tension. Composites called UHTCC should simultaneously possess the following characteristics [2]: it is reinforced with moderate short fiber with the corresponding volume fraction less than 2.5%; the hardened composites exhibits significant pseudo strain-hardening and multiple-cracking behaviors with tensile strain capability above 3%; moreover, it can effectively control the crack width to below 100 lm even when the strain achieves its maximum value. Polyvinyl alcohol (PVA) fiber is usually used to produce UHTCC. In the past, such PVA fiber reinforced cementitious composites, with the characteristics of pseudo strain-hardening and multiplecracking under uniaxial tension, was first studied in the USA, then in Japan, some European countries and China. Different names were given according to the different specific properties and the different emphasis, such as engineered cementitious

Abbreviations: ECC, engineered cementitious composites; FRC, fiber reinforced concrete; LVDT, linear variable differential transformer; PVA, polyvinyl alcohol; PVA-FRCC, PVA fiber reinforced cement composites, SEN, single edge notch; SHCC, strain hardening cementitious composites; UHTCC, ultra-high toughness cementitious composites. ⇑ Corresponding author at: Institute of Advanced Engineering Structures and Materials, Zhejiang University, Hangzhou 310058, China. Tel.: +86 571 88208676/88981950. E-mail address: [email protected] (S. Xu). 0013-7944/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engfracmech.2012.09.007

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Nomenclature a A b B JIC JIF P Pini IU S Vf DA r~ Y

crack length of single cracking fracture crack covering area of multiple cracking fracture net section depth of a fracture specimen width of a fracture specimen J-integral value at the initial cracking point J-integral value when the failure crack begins to localize applied load initial cracking load of UHTCC total area under the load–load-line displacement curve volume fraction of PVA fiber in UHTCC increment of crack covering area equivalent yield stress of UHTCC

composites (ECCs) [3], strain hardening cementitious composites (SHCCs) [4], PVA fiber reinforced cement composites (PVA– FRCC) [5], and UHTCC [2]. Different from concrete’s quasi-brittle fracture, ‘‘ductile fracture’’ happens for UHTCC [4], which involves extensive offcrack-plane inelastic energy absorption. Moreover, the total composite fracture energy can be divided into two parts: an off-crack-plane matrix-cracking component and an on-crack-plane fiber-bridging component. Maalej et al. [3] took an experimental study on the effect of fiber volume fraction on the off-crack-plane fracture energy of ECCs. The results showed that the off-crack-plane fracture energy increased with fiber volume fractions in a logarithmic fashion, and it exceeded the bridging fracture energy produced on the main fracture plane. Kabele and Horii [6] proposed a simple analytical model for fracture analysis of ECCs. In their research, ECC was idealized as a homogenous and continuous material, and a discrete crack model was used for localized cracks. Kabele and Li [7] emphasized the off-crack-plane fracture energy and calculated the composite fracture energy of SHCC by means of finite element analysis on crack growth under small-scale yielding conditions. The investigations before were mainly focusing on the fracture energy, especially the off-crack-plane fracture energy. However, in this paper, the authors attempted to propose a new method to evaluate the fracture performance of UHTCC. For ductile fracture, the linear elastic fracture mechanics was inapplicable. Therefore, the J-integral in nonlinear fracture mechanics was employed. Three-point flexural test was adopted here, and the deformation process and crack growth were discussed. The JR resistance curve was recorded and analyzed. Moreover, referring to the double-K fracture criterion of concrete [8], a double-J fracture criterion for UHTCC was introduced. 2. Experiment research 2.1. Experiment program Three-point flexural test was adopted in this paper. Specimens were cast with cementitious binders, fine sand, water, super plasticizer and PVA fiber. The superplasticizer was high-efficient polycarboxylate type, produced by Dalian Sika Company, China. The grain size of the fine sand was among 100–200 mesh. The proportion of the matrix was: cementitious binders:water:fine sand = 1:0.24:0.6. The PVA fiber was KURALON K-II REC 15, which was produced by Japan Kuraray Company. Table 1 showed the corresponding properties of PVA fiber. As has been studied, in order to achieve excellent performance, the corresponding volume fraction of PVA fiber in UHTCC is more than 1.5% and less than 2.5%, and usually 2.0% [2]. As a result, three fractions, 1.5%, 2.0% and 2.5%, were adopted in the experiment. The dimension of all specimens was 400 mm  100 mm  100 mm, and 5–6 specimens prepared for each fraction. All the specimens were demolded 24 h after casting and were standard cured for 28 days. Then, a single edge notch (SEN) was cut on every specimen, with the depth of 40 mm. Tests were conducted with a 250-kN MTS testing machine. Displacement control was employed here with a constant rate of 0.05 mm/min. The experimental setup is shown in Fig. 1. The support span was 300 mm, with the load applied vertically at the middle span. Two p gauges were employed to measure the CTOD and CMOD. In order to determine the initial crack load, one strain gauge of 10 mm and four strain gauges of 2 mm were applied, and they were symmetrically pasted on the side surface, along the horizontal line at the SEN tip. Meanwhile, one linear variable differential transfomer (LVDT) was used to measure the midspan deflection. It was mounted on a steel frame fixing on the supporting lines, so that the midspan Table 1 Properties of PVA fiber. Length (mm)

Diameter (lm)

Nominal strength (MPa)

Fiber elongation (%)

Young’s modulus (GPa)

Density (kg m3)

12

40

1600

7

42

1.3

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P

P

square grids with length of 10mm

LVDT strain gauges of 2 mm

strain gauge of 10 mm

π gauge π gauge

Fig. 1. Test setup and measuring method/mm.

Vf -1.5%

Vf -2.0%

Vf -2.5%

Fig. 2. Side surfaces of UHTCC specimens after failure.

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deflection measured by this LVDT was the relative displacement between the loading point and supporting points. A crevice width finder was applied to monitor the crack width. 2.2. Experiment results and discussion 2.2.1. Cracking pattern For UHTCC specimens, except for the localized crack, multiple macrocracks with small width, less than 80 lm, were generated near the SEN tip during the experiments. All cracks formed a similar rugby shape, as shown in Fig. 2, in which, Vf represents the volume fraction of PVA fiber. According to the previous investigation, SHCC fracture specimens had several cracking patterns [1]: distributed straight crack, curving crack and blunt crack. However, in this paper, the cracking patterns for UHTCC were either curving crack or blunt crack. The localized failure crack was accompanied by the obvious multiple macrocracks, and the failure section was rather unsmooth due to the effect of PVA fibers. The curved failure face would increase the fracture surface and reduced the stress concentration at the notch tip. Furthermore, from Fig. 2, obviously more cracks were produced when the fiber fraction increased. 2.2.2. Deformation evolution Because of the character of multiple cracking, the fracture deformation of UHTCC exhibited ductile phenomenon. Figs. 3 and 4 show the average P-CMOD curves and the average P-midspan deflection curves of the three UHTCCs. As can be seen, the multiple cracking property greatly enhanced the specimen’s deformation capability. In this experiment, the average ultimate flexural loads for UHTCC specimens with fiber volume fractions of 1.5%, 2.0% and 2.5% were measured as 7.09 kN, 7.95 kN and 10.60 kN, respectively. Moreover, it was observed that the deformation capacity increased when more fibers were added. From these two diagrams, at first, the CMOD and midspan defection curves basically coincided with each other, while after the generation of macrocracks, the slopes of these curves declined, more obviously with the increase of fiber fraction; also, the deformation capacity increased with the increase of fiber fraction. That is to say, the greater bridging effect and

12

Vf =2.5% 10

Vf =2.0%

Load /kN

8

Vf increases

Vf =1.5%

6

4

2

0 0

2

4

6

8

CMOD /mm Fig. 3. Average P-CMOD curves of UHTCC specimens.

12

Vf =2.5% 10

Vf =2.0%

Load /kN

8

Vf increases

Vf =1.5%

6

4

2

0 0

1

2

3

4

5

6

Midspan deflection /mm Fig. 4. Average P-midspan deflection curves of UHTCC specimens.

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D

Load /kN

C

B

E

A A

F O

CMOD /mm Fig. 5. Six stages of the fracture deformation.

4.0

PIUini

Load /kN

3.5

Vf =2.0% Vf =1.5% Vf =2.5%

3.0

2.5

2.0 0

50

100

150

200

250

300

-6

Stain /10

Fig. 6. Measured results of initial cracking loads.

5

Vf =1.5% Vf =2.0%

4

Vf =2.5% Load /kN

3

Vf increases

2

PIUini 1

1

0

0

300

600

Strain /10

900

1200

1500

−6

Fig. 7. P-horizontal strain curves.

more cracks of specimens with a larger fiber fraction would lead to a higher capacity of load carrying and deformation. All UHTCCs exhibited ductile fracture phenomena, and the reinforcing effect of UHTCC with Vf = 2.5% was clearly better than the other two UHTCCs. Different from the results of uniaxial tensile experiment on double edge notched PVA-FRCC specimens, carried out by Nelson et al. [5], here, the fracture deformation process of three-point flexural SEN specimens was divided into six stages: I, Linear elastic stage; II, Nonlinear deformation stage resulted from microcracks; III, Crack formation stage; IV, Stable growth stage; V, The failure crack localizing stage; VI, Failure stage, as is shown in Fig. 5. Firstly, in stage I, no microcrack appeared, and the applied load was carried by the matrix, so the effect of PVA fibers was ignored. Secondly, stage II was characterized by the nonlinear deformation. In this stage, sub-critical cracks developed in the frontal zone of the notch tip, and PVA fibers began to work, while the applied load was borne by both the fibers and the matrix. Due to the microcracks, the stiffness of specimens declined, leading to a slow increasing trend of the P-deformation curves. Thirdly, as the applied load increased,

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macrocracks initiated. However, because that the applied load was too small to develop excessive cracks, only several cracks (usually less than five) were produced in this stage. The development of these macrocracks led to the rapid development of specimens’ deformation. After that, in the stable developing stage, many new cracks were generated with old cracks continued to develop, and P-deformation curves developed stably. Next, when the cracks became saturated, a localized crack, which was the failure crack, was formed. The applied load was taken mainly by the fibers across the localized crack surface, and it decreased slowly. Furthermore, with this localized crack became wider and wider, the effect of matrix disappeared gradually. Finally, when the localized crack propagated through the entire specimen, the applied load was entirely borne by the fibers. With the failure crack continued to develop, the fibers either pulled out or ruptured, resulting to a fracture failure of the specimen.

Fig. 8. Relation between deflection and CMOD.

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2.2.3. Initial cracking load The fracture development is rather complicated for UHTCC before initial cracking. Original flaws existing in these composites lead to stress concentration in the frontal zones under the applied load. With the load increases, micrcracks would form and develop in the frontal zones of the flaws. When these microcracks become saturated, an initial crack is generated. For UHTCC, the initial cracking load of the composites is expressed by P ini IU . As stated above, five strain gauges were used to measure the initial cracking loads. Before cracking, the strain would grow with the increase of the applied load. However, once there produced a macrocrack, the strains near the two sides of the crack surface retract due to the release of stress. As a result, in the P-strain curve, the strain decreases while the load continues to grow at the initial cracking point. The corresponding load is decided as the initial cracking load. The measured results of the initial cracking loads for UHTCC specimens are shown in Fig. 6. The initial cracking loads were located between 2.2 kN and 3.5 kN. And the average values for specimens with fiber volume fractions of 1.5%, 2.0% and 2.5% were 2.51 kN, 2.89 kN and 3.01 kN, respectively. Clearly, these values increased when more fibers were added into the composites. For the reason that, when microcracks emerged in the composites, the bridging effect of PVA fibers turned up, which delayed the development of macrocracks, meanwhile, this delaying effect would increase with the increase of fiber fractions. In order to further study the effect of PVA fibers before cracking, Fig. 7 reveals the average evolution curves of the lateral strain in front of the notch tip, which were recorded by the 10 mm strain gauges. From this diagram, at the end of linear elastic stage, the strain values for these three UHTCCs differed a little from each other, all around 100 microstrains (line 1 in Fig. 7), indicating that the fiber fraction had a small effect on the deformation of UHTCCs before microcracking. However, in the nonlinear deflection stage, the lateral strains increased much quickly for UHTCC of Vf = 1.5%, while the other two UHTCCs of Vf = 2.0% and Vf = 2.5% had a similar developing trend. At the end point of the nonlinear elastic stage, the average strains for UHTCC specimens with fiber volume fractions of 1.5%, 2.0% and 2.5% were about 530, 510 and 500 microstrains, decreasing slowly with the increase of fiber volume fractions. 3. Evaluating ductile fracture performance 3.1. Double J-integral fracture criterion As discussed above, ductile fracture existed for UHTCC, due to its character of multiple cracking. Therefore, nonlinear fracture mechanics was employed to evaluate the fracture toughness of this material, for that linear elastic fracture mechanics was inapplicable. In nonlinear fracture mechanics, J-integral is a key parameter to calculate the fracture energy. According to the investigation of ASTM-24 committee [9], for fracture specimens under three-point flexure, the estimate formula for J integral is as follows:

J¼

2S Bb

ð1Þ

where S represents the total area under the load–load-line displacement curve; B is the width of the specimen; and b is the net depth of the fracture section. In this paper, this equation was employed to calculate the J integral of UHTCC specimens. However, in the calculation of fracture toughness, CMOD is more widely used. Hence, P-CMOD curves were employed here for J-integral calculation. Then the relation between the load-line displacement and CMOD had to be constructed. Fig. 8 shows that the load-line displacement, which was the midspan deflection in three-point flexural test, kept a good linear relationship with the CMOD for the tested beams. In this diagram, the fine lines are the tested results, while the heavy lines represent the average relation. Through linear fitting, load-line displacement of the specimens for the three UHTCCs

10

Initial cracking value, JIC Failure value, JIF

J-integral /(kJ/m

2

8

6

4

2

0 1.0

1.5

2.0

2.5

PVA fiber volume fraction /% Fig. 9. Average values of JIC and JIF of UHTCC specimens.

3.0

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was about 70% of CMOD. The P-CMOD curves, substituting for the P-deflection curves, could be used to calculate the composite fracture energy and fiber bridging fracture energy. For UHTCC, the ductile stage began to develop when the initial crack emerged, meanwhile, the failure stage turned up when the failure crack was localized. Therefore, a double J-integral fracture criterion was proposed here to evaluate the fracture property of UHTCC, referring to the double-K fracture criterion of concrete [8,10]. That is, two J integral values, JIC and JIF, were introduced here, with the former one used to represent starting point of the ductile stage while the latter one used to express the starting point of the failure stage. When J = JIC, the first macrocrack initiated; J > JIF, the localized failure crack developed; when J 6 JIF, the material was in a safe state. For the specimens in this paper, the average values of JIC and JIF, calculated by Eq. (1) with the average experimental curves of P-CMOD shown in Fig. 3, are shown in Fig. 9. From this diagram, it is clearly seen that the J-integral values at initial cracking point, JIC, were very small, nearly equaled to zero, while the J-integral values at the failure crack localizing point, JIF, which were the safe boundaries, increased obviously with the increase of fiber volume fractions. 15

Vf=1.5% 12

J-integral / kJ/m

2

Line 3

The failure crack began to localize

9

6

JIF Line2

3

The first crack began to initiate

Line 1 JIC 0

3

6

9

12

A /cm

Marocracking area,

15

2

24

J-integral / kJ/m

2

20

Vf=2.0%

16

The failure crack began to localize

Line 3

12

The first crack began to initiate

8

JIF 4

Line2

Line 1

JIC 0

4

8

12

Marocracking area,

16

A /cm

20

2

28

Vf=2.5%

J-integral / kJ/m

2

24 20

The failure crack began to localize

16

Line 3

12

JIF

8

Line2

4

The first crack began to initiate

Line 1

JIC 0

4

8

12

Marocracking area,

16

A /cm

20

24

2

Fig. 10. Relations between J-integral and increment of macrocracking area, DA.

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JIC / kJ/m

2

0.04

0.03

0.02

0.01

0.00 1.0

Average result from P-CMOD curves Fitted vaule from J- A relations Average result of fitted values 1.5

2.0

2.5

3.0

Fiber volume fraction /%

(a) JIC 10

6

JIF / kJ/m

2

8

4

Average result from P-CMOD curves Fitted vaule from J- A relations Average result of fitted values

2

0 1.0

1.5

2.0

2.5

3.0

Fiber volume fraction /%

(b) JIF Fig. 11. Comparison of JIC and JIF obtained from fitted lines of J–DA and experimental P-CMOD curves.

Fig. 12. Schematic diagram of UHTCC’s JR resistance curve.

3.2. JR resistance curve JR resistance curve is the relationship between J-integral and the quantity of crack growth. Rather than a single crack for plain concrete, for UHTCC, numerous cracks generated with a curved shape. Therefore, the parameter for single crack growing length, a, is not suitable for UHTCC. In this paper, macrocrack covering area, A, is employed to describe the crack growth

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2.5x10

-3

2.0x10

-3

1.5x10

-3

1.0x10

-3

5.0x10

-4

Equivalent yielding stress, σY / N/mm

3

3.0x10

Calculated value Average result

0.0 1.0

1.5

2.0

2.5

3.0

Fiber volume fraction /% ~ Y , from JR curves. Fig. 13. Calculated results of equivalent yield stress, r

of UHTCC. From the fracture mechanism, the composite fracture energy is released by two ways: matrix cracking and fiber bridging. And its character of multiple cracking has a positive effect on slowing down the crack propagation rate. For the reason that, when a new crack happens, on the one hand, the stress intensity factors for the former cracks would decrease, which lower their propagation speed; on the other hand, some fracture energy is released through fiber bridging happening at the new crack, which would reduce the energy absorbed by matrix cracking. During the experiment, it is noticed that, the crack of UHTCC was a multidimensional variable, including the information of number, shape, width, length and so on. As a result, it was rather difficult to measure the growth of each fatigue crack. However, during the experiment it was observed that the width of each crack was approximately less than 80 lm before the failure crack began to localize, meaning a safe state of these cracks. Therefore, in this paper, before the failure stage, the width of each fatigue crack was ignored, only its shape and length were considered as a whole variable. Then, the macrocracking area, A, was selected as a parameter to describe the crack propagation quantity of UHTCC. It equaled to the total area of the elements which are crossed by the cracks, when the surface was divided into numerous square elements, with the side length was small enough to ignore its influence on counting the macrocracking area. In order to obtain the macrocracking area, square grids with the length of 10 mm were labeled in front zone of the notch, on the side surface of the specimen. The macrocraking area which was occupied by the peripheral cracks was recorded with the help of these grids. Since cracking around the notch tip was three dimensional, the cracking area on the surface and that on the back side were usually different. However, in the static experiment, due to the application of LVDT, it was difficult to measure the cracking area on this surface. As a result, only the cracking area on the back side was measured, with ignoring the difference between the two side surfaces. The real-time J-integral values were computed by Eq. (1) too, with the average experimental curves of P-CMOD shown in Figs. 3 and 10 shows the relationship between the increment of macrocracking area, DA, and the J-integral values, only one specimen was selected for each fiber fraction. Obviously, three-linear relationship existed between them, with the two points of initial cracking and crack localizing as the separations. In the first stage (line 1 in Fig. 10), no visible crack was formed, and the J integral had a slightly increase while DA kept zero. In the second stage (line 2 in Fig. 10), multiple cracks were generated consecutively, which led to the clear increase of J-integral and DA. In the third stage (line 3 in Fig. 10), the failure crack developed quickly after localizing. Here DA was no longer suitable for describing crack propagation due to the continually growth of the failure crack. However, as this stage was not allowed in practical engineering, DA was still employed to represent crack growth for simplicity. Then, there are two methods to obtain JIC and JIF. One is to calculate JIC and JIF on the basis of the experimental P-CMOD curves, and the other is to fit JIC and JIF with JR curves. The results with the two methods were very similar, according to Fig. 11. As a result, the JR curve can be applied to judge the fracture state of UHTCC. Fig. 12 shows the simplified model of UHTCC’s JR curve, and the relationship between J and DA for the former two stages is regressed as follows:

(

DA ¼

0

0 6 J 6 J IC

JJ IC ~Y 2r

J IC < J 6 J IF

ð2Þ

~ Y is a constant. Instead of crack length, a, macrocracking area, A, is a two dimensional parameter, with mm2 as its where r ~ Y is N/mm3. According to the experimental JR curves and Eq. (2), the calculated unit rather than mm. Therefore, the unit for r ~ Y are shown in Fig. 13. As can be seen, r ~ Y grew mildly with the increase of fiber fractions. Obviously, if r ~ Y was results of r known the crack growth quantity could be calculated with the value of J-integral with Eq. (2).

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4. Conclusions This paper presented a three-point flexural experimental study on the ductile fracture performance of UHTCCs with the PVA fiber volume fractions of 1.5%, 2.0% and 2.5%. The following conclusions were obtained, (1) All the UHTCC specimens with SEN had ductile fracture deformation phenomenon under three-point flexure. Except for the localized crack, multiple macrocracks with small width, which was oval-shaped, were generated near the notch tip and spread to the loading point. The cracking patterns were either curving crack or blunt crack. (2) A double J-integral criterion was proposed to evaluate the fracture state of UHTCC. When J = JIC, the first macrocrack began to initiate and the ductile fracture stage started; J > JIF, the failure crack was localized and the failure fracture stage developed; and JIC < J 6 JIF, the deformation developed stably; moreover, when J 6 JIF, the material was in a safe state for application. (3) The macrocrack covering area, A, was suitable to describe the crack growth of UHTCC. Based on the experimental results, the JR curve of UHTCC had three stages, with the dividing points of initial cracking and crack localizing. Furthermore, the linear relation of J-integral and DA in the stable stage indicated that equal fracture energy was required for equal quantity of crack growth. ~ Y , was applied to express the relation of J-integral and DA in the stable stage. When r ~Y (4) The equivalent yield stress, r was known, the quantity of crack growth could be calculated with J-integral values.

Acknowledgements The authors gratefully acknowledge the financial supports provided by the National Natural Science Foundation of China (Project 51008270), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Project 20100101120058), and the Key Science and Technology Innovation Team of Zhejiang Province of China (Project 2010R50034). References [1] Kawamata A, Mihashi H, Kaneko Y, Kirikoshi K. Controlling fracture toughness of matrix for ductile fiber reinforced cementitious composites. Engng Fract Mech 2002;69:249–65. [2] Li HD, Xu SL, Leung CKY. Tensile and flexural properties of ultra high toughness cementitious composite. J Wuhan Univ of Technol (Mater Sci Ed) 2009;24:677–83. [3] Maalej M, Hashida T, Li VC. Effect of fiber volume fraction on the off-crack-plane fracture energy in strain-hardening engeered cementitious composites. J Am Ceram Soc 1995;78:3369–75. [4] Li VC, Hashida T. Engineering ductile fracture in brittle-matrix composites. J Mater Sci Lett 1993;12:898–901. [5] Nelson PK, Li VC, Kamada T. Fracture toughness of microfiber reinforced cement composites. J Mater Civil Engng September/October 2002:391. [6] Kabele P, Horii H. Analysis model for fracture behaviors of pseudo strain-hardening cementitious composites. J Mater Conrc Struct Pavements 1996;30:209–19. [7] Kabele P, Li VC. Fracture energy of strain-hardening cementitious composites. Fracture mechanics of concrete structures. In: Proceedings FRAMCOS3. Freiburg: AEDIFICATIO Publishers; 1998. pp. 487–498. [8] Xu SL, Reinhardt HW. A simplified method for determining double-K fracture parameters for three-point bending tests. Int J Fract 2000;104:181–209. [9] Latzko DGH. Post-yield fracture mechanics. London: Applied Science Publishers; 1979. [10] Xu SL, Zhao GF. A double-K fracture criterion for the crack propagation in concrete structures. China Civil Engng J 1992;25:32–8 [in Chinese].