Research Article Mechanical Behavior of Steel Fiber

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013, Article ID 945309, 5 pageshttp://dx.doi.org/10.1155/2013/945309

Research ArticleMechanical Behavior of Steel Fiber Reinforced ConcreteInvestigated by One-Stage Light Gas Gun Experiment

Guoping Jiang1,2 and Xinhong Wu2

1 School of Engineering, Fujian Jiangxia University, Fuzhou, Fujian 350108, China2 China Hydropower Eight Innings Base Station, Changsha, Hunan 410082, China

Correspondence should be addressed to Guoping Jiang; [email protected]

Received 23 May 2013; Revised 29 July 2013; Accepted 1 August 2013

Academic Editor: Peter Majewski

Copyright © 2013 G. Jiang and X. Wu.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents shock Hugoniot compression data for several concrete materials obtained from flat plate impact experiments.Themanganin pressure gaugewas used tomeasure the pressure-time curves of the samples.Thephysical quantities were all obtainedby the Lagrange method. Moreover, it is observed from the measured pressure-time curves that the rate sensitivity of the dynamicresponse to the concrete is not negligible. Based on the polynomial Gruneisen equation, the parameters of the equation of state ofconcrete were obtained. The steel fiber reinforcement effect was analyzed.

1. Introduction

Steel fiber reinforced concrete (SFRC) is one of the mainstructural materials widely used in industry and civil build-ings. Plain concrete is a brittle material which has a low straincapacity and tensile strength. The discontinuous steel fibersdistributed randomly have the role of bridging across thecracks which might develop, and they provide some postcracking “ductility.” In particular, SFRC has been widely usedin the construction of protective structures such as protectiveshelters, offshore structures, containments, and nuclear reac-tors. It may be subjected to loadings such as earthquake, blastloading. For reliable design andmodelling of SFRC structuressubjected to impact loadings, it is imperative and critical tounderstand the dynamic properties of SFRC.

The two most used setups to study the dynamic behaviorof concrete are the Split-Hopkinson pressure bar (SHPB) andthe one-stage light gas gun. The dynamic behavior of con-crete-like materials at strain rates between 101 and 102 s−1 isoften studied in the SHPB experiment, while at high strainrates between 103 and 105 s−1, the behavior is often studied inthe one-stage light gas gun experiment [1, 2]. The high pres-sure equation of state is often investigated by the one-stagelight gas gun experiment because the nonlinear pressure-density curve usually cannot be obtainedwith the range of thestrain rate in SHPB experiments.With strain rates of 2×10−4/s

and 8 × 10−6/s, compressive tests of concrete were firstlycarried out by Abrams (1917) [3]. Since then, the study of thedynamic mechanical properties of concrete has been a focusof many researchers. The equation of state (EOS, Hugoniot)properties of concrete are determined with two independentand different shock compression experiments byGebbeken etal. [1]. A number of experimental data from the SHPB experi-ments and the three-dimensional simulation of the specimenunder transient loading are presented in the study of thedynamic performance of high strength concrete by Abdullahet al. [4].The three-dimensional simulation of concrete undertransient loading and the experiment are all studied by Grady[5], Rinehart and Welch [6], Buzaud et al. [7], Shi and Wang[8], and Shao-hua et al. [9]. The dynamic mechanical per-formance of SFRC is much more complicated than that ofconcrete. Currently, there are few studies on the mechanicalperformance of SFRC by plate impact experiments.

In this paper, the impact experiments of SFRC are proc-essed by one-stage light gas gunwith different strain rates.Thepressure gauges are embedded in the target by which the volt-age (mv)-time (𝜇s) signals are recorded. Other parameters ofthematerial such as strain, volume, and the speed of the shockwave, are obtained using propagation theory of the shockwave. The high pressure Equation of State is investigated andthe effect of steel fiber reinforcement is analyzed.

2 Advances in Materials Science and Engineering

Table 1: The mix proportion of concrete.

Steel fiber (%) Cement (%) Flyash (%) Silica fume (%) Cement-sandratio

Water-cementratio

Water-binderratio

Admixture (%)

V0 0 55 35 10 1 : 1.2 0.30 0.165 5.0V3 3 55 35 10 1 : 1.2 0.30 0.165 5.5V5 5 55 35 10 1 : 1.2 0.30 0.165 5.5

7

6

5

43

2

1

(1) Projectile assembly(2) Light chamber(3) Flyer(4) Target

(5) Base board(6) Managanin gauges(7) Recycle bin

Figure 1: Setup of the experiment.

2. Experimental Investigations

The experiments are carried out with the one-stage light gasgun which is the main experimental setup for testing con-crete. A sketch of the experiment is presented in Figure 1.Thediameter of the gun is 100mm and the length is 15m. Theimpact velocities are in the range of 100m/s to 1000m/s. Thelevel of impact is no more than 10−3 rad. The experimentalstrain rates range from 103/s to 105/s. The flyer and the targetaremade out of the samematerial.The compositions of SFRCspecimens are all listed in Table 1. The 425 type of Portlandcement made in China is chosen. There are no aggregatesused in the specimens. There are in total three gauges placedin the target to receive the impact signal. The flyer is atthe front of the projectile assembly. The dimensions of thespecimens are all 𝜙 92 × 8mm with the aim to reduce theinfluence of rarefactionwaves.Themanganin pressure gaugesrecord the voltage-time signals fromwhich the pressure-timecurves can be obtained through (1).The pressure-time curvesare shown in Figure 2 with the same impact velocity butdifferent fiber content V

𝑓(volume content of the steel fibers).

They are Consider the typical curves of concrete subjected toimpact loading.

𝜎 = (0.3252 ± 0.0679)

+ (40.2733 ± 0.4164) (Δ𝑅

𝑅) (1.5 ∼ 12.67GPa) ,

𝜎 = (0.0014 ± 0.0055)

+ (51.4697 ± 0.2773) (Δ𝑅

𝑅) (0 ∼ 1.5GPa) .

(1)

As shown in Figure 2, with the same impact velocity,the load-carrying capacities of steel fiber reinforced concreteincrease greatly with reinforcement ratio, as indicated by theenhancement in the peak pressure.The Lagrange positions oftest concrete specimens are 0, 8mm, and 16mm, respectively.

3. Investigations Using the Lagrange Method

The Lagrange method [10–13] is often used to investigate thedynamic performance of concrete with the gas gun exper-iment. The conservation equations in two dimensions arelisted as follows:

𝑢 = 𝑢1−1

𝜌0

∫

𝑡2

𝑡1

(𝑙

𝑙0

)

2

(𝜕𝑃

𝜕ℎ)

𝑡

𝑑𝑡,

V = V1+ ∫

𝑡2

𝑡1

(𝑙

𝑙0

)

2

(𝜕𝑢

𝜕ℎ)

𝑡

𝑑𝑡,

𝐸 = 𝐸1− ∫

𝑡2

𝑡1

𝑃 (𝑡) (𝜕V

𝜕𝑡)

ℎ

𝑑𝑡,

(2)

where 𝜌0is the density, 𝑢 is the particle velocity, 𝑢

1is the

particle velocity of shock front, V is the relatively specificvolume, V

1is the relative specific volume of shock front, 𝐸 is

the internal energy per unit volume, 𝐸1is the internal energy

per unit volume of shock front, and 𝑡1, 𝑡2are the start time

and end time, respectively. ℎ is the Lagrangian position, 𝑙 isthe radial displacement, 𝑙

0is the length of sensitive part, and

𝑙/𝑙0is the relative radial displacement.For the aim of preventing loss of the useful information

in the integral along the isochrone lines, the path and tracelines are adopted in the Lagrange method investigations.Thecharacteristic points such as the peak point of the plastic waveand the end point of elastic wave in the pressure-time curvesare connected to establish the path lines.

Along the path lines and particle lines, the integral alongisochrone lines changes as follows:

(𝜕𝑝

𝜕ℎ)

𝑡

= (𝑑𝑝

𝑑ℎ)

𝑗

− (𝜕𝑝

𝜕𝑡)

ℎ

(𝑑𝑡

𝑑ℎ)

𝑗

,

(𝜕𝑢

𝜕ℎ)

𝑡

= (𝑑𝑢

𝑑ℎ)

𝑗

− (𝜕𝑢

𝜕𝑡)

ℎ

(𝑑𝑡

𝑑ℎ)

𝑗

.

(3)

Advances in Materials Science and Engineering 3

4 5 6 7 8 9 10 11 12 13 14

0

500

1000

1500

2000

2500

P(M

pa)

T (𝜇s)

0

8mm16mm

(a) V𝑓 = 0

6 8 10 12 14 16 18

0

500

1000

1500

2000

2500

P(M

pa)

T (𝜇s)

0

8mm16mm

(b) V𝑓 = 5%

Figure 2: Experimental stress-time curves.

Substituting (3) into (2), we get

𝑢 = 𝑢1−1

𝜌0

∫

𝑡2

𝑡1

(𝑙

𝑙0

)

2

[(𝜕𝑃

𝜕ℎ)

𝑗

− (𝜕𝑃

𝜕𝑡)

ℎ

(𝜕𝑡

𝜕ℎ)

𝑗

]𝑑𝑡,

V = V1+ ∫

𝑡2

𝑡1

(𝑙

𝑙0

)

2

[(𝜕𝑢

𝜕ℎ)

𝑗

− (𝜕𝑢

𝜕𝑡)

ℎ

(𝜕𝑡

𝜕ℎ)

𝑗

]𝑑𝑡,

𝐸 = 𝐸1− ∫

𝑡2

𝑡1

𝑃 (𝑡) (𝜕V

𝜕𝑡)

ℎ

𝑑𝑡.

(4)

The concrete (V𝑓= 0) is an example of the Lagrange

method investigations here. The 𝑢(𝑡), V(𝑡), and 𝑒(𝑡) curvescan all be obtained from (2), with the integral along the pathand trace lines of 𝑝(𝑡) curves. The strain rate can be obtained(Figure 3) from the Lagrange method.

The engineering strains are obtained (5).With (5) and the𝑝(𝑡) curves, the relationships between strain and pressure areall obtained (Figure 4). Consider

𝜀 =(V0− V)

V0

. (5)

4. The Grüneisen EOS Investigations

The high pressure equation of state for isotropic materialsusually defines the pressure as a function of specific volume V(or density, 𝜌) and specific internal energy 𝑒. Experimentalshock Hugoniot relationships are widely used as referencestate data extrapolating to other high-temperature thermo-dynamic and high-pressure states. The Gruneisen EOS is

3 4 5 6 7 8 9 10 11 12 13 14−0.011

−0.010

−0.009

−0.008

−0.007

−0.006

−0.005

−0.004

−0.003

−0.002

−0.001

0.000

0.001

Stra

in ra

te (1

/(𝜇

s))

T (𝜇s)

0

8mm16mm

Figure 3: Strain rate time curves.

widely used in the constitutive model of concrete such asin Johnson-Holmquist-Concrete, Mie-Gruneisen, and John-son-Holmquist-Ceramics.

The relationship of 𝑃-𝜇 in Gruneisen EOS is defined asfollows:

𝑃 = 𝐴1𝜇 + 𝐴

2𝜇2+ 𝐴3𝜇3, (6)

where 𝐴1, 𝐴2, and 𝐴

3are the material coefficients, 𝑢 = 𝜌/𝜌

0.

Consider

𝑃 = (1 − 𝜑) 𝑃𝑠[(1 − 𝜑)𝑉, 𝑒] , (7)

4 Advances in Materials Science and Engineering

1600

1400

1200

1000

800

600

400

200

0

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Strain

P(M

pa)

0

8mm16mm

Figure 4: Stress-strain curves.

0 100 200 300 400 500 600 700 8002400

2800

3200

3600

4000

4400

D (m

/s)

U (m/s)

The C100 concreteThe C30 concrete

Figure 5:𝐷-𝑈 curves of concrete.

where 𝜑 = 𝜑0𝜀𝑎(

∘

𝜀∗)𝑏, 𝑎, 𝑏 are the material contents. 𝜀∗ (=

∘

𝜀/∘

𝜀0), ∘𝜀0= 1/𝑠. With Figure 4 and (6) and (7), we can obtain

𝐴1= 46.4GPa; 𝐴

2= −195GPa; 𝐴

3= 416.6GPa.

(8)

The same method can be used to study the different steelfiber content in concrete. Consider

𝐴1= 43.4GPa; 𝐴

2= −196GPa;

𝐴3= 436.6GPa, V

𝑓= 3%;

𝐴1= 29.6GPa; 𝐴

2= −252GPa;

𝐴3= 616.6GPa, V

𝑓= 5%.

(9)

0

200 400 600 800 10000

2

4

6

8

10

12

P(G

pa)

U (m/s)

The C100 concreteThe C30 concrete

Figure 6: 𝑃-𝑈 curves of concrete.

There is also another method which can determine theEOS. Consider

𝑈 =1

2𝑢, (10)

where 𝑈 is the mass velocity and 𝑢 is the projectile velocity.The relationship of the shock wave velocity𝐷 and particle

𝑈 is often linear (Figure 5).

𝐷 = 𝑎 + 𝑏𝑈, (11)

where 𝑎 and 𝑏 are the material parameters. Fitting a linethrough the discrete points, we can obtain the 𝑎 and 𝑏 values.For C100, 𝑎 = 2713, 𝑏 = 2.368. For C30, 𝑎 = 2420, 𝑏 = 1.31.The shock wave pressure 𝑃 and the particle velocity 𝑈 havethe following relationship:

𝑃 = 𝜌𝐷𝑈 = 𝜌 (𝑎 + 𝑏𝑈)𝑈. (12)

So the 𝑃-𝑈 Hugoniot curve can be obtained (Figure 6).It is noted that 𝑃-𝑈 Hugoniot curve is not stress-strainrelationship.

The mass conservation equation is

𝜌1(𝐷1− 𝑢) = 𝜌

2(𝐷2− 𝑢) . (13)

Substituting (11) and (12) into (13), we get

𝑃 =𝑎2(V0− V)

(𝑏 − 1)2V2[𝑏/ (𝑏 − 1) − V

0/V]2. (14)

The 𝑃-𝜇 curves can be calculated (Figure 7, C30 is theexample). Considering (6), we get𝐴

1= 19.09,𝐴

2= −156.26,

and 𝐴3= 999.655 for C30 concrete.

Advances in Materials Science and Engineering 5

0.00 0.05 0.10 0.15 0.20 0.25

0

1

2

3

4

5

6

Experiment dataEOS data

P(G

pa)

𝜇

Figure 7: 𝑃-𝜇 curve for concrete.

5. Conclusions

The dynamic characteristics of steel fiber reinforced concreteare very complicated. In this paper, the impact experiments ofsteel fiber reinforced concrete with different steel fiber ratiosare processed by one-stage light gas gun.The pressure gaugesare embedded in the target and the voltage-time signals arerecorded. The stress-strain relationship curves are obtainedby the Lagrangianmethod.The parameters of the equation ofstate are all obtained. The steel fiber reinforced effect can beseen from the Gruneisen EOS parameters. It is shown thatwhen the steel fiber content is less than 3%, the effect of theEOS is unconspicuous. When the steel fiber content is morethan 3%, the effect of the EOS is more conspicuous.

Acknowledgments

Thisworkwas supported by the project of ChinaHydropowerEight innings (20133), Housing and Urban-Rural Construc-tion Bureau (2012-k4-17), and theGuangzhouBaiyunDistrictScience and Technology Project (Grant no. 2012-KZ-58).

References

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[2] S. Pattofatto, A. Poitou, H. Tsitsiris, and H. Zhao, “A new testingmethod to investigate the compacting behaviour of fresh con-cretes under impact loading,” Experimental Mechanics, vol. 46,no. 3, pp. 377–386, 2006.

[3] D. A. Abrams, “Effect of rate of application of load on the com-pressive strength of concrete,” Journal of ASTM, vol. 17, part 2,pp. 364–377, 1917.

[4] H. Abdullah, S. Husin, H. Umar, and S. Rizal, “The dynamicproperties behavior of high strength concrete under differentstrain rate,” in 3rd International Conference on ExperimentalMechanics and 3rd Conference of the Asian Committee on Exper-imental Mechanics, Proceedings of SPIE, pp. 56–59, December2004.

[5] D. Grady, “Shock equation of state properties of concrete,’ inProceedings of the 1996 4th International Conference on Struc-tures Under Shock and Impact IV (SUSI ’96), pp. 405–414, Com-putational Mechanics, July 1996.

[6] E. J. Rinehart andC. R.Welch, “Material properties testing usinghigh explosives,” International Journal of Impact Engineering,vol. 17, no. 7, pp. 673–684, 1995.

[7] E. Buzaud, D. Don, S. Chapelle et al., “Perforation studies intoMB50 concrete slabs,” in Proceedings of the 9th internationalSymposium on Interaction of the Effects of Munitions with Struc-tures, pp. 407–414, Berlin, Germany, 1999.

[8] S. Shi and L.Wang, “Discussion on dynamicmechanical behav-iors of cement mortar under quasi-one and one dimensionalstrain states,” Chinese Journal of Rock Mechanics and Engineer-ing, vol. 20, no. 3, pp. 327–331, 2001.

[9] Y. Shao-hua, Q. Qi-hu, Z. Zao-sheng et al., “Experimental studyof equation of state for high-strength concrete and high-strength fiber concrete,” Journal of PLAUniversity of Science andTechnology, vol. 1, no. 6, pp. 49–53, 2000.

[10] Z. Zhang, S. Huan, and F. Lu, “Lagrangian analysis of flow fieldin detonation reaction zone of high explosives,” Expolosion andShock Waves, vol. 16, no. 3, pp. 271–277, 1996.

[11] N. Jiangguo, “Investigation on impact behavior of concrete,”Chinese Journl ofTheoretical and Applied Mechanics, vol. 38, no.2, pp. 199–204, 2006.

[12] D. Chuen-tong, Z. Bao-ping, and J. Chun-lan, “A study on thedynamic constitutive relationship of metallic copper markinguse of lagrange analysis technique,” Sichuan Ordnance Journal,no. 1, pp. 54–59, 1995.

[13] H. Shi, “Consistency among reaction rate equation and stateequations of solid explosive,” Journal of Hunan University, vol.30, no. 3, pp. 15–18, 2003.

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Research Article Mechanical Behavior of Steel Fiber