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Measurement of elastic constants of limited-size piezoelectric ceramic sample by ultrasonic method
Measurement 42 (2009) 1214–1219
Contents lists available at ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Measurement of elastic constants of limited-size piezoelectric ceramic sample by ultrasonic method Guorong Song *, Cunfu He, Zenghua Liu, Yao Huang, Bin Wu College of Mechanical Engineering & Applied Electronics Technology, Beijing University of Technology, No. 100# Pingle Yuan, Chaoyang District, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 17 October 2006 Received in revised form 9 February 2009 Accepted 23 April 2009 Available online 3 May 2009
Keywords: Elastic constant Limited-size sample Line-focus PVDF transducer Piezoelectric ceramic NDT
a b s t r a c t Measurement of material elastic constants for limited-size samples by ultrasonic method is described and validated. Based on acoustic microscope technology, the material elastic constants are determined by longitudinal wave and leak surface wave velocities simultaneously measured by developed ultrasonic system with a line-focus PVDF transducer. In this paper, the elastic constants of limited-size Cr2O3 doping on 0.2 PZN–0.8 PZT piezoelectric ceramic wafer sample are determined and the measurement errors are analyzed. The experimental results show this ultrasonic system can be used for measurement of material elastic constants for limited-size samples with high measurement precision, and the relative errors for Poisson’s ratio and Young’s module measurements are, respectively, less than 1% and 3%. It can satisfy the requirement of engineering and science research. Furthermore, it is suitable to measure elastic constants of both isotropic and anisotropic materials by ultrasonic method with help of developed ultrasonic system. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Piezoelectric ceramics are important advanced functional materials because they can achieve mutual energy transformation between mechanical energy and electric energy. The piezoelectric ceramic materials are widely applied to modern electric field as basic materials of many new electronic and micro-electronic components due to their stable function, high electromechanical performance, etc. In order to satisfy different application needs, extensive studies were widely conducted for improving the performance of piezoelectric ceramics [1–4]. However, due to the special processing technology, that is, directly molding, piezoelectric ceramics made in experiment have the characteristic of limited-size, therefore, it is very difficult to obtain the elastic constants or mechanical performance of this kind of materials. A lot of traditional testing methods and equipments are not applicable to the mechanical test-
* Corresponding author. Tel.: +86 010 67391601; fax: +86 010 67391617. E-mail address: [email protected] (G. Song). URL: http://www.bjut.edu.cn (G. Song). 0263-2241/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2009.04.008
ing on the micro-scale. In recent years, a number of mechanical testing methods are developed for measuring the mechanical properties of micro-structures [5–8]. In order to analyze the mechanical characteristics of developed materials, the representative testing methods are microtensile testing, nano-indentation, bulge testing, microbeam bending testing, etc. However, the methods are used mostly to measure the film or nano materials, and furthermore, some of the equipments are quite expensive. In this paper, the elastic constants of a 9.82 mm diameter and 1.24 mm thickness wafer piezoelectric doping ceramic sample is measured by the ultrasonic system with a line-focus PVDF transducer developed by ourselves. The measurement result is quite satisfactory with high precision, and this system is reliable for characterization and measurement of the elastic constants of materials, especially for the limited-size materials prepared at experiments. 2. Piezoelectric ceramic sample In this paper, chosen measurement sample is a piece of piezoelectric ceramic supplied by the authors of reference
G. Song et al. / Measurement 42 (2009) 1214–1219
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However, the piezoelectric ceramic is an elastic material and the elastic constants are its important parameters characterizing the elastic behaviors of piezoelectric material. Thus, developing a new, facility, low cost nondestructive testing method has an important meaning for these new especially materials. 3. Measuring system and method
Fig. 1. PZN–PZT piezoelectric ceramic sample.
[4]. It is 0.2Pb (Zn1/3Nb2/3)O3–0.8Pb (Ti0.5Zr0.5)O3 (PZN– PZT) piezoelectric ceramic modified with Cr2O3 prepared by low-temperature firing. As shown in Fig. 1, the sample is a round slice with small-size and its diameter is 9.82 mm and thickness is 1.24 mm. Lu et al. [4] studied the effects of Cr2O3 addition on the micro-structure of 0.2PZN–0.8PZT ceramics, and analyzed the electric properties and piezoelectric properties, including dielectric constant ec, electricity losing tan d, piezoelectric constant d33, electromechanical coupling coefficient kp and mechanical factor Qm, with the increase of Cr2O3 content. But the elastic performance of the piezoelectric ceramic sample was not analyzed due to the limited-size of sample by using conventional testing methods.
The ultrasonic measurement system of elastic constants for small-size samples is based on acoustic microscope technology [9]. The material elastic constants are determined by measuring longitudinal wave velocity CL and leak surface wave velocity VR simultaneously by the system with a line-focus PVDF transducer developed by ourselves. This is one of nondestructive testing methods. The experimental system makes up of a line-focus PVDF transducer, a four-axis moving exactly framework, a NI PXI bus embedded controller, an ultrasonic pulser/receiver (Parametrics 5800) and a computer, as shown in Fig. 2. The line-focus transducer is a lens-less concave cylindrical transducer with a thin piezoelectric polyvinylidene fluoride (PVDF) film. The line-focus PVDF transducer is 26.15 mm in focal length and a bandwidth centered around 6 MHz. When the measurement system works, the pulse/receiver firstly excites the PVDF film and then a transient pulsed wave is launched into a coupling fluid which is usually water and focuses the acoustic waves on a focal plane. When a sample is placed right under this transducer, the incident wave is reflected back to the line-focus PVDF film by the sample surface and a reflected electric signal is generated.
Fig. 2. Experimental system set-up.
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G. Song et al. / Measurement 42 (2009) 1214–1219
D
Line-Focus Transducer
Coupling Fluid
Sample
B
t
D
θR
R
R
Where D—Directly reflected wave R—Leaky surface wave B—Bottom reflected longitudinal wave Focal Plane
Fig. 3. Line-focus PVDF transducer and its measurement method.
During testing, firstly the sample surface is aligned with focal plane, the acoustic waves launched by line-focus PVDF transducer are reflected by the sample surface and received and converted into the echo wave voltage signal, V, by transducer. And then, as the transducer moves downward, the focal plane is deflected from the sample surface, also known as defocusing. The line-focus PVDF transducer and its measurement method are schematically depicted in Fig. 3. The wave beams incidenting at the Rayleigh angle are converted into surface waves. After propagating a certain distance along a sample surface, the surface waves leak into coupling fluid and re-radiate back to the PVDF transducer. In this case, the echo voltage signals consist of not only the directly reflected waves D and bottom reflected longitudinal waves B, but also the leaky surface waves R. This line-focus PVDF transducer can differentiate the arrival times of the directly reflected wave, the leaky surface wave, as well as other possible echo arrivals with considerable accuracy when the sample is moved inside the focal region of the transducer. When the defocus distance z is varied continuously, a series of echo voltage signals V(t) can be obtained. The leaky surface wave velocity can be determined by measuring the line relationship (slope m = Dz/Dt) of the time interval between the leaky surface wave arrival R and directly reflected wave arrival D with the defocusing distance z. When m is obtained, the leaky surface wave velocity can be evaluated from the following equation [10]:
"
2 #12
VW VR ¼ VW 1 1 2m
ð1Þ
where VW is the longitudinal wave velocity in the coupling fluid which is usually water, then here, VW is the wave velocity in water. The longitudinal wave velocity can be obtained by detecting the thickness of the sample and the time interval between the directly reflected waves arrival D and bottom reflected wave arrival B
2h CL ¼ tB tD
ð2Þ
where h is the thickness of the sample, tB tD is the time interval between the directly reflected waves D and bottom reflected longitudinal waves B. According to the elastic dynamics theory, the ultrasonic longitudinal wave velocity CL and surface wave velocity CR are depended on the density q and elastic constants of the material. Material elastic constants include Poisson’s ratio t, Young’s modulus E, shear modulus G and volume modulus K. Therefore, the ultrasonic longitudinal wave velocity CL and surface wave velocity CR can be expressed as
sffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 1t q ð1 þ tÞð1 2tÞ sffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:89 þ 1:12t E 1 CR ¼ 1þt q 2ð1 þ tÞ CL ¼
ð3Þ ð4Þ
According to (3) and (4), we can obtain
2:5088C 2L =C 2R 2 t3 þ 2:6432C 2L =C 2R 2 t2 0:4350C 2L =C 2R 2 t 0:7569C 2L =C 2R 2 ¼ 0
ð5Þ
It can be seen that Poisson’s ratio t can be resolved when both longitudinal wave and surface wave velocities are measured. Furthermore, other elastic constants can be determined as formula in the case that the density is known
C 2L ð1 þ tÞð1 2tÞq 1t E K¼ 3ð1 2tÞ E G¼ 2ð1 þ tÞ E¼
ð6Þ ð7Þ ð8Þ
4. Results and analysis 4.1. Measurement results To sum up all the considerations mentioned above, in this paper, the elastic constants of a 9.82 mm diameter and 1.24 mm thickness wafer piezoelectric doping ceramic
G. Song et al. / Measurement 42 (2009) 1214–1219
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Fig. 4. Experimental waveforms from the 0.2PZN–0.8PZT + Cr2O3 sample in the different defocus distance z.
sample mentioned above is measured by this system. In the testing, the defocus distance z is increased continuously with 0.2 ± 0.004 mm/step. After several time testing, the representative experimental waveforms at the different defocus distance z are shown in Fig. 4. Fig. 5 gives experimental results obtained by varying defocusing distance z. Fig. 5(a) is the curve of echo wave signal superposed by wave voltage offsetting at the different defocusing distance, and Fig. 5(b) shows the fitting curve of the directly reflected wave D, leaky surface wave R and bottom reflected longitudinal wave B with propagation time at different defocusing distances, respectively. When the thickness h of sample is 1.24 ± 0.02 mm, then, the slope m and the time interval tB tD between the di-
rectly reflected waves D and bottom reflected longitudinal waves B can be obtained
m ¼ 4703:5 m=s; t B t D ¼ 0:4724 ls When the longitudinal wave velocity Vw is 1480 m/s in water, then from (1) and (2) yields
V R ¼ 2748:7 m=s C L ¼ 5249:4 m=s Substituted into (5), hence
t ¼ 0:262 The volume density of ceramic is q = 7.4 103 kg/m3 through the Archimedean principle of buoyancy, and other elastic constants can be obtained from (6)–(8), respectively
Fig. 5. Experimental results obtained by varying defocusing distance z.
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Fig. 6. The measurement result for PZT specimen by using ultrasonic method at 22, 30 and 40 °C.
E ¼ 165:982 GPa G ¼ 65:762 GPa K ¼ 116:234 GPa
where C2
3
2
2:5088t 2:6432t þ0:4350tþ0:7569 Dt L ¼ tB tD ; b ¼ C L2 ; f ðt; bÞ ¼ 3ð2:5088b2Þ t2 þ2ð2:6432b2Þtð0:4350b2Þ ;
4.2. Measurement error analysis In this paper, the error for measuring elastic constants of limited-size piezoelectric ceramic by using ultrasonic measurement system depends mainly on the error of measuring longitudinal wave velocity and surface wave velocity. Furthermore, the error of measuring velocity is mainly depended on length error, as known the error of sample thickness h, the error of defocusing distance z and the error of time interval. For analyzing the error for measuring elastic constants t, E, assume that dq = 0. That is to say, the error caused by density can be neglected, and consider that the time for measuring elastic constants is very short every time, the error caused by the longitudinal wave velocity changes with the temperature in water can be ignored, then dVW = 0. According to Eqs. (1), (2), (5) and (6), the maximum of the error for measuring Poisson’s ratio and Young’s modulus can be obtained as following:
" # b jdhj h dt ¼ 4 f ðt; bÞ þ jdðDtL Þj C L DtL Dt 2L V 2W VW 1 b f ðt; bÞ 2 CR m 2m " 2 #32 VW z jdzj 1 1 þ 2 jdðDtÞj ð9Þ 2m Dt Dt " # qð1 þ tÞð1 2tÞ 2qtðt 2Þ CL þ b C L f ðt; bÞ dE ¼ 4 1t ð1 tÞ2 " # jdhj h 2qtðt 2Þ 2 þ jdðDtL Þj b CR Dt L Dt 2L ð1 tÞ2 " 2 #32 V 2W VW VW f ðt; bÞ 2 1 1 1 m 2m 2m z jdzj þ ð10Þ jdðDtÞj Dt Dt 2
R
dE; dh; dz; dðDt L Þ and d(Dt) are the errors of Young’s modulus E, sample thickness h, defocusing distance z, time intervals DtL and Dt, respectively. During the testing, jdðDtL Þj ¼ jdðDtÞj ¼ 0:4 ns, jdhj ¼ 0:02 mm and jdzj ¼ 0:004 mm are substituted into (9) and (10), then yields
dt ¼ 0:00247 dE ¼ 4:694 GPa
dt
t
¼ 0:876%
dE ¼ 2:801% E
The relative errors for Poisson’s ratio and Young’s module measurements are, respectively, less than 1% and 3%. It can satisfy the need of engineering and science research. 4.3. The influence of temperature for measurement results In the measurement error analysis of PZT specimen, the error of the velocity change of longitudinal wave induced by the temperature change of water is ignored. The experiment is designed for discussing if this error can be ignored. The PZT specimen is measured at its circumstances with different temperature (22, 30 and 40 °C). It is measured five times at each temperature, and the mean of obtained measurement values is recorded. The statistical results are shown in Fig. 6. It is shown that the velocities of longitudinal wave and leak surface waves become higher, measurement value of Young’s module increases, and the Poisson’s ratio changes inconspicuous when the temperature changes from 22 to 40 °C. When the temperature is changed from 22 to 40 °C, the measurement values of material Young’s module at 30 and 40 °C increase 0.76% and 3.64% than at 22 °C, respectively. Result shows the increased amplitude of the measurement values of material elastic constant is less than 4% when the temperature increases 45%. The experimental results show the error induced by the temperature change of water can be ignored because the
G. Song et al. / Measurement 42 (2009) 1214–1219
specimen is measured within a short time which is only about 1 min.
5. Conclusion Measurement of material elastic constants by ultrasonic method has the advantage of requiring small-size sample and nondestructive testing. It can be applied to some special situations that conventional methods are not suitable. Especially, it is much suitable for measuring new materials of directly molding by special technologies, small-size, high cost, etc. Because the total measurement time is short, the influence of measurement results brought by the temperature change of water is very small, it can be ignored. The error analysis results show that this measurement system has high measurement precision, the relative errors for Poisson’s ratio and Young’s module measurements are, respectively, less than 1% and 3%. It can satisfy the need of engineering and science research. This measurement method and system can provide powerful technological assistance for characterizing and measuring material elastic constants of limited-size samples obtained from the experiment of material preparation. And besides, the line-focus PVDF transducer can launch surface acoustic waves in a specific direction, which is perpendicular to the focus line; it can be further applied for study of the anisotropic material.
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Acknowledgements The work presented in this paper is supported by the National Natural Science Foundation of China (Nos. 10372009 and 10827201). References [1] S.H. Lee, C.B. Yoon, S.B. Seo, et al., Effect of lanthanum on the piezoelectric properties of lead zirconate titanate-lead zinc niobate ceramics, J. Mater. Res. 18 (8) (2003) 1701–1765. [2] X.Y. Sun, J. Chen, R.B. Yu, et al., BiScO3 doped (Na0.5K0.5)NbO3 leadfree piezoelectric ceramics, J. Am. Ceram. Soc. 92 (1) (2009) 130–132. [3] Y.D. Hou, P.X. Lu, M.K. Zhu, et al., Effect of Cr2O3 addition on the structure and electrical properties of Pb((Zn1/3Nb2/3)0.20(Zr0.50Ti0.50)0.80)O3 ceramics, Mater. Sci. Eng. B 116 (1) (2005) 104–108. [4] Pengxian Lu, Yudong Hou, Mankang Zhu, Hui. Yan, Effects of Cr2O3 doping on microstructure and piezoelectric properties of 0.2PZN– 0.8PZT ceramics, J. Funct. Mater. Dev. 11 (3) (2005) 303–307. [5] Caijun Su, Hao Wu, Guo Zhanshe, et al., Mechanical testing methods of micro structures, Chin. J. Exp. Mech. 20 (3) (2005) 441–447. [6] Huimin Xie, Satoshi Kishimoto, Anand Asundi, Norio Shinya, In-plane deformation measurement using the atomic force microscope moiré method, Nanotechnology 11 (1) (2000) 24–29. [7] Xuefeng Yao, Hsien-Yang Yeh, H.P. Zhao, Dynamic response and fracture characterization of polymer/clay nanocomposites with mode-I crack, J. Compos. Mater. 39 (16) (2005) 1487–1496. [8] Xuefeng Yao, Hsien-Yang Yeh, D. Zhou, et al., The structural characterization and properties of SiO2/epoxy nanocomposite, J. Compos. Mater. 40 (4) (2006) 371–381. [9] Guorong Song, Cunfu He, Xiaoling Wei, et al., Development of an ultrasonic system of measuring materials elastic constants for small samples, Chin. J. Sci. Instrum. 27 (9) (2006) 1012–1015. [10] D. Xiang, N.N. Hsu, G.V. Blessing, Simplified ultrasonic immersion technique for materials evaluation, Mater. Eval. 56 (7) (1998) 854–859.
Contents lists available at ScienceDirect
Measurement journal homepage: www.elsevier.com/locate/measurement
Measurement of elastic constants of limited-size piezoelectric ceramic sample by ultrasonic method Guorong Song *, Cunfu He, Zenghua Liu, Yao Huang, Bin Wu College of Mechanical Engineering & Applied Electronics Technology, Beijing University of Technology, No. 100# Pingle Yuan, Chaoyang District, Beijing 100124, China
a r t i c l e
i n f o
Article history: Received 17 October 2006 Received in revised form 9 February 2009 Accepted 23 April 2009 Available online 3 May 2009
Keywords: Elastic constant Limited-size sample Line-focus PVDF transducer Piezoelectric ceramic NDT
a b s t r a c t Measurement of material elastic constants for limited-size samples by ultrasonic method is described and validated. Based on acoustic microscope technology, the material elastic constants are determined by longitudinal wave and leak surface wave velocities simultaneously measured by developed ultrasonic system with a line-focus PVDF transducer. In this paper, the elastic constants of limited-size Cr2O3 doping on 0.2 PZN–0.8 PZT piezoelectric ceramic wafer sample are determined and the measurement errors are analyzed. The experimental results show this ultrasonic system can be used for measurement of material elastic constants for limited-size samples with high measurement precision, and the relative errors for Poisson’s ratio and Young’s module measurements are, respectively, less than 1% and 3%. It can satisfy the requirement of engineering and science research. Furthermore, it is suitable to measure elastic constants of both isotropic and anisotropic materials by ultrasonic method with help of developed ultrasonic system. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Piezoelectric ceramics are important advanced functional materials because they can achieve mutual energy transformation between mechanical energy and electric energy. The piezoelectric ceramic materials are widely applied to modern electric field as basic materials of many new electronic and micro-electronic components due to their stable function, high electromechanical performance, etc. In order to satisfy different application needs, extensive studies were widely conducted for improving the performance of piezoelectric ceramics [1–4]. However, due to the special processing technology, that is, directly molding, piezoelectric ceramics made in experiment have the characteristic of limited-size, therefore, it is very difficult to obtain the elastic constants or mechanical performance of this kind of materials. A lot of traditional testing methods and equipments are not applicable to the mechanical test-
* Corresponding author. Tel.: +86 010 67391601; fax: +86 010 67391617. E-mail address: [email protected] (G. Song). URL: http://www.bjut.edu.cn (G. Song). 0263-2241/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2009.04.008
ing on the micro-scale. In recent years, a number of mechanical testing methods are developed for measuring the mechanical properties of micro-structures [5–8]. In order to analyze the mechanical characteristics of developed materials, the representative testing methods are microtensile testing, nano-indentation, bulge testing, microbeam bending testing, etc. However, the methods are used mostly to measure the film or nano materials, and furthermore, some of the equipments are quite expensive. In this paper, the elastic constants of a 9.82 mm diameter and 1.24 mm thickness wafer piezoelectric doping ceramic sample is measured by the ultrasonic system with a line-focus PVDF transducer developed by ourselves. The measurement result is quite satisfactory with high precision, and this system is reliable for characterization and measurement of the elastic constants of materials, especially for the limited-size materials prepared at experiments. 2. Piezoelectric ceramic sample In this paper, chosen measurement sample is a piece of piezoelectric ceramic supplied by the authors of reference
G. Song et al. / Measurement 42 (2009) 1214–1219
1215
However, the piezoelectric ceramic is an elastic material and the elastic constants are its important parameters characterizing the elastic behaviors of piezoelectric material. Thus, developing a new, facility, low cost nondestructive testing method has an important meaning for these new especially materials. 3. Measuring system and method
Fig. 1. PZN–PZT piezoelectric ceramic sample.
[4]. It is 0.2Pb (Zn1/3Nb2/3)O3–0.8Pb (Ti0.5Zr0.5)O3 (PZN– PZT) piezoelectric ceramic modified with Cr2O3 prepared by low-temperature firing. As shown in Fig. 1, the sample is a round slice with small-size and its diameter is 9.82 mm and thickness is 1.24 mm. Lu et al. [4] studied the effects of Cr2O3 addition on the micro-structure of 0.2PZN–0.8PZT ceramics, and analyzed the electric properties and piezoelectric properties, including dielectric constant ec, electricity losing tan d, piezoelectric constant d33, electromechanical coupling coefficient kp and mechanical factor Qm, with the increase of Cr2O3 content. But the elastic performance of the piezoelectric ceramic sample was not analyzed due to the limited-size of sample by using conventional testing methods.
The ultrasonic measurement system of elastic constants for small-size samples is based on acoustic microscope technology [9]. The material elastic constants are determined by measuring longitudinal wave velocity CL and leak surface wave velocity VR simultaneously by the system with a line-focus PVDF transducer developed by ourselves. This is one of nondestructive testing methods. The experimental system makes up of a line-focus PVDF transducer, a four-axis moving exactly framework, a NI PXI bus embedded controller, an ultrasonic pulser/receiver (Parametrics 5800) and a computer, as shown in Fig. 2. The line-focus transducer is a lens-less concave cylindrical transducer with a thin piezoelectric polyvinylidene fluoride (PVDF) film. The line-focus PVDF transducer is 26.15 mm in focal length and a bandwidth centered around 6 MHz. When the measurement system works, the pulse/receiver firstly excites the PVDF film and then a transient pulsed wave is launched into a coupling fluid which is usually water and focuses the acoustic waves on a focal plane. When a sample is placed right under this transducer, the incident wave is reflected back to the line-focus PVDF film by the sample surface and a reflected electric signal is generated.
Fig. 2. Experimental system set-up.
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G. Song et al. / Measurement 42 (2009) 1214–1219
D
Line-Focus Transducer
Coupling Fluid
Sample
B
t
D
θR
R
R
Where D—Directly reflected wave R—Leaky surface wave B—Bottom reflected longitudinal wave Focal Plane
Fig. 3. Line-focus PVDF transducer and its measurement method.
During testing, firstly the sample surface is aligned with focal plane, the acoustic waves launched by line-focus PVDF transducer are reflected by the sample surface and received and converted into the echo wave voltage signal, V, by transducer. And then, as the transducer moves downward, the focal plane is deflected from the sample surface, also known as defocusing. The line-focus PVDF transducer and its measurement method are schematically depicted in Fig. 3. The wave beams incidenting at the Rayleigh angle are converted into surface waves. After propagating a certain distance along a sample surface, the surface waves leak into coupling fluid and re-radiate back to the PVDF transducer. In this case, the echo voltage signals consist of not only the directly reflected waves D and bottom reflected longitudinal waves B, but also the leaky surface waves R. This line-focus PVDF transducer can differentiate the arrival times of the directly reflected wave, the leaky surface wave, as well as other possible echo arrivals with considerable accuracy when the sample is moved inside the focal region of the transducer. When the defocus distance z is varied continuously, a series of echo voltage signals V(t) can be obtained. The leaky surface wave velocity can be determined by measuring the line relationship (slope m = Dz/Dt) of the time interval between the leaky surface wave arrival R and directly reflected wave arrival D with the defocusing distance z. When m is obtained, the leaky surface wave velocity can be evaluated from the following equation [10]:
"
2 #12
VW VR ¼ VW 1 1 2m
ð1Þ
where VW is the longitudinal wave velocity in the coupling fluid which is usually water, then here, VW is the wave velocity in water. The longitudinal wave velocity can be obtained by detecting the thickness of the sample and the time interval between the directly reflected waves arrival D and bottom reflected wave arrival B
2h CL ¼ tB tD
ð2Þ
where h is the thickness of the sample, tB tD is the time interval between the directly reflected waves D and bottom reflected longitudinal waves B. According to the elastic dynamics theory, the ultrasonic longitudinal wave velocity CL and surface wave velocity CR are depended on the density q and elastic constants of the material. Material elastic constants include Poisson’s ratio t, Young’s modulus E, shear modulus G and volume modulus K. Therefore, the ultrasonic longitudinal wave velocity CL and surface wave velocity CR can be expressed as
sffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 1t q ð1 þ tÞð1 2tÞ sffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:89 þ 1:12t E 1 CR ¼ 1þt q 2ð1 þ tÞ CL ¼
ð3Þ ð4Þ
According to (3) and (4), we can obtain
2:5088C 2L =C 2R 2 t3 þ 2:6432C 2L =C 2R 2 t2 0:4350C 2L =C 2R 2 t 0:7569C 2L =C 2R 2 ¼ 0
ð5Þ
It can be seen that Poisson’s ratio t can be resolved when both longitudinal wave and surface wave velocities are measured. Furthermore, other elastic constants can be determined as formula in the case that the density is known
C 2L ð1 þ tÞð1 2tÞq 1t E K¼ 3ð1 2tÞ E G¼ 2ð1 þ tÞ E¼
ð6Þ ð7Þ ð8Þ
4. Results and analysis 4.1. Measurement results To sum up all the considerations mentioned above, in this paper, the elastic constants of a 9.82 mm diameter and 1.24 mm thickness wafer piezoelectric doping ceramic
G. Song et al. / Measurement 42 (2009) 1214–1219
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Fig. 4. Experimental waveforms from the 0.2PZN–0.8PZT + Cr2O3 sample in the different defocus distance z.
sample mentioned above is measured by this system. In the testing, the defocus distance z is increased continuously with 0.2 ± 0.004 mm/step. After several time testing, the representative experimental waveforms at the different defocus distance z are shown in Fig. 4. Fig. 5 gives experimental results obtained by varying defocusing distance z. Fig. 5(a) is the curve of echo wave signal superposed by wave voltage offsetting at the different defocusing distance, and Fig. 5(b) shows the fitting curve of the directly reflected wave D, leaky surface wave R and bottom reflected longitudinal wave B with propagation time at different defocusing distances, respectively. When the thickness h of sample is 1.24 ± 0.02 mm, then, the slope m and the time interval tB tD between the di-
rectly reflected waves D and bottom reflected longitudinal waves B can be obtained
m ¼ 4703:5 m=s; t B t D ¼ 0:4724 ls When the longitudinal wave velocity Vw is 1480 m/s in water, then from (1) and (2) yields
V R ¼ 2748:7 m=s C L ¼ 5249:4 m=s Substituted into (5), hence
t ¼ 0:262 The volume density of ceramic is q = 7.4 103 kg/m3 through the Archimedean principle of buoyancy, and other elastic constants can be obtained from (6)–(8), respectively
Fig. 5. Experimental results obtained by varying defocusing distance z.
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Fig. 6. The measurement result for PZT specimen by using ultrasonic method at 22, 30 and 40 °C.
E ¼ 165:982 GPa G ¼ 65:762 GPa K ¼ 116:234 GPa
where C2
3
2
2:5088t 2:6432t þ0:4350tþ0:7569 Dt L ¼ tB tD ; b ¼ C L2 ; f ðt; bÞ ¼ 3ð2:5088b2Þ t2 þ2ð2:6432b2Þtð0:4350b2Þ ;
4.2. Measurement error analysis In this paper, the error for measuring elastic constants of limited-size piezoelectric ceramic by using ultrasonic measurement system depends mainly on the error of measuring longitudinal wave velocity and surface wave velocity. Furthermore, the error of measuring velocity is mainly depended on length error, as known the error of sample thickness h, the error of defocusing distance z and the error of time interval. For analyzing the error for measuring elastic constants t, E, assume that dq = 0. That is to say, the error caused by density can be neglected, and consider that the time for measuring elastic constants is very short every time, the error caused by the longitudinal wave velocity changes with the temperature in water can be ignored, then dVW = 0. According to Eqs. (1), (2), (5) and (6), the maximum of the error for measuring Poisson’s ratio and Young’s modulus can be obtained as following:
" # b jdhj h dt ¼ 4 f ðt; bÞ þ jdðDtL Þj C L DtL Dt 2L V 2W VW 1 b f ðt; bÞ 2 CR m 2m " 2 #32 VW z jdzj 1 1 þ 2 jdðDtÞj ð9Þ 2m Dt Dt " # qð1 þ tÞð1 2tÞ 2qtðt 2Þ CL þ b C L f ðt; bÞ dE ¼ 4 1t ð1 tÞ2 " # jdhj h 2qtðt 2Þ 2 þ jdðDtL Þj b CR Dt L Dt 2L ð1 tÞ2 " 2 #32 V 2W VW VW f ðt; bÞ 2 1 1 1 m 2m 2m z jdzj þ ð10Þ jdðDtÞj Dt Dt 2
R
dE; dh; dz; dðDt L Þ and d(Dt) are the errors of Young’s modulus E, sample thickness h, defocusing distance z, time intervals DtL and Dt, respectively. During the testing, jdðDtL Þj ¼ jdðDtÞj ¼ 0:4 ns, jdhj ¼ 0:02 mm and jdzj ¼ 0:004 mm are substituted into (9) and (10), then yields
dt ¼ 0:00247 dE ¼ 4:694 GPa
dt
t
¼ 0:876%
dE ¼ 2:801% E
The relative errors for Poisson’s ratio and Young’s module measurements are, respectively, less than 1% and 3%. It can satisfy the need of engineering and science research. 4.3. The influence of temperature for measurement results In the measurement error analysis of PZT specimen, the error of the velocity change of longitudinal wave induced by the temperature change of water is ignored. The experiment is designed for discussing if this error can be ignored. The PZT specimen is measured at its circumstances with different temperature (22, 30 and 40 °C). It is measured five times at each temperature, and the mean of obtained measurement values is recorded. The statistical results are shown in Fig. 6. It is shown that the velocities of longitudinal wave and leak surface waves become higher, measurement value of Young’s module increases, and the Poisson’s ratio changes inconspicuous when the temperature changes from 22 to 40 °C. When the temperature is changed from 22 to 40 °C, the measurement values of material Young’s module at 30 and 40 °C increase 0.76% and 3.64% than at 22 °C, respectively. Result shows the increased amplitude of the measurement values of material elastic constant is less than 4% when the temperature increases 45%. The experimental results show the error induced by the temperature change of water can be ignored because the
G. Song et al. / Measurement 42 (2009) 1214–1219
specimen is measured within a short time which is only about 1 min.
5. Conclusion Measurement of material elastic constants by ultrasonic method has the advantage of requiring small-size sample and nondestructive testing. It can be applied to some special situations that conventional methods are not suitable. Especially, it is much suitable for measuring new materials of directly molding by special technologies, small-size, high cost, etc. Because the total measurement time is short, the influence of measurement results brought by the temperature change of water is very small, it can be ignored. The error analysis results show that this measurement system has high measurement precision, the relative errors for Poisson’s ratio and Young’s module measurements are, respectively, less than 1% and 3%. It can satisfy the need of engineering and science research. This measurement method and system can provide powerful technological assistance for characterizing and measuring material elastic constants of limited-size samples obtained from the experiment of material preparation. And besides, the line-focus PVDF transducer can launch surface acoustic waves in a specific direction, which is perpendicular to the focus line; it can be further applied for study of the anisotropic material.
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Acknowledgements The work presented in this paper is supported by the National Natural Science Foundation of China (Nos. 10372009 and 10827201). References [1] S.H. Lee, C.B. Yoon, S.B. Seo, et al., Effect of lanthanum on the piezoelectric properties of lead zirconate titanate-lead zinc niobate ceramics, J. Mater. Res. 18 (8) (2003) 1701–1765. [2] X.Y. Sun, J. Chen, R.B. Yu, et al., BiScO3 doped (Na0.5K0.5)NbO3 leadfree piezoelectric ceramics, J. Am. Ceram. Soc. 92 (1) (2009) 130–132. [3] Y.D. Hou, P.X. Lu, M.K. Zhu, et al., Effect of Cr2O3 addition on the structure and electrical properties of Pb((Zn1/3Nb2/3)0.20(Zr0.50Ti0.50)0.80)O3 ceramics, Mater. Sci. Eng. B 116 (1) (2005) 104–108. [4] Pengxian Lu, Yudong Hou, Mankang Zhu, Hui. Yan, Effects of Cr2O3 doping on microstructure and piezoelectric properties of 0.2PZN– 0.8PZT ceramics, J. Funct. Mater. Dev. 11 (3) (2005) 303–307. [5] Caijun Su, Hao Wu, Guo Zhanshe, et al., Mechanical testing methods of micro structures, Chin. J. Exp. Mech. 20 (3) (2005) 441–447. [6] Huimin Xie, Satoshi Kishimoto, Anand Asundi, Norio Shinya, In-plane deformation measurement using the atomic force microscope moiré method, Nanotechnology 11 (1) (2000) 24–29. [7] Xuefeng Yao, Hsien-Yang Yeh, H.P. Zhao, Dynamic response and fracture characterization of polymer/clay nanocomposites with mode-I crack, J. Compos. Mater. 39 (16) (2005) 1487–1496. [8] Xuefeng Yao, Hsien-Yang Yeh, D. Zhou, et al., The structural characterization and properties of SiO2/epoxy nanocomposite, J. Compos. Mater. 40 (4) (2006) 371–381. [9] Guorong Song, Cunfu He, Xiaoling Wei, et al., Development of an ultrasonic system of measuring materials elastic constants for small samples, Chin. J. Sci. Instrum. 27 (9) (2006) 1012–1015. [10] D. Xiang, N.N. Hsu, G.V. Blessing, Simplified ultrasonic immersion technique for materials evaluation, Mater. Eval. 56 (7) (1998) 854–859.