# Stress-Strain Relationship of Steel Fiber Reinforced Alkali Activated Slag Concrete under Static Compression

Authors: Xiaohui Yuan; Huiting Guan; Yanyu Shi
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

Stress-strain curve can accurately reflect the mechanical behavior of materials, and it is very important for structural design and nonlinear numerical analysis. Some cube and prism specimens were made to investigate the physical and mechanical properties of steel fiber reinforced alkali activated slag concrete (AASC); test results show that the strength, Young’s Elastic Modulus, and Poisson’s ratio all increase with the increase of steel fiber content. The steel fiber reinforced AASC shows an excellent postcracking behavior. Damage evolution parameter (D) was used to describe the formation and propagation of cracks, and continuum damage evolution model of steel fiber reinforced AASC was established by Weibull and Cauchy distribution. The establishing model can well describe the geometric characteristics of the key points of the concrete materials stress-strain curve. Finally, the accuracy of the model was verified by comparing the test stress-strain relationship curve of steel fiber reinforced AASC.


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## Body

## 1. Introduction

Alkali activated ground granulated blast-furnace slag (GGBS) is a kind of green building material that has shown great potential in replacing traditional Portland cement [1–3]. Meanwhile, GGBS is used as the cementing material of alkali activated slag concrete (AASC) to effectively reduce energy consumption and carbon dioxide emissions [4, 5]. The AASC has excellent characteristics such as high early strength, fast setting, high acid resistance, low permeability, and so on. However, the superior performance of AASC has not brought its wide application. First, although there are relevant industrial standards for GGBS [6, 7], it is only used as an additive in the actual use process [8]. Secondly, the phase composition and chemical composition of slag vary greatly in different areas, which has a great impact on the strength of AASC [9]. In addition, the higher dry shrinkage performance of AASC compared to the ordinary Portland cement concrete can lead to more shrinkage cracks in the actual use process, which also limits its application in practical engineering [10, 11].The drying shrinkage performance of AASC is an issue which has received a lot of attention in recent years. Adding expansion agent to the AASC is an effective way to improve its dry shrinkage cracking. The hydration of expansive agent can produce expansion products which can compensate concrete shrinkage. Jin and Al-Tabbaa [12] studied the shrinkage of Na2CO3 activated slag and found its drying shrinkage can be successfully reduced by adding active magnesia powder. Fang et al. [13] found that adding 4%–8% light burned magnesia in AASC can significantly reduce its shrinkage. Yuan et al. [14] found that calcium sulphoaluminate calcium oxide expansive agent has outstanding effect on improving the drying shrinkage of AASC. In addition, the crack resistance of concrete can be effectively improved by adding fiber. The fiber reinforcement effect on concrete can be provided by steel, glass, and organic polymer [15]. Previous studies have shown that adding an appropriate amount of steel fiber into the concrete will prevent the generation and expansion of internal cracks in concrete, because the tensile stress caused by drying shrinkage is taken by the steel fiber, so effectively improving the crack resistance of concrete [16, 17]. Meanwhile, the anticracking performance of steel fiber is also attributed to its bridging effect through the crack surface [18, 19]. Steel fiber reinforced concrete is widely used in civil engineering due to its superior mechanical properties [20, 21]. There are also relevant reports on fiber reinforced AASC in recent years [22, 23].Relevant scholars had also carried out a lot of research on the application of steel fiber reinforced concrete materials in members, which provides a good theoretical support for the popularization and application of steel fiber reinforced concrete. Chalioris [24] investigated the mechanical property of shear-critical reinforced concrete beams incorporated with steel fibers. Test results show that the addition of steel fibers makes the steel fiber reinforced concrete beam have good shear strength, toughness, and higher limit displacement in failure, that is because the steel fibers improved formation of cracks and enhanced energy dissipation capacities. The shape of the steel fibers will have a great impact on the mechanical performance of steel fiber reinforced concrete beams. For example, the beams reinforced with the hooked-end fibers had up to 38% higher shear strengths than those containing the crimped fibers [25]. Meanwhile, beams with higher amount of steel fibers will show better mechanical performance than that of lower content [26, 27]. In order to evaluate the performance of steel fiber reinforced concrete structure more accurately, simplified diverse embedment model was presented, which can be implanted on a nonlinear analysis for high-performance fiber reinforced concrete members or structures [28]. For steel fiber reinforced AASC, in order to be widely used in structure the same as the steel fiber reinforced common concrete, there must be a response material mathematical model.As known to all, the stress-strain relationship of materials is the key of structural design and numerical analysis. However, the failure mechanism of steel fiber reinforced concrete is extremely complex due to the existence of internal microcracks and holes. To simplify the problem, damage parameters can be used to describe the formation, propagation, and failure process of microcracks and microporous holes [29, 30]. Classical damage theory constitutive models have been proposed according to the mechanical properties, crack formation, and propagation mechanism and durability of concrete, such as Loland damage model [31], Mazars damage model [32], and Krajcinovic damage model [33]. The Loland damage model holds that the damage of materials exists all the time during the whole process of stress failure. Mazars damage model consider that there is only initial damage or no damage before the peak strain, which is because the stress-strain curve is linear before the peak, the real damage occurs after the peak stress. Krajcinovic damage model introduces the classical spring model into composite materials, modifies the material properties on microelements, and gives the physical meaning of randomness and nonlinearity of concrete materials more systematically. According to the definition of damage variable, if the spring element has displacement, damage occurs.The influence of steel fiber content on compressive properties of AASC was investigated by cube and prism compressive strength test. Young’s Elastic Modulus, Poisson’s ratio, and stress-strain relationship of steel fiber reinforced AASC are obtained by standard test method. Then according to the test results, statistical theory, and the characteristics of stress-strain curve, Weibull and Cauchy distribution are used for damage parameters before and after the peak stress to establish the continuum damage evolution model of the steel fiber reinforced AASC.


## 2. Materials and Methods

### 2.1. Materials

The GGBS is a water quenched slag from Xinxing cement factory in Xinxiang with the specific density of 3.27 g/cm3 and specific surface area of 854.6 m2/kg. The fly ash is produced by Henan Yulian energy factory with density of 2.187 g/cm3. The coarse aggregates are crashed limestone gravel, with the size between 5 and 20 mm. The fine aggregates are clean river sand with a fineness modulus of 2.99. Corrugated steel fibers with the length of 36 mm and effective diameter of 1.08 mm are used, and the tensile mechanical properties of the used fibers are 700 MPa. The shape of the corrugated steel fibers is shown in Figure 1.Figure 1
The shape of the corrugated steel fibers.The alkali activator solution mix proportion is listed in Table1, which is composed of water glass solution, chemically pure NaOH, and tap water (extra water was added to prepare the activator solution). The water glass solution has a modulus (n, SiO2 to Na2O molar ratio) of 2.93 containing 35.79% solid, and the solid contains 25.29% Na2O and 74.02% SiO2 (in mass). Sodium hydroxide was added to the water glass to change its modulus to 1.63.Table 1
Composition of the alkali activator solution for the AAS concrete.
Mass ratioMoisture content (%)Water glass solutionNaOHWater10010.3690.3573.5The steel fiber reinforced AASC mix proportion is shown in Table2. Different volume contents of steel fibers (0%, 0.5%, 0.9%, 1.0%, 1.1%, 1.2%, 1.3%, and 1.4%) are prepared for the specimens to investigate their influence on strength and stress-strain relationship.Table 2
Mix proportion design of steel fiber reinforced AASC.
Specimen IDMix proportion of the concrete (kg/m3)Solution to powder ratioWater to solid ratioSteel fiber content (%)GGBSFly ashFine aggregateCoarse aggregateActivator solutionSteel fiberSAC12801206151260298.500.750.550.0SAC242.390.5SAC375.360.9SAC484.781.0SAC594.201.1SAC6103.621.2SAC7113.041.3SAC8122.461.4


### 2.2. Experiment Methods

The steel fiber reinforced AASC is prepared firstly by mixing the sand, gravel, and steel fiber for 60 seconds, followed by adding the GGBS and fly ash and mixing for 60 second, then adding the activator solution, and continuing the mixing for 60 seconds. Specimens of size 100 mm × 100 mm × 100 mm are prepared for compressive strength test. Specimens of size 150 mm × 150 mm × 300 mm are prepared for Young’s Elastic Modulus, Poisson’s ratio, and stress-strain relationship test.Specimens of each group containing three cube specimens and three prism specimens are cast. There are all twenty-four cube specimens and twenty-four prism specimens, respectively. The specimens are removed from moulds after being cured for 24 hours and then cured in standard curing box (20 ± 2°C (RH ≥ 95%) for 27 d.The cube compressive strength (fcu) of steel fiber reinforced AASC is tested with a compressing machine (Model JES-2000) under the standard for test method of mechanical properties on ordinary concrete (GB/50081-2019) [34]. The stress-strain relationship of steel fiber reinforced AASC is tested with a microcomputer controlled electrohydraulic servo universal testing machine (Model WAW-3000).Figure2 is the experiment setup for stress-strain relationship test. Displacement meters of dial gauges one and two are installed symmetrically on both sides of the specimens, which are used to test the longitudinal displacement of the specimens under compression. The transverse strain gauges a and b are pasted in the middle of the specimen, which are used to test the transverse strain of the specimens under compression. Then the strain and μ can be calculated from the test data. The elastic modulus test method can refer to the standard for test method of mechanical properties on ordinary concrete (GB/50081-2019) [31].Figure 2
Experiment setup test.Before loading, several times of preloading were carried out, and the position of the specimen was adjusted according to the data of dial indicator and strain gauge to ensure that the specimen was in the state of axial compression. Start formal loading when the relative reading difference of strain gauge and dial indicator on both sides is within 5%. The test load is controlled by displacement, and the loading rate was 0.01 mm/s.


## 2.1. Materials

The GGBS is a water quenched slag from Xinxing cement factory in Xinxiang with the specific density of 3.27 g/cm3 and specific surface area of 854.6 m2/kg. The fly ash is produced by Henan Yulian energy factory with density of 2.187 g/cm3. The coarse aggregates are crashed limestone gravel, with the size between 5 and 20 mm. The fine aggregates are clean river sand with a fineness modulus of 2.99. Corrugated steel fibers with the length of 36 mm and effective diameter of 1.08 mm are used, and the tensile mechanical properties of the used fibers are 700 MPa. The shape of the corrugated steel fibers is shown in Figure 1.Figure 1
The shape of the corrugated steel fibers.The alkali activator solution mix proportion is listed in Table1, which is composed of water glass solution, chemically pure NaOH, and tap water (extra water was added to prepare the activator solution). The water glass solution has a modulus (n, SiO2 to Na2O molar ratio) of 2.93 containing 35.79% solid, and the solid contains 25.29% Na2O and 74.02% SiO2 (in mass). Sodium hydroxide was added to the water glass to change its modulus to 1.63.Table 1
Composition of the alkali activator solution for the AAS concrete.
Mass ratioMoisture content (%)Water glass solutionNaOHWater10010.3690.3573.5The steel fiber reinforced AASC mix proportion is shown in Table2. Different volume contents of steel fibers (0%, 0.5%, 0.9%, 1.0%, 1.1%, 1.2%, 1.3%, and 1.4%) are prepared for the specimens to investigate their influence on strength and stress-strain relationship.Table 2
Mix proportion design of steel fiber reinforced AASC.
Specimen IDMix proportion of the concrete (kg/m3)Solution to powder ratioWater to solid ratioSteel fiber content (%)GGBSFly ashFine aggregateCoarse aggregateActivator solutionSteel fiberSAC12801206151260298.500.750.550.0SAC242.390.5SAC375.360.9SAC484.781.0SAC594.201.1SAC6103.621.2SAC7113.041.3SAC8122.461.4


## 2.2. Experiment Methods

The steel fiber reinforced AASC is prepared firstly by mixing the sand, gravel, and steel fiber for 60 seconds, followed by adding the GGBS and fly ash and mixing for 60 second, then adding the activator solution, and continuing the mixing for 60 seconds. Specimens of size 100 mm × 100 mm × 100 mm are prepared for compressive strength test. Specimens of size 150 mm × 150 mm × 300 mm are prepared for Young’s Elastic Modulus, Poisson’s ratio, and stress-strain relationship test.Specimens of each group containing three cube specimens and three prism specimens are cast. There are all twenty-four cube specimens and twenty-four prism specimens, respectively. The specimens are removed from moulds after being cured for 24 hours and then cured in standard curing box (20 ± 2°C (RH ≥ 95%) for 27 d.The cube compressive strength (fcu) of steel fiber reinforced AASC is tested with a compressing machine (Model JES-2000) under the standard for test method of mechanical properties on ordinary concrete (GB/50081-2019) [34]. The stress-strain relationship of steel fiber reinforced AASC is tested with a microcomputer controlled electrohydraulic servo universal testing machine (Model WAW-3000).Figure2 is the experiment setup for stress-strain relationship test. Displacement meters of dial gauges one and two are installed symmetrically on both sides of the specimens, which are used to test the longitudinal displacement of the specimens under compression. The transverse strain gauges a and b are pasted in the middle of the specimen, which are used to test the transverse strain of the specimens under compression. Then the strain and μ can be calculated from the test data. The elastic modulus test method can refer to the standard for test method of mechanical properties on ordinary concrete (GB/50081-2019) [31].Figure 2
Experiment setup test.Before loading, several times of preloading were carried out, and the position of the specimen was adjusted according to the data of dial indicator and strain gauge to ensure that the specimen was in the state of axial compression. Start formal loading when the relative reading difference of strain gauge and dial indicator on both sides is within 5%. The test load is controlled by displacement, and the loading rate was 0.01 mm/s.


## 3. Results

The typical failure patterns for prismatic specimen of steel fiber reinforced AASC are shown in Figure3. There are many splitting tensile failure surfaces in the specimens without steel fiber, which shows obvious brittle features. The failure patterns of the specimens with steel fiber are mainly shear failure, which shows good ductility. Steel fibers marked in the red circle of Figure 3 can be seen clearly between the cracks; its bridging function effectively worked to prevent crack propagation in the process of loading.Figure 3
Typical failure patterns for prismatic specimen.The function of steel fiber in concrete substrate mainly depends on the bonding property between fiber and concrete, fiber content, and effective diameter. The function of steel fiber is characterized by steel fiber characteristic parameters (λf) [35]. λf = βVf(lf/df) = 40Vf, where Vf is volume fraction of steel fiber. lf is the length of steel fiber, df is the effective diameter of steel fiber, β is bond coefficient of steel fiber, when the steel fiber is corrugated type, and β value is 1.2 [36]. The test results are shown in Table 3.Table 3
Compressive strength, Young’s Elastic Modulus, Poisson’s ratio elastic modulus, peak strain, and ultimate strain test results.
Specimen IDVf (%)λfεc,rεcufcu (MPa)fc,r (MPa)E (GPa)μkSAC1000.001680.0022145.240.225.50.1481.32SAC20.50.200.001810.0024553.443.825.90.1711.35SAC30.90.360.001880.0025555.246.627.20.1891.36SAC41.00.400.001910.0026456.047.527.50.1991.38SAC51.10.440.002110.0032858.553.827.70.2021.55SAC61.20.480.002200.0040160.755.028.70.2101.82SAC71.30.520.002430.0044064.659.230.40.2191.81SAC81.40.560.003190.0058270.067.630.60.2261.82εc,r is peak strain, εcu is ultimate strain, fcu is the cube compressive strength, and fc,r is the axial compressive strength. The data of εc,r, εcu, fcu, fc,r, E, and μ in Table 3 are averaged data.It can be seen from the results of Table3 that the cube compressive strength and axial compressive strength increase with the increase of steel fiber content. When the content of steel fiber increases from 0% to 1.4%, the cube compressive strength and axial compressive strength increase from 37.0 MPa and 30.5 MPa to 70.8 MPa and 67.6 MPa, respectively; the strength increases more than 2 times. The results indicate that the reinforcement effect of steel fiber to AASC is obvious. This is mainly because the drying shrinkage of AASC is large, which will make a better adhesion force between steel fiber and AASC. The bridging effect through the crack surface is enhanced, steel fiber is hard to be pulled out, and then the ultimate strength is improved.Figure4 shows the relationship of cube compressive strength and prismatic compressive strength of steel fiber reinforced AASC with different steel fiber characteristic parameters. The fcu,0 and fcr,0 are the cube compressive strength and axial compressive strength of AASC without steel fiber. It shows a linear growth relationship in general. The influence coefficients of characteristic parameters on strength are 0.83 and 0.90, respectively. The results show that the influence of steel fiber on the strength of prism is greater than that of cube.Figure 4
Relationship ofλf with (a) fcu and (b) fc,r.
(a)(b)Figure5 shows the relationship of Young’s Elastic Modulus and Poisson’s ratio with steel fiber characteristic parameters. E0 and μ0 are Young’s Elastic Modulus and Poisson’s ratio of AASC without steel fiber. It shows that Young’s Elastic Modulus and Poisson’s ratio of steel fiber reinforced AASC increase with the increase of the characteristic parameters of the steel fiber. The results indicate that the toughness of AASC is improved gradually with the increase of steel fiber content.Figure 5
Relationship ofλf with (a) μ and (b) E.
(a)(b)According to the code for design of concrete structures (GB 50010-2010) [37], the ultimate strain εcu is the corresponding strain on the descending section of the stress-strain curve when the residual strength is reduced to 0.5 fc,r. The test values of peak strain εc,r and ultimate strain εcu are listed in Table 3. k is toughening effect factor of steel fiber, k = εcu/εc,r. Figure 6 shows the relationship of k with λf. k0 is the toughening effect factor without steel fiber. The k increase from 1.32 to 1.82, an increase 1.5 times when the content of steel fiber increases from 0% to 1.4%. The value of k affects the descending section of the stress-strain curve. The larger the k value, the more gentle the descending section, and the better the ductility of the material [37].Figure 6
Relationship ofk with λf.


## 4. Damage Constitutive Model

### 4.1. Damage Theory of Continuum

The difference of mechanical properties for concrete materials is mainly because of the different development rules of initial microcracks in the process of stress. The growth of microcracks in brittle materials usually ranges from 0.01 mm to 1 mm in width [38]. The main macrocrack gradually forms with the expansion of cracks, then eventually leading to concrete failure. Damage theory can be used to describe the nonlinear behavior of a progressive microcracking process of concrete [39, 40]. The complex discrete failure process of concrete can be described by damage evolution parameter (D) in continuum damage mechanics. D can be defined by(1)D=ADA,where A is the total area unit of concrete, and AD is the cross-sectional area of damage zone. D values range from 0 to 1. When D = 0, the concrete material element is without damage. When D = 1, it means that the concrete unit is completely damaged.Concrete is a quasi-brittle material; its damage process can be described by the strain produced by the load, so theD can be defined by [41].(2)dD=fεdε,ε=ζ,dε=dζ>0,0,ε<ζordε<0,where ζ > 0 is a variable threshold. Some scholars consider that the cut-off point from elastic damage to plastic damage should be 0.7 εc,r [42, 43], while others think that microcracks exist from the beginning, and damage will develop once loaded [38, 44]. In this study, it is considered that the damage always exists from the beginning of loading. F(x) is an assumed function; f(ε) is the density function of ε in the random damage probability model. If the initial damage condition is D = ε = ζ = 0, then the damage variable D is the distribution function of ε, as is shown in the following equation:(3)D=Fx=f0εxdx.Different forms of damage evolution functionF(x) can be selected for different damage mechanisms. Because the strength distribution of steel fiber AASC obeys Weibull statistical distribution [45, 46], the D of steel fiber AASC can be defined by(4)D=Fε=1−exp−εηm,where ε is the strain of steel fiber AASC, η is shape parameter, and m is scale parameter and its size reflects the toughness of the materials. Using Weibull distribution to describe the stress-strain curve of concrete, the ascending section is in good agreement [38, 42, 43, 46]. However, large differences are also shown in the descending section of the curve in some studies [44, 46]. Wu [46] compared the experimental and theoretical results and concluded that when the strain exceeds 2.5 times the peak strain, the theoretical curve will enter the complete failure state quickly.Comparing the existing concrete stress-strain curves, it can be found that the geometric characteristics of the key points of the curve are consistent although the shape of each curve is still different. The geometric characteristics of the stress-strain curve descending section are comprehensively described in the code for design of concrete [37], which considered that D obeys Cauchy distribution in the descending section of the stress-strain curve. The distribution function is shown in the following equation:(5)D=Fε=1−βαε/εc,r−12+ε/εc,r,where α is shape parameter, and its size affects the shape of the descending segment, and β is constant related to strength of concrete.


### 4.2. Continuum Damage Evolution Model

In this part, one continuum damage evolution model is developed to describe the microcrack development and failure process of steel fiber reinforced AASC. The Weibull distribution is used for the damage variable in the ascending section of the stress-strain curve, and the Cauchy distribution is used for the descending section of the stress-strain curve in this model.According to continuum damage mechanics, the stress-strain relationship can be expressed by(6)σ=1−DEε.Substituting equations (4) and (5) in equation (6) yields the stress-strain relationship of steel fiber reinforced AASC as follows:(7)σ=1−DEε=Eεexp−εηm,0≤ε≤εpk,Eεβαε/εc,r−12+ε/εc,r,εpk<ε..Derivingε on both sides of equation (7) (a), we get(8)dσdε=Eexp−εηm1−mεηm.According to the geometric characteristics of ascending section of stress-strain curve of concrete materials, equation (8) satisfies the following conditions:①ε=0, dσ/dε=E.②ε=0, σ=0.③ε=εc,r, dσ/dε=0.④ε=εc,r, σc,r=fc,r.Substituting conditions ① and ② in equation (8), we find the condition is tenable. Substituting condition ③ in equation (8), we get(9)Eexp−εc,rηm1−mεc,rηm=0.ForE≠0, exp−εc,r/ηm≠0, so we get(10)1−mεc,rηm=0.From equation (9) then we get(11)η=εc,r1/m1/m.Substituting condition ④ in equation (8), we get(12)σc,r=Eεexp−1m.Double logarithms are taken on both sides of equation (12) and we get(13)m=1lnEεc,r/σc,r.Substituting equations (11) and (13) in equation (7) (a) we can get the damage evolution equation of the ascending section of the steel fiber reinforced AASC stress-strain curve.In some studies, equation (7) (a) is the whole damage evolution equation of concrete stress-strain curve [38, 43, 44, 46]. From equation (7) (a) and parameters η and m we can see that if we know Young’s Elastic Modulus, peak strain, and peak stress of the concrete, the whole stress-strain curve will be obtained. The theory does not consider the geometric characteristics of the concrete stress-strain descending section. In fact, there is an inflection point and a maximum curvature point in the descending section of concrete stress-strain curve, and the curve converges to the coordinate axis but does not intersect with the coordinate axis when the strain tends to infinity [37].Because the ascending and descending sections of the stress-strain curve are continuous at the highest point, the boundary condition ④ is still valid for equation (7) (b). Substituting the condition ④ in equation (7) (b), we get(14)β=σc,rEεc,r.α is a parameter related to the shape of the descent segment. The ultimate strain εcu is the strain when the residual strength is reduced 0.5σc,r in the descending section of the stress-strain curve. Suppose ε = εcu = kεc,r, where k > 1. Substituting the condition in equation (14) we get(15)α=kk−12.Substituting equations (14) and (15) to equation (7) (b) we can get the damage evolution equation of the descending section of the steel fiber reinforced AASC stress-strain curve.


### 4.3. Discussion aboutm and α

Figure7 shows the relationship between strain rates (ratio of strain to peak strain) and D. Figure 7(a) shows the shape diagram of D curve under different values of m in the ascending section of the stress-strain curve. The results show that the damage increases with the increase of strain rate and the larger the value of m the smaller the initial damage. When m is 2, and the strain rate value is 1, the D is close to 0.4. But when m increases to 16 the D will reduce to near 0.05, indicating that it is more favorable to delay the damage of concrete by increasing the value of m. Figure 7(b) shows the shape diagram of D curve under different values of α in the descending section of the stress-strain curve. The results show that when the strain rate value is 1, the D is the same which is because the D is mainly determined by Young’s Elastic Modulus, peak stress, and peak strain of the concrete. This indicates that if the material is more brittle, the damage rate is greater, and α value is larger too. From the above analysis we know that with the bigger value of m and smaller value of α the materials will be more tough.Figure 7
Relationship between strain rates andD under different value of (a) m and (b) α.
(a)(b)Figure8 shows the relationship between m, α, and λf. When λf is 0, the corresponding m and α values of steel fiber reinforced AASC are 3.89 and 12.18, respectively. With the increase of steel fiber content, m increases and α decreases. When λf increases to 0.56, m increases to 16.52 and α decreases to 2.68. The toughness performance of steel fiber reinforced AASC is obviously improved with the increase of steel fiber content. The main reason is because the bridging microcracks ability of steel fiber in unit volume increases gradually with the increase of steel fiber content [42].Figure 8
Relationship ofλf with (a) m and (b) α.
(a)(b)


### 4.4. Verification and Discussion

Through experimental data in Table3, key parameters of AASC with different steel fiber characteristic parameters can be calculated form equation (11) to equation (15), as is shown in Table 4, and then the theoretical stress-strain curve of steel fiber reinforced AASC will be get from equation (7).Table 4
Key parameter of AASC with different steel fiber characteristic parameters.
Specimen IDηmαβSAC10.001723.8912.180.7732SAC20.002293.9411.620.7757SAC30.0021313.2711.420.9274SAC40.0022214.237.620.9322SAC50.0025015.054.830.9357SAC60.0027412.932.910.9256SAC70.0028916.162.750.9400SAC80.0037816.522.680.9413Figure9 shows the comparison between theoretical and experimental stress-strain curve of steel fiber reinforced AASC. As can be seen the theoretical curve is in good agreement with the experimental curve in the case of small strain; the large error is the case of big strain which is because the uncertainty of damage increases. The range of D variable is 0-1. The D curve almost coincides with the coordinate axis in the elastic section; the D in this stage is mainly the initial microcracks. After entering the plastic section, D increases gradually, and damage rate reaches the maximum value when the strain reaches to peak strain. Then the D growth rate reduces with the increase of strain, and the D value is getting close to 1. The point of D curve away from the coordinate axis moves back gradually with the increase of steel fiber content, which indicates that the addition of steel fiber can delay damage. From the D curves we can see the D value gradually decreases with the increase of steel fiber content under the same strain condition, which is mainly because the steel fiber crossing the crack prevents the propagation of microcrack.Figure 9
Comparison of experimental stress-strain curves for (a) SAC1; (b) SAC2; (c) SAC3; (d) SAC4; (e) SAC5; (f) SAC6; (g) SAC7; (h) SAC8 with the theoretical constitutive model.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4.1. Damage Theory of Continuum

The difference of mechanical properties for concrete materials is mainly because of the different development rules of initial microcracks in the process of stress. The growth of microcracks in brittle materials usually ranges from 0.01 mm to 1 mm in width [38]. The main macrocrack gradually forms with the expansion of cracks, then eventually leading to concrete failure. Damage theory can be used to describe the nonlinear behavior of a progressive microcracking process of concrete [39, 40]. The complex discrete failure process of concrete can be described by damage evolution parameter (D) in continuum damage mechanics. D can be defined by(1)D=ADA,where A is the total area unit of concrete, and AD is the cross-sectional area of damage zone. D values range from 0 to 1. When D = 0, the concrete material element is without damage. When D = 1, it means that the concrete unit is completely damaged.Concrete is a quasi-brittle material; its damage process can be described by the strain produced by the load, so theD can be defined by [41].(2)dD=fεdε,ε=ζ,dε=dζ>0,0,ε<ζordε<0,where ζ > 0 is a variable threshold. Some scholars consider that the cut-off point from elastic damage to plastic damage should be 0.7 εc,r [42, 43], while others think that microcracks exist from the beginning, and damage will develop once loaded [38, 44]. In this study, it is considered that the damage always exists from the beginning of loading. F(x) is an assumed function; f(ε) is the density function of ε in the random damage probability model. If the initial damage condition is D = ε = ζ = 0, then the damage variable D is the distribution function of ε, as is shown in the following equation:(3)D=Fx=f0εxdx.Different forms of damage evolution functionF(x) can be selected for different damage mechanisms. Because the strength distribution of steel fiber AASC obeys Weibull statistical distribution [45, 46], the D of steel fiber AASC can be defined by(4)D=Fε=1−exp−εηm,where ε is the strain of steel fiber AASC, η is shape parameter, and m is scale parameter and its size reflects the toughness of the materials. Using Weibull distribution to describe the stress-strain curve of concrete, the ascending section is in good agreement [38, 42, 43, 46]. However, large differences are also shown in the descending section of the curve in some studies [44, 46]. Wu [46] compared the experimental and theoretical results and concluded that when the strain exceeds 2.5 times the peak strain, the theoretical curve will enter the complete failure state quickly.Comparing the existing concrete stress-strain curves, it can be found that the geometric characteristics of the key points of the curve are consistent although the shape of each curve is still different. The geometric characteristics of the stress-strain curve descending section are comprehensively described in the code for design of concrete [37], which considered that D obeys Cauchy distribution in the descending section of the stress-strain curve. The distribution function is shown in the following equation:(5)D=Fε=1−βαε/εc,r−12+ε/εc,r,where α is shape parameter, and its size affects the shape of the descending segment, and β is constant related to strength of concrete.


## 4.2. Continuum Damage Evolution Model

In this part, one continuum damage evolution model is developed to describe the microcrack development and failure process of steel fiber reinforced AASC. The Weibull distribution is used for the damage variable in the ascending section of the stress-strain curve, and the Cauchy distribution is used for the descending section of the stress-strain curve in this model.According to continuum damage mechanics, the stress-strain relationship can be expressed by(6)σ=1−DEε.Substituting equations (4) and (5) in equation (6) yields the stress-strain relationship of steel fiber reinforced AASC as follows:(7)σ=1−DEε=Eεexp−εηm,0≤ε≤εpk,Eεβαε/εc,r−12+ε/εc,r,εpk<ε..Derivingε on both sides of equation (7) (a), we get(8)dσdε=Eexp−εηm1−mεηm.According to the geometric characteristics of ascending section of stress-strain curve of concrete materials, equation (8) satisfies the following conditions:①ε=0, dσ/dε=E.②ε=0, σ=0.③ε=εc,r, dσ/dε=0.④ε=εc,r, σc,r=fc,r.Substituting conditions ① and ② in equation (8), we find the condition is tenable. Substituting condition ③ in equation (8), we get(9)Eexp−εc,rηm1−mεc,rηm=0.ForE≠0, exp−εc,r/ηm≠0, so we get(10)1−mεc,rηm=0.From equation (9) then we get(11)η=εc,r1/m1/m.Substituting condition ④ in equation (8), we get(12)σc,r=Eεexp−1m.Double logarithms are taken on both sides of equation (12) and we get(13)m=1lnEεc,r/σc,r.Substituting equations (11) and (13) in equation (7) (a) we can get the damage evolution equation of the ascending section of the steel fiber reinforced AASC stress-strain curve.In some studies, equation (7) (a) is the whole damage evolution equation of concrete stress-strain curve [38, 43, 44, 46]. From equation (7) (a) and parameters η and m we can see that if we know Young’s Elastic Modulus, peak strain, and peak stress of the concrete, the whole stress-strain curve will be obtained. The theory does not consider the geometric characteristics of the concrete stress-strain descending section. In fact, there is an inflection point and a maximum curvature point in the descending section of concrete stress-strain curve, and the curve converges to the coordinate axis but does not intersect with the coordinate axis when the strain tends to infinity [37].Because the ascending and descending sections of the stress-strain curve are continuous at the highest point, the boundary condition ④ is still valid for equation (7) (b). Substituting the condition ④ in equation (7) (b), we get(14)β=σc,rEεc,r.α is a parameter related to the shape of the descent segment. The ultimate strain εcu is the strain when the residual strength is reduced 0.5σc,r in the descending section of the stress-strain curve. Suppose ε = εcu = kεc,r, where k > 1. Substituting the condition in equation (14) we get(15)α=kk−12.Substituting equations (14) and (15) to equation (7) (b) we can get the damage evolution equation of the descending section of the steel fiber reinforced AASC stress-strain curve.


## 4.3. Discussion aboutm and α

Figure7 shows the relationship between strain rates (ratio of strain to peak strain) and D. Figure 7(a) shows the shape diagram of D curve under different values of m in the ascending section of the stress-strain curve. The results show that the damage increases with the increase of strain rate and the larger the value of m the smaller the initial damage. When m is 2, and the strain rate value is 1, the D is close to 0.4. But when m increases to 16 the D will reduce to near 0.05, indicating that it is more favorable to delay the damage of concrete by increasing the value of m. Figure 7(b) shows the shape diagram of D curve under different values of α in the descending section of the stress-strain curve. The results show that when the strain rate value is 1, the D is the same which is because the D is mainly determined by Young’s Elastic Modulus, peak stress, and peak strain of the concrete. This indicates that if the material is more brittle, the damage rate is greater, and α value is larger too. From the above analysis we know that with the bigger value of m and smaller value of α the materials will be more tough.Figure 7
Relationship between strain rates andD under different value of (a) m and (b) α.
(a)(b)Figure8 shows the relationship between m, α, and λf. When λf is 0, the corresponding m and α values of steel fiber reinforced AASC are 3.89 and 12.18, respectively. With the increase of steel fiber content, m increases and α decreases. When λf increases to 0.56, m increases to 16.52 and α decreases to 2.68. The toughness performance of steel fiber reinforced AASC is obviously improved with the increase of steel fiber content. The main reason is because the bridging microcracks ability of steel fiber in unit volume increases gradually with the increase of steel fiber content [42].Figure 8
Relationship ofλf with (a) m and (b) α.
(a)(b)


## 4.4. Verification and Discussion

Through experimental data in Table3, key parameters of AASC with different steel fiber characteristic parameters can be calculated form equation (11) to equation (15), as is shown in Table 4, and then the theoretical stress-strain curve of steel fiber reinforced AASC will be get from equation (7).Table 4
Key parameter of AASC with different steel fiber characteristic parameters.
Specimen IDηmαβSAC10.001723.8912.180.7732SAC20.002293.9411.620.7757SAC30.0021313.2711.420.9274SAC40.0022214.237.620.9322SAC50.0025015.054.830.9357SAC60.0027412.932.910.9256SAC70.0028916.162.750.9400SAC80.0037816.522.680.9413Figure9 shows the comparison between theoretical and experimental stress-strain curve of steel fiber reinforced AASC. As can be seen the theoretical curve is in good agreement with the experimental curve in the case of small strain; the large error is the case of big strain which is because the uncertainty of damage increases. The range of D variable is 0-1. The D curve almost coincides with the coordinate axis in the elastic section; the D in this stage is mainly the initial microcracks. After entering the plastic section, D increases gradually, and damage rate reaches the maximum value when the strain reaches to peak strain. Then the D growth rate reduces with the increase of strain, and the D value is getting close to 1. The point of D curve away from the coordinate axis moves back gradually with the increase of steel fiber content, which indicates that the addition of steel fiber can delay damage. From the D curves we can see the D value gradually decreases with the increase of steel fiber content under the same strain condition, which is mainly because the steel fiber crossing the crack prevents the propagation of microcrack.Figure 9
Comparison of experimental stress-strain curves for (a) SAC1; (b) SAC2; (c) SAC3; (d) SAC4; (e) SAC5; (f) SAC6; (g) SAC7; (h) SAC8 with the theoretical constitutive model.
(a)(b)(c)(d)(e)(f)(g)(h)


## 5. Conclusions

This paper studies the compress strength, Young’s Elastic Modulus, Poisson’s ratio, and the stress-strain curve of steel fiber reinforced AASC, and based on the acquired results one damage statistical mathematics model for steel fiber reinforced AASC is proposed. The main conclusions from the present study are as follows:(1)
Steel fiber has a good effect on strengthening and toughening of AASC. Adding steel fiber into AASC can effectively improve its strength. The bridging effect of steel fiber is more and more obvious with the increase of content.(2)
The established continuum damage evolution model for steel fiber reinforced AASC can well describe the whole destruction process.(3)
Them and α are important parameters affecting the stress-strain relationship. Damage always exists in concrete, and the damage rate reaches the maximum at the peak strain.



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*Source: 7951646-2021-11-30.xml*