# Experimental Research on Fracture Characteristics of Reactive Powder Concrete in Different Volume Content of Steel Fiber

Authors: Wenzhen Wang; Haixiao Lin; Xiaogang Wu; Yuyao Zhao; Ziyu Hu; Pengshuai Wang; Menglong Zhou
Journal: Geofluids
 (2022)

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## Abstract

The fracture characteristics of reactive powder concrete in different volume content of steel fiber are researched in this paper, combining with the photoelastic coating method and the mathematical analytical software MATLAB. More precisely, the wholly integrated initiation, stable propagation, and final failure stages of precast cracks in three-point bending specimens of reactive powder concrete in different volume content of steel fiber are directly recorded and systematically analyzed. Besides, the photoelastic fringe distribution graphs from the vicinity of the precast cracks of specimens are obtained. Based on the experimental results and fracture mechanics theory, the mechanical property of reactive powder concrete, fracture energy, ductility index, initial fracture toughness, and unstable fracture toughness are quantitatively and qualitatively analyzed. The research achievements indicate that there is a stable crack propagation process before the instability failure of the three-point bending beam specimen of reactive powder concrete. The fracture process of the structure includes three stages, namely, the crack initiation, stable propagation, and instability failure. The order of the photoelastic fringe, critical effective crack length, crack initiation load, and maximum load of test parameters in the three-point bending beam specimen of reactive powder concrete increases with the increase of the fiber volume ratio. Based on the numerical treatment for the whole curves of each specimen group, the softening curve and the double fracture parameters of reactive powder concrete are obtained. Besides, the calculated critical effective joint length, initiation toughness, toughness increment, and unstable fracture toughness increase with the increase of the steel fiber volume ratio, by analyzing the measured initiation load, maximum load, and critical effective joint length. The research results can be treated as an important basis and reference for the engineering design and safety assessment of reactive powder concrete.


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## Body

## 1. Introduction

Reactive powder concrete is a new type of cement-based composite material with ultra-high strength, high durability, and low brittleness. It was first developed by French Richard and others in the 1990s [1, 2]. Based on the most compact packing theory, reactive powder concrete removes the coarse aggregate, takes fine quartz sand as the aggregate, and adds a high efficiency water reducing agent and an appropriate amount of steel fiber, thereby improving the compressive strength and toughness of concrete by reducing internal defects [3, 4]. Compared with ordinary concrete, reactive powder concrete has higher specific strength, which effectively reduces the dead weight of the structure and saves the amount of materials [5, 6]. Besides, reactive powder concrete has the characteristics of high brittleness of ordinary concrete and achieves the effect of strengthening and toughening by adding fiber [7, 8].At present, reactive powder concrete is becoming a hot research issue at home and abroad in recent years, because of its superior performance [9, 10]. According to the research results of Bonneau et al. [11], the reactive powder concrete has ultra-high compressive strength and durability, compared with ordinary concrete through experimental research. Cheyrezy et al. [12] studied the variation law of pore diameter and porosity of reactive powder concrete under different curing systems and obtained the conclusion that high-temperature autoclave curing significantly reduces the porosity of reactive powder concrete. Ahmad et al. [13] studied the influence law of three key factors on the performance of reactive powder concrete by optimizing sand gradation, namely, cement content, water binder ratio, and silica fume. The test results indicate that three key factors have a significant impact on the fluidity and strength of reactive powder concrete, and the regression equation of mechanical properties of reactive powder concrete is obtained.It is obvious that fracture characteristics of reactive powder concrete in different volume content of steel fiber are not researched systematically. Therefore, the experimental research on fracture characteristics of reactive powder concrete in different volume content of steel fiber is researched in this paper. The research results can be treated as an important basis and reference for the engineering design and safety assessment of reactive powder concrete.


## 2. Preparation of Experimental Specimens

In this paper, three-point bending beam specimens are tested in the laboratory. According to the change of the fiber volume content, the specimens are divided into four groups, with three specimens in each group. The volume content of steel fiber in the specimen is 0%, 1.0%, 2.0%, and 3.0%. Meanwhile, the corresponding numbers are 0, 1, 2, and 3. The numbers of three-point bending beam specimens are shown in Table1.Table 1
Serial number of specimens.
Specimen number0123Volume content of steel fiber0%1.0%2.0%3.0%The number of specimens3(0-1, 0-2, 0-3)3(1-1, 1-2, 1-3)3(2-1, 2-2, 2-3)3(3-1, 3-2, 3-3)The specific preparation method and test steps of the three-point bending beam specimen are set. (1) According to the mixing proportion, the raw materials are weighed, and the cement, silica fume, quartz powder, and quartz sand are poured into the mortar mixer. (2) When adding steel fiber, the steel fiber is slowly and evenly sprinkled into the mixer during the mixing process. (3) The half consumption of water dissolved in a water-reducing agent is added and stirred for about three minutes. (4) The other half consumption of the water is poured and stirred for about six minutes. (5) The mixed reactive powder concrete is poured into the triple mold placed on the shaking table coated with a release agent in advance. Half of the reactive powder concrete is poured in the mold firstly, and the shaking table is started, vibrating for one minute; then, the other half of the reactive powder concrete is poured and undergoes vibration for one minute. (6) For the initial cracks of the specimen, the grooves are set in advance inside the test partition, and the thin iron sheet is inserted vertically into the grooves in advance when the steel fiber concrete is poured. After pouring and curing the test piece for more than 24 hours, the thin iron sheet is pulled out to form a reserved seam. (7) After the reactive powder concrete specimens are formed, the reactive powder concrete specimens are moved into the curing room (temperature20±3°C, relative humidity above 90%), and the mold is removed after the specimens are cured for 24 hours. Then, the reactive powder concrete specimens are immediately placed in a 90±2°C hot water curing box for curing. Finally, the reactive powder concrete specimens are taken out after 72 hours of heat curing, and then, the specimens are cooled to room temperature.The key equipment employed to test the reactive powder concrete includes the UJZ-15 mortar blending machine, digital control magnetic vibrating table (the vibration frequency is 2860 times/min), YH-20B standard constant temperature and humidity curing box, and HJ-84 concrete accelerated curing box, shown in Figure1.Figure 1
Testing equipment.
(a)
UJZ-15 mortar blending machine(b)
Digital control magnetic vibrating table(c)
YH-20B standard constant temperature and humidity curing box(d)
HJ-84 concrete accelerated curing box


## 3. Photoelastic Patch Method Experiment

### 3.1. Experimental Principle

The photoelastic patch method extends the traditional photoelastic model test to measure the surface strain field distribution on a 2D and 3D opaque model made of structural material [14]. And the elastic and plastic strains of the structure surface are measured by the photoelastic patch method. The patch is a thin layer of birefringent material, deforming with the deformation of the plane or curved structural surface of the structure, which is used as a sensing element to analyze the test result [15].Therefore, the photoelastic patch method is employed to directly observe the whole process of prefabricated crack initiation, stable expansion, and instability failure of the three-point bending beam specimen of reactive powder concrete, which is of great importance for further research on the fracture mechanism and fracture characteristics of steel fiber-reinforced concrete [16].


### 3.2. Experimental Equipment

The experiment adopts the Photoelastic Division patch produced by the Measurements Group Inc., and the thickness of the patch is3.02±0.05mm. The photoelastic patch is closely attached to the surface of the object to be tested, in order to accurately reflect the stress state of the object [17]. A whole patch is attached to the three-point bending beam specimen of the reactive powder concrete and a crack of the same length as the specimen is prefabricated in the middle of the patch. The sticking steps are set: (1) filling the surface of the test piece and grinding it with sandpaper to achieve the superior flatness of test piece; (2) mixing the viscous material in proportion and spreading it evenly on the surface of the prepared test piece; (3) pasting the patch; (4) after the patch is pasted on the test piece, squeezing the glue evenly and smoothly sealing the glue at the edge of the patch.The key equipment employed in the experiment includes the loading equipment of a servo fatigue testing machine and the corresponding computer data acquisition system. The maximum load range of the testing machine is 200 kN, and the displacement control is adopted in the loading control method. The reflective photoelastic instrument consists of a polarizing environment, a light source, a digital camera, a filter, a bracket, and a device fixing table, shown in Figure2.Figure 2
Composition of reflective photoelastic instrument.
(a)
Overall instrument picture(b)
Local instrument picture


### 3.3. Experimental Results

The isoline fringe patterns displayed by the photoelastic patch method under various loads are photographed for each specimen in the test. The test specimens are tested for the photoelastic patch in the process of crack initiation and crack expansion under various loads. The images of specimen 0-2, specimen 1-2, specimen 2-1, and specimen 3-3 are shown in Figures3, 4, 5, and 6, respectively.Figure 3
Crack propagation displayed by coatings (specimen 0-2).
(a)
P=0.73kN(b)
P=1.67kNFigure 4
Crack propagation displayed by coatings (specimen 1-2).
(a)
P=2.16kN(b)
P=4.47kNFigure 5
Crack propagation displayed by coatings (specimen 2-1).
(a)
P=2.41kN(b)
P=4.82kN(c)
P=7.29kNFigure 6
Crack propagation displayed by coatings (specimen 3-3).
(a)
P=2.59kN(b)
P=4.88kN(c)
P=8.20kN(d)
P=10.41kNThe steel fibers prevent the development of defects such as microcracks in the concrete, delay the occurrence of initial cracks, and improve the initial crack strength in the reactive powder concrete specimens mixed with steel fibers. Although the matrix has been cracked, the steel fibers bridging at the cracks begin to bear the stress transmitted by the interface between the reactive powder concrete matrix and the steel fiber, so that the specimens continue to bear larger loads and produce elastic-plastic deformation.With the increase of the load, the length and width of the microcracks on the surface of the reactive powder concrete continues to increase, and some new microcracks occur near the original cracks. These new cracks continue to expand and connect with the original cracks to form the key failure cracks. When the load increases to the peak value, the cracks on the surface of the reactive powder concrete expand widely. Even if the load does not increase, the cracks continue to expand and start from close to the peak value. Meanwhile, the sound of the steel fibers being pulled out from the specimen is clearly heard in the test.After entering the residual stress stage after the peak, the deflection increases greatly and the load decreases slowly. When the specimen loses its bearing capacity, a small amount of steel fibers is still connected in the section. Besides, the fiber occlusion area extends to the upper edge, and the fracture is rough. The steel fibers at the fracture of the specimens are pulled out due to the bond failure, and the phenomenon of fiber breaking rarely occurs.


## 3.1. Experimental Principle

The photoelastic patch method extends the traditional photoelastic model test to measure the surface strain field distribution on a 2D and 3D opaque model made of structural material [14]. And the elastic and plastic strains of the structure surface are measured by the photoelastic patch method. The patch is a thin layer of birefringent material, deforming with the deformation of the plane or curved structural surface of the structure, which is used as a sensing element to analyze the test result [15].Therefore, the photoelastic patch method is employed to directly observe the whole process of prefabricated crack initiation, stable expansion, and instability failure of the three-point bending beam specimen of reactive powder concrete, which is of great importance for further research on the fracture mechanism and fracture characteristics of steel fiber-reinforced concrete [16].


## 3.2. Experimental Equipment

The experiment adopts the Photoelastic Division patch produced by the Measurements Group Inc., and the thickness of the patch is3.02±0.05mm. The photoelastic patch is closely attached to the surface of the object to be tested, in order to accurately reflect the stress state of the object [17]. A whole patch is attached to the three-point bending beam specimen of the reactive powder concrete and a crack of the same length as the specimen is prefabricated in the middle of the patch. The sticking steps are set: (1) filling the surface of the test piece and grinding it with sandpaper to achieve the superior flatness of test piece; (2) mixing the viscous material in proportion and spreading it evenly on the surface of the prepared test piece; (3) pasting the patch; (4) after the patch is pasted on the test piece, squeezing the glue evenly and smoothly sealing the glue at the edge of the patch.The key equipment employed in the experiment includes the loading equipment of a servo fatigue testing machine and the corresponding computer data acquisition system. The maximum load range of the testing machine is 200 kN, and the displacement control is adopted in the loading control method. The reflective photoelastic instrument consists of a polarizing environment, a light source, a digital camera, a filter, a bracket, and a device fixing table, shown in Figure2.Figure 2
Composition of reflective photoelastic instrument.
(a)
Overall instrument picture(b)
Local instrument picture


## 3.3. Experimental Results

The isoline fringe patterns displayed by the photoelastic patch method under various loads are photographed for each specimen in the test. The test specimens are tested for the photoelastic patch in the process of crack initiation and crack expansion under various loads. The images of specimen 0-2, specimen 1-2, specimen 2-1, and specimen 3-3 are shown in Figures3, 4, 5, and 6, respectively.Figure 3
Crack propagation displayed by coatings (specimen 0-2).
(a)
P=0.73kN(b)
P=1.67kNFigure 4
Crack propagation displayed by coatings (specimen 1-2).
(a)
P=2.16kN(b)
P=4.47kNFigure 5
Crack propagation displayed by coatings (specimen 2-1).
(a)
P=2.41kN(b)
P=4.82kN(c)
P=7.29kNFigure 6
Crack propagation displayed by coatings (specimen 3-3).
(a)
P=2.59kN(b)
P=4.88kN(c)
P=8.20kN(d)
P=10.41kNThe steel fibers prevent the development of defects such as microcracks in the concrete, delay the occurrence of initial cracks, and improve the initial crack strength in the reactive powder concrete specimens mixed with steel fibers. Although the matrix has been cracked, the steel fibers bridging at the cracks begin to bear the stress transmitted by the interface between the reactive powder concrete matrix and the steel fiber, so that the specimens continue to bear larger loads and produce elastic-plastic deformation.With the increase of the load, the length and width of the microcracks on the surface of the reactive powder concrete continues to increase, and some new microcracks occur near the original cracks. These new cracks continue to expand and connect with the original cracks to form the key failure cracks. When the load increases to the peak value, the cracks on the surface of the reactive powder concrete expand widely. Even if the load does not increase, the cracks continue to expand and start from close to the peak value. Meanwhile, the sound of the steel fibers being pulled out from the specimen is clearly heard in the test.After entering the residual stress stage after the peak, the deflection increases greatly and the load decreases slowly. When the specimen loses its bearing capacity, a small amount of steel fibers is still connected in the section. Besides, the fiber occlusion area extends to the upper edge, and the fracture is rough. The steel fibers at the fracture of the specimens are pulled out due to the bond failure, and the phenomenon of fiber breaking rarely occurs.


## 4. Fracture Mechanism of Reactive Powder Concrete Based on Double-K Fracture Criterion

### 4.1. Determination of Double-K Fracture Parameters

Because there is a stable crack propagation stage before the unstable fracture of concrete, the actual crack lengtha of the specimen before the unstable fracture is greater than the prefabricated joint length a0. If the stable crack propagation length Δac before the unstable fracture is recorded, then
(1)ac=a0+Δac.According to the linear progressive superposition principle, the critical effective crack lengthac is calculated by calculating the linear elastic fracture mechanical equation. When the external load P of the three-point bending beam reaches the maximum load Pmax, the corresponding critical effective crack length ac is determined by the following equation:
(2)CMODc=6PmaxSacth2EV1α,where CMODc is the crack opening displacement; E is the elastic modulus; S, t, and h are beam span, beam thickness, and beam height, respectively; the expression of V1α is V1α=0.76−2.28α+3.87α2−2.04α3+0.66/1−α2; and H0 is the blade thickness.The elastic modulus is calculated by equation (3), according to the initial elastic flexibility of a curve.
(3)E=24thcia0V1α0.There is a stable expansion stage of key cracks before the unstable expansion of concrete. When the crack opening displacementω is less than ω0, the stress is transferred, which is called the closing force. The stress intensity factor of the crack tip is
(4)KI=KIP−KIσ,where KIP is the stress intensity factor caused by concentrated force P.The three-point bending beam specimenKIp is directly calculated by the following linear elastic fracture mechanical equation:
(5)KIP=3PS2h2taf2ah,where
(6)f2ah=1.99−a/h1−a/h2.15−3.93a/h+2.7a/h21+2a/h1−a/h3/2.The calculation equation (7) of the stress intensity factor generated by the closing force is obtained.
(7)KIσ=∫a0a2σxFxa,ah/πadx.An approximate solution form of integral formula (7) is
(8)KIcc=−ZUe,V0Vc2PeπacFUe,Vc.It is obtained by equation (4) that when the crack is in the critical state of crack initiation, equation (9) is satisfied.
(9)KIcini=KIcun−KIcc.The instability fracture toughness is obtained by substituting the maximum loadPmax measured in the test and the ac calculated by equation (2) into equation (5). The crack initiation toughness is the stress intensity factor corresponding to the load Pini at the crack initiation of concrete. ac is substituted into equation (8) to obtain KIcc and then substituted into equation (9) to obtain crack initiation toughness.


### 4.2. Application Results of Fracture Criteria

According to the above derivation,α and β under each parameter are obtained, shown in Table 2.Table 2
Related indexes of softening curve for reactive powder concrete.
Specimen numberVffts (MPa)ft (MPa)λGf (KJ/m2)ω0 (mm)αβ0-109.698.2480.1500.1460.710-0.0020-20.1500.1460-30.1540.149Average value0.1510.1471-11%16.8014.28713.7657.1840.782-0.0051-210.2095.3281-310.3105.381Average value11.4285.9642-12%29.6424.68621.0615.7840.942-0.0252-225.5257.0102-318.5525.095Average value21.7135.9633-13%34.5429.36523.0294.6850.974-0.0253-230.4336.1913-325.8865.266Average value26.4495.381Takingα and β as the function of fiber volume content Vf, the polynomial model is adopted and fitted with MATLAB software. The fitting results are obtained in the following equations:
(10)α=−1.011Vf2+1.254Vf+0.699,(11)β=0.085Vf2−0.114Vf.The relationship offt=0.85fts is satisfied in steel fiber-reinforced concrete with different fiber volume content. The σ−ω softening curve is obtained, and the softening curve of reactive powder concrete without fiber is the most concave. The variations of the initiation load Pini and maximum load Pmax with fiber volume content are obtained, shown in Table 3.Table 3
Fracture energy and related indexes of reactive powder concrete.
Specimen numberPmax (kN)δmax (mm)mg (N)Gf (KJ×m−2)Du (m−1)lch (m)ft (MPa)0-11.590.215.960.1500.0940.0698.240-21.670.205.990.1500.0900.0660-31.490.236.010.1540.1030.076Average value1.580.215.990.1510.0960.0711-14.6410.216.0813.7652.9670.72714.281-24.477.866.1010.2092.2840.5601-34.388.106.1210.3102.3540.577Average value4.508.726.1011.4282.5350.6222-17.297.786.2521.0612.8890.23724.682-28.038.936.1725.5253.1790.2612-36.677.246.2218.5522.7810.228Average value7.337.986.2121.7132.9500.2423-19.676.546.3623.0292.3820.13829.363-212.007.266.3430.4332.5360.1473-310.416.966.3925.8862.4870.144Average value10.696.926.3626.4492.4680.143The initiation loadPini and maximum load Pmax of each group of reactive powder concrete specimens gradually increase with the increase of fiber volume content.


#### 4.2.1. Fracture EnergyGf

The fracture energy of the reactive powder concrete specimen without fiber is 0.151 KJ·m−2. After the fiber is added, the fracture energy increases by an order of magnitude. Taking the specimen with the lowest fiber content Vf=1.0% as an example, the fracture energy reaches 11.428 KJ·m−2, which is 76 times higher than that at Vf=0%. Since the fracture energy increases with the increase of fiber content, the fracture energy of the reactive powder concrete specimen at Vf=2.0% and Vf=3.0% is respectively 1.9 times and 2.3 times that at Vf=1%.


#### 4.2.2. Ductility IndexDu

The ductility indexDu is defined as the ratio the fracture energy Gf and load peak pmax in deflection curve.
(12)Du=GfPmax.The ductility index is employed to characterize the deformation resistance of reactive powder concrete. The greater the ductility index, the stronger the deformation resistance of the material. The ductility index of the reactive powder concrete without fiber is 0.096 m−1. After the fiber is added, the ductility index increases in the order of magnitude. Taking the specimen with the lowest fiber content Vf=1.0% as an example, the ductility index reaches 2.535 m−1, which is 26 times higher than that at Vf=0%.The effect of steel fiber on improving the ductility index of reactive powder concrete is obvious. When the fiber content exceeds 1%, the increase of the ductility index is small. The ductility index of the reactive powder concrete atVf=2.0% is 2.950 m−1, which is 0.415 m−1 compared with that at Vf=1%. The ductility index of the reactive powder concrete reaches the maximum at Vf=2.0%. When Vf=3.0%, the ductility index decreases with the increase of the fiber volume content, and the ductility index is 2.468 m−1. Because the peak strength of the reactive powder concrete increases with the increase of fiber volume content, especially at Vf=3.0%, the peak load reaches the maximum. However, the brittleness of the material increases with the increase of strength, and the P−δ curve becomes significantly steeper.


#### 4.2.3. Characteristic Lengthlch

The characteristic lengthL is employed to characterize the brittleness of concrete, which is defined in the following equation:
(13)lch=Gf×Eft2,whereE is the elastic modulus of concrete and ft is the tensile strength of concrete.The smaller the characteristic length, the greater the brittleness of the material. The characteristic length of each test piece is shown in Table3. The characteristic length of the reactive powder concrete without fiber is 0.071 m. After the fiber is added, the characteristic length increases. Taking the specimen with fiber content Vf=1.0% as an example, the characteristic length reaches 0.622 m. The characteristic length decreases with the increase of the fiber content. Besides, the characteristic length of reactive powder concrete is 0.242 m and 0.143 m at Vf=2.0% and Vf=3.0%, respectively.


## 4.1. Determination of Double-K Fracture Parameters

Because there is a stable crack propagation stage before the unstable fracture of concrete, the actual crack lengtha of the specimen before the unstable fracture is greater than the prefabricated joint length a0. If the stable crack propagation length Δac before the unstable fracture is recorded, then
(1)ac=a0+Δac.According to the linear progressive superposition principle, the critical effective crack lengthac is calculated by calculating the linear elastic fracture mechanical equation. When the external load P of the three-point bending beam reaches the maximum load Pmax, the corresponding critical effective crack length ac is determined by the following equation:
(2)CMODc=6PmaxSacth2EV1α,where CMODc is the crack opening displacement; E is the elastic modulus; S, t, and h are beam span, beam thickness, and beam height, respectively; the expression of V1α is V1α=0.76−2.28α+3.87α2−2.04α3+0.66/1−α2; and H0 is the blade thickness.The elastic modulus is calculated by equation (3), according to the initial elastic flexibility of a curve.
(3)E=24thcia0V1α0.There is a stable expansion stage of key cracks before the unstable expansion of concrete. When the crack opening displacementω is less than ω0, the stress is transferred, which is called the closing force. The stress intensity factor of the crack tip is
(4)KI=KIP−KIσ,where KIP is the stress intensity factor caused by concentrated force P.The three-point bending beam specimenKIp is directly calculated by the following linear elastic fracture mechanical equation:
(5)KIP=3PS2h2taf2ah,where
(6)f2ah=1.99−a/h1−a/h2.15−3.93a/h+2.7a/h21+2a/h1−a/h3/2.The calculation equation (7) of the stress intensity factor generated by the closing force is obtained.
(7)KIσ=∫a0a2σxFxa,ah/πadx.An approximate solution form of integral formula (7) is
(8)KIcc=−ZUe,V0Vc2PeπacFUe,Vc.It is obtained by equation (4) that when the crack is in the critical state of crack initiation, equation (9) is satisfied.
(9)KIcini=KIcun−KIcc.The instability fracture toughness is obtained by substituting the maximum loadPmax measured in the test and the ac calculated by equation (2) into equation (5). The crack initiation toughness is the stress intensity factor corresponding to the load Pini at the crack initiation of concrete. ac is substituted into equation (8) to obtain KIcc and then substituted into equation (9) to obtain crack initiation toughness.


## 4.2. Application Results of Fracture Criteria

According to the above derivation,α and β under each parameter are obtained, shown in Table 2.Table 2
Related indexes of softening curve for reactive powder concrete.
Specimen numberVffts (MPa)ft (MPa)λGf (KJ/m2)ω0 (mm)αβ0-109.698.2480.1500.1460.710-0.0020-20.1500.1460-30.1540.149Average value0.1510.1471-11%16.8014.28713.7657.1840.782-0.0051-210.2095.3281-310.3105.381Average value11.4285.9642-12%29.6424.68621.0615.7840.942-0.0252-225.5257.0102-318.5525.095Average value21.7135.9633-13%34.5429.36523.0294.6850.974-0.0253-230.4336.1913-325.8865.266Average value26.4495.381Takingα and β as the function of fiber volume content Vf, the polynomial model is adopted and fitted with MATLAB software. The fitting results are obtained in the following equations:
(10)α=−1.011Vf2+1.254Vf+0.699,(11)β=0.085Vf2−0.114Vf.The relationship offt=0.85fts is satisfied in steel fiber-reinforced concrete with different fiber volume content. The σ−ω softening curve is obtained, and the softening curve of reactive powder concrete without fiber is the most concave. The variations of the initiation load Pini and maximum load Pmax with fiber volume content are obtained, shown in Table 3.Table 3
Fracture energy and related indexes of reactive powder concrete.
Specimen numberPmax (kN)δmax (mm)mg (N)Gf (KJ×m−2)Du (m−1)lch (m)ft (MPa)0-11.590.215.960.1500.0940.0698.240-21.670.205.990.1500.0900.0660-31.490.236.010.1540.1030.076Average value1.580.215.990.1510.0960.0711-14.6410.216.0813.7652.9670.72714.281-24.477.866.1010.2092.2840.5601-34.388.106.1210.3102.3540.577Average value4.508.726.1011.4282.5350.6222-17.297.786.2521.0612.8890.23724.682-28.038.936.1725.5253.1790.2612-36.677.246.2218.5522.7810.228Average value7.337.986.2121.7132.9500.2423-19.676.546.3623.0292.3820.13829.363-212.007.266.3430.4332.5360.1473-310.416.966.3925.8862.4870.144Average value10.696.926.3626.4492.4680.143The initiation loadPini and maximum load Pmax of each group of reactive powder concrete specimens gradually increase with the increase of fiber volume content.


### 4.2.1. Fracture EnergyGf

The fracture energy of the reactive powder concrete specimen without fiber is 0.151 KJ·m−2. After the fiber is added, the fracture energy increases by an order of magnitude. Taking the specimen with the lowest fiber content Vf=1.0% as an example, the fracture energy reaches 11.428 KJ·m−2, which is 76 times higher than that at Vf=0%. Since the fracture energy increases with the increase of fiber content, the fracture energy of the reactive powder concrete specimen at Vf=2.0% and Vf=3.0% is respectively 1.9 times and 2.3 times that at Vf=1%.


### 4.2.2. Ductility IndexDu

The ductility indexDu is defined as the ratio the fracture energy Gf and load peak pmax in deflection curve.
(12)Du=GfPmax.The ductility index is employed to characterize the deformation resistance of reactive powder concrete. The greater the ductility index, the stronger the deformation resistance of the material. The ductility index of the reactive powder concrete without fiber is 0.096 m−1. After the fiber is added, the ductility index increases in the order of magnitude. Taking the specimen with the lowest fiber content Vf=1.0% as an example, the ductility index reaches 2.535 m−1, which is 26 times higher than that at Vf=0%.The effect of steel fiber on improving the ductility index of reactive powder concrete is obvious. When the fiber content exceeds 1%, the increase of the ductility index is small. The ductility index of the reactive powder concrete atVf=2.0% is 2.950 m−1, which is 0.415 m−1 compared with that at Vf=1%. The ductility index of the reactive powder concrete reaches the maximum at Vf=2.0%. When Vf=3.0%, the ductility index decreases with the increase of the fiber volume content, and the ductility index is 2.468 m−1. Because the peak strength of the reactive powder concrete increases with the increase of fiber volume content, especially at Vf=3.0%, the peak load reaches the maximum. However, the brittleness of the material increases with the increase of strength, and the P−δ curve becomes significantly steeper.


### 4.2.3. Characteristic Lengthlch

The characteristic lengthL is employed to characterize the brittleness of concrete, which is defined in the following equation:
(13)lch=Gf×Eft2,whereE is the elastic modulus of concrete and ft is the tensile strength of concrete.The smaller the characteristic length, the greater the brittleness of the material. The characteristic length of each test piece is shown in Table3. The characteristic length of the reactive powder concrete without fiber is 0.071 m. After the fiber is added, the characteristic length increases. Taking the specimen with fiber content Vf=1.0% as an example, the characteristic length reaches 0.622 m. The characteristic length decreases with the increase of the fiber content. Besides, the characteristic length of reactive powder concrete is 0.242 m and 0.143 m at Vf=2.0% and Vf=3.0%, respectively.


## 4.2.1. Fracture EnergyGf

The fracture energy of the reactive powder concrete specimen without fiber is 0.151 KJ·m−2. After the fiber is added, the fracture energy increases by an order of magnitude. Taking the specimen with the lowest fiber content Vf=1.0% as an example, the fracture energy reaches 11.428 KJ·m−2, which is 76 times higher than that at Vf=0%. Since the fracture energy increases with the increase of fiber content, the fracture energy of the reactive powder concrete specimen at Vf=2.0% and Vf=3.0% is respectively 1.9 times and 2.3 times that at Vf=1%.


## 4.2.2. Ductility IndexDu

The ductility indexDu is defined as the ratio the fracture energy Gf and load peak pmax in deflection curve.
(12)Du=GfPmax.The ductility index is employed to characterize the deformation resistance of reactive powder concrete. The greater the ductility index, the stronger the deformation resistance of the material. The ductility index of the reactive powder concrete without fiber is 0.096 m−1. After the fiber is added, the ductility index increases in the order of magnitude. Taking the specimen with the lowest fiber content Vf=1.0% as an example, the ductility index reaches 2.535 m−1, which is 26 times higher than that at Vf=0%.The effect of steel fiber on improving the ductility index of reactive powder concrete is obvious. When the fiber content exceeds 1%, the increase of the ductility index is small. The ductility index of the reactive powder concrete atVf=2.0% is 2.950 m−1, which is 0.415 m−1 compared with that at Vf=1%. The ductility index of the reactive powder concrete reaches the maximum at Vf=2.0%. When Vf=3.0%, the ductility index decreases with the increase of the fiber volume content, and the ductility index is 2.468 m−1. Because the peak strength of the reactive powder concrete increases with the increase of fiber volume content, especially at Vf=3.0%, the peak load reaches the maximum. However, the brittleness of the material increases with the increase of strength, and the P−δ curve becomes significantly steeper.


## 4.2.3. Characteristic Lengthlch

The characteristic lengthL is employed to characterize the brittleness of concrete, which is defined in the following equation:
(13)lch=Gf×Eft2,whereE is the elastic modulus of concrete and ft is the tensile strength of concrete.The smaller the characteristic length, the greater the brittleness of the material. The characteristic length of each test piece is shown in Table3. The characteristic length of the reactive powder concrete without fiber is 0.071 m. After the fiber is added, the characteristic length increases. Taking the specimen with fiber content Vf=1.0% as an example, the characteristic length reaches 0.622 m. The characteristic length decreases with the increase of the fiber content. Besides, the characteristic length of reactive powder concrete is 0.242 m and 0.143 m at Vf=2.0% and Vf=3.0%, respectively.


## 5. Conclusions

The fracture performance of the reactive powder concrete is experimentally studied by the fracture test of a precast cracked three-point bending beam with different fiber volume content in this paper, and the conclusions are obtained as follows.(1)
The whole process of crack generation and propagation of the three-point bending beam specimen of reactive powder concrete is obtained by the photoelastic patch method. Before the instability failure of the three-point bending beam specimen of the reactive powder concrete, there is a stable crack propagation process. The fracture process of the structure includes three stages, namely, the crack initiation, stable propagation, and instability failure. Besides, the order of photoelastic fringe, the critical effective crack length, the crack initiation load, and the maximum load of test parameters increase with the increase of fiber content in the test image of the beam specimen of reactive powder concrete(2)
According to the stripe image obtained by the photoelastic patch method, the stress distribution information of the three-point bending beam specimen of reactive powder concrete in the whole field is obtained, and the characteristics of stress distribution are intuitively displayed. By analyzing the test results, it is obvious that the stress distribution has obvious rationality and regularity. Through the numerical treatment of the whole curves of each group of specimens, the softening curve of reactive powder concrete is obtained(3)
Based on the test data in this paper, the double fracture parameters of reactive powder concrete specimens are obtained. Besides, the fracture energy, ductility index, and characteristic length of reactive powder concrete specimens with different fiber volume content are obtained. Besides, the calculated critical effective joint length, initiation toughness, toughness increment, and unstable fracture toughness increase with the increase of the steel fiber volume ratio, by analyzing the measured initiation load, maximum load, and critical effective joint length



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*Source: 1932642-2022-05-19.xml*