Effect of specimen size on dynamic compressive properties of fiber-reinforced reactive powder concrete at high strain rates

Construction and Building Materials 194 (2019) 71–82

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Effect of specimen size on dynamic compressive properties of fiber-reinforced reactive powder concrete at high strain rates Shaojun Cao a,b, Xiaomeng Hou a,b,⇑, Qin Rong a, Wenzhong Zheng a,b, Muhammad Abid a,b, Gang Li a,b a

Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China Key Lab of the Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin 150090, China b

h i g h l i g h t s
SHPB impact tests are conducted on hybrid fiber-reinforced RPC.
The DIF of RPC strength with bigger specimen is greater than smaller specimen.
RPC specimen size has no visible effect on critical strain DIF, elastic modulus DIF, and energy absorption.
The critical damage of RPC specimens with greater size is larger than that with smaller size.

a r t i c l e

i n f o

Article history: Received 4 July 2018 Received in revised form 27 October 2018 Accepted 2 November 2018 Available online 8 November 2018 Keywords: Size effect Split Hopkinson pressure bar (SHPB) Reactive powder concrete (RPC) Dynamic compressive properties Dynamic increase factor (DIF)

a b s t r a c t This study presents an experimental study of similar reactive powder concrete (RPC) cylinders with different sizes subjected to axial impact loading. Dynamic compressive experiments were conducted on two dimensional RPC specimens using split Hopkinson pressure bar device. 54 hybrid fiber-reinforced RPC samples with 2% steel fiber and 0.2% polypropylene (PP) fibers contents by volume were characterized at strain rates from 107 to 356 s1. Results from this and previous studies are analyzed to comprehensively investigate the effect of RPC specimen size on dynamic compressive performance. A series of quantitative analyses examining the relationship between dynamic properties and specimen size, strain rate, and steel fiber were conducted. Dynamic increase factor (DIF) of RPC compressive strength for specimens with larger diameters was consistently greater than RPC specimens with smaller diameters when the length to diameter ratio (l/d) was constant. RPC specimen size exhibited no visible effect on the critical strain DIF. Critical strain DIF increased with increasing strain rate and decreasing steel fiber addition. Critical damage variables of RPC specimens of greater size were higher than that of smaller sizes. Under identical strain rates, RPC specimen size had no visible effect on the DIF of Young’s modulus or energy absorption. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction Reactive powder concrete (RPC) is primarily used ultra-high performance concrete (UHPC) [1,2]. RPC has excellent mechanical properties such as ultra-high strength, high fracture capacity, and excellent durability [3]. RPC has great potential for use in longspan bridges, military engineering, and protective buildings [4]. Over the past few decades, some studies [5–8] investigated the mechanical properties of RPC. Under highly dynamic conditions, the correlation of strain-rate to material behavior and hydrostatic ⇑ Corresponding author at: Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China. E-mail address: [email protected] (X. Hou). https://doi.org/10.1016/j.conbuildmat.2018.11.024 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.

pressure causes a strikingly different material response compared to quasi-static conditions. Therefore, it is important to determine the dynamic behavior of RPC because of its wide application in protective engineering. Experimental studies on RPC behavior primarily examined RPC dynamic mechanical properties [9–21]. Ren et al. [9], Rong et al. [10] and Wang et al. [11] investigated the dynamic compressive properties of steel fiber-reinforced RPC (SFRPC) at the strain rate of 1–100 s1 using split Hopkinson pressure bar (SHPB) device. The presence of the strain rate has an obviously positive influence on the RPC compressive performances. In addition, the impact resistance capacity of RPC improved with increasing steel fiber content. Further, Ju et al. [12] studied the dynamic mechanical performances of SFRPC with five steel fiber volumetric fractions of 0,

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Nomenclature The following symbols are used in this paper: rs the average axial stress of RPC specimen es the average strain of RPC specimen e_ s strain rate of RPC specimen the cross-sectional area of pressure bars Ab As the cross-sectional area of RPC specimens Eb the elastic modulus of pressure bars c the elastic wave pulse velocity Ls original length of cylinder eI the strain of the incident pulse eR the reflected pulse eT the transmitted pulse zb the acoustic impedance of the incident bar Ab the section area of the incident bar qb the density of the incident bar the wave speed of the incident bar cb zs the acoustic impedance of the specimen As the section area of the specimen qs the density of the specimen the wave speed of the specimen cs l the height of RPC specimen d the diameter of RPC specimen v the Poisson’s ratio of the material rm dynamic compressive strength of RPC DIF dynamic increasing factor em critical strain of RPC

1%, 1.5%, 2%, and 3%, respectively, under strain rates at the range of 20–105 s1. Results indicated that when the fiber volume fraction was less than 1.75%, the dynamic compressive strength increased with fiber proportion under a constant strain rate. In contrast, when the fiber volume fraction was more than 1.75%, the dynamic compressive strength had a tendency to decrease with increasing fiber volume. Zhang et al. [13] performed axial impact tests on UHPC with steel fiber volumes of 0, 1%, 2%, 3%, and 4% at the strain rates ranging from 10 to 114.7 s1. Test results revealed that UHPC obviously presented an enhancement in the dynamic strength. The peak stress increased rapidly with increasing strain rate. Steel fiber can improve the dynamic strength as well as energy absorption capability of UHPC. Jiao et al. [14] investigated RPC dynamic behavior at the strain rate ranging from 30 to 95 s1. Results revealed that the dynamic increase factor (DIF) of plain RPC (PRPC) compressive strength was larger than that of SFRPC. Based on the SHPB apparatus, Hou et al. [20,21] performed dynamic compression tests on three kinds of SFRPC. Based on test data, a theoretical formula was established to calculate the dynamic elastic modulus, and a damage-softening model applicable to the dynamic stress-strain relationship of SFRPC at high strain rates was proposed. However, existing studies referring to RPC dynamic properties commonly analyzed on RPC specimens of varying sizes at the range of 10–35 mm in height and 30–75 mm in diameter [9–21]. Some studies [22–28] had already reported that specimen size has a significant effect on static and dynamic properties, e.g., elastic modulus and compressive strength, of concrete materials. Elfahal and Krauthammer et al. [22,23] carried out numerical analyses and experimental studies on geometrically similar normal strength concrete (NSC) and high strength concrete (HSC) cylinders with different dimensions subjected to impact. Results revealed a time-dependent size effect phenomenon under dynamic conditions, which was different from the known static size effect. Based on SHPB tests and numerical simulations, Hao et al. [24–26] investigated the influence of aggregates, end friction confinement, and

Ed Es Dc W DIFT DIF e_ DIF l DIFi fd fs f e_

Df e_ Df l Df i

v l

½e_  Vf D E0

ef

dynamic elastic modulus static elastic modulus critical damage variable energy absorption DIFs of compressive strength from the tests DIFs of compressive strength from strain rate effect DIFs of compressive strength from end friction confinement DIFs of compressive strength from lateral inertia confinement the dynamic compressive strength the quasi-static compressive strength the dynamic compressive strength enhancement due to strain-rate effect dynamic compressive strength increments due to strain rate effect dynamic compressive strength increments due to end friction confinement dynamic compressive strength increments due to lateral inertia confinement the end friction confinement coefficient dynamic friction coefficient during SHPB tests the critical strain rate steel fiber content damage variable the initial elastic modulus final ultimate compressive strain

lateral inertia confinement on dynamic concrete compressive strengths under impact. Results indicated that the end friction confinement effect is sensitive to the strain rate and specimen geometry. A DIF equation for concrete material strength that removes the effect of friction confinement effect was proposed. Lee et al. [27] experimentally studied the effects of cylinder size on empirical equations referring to the relationship between compressive strength and dynamic elastic modulus of NSC. Results indicated that the cylinder size did not influence the test results including compressive strength, experimental variability, and elastic modulus when the concrete strength is less than 40 MPa. Cylinder size does have a substantial effect when the strength of concrete greater than 40 MPa. Based on SHPB experiments, Wang et al. [28] investigated the effect of sample size on the static strength and DIF of fiber-reinforced HSC. Results indicated that the European code Comité Européen du Béton (CEB) [29] equation significantly overestimated experimental DIF values for fiber-reinforced HSC. Nevertheless, those existing studies were not sufficient to determine the effect of size for RPC elements subjected to dynamic loading conditions. Quantitative analysis of the effect of size on concrete dynamic properties is rare, and the effect of size on RPC dynamic properties, e.g., critical strain, energy absorption ability, and damage level, has not been studied yet. To overcome current limitations, this study presents SHPB experiments on RPC cylinders adding 2% steel fiber and 0.2% polypropylene (PP) fiber [30,31] by volume (SPFRPC) under high strain rates ranging from 107 to 356 s1. The characteristics of RPC dynamic properties (dynamic compressive strength, dynamic deformation, dynamic elastic modulus, damage level, and energy absorption) were investigated. The influence of RPC sample size on RPC dynamic properties is of great importance. The effects of specimen size, steel fiber content, and strain rate on the dynamic properties of RPC are quantitatively investigated.

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2. Specimen preparation A series of RPC samples for SHPB experiments were prepared according to the RPC ingredients and preparation procedure. The mixing ratio and ingredients of RPC, chemical contents of RPC raw materials including ordinary Portland cement (OPC), silica fume (SF), and ground granulated blast furnace slag (GGBS), and characteristics and mechanical properties of fibers including PP fiber and steel fiber are given in Ref. [20,21]. RPC raw materials particle size distributions are given in Table 1. The steel fibers have a diameter of 0.22 mm and a length of 13 mm on an average. The elastic modulus and tensile strength of steel fibers were 200 GPa and 2250 MPa, respectively. Besides, PP fibers with the length of 18 to 20 mm and the average diameter of 45 lm were used. The tensile strength and elastic modulus of the PP fibers were 500 MPa and 3.85 GPa, respectively. The PP fibers had a melting point of 165 °C and density of 0.91 gcm2. The RPC casting procedure in this study is described in [20,21]. The so-obtained RPC samples were first cured at ambient temperature for 24 h and then cured under 90 °C heated vapor for additional three days. To meet the SHPB test requirements, specimens were trimmed to the following dimension: six U36 mm  36 mm (diameter by height) RPC cylinders, six U75 mm  75 mm RPC cylinders, and six 100 mm  100 mm  100 mm cubes were prepared for quasi-static testing. 24 U36 mm  17.5 mm RPC cylindrical samples and 30 U75 mm  37.5 mm RPC cylinders were prepared for axial impact testing. The two ends of RPC samples were polished to ensure smooth contact between the specimen and test apparatus. 3. Quasi-static compression tests Prior to impact testing, a series of quasi-static compression tests were performed on 12 RPC cylinders and six RPC cubes (Table 2). The coefficient of variation (COV) of static compressive strength for RPC specimens with different sizes is presented in Table 2. In the Chinese standard code of Reactive Powder Concrete (GB/T 31387-2015) [32], the compressive strength of RPC is divided into five grades, which is RPC100, RPC120, RPC140, RPC160 and RPC180, respectively. According to this, the RPC in this study could meet the requirement category of RPC compressive strength. As the size of RPC samples increases with a constant heightdiameter ratio, the static compressive strength decreases, which is consistent with existing researches [22–28]. When the RPC specimen size changes from U36  36 mm to U75  75 mm, static compressive strength decreases 8.8%. 4. Split Hokinson pressure bar compression experiments The SHPB experiment is frequently used to measure the dynamic behavior of concrete materials under high strain rates at the range of 102–104 s1 [33]. The U36 mm17.5 mm RPC specimens were tested on a 40-mm diameter SHPB system at Harbin Institute of Technology. This SHPB device is comprised of an incident bar and transmission bar, both with 40 mm in diameter (Fig. 1). The U75 mm  37.5 mm RPC specimens were tested on a

100-mm diameter SHPB system at the Impact Dynamics Laboratory, Hunan University. This SHPB device is comprised of 6000 mm input bar and 4000 mm transmission bar (Fig. 1). The diameter of this SHPB device is 100 mm. The material of the SHPB bars is Aluminum. Strain gages were placed on the input and transmission bars. The RPC cylinder is placed between the input bar and transmission bar. Fig. 2 illustrates the signals obtained from the strain gages stuck on the input and transmission bars. During test, the compressive pulse is first generated by the striker. Next, the compressive stress pulse is transmitted to the input bar and is recorded by the strain gauges, forming the incident wave. Then, the stress pulse traversed the input bar and impacted on the RPC cylinder. A portion of the stress pulse was transmitted across the specimen into the transmitted bar, forming the transmitted wave. The other part was reflected back into the incident bar, forming the reflected wave. The split bar was based on one-dimensional elastic wave propagation theory [34–36]. The following strain measurements were utilized to obtain the time histories of the stress, strain and strain rate during impact:

rs ¼

Ab Eb ðeI þ eR þ eT Þ 2As

es ¼

c Ls

e_ s ¼

c ðeI  eR  eT Þ Ls

Z

t 0

ð1Þ

ðeI  eR  eT Þ

ð2Þ ð3Þ

where rs, es, and e_ s are the average stress, calculated strain, and strain rate, respectively. In addition, Ab and As represent the crosssectional area of the pressure bars and RPC specimens, respectively. c represents the elastic wave pulse velocity. Eb is the elastic modulus of the pressure bars, and Ls denotes the original length of specimen. Furthermore, eI, eR, and eT represent the strain of the incident pulse, reflected pulse, and transmission pulse, respectively. The bar material and the specimen material have to fit to each other, that is, they need to have a similar susceptibility to wave propagation [37]. The acoustic impedance of materials can be calculated as follows [37–39]:

zb ¼ Ab  qb  cb zs ¼ As  qs  cs where z, A, q, and c are the acoustic impedance, the section area, the density, and the wave speed. The subscripts of b and s represent the incident bar and the RPC specimen, respectively. The wave speed in incident bar and RPC specimen can be expressed as cb = 5052 m/s [40] and cs = 7000 m/s [41], respectively. From calculation, it can be found that the error of the acoustic impedance of these two materials is smaller than 4.0%, which means the acoustic impedances of these two materials are almost the same. According to previous related research conclusions using numerical simulation [42], a piece of 6 mm  6 mm  1 mm (length by width by thickness) lead shaper [42] was attached to the striker at the surface between the striker and input bar to

Table 1 Particle size distributions of RPC raw materials. Material types

Particle size distributions (lm) <0.1

<0.2

<0.5

<1.0

<5.0

<10

<50

<100

OPC (%) SF (%) GGBS (%)

0.00 15.64 0.00

0.00 39.23 0.01

0.25 70.20 2.42

2.57 92.29 7.12

23.41 99.30 39.87

40.16 99.77 64.76

95.05 100 98.74

100 100 100

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Table 2 Results of quasi-static test. No.

Fiber Types and dosage

Number of Specimens

Dimension of Specimens/mm

Compressive strength Average/MPa

COV

I II III

2% Steel fiber + 0.2 %PP fiber

6 6 6

U36  36 U75  75

100.1 93.8 115.1

0.049 0.027 0.021

Gas Gun

100  100  100

Input Bar

Strain Gauges

RPC Specimen

Output Bar

Strain Amplifier

Absorption Bar

Strain Gauges

Velometer Computer

Digital Oscilloscope

a) Principle diagram

Transmission bar

Transmission bar

Incident bar Incident bar Striker

Striker

b) 40-mm SHPB apparatus

c) 100-mm SHPB apparatus

Fig. 1. Details of SHPB experimental apparatus.

Voltage (V)

2

fiber was tested for its dynamic compressive properties. PP fiber melts at 160–170 °C. The addition of PP fiber is to obtain a higher fire resistance. Under intense heat, some channels will be produced in RPC. Water vapor can escape the concrete matrix through these channels, preventing RPC from experiencing fire-induced spalling. In a SHPB test, the sample must be thin enough for a uniform stress state to rapidly form along the height side of the sample. Davies [46] suggested that the optimal length to diameter ratio (l/d) is (3v/4)0.5, where v represents the Poisson’s ratio of the material. In this paper, the optimal l/d is 0.40. Samples were prepared as U36 mm  17.5 mm and U75 mm  37.5 mm cylinders, with aspect ratios of 0.48 and 0.50.

Incident Transmitted

1 0 Reflected -1 -2 -0.5

Incident Bar Signal Transmitted Bar Signal

5. Results and discussion

0.0

0.5 1.0 Time (ms)

1.5

2.0

Fig. 2. Recorded signals in 100-mm-diameter SHPB test at 190 s1.

achieve the stress equilibrium in this study. Fig. 3 certifies that the stress equilibrium in RPC specimens with the dimension of U75 mm  37.5 mm was achieved using the pulse shaper technique during the dynamic impact test [43–45]. In order to determine the effect of size on RPC dynamic properties, the hybrid fiber-reinforced RPC with 2% steel fiber and 0.2% PP

Results of each experiment including dynamic compressive strength (rm), DIF of compressive strength, critical strain (em), DIF of elastic modulus (Ed/Es), energy absorption (W), and critical damage variable (Dc) are listed in Tables 3 and 4. Six identical samples for each impact strain-rate group were tested. Three of each sample type were chosen and listed in Tables 3 and 4. 5.1. Dynamic compressive strength Factors which may influence the concrete dynamic compressive strength include the specimen l/d, which is described as the friction

75

30

30

0

0

-30

Stress (MPa)

Stress (MPa)

S. Cao et al. / Construction and Building Materials 194 (2019) 71–82

-60 -90 Transmitted Stress Incident+Reflected Stress

-120 -150

-200

0

200

400 600 Time ( s)

800 1000

-30 -60 -90 -120 Transmitted Stress Incident+Reflected Stress

-150 -180

-200

0

a) 190 s-1

200 400 Time ( s)

600

b) 356 s-1

Fig. 3. Stress equilibrium check in 100-mm-diameter SHPB test.

Table 3 SHPB results of U36  17.5 mm samples. Strain-rate group/s1

Strain rate e_ /s1

Dynamic Strength rm/MPa

Dynamic Strength DIF

Critical strain.em/103 e

Ed/Es

Critical damage.Dc

Energy absorption W/MJ/m3

173

170 200 148

101.8 99.8 117.0

1.02 1.00 1.17

3.72 3.61 3.98

1.41 1.60 1.26

0.397 0.407 0.426

3.54 3.46 3.15

223

228 233 209

120.0 100.6 147.1

1.20 1.01 1.47

4.44 3.82 5.32

1.77 1.79 1.71

0.381 0.348 0.406

3.89 3.70 3.07

278

275 278 281

170.4 185.0 184.7

1.70 1.85 1.85

5.23 5.16 5.09

1.93 1.94 1.96

0.416 0.455 0.458

4.49 4.23 4.53

304

304 317 292

233.8 249.8 205.1

2.34 2.50 2.05

5.26 5.31 5.18

2.49 2.54 2.45

0.537 0.555 0.489

4.95 5.62 5.53

Table 4 SHPB results of U75  37.5 mm samples. Strain-rate group/s1

Strain rate e_ /s1

Dynamic Strength rm/MPa

Dynamic Strength DIF

Ed/Es

Critical damage Dc

Energy absorption W/MJ/m3

118

126 107 120

74.2 84.4 67.0

0.79 0.90 0.71

3.73 4.29 3.46

1.33 1.26 1.32

0.482 0.506 0.512

3.528 3.738 2.835

193

197 190 193

123.0 131.1 153.9

1.31 1.40 1.64

4.31 3.90 5.08

1.51 1.47 1.49

0.643 0.580 0.621

3.990 5.030 4.599

237

233 245 233

148.7 135.6 137.9

1.59 1.45 1.47

3.38 5.40 5.44

1.70 1.73 1.68

0.466 0.662 0.671

5.838 3.854 5.093

278

277 275 280

133.0 173.7 172.4

1.42 1.85 1.84

5.73 6.83 6.81

2.19 2.18 2.20

0.699 0.682 0.684

5.166 5.660 5.051

349

356 344 348

174.0 160.0 157.0

1.86 1.71 1.67

3.58 6.44 6.51

2.97 2.90 2.91

0.482 0.703 0.708

4.452 5.975 5.775

Critical strain

em/103 e

on the end surfaces, the sample diameter, which is described as the lateral inertia confinement, and the real strain rate sensitivity of concrete materials [47]. When the dynamic compressive strength is investigated, structural factors including the lateral inertia confinement and the end friction confinement are intensively studied [24]. The friction at specimen-apparatus interfaces, known as end friction confinement, constrains the lateral deformation of RPC samples. The lateral deformation results in a backward inertial force that limits the lateral deformation of RPC specimens, which is recognized as the lateral inertia confinement (Fig. 4) [24].

The DIF of RPC compressive strength from SHPB impact testing can be expressed as [26]:

  DIF T ¼ f d =f s ¼ f e_ þ Df l þ Df i =f s ¼ DIF e_ þ DIF l þ DIF i

ð4Þ

where DIFT, DIF e_ , DIF l , and DIFi are DIFs of compressive strength from the tests, strain rate effect, lateral inertia confinement, and end friction confinement, respectively. fs and fd are quasi-static and dynamic compressive strength, respectively. In addition, f e_ ¼ Df e_ þ f s is the dynamic compressive strength enhancement

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S. Cao et al. / Construction and Building Materials 194 (2019) 71–82

2.5

DIF

2.0

36-17.5 mm 75-37.5 mm

1.5 1.0 0.5

Fig. 4. End friction confinements and Lateral inertia.

10 due to strain-rate. Df e_ , Df l , and Df i are dynamic compressive strength increments due to strain rate, end friction confinement, and lateral inertia confinement, respectively.

5.1.1. Size effect on dynamic compressive strength From test results in Tables 3 and 4, the comparison of dynamic compressive strength between RPC samples with different sizes is obtained and illustrated in Fig. 5. Generally, the RPC dynamic compressive strengths under the high strain rate at the range of 107–356 s1 are higher than static strengths. This is because the duration of dynamic loading during the SHPB impact is short. The stress in the RPC matrix was improved to achieve the improvement of energy absorption. In addition, the dynamic compressive strength for SPFRPC samples with a diameter of 75 mm is consistently greater than SPFRPC samples with a diameter of 36 mm. This phenomenon will be discussed later.

100 Strain rate/s-1

1000

Fig. 6. DIF of compressive strength of SPFRPC specimens with different sizes.

5.1.2.1. End friction confinement. Using mesoscale model considering the concrete raw materials, a series of numerical simulations for the effects of end friction confinement on concrete material DIF were performed by Hao et al. [25]. An equation to remove the influence of end friction confinement on DIF considering the l/d ratio, friction coefficient, and strain rate as variables was proposed by Hao et al. [25]:

v¼

DIF l¼0 2 ¼ e3:052410 ðl=dÞ0:2494l0:1043loge_ þ0:1563 DIF l>0

ð5Þ

where v is the end friction confinement coefficient. l is dynamic friction coefficient during SHPB testing. Applying l = 0.1 in Eq. (5), the best fit can be obtained [25]. The value range for individual parameters can be expressed as follows:

0:5  l=d  2:0; 0:0  l  0:5 and 10 s1  e_  600 s1 5.1.2. Effect of size on compressive strength DIF The effect of strain rate on the compressive strength DIF for SFRPC with various sizes in this study is shown in Fig. 6. With identical l/d, DIFs of SPFRPC samples with a large diameter are consistently greater than SPFRPC samples with small diameter [47,48]. To further investigate the effect of structural size on compressive strength DIF, the end friction confinement and lateral inertia confinement are considered.

5.1.2.2. Lateral inertia confinement. To investigate the influence of lateral inertia confinement on RPC dynamic compressive strength, the compressive strength DIF for SPFRPC samples with various diameters are compared after removing end friction confinement using Eq. (5) (Fig. 7). Lateral inertia confinement has a significant influence on the DIF of SFRPC compressive strength. Nevertheless, test data in this study are slightly insufficient to propose a

2.5

250

36-17.5 mm 75-37.5 mm

2.0 1.5

150

DIF

Strength/MPa

200

36 mm 75 mm Fitting curve Fitting curve

1.0

100 0.5

50 100

150

200 250 300 Strain rate/s-1

350

Fig. 5. Dynamic compressive strength for RPC of different sizes.

0.0 10

100 Strain rate/s-1

1000

Fig. 7. DIF of compressive strength after removing end friction confinement.

S. Cao et al. / Construction and Building Materials 194 (2019) 71–82

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theoretical equation to calculate the DIF for RPC compressive strength considering the effect of structural size. To overcome this limitation, some existing test data from RPC impact compression tests [9–21] are taken into account. Only existing DIFs for PRPC and RPC with 2% steel fiber by volume (SFRPC2) compressive strengths are considered after removing end friction confinement (Fig. 8). Lateral inertia confinement has a significant effect on the DIF of compressive strength for PRPC and SFRPC. Specifically, the influence on PRPC is higher than SFRPC, which is attributed to the decreased sensitivity of steel fiber to strain rate compared with RPC. Thus, adding steel fiber can decrease the total strain-rate sensitivity of SFRPC. In this paper, a fitting formula for the relationship between the strain rate and dynamic compressive strength considering the effect of structural lateral inertia confinement is determined on the basis of experimental results in this contribution and existing publications [9–21]:

where ½e_  is the critical strain-rate. The strain-rate effect was neglected when the strain rate is less than ½e_ . e_ s = 1 s1 is utilized to achieve a dimensionless of solution. The values for a and b are as follows:

 b e_ DIF e_ þ DIF i ¼ a ½e_   e_  317s _es

DIF i ¼ f d =f s ¼ Df i =f s ð6-aÞ

- 1

DIF e_ þ DIF i  1

ð6-bÞ

2.5 2.0

DIF

1.5

30 mm [15] 30 mm [16] 36 mm [20-21] 36.5 mm [18] 50 mm [11] 50 mm [17] 56 mm [12] 70 mm [10] 70 mm [14] 70 mm [16] 75 mm [13]

1.0 0.5 0.0 10

100 Strain rate/s-1

DIF

1.5 30 mm[15] 30 mm[16] 36 mm [20-21] 50 mm[11] 50 mm[17] 56 mm[12] 60 mm[9] 60 mm[19] 70 mm[16] 75 mm[13]

100 Strain rate/s-1

ð7-bÞ

Eq. (6) considers effects of both lateral inertia confinement and real strain-rate on RPC dynamic compressive strength. However, the misinterpretation of confinement enhancement as strain-rate enhancement in SHPB tests leads to a non-conservative design or analysis of a concrete structure against impact or blast loading [49]. Previous researches show that the inertia effect cannot be removed by adjusting the specimen geometry [47,50]. For strain rate independent materials, i.e., Df e_ ¼ 0:0, a DIF is only resulted from lateral inertia confinement [26] as follows:

ð8Þ

where DIFi is the DIF obtained from simulations without considering the effect of strain rate. The true material DIF is then

DIF e_ ¼ f d =f s ¼ ðDf e_ þ Df s Þ=f s ¼ DIF T  DIF i  DIF l

ð9Þ

Using the computer code AUTODYN with user defined subroutines, Hao et al. [26] conducted numerical simulations of SHPB tests for concrete and rock specimens at different strain rates to determine the DIF attributed to lateral inertia confinement by setting DIF equal to 1.0 and neglecting end friction. In this case, the numerically obtained strength increment of the tested material is attributed purely to lateral inertia confinement (see Fig. 9). However, Hao et al. [26] only carried out numerical simulations on 100  100 mm concrete specimens. It is not proper to take the lateral inertia confinement of 100  100 mm concrete specimens to represent the lateral inertia confinement of all concrete samples since lateral inertia confinement is specimen size dependent [26]. Therefore, since no sufficient information is available to separate the DIFs of RPC materials from different specimen sizes, in the present study, a DIF formula which considers both the effect of real material strain rate and lateral inertia confinement is proposed.

A series of strain pulses and failure patterns with varying impact strain rates for 40-mm-diamater SHPB tests and 100-mmdiameter SHPB tests, respectively are illustrated in Figs. 10 and 11. The dynamic stress-strain relationship varies with average strain rate for these two types of tests (Fig. 12).

2.0

0.0 10

  ð1:428Þ b ¼ 88:686 þ 389:2  V f  d

5.2. Compressive deformation

2.5

0.5

ð7-aÞ

1000

a) PRPC

1.0

    a ¼ 0:0125  0:031  V f  d  0:33 þ 0:144  V f

1000

b) SFRPC2 Fig. 8. DIF of RPC with different diameters after removing end friction confinement [9–21].

5.2.1. Effect of size on dynamic compressive deformation The incident pulse waves and reflected waves of U36  17.5 mm SFRPC specimens are quite smaller than that of U75  37.5 mm SFRPC specimens under approximately identical strain rate (Figs. 10a and 11a). The range of input and transmission pulses increase with strain rate for RPC specimens with different dimensions (see Fig. 13). SPFRPC specimens with dimensions of U36  17.5 mm break into large pieces at 173 s1, and break into small pieces over 223 s1 (Fig. 10b). Therefore, the fracture strain-rate of SPFRPC with dimensions of U36  17.5 mm is approximately 173 s1. SPFRPC specimens with dimensions of U75  37.5 mm crack at 118 s1, and break into small pieces at strain rates greater than 193 s1 (Fig. 11b). Therefore, the fracture strain-rate of SPFRPC with dimensions of U75  37.5 mm is approximately 150 s1. The critical strain em in two dimensions for SPFRPC specimens at different strain rates is shown in Fig. 12. Critical strain is defined

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U75  37.5 mm SPFRPC specimens. It seems that the critical strain for SFPRC specimens with different dimensions has the same increase rate. In order to further investigate and validate this phenomenon, more test data are collected from existing publications [9–21]. The relationship between the increase rate of critical strain and specimen dimensions will be discussed later. 5.2.2. Effect of size on DIF for compressive deformation Test results are collected from existing publications [9–21], and parts of these results referring to PRPC and SFRPC2 are selected. The relationship between the DIF of critical strain for PRPC and SFRPC2 and strain rate is shown in Fig. 14. RPC specimen size has no visible effect on critical strain DIF. An empirical formula for the critical strain DIF is proposed as follows:

DIF em ¼ ½0:0125  expðV f =0:1822Þ  0:0055  e_ þ 1

ð10Þ

The DIF of critical strain is associated with steel fiber addition and strain rate, regardless of specimen dimension (Fig. 12). On the one hand, the critical strain DIF increases with strain rate, which is consistent with Chen et al. [51] and Wastein et al. [52]. When subjected to an identical strain rate, the critical strain DIF decreases with increasing steel fiber addition (Fig. 14) [21].

a) PRPC

5.3. Dynamic elastic modulus The dynamic elastic modulus Ed varies under various strain rates [29]. In order to comprehensively understand the effect of specimen size on dynamic elastic modulus, the relationship between dynamic elastic modulus and strain rate for RPC specimens with different sizes needs to be examined. However, due to the variety of RPC mixture proportions and data processing methods, the elastic modulus of samples under various SHPB test conditions [9–21] is not consistent.

b) SFRPC2 Fig. 9. Test results and empirical formula for DIF [9–21].

as the strain when the stress reaches peak stress. There are differing interpretations regarding how critical strain varies with strainrate [51,52]. Under identical strain rates, the critical strain value for U36  17.5 mm SPFRPC specimens is larger than for

5.3.1. Effect of size on dynamic elastic modulus Since the randomness of elastic modulus in different experiments of existing publications, the CEB recommendation [29] suggests a functional relationship between the DIF of elastic modulus (Ed/Es) and strain rate. In that case, the relationship between DIF of elastic modulus and strain rate for SPFRPC with different sizes is examined in this research and illustrated in Fig. 15. Fig. 15 illustrates the Ed/Es for SPFRPC with different specimen dimensions under strain rates ranging from 107 to 356 s1 in this

Strains of bars/×10-3

2 ε =173s −1

1

0

ε =148s−1 ε =233s−1 ε =281s −1 ε =304s−1

-1

-2 0.6

0.8

1.0 1.2 Time/ms

a) strain pulses

ε =223s−1

1.4 ε =278s−1

b) failure patterns

Fig. 10. SHPB test results of U36  17.5 mm SPFRPC specimens.

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3 ε =118s −1

Strains of bars/×10-3

2 1 0

ε =120s −1 ε =190s −1 ε =233s −1 ε =277s−1 ε =356s−1

-1 -2 -3 -0.5

0.0

0.5

ε =193s −1

1.0 Time/ms

1.5

2.0

ε =278s −1

b) failure patterns

a) strain pulses

Fig. 11. SHPB test results of U75  37.5 mm SPFRPC specimens.

250

250

Stress/MPa

200 150 100 50 0

200 Stress/MPa

ε =148s−1 ε =209s −1 ε =278s−1 ε =304s−1

ε =190s−1 ε =233s−1 ε =277s −1 ε =356s −1

150 100 50

0.00

0.01

0.02 0.03 Strain/ a) 36×17.5 mm

0.04

0

0.00

0.01

0.02 0.03 Strain/ b) 75×37.5 mm

0.04

Fig. 12. Stress–strain curves of SPFRPC specimens with different sizes.

Critical strain/×10-3

8 6 4 36-17.5 mm 75-37.5 mm

2 0

100

150

200 250 300 Strain rate/s-1

350

400

Fig. 13. Critical strain of SPFRPC specimens with different sizes.

study. Specimen dimension has no visible effect on Ed/Es when subjected to equal strain rates (Fig. 15). To further validate this finding, more test data are collected from existing publications [9–21]. The relationship between the DIF for elastic modulus, RPC specimen dimensions, and strain rate will be discussed later.

Fig. 14. Critical strain DIF for SFRPC specimens with different sizes [9–21].

5.3.2. Effect of size on DIF of elastic modulus Test results are collected from existing publications [9–21]. Fig. 16 shows the DIF for elastic modulus for some existing SHPB tests of SFRPC under strain rates no larger than 280 s1. Steel fiber content ranges from 0 to 5%. Ed/Es has a good relationship with

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3.0 36-17.5 mm 75-37.5 mm

Ed/Es

2.5 2.0 1.5 1.0 0

50 100 150 200 250 300 350 400 Strain rate/s-1 Fig. 15. Size effect on DIF of elastic modulus.

steel fiber content Vf and strain rate. RPC specimen dimensions have no visible effect on Ed/Es (Fig. 16). Based on the relationship of Ed/Es, steel fiber, and strain rate in Fig. 16, a theoretical formula is proposed using linear fitting approach (Eq. (11)). The comparison of existing test results and Eq. (11) is illustrated in Fig. 16. Eq. (11) is in good agreement with existing test results.

Ed =Es ¼ ð0:0061  0:05  V f Þ  e_ þ 1

ð11Þ

5.4. Damage level The crushing level of U75  37.5 mm SPFRPC specimens is higher than that of U36  17.5 mm SPFRPC specimens under roughly identical strain rate (Figs. 10b and 11b). A damage variable D was introduced to describe the damage level of RPC specimens after impact [53–55]:

( D¼

0 1  Er0 e

e¼0 for e > 0

for

ð12Þ

where e and r are the strain and stress of RPC samples, respectively. E0 is the original elastic modulus. The RPC specimen will lose all bearing capability when D equals 1 [53]. By analogy to critical strain at peak stress, the damage at this point is called the critical damage

Dc [56]. The relationship between critical damage Dc and strain rate for SPFRPC samples with different dimensions is shown in Fig. 17. The critical damage of SPFRPC specimens with larger dimension is higher than SPFRPC specimens with smaller dimension (Fig. 17). RPC specimen size has a significant effect on critical damage. To further validate this phenomenon, more test results in existing publications are organized. In order to comprehensively examine the relationship among the critical damage, RPC specimen dimensions, and strain rate, test results are collected from existing publications [9–21]. Portions of these results referring to PRPC and SFRPC2 are selected. Fig. 18 illustrates the relationship among critical damage, strain rate, and specimen dimension for some existing SHPB tests [9–21] of PRPC and SFRPC2, respectively. There is a clear relationship among critical damage, strain rate, and specimen dimension (Fig. 18). For RPC specimens of the same dimensions, the critical damage variables increase with strain-rate. The relation of critical damage is proportional to specimen dimension when subjected to identical strain rates (Fig. 18). Based on the relationship between critical damage, strain rate, and specimen dimension in Fig. 18, a theoretical formula is proposed using a linear fitting approach (Eq. (13)). The comparison of existing test results and Eq. (13) reveals that Eq. (13) is in good agreement with test results. 0:6 Dc ¼ ½ð0:525  1:25  V f Þ  D þ ð8:42  271  V f Þ  103  ðe_ Þ

5.5. Energy absorption Energy absorption capability W is one of most important properties for RPC dynamic behavior. The definition of the energy absorption capacity during compressive deformation is the amount of energy needed for the deformation of RPC and can be calculated as follows [57]:

Z

e

W ¼ 0

rðeÞde

ð14Þ

The energy absorption of RPC can be calculated by calculating the area under stress-strain curves from 0 up to the final ultimate compressive strain ef [20]. Fig. 19 illustrates the energy absorption for SPFRPC with different specimen dimensions under strain rates ranging from 107 to 356 s1. SPFRPC specimen dimensions have no visible effect on energy absorption (Fig. 19). Fig. 20 shows the energy absorption for SFRPC

1.0 36-17.5 mm 75-37.5 mm

0.8

Dc

0.6 0.4 0.2 0.0

Fig. 16. Comparison of existing test data and theoretical formula [9–21].

ð13Þ

0

50

100

150 200 Strain rate/s-1

250

300

Fig. 17. Size effect on critical damage Dc of SPFRPC specimens.

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a) PRPC Fig. 20. The relationship among energy absorption, steel fiber, and strain rate [9– 21].

A theoretical formula for energy absorption capability is proposed (Eq. (15)). Comparison of existing test results and Eq. (15) shows that Eq. (15) is in agreement with existing test results, indicating that the energy absorption of RPC structures can be represented by that of RPC specimens (Fig. 20).

W ¼ ð0:11 þ 3:5  V f Þ  ðe_ Þ

0:6

ð15Þ

6. Conclusions

b) SFRPC2 Fig. 18. The relationship among critical damage, strain rate, and specimen dimension [9–21].

8

W/MJ/m3

6 4 36-17.5 mm 75-37.5 mm

2 0

0

50 100 150 200 250 300 350 400 Strain rate/s-1

Fig. 19. Energy absorption of SPFRPC with different specimen dimensions.

In this study, axial impact tests are conducted using a SHPB device on two-dimensional hybrid fiber-reinforced RPC with 2% steel fiber and 0.2% PP fiber under strain rates from 107 to 356 s1 to investigate the effect of specimen size on RPC dynamic compressive performance. Through analyzing test results in this study and existing publications [9–21], the following conclusions can be drawn: (1) With identical length to diameter ratio, the DIFs of RPC specimens with larger diameter are consistently greater than RPC specimens with smaller diameter. The influence of specimen size on plain RPC is greater than on fiber-reinforced RPC. (2) The size of RPC specimen has no visible effect on critical strain DIF, the DIF of elastic modulus, and the energy absorption. (3) The critical damage of RPC specimens of greater size is larger than SPFRPC specimens with smaller size. (4) Based on nonlinear regression analyses, the equations to calculate the DIF of compressive strength, the DIF of critical strain, the DIF of elastic modulus, the damage variable, and the energy absorption are proposed under the consideration of the strain rate, steel fiber dosage, and the specimen size. Conflicts of interest statement The authors whose names are listed immediately below certify that they have no conflict of interest. Acknowledgements

with different dimensions under strain rates below 280 s1. RPC specimen dimensions have no visible effect on the energy absorption (Fig. 20).

The authors would like to acknowledge the National Natural Science Foundation of China (No. 51578184, No. 51408167), the

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