Dynamic mechanical properties of hybrid fiber reinforced concrete

Abstract

In this study, the dynamic mechanical properties of hybrid fiber reinforced concrete (HFRC) are analyzed with respect to failure mode, dynamic increase factor (DIF), and peak strain by means of a SHPB testing apparatus. The factors that influence the dynamic mechanical properties include fiber type and fiber content. It is concluded that the best dynamic mechanical properties of fibers are CS-PHFRC at medium and low strain rates and AS-PHFRC at a high strain rate. Within a certain range, the higher the fiber content is, the larger the DIF of the corresponding HFRC and the more obvious the increase in dynamic compressive strength. AS-CSHFRC improves the dynamic compressive deformability of the HFRC. The polypropylene fiber causes plasticity, as shown in the failure mode of concrete. The Ottosen nonlinear elastic model, modified by introducing the damage factor, can better describe the dynamic mechanical properties of HFRC.

Keywords

Hybrid fiber reinforced concrete fiber type fiber content dynamic increase factor constitutive model

Introduction

In recent years, many scholars have focused on hybrid fiber reinforced concrete (HFRC), with the expectation that the mixing of different fiber types complements each other’s advantages, thus resulting in an HFRC with comprehensive performance, able to satisfy the needs of different construction projects. Overseas (Altun et al., 2013; Bangi and Horiguchi, 2011; Chi et al., 2014; Eethar and Ramli, 2011; Hameed et al., 2010; Köksal et al., 2012; Mitsui et al., 2010; Mohammed et al., 2011; Soea et al., 2013) and domestic (Guo-dong et al., 2013; Liu and Xu, 2012, 2013; Xiao-kai and Diao, 2012; Zhi and Lu, 2011) research has mainly concentrated on high and low elastic modulus hybrid fibers. The low elastic modulus fiber can restrict crack propagation in concrete, so its toughness, durability, and impact resistance can be improved. Therefore, low elastic modulus fibers such as polypropylene fiber (PF) have been widely used. Research on the mix of high elastic modulus steel fiber (SF) and low elastic modulus PF has found that the SF can reinforce concrete, whereas the PF can improve toughness and the post-cracking strain capacity of concrete.

In this study, the object is the HFRC with strength grade C50, and three types of HFRC are selected according to the current research directions. They are the end-hook and corrugated hybrid fiber reinforced concrete (AS-CSHFRC), the hooked SF and PF hybrid fiber concrete (AS-PHFRC), and the corrugated SF and PF hybrid fiber reinforced concrete (CS-PHFRC). This experimental study on the static and dynamic mechanical properties of the HFRC is carried out by means of the MTS816 statics experiment system and the separating SHPB experiment system. At the same time, the plain and uni-doped fiber concretes are designed as control experiments. Emphasis is laid on research of the dynamic mechanical properties of HFRC at different strain rates, and the dynamic constitutive model of HFRC is established. It provides experimental and theoretical support for research and applications of the HFRC.

Design of experimental program

In this experiment, three types of HFRC are designed, namely, AS-CSHFRC, AS-PHFRC, and CS-PHFRC. The ASF and CSF volume fractions are assigned with 0.5%, 1%, and 1.5%, respectively, while the PF volume fraction is assigned with 0.1%, 0.2%, and 0.3%. For each type of HFRC, nine groups of tests are designed, and thus, the total number of tests is 27. Nonstandard test cubes with size of 100 × 100 × 100 mm are used as the static compression resistance specimens, and three specimens are used in each group. Test cylinders of Φ73.5 × 36.5 mm are used as the SHPB specimens, and nine specimens are used in each group. The number of experiments in each of the three types of concrete is as follows (Tables 1 –3).

Table 1.

Grouping of AS-CSHFRC experiment.

Experiment no.	ASF content (%)	CSF content (%)
P-1	0.5	0.5
P-2	0.5	1
P-3	0.5	1.5
P-4	1	0.5
P-5	1	1
P-6	1	1.5
P-7	1.5	0.5
P-8	1.5	1
P-9	1.5	1.5

Notes: Hereinafter, fiber content is volume fraction, namely, the percent of fiber volume in the total concrete volume, indicated with $V_{f}$ .

Table 2.

Grouping of AS-PHFRC experiment.

Experiment No.	ASF content (%)	PF content (%)
Q-1	0.5	0.1
Q-2	0.5	0.2
Q-3	0.5	0.3
Q-4	1	0.1
Q-5	1	0.2
Q-6	1	0.3
Q-7	1.5	0.1
Q-8	1.5	0.2
Q-9	1.5	0.3

Table 3.

Grouping of CS-PHFRC experiment.

Experiment No.	CSF content (%)	PF content (%)
R-1	0.5	0.1
R-2	0.5	0.2
R-3	0.5	0.3
R-4	1	0.1
R-5	1	0.2
R-6	1	0.3
R-7	1.5	0.1
R-8	1.5	0.2
R-9	1.5	0.3

In this study, in order to compare the effect of the hybrid fiber upon the mechanical behavior of concrete more intuitively, a control experiment with uni-doped fiber concrete is additionally designed. In the case of the invariable mix proportion of concrete, single-fiber concrete of three different types is contrasted with plain concrete and 16 groups of tests are implemented. In Table 4, O-1 indicates the plain concrete, O-2 to O-7 indicate ASFRC, O-8 to O-13 indicate CSFRC, and O-14 to O-16 indicate PFRC. The details are listed in Table 4.

Table 4.

Control experiments of uni-doped fiber concrete.

Experiment no.	ASF content (%)	CSF content (%)	PF content (%)
O-1	0	0	0
O-2	0.5	0	0
O-3	1	0	0
O-4	1.5	0	0
O-5	2	0	0
O-6	2.5	0	0
O-7	3	0	0
O-8	0	0.5	0
O-9	0	1	0
O-10	0	1.5	0
O-11	0	2.0	0
O-12	0	2.5	0
O-13	0	3.0	0
O-14	0	0	0.1
O-15	0	0	0.2
O-16	0	0	0.3

Specimen preparation

Experimental raw material

In this experiment, the raw material of the HFRC includes the following parts:

(1) Cement: local complex Portland cement in Xuzhou city, grade: P.C 42.5.

(2) Course aggregate: local gravel in Xuzhou city, grain size of 10–20 mm.

(3) Fine aggregate: local building sand is used and its calculated fineness modulus is 2.6. Considering all indicators, it meets the requirements for Class II building sand. The grain composition is shown in Table 5.

(4) Water: tap water in Xuzhou city

(5) SF: the two types of SF used in this experiment are produced by Shandong Lubang Steel Fiber Co., Ltd., as shown in Figures 1(a) and (b).

(6) PF: sheaf-like monofil PF produced by Shanghai Meimengjia Chemical Technology Co. Ltd., as shown in Figure 1(c).

(7) Water reducer: the efficient water reducer produced in Xuzhou city.

Table 5.

Grain composition of sand.

Screen size (mm)		4.75	2.36	1.18	0.6	0.3	0.15
Accumulative screen residue (%)	Zone II standard	0–10	0–25	10–50	41–70	70–92	90–100
Accumulative screen residue (%)	Measured value	3.5	15.8	30.5	53.7	75.2	93.4

Figure 1.

Steel fiber (SF) and polypropylene fiber (PF) used in this experiment (a) AS-SF, (b) CS-SF, (c) PF.

Table 6 provides the main performance parameters of the two types of SF and PF used in the experiment.

Table 6.

Main performance parameters of fibers.

Fiber type	Length (mm)	Diameter (mm)	Density (g/cm³)	Tensile strength (MPa)	Elasticity modulus (GPa)
AS-SF	30–35	0.3–1.2	7.86	≥600	200
CS-SF	30–35	0.4–0.7	7.86	≥1200	200
Polypropylene fiber	9	0.05	0.91	≥250	3.79

Determination of mix proportion

According to the design requirements of mix proportion, the absolute volume method is utilized for calculation. The consumption of every component in 1 m³ concrete is calculated as shown in Figure 7 (Table 7).

Table 7.

Mix proportion of concrete.

Ingredient	Cement	Sand	Stone	Water	Water reducer
Mass (kg/m³)	486	645	1099	170	10

Notes: The design strength grade of concrete is C50; the water-binder ratio is 0.36 and the sand percentage is 37%.

Fabricating and curing of specimens

On the basis of the experimental design requirements, two different sets of molds are used to fabricate the specimens. The static compression resistance specimens are fabricated by means of a self-made nonstandard cubic mold with side length of 100 mm, and three specimens can be simultaneously manufactured. The SHPB dynamic specimens are fabricated by means of a self-made cylindrical mold with dimensions of Φ73.5 × 36.5 mm, and six specimens can be simultaneously manufactured. The two sets of molds are shown in Figures 2 and 3.

Figure 2.

Main process of specimen fabrication (a) Material weighing, (b) Stirring, (c) Vibrating, (d) Specimen shaping, (e) Specimen curing, (f) Specimens before testing.

Figure 3.

Specimen grinding and grinding effect.

After the raw materials are weighed and stirred, a small amount of mixture is taken with a little shovel and added into the prepared mold. The mold is placed on the vibrating table plate; after the vibrating table is started and materials are compacted, the mixing materials are continuously added until the mold is filled; when the cement paste overflows the specimen surface, a trowel is used to smooth the surface. Next, the molded specimens are placed at the standard indoor temperature for curing; after 24 h, the mold is removed and the specimen is numbered. The demolded specimens are put in a standard concrete curing cabinet for 28 days; the temperature in the cabinet is controlled at 20 ± 2°C and the relative humidity is set at more than 95%. The main process for fabricating the specimens is shown in Figure 2.

In the SHPB experiment, the requirement for the specimens is relatively higher, and thus, the specimens need smoothing. The grinder machine used is the SHM-200 double-face stone grinding machine manufactured by Xinguang Machinery Factory; the diameter of its grinding disc is 200 mm and the dimensions of the largest grinded specimen are 150 × 150 × 150 mm. The grinding process of the specimens and its results are shown in Figure 3.

Static compressive and dynamic impact experiments

This static experiment is performed in the Rock and Soil Mechanics Key Laboratory of China University of Mining and Technology. The MTS816 statics experiment system is adopted to carry out the static compressive strength experiment of the HFRC. The system is mainly composed of the host machine, multi-channel controller, hydraulic oil source, and main control computer. Its maximum axial compressive force is 1459 kN, maximum axial tensile force is 961 kN, and maximum stroke of the actuator is 100 mm. With this system, a variety of tests can be conducted under the multi-filed coupling of stress, seepage, and temperature fields (see Figure 4).

Figure 4.

MTS816 Statics experiment system.

In the dynamic experiment, the SHPB experiment apparatus of Φ74 mm is used, and its structure principle and field apparatus are shown in Figure 5. The power-loading system is driven by compressed nitrogen, its pressure lever is made of a spring steel bar with a length of 800 mm, the repetitive measurement deviation of the velocity is less than 5%, the signal is acquired by a DH5960 super-dynamic signal testing and analyzing system, and a self-programmed software is used for information processing. The stress–strain relationship of the material can be obtained and its mechanical property parameters can be measured. Further, the failure process and state of specimens can be acquired.

Figure 5.

SHPB Experimental Apparatus (a) Pressure lever system, (b) Velocity measuring device, (c) Strain gauge, (d) Data acquisition and processing system.

Analysis of experimental results

The experimental results are listed in Table 8. Owing to space limitations, only the partial measured average strain rates, peak strain values, peak stress values, and the dynamic increase factor (DIF, the ratio of peak stress to static-load compressive strength) are included. Each data are the average of the three experimental data.

Table 8.

SHPB experimental data of AS-CSHFRC specimen group.

Specimen No.	Air pressure (MPa)	Average strain rate (s⁻¹)	Peak strain	Peak stress (MPA)	Static-load compressive strength (MPa)	Dynamic increase factor (DIF)
P-1	0.3	61.24	0.0053	55.63	63.75	0.87
	0.35	65.38	0.0064	63.91		1.00
	0.4	76.74	0.0078	69.35		1.09
P-2	0.3	67.42	0.0064	63.24	65.88	0.96
	0.35	75.27	0.0091	70.51		1.07
	0.4	83.71	0.0097	75.47		1.15
P-3	0.3	54.37	0.0065	71.81	66.44	1.08
	0.35	61.12	0.0089	76.38		1.15
	0.4	76.85	0.0098	80.90		1.21
P-4	0.3	57.57	0.0078	62.57	65.96	0.95
	0.35	68.81	0.0094	69.47		1.05
	0.4	74.27	0.0108	78.20		1.19
P-5	0.3	58.95	0.0084	72.35	67.53	1.07
	0.35	71.75	0.0101	78.42		1.16
	0.4	73.79	0.0102	86.54		1.28
P-6	0.3	58.07	0.0076	75.10	65.74	1.14
	0.35	71.85	0.0082	80.46		1.22
	0.4	74.37	0.0095	89.67		1.36
P-7	0.3	61.42	0.0071	70.25	67.34	1.04
	0.35	68.71	0.0081	76.87		1.14
	0.4	74.53	0.0093	86.37		1.28
P-8	0.3	58.74	0.0077	74.64	66.21	1.12
	0.35	67.92	0.0081	81.69		1.23
	0.4	69.20	0.0099	90.74		1.37
P-9	0.3	58.74	0.0062	73.63	64.97	1.13
	0.35	66.24	0.0083	80.49		1.24
	0.4	78.83	0.0090	86.35		1.33

Also because of the space limitations, for the plain concrete, AS-CSHFRC, AS-PHFRC, and CS-PHFRC, only one group of specimens is included in the example to illustrate the stress–strain curves under three different air pressures. These are shown in Figure 6.

Figure 6.

Stress–strain curves of specimens under three different air pressures (a) Stress–strain curve of plain concrete specimens in sub-group O-1, (b) Stress–strain curve of specimens in sub-group P-5, (c) Stress–strain curve of specimens in sub-group Q-5, (d) Stress–strain curve of specimens in sub-group R-5.

Strain rate effect of hybrid fiber reinforced concrete

It can be seen from the obtained data that the hybrid fiber mix significantly improves the dynamic mechanical property of concrete. Especially, the dynamic compressive strength is clearly enhanced under different strain rates. For example, for the specimen in sub-group Q-9, the dynamic compressive strength at 0.4 MPa is increased by 58% relative to the static compressive strength. In addition, it can also be seen from the stress–strain curve that HFRC has a distinct enhancement effect in strain rate. Under dynamic impact load, as the strain rate and the air pressure rise, the peak stress and peak strain show enhancement effects on the strain rate, in a different degree.

Relationship between peak stress and peak strain rate

For the three different types of HFRC (AS-CSHFRC, AS-PHFRC, and CS-PHFRC), three sets of tests for each type are chosen so as to study the relationship between the peak stress and the strain rate when each group of specimens are impacted under different air pressures, as shown in Figure 7.

Figure 7.

Relationship curve between peak stress and strain rate of specimens (a) AS-CSHFRC specimen group, (b) AS-PHFRC specimen group, (c) CS-PHFRC specimen group.

It can be seen from the figure that the peak stress is positively correlated with the strain rate and increases as the strain rate goes up. Researches show that realization of the HFRC’s strain rate effect needs a critical strain rate, but in this experiment, the average strain rate ranges from 54.37 s⁻¹ to 86.93 s⁻¹ and they lie in the strain-rate sensitive area. Therefore, the peak stress values of AS-CSHFRC, AS-PHFRC, and CS-PHFRC display significant enhancement effect in the strain rate. From Figure 7, within the same strain rate, the growth of compressive strength of AS-CSHFRC and AS-PHFRC are obvious than that of CS-PHFRC. So three kinds of hybrid fiber’s contribution to the dynamic performance of concrete can be compared.

Relationship between peak strain and strain rate

Similarly, three groups of specimens of each type are chosen for the experiment to study the relationship between peak strain and strain rate when a different air pressure impact is applied to each of the groups. The selected specimen groups are the same as those above. The relationship between the peak strain and strain rate of each group of specimens is shown in Figure 8.

Figure 8.

Relationship curve between peak strain and strain rate of specimens (a) AS-CSHFRC specimen group, (b) AS-PHFRC specimen group, (c) CS-PHFRC specimen group.

It can be seen from Figure 8 that the peak strain is positively correlated with the strain rate and increases as the strain rate grows; thus, the strain rate enhancement effect is shown.

Effects of fiber type and fiber content on dynamic mechanical properties

Dynamic increase factor

At a given strain rate, the ratio of the dynamic compressive strength of concrete and its static compressive strength is the DIF. It can be seen from the obtained data that the DIF values are different for different fiber types and fiber contents. In order to analyze better the influence of three different types of HFRC (AS-CSHFRC, the AS-PHFRC, and CS-PHFRC) and different fiber contents on the DIF, in Figure 9, the broken line graphs of the corresponding DIF values of each group are drawn under air pressures of 0.3 MPa, 0.35 MPa, and 0.4 MPa.

Figure 9.

DIF values of each group of specimens under three types of pressures (a) Under 0.3 MPa, (b) Under 0.35 MPa, (c) Under 0.4 MPa.

It can be seen from Figure 9(a) that, under the air pressure of 0.3 MPa, the DIF values of the HFRC in Group R are obviously higher than those in Group P and in Group Q. After calculation, the average DIF values corresponding to the three groups of specimens (Group P, Q, and R) are 1.04, 0.97, and 1.14, respectively. This reveals that the promotion effects of three groups of HFRC upon the dynamic compressive strength under 0.3 MPa can be ranked in descending order, namely, R>P>Q. Similarly, it can be seen from Figure 9(b) that the sequence under 0.35 MPa is R>Q>P. Figure 9(c) shows that the sequence under 0.4 MPa is Q>R>P. As the strain rate rises, the promotion effect of specimens in Group Q upon the DIF values becomes more obvious, increasing its rank from the last to the first when air pressure varies from 0.3 MPa to 0.4 MPa, while the promotion effects of specimens in Group R and Group Q gradually decrease.

In conclusion, the fiber type has a large effect on the DIF value of the HFRC. At medium and low strain rates (0.3 MPa, 0.35 MPa), the corresponding DIF value of CS-PHFRC is the largest, and its promotion effect on the dynamic compressive strength of the HFRC is the most obvious. Therefore, at medium and low strain rates, the best HFRC is CS-PHFRC. At a high strain rate (0.4 MPa), the corresponding DIF value of AS-PHFRC is the largest and its promotion effect on the dynamic compressive strength is the most distinct. Thus, the best HFRC is AS-PHFRC at a high strain rate.

Next, we analyze the effect of the fiber content on the DIF value of the HFRC. It can be seen from Figure 9(a) that, with air pressure of 0.3 MPa, the DIF values corresponding to the same fiber type are obviously different, but this is related to the different fiber contents of each group of concrete specimens. In Group P, the DIF value ranges from 0.87 to 1.14 and the largest DIF value is that of Sub-group P-6. In Group Q, the DIF value varies from 0.82 to 1.15 and the largest DIF value is in Sub-group Q-9. In Group R, the DIF value varies from 0.98 to 1.38 and the largest DIF value is in Sub-group R-9. Considering fiber type, the best is CS-PHFRC, while the best fiber content is 1.5% CSF mixed with 0.3% PF. In a similar way, in Figure 9(b), under the air pressure of 0.35 MPa, the best is CS-PHFRC considering fiber type, while the best fiber content is 1.5% CSF mixed with 0.3% PF. In Figure 9(c), under the air pressure of 0.4 MPa, the best is AS-PHFRC considering fiber type, while the best fiber content is 1.5% ASF mixed with 0.3% PF.

In summary, the fiber content has a large effect on the DIF value of the HFRC and, within a certain range, the higher the fiber content is, the larger the DIF value of the corresponding HFRC and the more obvious the promotion effect on the dynamic compressive strength.

Peak strain

In order to analyze the effects of different hybrid fiber types and different fiber content upon the peak strain, broken line graphs of the peak strain values corresponding to each group of specimens under the air pressures of 0.3 MPa, 0.35 MPa, and 0.4 MPa are drawn, as illustrated in Figure 10.

Figure 10.

Peak strains of each group of specimens under three air pressures (a) 0.3 MPa, (b) 0.35 MPa, (c) 0.4 MPa.

Figure 10(a) shows that, under the air pressure of 0.3 MPa, the peak strains of the HFRC in Group P are obviously higher than those in Group Q and in Group R. After calculation, the average DIF values corresponding to three groups of specimens (Group P, Q, and R) are 0.007, 0.0044, and 0.0023, respectively. This reveals that the dynamic compressive deformability of the three groups of HFRC under 0.3 MPa air pressure can be ranked in descending order, namely, P>Q>R. Similarly, it can be seen from Figure 10(b) that the dynamic compressive deformability of the three groups of HFRC under 0.35 MPa air pressure can be ranked in the order P>Q>R. Figure 10(c) shows that their dynamic compressive deformability under 0.4 MPa can be ranked in the order P>Q>R.

In conclusion, under the three air pressures, the peak strain of AS-CSHFRC is the largest and its dynamic compressive deformability is the best. Therefore, the best fiber concrete is AS-CSHFRC.

Next, we analyze the effect of the fiber content on the peak strains of the HFRC. It can be seen from Figure 10(a) that, with the air pressure of 0.3 MPa, the peak strains of Group P vary from 0.0053 to 0.0084, and the largest peak strain is in Sub-group P-5. The peak strains of Group Q vary from 0.0021 to 0.0091, and the largest peak strain is in Sub-group Q-3. The peak strains of Group R vary from 0.0012 to 0.0044, and the largest peak strain is in Sub-group R-2. Considering fiber types, the best HFRC is AS-CSHFRC, while the best fiber content is 1% ASF mixed with 1% CPF. In Figure 10b, under the air pressure of 0.35 MPa, the best concrete is AS-CSHFRC considering fiber type, while the best fiber content is 1% ASF mixed with 1% CSF. Figure 10c shows that, under the air pressure of 0.4 MPa, the best concrete is AS-CSHFRC considering fiber type, while the best fiber content is 1% ASF mixed with 0.5% CSF.

Summarizing at different strain rates, the peak strain of AS-CSHFRC has small fluctuation as the fiber content changes, and it basically keeps a high dynamic compressive deformability. Relative to AS-PHFRC and CS-PHFRC, its dynamic compressive deformability is improved to a larger degree.

Analysis of failure mode

The SHPB experiment on the HFRC is carried out to collect the failure modes of each group of specimens under impact load at different air pressures. The failure modes of the plain concrete, AS-CSHFRC, AS-PHFRC, and CS-PHFRC under the air pressures of 0.3 MPa, 0.35 MPa, and 0.4 MPa are shown in Figures 11 –14, respectively. AS-CSHFRC in Sub-group P-2, AS-PHFRC in Sub-group Q-4, and CS-PHFRC in Sub-group R-5 are taken as examples.

Figure 11.

Failure modes of plain concrete 0.3 MPa, 0.35 MPa, 0.4 MPa.

Figure 12.

Failure modes of concrete in Sub-group P-2 0.3 MPa, 0.35 MPa, 0.4 MPa.

Figure 13.

Failure modes of concrete in Sub-group Q-4 0.3 MPa, 0.35 MPa, 0.4 MPa.

Figure 14.

Failure modes of concrete in Sub-group R-5 0.3 MPa, 0.35 MPa, 0.4 MPa.

Compared with the plain concrete specimens, the failure modes of AS-CSHFRC, AS-PHFRC, and CS-PHFRC are relatively better. As the air pressure increases, the failure modes of each group of specimens show a distinct strain rate effect.

The specimens of sub-group P-2 in Figure 12 are taken as example to show the failure modes of AS-CSHFRC under different air pressures. It can be seen from the figure that a small amount of fragments are spalled under a low air pressure. Under a medium air pressure, more fragments are spalled. Under a high air pressure, the failure mode with residual core occurs. After failure, there is still steel fibrous connection between fragments and the SF is pulled or snapped from the concrete matrix; this shows that the SF improves the failure modes of concrete.

In Figure 13, the specimens of Sub-group P-2 are taken as example to show the failure modes of AS-PHFRC under different air pressures. It can be seen from Figure 13 that, under a low air pressure, a small amount of corner fragmentation occurs and a flocculent PF is revealed. Under a medium air pressure, the corner fragmentation becomes more serious and large fragments are spalled. Under a high air pressure, two pieces of penetrating cracks appear on the specimen surface as fragments are spalled.

In Figure 14, the specimens of Sub-group R-5 are taken as example to show the failure modes of CS-PHFRC under different air pressures. It can be seen from Figure 14 that, under a low air pressure, the corner fragmentation is not severe. Under a medium air pressure, there are small spalled fragments and fine cracks occur on the specimen surface. Under a high air pressure, more fragments are spalled on the specimens and the PF is exposed at the breakage.

In conclusion, it can be seen from the failure modes that the failure degree of the HFRC specimens is slighter than that of the plain concrete specimens. The integrity of the specimens after failure is better kept and their ability to resist impact load is stronger. Regarding the concretes with different fiber types, AS-PHFRC and CS-PHFRC have a smaller failure degree than AS-CSHFRC. This shows that PF can improve the failure mode of concrete under impact load and plastic failure characteristics are displayed to some extent.

Establishment of constitutive model

The constitutive models established based on the principles and methods of viscoelasticity and viscoplasticity generally have complicated forms, with a large quantity of parameters introduced. This increases the calculation workload and degree of difficulty, as the parameters are difficult to calibrate and their precisions are limited. These problems greatly restrict the development of this kind of models. The dynamic constitutive model of concrete, based on the modified static constitutive model, is a simple and practical method. In this method, complex theories and mechanical mechanisms are not considered, and only the strain rate, loading rate, and damage factor are introduced into the static constitutive model. Therefore, their forms are relatively simple and the fitting effect obtained by using data is preferable; thus, it is extensively applied. Many scholars have used this method to establish dynamic constitutive models of concrete (Cao, 2011; Da-fu et al., 2013; Fan et al., 2010; Jing-yi et al., 2008; Ottosen, 1979). In this study, by modifying the Ottosen nonlinear elastic constitutive equation and introducing the damage factor based on the Loland model, the dynamic constitutive model of HFRC is built.

The equation of the Ottosen nonlinear elastic constitutive model (Ottosen, 1979) and the stress–strain curve are given as follows:

σ = f_{s} \frac{A_{1} (\frac{ε}{ε_{s}}) + (A_{2} - 1) {(\frac{ε}{ε_{s}})}^{2}}{1 + (A_{1} - 2) (\frac{ε}{ε_{s}}) + A_{2} {(\frac{ε}{ε_{s}})}^{2}}

(1)

where the compressive force direction of

σ

and

ε

is positive,

f_{s}

is the static peak stress of concrete, and

ε_{s}

is the static peak strain, which indicates the strain corresponding to the peak stress and it is the highest point in the stress–strain curve in Figure 10.

E_{0}

is the initial elastic module of concrete,

E_{c}

is the secant modulus when concrete stress reaches

f_{s}

, the factor

A_{1} = E_{0} / E_{c}

, and the factor

A_{2}

will have a large effect on the ascending branch and the descending branch of the stress–strain curve.

Besides, the damage factor based on the Loland model is considered in this study. On the basis of existing research results, the damage evolution law at the ascending branch of the stress–strain curve in the Loland damage model is extended and applied to the entire curve. Thus, the damage factor is expressed as follows:

D_{ε} = D_{0} + m ε^{n}

(2)

where

D_{ε}

is the damage factor in the damage evolution process of concrete (it is a variable and affected by initial damage, strain, and material),

D_{0}

indicates the initial damage of concrete material,

ε

is strain, and m and n are material constants. However, the damage factor is obtained from the results of the concrete tensile experiment, and thus, it has a certain limitation. According to existing research results, in order to use the damage factor in the damage evolution at tensile and compressive states, the crack closure factor h is brought in to modify the damage factor (Cui et al., 2001). The modified damage factor formula is given by

D = h D_{ε} = h D_{0} + m ε^{n}

(3)

where the crack closure factor h can simultaneously consider the concrete damage at the tensile and compressive states; thus, it may be regarded as a factor representing the material and it is influenced by the micro-defect shape and other factors. In the case of

σ \leq 0

, it is the compressive state, and

0 \leq h \leq 1

; in the case of

σ > 0

, it is the tensile state, and h equals to 1. The meanings of other parameters are the same as those in Formula (2).

On the basis of Formula (1), the equation of modified Ottosen nonlinear elastic constitutive model is derived as follows:

σ = σ_{p} \frac{A_{1} (\frac{ε}{ε_{p}}) + (A_{2} - 1) {(\frac{ε}{ε_{p}})}^{2}}{1 + (A_{1} - 2) (\frac{ε}{ε_{p}}) + A_{2} {(\frac{ε}{ε_{p}})}^{2}}

(4)

In Formula (4), when HFRC is subjected to impact compression, the dynamic peak stress $σ_{p}$ is used to replace the static peak stress $f_{s}$ , and the dynamic peak strain $ε_{p}$ is used in place of the static peak strain $ε_{s}$ . For a convenient calculation, Formula (4) is further simplified to Formula (5)

y = \frac{A_{1} x + (A_{2} - 1) x^{2}}{1 + (A_{1} - 2) x + A_{2} x^{2}}

(5)

where y is the ratio of the stress to the dynamic peak stress (in the case of y = 1, the stress is the largest) and x is the ratio of the strain to the dynamic peak strain (in the case of x < 1, the stress–strain curve is ascending; in the case of x > 1, the stress–strain curve is descending).

On the basis of the modified Ottosen nonlinear elastic constitutive model, the damage factor is introduced based on the Loland model. Considering that the damage of the concrete material mainly comes from a later evolution, Formula (3) is simplified to ignore the initial damage, namely, $D_{0} = 0$ .

D = k ε^{n}

(6)

Considering damage evolution, (1-D) is substituted into Formula (4) to get the dynamic constitutive equation of HFRC

σ = (1 - k ε^{n}) σ_{p} \frac{A_{1} (\frac{ε}{ε_{p}}) + (A_{2} - 1) {(\frac{ε}{ε_{p}})}^{2}}{1 + (A_{1} - 2) (\frac{ε}{ε_{p}}) + A_{2} {(\frac{ε}{ε_{p}})}^{2}}

(7)

The simplified dynamic constitutive equation of HFRC is given by

y = [1 - k {(ε_{p} x)}^{n}] \frac{A_{1} x + (A_{2} - 1) x^{2}}{1 + (A_{1} - 2) x + A_{2} x^{2}}

(8)

The formula has four parameters, namely, $A_{1}$ , $A_{2}$ , $k$ , and $n$ ; although the number of parameters is small, they have specific meanings.

Because of the limited length of this paper, only one set of SHPB experimental data has been selected for each AS-CSHFRC, AS-PHFRC, and CS-PHFRC.

The parameters obtained from each set of experimental fitting are shown in Table 9. By analyzing these fitting parameters, it is seen that the mean square error of each set of constitutive equation fitting results is small. This reveals that the constitutive equation can better fit the experimental data.

Table 9.

Fitting parameters of all groups of constitutive equations.

Specimen no.	Air pressure (MPa)	$A_{1}$	$A_{2}$	$k$	$n$	Mean square error
P-5	0.3	4.2666	7.5801	1.2206	2.7451	0.0226
	0.35	3.3811	0.8437	26,464,000	4.7681	0.0191
	0.4	3.2489	0.0238	1.0045	0.6137	0.0346
Q-5	0.3	20.3633	−5.2655	9409.5	2.3863	0.0520
	0.35	2.9455	3.7143	42,652,000	4.4866	0.0200
	0.4	1.5354	0.5881	3,812,200	5.3446	0.0115
R-5	0.3	14.3145	−3.9212	167,570	2.7005	0.0361
	0.35	12.4771	−0.9147	241.175	1.5789	0.0387
	0.4	1.8767	4.0558	35,329,000	4.4248	0.0181

Each set of fitting results of the experimental stress–strain curve is shown in Figure 15. On the one hand, the listed experimental curves coincide quite well with the fitted curves; furthermore, other unlisted experimental groups also have good fitting results. This demonstrates that the Ottosen nonlinear elastic model modified by introducing the damage factor can better describe the dynamic mechanical properties of HFRC. On the other hand, because the initial damage of concrete and the experimental error are ignored when simplifying the constitutive equation, the curve fitting produces a certain deviation.

Figure 15.

Fitting results of experimental stress–strain curves (a) Sub-group P-5, (b) Sub-group Q-5, (c) Sub-group R-5.

Conclusions

(1) The dynamic compressive strength of the HFRC has obvious rate dependence; the peak stress and peak strain are positively related to variation of the strain rate.

(2) Fiber type and fiber content have significant effects on the dynamic mechanical properties. The best fiber type is CS-PHFRC at medium and low strain rates, while the best at a high strain rate is AS-PHFRC.

(3) Fiber content has a great effect on the DIF value of the HFRC. Within a certain range, the higher the fiber content is, the larger the DIF value of the relevant HFRC and the more obvious the promotion effect on the dynamic compressive strength.

(4) The peak strain of AS-CSHFRC has small fluctuation as fiber content varies, and it still keeps its high dynamic compressive deformability. Compared with AS-PHFRC and CS-PHFRC, it improves to a larger degree the dynamic compressive deformability of the HFRC.

(5) Regarding the failure modes, AS-PHFRC and CS-PHFRC have less failure degree than AS-CSHFRC; this shows that PF can improve the failure mode of concrete under impact load, and plastic failure characteristics are displayed to some extent.

(6) The Ottosen nonlinear elastic model modified by introducing the damage factor can better describe the dynamic mechanical properties of HFRC.

Footnotes

Acknowledgments

The authors express special thanks to the fund of the National Natural Science Foundation of China (Nos.51978166, 51738011).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

ORCID iDs

Yong Zhang

Li Chen

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Experiment no.	ASF content (%)	CSF content (%)
P-1	0.5	0.5
P-2	0.5	1
P-3	0.5	1.5
P-4	1	0.5
P-5	1	1
P-6	1	1.5
P-7	1.5	0.5
P-8	1.5	1
P-9	1.5	1.5

Experiment No.	ASF content (%)	PF content (%)
Q-1	0.5	0.1
Q-2	0.5	0.2
Q-3	0.5	0.3
Q-4	1	0.1
Q-5	1	0.2
Q-6	1	0.3
Q-7	1.5	0.1
Q-8	1.5	0.2
Q-9	1.5	0.3

Experiment No.	CSF content (%)	PF content (%)
R-1	0.5	0.1
R-2	0.5	0.2
R-3	0.5	0.3
R-4	1	0.1
R-5	1	0.2
R-6	1	0.3
R-7	1.5	0.1
R-8	1.5	0.2
R-9	1.5	0.3

Experiment no.	ASF content (%)	CSF content (%)
P-1	0.5	0.5
P-2	0.5	1
P-3	0.5	1.5
P-4	1	0.5
P-5	1	1
P-6	1	1.5
P-7	1.5	0.5
P-8	1.5	1
P-9	1.5	1.5

Experiment No.	ASF content (%)	PF content (%)
Q-1	0.5	0.1
Q-2	0.5	0.2
Q-3	0.5	0.3
Q-4	1	0.1
Q-5	1	0.2
Q-6	1	0.3
Q-7	1.5	0.1
Q-8	1.5	0.2
Q-9	1.5	0.3

Experiment No.	CSF content (%)	PF content (%)
R-1	0.5	0.1
R-2	0.5	0.2
R-3	0.5	0.3
R-4	1	0.1
R-5	1	0.2
R-6	1	0.3
R-7	1.5	0.1
R-8	1.5	0.2
R-9	1.5	0.3

Experiment no.	ASF content (%)	CSF content (%)
P-1	0.5	0.5
P-2	0.5	1
P-3	0.5	1.5
P-4	1	0.5
P-5	1	1
P-6	1	1.5
P-7	1.5	0.5
P-8	1.5	1
P-9	1.5	1.5

Experiment No.	ASF content (%)	PF content (%)
Q-1	0.5	0.1
Q-2	0.5	0.2
Q-3	0.5	0.3
Q-4	1	0.1
Q-5	1	0.2
Q-6	1	0.3
Q-7	1.5	0.1
Q-8	1.5	0.2
Q-9	1.5	0.3

Experiment No.	CSF content (%)	PF content (%)
R-1	0.5	0.1
R-2	0.5	0.2
R-3	0.5	0.3
R-4	1	0.1
R-5	1	0.2
R-6	1	0.3
R-7	1.5	0.1
R-8	1.5	0.2
R-9	1.5	0.3