# Dynamic Compression Behavior and a Damage Constitutive Model of Steel Fibre Reinforced Self-Compacting Concrete

Authors: Liang Bian; Jian Ma; Jie Zhang; Ping Li
Journal: Advances in Materials Science and Engineering
 (2021)

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## Abstract

The dynamic behavior of steel fibre reinforced self-compacting concrete (SFRSCC) was investigated by using a split Hopkinson pressure bar (SHPB). SFRSCC specimens with two strength classes of about 40 MPa and 60 MPa were prepared. Different steel fibre volume fractions were used varying from 0.5% to 2.0%. The tested strain rate ranged from about 50 to 240 s−1. Signiﬁcant rate dependence was observed, and dynamic increase factor (DIF) was used to quantify the rate sensitivity. The results showed that both the matrix strength and fibre content had effect on the strain rate sensitivity of SFRSCC. A DIF formula was proposed for describing the dynamic strength of SFRSCC at high strain rates, and a dynamic damage constitutive model was derived to describe the stress-strain relationship of SFRSCC. The parameters in the model were determined by fitting the experimental data. Good consistency between theoretical curves and experimental data was obtained.


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## Body

## 1. Introduction

Concrete structures may be exposed to impact loading conditions during their functional life, such as moving vehicles impacts, violent earthquakes, bomb blasts, and missile attacks [1–3]. However, plain concrete exhibits insufficient capacity to resist impact loads due to its low tensile strength and poor resistance to cracking. Adding fibres can improve the energy absorption capacity of the matrix and enhancing the mechanical properties and ductility of concrete [4–6]. Among the fibres, the most commonly used are the steel fibres which provide reinforcement by bridging the cracks in concrete under various loads. It was also found that concrete reinforced with steel fibres showed better impact resistance than other fibres such as polypropylene (PP) fibres or polyethylene (PE) fibres [7, 8]. Steel fibre reinforced concrete is being widely used in civil and military structural applications including road pavements, bridges, channel lining, offshore structures, and military infrastructures [9, 10].Although fibre reinforced concrete has its advantages in improving the impact resistance, it must be noted that the addition of fibres will reduce the workability of concrete and increase the difficulty in vibration work [11, 12]. In addition, the decrease of workability may adversely affect the uniformity of fibre distribution in concrete, which may cause a detrimental effect on the mechanical performance of fibre reinforced concrete [13]. One of the feasible solutions would be using self-compacting concrete (SCC) to enhance the fluidity of the matrix. SCC is a special concrete mixture which can consolidate under its own weight without extra vibration [14]. The mix of steel fibres and SCC is commonly referred to as steel fibre reinforced self-compacting concrete (SFRSCC).There have been conducted a number of studies on the static behavior of SFRSCC [15–18]. The mechanical properties of SFRSCC are influenced by a great number of parameters including the content and type of cement, aggregate content and size, properties (shape, length, and diameter) of steel fibres, and volume fraction of steel fibres [19–21]. There are still some disagreements in understanding of mechanical properties of SFRSCC. It is difficult to compare the results obtained by different researchers because of the different adopted mix-design and test procedures [22]. In case of advanced design using computer programs, it is necessary to know exact mechanical properties. Thus, a procedure for identifying mechanical and fracture properties of SFRSCC by inverse analysis was presented in [23].It has been shown that concrete properties such as compressive strength and tensile strength are rate dependent [24–26]. Therefore, the mechanical properties at quasi-static state cannot be applied to structures when dynamic effects are significant [27]. Characterization of these properties under dynamic loading conditions is a necessity. Compared with the quasi-static tests, the dynamic tests are more difficult to conduct, and many additional conditions have to be considered to obtain the real response of SFRSCC to high strain rates [28]. The main devices used to imitate the different strain rates are drop hammer machines and the split Hopkinson pressure bar (SHPB). Kantar et al. [29] presented a study on the impact dynamics of SFRSCC mixes with different fibre contents using a drop-weight testing apparatus. It was found that the impact resistances and the energy dissipation capacities of the SFRSCC plates can be significantly higher than that of pure SCC plates by 6–14 times. Ruiz et al. [27] studied the dynamic mixed-mode fracture of SFRSCC using three-point bending tests. Results showed that the peak load increased with the increase of loading velocity and fibre content. Abid et al. [30] conducted an experimental work to evaluate the performance of SFRSCC under repeated impact loading. The inclusion of micro-steel fibre was found to increase the impact resistance ranging from 150 to 860% compared to plain samples. Li et al. [31] investigated the dynamic properties of self-compacting concrete with 0.5%, 0.75%, and 1.0% steel fibres using a split Hopkinson pressure bar. Results showed that the impact properties including failure modes, peak stress, peak strain, and elastic modulus were all significantly influenced by strain rate and steel fibre content. Steel fibre reinforced SCC showed a more remarkable strain rate effect than that of steel fibre reinforced normal concrete.It can be seen that the dynamic mechanical properties of SFRSCC are more complex than those under static conditions, and there are few studies on the dynamic damage constitutive model of SFRSCC. It is well known that the failure mechanism of brittle material is complicated since flaws such as cracks and voids commonly exist within the material. These discontinuities have significant effects on the damage evolution and failure characteristics of SFRSCC. It is crucial to develop a model that can describe the damage evolution of SFRSCC under dynamic loading for theoretical and numerical analysis. Burlion et al.[32] assumed that the microvoid evolution is controlled by the plastic volumetric strain, and they proposed a compression damage model for concrete. Johnson et al. [33] also proposed a damage model described by equivalent plastic strain and plastic volumetric strain of concrete. Although the simulation results using the above models were in good agreement with the experiments in some impact problems, these empirical damage models did not give an appropriate definition of damage from the microscopic point of view.In this study, two strength classes of SFRSCC (40 MPa and 60 MPa) were designed. Dynamic compression tests were carried out to investigate the influence of loading rate, matrix strength, and fibre content on the dynamic mechanical properties of SFRSCC. A dynamic damage constitutive model of SFRSCC was proposed based on the equivalent microvoid hypothesis [34], and the relevant parameters in the model were determined by fitting the experimental data.


## 2. Materials and Methods

### 2.1. Mix Proportions

Two types of self-compacting concrete mixtures, with static uniaxial compressive strengths of about 40 MPa and 60 MPa, were designed. Four quantities of steel fibres, 0.5%, 1.0%, 1.5%, and 2.0% in volume ratio, were added to each type of SCC mixture. MC-FX and HC-FX are used for coding for the different mixtures. MC-FX stands for medium strength (40 MPa) SCC, andX is the steel fibre volume fraction in percent. HC-FX stands for high strength (60 MPa) SCC with the steel fibre volume fraction of X percent. The material composition and mix proportions are given in Table 1. The steel fibres, coarse granite aggregate with a maximum size of 16 mm, and natural sand with average fineness modulus of 2.39 are shown in Figure 1. The diameter and length of the steel fibres are 0.2 mm and 10 mm, respectively, the yield strength of the steel fibres is 780 MPa, and the tensile strength is 2800 MPa.Table 1
Mix proportions of SFRSCC (kg/m3).
MixCementSandCoarse aggregateFly ashWaterSuperplasticizerSteel fibresMC-F0.54277248621071712.639MC-F1.04277218561071712.678MC-F1.54277148491071712.6117MC-F2.04277088431071712.0156HC-F0.5494714831491483.339HC-F1.0494696810491484.478HC-F1.5494678789491485.1117HC-F2.0494658778491483.6156Figure 1
Materials used in this study: (a) steel fibres; (b) coarse aggregate; (c) sand.
(a)(b)(c)


### 2.2. Mechanical and Rheological Properties

The fluidity and rheology of the fresh SFRSCC mixtures were evaluated by slump flow and L-box tests following the EFNARC guidelines [35]. The measured values for dm, the largest diameter of the circular spread for concrete to stop ﬂowing (Figure 2), and the ratio of the concrete height of the horizontal and vertical sections in L-box are listed in Table 2.Figure 2
Slump ﬂow of fresh SFRSCC.Table 2
Workability of SFRSCC.
MixSlump flowdm (mm)∗L-boxh2/h1#MC-F0.57950.91MC-F1.08450.91MC-F1.57500.93MC-F2.07100.90HC-F0.57550.89HC-F1.07500.93HC-F1.57600.93HC-F2.07650.94∗slump ﬂow values for proper SCC should be within 550–850 mm [35]; #h2/h1 values for proper SCC should be ≥0.8 [35].As for slump flow, all types of SFRSCC exhibited satisfactory results in the range of 710–845 mm, demonstrating acceptable filling ability for practical applications. In L-box tests, the values were also in a practically acceptable range: 0.89–0.94. During the tests, no indication of segregation was observed, and the mixtures maintained good homogeneity and cohesion.The quasi-static compression tests of SFRSCC were carried out on a hydraulic testing machine with a load capacity of 2000 kN. The specimens were 150 mm cubes at the age of 28 days. Brazilian disc tests were also conducted to obtain the split tensile strength using cylindrical specimens of dimension 70 mm × 35 mm (diameter × height). The basic mechanical properties of SFRSCC are listed in Table3. The quasi-static compressive strength of SFRSCC for the same type of mixture did not increase with the increase of fibres. Proper amount of steel fibre provides reinforcement for concrete. Higher fibers concentrations, however, may destroy the uniformity of concrete and cause a decrease of the compressive strength. As still more steel ﬁbers are included, the reinforcement effect provides a gradual increase of the strength [18].Table 3
Mechanical properties of SFRSCC (standard deviation in parentheses).
MixDensity (g/cm3)Elastic modulus (GPa)Compressive strength (MPa)Splitting tensile strength (MPa)MC-F0.52.22 (0.01)35.6 (2.3)49.8 (1.1)3.78 (0.25)MC-F1.02.27 (0.01)35.0 (1.7)43.8 (1.9)4.08 (0.15)MC-F1.52.37 (0.03)34.9 (1.1)46.6 (0.9)4.34 (0.11)MC-F2.02.41 (0.02)33.4 (1.5)48.8 (1.8)5.14 (0.65)HC-F0.52.38 (0.01)36.6 (1.1)60.1 (0.3)4.73 (0.10)HC-F1.02.39 (0.03)38.0 (1.3)69.6 (1.4)5.37 (0.24)HC-F1.52.41 (0.02)37.7 (2.1)64.9 (1.0)4.93 (0.18)HC-F2.02.42 (0.01)36.3 (1.3)66.1 (0.4)5.40 (0.40)


### 2.3. Dynamic Compression Tests

A 74 mm diameter split Hopkinson pressure bar (SHPB) [36] was used to conduct dynamic compression tests at strain rates between 50 and 240 s−1. The diameter of the specimen is 70 mm and the length is 35 mm. SHPB device consists of a gas gun, a 500 mm long striker bar, a 5.46 m long incident bar, a 3.48 m long transmission bar, and a buffer bar, all made of steel. The photo and schematic diagram of the device are shown in Figure 3. Brass discs placed at the end of the incident bar were used to improve the incident pulse. The rising time of loading pulse then can be increased to meet the requirement of stress uniformity. Also, the high frequency noise in the incident pulse can be reduced.Figure 3
The SHPB setup: (a) photo of the device; (b) schematic diagram of the device.
(a)(b)During the test, the striker launched by the gas gun impacted on the incident bar and generated a compressive pulse propagating in the bar. Due to the mismatching of wave impedances between the specimen and the incident/transmission bars, part of the incident pulse was reflected back to the incident bar, while the rest of the pulse was transmitted through the specimen to the transmission bar. The incident pulse and reflected pulse were recorded by two diametrically opposed strain gauges mounted on the incident bar, and the transmitted pulse was measured by another pair of strain gauges on the transmission bar. The typical incident, reflected, and transmitted pulses recorded in the tests are shown in Figure4.Figure 4
Recorded waveforms from dynamic compression test.According to the measured incident strainεi, reflected strain εr, and transmitted strain εt, the engineering stress σ, engineering strain ε, and engineering strain ε rate experienced by the specimen can be derived by the following equations [36]:(1)εt=c0L∫0tεit−εrt−εttdt,ε˙t=c0Lεit−εrt−εtt,σt=EA2Asεit+εrt+εtt,where E is the Young's modulus of the bar, A is the cross-sectional area of the incident/transmission bars, As is the cross-sectional area of the specimen, c0 is the longitudinal wave velocity in the bars, and L is the length of the specimen. If stress equilibrium is satisfied in the specimen, the following expression can be obtained:(2)εit+εrt=εtt.Substituting equation (2) into equation (1) yields(3)εt=2c0L∫0tεit−εttdt,ε˙t=2c0Lεit−εtt,σt=EAAsεtt.Due to the large diameter of the bar, the reflected wave dispersion is usually serious. Therefore, equation (3) is often used to obtain the stress-strain curve for test specimen.For SFRSCC specimens, the strain rate recorded in the SHPB test varies with time. Three different methods have been used to calculate the strain rate [37]: mean strain rate over the loading period, mean strain rate over a deﬁned stress interval, and strain rate at failure point. In this paper, calculating the strain rate over a defined period was adopted. The range includes the strain rates from the one speciﬁed to 40% of the maximum stress to the strain rate corresponding to the maximum stress, as shown in Figure 5.Figure 5
Determination of the strain rate in the specimen.


## 2.1. Mix Proportions

Two types of self-compacting concrete mixtures, with static uniaxial compressive strengths of about 40 MPa and 60 MPa, were designed. Four quantities of steel fibres, 0.5%, 1.0%, 1.5%, and 2.0% in volume ratio, were added to each type of SCC mixture. MC-FX and HC-FX are used for coding for the different mixtures. MC-FX stands for medium strength (40 MPa) SCC, andX is the steel fibre volume fraction in percent. HC-FX stands for high strength (60 MPa) SCC with the steel fibre volume fraction of X percent. The material composition and mix proportions are given in Table 1. The steel fibres, coarse granite aggregate with a maximum size of 16 mm, and natural sand with average fineness modulus of 2.39 are shown in Figure 1. The diameter and length of the steel fibres are 0.2 mm and 10 mm, respectively, the yield strength of the steel fibres is 780 MPa, and the tensile strength is 2800 MPa.Table 1
Mix proportions of SFRSCC (kg/m3).
MixCementSandCoarse aggregateFly ashWaterSuperplasticizerSteel fibresMC-F0.54277248621071712.639MC-F1.04277218561071712.678MC-F1.54277148491071712.6117MC-F2.04277088431071712.0156HC-F0.5494714831491483.339HC-F1.0494696810491484.478HC-F1.5494678789491485.1117HC-F2.0494658778491483.6156Figure 1
Materials used in this study: (a) steel fibres; (b) coarse aggregate; (c) sand.
(a)(b)(c)


## 2.2. Mechanical and Rheological Properties

The fluidity and rheology of the fresh SFRSCC mixtures were evaluated by slump flow and L-box tests following the EFNARC guidelines [35]. The measured values for dm, the largest diameter of the circular spread for concrete to stop ﬂowing (Figure 2), and the ratio of the concrete height of the horizontal and vertical sections in L-box are listed in Table 2.Figure 2
Slump ﬂow of fresh SFRSCC.Table 2
Workability of SFRSCC.
MixSlump flowdm (mm)∗L-boxh2/h1#MC-F0.57950.91MC-F1.08450.91MC-F1.57500.93MC-F2.07100.90HC-F0.57550.89HC-F1.07500.93HC-F1.57600.93HC-F2.07650.94∗slump ﬂow values for proper SCC should be within 550–850 mm [35]; #h2/h1 values for proper SCC should be ≥0.8 [35].As for slump flow, all types of SFRSCC exhibited satisfactory results in the range of 710–845 mm, demonstrating acceptable filling ability for practical applications. In L-box tests, the values were also in a practically acceptable range: 0.89–0.94. During the tests, no indication of segregation was observed, and the mixtures maintained good homogeneity and cohesion.The quasi-static compression tests of SFRSCC were carried out on a hydraulic testing machine with a load capacity of 2000 kN. The specimens were 150 mm cubes at the age of 28 days. Brazilian disc tests were also conducted to obtain the split tensile strength using cylindrical specimens of dimension 70 mm × 35 mm (diameter × height). The basic mechanical properties of SFRSCC are listed in Table3. The quasi-static compressive strength of SFRSCC for the same type of mixture did not increase with the increase of fibres. Proper amount of steel fibre provides reinforcement for concrete. Higher fibers concentrations, however, may destroy the uniformity of concrete and cause a decrease of the compressive strength. As still more steel ﬁbers are included, the reinforcement effect provides a gradual increase of the strength [18].Table 3
Mechanical properties of SFRSCC (standard deviation in parentheses).
MixDensity (g/cm3)Elastic modulus (GPa)Compressive strength (MPa)Splitting tensile strength (MPa)MC-F0.52.22 (0.01)35.6 (2.3)49.8 (1.1)3.78 (0.25)MC-F1.02.27 (0.01)35.0 (1.7)43.8 (1.9)4.08 (0.15)MC-F1.52.37 (0.03)34.9 (1.1)46.6 (0.9)4.34 (0.11)MC-F2.02.41 (0.02)33.4 (1.5)48.8 (1.8)5.14 (0.65)HC-F0.52.38 (0.01)36.6 (1.1)60.1 (0.3)4.73 (0.10)HC-F1.02.39 (0.03)38.0 (1.3)69.6 (1.4)5.37 (0.24)HC-F1.52.41 (0.02)37.7 (2.1)64.9 (1.0)4.93 (0.18)HC-F2.02.42 (0.01)36.3 (1.3)66.1 (0.4)5.40 (0.40)


## 2.3. Dynamic Compression Tests

A 74 mm diameter split Hopkinson pressure bar (SHPB) [36] was used to conduct dynamic compression tests at strain rates between 50 and 240 s−1. The diameter of the specimen is 70 mm and the length is 35 mm. SHPB device consists of a gas gun, a 500 mm long striker bar, a 5.46 m long incident bar, a 3.48 m long transmission bar, and a buffer bar, all made of steel. The photo and schematic diagram of the device are shown in Figure 3. Brass discs placed at the end of the incident bar were used to improve the incident pulse. The rising time of loading pulse then can be increased to meet the requirement of stress uniformity. Also, the high frequency noise in the incident pulse can be reduced.Figure 3
The SHPB setup: (a) photo of the device; (b) schematic diagram of the device.
(a)(b)During the test, the striker launched by the gas gun impacted on the incident bar and generated a compressive pulse propagating in the bar. Due to the mismatching of wave impedances between the specimen and the incident/transmission bars, part of the incident pulse was reflected back to the incident bar, while the rest of the pulse was transmitted through the specimen to the transmission bar. The incident pulse and reflected pulse were recorded by two diametrically opposed strain gauges mounted on the incident bar, and the transmitted pulse was measured by another pair of strain gauges on the transmission bar. The typical incident, reflected, and transmitted pulses recorded in the tests are shown in Figure4.Figure 4
Recorded waveforms from dynamic compression test.According to the measured incident strainεi, reflected strain εr, and transmitted strain εt, the engineering stress σ, engineering strain ε, and engineering strain ε rate experienced by the specimen can be derived by the following equations [36]:(1)εt=c0L∫0tεit−εrt−εttdt,ε˙t=c0Lεit−εrt−εtt,σt=EA2Asεit+εrt+εtt,where E is the Young's modulus of the bar, A is the cross-sectional area of the incident/transmission bars, As is the cross-sectional area of the specimen, c0 is the longitudinal wave velocity in the bars, and L is the length of the specimen. If stress equilibrium is satisfied in the specimen, the following expression can be obtained:(2)εit+εrt=εtt.Substituting equation (2) into equation (1) yields(3)εt=2c0L∫0tεit−εttdt,ε˙t=2c0Lεit−εtt,σt=EAAsεtt.Due to the large diameter of the bar, the reflected wave dispersion is usually serious. Therefore, equation (3) is often used to obtain the stress-strain curve for test specimen.For SFRSCC specimens, the strain rate recorded in the SHPB test varies with time. Three different methods have been used to calculate the strain rate [37]: mean strain rate over the loading period, mean strain rate over a deﬁned stress interval, and strain rate at failure point. In this paper, calculating the strain rate over a defined period was adopted. The range includes the strain rates from the one speciﬁed to 40% of the maximum stress to the strain rate corresponding to the maximum stress, as shown in Figure 5.Figure 5
Determination of the strain rate in the specimen.


## 3. Results and Discussion

### 3.1. Dynamic Stress-Strain Relationship and Rate Dependence of SFRSCC

The dynamic compressive stress-strain curves obtained in the SHPB tests are presented in Figures6 and 7 for medium strength and high strength SFRSCC, respectively. Significant rate dependence was observed for all types of SFRSCC. The compressive strength increased with strain rate, and the dynamic strength was much higher than the static value. The maximum peak stress achieved in the tests was up to 220 MPa for HC-F1.5, which was more than three times the static strength (64.9 MPa).Figure 6
Dynamic compressive stress-strain curves of medium strength SFRSCC: (a) MC-F0.5; (b) MC-F1.0; (c) MC-F1.5; (d) MC-F2.0.
(a)(b)(c)(d)Figure 7
Dynamic compressive stress-strain curves of high strength SFRSCC: (a) HC-F0.5; (b) HC-F1.0; (c) HC-F1.5; (d) HC-F2.0.
(a)(b)(c)(d)Dynamic increase factor (DIF) is usually used to evaluate the dynamic enhancement effect of concrete strength. DIF is defined as the ratio of dynamic strength to quasi-static strength of concrete. For dynamic compression, DIF =fc,d/fc,s, where fc,d is the compressive strength under dynamic loading and fc,s is the quasi-static compressive strength of concrete. So far, many empirical formulas have been proposed to describe the relationship between DIF and strain rate of normal concrete, such as CEB-FIP equation [38]:(4)DIF=ε˙ε˙00.014,ε˙≤30s−1,0.012ε˙ε˙01/3,ε˙>30s−1,where ε˙0=30×10−6s−1 is the reference strain rate. Figures 8 and 9 show the variation of DIF with strain rate for medium strength and high strength SFRSCC. The curve deﬁned by the empirical formula CEB-FIP [38] for normal concrete was also plotted for comparison. The experimental data of DIF for medium strength SFRSCC were all located above the CEB-FIP curve within the strain rate range of 53 s−1–184 s−1, as shown in Figure 8, indicating that the strain rate sensitivity of medium strength SFRSCC was higher than normal concrete. The DIF increased with increasing fibre content as well as increasing strain rate.Figure 8
Dynamic increase factor for medium strength SFRSCC.Figure 9
Dynamic increase factor for high strength SFRSCC.For high strength SFRSCC, the DIF values were close to the CEB-FIP curve in the strain rate range of 50 s−1–240 s−1, as shown in Figure 9, and the strain rate sensitivity was lower than that of medium strength SFRSCC. The DIF values also increased with the increase of strain rate, but the strain rate sensitivity tended to decrease when the fibre content was more than 1.5%.An empirical formula was proposed to describe the strain rate sensitivity of the SFRSCC. It assumed a linear increase in DIF with the logarithm of the strain rate:(5)DIF=A⋅log10ε˙ε˙0+B,where ε˙0=30×10−6s−1 and A and B are constants. A linear regression was carried out to obtain the values of A and B. The results are listed in Table 4. It was found that the slope of the DIF for medium strength SFRSCC decreased when the fibre content increased from 0.5% to 1.5%, but increased when the fibre content increased from 1.5% to 2.0%. The slope of the DIF for high strength SFRSCC increased when the fibre content increased from 0.5% to 1.5%, but decreased when the fibre content increased from 1.5% to 2.0%.Table 4
Constants in proposed DIF formula for SFRSCC.
MixABMC-F0.53.4126−19.9991MC-F1.02.6310−14.7267MC-F1.51.7937−8.8893MC-F2.04.9260−29.1629HC-F0.50.9176−3.9057HC-F1.01.5821−8.2046HC-F1.54.5814−28.1943HC-F2.01.6821−9.3132


### 3.2. Influence of the Matrix Strength and Fibre Content on the Behavior of SFRSCC

The experimental results showed that strain rate sensitivity of SFRSCC was affected by both the matrix strength and fibre content. From the microscopic point of view, the main failure mode of SFRSCC during compression is caused by the collapse of microvoids. For low strength SFRSCC, the structural effect is more significant due to high porosity, which makes the material more sensitive to strain rate. The influence of fibres on the DIF may be considered as the additional voids caused by fibres. Generally speaking, increasing of fibres leads to decrease of SFRSCC workability, which implies reduction in compaction levels of SFRSCC. However, in this study, the mix proportions of SFRSCC were optimized in order to satisfy the requirement of strength grade. The workability of SFRSCC did not decrease with the increase of fibre content. As can be seen from Table2, high strength SFRSCC with 2% fibres showed good fluidity, which result in low sensitive to strain rate.


### 3.3. Damage Constitutive Model of SFRSCC

Design and nonlinear analysis of structures taking into account the actual behavior of SFRSCC is typically based on the constitutive models [23]. In the present study, a damage constitutive model was developed based on the stress-strain relationship, as demonstrated in Figures 6 and 7. SFRSCC is a heterogeneous material which contains many flaws such as cracks and voids. These cracks and voids grow and develop under external loading, resulting in damage of material. To simplify the analysis, each microvoid and microcrack can be regarded as a spherical cavity with its maximum diameter. The total volume of material can be written as the sum of the total volume of all equivalent microvoids Vd and the total volume of solid parts Vs, i.e., V=Vs+Vd. Vd=∑vdi, where vdi represents the volume of the ith equivalent microvoid. The damage of SFRSCC then can be defined as [34](6)D=VdV.Thus,(7)V˙d=D˙V+V˙D,where the black dot above the variable represents the derivative of time, i.e., V˙d=dVd/dt. If the damage is assumed to develop with the material deformation due to compression and shearing, the relative growth rate of each microvoid can be assumed to be proportional to the rate of work [34]:(8)v˙divdi=a1W˙WB,where a1 is the material constant, W˙ is the rate of work, and WB is the normalization factor. Thus, for the whole material, we have(9)V˙dVd=a1W˙WB.By substituting equation (7) into equation (9) and using equation (6), the damage evolution equation can be obtained as follows:(10)D˙=a1DW˙WB−DV˙V.If the solid material is assumed to be incompressible, i.e.,V˙s=0, then(11)0=V˙s=V˙−V˙d=V˙−DV˙−D˙V,(12)orDV˙V=DD˙1−D.Substitution of equation (12) into equation (10) yields(13)D˙=a1D1−DW˙WB.For uniaxial compression,(14)D˙=a1D1−Dσε˙WB,where σ is the stress and ε˙ is the strain rate. For high strain rates, we assume that WB is a power function of strain rate:(15)WB=b1ε˙ε˙0γ,where b1 and γ are material constants and ε˙0=1s−1 is the reference strain rate.Equation (14) then can be expressed as follows:(16)D˙=aD1−Dσε˙ε˙/ε˙0γ,where a=a1/b1.The one-dimensional stress-strain relationship without damage can be expressed as(17)σ=Eε,where E, ε are Young’s modulus and strain. Considering the effect of damage softening, the stress-strain relationship can be written as(18)σ=Eε1−D.The differential form of equation (18) is(19)σ˙=E1−Dε˙−ED˙ε.Substitution of equation (16) into equation (19) yields(20)σ˙=E1−Dε˙1−aε˙0ε˙γDσε.A combination of equations (16) and (20) provides a differential form to describe the constitutive relationship of SFRSCC with damage under uniaxial compression. Two material parameters, a and γ, in the damage evolution model (equation (16)) can be determined by fitting the experimental data. A least square method was used to fit the stress-strain curves of SFRSCC. The optimal parameters a and γ in the damage model for different types of SFRSCC are listed in Table 5.Table 5
Material parameters in the damage model.
Mixa (MPa−1)γMC-F0.55.87 × 1052.62MC-F1.02.22 × 1041.94MC-F1.51.04 × 1010.41MC-F2.05.13 × 1042.16HC-F0.53.13 × 1010.52HC-F1.01.06 × 1020.84HC-F1.52.45 × 1052.38HC-F2.03.05 × 1021.20The stress-strain curves calculated based on the proposed damage model are shown in Figure10 and compared with the experimental curves. The theoretical curves well illustrated the rate effect on strength of SFRSCC and showed good consistency with the relevant experimental data which indicated that the proposed dynamic damage constitutive model was credible to describe the dynamic stress-strain relationship of SFRSCC under uniaxial compression.Figure 10
Comparison between experimental and theoretical stress-strain curves: (a) MC-F0.5; (b) MC-F1.0; (c) MC-F1.5; (d) MC-F2.0; (e) HC-F0.5; (f) HC-F1.0; (g) HC-F1.5; (h) HC-F2.0.
(a)(b)(c)(d)(e)(f)(g)(h)


## 3.1. Dynamic Stress-Strain Relationship and Rate Dependence of SFRSCC

The dynamic compressive stress-strain curves obtained in the SHPB tests are presented in Figures6 and 7 for medium strength and high strength SFRSCC, respectively. Significant rate dependence was observed for all types of SFRSCC. The compressive strength increased with strain rate, and the dynamic strength was much higher than the static value. The maximum peak stress achieved in the tests was up to 220 MPa for HC-F1.5, which was more than three times the static strength (64.9 MPa).Figure 6
Dynamic compressive stress-strain curves of medium strength SFRSCC: (a) MC-F0.5; (b) MC-F1.0; (c) MC-F1.5; (d) MC-F2.0.
(a)(b)(c)(d)Figure 7
Dynamic compressive stress-strain curves of high strength SFRSCC: (a) HC-F0.5; (b) HC-F1.0; (c) HC-F1.5; (d) HC-F2.0.
(a)(b)(c)(d)Dynamic increase factor (DIF) is usually used to evaluate the dynamic enhancement effect of concrete strength. DIF is defined as the ratio of dynamic strength to quasi-static strength of concrete. For dynamic compression, DIF =fc,d/fc,s, where fc,d is the compressive strength under dynamic loading and fc,s is the quasi-static compressive strength of concrete. So far, many empirical formulas have been proposed to describe the relationship between DIF and strain rate of normal concrete, such as CEB-FIP equation [38]:(4)DIF=ε˙ε˙00.014,ε˙≤30s−1,0.012ε˙ε˙01/3,ε˙>30s−1,where ε˙0=30×10−6s−1 is the reference strain rate. Figures 8 and 9 show the variation of DIF with strain rate for medium strength and high strength SFRSCC. The curve deﬁned by the empirical formula CEB-FIP [38] for normal concrete was also plotted for comparison. The experimental data of DIF for medium strength SFRSCC were all located above the CEB-FIP curve within the strain rate range of 53 s−1–184 s−1, as shown in Figure 8, indicating that the strain rate sensitivity of medium strength SFRSCC was higher than normal concrete. The DIF increased with increasing fibre content as well as increasing strain rate.Figure 8
Dynamic increase factor for medium strength SFRSCC.Figure 9
Dynamic increase factor for high strength SFRSCC.For high strength SFRSCC, the DIF values were close to the CEB-FIP curve in the strain rate range of 50 s−1–240 s−1, as shown in Figure 9, and the strain rate sensitivity was lower than that of medium strength SFRSCC. The DIF values also increased with the increase of strain rate, but the strain rate sensitivity tended to decrease when the fibre content was more than 1.5%.An empirical formula was proposed to describe the strain rate sensitivity of the SFRSCC. It assumed a linear increase in DIF with the logarithm of the strain rate:(5)DIF=A⋅log10ε˙ε˙0+B,where ε˙0=30×10−6s−1 and A and B are constants. A linear regression was carried out to obtain the values of A and B. The results are listed in Table 4. It was found that the slope of the DIF for medium strength SFRSCC decreased when the fibre content increased from 0.5% to 1.5%, but increased when the fibre content increased from 1.5% to 2.0%. The slope of the DIF for high strength SFRSCC increased when the fibre content increased from 0.5% to 1.5%, but decreased when the fibre content increased from 1.5% to 2.0%.Table 4
Constants in proposed DIF formula for SFRSCC.
MixABMC-F0.53.4126−19.9991MC-F1.02.6310−14.7267MC-F1.51.7937−8.8893MC-F2.04.9260−29.1629HC-F0.50.9176−3.9057HC-F1.01.5821−8.2046HC-F1.54.5814−28.1943HC-F2.01.6821−9.3132


## 3.2. Influence of the Matrix Strength and Fibre Content on the Behavior of SFRSCC

The experimental results showed that strain rate sensitivity of SFRSCC was affected by both the matrix strength and fibre content. From the microscopic point of view, the main failure mode of SFRSCC during compression is caused by the collapse of microvoids. For low strength SFRSCC, the structural effect is more significant due to high porosity, which makes the material more sensitive to strain rate. The influence of fibres on the DIF may be considered as the additional voids caused by fibres. Generally speaking, increasing of fibres leads to decrease of SFRSCC workability, which implies reduction in compaction levels of SFRSCC. However, in this study, the mix proportions of SFRSCC were optimized in order to satisfy the requirement of strength grade. The workability of SFRSCC did not decrease with the increase of fibre content. As can be seen from Table2, high strength SFRSCC with 2% fibres showed good fluidity, which result in low sensitive to strain rate.


## 3.3. Damage Constitutive Model of SFRSCC

Design and nonlinear analysis of structures taking into account the actual behavior of SFRSCC is typically based on the constitutive models [23]. In the present study, a damage constitutive model was developed based on the stress-strain relationship, as demonstrated in Figures 6 and 7. SFRSCC is a heterogeneous material which contains many flaws such as cracks and voids. These cracks and voids grow and develop under external loading, resulting in damage of material. To simplify the analysis, each microvoid and microcrack can be regarded as a spherical cavity with its maximum diameter. The total volume of material can be written as the sum of the total volume of all equivalent microvoids Vd and the total volume of solid parts Vs, i.e., V=Vs+Vd. Vd=∑vdi, where vdi represents the volume of the ith equivalent microvoid. The damage of SFRSCC then can be defined as [34](6)D=VdV.Thus,(7)V˙d=D˙V+V˙D,where the black dot above the variable represents the derivative of time, i.e., V˙d=dVd/dt. If the damage is assumed to develop with the material deformation due to compression and shearing, the relative growth rate of each microvoid can be assumed to be proportional to the rate of work [34]:(8)v˙divdi=a1W˙WB,where a1 is the material constant, W˙ is the rate of work, and WB is the normalization factor. Thus, for the whole material, we have(9)V˙dVd=a1W˙WB.By substituting equation (7) into equation (9) and using equation (6), the damage evolution equation can be obtained as follows:(10)D˙=a1DW˙WB−DV˙V.If the solid material is assumed to be incompressible, i.e.,V˙s=0, then(11)0=V˙s=V˙−V˙d=V˙−DV˙−D˙V,(12)orDV˙V=DD˙1−D.Substitution of equation (12) into equation (10) yields(13)D˙=a1D1−DW˙WB.For uniaxial compression,(14)D˙=a1D1−Dσε˙WB,where σ is the stress and ε˙ is the strain rate. For high strain rates, we assume that WB is a power function of strain rate:(15)WB=b1ε˙ε˙0γ,where b1 and γ are material constants and ε˙0=1s−1 is the reference strain rate.Equation (14) then can be expressed as follows:(16)D˙=aD1−Dσε˙ε˙/ε˙0γ,where a=a1/b1.The one-dimensional stress-strain relationship without damage can be expressed as(17)σ=Eε,where E, ε are Young’s modulus and strain. Considering the effect of damage softening, the stress-strain relationship can be written as(18)σ=Eε1−D.The differential form of equation (18) is(19)σ˙=E1−Dε˙−ED˙ε.Substitution of equation (16) into equation (19) yields(20)σ˙=E1−Dε˙1−aε˙0ε˙γDσε.A combination of equations (16) and (20) provides a differential form to describe the constitutive relationship of SFRSCC with damage under uniaxial compression. Two material parameters, a and γ, in the damage evolution model (equation (16)) can be determined by fitting the experimental data. A least square method was used to fit the stress-strain curves of SFRSCC. The optimal parameters a and γ in the damage model for different types of SFRSCC are listed in Table 5.Table 5
Material parameters in the damage model.
Mixa (MPa−1)γMC-F0.55.87 × 1052.62MC-F1.02.22 × 1041.94MC-F1.51.04 × 1010.41MC-F2.05.13 × 1042.16HC-F0.53.13 × 1010.52HC-F1.01.06 × 1020.84HC-F1.52.45 × 1052.38HC-F2.03.05 × 1021.20The stress-strain curves calculated based on the proposed damage model are shown in Figure10 and compared with the experimental curves. The theoretical curves well illustrated the rate effect on strength of SFRSCC and showed good consistency with the relevant experimental data which indicated that the proposed dynamic damage constitutive model was credible to describe the dynamic stress-strain relationship of SFRSCC under uniaxial compression.Figure 10
Comparison between experimental and theoretical stress-strain curves: (a) MC-F0.5; (b) MC-F1.0; (c) MC-F1.5; (d) MC-F2.0; (e) HC-F0.5; (f) HC-F1.0; (g) HC-F1.5; (h) HC-F2.0.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4. Conclusions

The dynamic compression tests of SFRSCC with two strength classes (40 MPa and 60 MPa) were carried out using a 74 mm diameter SHPB. The dynamic compressive stress-strain curves were obtained, and the effects of the matrix strength and fibre content on the strain rate sensitivity of SFRSCC were studied. Following conclusions can be drawn from the present study:(1)
SFRSCC exhibited significant rate sensitivity, whereby the dynamic compressive strength increased with the increase of strain rate. The strain rate sensitivity of SFRSCC was affected by both matrix strength and fibre content. Medium strength SFRSCC showed higher strain rate sensitivity than high strength SFRSCC in the strain rate range of 50 s−1–240 s−1. The DIF values of medium strength SFRSCC were increased with the increase of fibre content, but for high strength SFRSCC, the DIF values tended to decrease when the fibre content was more than 1.5%. The mix proportion and compaction levels of SFRSCC were considered to be the main factors affecting the strain rate sensitivity.(2)
An empirical formula was proposed to describe the strain rate sensitivity of the SFRSCC by assuming a linear increase in DIF with the logarithm of the strain rate.(3)
A damage constitutive model for SFRSCC was derived based on the equivalent microvoid hypothesis. The theoretical stress-strain curves calculated based on the proposed model could well illustrate the rate effect of SFRSCC and the damage softening process. Due to the few parameters involved, the model can be easily used to evaluate the dynamic compressive behavior of SFRSCC with reasonable accuracy.



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*Source: 8085949-2021-11-17.xml*