Stress-Strain Relationship of Steel Fiber Reinforced Alkali Activated Slag Concrete under Static Compression
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.
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 . Secondly, the phase composition and chemical composition of slag vary greatly in different areas, which has a great impact on the strength of AASC . 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  studied the shrinkage of Na2CO3 activated slag and found its drying shrinkage can be successfully reduced by adding active magnesia powder. Fang et al.  found that adding 4%–8% light burned magnesia in AASC can significantly reduce its shrinkage. Yuan et al.  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 . 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  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 . 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 . 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 , Mazars damage model , and Krajcinovic damage model . 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
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.
The alkali activator solution mix proportion is listed in Table 1, 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.
The steel fiber reinforced AASC mix proportion is shown in Table 2. 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.
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) . The stress-strain relationship of steel fiber reinforced AASC is tested with a microcomputer controlled electrohydraulic servo universal testing machine (Model WAW-3000).
Figure 2 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) .
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.
The typical failure patterns for prismatic specimen of steel fiber reinforced AASC are shown in Figure 3. 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.
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) . 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 . The test results are shown in Table 3.
It can be seen from the results of Table 3 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.
Figure 4 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 5 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.
According to the code for design of concrete structures (GB 50010-2010) , 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 .
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 . 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 bywhere 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 the D can be defined by .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:
Different forms of damage evolution function F(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 bywhere 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  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 , 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: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
Substituting equations (4) and (5) in equation (6) yields the stress-strain relationship of steel fiber reinforced AASC as follows:
Deriving ε on both sides of equation (7) (a), we get
According to the geometric characteristics of ascending section of stress-strain curve of concrete materials, equation (8) satisfies the following conditions: ① , . ② , . ③ , . ④ , .
Substituting conditions ① and ② in equation (8), we find the condition is tenable. Substituting condition ③ in equation (8), we get
For , , so we get
From equation (9) then we get
Substituting condition ④ in equation (8), we get
Double logarithms are taken on both sides of equation (12) and we get
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 .
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
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
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 about m and α
Figure 7 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 8 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 .
4.4. Verification and Discussion
Through experimental data in Table 3, 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).
Figure 9 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.
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)The m 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.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare no conflicts of interest.
Conceptualization was done by Yuan X. H.; methodology was developed by Yuan X. H.; validation was performed by Yuan X. H.; investigation was performed by Chen X. Y.; data curation was performed by Shi Y. Y; original draft preparation was done by Shi Y. Y; review and editing were done by Yuan X. H.; visualization was performed by Yuan X. H.; supervision was done by Chen X. Y.
T. Bakharev, “Resistance of geopolymer materials to acid attack,” Cement and Concrete Research, vol. 35, no. 7, pp. 658–670, 2005.View at: Publisher Site | Google Scholar
P. Duxson, A. Fernández-Jimfénez, and J. L. Provis, “Geopolymer technology: the current state of the art,” Journal of Materials Science, vol. 42, no. 12, pp. 2917–2933, 2006.View at: Publisher Site | Google Scholar
C. Shi, A. F. Jiménez, and A. Palomo, “New cements for the 21st century: the pursuit of an alternative to Portland cement,” Cement and Concrete Research, vol. 41, no. 7, pp. 750–763, 2011.View at: Publisher Site | Google Scholar
S. A. Bernal, J. L. Provis, V. Rose, and R. M. Gutiérrez, “High-resolution X-ray diffraction and fluorescence microscopy characterization of alkali-activated slag-metakaolin binders,” Journal of the American Ceramic Society, vol. 96, no. 6, pp. 1951–1957, 2013.View at: Publisher Site | Google Scholar
Y. J. Zhang and Q. Chai, “Alkali-activated blast furnace slag-based nanomaterial as a novel catalyst for synthesis of hydrogen fuel,” Fuel, vol. 115, no. 7, pp. 84–87, 2014.View at: Publisher Site | Google Scholar
Standardization Administration of China, Ground Granulated Blast Furnace Slag Used for Cement and Concrete, China Standards Press, Inc, Beijing, China, 2008.
Standardization Administration of China, Test Method for Activity of Industrial Waste Slag Used as Addition to Cement, China Standards Press, Inc, Beijing, China, 2005.
Y. Gao, J. Xu, and E. Bai, “Static and dynamic mechanical properties of high early strength alkali activated slag concrete,” Ceramics International, vol. 41, no. 10, pp. 12901–12909, 2015.View at: Publisher Site | Google Scholar
D. M. Roy, “Alkali-activated cements opportunities and challenges,” Cement and Concrete Research, vol. 29, no. 2, pp. 249–254, 1999.View at: Publisher Site | Google Scholar
W. Chen, H. J. H. Brouwers, and Z. H. Shui, “Three-dimensional computer modeling of slag cement hydration,” Journal of Materials Science, vol. 42, no. 23, pp. 9595–9610, 2007.View at: Publisher Site | Google Scholar
A. A. Melo Neto, M. A. Cincotto, and W. Repette, “Drying and autogenous shrinkage of pastes and mortars with activated slag cement,” Cement and Concrete Research, vol. 38, no. 4, pp. 565–574, 2007.View at: Google Scholar
F. Jin and A. Al-Tabbaa, “Strength and drying shrinkage of slag paste activated by sodium carbonate and reactive MgO,” Construction and Building Materials, vol. 81, no. 1, pp. 58–65, 2015.View at: Publisher Site | Google Scholar
Y. Fang, J. Liu, and Y. Chen, “Effect of magnesia on properties and microstructure of alkali-activated slag cement,” Water Science and Engineering in, vol. 4, no. 4, pp. 463–469, 2011.View at: Publisher Site | Google Scholar
X. Yuan, W. Chen, and Z. Lu, “Shrinkage compensation of alkali-activated slag concrete and microstructural analysis,” Construction and Building Materials, vol. 66, no. 3, pp. 422–428, 2014.View at: Publisher Site | Google Scholar
T. L. Teng, Y. A. Chu, and F. A. Chang, “Calculating the elastic moduli of steel-fiber reinforced concrete using a dedicated empirical formula,” Computational Materials Science, vol. 31, no. 3, pp. 337–346, 2004.View at: Publisher Site | Google Scholar
L. Liao, A. De la Fuente, and S. Cavalaro, “Design procedure and experimental study on fibre reinforced concrete segmental rings for vertical shafts,” Materials & Design, vol. 92, no. 6, pp. 590–601, 2016.View at: Publisher Site | Google Scholar
N. Mohamed, A. M. Soliman, and M. L. Nehdi, “Mechanical performance of full-scale precast steel fibre-reinforced concrete pipes,” Engineering Structures, vol. 84, no. 11, pp. 287–299, 2014.View at: Publisher Site | Google Scholar
X. Shi, P. Park, and Y. Rew, “Constitutive behaviors of steel fiber reinforced concrete under uniaxial compression and tension,” Construction and Building Materials, vol. 233, no. 5, pp. 117–316, 2020.View at: Publisher Site | Google Scholar
S. Zhang, C. Zhang, and L. Liao, “Investigation into the effect of fibre distribution on the post-cracking tensile strength of SFRC through physical experimentation and numerical simulation,” Construction and Building Materials, vol. 248, no. 1, pp. 118–433, 2020.View at: Publisher Site | Google Scholar
I. L. Larsen and R. T. Thorstensen, “The influence of steel fibres on compressive and tensile strength of ultra high performance concrete: a review,” Construction and Building Materials, vol. 256, no. 3, pp. 119–459, 2020.View at: Publisher Site | Google Scholar
Y. Zhang, Y. Zhu, and S. Qu, “Improvement of flexural and tensile strength of layered-casting UHPC with aligned steel fibers,” Construction and Building Materials, vol. 251, no. 2, pp. 118–893, 2020.View at: Publisher Site | Google Scholar
Y. Lee, J. Choi, and H. Kim, “Effects of a defoamer on the compressive strength and tensile behavior of alkali-activated slag-based cementless composite reinforced by polyethylene fiber,” Composite Structures, vol. 172, no. 6, pp. 166–172, 2017.View at: Publisher Site | Google Scholar
Ü. Yurt, “An experimental study on fracture energy of alkali activated slag composites incorporated different fibers,” Journal of Building Engineering, vol. 32, pp. 101–519, 2020.View at: Publisher Site | Google Scholar
C. E. Chalioris, “Steel fibrous RC beams subjected to cyclic deformations under predominant shear,” Engineering Structures, vol. 49, no. 4, pp. 104–118, 2013.View at: Publisher Site | Google Scholar
K. Jain and B. Singh, “Deformed steel fibres as minimum shear reinforcement-An investigation,” Structures in, vol. 7, no. 8, pp. 126–137, 2016.View at: Publisher Site | Google Scholar
S. W. Choi, J. Choi, and S. C. Lee, “Probabilistic analysis for strain-hardening behavior of high-performance fiber-reinforced concrete,” Materials, vol. 12, no. 15, pp. 1–24, 2019.View at: Publisher Site | Google Scholar
V. K. Kytinou, C. E. Chalioris, and C. G. Karayannis, “Analysis of residual flexural stiffness of steel fiber-reinforced concrete beams with steel reinforcement,” Materials, vol. 13, no. 12, pp. 1–32, 2020.View at: Publisher Site | Google Scholar
S. W. Choi, J. Choi, and S. C. Lee, “Effect of steel fibers on the hysteretic performance of concrete beams with steel reinforcement-tests and analysis,” Materials, vol. 13, no. 13, pp. 1–15, 2020.View at: Publisher Site | Google Scholar
D. Krajcinovic and M. G. Silva, “Statistical aspects of the continuous damage theory,” International Journal of Solids and Structures, vol. 18, no. 7, pp. 551–562, 1982.View at: Publisher Site | Google Scholar
Z. Yi, L. Hualin, and L. Gensheng, “A statistical constitutive model for rock continuous damage,” Journal of the University of Petroleum, vol. 28, no. 3, pp. 37–40, 2004.View at: Publisher Site | Google Scholar
K. E. Loland, “Continuous damage model for load-response estimation of concrete,” Cement and Concrete Research, vol. 10, no. 3, pp. 395–402, 1980.View at: Publisher Site | Google Scholar
J. Mazars and J. Lemaitre, “Application of continuous damage mechanics to strain and fracture behavior of concrete,” Application of Fracture Mechanics to Cementitious Composites, vol. 94, pp. 507–520, 1985.View at: Publisher Site | Google Scholar
D. Krajcinovic, “Constitutive equations for damaging materials,” Journal of Applied Mechanics, vol. 50, no. 2, pp. 355–360, 1983.View at: Publisher Site | Google Scholar
Ministry of Housing and Urban-Rural Development of the People's Republic of China, Standard for Test Methods of Concrete Physical and Mechanical Properties, China Building Industry Press, Inc, Beijing, China, 2019.
R. Narayanan and I. Y. S. Darwish, “Use of steel fibers as shear reinforcement,” ACI Structural Journal, vol. 84, no. 3, pp. 216–227, 1987.View at: Publisher Site | Google Scholar
S. Altaan, “Deformation and ultimate strength in flexure of reinforced concrete beams made with steel fiber concrete,” Journal of the American Concrete Institute, vol. 78, no. 5, pp. 395–405, 1981.View at: Publisher Site | Google Scholar
Ministry of Housing and Urban-Rural Development of the People's Republic of China, Code for Design of Concrete Structures, China Building Industry Press, Inc, Beijing, China, 2010.
Z. Wang, Y. Liu, and R. F. Shen, “Stress–strain relationship of steel fiber-reinforced concrete under dynamic compression,” Construction and Building Materials, vol. 22, no. 5, pp. 811–819, 2008.View at: Publisher Site | Google Scholar
R. Huang, L. Meng, and J. Qin, “Hydrostatic pressure–volumetric strain relationship of steel fiber-reinforced concrete under different stress conditions,” Construction and Building Materials, vol. 259, no. 12, pp. 119–823, 2020.View at: Publisher Site | Google Scholar
J. Lv, T. Zhou, and Q. Du, “Evaluation of stress-strain behavior of self-compacting rubber lightweight aggregate concrete under uniaxial compression loading,” Materials, vol. 12, no. 24, pp. 1–17, 2019.View at: Publisher Site | Google Scholar
W. Cao, H. Zhao, and L. Zhang, “Damage statistical softening constitutive model for rock considering effect of damage thereshold and its papameters determination method,” Chinese Journal of Rock Mechanics and Engineering, vol. 27, no. 6, pp. 1148–1154, 2008.View at: Publisher Site | Google Scholar
L. Xu, C. Li, and B. Li, “Investigation on 1D elasto-plastic constitutive model of steel fiber reinforced concrete under uniaxial cyclic compression,” China Civil Engineering Journal, vol. 51, no. 11, pp. 77–87, 2018.View at: Google Scholar
Y. Zhongbao, L. Yongchi, and Z. Kai, “A constitutive relation of steel fiber reinforced concrete under low strain rate,” Journal of Experimental Mechanics, vol. 34, no. 2, pp. 284–288, 2019.View at: Publisher Site | Google Scholar
X. Ning and Y. Ding, “Effect of steel fiber on the damage constitutive model of concrete under unixial compression,” Journal of Building Materials, vol. 18, no. 2, pp. 214–220, 2015.View at: Publisher Site | Google Scholar
W. Wellbull, “A Statistical distribution function of wide applicability,” Journal of Applied Mechanics, vol. 18, no. 3, pp. 293–297, 1951.View at: Publisher Site | Google Scholar
Z. Wu, “Investigation of concrete tensive and compressive constitutive model based on damage mechanics,” Water Resources and Hydropower Engineering, vol. 37, no. 11, pp. 58–63, 1995.View at: Google Scholar