Ultrasonics 54 (2014) 867–873
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Measurements of Young’s and shear moduli of rail steel at elevated temperatures Yuanye Bao a, Haifeng Zhang a,⇑, Mehdi Ahmadi a, Md Afzalul Karim a, H. Felix Wu b a b
Department of Engineering Technology, University of North Texas, Denton, TX 76207, United States Office of Research and Economic Development, University of North Texas, Denton, TX 76207, United States
a r t i c l e
i n f o
Article history: Received 10 August 2013 Received in revised form 21 October 2013 Accepted 25 October 2013 Available online 5 November 2013 Keywords: Rail steel Elevated temperatures Young’s moduli Shear moduli Sonic resonance method
a b s t r a c t The design and modelling of the buckling effect of Continuous Welded Rail (CWR) requires accurate material constants, especially at elevated temperatures. However, such material constants have rarely been found in literature. In this article, the Young’s moduli and shear moduli of rail steel at elevated temperatures are determined by a new sonic resonance method developed in our group. A network analyser is used to excite a sample hanged inside a furnace through a simple tweeter type speaker. The vibration signal is picked up by a Polytec OFV-5000 Laser Vibrometer and then transferred back to the network analyser. Resonance frequencies in both the flexural and torsional modes are measured, and the Young’s moduli and shear moduli are determined through the measured resonant frequencies. To validate the measured elastic constants, the measurements have been repeated by using the classic sonic resonance method. The comparisons of obtained moduli from the two methods show an excellent consistency of the results. In addition, the material elastic constants measured are validated by an ultrasound test based on a pulse-echo method and compared with previous published results at room temperature. The measured material data provides an invaluable reference for the design of CWR to avoid detrimental buckling failure. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Continuously Welded Rails (CWRs), or ribbon rails, refers to rails that are welded together to form one uninterrupted rail that may be several kilometers long. For this form of rails, high compressive and tensile forces can be created due to a thermal expansion of the rail steel. The temperature-induced loads may cause a dangerous derailment, resulting in huge economic and human loss [1]. Modeling and monitoring approaches must be adopted to predict the failure stress and prevent derailments. Such efforts have been made in the past. A predictive model accounting for all the important parameters influencing track buckling was proposed by the US Department of Transportation [2]. A consistent nonlinear governing equation that describes the track buckling and a generalization of this analysis have been conducted by Kerr [3,4]. Furthermore, Nakamura et al. conclude that a decrease in lateral ballast resistance force could be prevented by applying adequate countermeasures against track buckling [5]. Meanwhile, a monitoring system has been proposed by Loveday to detect the axial stress in rails to prevent bucking [6]. A technique for in situ measurement of rail Neutral Temperature (a temperature at which the length of the rail fixed to the track is equal to the length of the
⇑ Corresponding author. Tel.: +1 9403698266. E-mail address: [email protected] (H. Zhang). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.10.015
unfixed rail) and detection of incipient buckling in continuous welded rail has been developed at UC San Diego [7]. The mechanical behavior of rail steel at elevated temperatures is not only the foundation for the modeling approaches but also is important for the monitoring technique. However, very few temperature-dependent rail steel properties have been found in literature. Past research work on rail steel elastic constants has mostly emphasized on the nonlinear constant instead of the linear constant [8–10]. Various experimental techniques have been used to measure the Young’s moduli and shear moduli at elevated temperatures for other types of steel such as cold formed steel. Chen and Young used universal testing machine to test heated cold-formed steel specimen [11]. A similar tensile test technique is used to measure not only the curves of elastic modulus but also the yield strength obtained at different strain levels by Kankanamge and Mahendran [12]. However, traditional techniques such as the tensile test method require expensive and bulky equipment such as universal test machine. Therefore, a test method that is economic, accurate, and reliable is needed to measure the Young’s moduli and shear moduli of rail steel and other type of steel at elevated temperatures. Such a test method can be found in the ASTM E1875-08 standard [13]. In this method, a frequency synthesizer or function generator is used to excite a speaker connected to the sample inside a furnace through a hanging wire, and the vibration of the sample can be picked up by a crystal cartridge through a wire connected to the other end of the sample. The resonance frequency can be
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determined as a function of the elevated temperature. From this, the mechanical properties can be determined as a function of temperature. This method has been used to measure the elastic properties of cold-rolled stainless steel [14]. In this article, we have utilized a new sonic resonance method developed by our group [15] to replace the standard sonic resonance method to measure the Young’s modulus and shear modulus of rail steel at elevated temperatures. The uniqueness of our method lies in two aspects: (1) the use of a network analyzer as the excitation source rather than a function generator, and (2) the use of a high accuracy laser vibrometer to detect the vibration of the beam instead of using a crystal cartridge. The advantages over the standard test method are low noise, high accuracy vibration measurement, smooth excitation, and less influence by the heat source. To validate the measured elastic constants, the Young’s moduli and shear moduli were measured for the same rail steel using both the standard sonic resonance method at elevated temperatures and ultrasonic pulseecho test method at room temperature. The comparison among these three test methods shows a good agreement. 2. Theoretical background The resonant frequency is determined by moduli and density of the material. Calculation can be conducted to obtain Young’s moduli of specimen if the resonant frequency is known based on Eq. (1). For a rectangular bar, 2
mf f E ¼ 0:9465 b
!
! L3 T1; t3
G¼
bt
B ; 1þA
b
ð3Þ
ð4Þ
t
þb t B¼ 3 6 4 bt 2:52 bt þ 0:21 bt
2 fT 1 ; 1 þ aDT f0
ð7Þ
The Poisson’s ratio has been calculated in both groups of experiment given in Tables 3 and 4. 3. Material and test methods 3.1. The specimen preparation A sample of rail steel is cut into a thin, rectangular bar. The surface has been polished to reach the best parallelism, and the dimensions are shown in Table 1. The averaged dimensions used in the calculation were measured at five different points with an electric caliper. The weight of the bar is scaled and given in Table 1. 3.2. Isotropy test A rail steel cube cut from same material is used to demonstrate the isotropy of this material by ultrasound method. Ultrasound speeds have been determined in three different directions. Table 2 shows the test result. The ultrasound speeds are close in each direction which is evidence that the material is isotropic.
The schematic for the experiment is shown in Fig. 1. Fig. 1(a) shows the setup for the new sonic resonance method which uses laser to detect the displacement of the specimen, and Fig. 1(b) shows the setup for the standard sonic test method – cartridge method. In both methods, a network analyzer is used to send an excitation signal to speaker through a 386 low voltage audio amplifier. The amplifier circuit can be found in a recent publication [15]. The sample
Sample Length (mm) Width (mm) Thickness (mm) Mass (g)
88.03 15.05 1.48 15.12
Table 2 Ultrasound velocity in different direction. Direction
Shear wave velocity (m/s)
Longitudinal wave velocity (m/s)
100 010 001
3222.0 3224.5 3234.9
5941.2 5936.1 5934.0
ð5Þ
Eq. (6) is used to get the moduli under elevated temperature:
MT ¼ M0
E 1 2G
Table 1 Dimensions of steel samples.
where G is the shear modulus in Pa, ft is the fundamental torsional frequency, and b, L, t, and m are given in Eq. (1). The equations for A and B are shown as:
2 3 0:5062 0:8776 bt þ 0:3504 bt 0:0078 bt A¼ ; b b2 12:03 t þ 9:892 t
ð2Þ
According to torsional frequency with the specimen’s mass and geometry, G can be calculated through following equation: 2 4Lmf t
l¼
3.3. Experimental apparatus
ð1Þ
where E is the Young’s modulus in Pa, m is the mass in grams, ff is the fundamental flexure frequency in Hz, b, L, and t are the width, length and thickness in mm and T1 is the shape factor given in Eq. (2). For L/t > 20:
2 t T 1 ¼ 1 þ 6:585 L
The Poisson’s ratio is given in following equation [14]:
ð6Þ
where MT is the constant at temperature T, M0 is the constant at room temperature, fT is the resonant frequency at temperature T in Hz, f0 is the resonant frequency at room temperature, a is the average linear thermal expansion coefficient, and DT is the difference between the test temperature and room temperature. 12.0 106/°C will be used for the coefficient of linear expansion for low-carbon based steel [16].
Table 3 The experimental data measured by Laser Vibrometer method. T (°C)
Frequency (Hz)flexural mode
E (GPa)
T (°C)
Frequency (Hz) Torsional mode
G (GPa)
l
22 50 100 150 201 250 300 351
1019.0 1016.5 1010.0 1003.0 997.5 988.5 980.3 969.5
208.18 207.16 204.52 201.69 199.49 195.90 192.65 188.45
23 50 100 150 201 250 301 350
3512.1 3505.2 3485.5 3451.6 3434.8 3404.3 3370.1 3333.0
77.43 77.12 76.24 74.76 74.03 72.74 71.29 69.73
0.34 0.34 0.34 0.35 0.35 0.35 0.35 0.35
Y. Bao et al. / Ultrasonics 54 (2014) 867–873
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Table 4 The experimental data measured by the conventional crystal cartridge method. T (°C)
Frequency (Hz)-flexural mode
E (GPa)
T (°C)
Frequency (Hz)-flexural mode
G (GPa)
l
22 51 100 151 200 251 301 350
1019.0 1015.3 1010.8 1003.8 996.8 988.2 978.9 970.1
208.18 206.67 204.84 202.02 199.21 195.79 192.12 188.68
22 51 101 150 200 251 300 350
3513.3 3507.6 3485.6 3454.6 3428.0 3396.3 3365.0 3328.0
77.49 77.24 76.27 74.92 73.77 72.41 71.09 69.53
0.34 0.34 0.34 0.35 0.35 0.35 0.35 0.36
(rectangular bar) is suspended to the speaker through a fire retardant embroidery thread that has a temperature limit of 371 °C. Fig. 2 shows a picture of the new sonic test method [15]. The major difference between the new sonic resonance and standard sonic resonance method lies in the detection method. The new sonic resonance method uses laser to detect the vibration, while the traditional sonic resonance method uses a crystal cartridge to detect the vibration through a high temperature wire. To measure the Young’s moduli and shear moduli, the sample must vibrate at different modes. Therefore, the sample is hanged in two configurations: torsional mode (Fig. 3(a)) and flexural mode (Fig. 3(b)) [15]. 3.4. Experiment procedure The excitation signal from the Bode 100 network analyzer is swept from 100 Hz to 10,000 Hz. After the clear peaks of the response graphs at room temperature have been obtained, the temperature is increased from room temperature up to 350 °C slowly at 50 °C/h to reduce transient thermal effects. Both torsional mode and flexural mode have been tested with both Laser Vibrometer method and Crystal cartridge method. The obtained resonant frequencies are used to calculate Young’s and Shear modulus at elevated temperatures. 4. Results 4.1. Young’s modulus and shear modulus determined by the new Laser Vibrometer Method at elevated temperatures Figs. 4 and 5 show the measured frequency spectrums for torsional mode and flexural mode at room temperature separately.
Fig. 2. A picture of the new sonic resonance method. A polytech laser vibrometer is used to detect the vibration of the sample remotely, and the sample is suspended to a speaker and fixed beam as right side [15].
Table 3 lists the measured resonance frequencies at elevated temperatures and the determined Young’s modulus, shear modulus, and Poisson’s ratio based on Eqs. (1)–(5). Figs. 6 and 7 show the plotted Young’s modulus and shear modulus vs. temperatures obtained by laser method. As can be seen from these figures, the elastic constants and resonance frequencies decrease as a parabolic curve at elevated temperature. Additionally, the Poisson’s ratio is subject to small changes at the elevated temperature. 4.2. Young’s moduli and shear modulus determined by the conventional crystal cartridge method at elevated temperatures The measurement has been repeated by using the conventional crystal cartridge method described in Section 3.3. Table 4 lists the measured resonance frequencies at elevated temperatures and the determined Young’s modulus, shear modulus, and Poisson’s ratio based on Eqs. (1)–(5). Figs. 8 and 9 show the plotted Young’s modulus and shear modulus vs. temperatures obtained by the conventional crystal cartridge method. The measurement results show a good agreement with the new sonic resonance method-laser vibrometer method.
Fig. 1. (a) Schematic for the laser method; (b) Schematic for the standard cartridge method.
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210
E (GPa)
205
200
195
190
185 0
50
100
150
200
250
300
350
Temperature (°C) Fig. 6. Young’s modulus vs. temperatures by laser method. Fig. 3. (a) Torsion mode; (b) Flexural mode.
78 77 -66 76
-67
75
G (Gpa)
-68
Gain (dB)
-69 -70 -71
74 73 72
-72
71
-73 70 -74 69
-75
0
-76 3460 3470 3480 3490 3500 3510 3520 3530 3540 3550
Frequency (Hz) Fig. 4. Frequency spectrum for torsional mode at room temperature.
-74
-75
Gain (dB)
-76
-77
-78
-79
-80 1010 1012 1014 1016 1018 1020 1022 1024 1026 1028 1030
Frequency (Hz) Fig. 5. Frequency spectrum for flexural mode at room temperature.
50
100
150
200
250
300
350
Temperature (°C) Fig. 7. Shear modulus vs. temperature by laser method.
4.3. Young’s moduli and shear modulus determined by ultrasound test method at room temperature The experiment setup by ultrasound test method is shown schematically in Fig. 10. The rail steel cubic sample with geometry of 20 20 20 mm and the attached transducer are placed in an Instron 3119 heating chamber in order to maintain a temperature of 25.0 °C ± 0.10. The oscilloscope that we used was the Wavemaster 804Zi-A series with a 40GS/s sampling rate, and the pulser receiver was the Olympus Model 5072PR. Two different types of transducers are used V121 (7.5/.25)-633847 – contact transducer and V156 (5/25//)-761131 – Normal Incidence Shear Wave transducer. In the experiment, pulse-echo method was used [17] to measure the ultrasonic wave velocity of the sample at room temperature. The basic quantity that was measured was the delay time for waves making a round trip within the specimen. A single transducer in contact with the specimen was used to send and receive ultrasonic waves and a cross-correlation program based on MATLAB was developed to measure the delay time. The cross-correlation method could measure the time delays between pairs of ultrasonic pulse-echo signals and could minimize the measuring error [18]. Fig. 11 shows typical longitudinal waveforms where the first wave peak was the trigger and the measurement was
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210
2 1.5
205
Amplitude (Volt)
E (GPa)
1 200
195
0.5 0 -0.5 -1
190 -1.5 185 0
50
100
150
200
250
300
-2 -5
350
0
5
Temperature (°C)
10
15
20 x 10-6
Time (Second)
Fig. 8. Young’s modulus vs. temperature by cartridge method.
Fig. 11. Longitudinal waveforms.
78 77 2 76 1.5 1
74
Amplitude (Volt)
G (GPa)
75
73 72 71 70 69
0.5 0 -0.5 -1
0
50
100
150
200
250
300
-1.5
350
Temperature (°C)
-2 5
5.5
6
Fig. 9. Shear modulus vs. temperature by cartridge method.
6.5
7
7.5
8 x 10-6
Time (Second) Fig. 12. First echo.
1.5
Fig. 10. Experiment setup for ultrasound method.
taken between the second and the third peaks. The cross-correlation method was applied for the measurement of the delay time for waves making a round trip within the specimen. Figs. 12 and 13 show the specific area of first echo and second echo. The Young’s moduli determined by the ultrasound test is 211 GPa, the shear moduli is 81.84 GPa and Poisson’s ratio is 0.289. 4.4. Uncertainty analysis
Amplitude (Volt)
1
0.5
0
-0.5
-1
-1.5 1.15
1.2
1.25
1.3
1.35
1.4
1.45
Time (Second) Table 5 shows the measurands, related uncertainties and their units and symbols during this research. According to SM&T [19],
Fig. 13. Second echo.
1.5
1.55
1.6 -5
x 10
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Y. Bao et al. / Ultrasonics 54 (2014) 867–873
Table 5 Measurands, measurements, their units and symbols. Measurands
Symbol
Units
Uncertainty
Mass of the bar Width of the bar Length of the bar Thickness of the bar Fundamental resonant frequency of a rectangular bar in flexure Fundamental resonant frequency of a rectangular bar in torsion Test temperature Dynamic Young’s modulus Dynamic shear modulus Poisson’s ratio
m b L t ff
g mm mm mm Hz
±0.01 g ±0.01 mm ±0.01 mm ±0.01 mm ±0.1 Hz
ft
Hz
±0.1 Hz
T E G
°C GPa GPa dimensionless
±1 °C ±8.45 GPa ±1.05 GPa ±0.015
Table 7 Comparisons for the shear moduli determined by different methods at room temperature. Property
Methods
G (GPa)
Laser method
Cartridge method
Ultrasound method
77.43
77.49
81.84
210
l
Laser Cartridge
205
2 X @Y UðYÞ2 ¼ ; Uðxi Þ: @xi
E (GPa)
the coverage factor k = 2 is adapted to provide a level of confidence of approximately 95%. The uncertainties of Y is a function of measurands, x1, x2, x3,. . ., and each xi is subject to uncertainty u (x1) from Eqs. (1)–(5), as shown in Eq. (7).
200
195
ð8Þ 190
As shown in Table 5, both of the uncertainties of moduli only ±8.45 GPa and ±1.05 GPa are small enough to guarantee the accuracy of this experiment.
185
0
50
100
150
200
250
300
350
400
Temperature (°C) Fig. 14. Young’s modulus comparison.
5. Discussion and comparison 5.1. Comparisons of the measured modulus with others at room temperature 78
Laser
Table 6 compares the Young’s moduli determined through the new sonic resonance method, conventional cartridge method and ultrasound test method. It also compares the measurement result from peers’ work. It can be seen that the laser method and cartridge method shows a very good agreement, while the Young’s moduli determined by ultrasound test method exhibits a 1% difference, which is within the experiment error. The comparison with peers’ results [1,20–23], shows a fairly well agreement as well. Table 7 compares the measured shear moduli of rail steel at room temperature using the three methods. Similar to the Young’s moduli measurements, the laser method and cartridge method show an excellent agreement while the ultrasound method has a 4% difference. Due to the lack of existing data for the shear moduli at elevated temperature, we could not compare our result with peers’ works. Results of Poisson’s ratio in Tables 3 and 4 are not agreed with the normally expected one as 0.3. In consideration of the 4% difference of shear moduli shown in Table 7, it tends to indicate this distinction between dynamic shear moduli and static shear moduli which causes the disagreement of Poisson’s ratio. 5.2. Comparison of moduli between these two sonic test methods at elevated temperature Figs. 14 and 15 show the comparison of the Young’s moduli and shear moduli at elevated temperature determined by the new sonic resonance method and the conventional cartridge method.
77
Cartridge
76
G (GPa)
75 74 73 72 71 70 69 0
50
100
150
200
250
300
350
Temperature (°C) Fig. 15. Shear modulus comparison.
The comparison shows a good agreement between these two methods. As mentioned by Sandor [24], the two moduli decrease at elevated temperatures, which make buckling easier to happen at high temperature [2]. This trend should definitely be considered during the design of rail structure to alleviate/prevent buckling of rail track.
Table 6 Comparison of the Young’s modulus determined by different methods at room temperature. Property
E (GPa)
Method Laser method
Cartridge method
Ultrasound method
[1]
[17]
[18]
[19]
[20]
208
208
211
210
207
210
207
206
Y. Bao et al. / Ultrasonics 54 (2014) 867–873
6. Summary This paper describes a new sonic resonance method to measure the Young’s moduli and Shear moduli of rail steel at elevated temperatures. The measurement methodology has been validated by the conventional cartridge method and ultrasound test method. Compared with conventional cartridge method, the new sonic method is economic because the head of cartridge is fragile and easy to be broken which will cost more for continuous experiments. And the cartridge head also limits the size of the specimen. In addition, this new noncontact sonic method has a better reliability by reducing effects from heat resource which is unavoidable in conventional cartridge method. Compared with the ultrasound test method, this method excels in the convenience of measurements of material properties at high temperatures, as the ultrasound transducer suffers the inevitable phase transition at high temperatures and it is difficult to utilize a buffer rod to avoid the influence of high temperature to the transducer. The measured material constants of rail steel at elevated temperature provide an important reference for the design of the Continuous Weld Rail to avoid buckling. References [1] N.H. Lim, N.H. Park, Y.J. Kang, Stability of continuous welded rail track, Comput. Struct. 81 (2003) 2219–2236. [2] A. Kish, G. Samavedam, Track Buckling Prevention: Theory, Safety Concepts, and Applications, DOT/FRA/ORD-13/16, 2013. [3] A.D. Kerr, Analysis of thermal track buckling in the lateral plane, Acta Mech. 30 (1978) 17–50. [4] AD. Kerr, An improved analysis for thermal track buckling, Int. J. Non-linear Mech. 15 (1980) 99–114. [5] T. Nakamura, E. Sekine, Y. Shirae, Assessment of a seismic performance of ballasted track with large-scale shaking table tests, Quarterly Rep of RTRI 52 (2011) 156–162.
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[6] P.W. Loveday, Guided wave inspection and monitoring of railway track, J. Nondestruct. Eval. 31 (2012) 303–309. [7] R. Phillips, Nonlinear guided waves in continuously welded rails for buckling prediction, AIP Conf. Proc. 1335 (2011) 314–321. [8] D.M. Egle, D.E. Bray, Measurement of acoustoelastic and third-order elastic constants of rail steel, J. Acoust. Soc. Am. 60 (1976) 741–744. [9] R.L. Weaver, Vibration Measurement of Rail Stress, HSR-IDEA Project 48, 2006. [10] A.M. Karim, H.F. Zhang, Measurement of nonlinear elastic constants of rail steel, Proc. SPIE. 8347 (2012) 83472U1-9. [11] J. Chen, B. Young, Experimental investigation of cold-formed steel material at elevated temperatures, Thin-Walled Struct. 45 (2007) 96–110. [12] N.D. Kankanamge, M. Mahendran, Mechanical properties of cold-formed steels at elevated temperatures, Thin-Walled Struct. 49 (2011) 26–44. [13] ASTM-E 1875-08 Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Sonic Resonance, Available at . [14] B. Ahn, S.S. Lee, U.S. Min, Elastic constants determination of thin cold-rolled stainless steels by dynamic elastic modulus measurements, J. Mater. Sci. 33 (1998) 687–692. [15] M. Ahmadi, H.F. Zhang, C. Yu, J. Wahrmund, Determining elastic and shear modulae of cold-formed steel at elevated temperatures using a new sonic resonance method, Nondestruct. Test Eval. DOI:http://dx.doi.org/10.1080/ 10589759.2013.823609. [16] P.P. Benham, F.V. Warnock, Mechanics of Solids and Strengths of Materials, Pitman Publication, London, 1965. [17] R.E. Green Jr., Ultrasonic Investigation of Mechanical Properties, Acad Press, New York, NY, 1973. [18] D.R. Hull, H.E. Kautz, A. Vary, Measurement of ultrasonic velocity using phaseslope and cross-correlation methods, Mater. Eval. 43 (1985) 1455–1460. [19] C.K. Bullough, The Determination of Uncertainties in Dynamic Young’s Modulus, Stand Meas & Test Proj No. SMT4-CT97-2165, 2000. [20] AS 1085.1.Standards-Australia Railway Track Materials, Part 1: Steel Rails. Australia, 2002. [21] G. Baumann, H.J. Fecht, S. Liebelt, Formation of white-etching layers on rail treads, Wear 191 (1996) 133–140. [22] D.L. McDowell, Stress state dependence of cyclic ratchetting behavior of two rail steels, Int. J. Plast. 11 (1995) 397–421. [23] F. Bucher, A.I. Dmitriev, M. Ertz, K. Knothe, V.L. Popov, S.G. Psakhie, E.V. Shilko, Multiscale simulation of dry friction in wheel/rail contact, Wear 261 (2006) 874–884. [24] B.I. Sandor, Strengths of Materials, Prentice Hall, Upper Saddle River, New Jersey, 1978.
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Measurements of Young’s and shear moduli of rail steel at elevated temperatures Yuanye Bao a, Haifeng Zhang a,⇑, Mehdi Ahmadi a, Md Afzalul Karim a, H. Felix Wu b a b
Department of Engineering Technology, University of North Texas, Denton, TX 76207, United States Office of Research and Economic Development, University of North Texas, Denton, TX 76207, United States
a r t i c l e
i n f o
Article history: Received 10 August 2013 Received in revised form 21 October 2013 Accepted 25 October 2013 Available online 5 November 2013 Keywords: Rail steel Elevated temperatures Young’s moduli Shear moduli Sonic resonance method
a b s t r a c t The design and modelling of the buckling effect of Continuous Welded Rail (CWR) requires accurate material constants, especially at elevated temperatures. However, such material constants have rarely been found in literature. In this article, the Young’s moduli and shear moduli of rail steel at elevated temperatures are determined by a new sonic resonance method developed in our group. A network analyser is used to excite a sample hanged inside a furnace through a simple tweeter type speaker. The vibration signal is picked up by a Polytec OFV-5000 Laser Vibrometer and then transferred back to the network analyser. Resonance frequencies in both the flexural and torsional modes are measured, and the Young’s moduli and shear moduli are determined through the measured resonant frequencies. To validate the measured elastic constants, the measurements have been repeated by using the classic sonic resonance method. The comparisons of obtained moduli from the two methods show an excellent consistency of the results. In addition, the material elastic constants measured are validated by an ultrasound test based on a pulse-echo method and compared with previous published results at room temperature. The measured material data provides an invaluable reference for the design of CWR to avoid detrimental buckling failure. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Continuously Welded Rails (CWRs), or ribbon rails, refers to rails that are welded together to form one uninterrupted rail that may be several kilometers long. For this form of rails, high compressive and tensile forces can be created due to a thermal expansion of the rail steel. The temperature-induced loads may cause a dangerous derailment, resulting in huge economic and human loss [1]. Modeling and monitoring approaches must be adopted to predict the failure stress and prevent derailments. Such efforts have been made in the past. A predictive model accounting for all the important parameters influencing track buckling was proposed by the US Department of Transportation [2]. A consistent nonlinear governing equation that describes the track buckling and a generalization of this analysis have been conducted by Kerr [3,4]. Furthermore, Nakamura et al. conclude that a decrease in lateral ballast resistance force could be prevented by applying adequate countermeasures against track buckling [5]. Meanwhile, a monitoring system has been proposed by Loveday to detect the axial stress in rails to prevent bucking [6]. A technique for in situ measurement of rail Neutral Temperature (a temperature at which the length of the rail fixed to the track is equal to the length of the
⇑ Corresponding author. Tel.: +1 9403698266. E-mail address: [email protected] (H. Zhang). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.10.015
unfixed rail) and detection of incipient buckling in continuous welded rail has been developed at UC San Diego [7]. The mechanical behavior of rail steel at elevated temperatures is not only the foundation for the modeling approaches but also is important for the monitoring technique. However, very few temperature-dependent rail steel properties have been found in literature. Past research work on rail steel elastic constants has mostly emphasized on the nonlinear constant instead of the linear constant [8–10]. Various experimental techniques have been used to measure the Young’s moduli and shear moduli at elevated temperatures for other types of steel such as cold formed steel. Chen and Young used universal testing machine to test heated cold-formed steel specimen [11]. A similar tensile test technique is used to measure not only the curves of elastic modulus but also the yield strength obtained at different strain levels by Kankanamge and Mahendran [12]. However, traditional techniques such as the tensile test method require expensive and bulky equipment such as universal test machine. Therefore, a test method that is economic, accurate, and reliable is needed to measure the Young’s moduli and shear moduli of rail steel and other type of steel at elevated temperatures. Such a test method can be found in the ASTM E1875-08 standard [13]. In this method, a frequency synthesizer or function generator is used to excite a speaker connected to the sample inside a furnace through a hanging wire, and the vibration of the sample can be picked up by a crystal cartridge through a wire connected to the other end of the sample. The resonance frequency can be
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determined as a function of the elevated temperature. From this, the mechanical properties can be determined as a function of temperature. This method has been used to measure the elastic properties of cold-rolled stainless steel [14]. In this article, we have utilized a new sonic resonance method developed by our group [15] to replace the standard sonic resonance method to measure the Young’s modulus and shear modulus of rail steel at elevated temperatures. The uniqueness of our method lies in two aspects: (1) the use of a network analyzer as the excitation source rather than a function generator, and (2) the use of a high accuracy laser vibrometer to detect the vibration of the beam instead of using a crystal cartridge. The advantages over the standard test method are low noise, high accuracy vibration measurement, smooth excitation, and less influence by the heat source. To validate the measured elastic constants, the Young’s moduli and shear moduli were measured for the same rail steel using both the standard sonic resonance method at elevated temperatures and ultrasonic pulseecho test method at room temperature. The comparison among these three test methods shows a good agreement. 2. Theoretical background The resonant frequency is determined by moduli and density of the material. Calculation can be conducted to obtain Young’s moduli of specimen if the resonant frequency is known based on Eq. (1). For a rectangular bar, 2
mf f E ¼ 0:9465 b
!
! L3 T1; t3
G¼
bt
B ; 1þA
b
ð3Þ
ð4Þ
t
þb t B¼ 3 6 4 bt 2:52 bt þ 0:21 bt
2 fT 1 ; 1 þ aDT f0
ð7Þ
The Poisson’s ratio has been calculated in both groups of experiment given in Tables 3 and 4. 3. Material and test methods 3.1. The specimen preparation A sample of rail steel is cut into a thin, rectangular bar. The surface has been polished to reach the best parallelism, and the dimensions are shown in Table 1. The averaged dimensions used in the calculation were measured at five different points with an electric caliper. The weight of the bar is scaled and given in Table 1. 3.2. Isotropy test A rail steel cube cut from same material is used to demonstrate the isotropy of this material by ultrasound method. Ultrasound speeds have been determined in three different directions. Table 2 shows the test result. The ultrasound speeds are close in each direction which is evidence that the material is isotropic.
The schematic for the experiment is shown in Fig. 1. Fig. 1(a) shows the setup for the new sonic resonance method which uses laser to detect the displacement of the specimen, and Fig. 1(b) shows the setup for the standard sonic test method – cartridge method. In both methods, a network analyzer is used to send an excitation signal to speaker through a 386 low voltage audio amplifier. The amplifier circuit can be found in a recent publication [15]. The sample
Sample Length (mm) Width (mm) Thickness (mm) Mass (g)
88.03 15.05 1.48 15.12
Table 2 Ultrasound velocity in different direction. Direction
Shear wave velocity (m/s)
Longitudinal wave velocity (m/s)
100 010 001
3222.0 3224.5 3234.9
5941.2 5936.1 5934.0
ð5Þ
Eq. (6) is used to get the moduli under elevated temperature:
MT ¼ M0
E 1 2G
Table 1 Dimensions of steel samples.
where G is the shear modulus in Pa, ft is the fundamental torsional frequency, and b, L, t, and m are given in Eq. (1). The equations for A and B are shown as:
2 3 0:5062 0:8776 bt þ 0:3504 bt 0:0078 bt A¼ ; b b2 12:03 t þ 9:892 t
ð2Þ
According to torsional frequency with the specimen’s mass and geometry, G can be calculated through following equation: 2 4Lmf t
l¼
3.3. Experimental apparatus
ð1Þ
where E is the Young’s modulus in Pa, m is the mass in grams, ff is the fundamental flexure frequency in Hz, b, L, and t are the width, length and thickness in mm and T1 is the shape factor given in Eq. (2). For L/t > 20:
2 t T 1 ¼ 1 þ 6:585 L
The Poisson’s ratio is given in following equation [14]:
ð6Þ
where MT is the constant at temperature T, M0 is the constant at room temperature, fT is the resonant frequency at temperature T in Hz, f0 is the resonant frequency at room temperature, a is the average linear thermal expansion coefficient, and DT is the difference between the test temperature and room temperature. 12.0 106/°C will be used for the coefficient of linear expansion for low-carbon based steel [16].
Table 3 The experimental data measured by Laser Vibrometer method. T (°C)
Frequency (Hz)flexural mode
E (GPa)
T (°C)
Frequency (Hz) Torsional mode
G (GPa)
l
22 50 100 150 201 250 300 351
1019.0 1016.5 1010.0 1003.0 997.5 988.5 980.3 969.5
208.18 207.16 204.52 201.69 199.49 195.90 192.65 188.45
23 50 100 150 201 250 301 350
3512.1 3505.2 3485.5 3451.6 3434.8 3404.3 3370.1 3333.0
77.43 77.12 76.24 74.76 74.03 72.74 71.29 69.73
0.34 0.34 0.34 0.35 0.35 0.35 0.35 0.35
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Table 4 The experimental data measured by the conventional crystal cartridge method. T (°C)
Frequency (Hz)-flexural mode
E (GPa)
T (°C)
Frequency (Hz)-flexural mode
G (GPa)
l
22 51 100 151 200 251 301 350
1019.0 1015.3 1010.8 1003.8 996.8 988.2 978.9 970.1
208.18 206.67 204.84 202.02 199.21 195.79 192.12 188.68
22 51 101 150 200 251 300 350
3513.3 3507.6 3485.6 3454.6 3428.0 3396.3 3365.0 3328.0
77.49 77.24 76.27 74.92 73.77 72.41 71.09 69.53
0.34 0.34 0.34 0.35 0.35 0.35 0.35 0.36
(rectangular bar) is suspended to the speaker through a fire retardant embroidery thread that has a temperature limit of 371 °C. Fig. 2 shows a picture of the new sonic test method [15]. The major difference between the new sonic resonance and standard sonic resonance method lies in the detection method. The new sonic resonance method uses laser to detect the vibration, while the traditional sonic resonance method uses a crystal cartridge to detect the vibration through a high temperature wire. To measure the Young’s moduli and shear moduli, the sample must vibrate at different modes. Therefore, the sample is hanged in two configurations: torsional mode (Fig. 3(a)) and flexural mode (Fig. 3(b)) [15]. 3.4. Experiment procedure The excitation signal from the Bode 100 network analyzer is swept from 100 Hz to 10,000 Hz. After the clear peaks of the response graphs at room temperature have been obtained, the temperature is increased from room temperature up to 350 °C slowly at 50 °C/h to reduce transient thermal effects. Both torsional mode and flexural mode have been tested with both Laser Vibrometer method and Crystal cartridge method. The obtained resonant frequencies are used to calculate Young’s and Shear modulus at elevated temperatures. 4. Results 4.1. Young’s modulus and shear modulus determined by the new Laser Vibrometer Method at elevated temperatures Figs. 4 and 5 show the measured frequency spectrums for torsional mode and flexural mode at room temperature separately.
Fig. 2. A picture of the new sonic resonance method. A polytech laser vibrometer is used to detect the vibration of the sample remotely, and the sample is suspended to a speaker and fixed beam as right side [15].
Table 3 lists the measured resonance frequencies at elevated temperatures and the determined Young’s modulus, shear modulus, and Poisson’s ratio based on Eqs. (1)–(5). Figs. 6 and 7 show the plotted Young’s modulus and shear modulus vs. temperatures obtained by laser method. As can be seen from these figures, the elastic constants and resonance frequencies decrease as a parabolic curve at elevated temperature. Additionally, the Poisson’s ratio is subject to small changes at the elevated temperature. 4.2. Young’s moduli and shear modulus determined by the conventional crystal cartridge method at elevated temperatures The measurement has been repeated by using the conventional crystal cartridge method described in Section 3.3. Table 4 lists the measured resonance frequencies at elevated temperatures and the determined Young’s modulus, shear modulus, and Poisson’s ratio based on Eqs. (1)–(5). Figs. 8 and 9 show the plotted Young’s modulus and shear modulus vs. temperatures obtained by the conventional crystal cartridge method. The measurement results show a good agreement with the new sonic resonance method-laser vibrometer method.
Fig. 1. (a) Schematic for the laser method; (b) Schematic for the standard cartridge method.
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210
E (GPa)
205
200
195
190
185 0
50
100
150
200
250
300
350
Temperature (°C) Fig. 6. Young’s modulus vs. temperatures by laser method. Fig. 3. (a) Torsion mode; (b) Flexural mode.
78 77 -66 76
-67
75
G (Gpa)
-68
Gain (dB)
-69 -70 -71
74 73 72
-72
71
-73 70 -74 69
-75
0
-76 3460 3470 3480 3490 3500 3510 3520 3530 3540 3550
Frequency (Hz) Fig. 4. Frequency spectrum for torsional mode at room temperature.
-74
-75
Gain (dB)
-76
-77
-78
-79
-80 1010 1012 1014 1016 1018 1020 1022 1024 1026 1028 1030
Frequency (Hz) Fig. 5. Frequency spectrum for flexural mode at room temperature.
50
100
150
200
250
300
350
Temperature (°C) Fig. 7. Shear modulus vs. temperature by laser method.
4.3. Young’s moduli and shear modulus determined by ultrasound test method at room temperature The experiment setup by ultrasound test method is shown schematically in Fig. 10. The rail steel cubic sample with geometry of 20 20 20 mm and the attached transducer are placed in an Instron 3119 heating chamber in order to maintain a temperature of 25.0 °C ± 0.10. The oscilloscope that we used was the Wavemaster 804Zi-A series with a 40GS/s sampling rate, and the pulser receiver was the Olympus Model 5072PR. Two different types of transducers are used V121 (7.5/.25)-633847 – contact transducer and V156 (5/25//)-761131 – Normal Incidence Shear Wave transducer. In the experiment, pulse-echo method was used [17] to measure the ultrasonic wave velocity of the sample at room temperature. The basic quantity that was measured was the delay time for waves making a round trip within the specimen. A single transducer in contact with the specimen was used to send and receive ultrasonic waves and a cross-correlation program based on MATLAB was developed to measure the delay time. The cross-correlation method could measure the time delays between pairs of ultrasonic pulse-echo signals and could minimize the measuring error [18]. Fig. 11 shows typical longitudinal waveforms where the first wave peak was the trigger and the measurement was
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210
2 1.5
205
Amplitude (Volt)
E (GPa)
1 200
195
0.5 0 -0.5 -1
190 -1.5 185 0
50
100
150
200
250
300
-2 -5
350
0
5
Temperature (°C)
10
15
20 x 10-6
Time (Second)
Fig. 8. Young’s modulus vs. temperature by cartridge method.
Fig. 11. Longitudinal waveforms.
78 77 2 76 1.5 1
74
Amplitude (Volt)
G (GPa)
75
73 72 71 70 69
0.5 0 -0.5 -1
0
50
100
150
200
250
300
-1.5
350
Temperature (°C)
-2 5
5.5
6
Fig. 9. Shear modulus vs. temperature by cartridge method.
6.5
7
7.5
8 x 10-6
Time (Second) Fig. 12. First echo.
1.5
Fig. 10. Experiment setup for ultrasound method.
taken between the second and the third peaks. The cross-correlation method was applied for the measurement of the delay time for waves making a round trip within the specimen. Figs. 12 and 13 show the specific area of first echo and second echo. The Young’s moduli determined by the ultrasound test is 211 GPa, the shear moduli is 81.84 GPa and Poisson’s ratio is 0.289. 4.4. Uncertainty analysis
Amplitude (Volt)
1
0.5
0
-0.5
-1
-1.5 1.15
1.2
1.25
1.3
1.35
1.4
1.45
Time (Second) Table 5 shows the measurands, related uncertainties and their units and symbols during this research. According to SM&T [19],
Fig. 13. Second echo.
1.5
1.55
1.6 -5
x 10
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Table 5 Measurands, measurements, their units and symbols. Measurands
Symbol
Units
Uncertainty
Mass of the bar Width of the bar Length of the bar Thickness of the bar Fundamental resonant frequency of a rectangular bar in flexure Fundamental resonant frequency of a rectangular bar in torsion Test temperature Dynamic Young’s modulus Dynamic shear modulus Poisson’s ratio
m b L t ff
g mm mm mm Hz
±0.01 g ±0.01 mm ±0.01 mm ±0.01 mm ±0.1 Hz
ft
Hz
±0.1 Hz
T E G
°C GPa GPa dimensionless
±1 °C ±8.45 GPa ±1.05 GPa ±0.015
Table 7 Comparisons for the shear moduli determined by different methods at room temperature. Property
Methods
G (GPa)
Laser method
Cartridge method
Ultrasound method
77.43
77.49
81.84
210
l
Laser Cartridge
205
2 X @Y UðYÞ2 ¼ ; Uðxi Þ: @xi
E (GPa)
the coverage factor k = 2 is adapted to provide a level of confidence of approximately 95%. The uncertainties of Y is a function of measurands, x1, x2, x3,. . ., and each xi is subject to uncertainty u (x1) from Eqs. (1)–(5), as shown in Eq. (7).
200
195
ð8Þ 190
As shown in Table 5, both of the uncertainties of moduli only ±8.45 GPa and ±1.05 GPa are small enough to guarantee the accuracy of this experiment.
185
0
50
100
150
200
250
300
350
400
Temperature (°C) Fig. 14. Young’s modulus comparison.
5. Discussion and comparison 5.1. Comparisons of the measured modulus with others at room temperature 78
Laser
Table 6 compares the Young’s moduli determined through the new sonic resonance method, conventional cartridge method and ultrasound test method. It also compares the measurement result from peers’ work. It can be seen that the laser method and cartridge method shows a very good agreement, while the Young’s moduli determined by ultrasound test method exhibits a 1% difference, which is within the experiment error. The comparison with peers’ results [1,20–23], shows a fairly well agreement as well. Table 7 compares the measured shear moduli of rail steel at room temperature using the three methods. Similar to the Young’s moduli measurements, the laser method and cartridge method show an excellent agreement while the ultrasound method has a 4% difference. Due to the lack of existing data for the shear moduli at elevated temperature, we could not compare our result with peers’ works. Results of Poisson’s ratio in Tables 3 and 4 are not agreed with the normally expected one as 0.3. In consideration of the 4% difference of shear moduli shown in Table 7, it tends to indicate this distinction between dynamic shear moduli and static shear moduli which causes the disagreement of Poisson’s ratio. 5.2. Comparison of moduli between these two sonic test methods at elevated temperature Figs. 14 and 15 show the comparison of the Young’s moduli and shear moduli at elevated temperature determined by the new sonic resonance method and the conventional cartridge method.
77
Cartridge
76
G (GPa)
75 74 73 72 71 70 69 0
50
100
150
200
250
300
350
Temperature (°C) Fig. 15. Shear modulus comparison.
The comparison shows a good agreement between these two methods. As mentioned by Sandor [24], the two moduli decrease at elevated temperatures, which make buckling easier to happen at high temperature [2]. This trend should definitely be considered during the design of rail structure to alleviate/prevent buckling of rail track.
Table 6 Comparison of the Young’s modulus determined by different methods at room temperature. Property
E (GPa)
Method Laser method
Cartridge method
Ultrasound method
[1]
[17]
[18]
[19]
[20]
208
208
211
210
207
210
207
206
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6. Summary This paper describes a new sonic resonance method to measure the Young’s moduli and Shear moduli of rail steel at elevated temperatures. The measurement methodology has been validated by the conventional cartridge method and ultrasound test method. Compared with conventional cartridge method, the new sonic method is economic because the head of cartridge is fragile and easy to be broken which will cost more for continuous experiments. And the cartridge head also limits the size of the specimen. In addition, this new noncontact sonic method has a better reliability by reducing effects from heat resource which is unavoidable in conventional cartridge method. Compared with the ultrasound test method, this method excels in the convenience of measurements of material properties at high temperatures, as the ultrasound transducer suffers the inevitable phase transition at high temperatures and it is difficult to utilize a buffer rod to avoid the influence of high temperature to the transducer. The measured material constants of rail steel at elevated temperature provide an important reference for the design of the Continuous Weld Rail to avoid buckling. References [1] N.H. Lim, N.H. Park, Y.J. Kang, Stability of continuous welded rail track, Comput. Struct. 81 (2003) 2219–2236. [2] A. Kish, G. Samavedam, Track Buckling Prevention: Theory, Safety Concepts, and Applications, DOT/FRA/ORD-13/16, 2013. [3] A.D. Kerr, Analysis of thermal track buckling in the lateral plane, Acta Mech. 30 (1978) 17–50. [4] AD. Kerr, An improved analysis for thermal track buckling, Int. J. Non-linear Mech. 15 (1980) 99–114. [5] T. Nakamura, E. Sekine, Y. Shirae, Assessment of a seismic performance of ballasted track with large-scale shaking table tests, Quarterly Rep of RTRI 52 (2011) 156–162.
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[6] P.W. Loveday, Guided wave inspection and monitoring of railway track, J. Nondestruct. Eval. 31 (2012) 303–309. [7] R. Phillips, Nonlinear guided waves in continuously welded rails for buckling prediction, AIP Conf. Proc. 1335 (2011) 314–321. [8] D.M. Egle, D.E. Bray, Measurement of acoustoelastic and third-order elastic constants of rail steel, J. Acoust. Soc. Am. 60 (1976) 741–744. [9] R.L. Weaver, Vibration Measurement of Rail Stress, HSR-IDEA Project 48, 2006. [10] A.M. Karim, H.F. Zhang, Measurement of nonlinear elastic constants of rail steel, Proc. SPIE. 8347 (2012) 83472U1-9. [11] J. Chen, B. Young, Experimental investigation of cold-formed steel material at elevated temperatures, Thin-Walled Struct. 45 (2007) 96–110. [12] N.D. Kankanamge, M. Mahendran, Mechanical properties of cold-formed steels at elevated temperatures, Thin-Walled Struct. 49 (2011) 26–44. [13] ASTM-E 1875-08 Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Sonic Resonance, Available at . [14] B. Ahn, S.S. Lee, U.S. Min, Elastic constants determination of thin cold-rolled stainless steels by dynamic elastic modulus measurements, J. Mater. Sci. 33 (1998) 687–692. [15] M. Ahmadi, H.F. Zhang, C. Yu, J. Wahrmund, Determining elastic and shear modulae of cold-formed steel at elevated temperatures using a new sonic resonance method, Nondestruct. Test Eval. DOI:http://dx.doi.org/10.1080/ 10589759.2013.823609. [16] P.P. Benham, F.V. Warnock, Mechanics of Solids and Strengths of Materials, Pitman Publication, London, 1965. [17] R.E. Green Jr., Ultrasonic Investigation of Mechanical Properties, Acad Press, New York, NY, 1973. [18] D.R. Hull, H.E. Kautz, A. Vary, Measurement of ultrasonic velocity using phaseslope and cross-correlation methods, Mater. Eval. 43 (1985) 1455–1460. [19] C.K. Bullough, The Determination of Uncertainties in Dynamic Young’s Modulus, Stand Meas & Test Proj No. SMT4-CT97-2165, 2000. [20] AS 1085.1.Standards-Australia Railway Track Materials, Part 1: Steel Rails. Australia, 2002. [21] G. Baumann, H.J. Fecht, S. Liebelt, Formation of white-etching layers on rail treads, Wear 191 (1996) 133–140. [22] D.L. McDowell, Stress state dependence of cyclic ratchetting behavior of two rail steels, Int. J. Plast. 11 (1995) 397–421. [23] F. Bucher, A.I. Dmitriev, M. Ertz, K. Knothe, V.L. Popov, S.G. Psakhie, E.V. Shilko, Multiscale simulation of dry friction in wheel/rail contact, Wear 261 (2006) 874–884. [24] B.I. Sandor, Strengths of Materials, Prentice Hall, Upper Saddle River, New Jersey, 1978.
