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Rate-dependent constitutive models of S690 high-strength structural steel
Construction and Building Materials 198 (2019) 597–607
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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Rate-dependent constitutive models of S690 high-strength structural steel Xiaoqiang Yang a,b, Hua Yang b,c,⇑, Sumei Zhang d a
School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China c Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin 150090, China d School of Civil and Environment Engineering, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China b
h i g h l i g h t s Tests of S690 high-strength structural steel at high strain rates were conducted. Sensitivity of the flow stress to strain rate for S690 was investigated. DIF of yield stress was studied and a new Cowper-Symonds model was developed. Johnson-Cook model was fitted and modified, performing higher prediction accuracy.
a r t i c l e
i n f o
Article history: Received 12 July 2018 Received in revised form 20 November 2018 Accepted 30 November 2018 Available online 6 December 2018 Keywords: S690 SHPB High strain rate Dynamic increase factor Dynamic constitutive models
a b s t r a c t The demand of applying high-strength structural steel (HSSS) in engineering practice is progressively increasing, while insufficient research achievements of HSSS restrict its applications, especially for those under dynamic loadings. Therefore, to meet the increasingly demand of studies for HSSS at high strain rates, a grade of HSSS, S690 was experimentally investigated by quasi-static tensile tests and dynamic tests using a Split Hopkinson Pressure Bar (SHPB). The dynamic yield strength and stress-strain curves at various strain rates were obtained, which showed that S690 has obvious effects to the sensitivity of strain rate, while the existing models are not suitable for predicting its strain-rate effect. Further, the dynamic increase factor (DIF) of yield stress was determined and a new Cowper-Symonds model was fitted to predict the DIF precisely. To establish a rate-dependent constitutive model for S690, the Johnson-Cook (J-C) model was also adopted to fit true stress-strain curves and has been proved to perform acceptable prediction accuracy. By substituting the strain-rate parameter C, the standard J-C model was modified to improve its precision. These test results may provide fundamental data for its future studies, and the constitutive models may be applied to analyze the dynamical performance of S690 components. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction In recent decades, the interest on using high-strength structural steel (HSSS) in large-span, high-rise and heavily loaded modern engineering structures is increasingly on the rise. HSSS brings benefits in term of reducing material consumption and environmental protection, which is in line with the developing trend of green ⇑ Corresponding author at: Room 207, School of Civil Engineering, Harbin Institute of Technology, Huanghe Road #73, Nangang District, Harbin 150090, Heilongjiang Province, China. E-mail addresses: [email protected] (X. Yang), [email protected] (H. Yang), [email protected] (S. Zhang). https://doi.org/10.1016/j.conbuildmat.2018.11.285 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.
economy. However, the application of HSSS may cause a series of problems owing to the inadequate and incomplete theoretical and practical experiences, especially for those grand constructions subjected to extreme and dynamic loadings. The recent introduction of HSSS labeled with S690 included in EN 1993-1-12 [1] offers an ideal option for both steel structures and steel-concrete composite structures, of which yield strength is more than 690 MPa, while tensile to yield strength ratio and elongation should be not less than 1.05 and 10%, respectively. As a new-style high-strength steel material, HSSS with yield strength 690 MPa has been attracting extensive attention by researchers. From 2011, a series of studies on welding properties [2,3], fatigue
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behavior [4], mechanical and fracture properties of flame heat treatment [5], low temperature resistance [6] and post-fire mechanical properties [7–9] for 690 MPa structural steels have been conducted and proved that 690 MPa structural steels have acceptable mechanical properties subjected to those conditions. Based on the material researches, scholars have started studies on the performances of structural components. Li et al. [10,11] conducted experimental and numerical studies on compressive behaviors of Q690 welded box- and H-sections. Recently, the investigation on behaviors of S690 hot-finished high strength prestressed trusses under axial loadings was carried out by Wang et al. [12]. A recent study on the constitutive modelling of stress-strain curves for hot-rolled steels under quasi-static strain-rate loading was conducted by Yun and Gardner [13], covering S690 steel. Though previous studies into S690 have covered both material and member levels, knowledge of this material is still not far enough compared with those of common structural steels with normal yield strengths. The strain-rate effect of such steel has not yet been investigated resulting no specific rate-dependent constitutive model was proposed. The absence of rate-dependent model is inconsistent with the structural demands of resisting accidental actions, such as gas explosion, vehicle impact, terrorist attack, etc. Indeed, the structural design and assessment for accident safety over the serving duration could be vital especially for those structures employing high strength steel. It is a priority to clarify the dynamic properties and strain rate effect of such structural steel so as to analyze its further structural dynamical performances. In the last century, a series of investigations on the strain-rate effect of sheet steels and reinforcing steel bars with a wide range of strength from 180 MPa to 853 MPa at various high strain rates were conducted by researchers [14–16]. Based on test results, several equations had been proposed to characterize the dynamic increase factor (DIF) of yield stress and ultimate stress for these different types of steels, respectively. After 9/11 attacks, in 2005, to investigate the aircraft impact on the building, the dynamic properties of steels used in World Trade Center at high strain rates were researched by Luecke et al. [17], and a full set model was suggested for more precise prediction of DIF for different steels at high strain rate. Recently, by using MTS and Split Hopkinson Pressure Bar (SHPB), the dynamic properties of Q345 steel subjected to high temperature at various strain rates were studied by Yu et al. [18], and the results showed Q345 displays an obvious strain rate strengthening effect. Subsequently, in 2016, based on progressive collapse analysis, the dynamic tensile behavior of S355 steel under intermediate and high strain rates (5 s1–850 s1) by Forni et al. [19]. In 2017, the study on dynamic tensile tests of Q345 and Q420 steels at medium strain rates were performed by Chen et al. [20,21], and empirical constitutive models were developed to improve the accuracy in describing the dynamic properties for Q345 and Q420 respectively. Conclusion can be considered from above: the static properties of HSSS have been investigated widely, while there are few studies on the dynamic behaviors. Note that although previous researchers had presented several investigations on dynamical properties of high-strength sheet steels and reinforcing steel bars, and established equations to predict the DIF for such materials subjected to high strain rates, Fig. 1 illustrates that these models were still quite different when they were adopted to predict the DIF with yield stress of 690 MPa, varying 1.04–1.25 at strain rate 4000 s1. As a result, it may overrate or underrate the DIF if these existing models proposed before were used to predict the performance of 690 MPa steels directly. Moreover, there are no studies on the rate-dependent constitutive model for depicting the stress-strain curve of 690 MPa HSSS at high strain rate. Therefore, the grade of HSSS, S690, was investigated systematically in this paper. The
Fig. 1. Relationships of DIF versus strain rate for 690 MPa steel based on previous researches.
quasi-static tensile tests and dynamic SHPB tests were carried out to obtain the stress-strain relationships of S690 at quasistatic condition and high strain rates, using a universal electromechanical testing machine and a SHPB tester, respectively, which are both installed in Harbin Institute of Technology. The strain rate, dynamic yield stress and dynamic stress-strain curves were obtained and discussed, based on which three rate-dependent constitutive models were proposed to describe the dynamic behaviors of S690 at various strain rates precisely.
2. Experimental program 2.1. Materials and specimens In this study, a commercial S690QL hot-rolled steel plate was produced by Jiangyin Xingcheng Special Steel Works Corporation, China, of which chemical compositions and carbon equivalent value (CEV) were shown in Table 1, meeting the demands of EN 10025-6:2004 + A1:2009. The geometry of specimens and test method for quasi-static tests are according to ISO 6892-1:2016 and ISO 377:2013. The ratio of diameter/length for SHPB specimens was taken as 2.0, with 8.0 mm in diameter and 4.0 mm in length. The specimens were shown in Fig. 2.
2.2. Experimental setup 2.2.1. Quasi-static tension A universal electromechanical testing machine (Shimadzu AGX Plus 250 kN) was utilized to test all quasi-static tension specimens at room temperature. The engineering stress and strain were calculated by the tensile force and extension recorded by a force sensor and a 50 mm extensometer, respectively. During the elastic period, two pairs of strain gauges were stuck on the middle of both sides of specimen to record the longitudinal and transverse strains (see Fig. 2), which were utilized to calculate the Young’s modulus and Poisson’s ratio. For the quasi-static test, the strain rate was 0.00025 s1 (the crosshead separation rate was 0.9 mm/min and the parallel length is 60 mm). Two different directions – rolling direction 0° and transverse direction 90° were employed to machine the test coupons and three repeated experiments for each direction were carried out to ensure the reliability of test results.
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X. Yang et al. / Construction and Building Materials 198 (2019) 597–607 Table 1 Chemical compositions and CEV of S690QL steel (in weight %). C
Si
Mn
S
P
N
Ni
Mo
Cr
Cu
V
Ti
Nb
CEV
0.14
0.29
1.26
0.0006
0.014
0.0018
0.02
0.11
0.2
0.01
0.003
0.023
0.015
0.42
Fig. 2. Test specimens for: (a) quasi-static tension test and (b) SHPB test.
Fig. 3. Split Hopkinson Pressure Bar testing system: (a) SHPB devices; (b) specific details.
2.2.2. SHPB In this paper, an investigation of S690 high structural steel at high strain-rate was conducted by a SHPB tester, as shown in Fig. 3. An entire SHPB system contains a gas cabin, a strike bar, an incident bar, a transmission bar and an energy absorption setup, which are installed on a horizontal guide rail to guarantee that they are all on the same axis. When the gas chamber is inflated by the nitrogen gas cylinder to the designed pressure, the strike bar is propelled by high-pressure gas to impact the incident bar. The velocity of the strike bar will be recorded by the laser sensor before reaching the incident bar. A compressive elastic wave eI is generated and propagates through the incident bar after the strike bar impacts the incident bar. Once the inci-
dent wave (eI ) reaches the specimen that is sandwiched between the incident bar and transmission bar and lubricated by petroleum jelly, the wave reflects on the interface between incident bar and specimen to generate a reflected wave (eR ). At the same time, the specimen deforms, and the remaining wave passes to transmission bar, followed by a transmission wave (eT ) propagating through that. For this study, the length of strike bar is 200 mm, while incident and transmission are both 1200 mm. All the bars are 16 mm in diameter and they are all made of high strength alloy steel, whose density is 7850 kg/m3 and young’s modulus is 200 GPa, with 5000 m/s of the wave speed. These bars remain elastic over the entire period of test and strain gauges are stuck to the middle of
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both incident and transmission bars to record strain-wave histories (eI , eR and eT ) with a 5 MHz digital storing oscilloscope. According to the two-wave method [22], the engineering stress (reng ), engineering strain (eeng ) and strain rate (e_ ) in the function of time t can be calculated based on the data of strain waves through Eq. (1).
8 > r ðtÞ ¼ EAAs0 eT ðtÞ > < eng R eeng ðtÞ ¼ 2CL 0 0t eR ðtÞdt > > :_ eðtÞ ¼ 2CL 0 eR ðtÞ
ð1Þ
where E and A0 refer to the young’s modulus and cross-sectional area of alloy steel bar, C 0 is the speed of strain wave for alloy steel bar (5000 m/s), As and L are the cross-sectional area and length of specimen respectively. Further, true stress (rtrue ) and true strain (etrue ) can be calculated by the Eq. (2).
(
rtrue ¼ reng 1 þ eeng etrue ¼ ln 1 þ eeng
ð2Þ
In this study, six different gas pressures were determined to test steel specimens, based on which six different velocities of strike bar were recorded: 12.1 m/s, 16.6 m/s, 20.1 m/s, 25.6 m/s, 29.8 m/s and 33.3 m/s. For each impact velocity, three repeated experiments were carried out respectively to ensure repeatability of test results. 3. Results & discussion 3.1. Experimental results 3.1.1. Quasi-static tests Six uniaxial quasi-static tension tests of S690 high-strength steel for both 0° and 90° directions subjected to room temperature were carried out and the engineering stress-strain curves were presented in Fig. 4. It can be seen apparently that these tensile coupons showed good consistency for engineering stress-strain curves, and the tension properties were almost same at two different directions. The proof strength at 0.2% plastic strain was determined in this paper as the yield stress of S690 and the average tensile properties of S690 were summarized in Table 2. It showed that the tensile behaviors of S690 meet the demands of EN 1993-1-12. The engineering stress-strain curves of S690 exhibit a linearelastic part at the beginning of tests, followed by a short yield plateau and the strain hardening phenomenon subsequently when
Fig. 4. Engineering stress-strain curves of S690 under uniaxial quasi-static tension test.
plastic strain starts to grow. Compared with the tensile curves of S690 steel obtained by previous researchers [5,23,24], the strengths and plastic deforming performances are similar, while [24] shows a very long and obvious yield plateau for the stressstrain curve. Actually, the length of yield plateau is not an intrinsic property for one specific grade of steels and is known to depend on chemical composites, heat treatment, grain size, and test conditions, e.g. strain aging and loading rate, which can be concluded as the manufacturing process and strain history of material tests [13,25]. According to the review for high strength structural steels by Ban and Shi [26], in general, there is no visible yield plateau for most of HSSS with yield stress above 500 MPa. Therefore, it is common that different lengths of yield plateau were observed by different studies for those S690 steels. 3.1.2. SHPB tests 3.1.2.1. Raw strain waves. As mentioned above, the raw strain waves recorded by the strain gauges during the test can be analyzed to extract the stress-strain curve of the tested material further. For the SHPB tests, typical strain-time histories of eI , eR and eT at six terms of impact velocities were shown in Fig. 5(a). Based on theory of one-dimensional elastic wave, the assumption that specimen undergoes homogeneous deformation ensuring stress equilibrium is significant for the reliability of test results. Therefore, during the test, the forces on the front and back of interfaces of specimen described by two equations: F f ¼ EA0 ðeI þ eR Þ and F b ¼ EA0 eT should be almost equilibrated. Due to the same material for incident bar and transmission bar, it can be simplified to verify whether eI þ eR (eI and eR are at the opposite directions, as shown in Fig. 5(a)) is equal to eT . In the Fig. 5(b), note that although the time history of eI þ eR has a large vibration at the beginning, eT is basically close to the average of eI þ eR during the entire test at 20.1 m/s. For other impact velocities, all show the same phenomena. In this sense, stress equilibrium is mainly satisfied during the entire SHPB tests for this paper. 3.1.2.2. Strain rate. Most of metallic materials are sensitive to strain rate when subjected to quick loadings [27]. However, to obtain a completely constant strain rate directly during the entire period of loading for a SHPB test is mostly impossible. So, how to define the representative strain rate as precise as possible is important so as to depict the dynamic behavior of materials. For this study, a group of typical strain-rate time histories at 20.1 m/s were shown in Fig. 6(a) and the average strain-rate over the entire loading duration was calculated. Results show that the strain-rate time histories of three samples for three repeated tests are in good agreement and the average strain rate is relatively close to the time history for most of the time. Besides, another method suggested by Sun [28] was also adopted to define the strain rate of S690. Fitting a line through the linear part of the strain-time history curve, the strain rate was determined as the slope of fitted line shown in Fig. 6(b). In this sense, the strain rate at 20.1 m/s for linear fitting method is 1640 s1, which is so close to the mean strain rate for entire loading duration (1614 s1). Therefore, for convenience, in this paper, a mean strain rate over the total period of a SHPB test was defined as the representative strain rate. 3.1.2.3. Stress-strain curves. As described before, six terms of specimens under six different impact velocities by SHPB tester were tested and for each impact velocity, an average representative strain rate was determined. Based on which, all test results of S690 subjected to dynamic SHPB tests at wide strain rate ranged from 266 s1 to 4109 s1 were obtained in accordance with the Eq. (1). The engineering stress-strain curves of three repeated experiments was shown in Fig. 7 for each strain rate, which showed a good consistency. Besides, it can be noted that at lower
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X. Yang et al. / Construction and Building Materials 198 (2019) 597–607 Table 2 Tensile properties of S690 under uniaxial quasi-static tension tests. Grade
Yield stress
S690
722
r0:2 (N/mm2)
Tensile stress
ru (N/mm2)
761
Young’s modulus E(N/mm2)
Poisson’s ratio
Elongation
196,000
0.3
15%
Fig. 5. Typical strain waves of S690: (a) raw three-wave signals obtained from incident bar and transmission bar in SHPB testing, (b) comparison of eI þ
(a) Average strain rate over entire period
eR and eT at 20.1 m/s.
(b) Linear fitting method [28]
Fig. 6. Representative strain rate of S690 steel at impact velocity of 20.1 m/s.
strain rate, the stress-strain curve is shorter than that at higher strain rate. It is because that for lower strain rate, the impact velocity is lower so that the specimen performs less deformation due to its lower impact energy of strike bar. To make a clearer presentation of the difference of S690 at each strain rate, three engineering stress-strain curves were calculated in average and then used to be compared in Fig. 8(a). Further, all the engineering curves were transformed to true stress-strain curves by Eq. (2), which were also compared and shown in Fig. 8 (b). In particular, the computing method by Eq. (2) is no longer correct after necking of steel specimens, and the true stress-strain curve after necking cannot be recorded by the standard tensile test in this paper. As a result, the descent stage of true stress-strain curve was abandoned at 0.00025 s1 in Fig. 8(b). 3.1.2.4. Dynamic yield strength. Dynamic yield strength (rdy ) is an essential parameter characterizing the dynamical property for materials subjected to dynamic loadings. In practice, however, none of common or widely recognized definitions of dynamic yield stress for SHPB tests has been suggested by current standards. Due to the assumption that the specimen is in stress equilibrium during the test is not valid before it yields [17], the elastic part of stress-
strain curves for dynamic samples is not reliable, so that the 0.2% offset method is no longer suitable for determination of dynamic yield stress at high strain rates only based on the dynamic stress-strain curves which were derived from SHPB tests. Therefore, a method suggested by Sun [28] was adopted in this study to define the dynamic yield stress of S690. Actually, the dynamical modulus of elasticity is independent on strain rate, which is almost same as that in static condition in compliance with previous researches [17,29]. According to Sun’s research, truncating the elastic region and fitting a linear line for the strain hardening part of engineering stress-strain curve, the dynamic yield strength is the intersection point which can be determined by the 0.2% offset method using the quasi-static modulus of elasticity. The method for definition of dynamic yield stress for S690 was shown in Fig. 9. All the dynamic yield stresses at each strain rate were listed in Table 3 and the dynamic yield stresses versus strain rate curve was shown in Fig. 10.
3.2. DIF of yield stress Based on the results from Fig. 10, the strain-rate effect of yield stress for S690 is quite obvious. It can be seen that dynamic yield
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(a) 266s-1
(b) 1073s-1
(c) 1614s-1
(d) 2581s-1
(e) 3346s-1
(f) 4109s-1
Fig. 7. Engineering stress-strain curves of S690 for three repeated tests under different strain rates.
strength increases apparently, from 722 MPa to 1106 MPa when strain rate increases to 4109 s1. In this sense, a dynamic increase factor of yield stress (DIFy) expressed by dividing dynamic yield stress by quasi-static yield stress: DIFy = rdy =rsy was introduced to characterize the magnitude of strain-rate effect of yield stress. Each DIFy was calculated and listed in Table 4. To depict the tendency of the DIFy with the increasing strain rate, a Cowper-Symonds (C-S) model [30] was adopted in this paper. C-S model is a widely accepted dynamic constitutive model, which is simple, intuitive and accurate to characterize the DIF with strain rate. The standard C-S model can be expressed by the Eq. (3).
p rdy e_ ¼1þ rsy D
1
ð3Þ
where rsy is the yield stress at quasi-static, while rdy is the dynamic yield stress at the strain rate of e_ . D and p are the material coefficients. As one of the popular dynamic constitutive models, C-S model is widely accepted by many researchers so that they conducted studies on the dynamic behaviors of metallic materials and worked out several C-S models with fitted coefficients in the past years. Among them, D = 40.4 and p = 5 obtained experimentally from mild steels are extensively applied to predict the DIF of yield stress at high strain rate [31]. Therefore, for verifying its feasibility, the C-S model with these coefficients was used to predict the DIFy of S690 to estimate whether it can be applied directly for S690. Besides, D = 4945 and p = 2.696 obtained from test results for S355 steel [19] was also adopted to calculate the DIFy of S690 for comparison. However, it can be seen from Table 4 that the percent-
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Fig. 8. Comparison of engineering (a) and true (b) stress-strain curves for S690 under various strain rates from 0.00025 s1 to 4109 s1.
and steel bars with 690 MPa or similar yield strengths were added up to perform a comparison in Fig. 12. It demonstrates that these models proposed before may not be used to predict the performance of 690 MPa steels directly, because they will underrate the DIFy of S690, while the curve derived from the C-S model fitted in this paper has remarkably good prediction accuracy with test data. 3.3. J-C models
Fig. 9. Definition of dynamic yield stress for S690.
age of deviation between predicting results and test data are more than 110% and 15%, respectively. It means that the existing fitted coefficients for C-S model obtained from normal-strength steels are not suitable for S690, which will overrate the strain-rate effect of yield strength of S690. Based on the results as mentioned before, in this study, a new CS model was established by fitting the DIFy of S690 at different strain rates using the least square method through Eq. (4) in logarithmic coordinates.
rdy lnðe_ Þ ¼ pln 1 þ lnðDÞ
ð4Þ
rsy
As shown in Fig. 11, all the DIFy and strain rates have been transformed in logarithmic coordinates, and the elastic-linear fitting results showed that linearity of data is good, which means that the C-S model can predict the DIFy of S690 well. According to the fitting data, D = 18404 and p = 2.38 were taken out and the C-S model with such fitted coefficients was also compared with test data and other C-S models in Table 4, which shows much better agreement with test results than others. Moreover, other models obtained from previous studies [14–17] for high-strength steels
In the previous studies, scholars and researchers carried out the analyses of materials under dynamic loadings at high strain rates and worked out some equations to express their ratedependence of yield stress and ultimate stress. These equations and models gave the policy for them to consider the DIF of yield stress or ultimate stress when materials or structural components were subjected to quick loadings so that the prediction precision would be improved. So far, however, none of constitutive models has been developed for S690 to represent the complete stressstrain relationship under dynamic loadings. In this study, a Johnson-Cook (J-C) model [32] that is popularly and widely accepted for characterizing the true stress-strain curve of metallic materials is also adopted to describe the dynamic constitutive relationship of S690. 3.3.1. Standard J-C model A standard J-C model consists of three parts considering the strain hardening, strain-rate hardening and thermal softening, which is intuitive and without coupling effect. The stress can be expressed as:
r ¼ ðA þ Ben Þ 1 þ Clne_ ð1 T m Þ
ð5Þ
where r is the plastic stress, e is the plastic strain, e_ is the dimen sionless plastic strain rate(e_ ¼ e_ =e_ 0 , e_ 0 ¼ 0:00025 s1 ), T is the homologous temperature expressed as T ¼ ðT T r Þ=ðT m T r Þ, T is the test temperature, T r is the room temperature and T m is the melting temperature. A, B, C, m and n are the material constants. In this test, all the experiments are conducted at ambient temperature so that the J-C model can be simplified as Eq. (6):
r ¼ ðA þ Ben Þ 1 þ Clne_
ð6Þ
Table 3 Dynamic yield stresses of S690 at different strain rates. Strain rate e_
266 s1
1073 s1
1614 s1
2581 s1
3346 s1
4109 s1
rdy (N/mm )
841
954
985
1039
1060
1106
2
604
X. Yang et al. / Construction and Building Materials 198 (2019) 597–607
Fig. 10. Dynamic yield stress versus strain rate curve for S690. Fig. 11. Fitting data for C-S model.
In this sense, the dynamical stress-strain curves were described by multiplying the quasi-static curve (A þ Ben ) by a magnification factor (1 þ Clne_ ). To fit the J-C model, the true stress-strain curves of S690 under various strain rates were transformed to the true plastic curves by removing the elastic strain, as shown in Fig. 13. The true stress versus true plastic strain curves reported in Fig. 13 demonstrate that the strain hardening phenomena for S690 steel at different strain rates is almost in some tendency, based on which, the J-C model using a magnification factor (DIF) to multiply the quasi-static curve without coupling effect is suitable for characterizing this dynamic behavior. As discussed before, without coupling effect, coefficients A, B and n can be fitted only using the quasi-static test results. For the quasi-static condition, e_ ¼ 1, so 1 þ Clne_ ¼ 1. The J-C model n was simplified as: r ¼ A þ Be . A is the yield stress, while B and n represent the effect of strain strengthening obtained by fitting the quasi-static stress-strain curve. In this paper, A was taken as 722, B was 400 and n was 0.57. It can be seen clearly in Fig. 14 (a) that there is a good fit between quasi-static tensile stressstrain curve and the predicting curve derived from the fitted J-C model. After that, the parameter C can be gained through the fitting method in accordance with the dynamical SHPB test data by Eq. (7):
r e_ 1 ¼ Cln rs e_ 0
Fig. 12. Comparison among test data and various models.
ð7Þ
where rs ¼ A þ Ben , representing the stress at the strain rate of e_ 0 (e_ 0 ¼ 0:00025 s1 ), while r is the dynamic stress at high strain rate. For the standard J-C model, 1 þ Clne_ as an increase factor is significant to determine the strain-rate effect so that it is the essential parameter for the parameter C corresponding to the specific reference strain rate. Over the past years, couple of values of C describ-
Table 4 Comparison of DIFy for S690 under each strain rate with various C-S models.
e_ /s1
266 1073 1614 2581 3346 4109
Test
D = 40.4, p = 5 [31]
D = 4945, p = 2.696 [19]
D = 18404, p = 2.38
DIFtest
DIFpre
D/%
DIFpre
D/%
DIFpre
D/%
1.165 1.321 1.364 1.439 1.468 1.532
2.461 2.931 3.095 3.301 3.424 3.525
111.2 121.9 126.9 129.4 133.2 130.1
1.338 1.567 1.660 1.786 1.865 1.934
14.9 18.7 21.7 24.1 27.1 26.2
1.169 1.303 1.360 1.438 1.489 1.533
0.3 1.4 0.3 0.1 1.4 0.0
Fig. 13. True stress versus true plastic strain curves of S690 at various strain rates.
ing the dynamic properties of several mild steels were worked out by researchers, varying with different grade steels. For instance, C = 0.076 (e_ 0 ¼ 1 s1 ) [33] and C = 0.0331 (e_ 0 ¼ 0:001 s1 ) [20] were fitted for the J-C model based on the test results of mild steel with yield stress 217 MPa and Q345 structural steel respectively.
X. Yang et al. / Construction and Building Materials 198 (2019) 597–607
(a) 0.00025s-1
(b) 266s-1
(c) 1073s-1
(d) 1614s-1
(e) 2581s-1
(f) 3346s-1
605
(g) 4109s-1 Fig. 14. Comparison among experimental results and several models for S690 at each strain rate.
However, if those values were adopted to predict the dynamical stress-strain curves of S690 directly, these models will overrate the dynamic stress for S690, as shown in Fig. 14.
In this study, six terms of specimens at different strain rates had been chosen to fit the J-C model. Owing to the vibration of raw curves (Fig. 13), a fitted curve using a power function was adopted
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(a) stress-strain curves
(b) DIF
Fig. 15. Fitting method of DIF for J-C model at each strain rate.
Table 5 Parameters of J-C model for S690. Parameters
A/MPa
B/MPa
n
C
Standard J-C model Modified J-C model
722 722
400 400
0.57 0.57
0.021
m
0:0041e_
0:217
– –
Fig. 16. Fitting data of C for J-C model.
Fig. 17. Comparison between test data and modified J-C model in the engineering format.
to represent the original true stress versus true plastic strain curve at each high strain rate shown in Fig. 15(a). As mentioned in Section 3.1.2.3, the true stress-strain curve at quasi-static condition after necking has not been obtained, so that J-C model with fitted parameters was also used to predict the curve after necking for S690. Dividing each dynamical fitted curve at high strain rate by the quasi-static one, curves of DIF versus plastic strain were calculated and shown in Fig. 15(b). Then, an average dynamic increase factor (DIFave) was obtained for each strain rate. As discussed before, the engineering stress-strain curve is short at 266 s1 so that the average value DIFave may not be quite suitable for that condition, but DIFave at 266 s1 is still chosen to represent the DIF for the next analysis due to using the same data processing method in this study. Every DIFave was used to fit the value of C according to Eq. (7). After that, parameter C was calculated as 0.021 by averaging the obtained six values. With the data at disposal, all material parameters for standard J-C model were determined and listed in Table 5. Fig. 14 shows that the new J-C model with fitted coefficients in this paper has acceptable prediction accuracy for S690. 3.3.2. Modified J-C model Although the standard J-C model fitted before is suitable for characterizing dynamical constitutive relationship of S690 steel with acceptable accuracy, it will still slightly overrate the flow stress below 1073 s1, while underrate the flow stress over 1614 s1. Due to the constant strain-rate parameter C in standard J-C model, the value of C was fitted in average from test results. In practice, however, some errors will be induced by only choosing a single value or averaging all values directly as the value of C, since the value of C grows as increasing strain rate shown in Fig. 16. Therefore, the constant coefficient C ¼ 0:021 was substi0:217 tuted by a power function C ¼ 0:0041e_ in Fig. 16, which was more suitable for the increasing tendency of C as strain rate increased. As a result, a new modified J-C model was worked out in this paper, with all material coefficients listed in Table 5. Comparing with other J-C models in Fig. 14, the true stress-strain curve 0:217 derived from the modified J-C model with C ¼ 0:0041e_ has much better agreement with test data than that of others. The modified J-C model developed in this paper was based on true plastic strain and stress, which can be directly accepted in Abaqus, Ansys, etc. commercial Finite Element Analysis (FEA) software. However, the engineering stress-strain curve is also useful for structural analyses in some cases. By Eq. (2), the J-C model fitted in this study can be transformed to engineering curves before onset of necking, and the similar prediction accuracy was shown in Fig. 17.
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4. Conclusions The quasi-static tensile tests using the material testing machine and the SHPB tests by a SHPB system were conducted systematically in this paper for S690 to study its dynamical behaviors. All the specimens containing static tension tests and six terms of dynamic tests were tested three times to confirm the repeatability of test data with strain rate ranged from 0.00025 s1 to 4109 s1. According to the study, these following conclusions have been drawn: S690 is sensitive to the strain rate. The dynamic yield strength of S690 increases from 722 MPa to 1106 MPa as the strain rate rises to 4109 s1, while the strain strengthening phenomenon at high strain rate is almost same as that under quasi-static condition. A new C-S model was fitted to depict its DIF of yield stress. By fitting the DIFy obtained from test results, the two material coefficients for C-S model were determined as D = 18404 and p = 2.38, and the new C-S model has been proved that it performs much better prediction precision comparing with other models proposed before. The standard J-C model was accepted in this paper to fit the true stress-strain curve of S690 at various strain rates. Based on the test data, the parameters A, B, C and n were deeply analyzed and fitted, except for m that makes no sense for tests at room temperature. Moreover, the coefficient C = 0.0021 for standard J-C model was 0:217 substituted by a power function C ¼ 0:0041e_ , and a new modified J-C model was established to improve its prediction accuracy. Finally, this modified J-C model presented a higher predicting precision than standard J-C model with a constant parameter C. Conflict of interest None. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 51678194). The authors are grateful for the financial support from China Scholarship Council (Grant No. 201706120260). References [1] BS EN 1993-1-12, Design of Steel Structures - Part 1-12: Additional Rules for the Extension of EN 1993 up to Steel Grades S700, British Standards Institution, 2007. [2] L. Zhang, Y. Li, J. Wang, Q. Jiang, Effect of acicular ferrite on cracking sensibility in the weld metal of Q690+Q550 high strength steels, ISIJ Int. 51 (7) (2011) 1132–1136. [3] C. Fang, X. Meng, Q. Hu, F. Wang, H. Ren, H. Wang, et al., TANDEM and GMAW twin wire welding of Q690 steel used in hydraulic support, J. Iron Steel Res. Int. 19 (5) (2012) 79–85. [4] A.M.P. Jesus, R. Matos, B.F.C. Fontoura, C. Rebelo, L.S. Da Silva, M. Veljkovic, A comparison of the fatigue behavior between S355 and S690 steel grades, J. Constr. Steel Res. 79 (2012) 140–150.
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Contents lists available at ScienceDirect
Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat
Rate-dependent constitutive models of S690 high-strength structural steel Xiaoqiang Yang a,b, Hua Yang b,c,⇑, Sumei Zhang d a
School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China Key Lab of Structures Dynamic Behavior and Control of the Ministry of Education, Harbin Institute of Technology, Harbin 150090, China c Key Lab of Smart Prevention and Mitigation of Civil Engineering Disasters of the Ministry of Industry and Information Technology, Harbin Institute of Technology, Harbin 150090, China d School of Civil and Environment Engineering, Harbin Institute of Technology (Shenzhen), Shenzhen 518055, China b
h i g h l i g h t s Tests of S690 high-strength structural steel at high strain rates were conducted. Sensitivity of the flow stress to strain rate for S690 was investigated. DIF of yield stress was studied and a new Cowper-Symonds model was developed. Johnson-Cook model was fitted and modified, performing higher prediction accuracy.
a r t i c l e
i n f o
Article history: Received 12 July 2018 Received in revised form 20 November 2018 Accepted 30 November 2018 Available online 6 December 2018 Keywords: S690 SHPB High strain rate Dynamic increase factor Dynamic constitutive models
a b s t r a c t The demand of applying high-strength structural steel (HSSS) in engineering practice is progressively increasing, while insufficient research achievements of HSSS restrict its applications, especially for those under dynamic loadings. Therefore, to meet the increasingly demand of studies for HSSS at high strain rates, a grade of HSSS, S690 was experimentally investigated by quasi-static tensile tests and dynamic tests using a Split Hopkinson Pressure Bar (SHPB). The dynamic yield strength and stress-strain curves at various strain rates were obtained, which showed that S690 has obvious effects to the sensitivity of strain rate, while the existing models are not suitable for predicting its strain-rate effect. Further, the dynamic increase factor (DIF) of yield stress was determined and a new Cowper-Symonds model was fitted to predict the DIF precisely. To establish a rate-dependent constitutive model for S690, the Johnson-Cook (J-C) model was also adopted to fit true stress-strain curves and has been proved to perform acceptable prediction accuracy. By substituting the strain-rate parameter C, the standard J-C model was modified to improve its precision. These test results may provide fundamental data for its future studies, and the constitutive models may be applied to analyze the dynamical performance of S690 components. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction In recent decades, the interest on using high-strength structural steel (HSSS) in large-span, high-rise and heavily loaded modern engineering structures is increasingly on the rise. HSSS brings benefits in term of reducing material consumption and environmental protection, which is in line with the developing trend of green ⇑ Corresponding author at: Room 207, School of Civil Engineering, Harbin Institute of Technology, Huanghe Road #73, Nangang District, Harbin 150090, Heilongjiang Province, China. E-mail addresses: [email protected] (X. Yang), [email protected] (H. Yang), [email protected] (S. Zhang). https://doi.org/10.1016/j.conbuildmat.2018.11.285 0950-0618/Ó 2018 Elsevier Ltd. All rights reserved.
economy. However, the application of HSSS may cause a series of problems owing to the inadequate and incomplete theoretical and practical experiences, especially for those grand constructions subjected to extreme and dynamic loadings. The recent introduction of HSSS labeled with S690 included in EN 1993-1-12 [1] offers an ideal option for both steel structures and steel-concrete composite structures, of which yield strength is more than 690 MPa, while tensile to yield strength ratio and elongation should be not less than 1.05 and 10%, respectively. As a new-style high-strength steel material, HSSS with yield strength 690 MPa has been attracting extensive attention by researchers. From 2011, a series of studies on welding properties [2,3], fatigue
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behavior [4], mechanical and fracture properties of flame heat treatment [5], low temperature resistance [6] and post-fire mechanical properties [7–9] for 690 MPa structural steels have been conducted and proved that 690 MPa structural steels have acceptable mechanical properties subjected to those conditions. Based on the material researches, scholars have started studies on the performances of structural components. Li et al. [10,11] conducted experimental and numerical studies on compressive behaviors of Q690 welded box- and H-sections. Recently, the investigation on behaviors of S690 hot-finished high strength prestressed trusses under axial loadings was carried out by Wang et al. [12]. A recent study on the constitutive modelling of stress-strain curves for hot-rolled steels under quasi-static strain-rate loading was conducted by Yun and Gardner [13], covering S690 steel. Though previous studies into S690 have covered both material and member levels, knowledge of this material is still not far enough compared with those of common structural steels with normal yield strengths. The strain-rate effect of such steel has not yet been investigated resulting no specific rate-dependent constitutive model was proposed. The absence of rate-dependent model is inconsistent with the structural demands of resisting accidental actions, such as gas explosion, vehicle impact, terrorist attack, etc. Indeed, the structural design and assessment for accident safety over the serving duration could be vital especially for those structures employing high strength steel. It is a priority to clarify the dynamic properties and strain rate effect of such structural steel so as to analyze its further structural dynamical performances. In the last century, a series of investigations on the strain-rate effect of sheet steels and reinforcing steel bars with a wide range of strength from 180 MPa to 853 MPa at various high strain rates were conducted by researchers [14–16]. Based on test results, several equations had been proposed to characterize the dynamic increase factor (DIF) of yield stress and ultimate stress for these different types of steels, respectively. After 9/11 attacks, in 2005, to investigate the aircraft impact on the building, the dynamic properties of steels used in World Trade Center at high strain rates were researched by Luecke et al. [17], and a full set model was suggested for more precise prediction of DIF for different steels at high strain rate. Recently, by using MTS and Split Hopkinson Pressure Bar (SHPB), the dynamic properties of Q345 steel subjected to high temperature at various strain rates were studied by Yu et al. [18], and the results showed Q345 displays an obvious strain rate strengthening effect. Subsequently, in 2016, based on progressive collapse analysis, the dynamic tensile behavior of S355 steel under intermediate and high strain rates (5 s1–850 s1) by Forni et al. [19]. In 2017, the study on dynamic tensile tests of Q345 and Q420 steels at medium strain rates were performed by Chen et al. [20,21], and empirical constitutive models were developed to improve the accuracy in describing the dynamic properties for Q345 and Q420 respectively. Conclusion can be considered from above: the static properties of HSSS have been investigated widely, while there are few studies on the dynamic behaviors. Note that although previous researchers had presented several investigations on dynamical properties of high-strength sheet steels and reinforcing steel bars, and established equations to predict the DIF for such materials subjected to high strain rates, Fig. 1 illustrates that these models were still quite different when they were adopted to predict the DIF with yield stress of 690 MPa, varying 1.04–1.25 at strain rate 4000 s1. As a result, it may overrate or underrate the DIF if these existing models proposed before were used to predict the performance of 690 MPa steels directly. Moreover, there are no studies on the rate-dependent constitutive model for depicting the stress-strain curve of 690 MPa HSSS at high strain rate. Therefore, the grade of HSSS, S690, was investigated systematically in this paper. The
Fig. 1. Relationships of DIF versus strain rate for 690 MPa steel based on previous researches.
quasi-static tensile tests and dynamic SHPB tests were carried out to obtain the stress-strain relationships of S690 at quasistatic condition and high strain rates, using a universal electromechanical testing machine and a SHPB tester, respectively, which are both installed in Harbin Institute of Technology. The strain rate, dynamic yield stress and dynamic stress-strain curves were obtained and discussed, based on which three rate-dependent constitutive models were proposed to describe the dynamic behaviors of S690 at various strain rates precisely.
2. Experimental program 2.1. Materials and specimens In this study, a commercial S690QL hot-rolled steel plate was produced by Jiangyin Xingcheng Special Steel Works Corporation, China, of which chemical compositions and carbon equivalent value (CEV) were shown in Table 1, meeting the demands of EN 10025-6:2004 + A1:2009. The geometry of specimens and test method for quasi-static tests are according to ISO 6892-1:2016 and ISO 377:2013. The ratio of diameter/length for SHPB specimens was taken as 2.0, with 8.0 mm in diameter and 4.0 mm in length. The specimens were shown in Fig. 2.
2.2. Experimental setup 2.2.1. Quasi-static tension A universal electromechanical testing machine (Shimadzu AGX Plus 250 kN) was utilized to test all quasi-static tension specimens at room temperature. The engineering stress and strain were calculated by the tensile force and extension recorded by a force sensor and a 50 mm extensometer, respectively. During the elastic period, two pairs of strain gauges were stuck on the middle of both sides of specimen to record the longitudinal and transverse strains (see Fig. 2), which were utilized to calculate the Young’s modulus and Poisson’s ratio. For the quasi-static test, the strain rate was 0.00025 s1 (the crosshead separation rate was 0.9 mm/min and the parallel length is 60 mm). Two different directions – rolling direction 0° and transverse direction 90° were employed to machine the test coupons and three repeated experiments for each direction were carried out to ensure the reliability of test results.
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X. Yang et al. / Construction and Building Materials 198 (2019) 597–607 Table 1 Chemical compositions and CEV of S690QL steel (in weight %). C
Si
Mn
S
P
N
Ni
Mo
Cr
Cu
V
Ti
Nb
CEV
0.14
0.29
1.26
0.0006
0.014
0.0018
0.02
0.11
0.2
0.01
0.003
0.023
0.015
0.42
Fig. 2. Test specimens for: (a) quasi-static tension test and (b) SHPB test.
Fig. 3. Split Hopkinson Pressure Bar testing system: (a) SHPB devices; (b) specific details.
2.2.2. SHPB In this paper, an investigation of S690 high structural steel at high strain-rate was conducted by a SHPB tester, as shown in Fig. 3. An entire SHPB system contains a gas cabin, a strike bar, an incident bar, a transmission bar and an energy absorption setup, which are installed on a horizontal guide rail to guarantee that they are all on the same axis. When the gas chamber is inflated by the nitrogen gas cylinder to the designed pressure, the strike bar is propelled by high-pressure gas to impact the incident bar. The velocity of the strike bar will be recorded by the laser sensor before reaching the incident bar. A compressive elastic wave eI is generated and propagates through the incident bar after the strike bar impacts the incident bar. Once the inci-
dent wave (eI ) reaches the specimen that is sandwiched between the incident bar and transmission bar and lubricated by petroleum jelly, the wave reflects on the interface between incident bar and specimen to generate a reflected wave (eR ). At the same time, the specimen deforms, and the remaining wave passes to transmission bar, followed by a transmission wave (eT ) propagating through that. For this study, the length of strike bar is 200 mm, while incident and transmission are both 1200 mm. All the bars are 16 mm in diameter and they are all made of high strength alloy steel, whose density is 7850 kg/m3 and young’s modulus is 200 GPa, with 5000 m/s of the wave speed. These bars remain elastic over the entire period of test and strain gauges are stuck to the middle of
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both incident and transmission bars to record strain-wave histories (eI , eR and eT ) with a 5 MHz digital storing oscilloscope. According to the two-wave method [22], the engineering stress (reng ), engineering strain (eeng ) and strain rate (e_ ) in the function of time t can be calculated based on the data of strain waves through Eq. (1).
8 > r ðtÞ ¼ EAAs0 eT ðtÞ > < eng R eeng ðtÞ ¼ 2CL 0 0t eR ðtÞdt > > :_ eðtÞ ¼ 2CL 0 eR ðtÞ
ð1Þ
where E and A0 refer to the young’s modulus and cross-sectional area of alloy steel bar, C 0 is the speed of strain wave for alloy steel bar (5000 m/s), As and L are the cross-sectional area and length of specimen respectively. Further, true stress (rtrue ) and true strain (etrue ) can be calculated by the Eq. (2).
(
rtrue ¼ reng 1 þ eeng etrue ¼ ln 1 þ eeng
ð2Þ
In this study, six different gas pressures were determined to test steel specimens, based on which six different velocities of strike bar were recorded: 12.1 m/s, 16.6 m/s, 20.1 m/s, 25.6 m/s, 29.8 m/s and 33.3 m/s. For each impact velocity, three repeated experiments were carried out respectively to ensure repeatability of test results. 3. Results & discussion 3.1. Experimental results 3.1.1. Quasi-static tests Six uniaxial quasi-static tension tests of S690 high-strength steel for both 0° and 90° directions subjected to room temperature were carried out and the engineering stress-strain curves were presented in Fig. 4. It can be seen apparently that these tensile coupons showed good consistency for engineering stress-strain curves, and the tension properties were almost same at two different directions. The proof strength at 0.2% plastic strain was determined in this paper as the yield stress of S690 and the average tensile properties of S690 were summarized in Table 2. It showed that the tensile behaviors of S690 meet the demands of EN 1993-1-12. The engineering stress-strain curves of S690 exhibit a linearelastic part at the beginning of tests, followed by a short yield plateau and the strain hardening phenomenon subsequently when
Fig. 4. Engineering stress-strain curves of S690 under uniaxial quasi-static tension test.
plastic strain starts to grow. Compared with the tensile curves of S690 steel obtained by previous researchers [5,23,24], the strengths and plastic deforming performances are similar, while [24] shows a very long and obvious yield plateau for the stressstrain curve. Actually, the length of yield plateau is not an intrinsic property for one specific grade of steels and is known to depend on chemical composites, heat treatment, grain size, and test conditions, e.g. strain aging and loading rate, which can be concluded as the manufacturing process and strain history of material tests [13,25]. According to the review for high strength structural steels by Ban and Shi [26], in general, there is no visible yield plateau for most of HSSS with yield stress above 500 MPa. Therefore, it is common that different lengths of yield plateau were observed by different studies for those S690 steels. 3.1.2. SHPB tests 3.1.2.1. Raw strain waves. As mentioned above, the raw strain waves recorded by the strain gauges during the test can be analyzed to extract the stress-strain curve of the tested material further. For the SHPB tests, typical strain-time histories of eI , eR and eT at six terms of impact velocities were shown in Fig. 5(a). Based on theory of one-dimensional elastic wave, the assumption that specimen undergoes homogeneous deformation ensuring stress equilibrium is significant for the reliability of test results. Therefore, during the test, the forces on the front and back of interfaces of specimen described by two equations: F f ¼ EA0 ðeI þ eR Þ and F b ¼ EA0 eT should be almost equilibrated. Due to the same material for incident bar and transmission bar, it can be simplified to verify whether eI þ eR (eI and eR are at the opposite directions, as shown in Fig. 5(a)) is equal to eT . In the Fig. 5(b), note that although the time history of eI þ eR has a large vibration at the beginning, eT is basically close to the average of eI þ eR during the entire test at 20.1 m/s. For other impact velocities, all show the same phenomena. In this sense, stress equilibrium is mainly satisfied during the entire SHPB tests for this paper. 3.1.2.2. Strain rate. Most of metallic materials are sensitive to strain rate when subjected to quick loadings [27]. However, to obtain a completely constant strain rate directly during the entire period of loading for a SHPB test is mostly impossible. So, how to define the representative strain rate as precise as possible is important so as to depict the dynamic behavior of materials. For this study, a group of typical strain-rate time histories at 20.1 m/s were shown in Fig. 6(a) and the average strain-rate over the entire loading duration was calculated. Results show that the strain-rate time histories of three samples for three repeated tests are in good agreement and the average strain rate is relatively close to the time history for most of the time. Besides, another method suggested by Sun [28] was also adopted to define the strain rate of S690. Fitting a line through the linear part of the strain-time history curve, the strain rate was determined as the slope of fitted line shown in Fig. 6(b). In this sense, the strain rate at 20.1 m/s for linear fitting method is 1640 s1, which is so close to the mean strain rate for entire loading duration (1614 s1). Therefore, for convenience, in this paper, a mean strain rate over the total period of a SHPB test was defined as the representative strain rate. 3.1.2.3. Stress-strain curves. As described before, six terms of specimens under six different impact velocities by SHPB tester were tested and for each impact velocity, an average representative strain rate was determined. Based on which, all test results of S690 subjected to dynamic SHPB tests at wide strain rate ranged from 266 s1 to 4109 s1 were obtained in accordance with the Eq. (1). The engineering stress-strain curves of three repeated experiments was shown in Fig. 7 for each strain rate, which showed a good consistency. Besides, it can be noted that at lower
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X. Yang et al. / Construction and Building Materials 198 (2019) 597–607 Table 2 Tensile properties of S690 under uniaxial quasi-static tension tests. Grade
Yield stress
S690
722
r0:2 (N/mm2)
Tensile stress
ru (N/mm2)
761
Young’s modulus E(N/mm2)
Poisson’s ratio
Elongation
196,000
0.3
15%
Fig. 5. Typical strain waves of S690: (a) raw three-wave signals obtained from incident bar and transmission bar in SHPB testing, (b) comparison of eI þ
(a) Average strain rate over entire period
eR and eT at 20.1 m/s.
(b) Linear fitting method [28]
Fig. 6. Representative strain rate of S690 steel at impact velocity of 20.1 m/s.
strain rate, the stress-strain curve is shorter than that at higher strain rate. It is because that for lower strain rate, the impact velocity is lower so that the specimen performs less deformation due to its lower impact energy of strike bar. To make a clearer presentation of the difference of S690 at each strain rate, three engineering stress-strain curves were calculated in average and then used to be compared in Fig. 8(a). Further, all the engineering curves were transformed to true stress-strain curves by Eq. (2), which were also compared and shown in Fig. 8 (b). In particular, the computing method by Eq. (2) is no longer correct after necking of steel specimens, and the true stress-strain curve after necking cannot be recorded by the standard tensile test in this paper. As a result, the descent stage of true stress-strain curve was abandoned at 0.00025 s1 in Fig. 8(b). 3.1.2.4. Dynamic yield strength. Dynamic yield strength (rdy ) is an essential parameter characterizing the dynamical property for materials subjected to dynamic loadings. In practice, however, none of common or widely recognized definitions of dynamic yield stress for SHPB tests has been suggested by current standards. Due to the assumption that the specimen is in stress equilibrium during the test is not valid before it yields [17], the elastic part of stress-
strain curves for dynamic samples is not reliable, so that the 0.2% offset method is no longer suitable for determination of dynamic yield stress at high strain rates only based on the dynamic stress-strain curves which were derived from SHPB tests. Therefore, a method suggested by Sun [28] was adopted in this study to define the dynamic yield stress of S690. Actually, the dynamical modulus of elasticity is independent on strain rate, which is almost same as that in static condition in compliance with previous researches [17,29]. According to Sun’s research, truncating the elastic region and fitting a linear line for the strain hardening part of engineering stress-strain curve, the dynamic yield strength is the intersection point which can be determined by the 0.2% offset method using the quasi-static modulus of elasticity. The method for definition of dynamic yield stress for S690 was shown in Fig. 9. All the dynamic yield stresses at each strain rate were listed in Table 3 and the dynamic yield stresses versus strain rate curve was shown in Fig. 10.
3.2. DIF of yield stress Based on the results from Fig. 10, the strain-rate effect of yield stress for S690 is quite obvious. It can be seen that dynamic yield
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(a) 266s-1
(b) 1073s-1
(c) 1614s-1
(d) 2581s-1
(e) 3346s-1
(f) 4109s-1
Fig. 7. Engineering stress-strain curves of S690 for three repeated tests under different strain rates.
strength increases apparently, from 722 MPa to 1106 MPa when strain rate increases to 4109 s1. In this sense, a dynamic increase factor of yield stress (DIFy) expressed by dividing dynamic yield stress by quasi-static yield stress: DIFy = rdy =rsy was introduced to characterize the magnitude of strain-rate effect of yield stress. Each DIFy was calculated and listed in Table 4. To depict the tendency of the DIFy with the increasing strain rate, a Cowper-Symonds (C-S) model [30] was adopted in this paper. C-S model is a widely accepted dynamic constitutive model, which is simple, intuitive and accurate to characterize the DIF with strain rate. The standard C-S model can be expressed by the Eq. (3).
p rdy e_ ¼1þ rsy D
1
ð3Þ
where rsy is the yield stress at quasi-static, while rdy is the dynamic yield stress at the strain rate of e_ . D and p are the material coefficients. As one of the popular dynamic constitutive models, C-S model is widely accepted by many researchers so that they conducted studies on the dynamic behaviors of metallic materials and worked out several C-S models with fitted coefficients in the past years. Among them, D = 40.4 and p = 5 obtained experimentally from mild steels are extensively applied to predict the DIF of yield stress at high strain rate [31]. Therefore, for verifying its feasibility, the C-S model with these coefficients was used to predict the DIFy of S690 to estimate whether it can be applied directly for S690. Besides, D = 4945 and p = 2.696 obtained from test results for S355 steel [19] was also adopted to calculate the DIFy of S690 for comparison. However, it can be seen from Table 4 that the percent-
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Fig. 8. Comparison of engineering (a) and true (b) stress-strain curves for S690 under various strain rates from 0.00025 s1 to 4109 s1.
and steel bars with 690 MPa or similar yield strengths were added up to perform a comparison in Fig. 12. It demonstrates that these models proposed before may not be used to predict the performance of 690 MPa steels directly, because they will underrate the DIFy of S690, while the curve derived from the C-S model fitted in this paper has remarkably good prediction accuracy with test data. 3.3. J-C models
Fig. 9. Definition of dynamic yield stress for S690.
age of deviation between predicting results and test data are more than 110% and 15%, respectively. It means that the existing fitted coefficients for C-S model obtained from normal-strength steels are not suitable for S690, which will overrate the strain-rate effect of yield strength of S690. Based on the results as mentioned before, in this study, a new CS model was established by fitting the DIFy of S690 at different strain rates using the least square method through Eq. (4) in logarithmic coordinates.
rdy lnðe_ Þ ¼ pln 1 þ lnðDÞ
ð4Þ
rsy
As shown in Fig. 11, all the DIFy and strain rates have been transformed in logarithmic coordinates, and the elastic-linear fitting results showed that linearity of data is good, which means that the C-S model can predict the DIFy of S690 well. According to the fitting data, D = 18404 and p = 2.38 were taken out and the C-S model with such fitted coefficients was also compared with test data and other C-S models in Table 4, which shows much better agreement with test results than others. Moreover, other models obtained from previous studies [14–17] for high-strength steels
In the previous studies, scholars and researchers carried out the analyses of materials under dynamic loadings at high strain rates and worked out some equations to express their ratedependence of yield stress and ultimate stress. These equations and models gave the policy for them to consider the DIF of yield stress or ultimate stress when materials or structural components were subjected to quick loadings so that the prediction precision would be improved. So far, however, none of constitutive models has been developed for S690 to represent the complete stressstrain relationship under dynamic loadings. In this study, a Johnson-Cook (J-C) model [32] that is popularly and widely accepted for characterizing the true stress-strain curve of metallic materials is also adopted to describe the dynamic constitutive relationship of S690. 3.3.1. Standard J-C model A standard J-C model consists of three parts considering the strain hardening, strain-rate hardening and thermal softening, which is intuitive and without coupling effect. The stress can be expressed as:
r ¼ ðA þ Ben Þ 1 þ Clne_ ð1 T m Þ
ð5Þ
where r is the plastic stress, e is the plastic strain, e_ is the dimen sionless plastic strain rate(e_ ¼ e_ =e_ 0 , e_ 0 ¼ 0:00025 s1 ), T is the homologous temperature expressed as T ¼ ðT T r Þ=ðT m T r Þ, T is the test temperature, T r is the room temperature and T m is the melting temperature. A, B, C, m and n are the material constants. In this test, all the experiments are conducted at ambient temperature so that the J-C model can be simplified as Eq. (6):
r ¼ ðA þ Ben Þ 1 þ Clne_
ð6Þ
Table 3 Dynamic yield stresses of S690 at different strain rates. Strain rate e_
266 s1
1073 s1
1614 s1
2581 s1
3346 s1
4109 s1
rdy (N/mm )
841
954
985
1039
1060
1106
2
604
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Fig. 10. Dynamic yield stress versus strain rate curve for S690. Fig. 11. Fitting data for C-S model.
In this sense, the dynamical stress-strain curves were described by multiplying the quasi-static curve (A þ Ben ) by a magnification factor (1 þ Clne_ ). To fit the J-C model, the true stress-strain curves of S690 under various strain rates were transformed to the true plastic curves by removing the elastic strain, as shown in Fig. 13. The true stress versus true plastic strain curves reported in Fig. 13 demonstrate that the strain hardening phenomena for S690 steel at different strain rates is almost in some tendency, based on which, the J-C model using a magnification factor (DIF) to multiply the quasi-static curve without coupling effect is suitable for characterizing this dynamic behavior. As discussed before, without coupling effect, coefficients A, B and n can be fitted only using the quasi-static test results. For the quasi-static condition, e_ ¼ 1, so 1 þ Clne_ ¼ 1. The J-C model n was simplified as: r ¼ A þ Be . A is the yield stress, while B and n represent the effect of strain strengthening obtained by fitting the quasi-static stress-strain curve. In this paper, A was taken as 722, B was 400 and n was 0.57. It can be seen clearly in Fig. 14 (a) that there is a good fit between quasi-static tensile stressstrain curve and the predicting curve derived from the fitted J-C model. After that, the parameter C can be gained through the fitting method in accordance with the dynamical SHPB test data by Eq. (7):
r e_ 1 ¼ Cln rs e_ 0
Fig. 12. Comparison among test data and various models.
ð7Þ
where rs ¼ A þ Ben , representing the stress at the strain rate of e_ 0 (e_ 0 ¼ 0:00025 s1 ), while r is the dynamic stress at high strain rate. For the standard J-C model, 1 þ Clne_ as an increase factor is significant to determine the strain-rate effect so that it is the essential parameter for the parameter C corresponding to the specific reference strain rate. Over the past years, couple of values of C describ-
Table 4 Comparison of DIFy for S690 under each strain rate with various C-S models.
e_ /s1
266 1073 1614 2581 3346 4109
Test
D = 40.4, p = 5 [31]
D = 4945, p = 2.696 [19]
D = 18404, p = 2.38
DIFtest
DIFpre
D/%
DIFpre
D/%
DIFpre
D/%
1.165 1.321 1.364 1.439 1.468 1.532
2.461 2.931 3.095 3.301 3.424 3.525
111.2 121.9 126.9 129.4 133.2 130.1
1.338 1.567 1.660 1.786 1.865 1.934
14.9 18.7 21.7 24.1 27.1 26.2
1.169 1.303 1.360 1.438 1.489 1.533
0.3 1.4 0.3 0.1 1.4 0.0
Fig. 13. True stress versus true plastic strain curves of S690 at various strain rates.
ing the dynamic properties of several mild steels were worked out by researchers, varying with different grade steels. For instance, C = 0.076 (e_ 0 ¼ 1 s1 ) [33] and C = 0.0331 (e_ 0 ¼ 0:001 s1 ) [20] were fitted for the J-C model based on the test results of mild steel with yield stress 217 MPa and Q345 structural steel respectively.
X. Yang et al. / Construction and Building Materials 198 (2019) 597–607
(a) 0.00025s-1
(b) 266s-1
(c) 1073s-1
(d) 1614s-1
(e) 2581s-1
(f) 3346s-1
605
(g) 4109s-1 Fig. 14. Comparison among experimental results and several models for S690 at each strain rate.
However, if those values were adopted to predict the dynamical stress-strain curves of S690 directly, these models will overrate the dynamic stress for S690, as shown in Fig. 14.
In this study, six terms of specimens at different strain rates had been chosen to fit the J-C model. Owing to the vibration of raw curves (Fig. 13), a fitted curve using a power function was adopted
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(a) stress-strain curves
(b) DIF
Fig. 15. Fitting method of DIF for J-C model at each strain rate.
Table 5 Parameters of J-C model for S690. Parameters
A/MPa
B/MPa
n
C
Standard J-C model Modified J-C model
722 722
400 400
0.57 0.57
0.021
m
0:0041e_
0:217
– –
Fig. 16. Fitting data of C for J-C model.
Fig. 17. Comparison between test data and modified J-C model in the engineering format.
to represent the original true stress versus true plastic strain curve at each high strain rate shown in Fig. 15(a). As mentioned in Section 3.1.2.3, the true stress-strain curve at quasi-static condition after necking has not been obtained, so that J-C model with fitted parameters was also used to predict the curve after necking for S690. Dividing each dynamical fitted curve at high strain rate by the quasi-static one, curves of DIF versus plastic strain were calculated and shown in Fig. 15(b). Then, an average dynamic increase factor (DIFave) was obtained for each strain rate. As discussed before, the engineering stress-strain curve is short at 266 s1 so that the average value DIFave may not be quite suitable for that condition, but DIFave at 266 s1 is still chosen to represent the DIF for the next analysis due to using the same data processing method in this study. Every DIFave was used to fit the value of C according to Eq. (7). After that, parameter C was calculated as 0.021 by averaging the obtained six values. With the data at disposal, all material parameters for standard J-C model were determined and listed in Table 5. Fig. 14 shows that the new J-C model with fitted coefficients in this paper has acceptable prediction accuracy for S690. 3.3.2. Modified J-C model Although the standard J-C model fitted before is suitable for characterizing dynamical constitutive relationship of S690 steel with acceptable accuracy, it will still slightly overrate the flow stress below 1073 s1, while underrate the flow stress over 1614 s1. Due to the constant strain-rate parameter C in standard J-C model, the value of C was fitted in average from test results. In practice, however, some errors will be induced by only choosing a single value or averaging all values directly as the value of C, since the value of C grows as increasing strain rate shown in Fig. 16. Therefore, the constant coefficient C ¼ 0:021 was substi0:217 tuted by a power function C ¼ 0:0041e_ in Fig. 16, which was more suitable for the increasing tendency of C as strain rate increased. As a result, a new modified J-C model was worked out in this paper, with all material coefficients listed in Table 5. Comparing with other J-C models in Fig. 14, the true stress-strain curve 0:217 derived from the modified J-C model with C ¼ 0:0041e_ has much better agreement with test data than that of others. The modified J-C model developed in this paper was based on true plastic strain and stress, which can be directly accepted in Abaqus, Ansys, etc. commercial Finite Element Analysis (FEA) software. However, the engineering stress-strain curve is also useful for structural analyses in some cases. By Eq. (2), the J-C model fitted in this study can be transformed to engineering curves before onset of necking, and the similar prediction accuracy was shown in Fig. 17.
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4. Conclusions The quasi-static tensile tests using the material testing machine and the SHPB tests by a SHPB system were conducted systematically in this paper for S690 to study its dynamical behaviors. All the specimens containing static tension tests and six terms of dynamic tests were tested three times to confirm the repeatability of test data with strain rate ranged from 0.00025 s1 to 4109 s1. According to the study, these following conclusions have been drawn: S690 is sensitive to the strain rate. The dynamic yield strength of S690 increases from 722 MPa to 1106 MPa as the strain rate rises to 4109 s1, while the strain strengthening phenomenon at high strain rate is almost same as that under quasi-static condition. A new C-S model was fitted to depict its DIF of yield stress. By fitting the DIFy obtained from test results, the two material coefficients for C-S model were determined as D = 18404 and p = 2.38, and the new C-S model has been proved that it performs much better prediction precision comparing with other models proposed before. The standard J-C model was accepted in this paper to fit the true stress-strain curve of S690 at various strain rates. Based on the test data, the parameters A, B, C and n were deeply analyzed and fitted, except for m that makes no sense for tests at room temperature. Moreover, the coefficient C = 0.0021 for standard J-C model was 0:217 substituted by a power function C ¼ 0:0041e_ , and a new modified J-C model was established to improve its prediction accuracy. Finally, this modified J-C model presented a higher predicting precision than standard J-C model with a constant parameter C. Conflict of interest None. Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant No. 51678194). The authors are grateful for the financial support from China Scholarship Council (Grant No. 201706120260). References [1] BS EN 1993-1-12, Design of Steel Structures - Part 1-12: Additional Rules for the Extension of EN 1993 up to Steel Grades S700, British Standards Institution, 2007. [2] L. Zhang, Y. Li, J. Wang, Q. Jiang, Effect of acicular ferrite on cracking sensibility in the weld metal of Q690+Q550 high strength steels, ISIJ Int. 51 (7) (2011) 1132–1136. [3] C. Fang, X. Meng, Q. Hu, F. Wang, H. Ren, H. Wang, et al., TANDEM and GMAW twin wire welding of Q690 steel used in hydraulic support, J. Iron Steel Res. Int. 19 (5) (2012) 79–85. [4] A.M.P. Jesus, R. Matos, B.F.C. Fontoura, C. Rebelo, L.S. Da Silva, M. Veljkovic, A comparison of the fatigue behavior between S355 and S690 steel grades, J. Constr. Steel Res. 79 (2012) 140–150.
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