Experimental study of dynamic mechanical properties of reactive ...

SCIENCE CHINA Technological Sciences • RESEARCH PAPER •

September 2010 Vol.53 No.9: 2435–2449 doi: 10.1007/s11431-010-4061-x

Experimental study of dynamic mechanical properties of reactive powder concrete under high-strain-rate impacts JU Yang1,2*, LIU HongBin1, SHENG GuoHua1,3 & WANG HuiJie1 1

State Key Laboratory of Coal Resources and Safe Mining, Beijing Key Laboratory of Fracture and Damage Mechanics of Rocks and Concrete, China University of Mining and Technology, Beijing 100083, China; 2 Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Drive, AB T2N 1N4, Canada; 3 School of Resources and Civil Engineering, Northeastern University, Shenyang 110004, China Received January 2, 2010; accepted June 17, 2010

The dynamic mechanical properties of reactive powder concrete subjected to compressive impacts with high strain rates ranging from 10 to 1.1×102 s-1 were investigated by means of SHPB (split-Hopkinson-pressure-bar) tests of the cylindrical specimens with five different steel fiber volumetric fractions. The properties of wave stress transmission, failure, strength, and energy consumption of RPC with varied fiber volumes and impact strain rates were analyzed. The influences of impact strain rates and fiber volumes on those properties were characterized as well. The general forms of the dynamic stress-strain relationships of RPC were modeled based on the experimental data. The investigations indicate that for the plain RPC the stress response is greater than the strain response, showing strong brittle performance. The RPC with a certain volume of fibers sustains higher strain rate impact and exhibits better deformability as compared with the plain RPC. With a constant fiber fraction, the peak compressive strength, corresponding peak strain and the residual strain of the fiber-reinforced RPC rise by varying amounts when the impact strain rate increases, with the residual strain demonstrating the greatest increment. Elevating the fiber content makes trivial contribution to improving the residual deformability of RPC when the impact strain rate is constant. The tests also show that the fiber content affects the peak compressive strength and the peak deformability of RPC in a different manner. With a constant impact strain rate and the fiber fraction less than 1.75%, the peak compressive strength rises with an increasing fiber volume. The peak compressive strength tends to decrease as the fiber volume exceeds 1.75%. The corresponding peak strain, however, incessantly rises with the increasing fiber volume. The total energy Edisp that RPC consumed during the period from the beginning of impacts to the time of residual strains elevates with the fiber volume increment as long as the fiber fraction is not larger than 2%. It turns to decrease if the fiber volume exceeds 2%. The added fibers make various contributions to enhancing the capability of RPC to consume energy at different loading stages. If the fiber fraction is not larger than 2%, the added fibers make more contribution to enhancing the energy consumption ability of RPC in the period before the peak strain than in the period after the peak strain. The impact strain rate, however, distinctively affects the total energy that RPC consumed and the energy consumed in the different loading periods. The higher the impact strain rate, the more the energy consumed in the stages and therefore the higher the dynamic impact toughness. The empirical relationships of the peak compressive strength, corresponding peak strain, residual strain, total consumed energy and the energy consumed in the varied periods with the impact strain rate and the fiber fraction are derived. Four generalized forms of the dynamic impact stress-strain responses of RPC are formulated by normalizing stresses and strains as the generalized coordinates and by taking account of the influences of impact strain rates and fiber volumetric fractions. impact, reactive powder concrete (RPC), high strain rate, dynamic strength, energy consumption, dynamic stress-strain response Citation:

Ju Y, Liu H B, Sheng G H, et al. Experimental study of dynamic mechanical properties of reactive powder concrete under high-strain-rate impacts. Sci China Tech Sci, 2010, 53: 2435−2449, doi: 10.1007/s11431-010-4061-x

*Corresponding author (email: [email protected]) © Science China Press and Springer-Verlag Berlin Heidelberg 2010

tech.scichina.com

www.springerlink.com

2436

1

JU Yang, et al.

Sci China Tech Sci

Introduction

With rapid growth of world industry and wide application of innovative technology, the scale, intensity and complexity of project construction in the fields of resources exploitation, land transportation, hydroelectric engineering, civil engineering, energy utilization and underground storage have been continuously increasing. It has set more requirements and higher standards for concrete and its applications. Developing high-performance concrete with improved strength, toughness, durability, impact resistance and fatigue behavior has been of great importance and crucial for concrete constructions. Under the circumstances, a new type of cement-based composite material, i.e., reactive powder concrete (RPC), was developed in late 90s. Unlike ordinary strength and high-performance concrete, a few novel technologies including eliminating coarse aggregates, adding short steel fibers, mixing active fly ash, and curing under high temperature and pressure have been applied, resulting in RPC with improved hydratation, dense structure and enhanced steel-matrix bonding strength. With specific fabrication procedures, RPC achieved a static compressive strength between 200 and 800 MPa, a flexural strength around 140 MPa and a rupture energy capacity as much as 40 kJ/m2. RPC can sustain 500°C high temperature for two hours with a residual strength of 50% of the peak compressive capacity [1–17]. For these exclusive physical and mechanical performances, RPC has been widely promoted and applied in construction of highways, bridges, skyscrapers, oversized buildings, nuclear facilities, military engineering, etc. Consequently, RPC has attracted more and more research attentions from both academic and engineering sides. Most of present researches on RPC’s performance and applications mainly focus on its static behaviors. Few comprehensive experimental and theoretical analyses of RPC’s dynamic mechanical properties are reported. The dynamic strength, deformability, stress-strain relationship, difference between dynamic and static responses and the mechanisms that are responsible for the variation in RPC’s performance under impacts or blasting loads with high strain rates, specifically within 100–104 s−1, have not been thoroughly studied. As a consequence, it is intractable to conduct the safety design and reliability analysis of RPC’s structures under high-strain-rate impacts. It has impeded RPC’s applications in the relevant fields. In recent years, lots of efforts have been made to break off the situation. To name but a few, Tai [18, 19] investigated the dynamic strength, stress-strain response and failure of RPC200 under the impacts with strain rates between 78.5–1.23×102 s−1 and the penetrations with velocities within 27.0–1.04×102 m/s by means of SHPB compressive tests and flat panel penetrations, respectively. The effects of strain rates and steel fiber volumes on the dynamic properties and failures of RPC were probed. Tian et al. [20] ana-

September (2010) Vol.53 No.9

lyzed the dynamic response and the bearing capacity of RPC-filled steel tabular columns by using SHPB impact tests and numerical simulation. Wang et al. [21] investigated the influence of hydrostatic pressure and strain rate on the RPC dynamic strength with numerical modeling based on SHPB impact tests of RPC with different steel fiber volumes. The effects of the strain rates on the RPC’s dynamic strength were evaluated. In addition, the influences of steel fiber volume on the RPC’s dynamic strength and failures were examined as well. Lai et al. [22] tested the dynamic mechanical behaviors of RPC under the repeated impact loads using SHPB equipment and analyzed the effects of repeat number, impact mode and fiber volume fraction on the properties of RPC under repeated impact loads. Wang et al. [23] studied the properties of stress-strain responses of RPC with various steel fiber volumes under different strain rates by SHPB tests. It was shown that RPC was sensitive to the strain rate. Steel fibers enhanced the RPC’s dynamic compressive strength by limit amount. Bagheri et al. [24] compared the energy absorption indices of RPC under static and dynamic loads and found that the energy absorption of RPC under dynamic loads was greater than that of ordinary high-strength fiber-reinforced concrete. Ge et al. [25] studied the impact and penetration resistances of RPC with a steel fiber fraction of 5% by penetration and contact detonation experiments and validated the proposed formula of penetration depth. Fujikake et al. [26, 27] illustrated the effects of striker dropping height and strain rate on the flexural strength, tensile failure, tensile stress-strain response and tensile stress-crack opening displacement of RPC. A model for RPC’s strength taking account of the strain rate effects was established. Kuznetsov et al. [28] compared the blasting resistance of RPC with that of ordinary steel reinforced concrete by conducting panel blast tests and found out that the blasting resistance of RPC was much higher than ordinary reinforced concrete under the same explosion equivalent. Wang et al. [29] probed the nature and the influencing factors of RPC’s failure under penetration and derived a simplified penetration formula. It was shown that the penetration resistance of RPC is about three times as high as that of ordinary concrete. Toutlemonde et al. [30] showed the impact-resistant behaviors and the design method for RPC containers of radioactive materials under high strain rates by means of numerical simulation. Undoubtedly, these pioneer attempts have provided people with valuable references of understanding the dynamic mechanical properties and failure mechanisms of RPC under high strain rate impacts. Nevertheless, it should be noticed that diversity of research objectives, inconsistence in material ingredients, testing and measurement methods, and lack of experiences in dynamic experiments have brought in some problems to the researches of dynamic behavior of RPC. First, for instance, the investigations were insufficient in analysis of influencing factors and laws governing the responses, especially the

JU Yang, et al.

Sci China Tech Sci

influence of strain rates on stress-strain responses. Second, there was apparent discrepancy between the measured data and findings, resulting in difficulties to achieve a common view of the dynamic behaviors of RPC. Third, there was few insightful analyses of the nature of dynamic property of RPC and the intrinsic mechanism, which raised the concerns that the proposed models are complicated and have strong experience dependency and limitations. Apparently, there is a long way to go for well understanding and quantitatively describing the dynamic properties and the governing mechanisms of RPC under high-strain-rate impacts so as to meet the application requirements. In this paper, a few SHPB tests of RPC cylindrical specimens with five volumetric fractions of steel fibers are performed. The characteristics of wave transmission, fracture, strength and energy absorption of RPC under the impacts of strain rates ranging from 10–1.1×102 s−1 and various steel fiber volumetric fractions are investigated. The influences of strain rates and fiber fractions on the dynamic stress-strain responses and the energy absorption (or consumption) capacity of RPC are stressed. The dynamic constitutive relationships of RPC taking account of the influences of strain rates and fiber fractions are derived. The study attempts to provide a way for better understanding the dynamic behavior and its nature of RPC subject to high-strain-rate impacts and for facilitating engineering applications.

2 Outlines of SHPB test 2.1

Specimen preparation

Generally, reactive powder concrete (RPC) is categorized into two groups, i.e., RPC200 and RPC800, by its ingredients and fabrication techniques. RPC with a cubic or cylindrical compressive strength up to 140–230 MPa under quasistatic uniaxial compression refers to RPC200, while RPC with the compressive strength between 500–800 MPa refers to RPC800 [1–17, 31, 32]. Considering the disagreement in fabrication technique, specimen form and size, fiber type and testing method, none of the measurement and assessment methodologies for RPC’s compressive strengths have

2437


been accepted as a general standard. Referring to the ingredients and preparation procedure of RPC200, we prepared a series of RPC specimens for SHPB tests. Table 1 lists the material ingredients and mixing ratio for the specimens, where P.O42.5 Portland cement with 28-day static compressive strength of 56.4 MPa was employed. Fine quartz sands with diameters within 0.15–0.63 mm and SiO2 content of more than 90%, quartz powder with granular diameter of 45 μm, and Silica fly ash with granular diameters 0.1–0.2 μm and SiO2 content of more than 98% were mixed together. Sika superplasticizer with water reducing ratio of more than 30% and solid content of 37.2% was employed. Shear-type short steel fibers with diameter of 0.2–0.22 mm, length of 13 mm and tensile strength of 2900 MPa were embedded. Five patches of RPC cubic specimens with various fiber volumes were tested to determine their static compressive strengths in accordance with the procedures in refs. [7, 33]. Table 2 shows the results, which indicates that the prepared RPC specimens satisfy the requirements of RPC200. When producing the specimens, we first mixed the cement, quartz sand, quartz powder, and silica fly ash in a designated mixing ratio. Steel fibers were then added and blended uniformly (only for the fibered RPC). Half of the mixed water and superplasticizer were added and mixed for 3 min. The other half was finally added and mixed for another 3 min. The mixture was poured into a cylindrical mould with a diameter of Φ 56 mm and vibrated for 3 min on a vibrating table. The specimens were first cured in ambient temperature for 24 h then demolded and cured in a heated water box for additional 72 h. To meet the requirements of SHPB tests [34–36], all specimens were trimmed in Φ 56 mm×26 mm plates. The two ends of the specimens were polished to ensure smooth contact between the specimen and Hopkinson bars. 2.2

Impact tests and measurements

The SHPB test was performed on a variable-cross-section SHPB system in the State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology. The devices and setup are sketched in Figure 1. The input bar consists of an incident bar and a transition bar that

Table 1 The ingredients and mixing ratio for RPC specimens Gel composition (B)

Water-cement (W/B)

Cement (C)

Silica fly ash (SF/C)

Intergrades sand (S/C)

Fine sand (S/C)

Quartz powder (Qu/C)

Plasticizer (%)

Curing

0.19

1.0

0.28

0.75

0.37

0.39

2.0

90°C hot water 72 h

Table 2 The compressive strength of 100 mm×100 mm×100 mm cubic specimens Fiber volumetric fraction ρv (%)

0

1.0

1.5

2.0

3.0

Compressive strength (MPa)

126

160

162

165

167

Note: The result listed for every fiber volume is an average of the compressive strengths of three specimens. The curing and test method refers to refs. [7, 33, 36].

2438

JU Yang, et al.

Sci China Tech Sci


Figure 1 SHPB experimental setup. (a) Principle diagram; (b) photograph.

varies in its cross-section. It is totally 3200 mm long including 400 mm transition bar whose diameter varies from 37 to 74 mm. The transmitted bar is 74 mm in diameter and 1800 mm in length. In order to disclose the effects of fiber content on the dynamic stress-strain responses, a large-size specimen with a diameter of 56 mm was employed, which enabled a uniform distribution of the steel fibers in the specimen. Thus, we adopted a variable cross-section bar to achieve the purpose of elevating the impact speed of a large-size specimen. Stress wave propagation may induce two-dimensional dispersion effect in the bars, imposing a high-frequency oscillation upon the incident pulse. It remarkably impairs the uniformity of stress-strain response of brittle and concrete-like materials and greatly lowers the precision and reliability of experiment results [37–41]. According to the observed static mechanical properties of RPC [3, 7], a copper plate with diameter of 20 mm and thickness of 2.5 mm was mounted to the head of the incident bar as a wave shaper to extend the impulse rising time and reduce both the dispersion effect of incident wave and the inertial effect. The tests indicated that this measure effectively filtered the high-frequency components out of the incident wave and attained a better waveform [36]. Since the time that the wave propagation spent in a specimen was much shorter than the time spent in the bars, the wave bounced back and forth through a specimen many times. It seems to be reasonable to assume that the stresses homogeneously distribute in a specimen when it comes to failure. In order to ease the non-planar contact effect between bars and the specimen coming from the misalignment of system and the specimen flaws and to improve stress uniformity and data scattering, we used a universal joint in the SHPB system so that better alignment and contact could be achieved by the readjustment of the joint during the impact. In order to compare the dynamic stress-strain responses and energy absorption capability of RPC and reveal the mechanisms how the impact strain rate takes effect, we controlled the velocity of striker for each specimen of varied fiber content in such a way that the specimen would be not smashed into pieces. The specific velocity for the specimens was determined to be within 14.5–22.5 m/s,

equivalent to the strain rate of 10–1.1×102 s−1, after a series of trials [36]. Five steel fiber volumetric fractions (ρv), i.e., ρv=0%, 1.0%, 1.5%, 2.0% and 3.0%, respectively, were tested for RPC’s dynamic properties, where ρv=0% meant that no steel fibers were embedded. We refer RPC without steel fibers as the plain RPC, and RPC with varied volume of fibers as the fibered RPC, hereafter. To ensure the results comparable, we repeated 3–4 samples with identical fiber fraction for every impact strain rate.

3

Results and analyses

Figures 2–6 illustrate the transmitted and reflected stress waveforms and the failure patterns of RPC varying with the impact strain rate for the specific fiber volumetric fraction ρv. Figure 7 displays the stress waveforms varying with the fiber fraction ρv of RPC subjected to the specific impact strain rate. Figure 8 demonstrates the dynamic stress-strain curves varying with the impact strain rate for each specific fiber volume. Figure 9 plots the dynamic stress-strain curves varying with the fiber volume under each impact strain rate. From the experiments we found the followings. 1) The plain RPC specimen significantly fragmented when an impact load applied. The higher the impact strain rate, the greater the degree of fragmentation. Although no apparent change took place in the transmitted and reflected waveforms of the plain RPC with the varied strain rates, the amplitude of the transmitted pulse appeared to be much greater than that of the reflected wave (see Figure 2). It implies that the stress response is higher than the strain response for plain RPC, indicating the distinct characteristics of brittleness. The peak compressive strength and its corresponding strain increased by a small amount as the impact strain rate rose. The residual strain after the peak stress, however, increased remarkably (see Figures 8(a), 9(a) and 9(b)). For the plain RPC, the higher the strain rate, the larger the residual deformability (i.e., micro-plasticity and cracking), thus the more the energy dissipated and the severer the fragmentation.

JU Yang, et al.

Sci China Tech Sci


2439

Figure 2 Stress waveforms and failure patterns of the plain RPC varying with the impact strain rate. (a) Failure at ε =20 s−1; (b) failure at ε =34 s−1; (c) failure at ε =45 s−1; (d) failure at ε =57 s−1.

Figure 3 Stress waveforms and failure patterns of RPC with fiber fraction of 1% varying with the impact strain rate. (a) Failure at ε =35 s−1; (b) failure at ε =50 s−1; (c) failure at ε =70 s−1; (d) failure at ε =1.05×102 s−1.

Figure 4 Stress waveforms and failure patterns of RPC with fiber fraction of 1.5% varying with the impact strain rate. (a) Failure at ε =40 s−1; (b) failure at ε =60 s−1; (c) failure at ε =70 s−1; (d) failure at ε =0.95×102 s−1.


2440

JU Yang, et al.

Sci China Tech Sci



Figure 7 Stress waveforms varying with the fiber fraction ρv of RPC under the specific impact strain rate. (a) Average impact rate ε =55 s−1, for the plain RPC the average impact rate approximates ε =45 s−1; (b) average impact rate ε =72.5 s−1; (c) average impact rate ε =85 s−1; (d) average impact rate ε =1.02×102 s−1.

2) When the fiber fraction rose to 1%, the impact strain rate responsible for the fibered RPC fracture was much higher than that for the plain RPC, where as the strain rate was not larger than 70 s−1, the specimen was essentially intact except some edge areas (see Figure 3). Both the transmitted and the reflected waves gradually elevated their amplitudes with an increasing impact strain rate. The reflected amplitude increased by a larger amount, while the transmitted by a smaller amount. As the strain rate exceeded 70 s−1, the reflected wave exhibited multiple peaks. When the strain rate reached the level of 1.05×102 s−1, the second

peak amplitude of the reflected wave turned to be much higher than the peak amplitude of the transmitted. It means that adding fibers improves the compressive strength and its deformability to different degrees. In contrast, the post-peak deformability has considerable improvement. This can be testified by the dynamic stress-strain responses of RPC with fiber fraction of 1%, as shown in Figures 8(a)–(d). 3) When the fiber volume reached 1.5% or 2%, as compared with the plain RPC and the fibered of fraction of 1%, the RPC specimen fractured only around the edge areas and no fragmentation took place even if the strain rate was as

JU Yang, et al.

Sci China Tech Sci


2441

Figure 8 Dynamic stress-strain response curves varying with the impact strain rate for the specific fiber fraction ρv. (a) ρv=0%; (b) ρv=1%; (c) ρv=1.5%; (d) ρv=2%; (e) ρv=3%.

high as 1.0×102 s−1. It implies that the deformability of the fibered RPC under impacts was further enhanced by fibers. For this circumstance, the amplitude of the transmitted wave was found to be comparative to that of the reflected. Multiple peaks occurred in the reflected wave. For the specimen with fiber fraction of 2%, if the impact strain rate rose to 1.0×102 s−1, the amplitude of the second peak of the reflected wave turned to be higher than the peak amplitude of the transmitted. The fibered RPC maintained considerably large deformability after the peak stress. This can also be verified by the dynamic stress-strain responses of the fibered RPC with the fraction of 1.5% or 2% (see Figures 8(a)–(d)). 4) When the fiber fraction increased to 3%, the specimen remained intact except a few cracks near the edge areas. No fragmentation occurred under the every applied strain rate. The specimen with fiber volume of 3% did not display the apparent change in the amplitude of the reflected wave, as

compared with the other fibered specimens. Its reflected wave turned out to fluctuate by different degrees upon the strain rates. If the strain rate was lower than 70 s−1, the amplitude was smaller than that of the transmitted, meaning that the strain response was smaller than the stress response. If the strain rate surpassed 70 s−1, the amplitude was greater than that of the transmitted, indicating that the strain response was larger than the stress response. Large deformability and fracture came up to the specimen. 5) For any specific strain rate, as the fiber volume was more than 2%, the reflected amplitude tended to drop down, while the transmitted amplitude ascended with an increasing fiber volume (see Figure 7). This implies that RPC’s dynamic compressive strength turns to go down, while its corresponding deformability turns to go up if the fiber volume elevates to more than 2%. This seems to be testified by the dynamic stress-strain curves of the fibered RPC

2442

JU Yang, et al.

Sci China Tech Sci


Figure 9 Dynamic stress-strain response curves varying with the fiber fraction under the different impact strain rates. (a) Average impact rate ε =35 s−1; (b) average impact rate ε =55 s−1; (c) average impact rate ε =85 s−1; (d) average impact rate ε =1.02×102 s−1.

(see Figures 9(a)–(d)). To uncover the effects of fiber content and strain rate on RPC’s dynamic properties, we plot the peak compressive strength σd,p, corresponding peak strain εd,p and post-peak residual strain εres varying with the fiber content ρv and the strain rate ε in Figure 10. Here, the peak strain εd,p represents the strain corresponding to the peak compressive strength. The residual strain εres refers to the post-peak strain when the stress drops to 20% of the peak stress. In Figure 10, the black dots stand for the measured quantities of each RPC specimen, and the curved surfaces are located to fit these quantities. It is clear from the results that: (i) The impact strain rate ε significantly influences the strength and deformability of RPC. For a given fiber content, the peak compressive strength σd,p, the peak strain εd,p and the residual strain εres rise with an increasing strain rate. The largest increment comes up in the residual strain εres. (ii) The fiber fraction ρv influences the peak compressive strength σd,p and the peak deformability εd,p to different degrees. For a given strain rate, if the fiber fraction is less than 1.75%, the peak strength σd,p appears to increase with the increasing fiber volume. As the fiber fraction is over 1.75%, however, the peak strength σd,p tends to decrease gradually. The compressive strength reaches its maximum at the fiber fraction of 1.75%, approximately. In contrast to the property

of peak compressive strength, the peak strain εd,p rises consistently with the fiber volume increment for any given impact strain rate. (iii) The fiber volume takes trivial effects on the residual strain εres of RPC for any given strain rate. Considering the residual strain εres is set to measure RPC’s deformability when the stress drops to 20% of the peak value, this implies that increasing fiber volume cannot effectively improve the capability that RPC deforms after it fractures. On the contrary, the impact strain rate significantly affects RPC’s residual deformability. It should be noted that the dynamic properties of RPC aforementioned are limited to such a circumstance that the strike velocity (or impact strain rate, equivalently) was controlled in such a way that the specimen was not smashed to pieces. It is necessary to verify whether RPC conforms to the similar regularity of the dynamic strength and deformability if the applied strain rate overrides the range of 10–1.1×102 s−1. Considering the inconsistent performance of the peak strain εd,p and the residual strain εres varying with the fiber content ρv, to properly evaluate the contribution of fiber volume to improve the toughness of RPC (i.e., energy consumption capability), we calculated the energy consumed during the period from the beginning of impact until the peak strain, denoted as Epeak , the energy consumed during the period from the peak strain up to the residual strain,

JU Yang, et al.

Sci China Tech Sci


2443

Figure 10 Peak compressive strength, corresponding peak strain and residual strain of RPC varying with the fiber fraction and impact strain rate. (a) Peak strength; (b) peak strain; (c) residual strain. The black dots stand for the experimental data of the quantities, and the curved surface denotes fitting surface of the measured values.

denoted as Eres, and the total energy consumed from the beginning of impact until the residual strain, denoted as Edisp, from the measured stress-strain curves of RPC. Each energy quantity was taken as the area surrounded by the dynamic stress-strain curve from the start point until the end point of the specific period. Figure 11 illustrates the variation of dissipated energy with the fiber fraction and the strain rate during the period of concerns. The followings can be summed up from Figure 11. (i) For a given impact strain rate, the total consumed energy Edisp gradually rises with the increment of fiber fraction

if the fiber fraction is less than 2% (see Figure 11). It turns out to drop down as the fraction goes over 2%. It means that the fiber content plays a different role in improving impact-resistant deformability and toughness of RPC. (ii) The effects of steel fibers on improvement of energy consumption capability of RPC are different for the varied deformation periods. For instance, the energy Epeak consumed before the peak strain increases remarkably with the fiber volume increment (see Figure 11(a)). This energy, however, begins to drop gradually as long as the fiber volume exceeds 2%. The energy Eres dissipated during the post

2444

JU Yang, et al.

Sci China Tech Sci

peak period increases by a small amount with the increment of fiber volume (see Figure 11(b)). It implies that as long as the fiber content is not larger than 2%, fibers contribute more to improving the energy consumption capability in the period before the peak strain than to that in the period after the peak strain. The contribution of increasing fibers to enhancing the energy consumption turns out to be trivial as long as the fiber volume is more than 2%. The bigger the impact strain rate, the more apparent the characteristics. (iii) The impact strain rate ε has considerable effects on the total energy and the energy consumed in the varied periods as well. The higher the impact strain rate, the more


the energy consumed in each period, indicating a better dynamic toughness. For the sake of contrast and for easy applications, we formulate the empirical relationships of the quantities σd, p, εd, p, εres, Edisp, Epeak, Eres with the impact strain rate ε and the fiber volume ρv based on the experimental data as follows.

σ d,p = 1.31 × 10 2 + 0.38ε + 1.86 × 10 3 ρv − 6.42 × 10 4 ρv2 (MPa),

ε d,p = 7.32 + 5.32ε + 5.80 ρv − 0.57 ρv2 ,

(1) (2)

Figure 11 Energy consumption varies with fiber fraction and strain rate during the specific periods. (a) Epeak, energy consumed during the period from the beginning of impact up to the peak strain; (b) Eres, energy consumed during the period from the peak strain until the residual strain; (c) Edisp, total energy consumed from the beginning of impact until the residual strain. The black dots stand for the measured values and the curved surface represents the fitting surface for the measured data.

JU Yang, et al.

Sci China Tech Sci

ε res = 6.81 + 3.32ε − 9.02 ρv + 2.76 ρv2 ,

(3)

Edisp = 1.11 + 1.47ε + 4.71 × 10 3 ρv − 1.09 × 10 5 ρv2 (J),

(4)

Epeak = 14.90 + 0.48ε + 3.38 × 10 3 ρv − 7.81 × 10 4 ρv2 (J), (5) Eres = −13.79 + 0.99ε + 1.33 × 10 3 ρv − 3.05 × 10 4 ρv2 (J), (6)

where 0 ≤ ρv ≤ 3.0%, 10 ≤ ε ≤ 1.1×102 s−1. The formulas demonstrate that for a specific fiber fraction the dynamic quantity aforementioned of RPC linearly correlates to the impact strain rate ε , roughly. For a specific impact strain rate the quantities keeps a close quadratic non-linear relationship with the fiber volumetric fraction ρv.

4 Characteristics of dynamic stress-strain responses and modelling Stress-strain relationship not only is one of the major indexes for evaluation of the dynamic mechanical properties of RPC, but also underlies analysis and simulation of the dynamic responses of RPC structures. Comparing the failure patterns and stress-strain responses, one can identify that the characteristics of the dynamic responses of RPC closely relate to the impact strain rate and the fiber volume. 1) If the impact strain rate is low (i.e., ε ≤ 40 s−1) and no apparent fracture occurs in the specimen, the ascending portion of the stress-strain curve is convex toward the stress axis and the descending portion is convex toward the strain axis. An elastic strain release emerges in the descending portion. The starting point of the elastic strain release (i.e., point A in Figures 8 and 9) depends on the volume of steel fibers. The lower the fiber volume, the smaller the starting strain (see Figure 9(a)). For instance, the starting point for elastic stain release of the plain RPC is much smaller than that of the fibered RPC. This drops a clue of the mechanism that microcracking and viscous flow first take place in RPC’s matrix under impacts. Due to the effects of low impact rate and cracking resistance of steel fibers, there is no sufficient energy to raise the micro cracks. It implies that RPC’s deformability is primarily attributed to the increase in number of micro cracks and apparent viscous flow. The larger the fiber volume, the more evident the mechanism. When the wave stress lifts, part of irreversible microcracking and viscous flow forms the residual plastic deformation, while the other restores to induce the elastic strain release. The more the fiber content, the more the micro cracks and the more development of viscous flow inside. It leads to a late presence of the elastic strain release. 2) If a high impact strain rate applies (i.e., 60 < ε ≤ 1.1×102 s−1) and the specimen fractures markedly, the ascending part of the stress-strain curve appears to be convex toward the stress axis, and the descending part turns out to


2445

be concave toward the strain axis emerging an inflexion point (i.e., point B in Figures 8 and 9). For a given impact strain rate, the higher the fiber volume, the larger the stress of the inflexion point. The elastic strain release is not evident at this moment, which mainly emerges when the stress drops under 20%–25% of the peak stress (see Figures 9(c) and (d), since the elastic strain release occurs late and the amount is small, it is not marked in the figures). These features indicate that for a high strain rate more micro cracks emerge and coalesce in the matrix, resulting in the visible macro fracture cracks when the wave stress elevates. RPC deformability primarily results from the propagation and coalescence of the microcracks in the matrix and the pullout of steel fibers. As a consequence of the effect of fiber constraint, the stresses to drive and coalesce the microcracks rise when the fiber content increases. Coalescence of the microcracks and pull-out of the fibers induce a large unrecoverable residual deformation. The stiffness of RPC’s matrix gradually decreases, bringing the inflexion in the descending part of the stress-strain curve. Only a small part of deformation turns to be the elastic strain release. 3) As the strain rate falls within 40 < ε ≤ 60 s−1 and no apparent fracture takes place in the specimen, the ascending part of the stress strain curve appears to be convex toward the stress axis, while the descending portion turns out a straight line. Neither the elastic strain release, as in Case 1, nor the curve inflexion, as in Case 2, is observed (see Figures 8(a)–(c) and Figure 9(b)). This implies that the deformation mechanism of RPC lies between Cases 1 and 2. 4) It should be noted that the three patterns of stress-strain responses aforementioned emerges with the condition that the fiber volume satisfies ρv ≤ 2.0%. If the fiber volume surpasses 2%, the descending part of the stress-strain curve responds in a different manner (Figures 8(e) and 9). In Case 3, for instance, the descending curve of the specimen with fiber fraction of 2% or 3% turns out the inflexion and the elastic strain release (see Figure 9(b)). The inflexion segment appears to be very short. The curve turns to be convex to the strain axis shortly, which is similar to that of Case 1. For this case, the elastic strain release occurs late and the starting point of the strain release is greater than that in Case 1, close to the value in Case 2. It indicates that in spite of straying from the linear path the descending stress-strain curve responds in a manner between Cases 1 and 2. In addition, in Case 2, for example, the descending curve of the specimen with fiber volume of 2% or 3% turns out to be convex toward the strain axis, as is similar to the response in Case 1. The causes responsible for the deviations in the descending stress-strain responses, from the authors’ points of view, may be attributed to the limited functions of the added fibers to prevent the cement matrix from cracking and to constrain the growth and coalescence of microcracks when too many fibers are added. If too more steel fibers are embed-

2446

JU Yang, et al.

Sci China Tech Sci

ded, on the one hand, the fibers tend to bunch up, leading to a high nonuniform stress distribution and concentration in the matrix. As a consequence, the matrix is vulnerable to cracking or damage. The peak compressive strength is inclined to decrease and the stress-strain curve descends in the early stage. With the stress incessantly acting and RPC’s deformation increasing, on the other hand, a large amount of fibers can constrain to some extent the growth and coalescence of the microcracks. Higher forces and more energy are demanded to pull out the steel fibers. This brings a short inflexion stage in the descending stress-strain curve and a high corresponding stress. If the specimen is unloaded at this moment, the evident elastic strain release would take place and the stress-strain response would turn to be convex outward the strain axis. This deformation mechanism can explain the variations in the stress-strain responses when the steel fiber volume exceeds 2%. Based on the basic characteristics of the stress-strain curves, the dynamic stress-strain responses of RPC can be characterized by four models, namely, Types A, B, C and D, as shown in Figure 12, by taking account of the effects of impact strain rate and fiber content. Among them Type A is represented by the quad-linear model OMPNR, Type B by the tri-linear model OM′PR′, Type C by the quad-linear model OM″PN′R″ and Type D by the penta-linear model OM″′PN″N″′R″′. The lateral coordinates, ε/εd,p, refers to the ratio between the present strain ε of RPC subjected to the impact and the peak strain εd,p. We defined the ratio as the generalized strain. The vertical coordinate, σ/σd,p, refers to the ratio between the applied stress σ and the peak strength σd,p. It is defined as the generalized stress. The measure of normalizing stresses and strains to establish the general model of the dynamic stress-strain relationships is adopted to minimize the scattering data effects on the results and unveil the nature of the dynamic stress-strain responses of RPC. For simplifying the stress-strain model and easy application, the piecewise linear equations are employed to describe the four basic stress-strain responses. The generalized expression of each model is formulated as the followings. (I) Type A, the quad-linear model OMPNR, requiring ρv ≤ 2.0%, ε ≤ 40 s−1 . Its ascending portion consists of two linear segments with the origin O (0,0), the pre-peak knee point σm ⎞ ⎛ εm , M⎜ ⎟ and the peak point P (1,1). The linear equa⎝ ε d, p σ d, p ⎠ tions of the segments OM and MP are written as ⎧ ⎛ ε ⎞ σ = a1 + k1a ⎜ ⎪OM: ⎟⎟ , ⎜ σ d, p ⎪ ⎝ ε d, p ⎠ (a1 = 0; k1a > k2a > 0). (7a) ⎨ ⎛ ⎞ σ ε ⎪ a ⎪MP: σ = a2 + k2 ⎜⎜ ε ⎟⎟ , d, p ⎝ d, p ⎠ ⎩


Figure 12

Four basic models for the stress-strain relationships. Type A: quad-linear model, ρv ≤ 2.0%, ε ≤ 40 s −1 ; Type B: Tri-linear model,

ρv ≤ 2.0%, 40 < ε ≤ 60 s−1; Type C: quad-linear model with inflexion,

ρv ≤ 2.0%, 60 < ε ≤ 1.1 × 10 2 s−1 ;

and Type D: penta-linear model,

ρv > 2.0%, 40 ≤ ε ≤ 1.1 × 10 2 s−1 .

Its descending portion comprises another two linear segments with the peak point P (1,1), the post- peak inflex⎛ ε ⎛ εr σn ⎞ , ion point N ⎜ n , ⎟⎟ , and the ending point R ⎜ ⎜ε ⎝ ε d, p ⎝ d, p σ d, p ⎠

σr σ d, p

⎞ ⎟ . The residual stress abides by σ r ≤ 0.25σ d , p . The ⎠ linear equations of the segments PN and NR are expressed as

⎧ ⎛ ε ⎞ σ , = a3 + k3a ⎜ ⎪ PN: ⎜ ε ⎟⎟ σ d, p ⎪ ⎝ d, p ⎠ ⎨ ⎛ ε ⎞ σ ⎪ a ⎪NR: σ = a4 + k4 ⎜⎜ ε ⎟⎟ , d, p ⎝ d, p ⎠ ⎩

(k3a < 0, k4a > 0).

(7b)

(II) Type B, the tri-linear model OM′PR′, requiring ρv ≤ 2.0%, 40 < ε ≤ 60 s−1 . The ascending part consists of two straight lines with the origin O (0,0), the peak point P (1,1) and the pre-peak knee ⎛ ε m′ σ m′ ⎞ , point M′ ⎜ ⎟ . The linear segmental equations of ⎝ ε d, p σ d, p ⎠ OM' and M'P are expressed as ⎧ ⎛ ε ⎞ σ = b1 + k1b ⎜ , ⎪OM ′: ⎜ ε ⎟⎟ σ d, p ⎪ ⎝ d, p ⎠ ⎨ ⎛ ε ⎞ σ ⎪ b ⎪M ′P: σ = b2 + k2 ⎜⎜ ε ⎟⎟ , d, p ⎝ d, p ⎠ ⎩

(b1 = 0; k1b > k2b > 0). (8a)

Its descending part is represented by a straight line with σ r′ ⎞ ⎛ ε r′ , the peak point P (1,1) and the ending point R ′ ⎜ ⎟. ⎝ ε d, p σ d, p ⎠

JU Yang, et al.

Sci China Tech Sci


2447

The residual stress abides by σ r ′ ≤ 0.25σ d , p . Its linear

ual stress satisfies σ r ′′′ ≤ 0.25σ d , p . The tri-linear equations

equation is expressed as

of the segments PN″, N″N″′ and N″′R″′ are expressed as

PR ′:

⎛ ε σ = b3 + k3b ⎜ ⎜ε σ d, p ⎝ d, p

⎞ ⎟⎟ , ⎠

(k3b < 0).

(III) Type C, the quad-linear model OM″PN′R″, satisfying ρv ≤ 2.0%, 60 < ε ≤ 1.1 × 10 2 s−1 . Its ascending portion comprises two linear segments with ⎛ ε m′′ σ m′′ ⎞ , the origin O (0,0), the pre-peak knee point M ′′ ⎜ ⎟ ⎝ ε d, p σ d, p ⎠ and the peak point P (1,1). The bilinear equations of the segments OM″ and M″P can be formulated as ⎧ ⎛ ε ⎞ σ = c1 + k1c ⎜ ⎪OM ′′: ⎟, σ d, p ⎝ ε d, p ⎠ ⎪ ⎨ ε ⎞ σ ⎪ ′′ c⎛ ⎪M P: σ = c2 + k2 ⎜ ε d , p ⎟ , ⎝ ⎠ d, p ⎩

⎧ ⎛ ε ⎞ σ = d3 + k3d ⎜ ⎪ PN ′′: ⎟, σ d, p ⎝ ε d, p ⎠ ⎪ ⎪ ⎛ ε ⎞ σ ⎪ d d d = d 4 + k4d ⎜ ⎨N′′N′′′: ⎟ , (0 > k3 > k4 > k5 ). (10b) ε σ d p , ⎝ ⎠ d, p ⎪ ⎪ ε ⎞ ⎪N′′′R ′′′: σ = d + k d ⎛⎜ ⎟, 5 5 ε ⎪ σ ⎝ d, p ⎠ d, p ⎩

(8b)

(c1 = 0; k1c > k2c > 0). (9a)

The descending part contains two linear segments with the peak point P (1,1), the post-peak inflexion point σ n′ ⎞ ⎛ ε n′ ⎛ ε m′′ σ m′′ ⎞ , , N′ ⎜ ⎟ and the ending point R ′′ ⎜ ⎟. ⎝ ε d, p σ d, p ⎠ ⎝ ε d, p σ d, p ⎠ The residual stress requires σ r ′′ ≤ 0.25σ d , p . The bilinear

Regression analysis shows that the correlation indices between the equations and the measured data for each part of the four basic models are greater than 90%. It should be pointed out that each model contains a few coefficients to be determined. Among them there are seven coefficients for Types A and C, five for Type B and nine for Type D, respectively. For the equivalent test conditions and material ingredients, the coefficients of each model closely relate to the fiber fraction ρv and the impact strain rate ε. Since the models A, B, C and D comprise the piecewise linear segments with known convex and/or concave directions, i.e., the slope sign of the segmental line is known, the coefficients can be determined by impact tests. The determination procedure will be addressed separately.

equations of the segments PN'and N'R'' are written as ⎧ ⎛ ε ⎞ σ = c3 + k3c ⎜ ⎪ PN ′: ⎟, σ d, p ⎝ ε d, p ⎠ ⎪ (k3c < k4c < 0). (9b) ⎨ ε ⎞ ⎪ ′ ′′ σ c ⎛ ⎪N R : σ = c4 + k4 ⎜ ε d , p ⎟ , ⎝ ⎠ d, p ⎩ (IV) Type D, the penta-linear model OM″′PN″N″′R″′, requiring ρv > 2.0%, 40 ≤ ε ≤ 1.1 × 10 2 s−1 . Its ascending portion is represented by two linear segments with the origin O (0,0), the pre-peak knee point ⎛ ε m′′′ σ m′′′ ⎞ , M ′′′ ⎜ ⎟ and the peak point P (1,1). The linear ⎝ ε d, p σ d, p ⎠ equations of the segments OM″′ and M″′P can be written as ⎧ ⎛ ε ⎞ σ = d1 + k1d ⎜ ⎪OM ′′′: ⎟, σ d, p ⎝ ε d, p ⎠ ⎪ (d1 = 0; k1d > k2d > 0). (10a) ⎨ ε ⎛ ⎞ σ ⎪ ′′′ d ⎪M P: σ = d2 + k2 ⎜ ε d , p ⎟ , ⎝ ⎠ d, p ⎩ Its descending part comprises three straight lines with the peak point P(1,1), the first post-peak knee point ⎛ ε n′′ σ n′′ ⎞ ⎛ ε n′′′ , , N′′ ⎜ ⎟ , the second post-peak knee point N ′′′ ⎜ ⎝ ε d, p ⎝ ε d, p σ d, p ⎠ σ n′′′ ⎞ ⎛ ε r ′′′ σ r ′′′ ⎞ , and the ending point R ′′′ ⎜ ⎟ . The residσ d , p ⎟⎠ ⎝ ε d, p σ d, p ⎠

5

Conclusions

With regard to the prepared RPC materials and the specific SHPB impact conditions, we conclude the followings. (i) Plain RPC severely fragments when the impact load applies. The higher the impact strain rate, the more the energy consumed by the plain RPC and the greater the degree of fragmentation as well. The stress response appears to be greater than the strain response, showing a brittle dynamic characteristic of the plain RPC. Adding an apt amount of short steel fibers considerably raises the impact strain rate that RPC survives and the deformability of RPC at failure, as compared with the plain RPC. The characteristics of the reflected and transmitted waves of RPC vary with the steel fiber volume ρv and the impact strain rate ε. As long as ρv ≤ 2.0%, the transmitted and reflected pulses elevate their amplitudes with the increasing strain rate. The reflected wave introduces a higher increment in its amplitude and turns out multiple peaks. When the fiber volume fraction is over 2%, the transmitted wave amplitude fluctuates downhill gradually, while the reflected wave amplitude elevates continuously with the increasing fiber volume and impact strain rate. (ii) The impact strain rate significantly affects the impact resistance and deformability of RPC. For a specific fiber volume, the peak compressive strength, peak strain and re-

2448

JU Yang, et al.

Sci China Tech Sci

sidual strain increases by varied amount when the strain rate rises, among which the increment of the residual strain appears to be the largest. For a given impact strain rate, expanding the fiber volume makes little contribution to improving the residual deformability of RPC upon failure. The impact strain rate greatly influences the residual strain capability. (iii) The fiber volume has different effects on the peak compressive strength and the deformability of RPC. For a specific impact strain rate, the increasing fiber volume elevates the peak compressive strength σd,p of RPC when the volume fraction is less than 1.75%. If the fiber fraction exceeds 1.75%, however, the peak compressive strength σd,p turns out to decrease gradually. The peak compressive strength gets its maximum around the fiber fraction of 1.75%. In contrast with the peak strength, the peak strain εd,p rises incessantly with the increasing fiber volume for a given strain rate. (iv) For the identical impact strain rate, increasing fiber content can enhance the total energy Edisp of RPC consumed from beginning of impact until the time of residual strains. As the fiber volumetric fraction exceeds 2%, however, the total energy Edisp turns to decrease gradually. The contribution of fibers on the energy consumption capability of RPC varies during different loading periods. The energy Epeak consumed before the time of peak load increases remarkably with an increasing fiber volume. However, it tends to drop down as the fiber volume exceeds 2%. The energy Eres consumed after the peak load until the time of residual strains increases by a small amount as the fiber volume expands. It seems that as long as the fiber volume is less than 2% the embedded fibers contribute more to the energy consumption capacity of RPC in the period before the peak load than to those in period after the peak load. The higher the impact strain rate, the more significant the performance. On the contrary, the impact strain rate takes equally important effects on the total energy and the energy consumed in various stages. The larger the impact strain rate, the more the energy consumed and the better the dynamic impact toughness. (v) The empirical relationships between the peak compressive strength, peak strain, residual strain, total energy Edisp consumed until the time of residual strain, energy Epeak consumed before peak load and energy Eres consumed after peak load with the impact strain rate and the fiber volumetric fraction are formulated. It is shown that within a specific range of fiber volumes these quantities linearly relate to the impact strain rate approximately. For a specific range of impact strain rates the quantities are in a close non-linear quadratic relation with the steel fiber volume. (vi) The dynamic stress-strain response of RPC closely relates to the impact strain rate and the steel fiber volumetric fraction. By normalizing stresses and strains as the generalized stresses and strains, and taking account of the im-


pact strain rate ε and the fiber content ρv, we classify the dynamic stress-strain responses of RPC into four basic piecewise linear models. Among them Type A is a quad-linear model applicable to ρv ≤ 2.0%, ε ≤ 40 s−1, Type B a tri-linear model applicable to ρv ≤ 2.0%, 40< ε ≤ 60 s−1, Type C a quad-linear model applicable to ρv ≤ 2.0%, 60 < ε ≤ 1.1×102 s−1 and Type D a penta-linear model applicable to ρv > 2.0%, 40 ≤ ε ≤ 1.1×102 s−1. The coefficients in the models depend on the fiber fraction and the impact strain rate, which can be determined by impact tests. It should be addressed that the dynamic mechanical properties of RPC are concluded in accordance with the present material ingredients, experimental setup and measuring methods. It is necessary to verify whether the findings and conclusions apply to those RPC with the different ingredients, static mechanical properties and the strain rate exceeding the range of 10–1.1×102 s−1. This work was supported by the National Natural Science Foundation of China (Grant No. 50974125), the National Basic Research Project of China (“973” Project) (Grant Nos. 2010CB226804, 2002CB412705), the Natural Sciences and Engineering Research Council of Canada (PGS-D22006) and the Beijing Key Laboratory Projects.

1

2 3

4

5

6 7

8

9

10

11 12

13

Yazici H, Yardimci M Y, Aydin S, et al. Mechanical properties of reactive powder concrete containing mineral admixtures under different curing regimes. Construc Building Mater, 2009, 23: 1223– 1231 Liu C T, Huang J S. Fire performance of highly flowable reactive powder concrete. Construc Building Mater, 2009, 23: 2072–2079 Ju Y, Liu H B, Chen J, et al. Toughness and characterization of reactive powder concrete with ultra-high strength. Sci China Ser E-Tech Sci, 2009, 52: 1000–1018 Zhang Y S, Sun W, Liu S F, et al. Preparation of C200 green reactive powder concrete and its static-dynamic behaviors. Cem Concrete Comp, 2008, 30: 831–838 Lee M G, Wang Y C, Chiu C T. A preliminary study of reactive powder concrete as a new repair material. Construc Building Mater, 2007, 21: 182–189 Shaheen E, Shrive N G. Cyclic loading and fracture mechanics of Ductal concrete. Int J Fracture, 2007, 148: 251–260 Ju Y, Jia Y D, Liu H B, et al. Mesomechanism of steel fiber reinforcement and toughening of reactive powder concrete. Sci China Ser E-Tech Sci, 2007, 50: 815–832 Shaheen E, Shrive N G. Optimization of mechanical properties and durability of reactive powder concrete. ACI Mater J, 2006, 103: 444–451 Washer G, Fuchs P, Graybeal B A, et al. Ultrasonic testing of reactive powder concrete. IEEE Trans Ultras Ferroelec Freq Control, 2004, 51: 193–201 Bayard O, Plé O. Fracture mechanics of reactive powder concrete: Material modelling and experimental investigations. Engng Fract Mech, 2003, 70: 839–851 Blais P Y, Couture M. Precast, prestressed pedestrian bridge-world's first reactive powder concrete structure. PCI J, 1999, 44: 60–71 Bonneau O, Lachemi M, Dallaire E, et al. Mechanical properties and durability of two industrial reactive powder concretes. ACI Mater J, 1997, 94: 286–290 Dugat J, Roux N, Bernier G. Mechanical properties of reactive powder concretes. Mater Struct, 1996, 29: 233–240

JU Yang, et al.

14 15 16 17 18 19 20

21

22

23

24

25

26

27

28

Sci China Tech Sci

Roux N, Anreade C, Sanjuan M A. Experimental study of durability of reactive powder concretes. J Mater Civil Engng, 1996, 8: 1–6 Dowd W, Dauriac C E. Reactive powder concrete. Construc Specifier, 1996, 49: 47–52 Richard P, Cheyrezy M. Composition of reactive powder concretes. Cem Concrete Res, 1995, 25: 1501–1511 Richard P, Cheyrezy M. Reactive powder concrete with high ductility and 200-800 MPa compressive strength. ACI SP144, 1994, 507–518 Tai Y S. Uniaxial compression tests at various loading rates for reactive powder concrete. Theore Appl Fract Mech, 2009, 52: 14–21 Tai Y S. Flat ended projectile penetrating ultra-high strength concrete plate target. Theore Appl Fract Mech, 2009, 51: 117–128 Tian Z M, Wu P A, Jia J W. Dynamic response of RPC-filled steel tubular columns with high load carrying capacity under axial impact loading. Trans Tianjin Univer, 2008, 14: 441–449 Wang Y H, Wang Z D, Liang X Y, et al. Experimental and numerical studies on dynamic compressive behavior of reactive powder concretes. Acta Mech Solida Sinica, 2008, 21: 420–430 Lai J Z, Sun W, Rong Z D. Dynamic mechanical behavior of reactive powder concrete subjected to repeated impact (in Chinese). Explosion Shock Waves, 2008, 28: 532–538 Wang Y H, Liang X Y, Wang Z D, et al. Experimental study on the impact compressive behavior of reactive powder concrete (in Chinese). Eng Mech, 2008, 25: 167–172 Bagheri A R, Zanghane H. Energy absorption of reactive powder concrete (RPC) in static and impact loadings. Proc Annual ConfCanadian Soc Civil Engng: Where the Road Ends, Ingenuity Begins, 2007, 1: 503–510 Ge T, Pang Y F, Tan K K, et al. Study on resistance of reactive powder concrete to impact (in Chinese). J Rock Mech Eng, 2007, 26: 3553–3557 Fujikake K, Senga T, Ueda N, et al. Study on impact response of reactive powder concrete beam and its analytical model. J Advan Concr Tech, 2006, 4: 99–108 Fujikake K, Senga T, Ueda N, et al. Effects of strain rate on tensile behavior of reactive powder concrete. J Advan Concr Tech, 2006, 4: 79–84 Kuznetsov V, Rebentrost M, Waschl J. Strength and toughness of steel fiber reinforced reactive powder concrete under blast loading. Trans Tianjin Univ, 2006, 12: 70–74


29

30

31

32

33

34 35

36

37

38

39 40

41

2449

Wang D R, Ge T, Zhou Z P, et al. Investigation of calculation method for anti-penetration of reactive power steel fiber concrete (RPC) (in Chinese). Explos Shock Waves, 2006, 4: 46–52 Toutlemonde F, Sercombe J. FRC containers design to ensure impact performance. Proc the 1999 Struct Congress Struct Engng in the 21st Century, ASCE Structures Congress-Proceedings, 1999. 183–186 Yan G J, Yan G P, An M Z, et al. Experimental study on 200 MPa reactive powder concrete (in Chinese). J China Railway Society, 2004, 26: 116–119 Wu Y H, He Y B, Yang Y H. Investigation on RPC200 mechanical performance (in Chinese). J Fuzhou Univ (Natural Science), 2003, 31: 598–602 Liu H B. Experiment Investigation on Preparation Technology and Mechanics Performance for Reactive Powder Concrete (in Chinese). Dissertation of Masteral Degree. Beijing: China University of Mining and Technology, 2006 Lindholm U S. Some experiments with split Hopkinson pressure bar. J Mech Phys Solids, 1964, 12: 317–335 Davies E D H, Hunter S C. Dynamic compression testing of solids by method of split Hopkinson pressure bar. J Mech Phys Solids, 1963, 11: 155–179 Sheng G H. Dynamic Mechanical Properties of Reactive Powder Concrete under Impact Loading Dissertation (in Chinese). Dissertation of Masteral Degree. Beijing: China University of Mining and Technology, 2009 Frew D J, Forrestal M J, Chen W. Pulse shaping techniques for testing elastic-plastic materials with a split Hopkinson pressure bar. Exper Mech, 2005, 45: 186–195 Frew D J, Forrestal M J, Chen W. Pulse shaping techniques for testing brittle materials with a split Hopkinson pressure bar. Exper Mech, 2002, 42: 93–106 Lambert D E, Ross C A. Strain rate effects on dynamic fracture and strength. Int J Impact Engng, 2000, 24: 985–998 Zhao H. A study on testing techniques for concrete-like materials under compressive impact loading. Cem Concrete Comp, 1998, 20: 293–299 Franz C E, Follansbee P S, Wright W J. New experimental techniques with the split Hopkinson pressure bar. Proc the 8th Int Conf High Energy Rate Fabrication, Pressure Vessel and Piping Division, San Antonio, TX, 1984