The Application of The Variation Method For The Estimation of The Viscosity of Solid

Boris D.Zaitsev1, Guy Feuillard2, and Oumar Diallo2

1-Saratov Branch of Institute of Radio Engineering and Electronics of RAS, 38 Zelyonaya str., Saratov, 410019, Russia 2-Universite Francois Rabelais de Tours, Rue de la Chocolaterie, BP 3410, 41034 Blois Cedex France,

e-mail: zai-boris@yandex.ru

Abstract—During long time for the analysis of the mechanical oscillations of the solid samples with finite sizes and for determination of their material constants the variational method was widely used. However the calculations have been performed without considering the loss of the energy of the mechanical oscillations. In this paper the experimental study of the frequency dependence of the real and imaginary parts of the electrical admittance of the piezoelectric samples having the cube form was carried out in order to determine the resonant frequencies of the mechanical oscillations and estimate their Q-factors. Then we performed the theoretical analysis of these oscillations by the variation method and made the comparison of the theoretical data with experimental one. This allowed us to find the coefficients of viscosity, which was considered as only source of the mechanical loss for the samples of piezoceramics of PZ27, PMN34.5%PT, and PZ26.

Keywords – variation method; Rayleigh-Ritz approach; frequency dependencies of the electrical admittance of the piezoelectric sample; complex elastic constants; Q-factor; resonant frequency; viscosity coefficients; decrement of attenuation; piezoelectric ceramics; Legendre’s polynomial.

I. INTRODUCTION

By now the variation methods are widely used for the analysis of the mechanical oscillations of the solid samples with finite sizes and for determination of their material constants [1-8]. Their basis is Rayleigh-Ritz method which suggests the representation of the sought functions (the mechanical displacement and electrical potential) as a linear combination of the basic functions with unknown coefficients. By the use of this method one can determine the frequencies of the natural mechanical oscillations of the samples and the distribution of the amplitudes of the mechanical displacement and electrical potential in the sample under study. The possibilities of the variational method have been shown for the samples of the parallelepiped forms [2,4-8] and of the circle disks [1,3]. In the first and second cases as the basic functions Legendre polynomial and Bassel functions, respectively, were used. The possibility of considering the presence of the electrodes on some sides of the sample was shown and the expressions for the cases when electrodes are electrically disconnected and shorted were obtained. The possibility of

finding all elastic, piezoelectric and dielectric constants of the samples were demonstrated on the examples of olivine [5], quartz [6], and monocrystal PZN-12%PT [8]. However in all papers the calculations have been performed without considering the loss of the energy of the mechanical oscillations because till recently it has been considered that the Hamilton’s variational principle cannot be formulated for dissipative systems [9,10]. But by now the possibility of the formulation of such principle has been demonstrated for the dissipative systems, in particular for dissipative hydrodynamics [10]. In the present paper the first attempt to apply the variational method for the estimation of the viscosity coefficients of the solid was undertaken. For this at first theexperimental study of the frequency dependence of the real and imaginary parts of the electrical admittance of the piezoelectric samples having the cube form was carried out in order to determine the resonant frequencies of the mechanical oscillations and estimate their Q-factors. Then we performed the theoretical analysis of these oscillations by the variational method and made the comparison of the theoretical data with experimental one. This allowed us to find the coefficients of viscosity, which was considered as only source of the mechanical loss.

II. THE EXPERIMENT Thus we carried out the measurement of the frequency

dependencies of real and imaginary parts of the electrical admittance of the samples of various types of piezoceramics with the help of the impedance analyzer 4395A. For this the samples of cube form with two thin metal electrodes on the edges which are perpendicular to the polar axis X3 were placed into the special arrangement providing the electrical contact by means the pressure electrodes. The special micrometer device allowed to ensure the same pressing force of the electrodes independently on the size of the sample in order to keep all calibration sets during the measurements. We carried out the experiments with numerous samples and types of the piezoceramics.

Fig. 1 presents the frequency dependencies of the real parts of the electrical admittance of the samples of ceramics PZ27 (a), PMN34.5%PT (b), and PZ26 (c). These dependencies allowed to determine the resonant frequencies of the mechanical oscillations and their values of Q –factor at the level 0.707 respectively to the maximum value. This data and also the sizes of the samples are presented in the Table 1.

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10.1109/ULTSYM.2012.0436

TABLE 1

Material Sizes, mm3

Resonant frequency,

kHz Q-factor

PZ27 4×4×4 299462.5 100 PMN34.5%PT 10×10×10 126668.75 579

PZ26 15×15×15 90697.969 2738

0.000

0.001

0.002

0.003

0.004

280 290 300 310 320

Frequency (kHz)

Re

G (

S)

0.000

0.010

0.020

0.030

0.040

124 125 126 127 128 129

Frequency (kHz)

Re

G (S

)

0

0.02

0.04

0.06

90.2 90.4 90.6 90.8 91 91.2

Frequency (KHz)

Re

G (S

)

Fig. 1. The real part of the electrical admittance of the ceramic samples of PZ27 (a), PMN34.5%PT (b), and PZ26 (c)

III. THE THEORETICAL ANALYSIS AND COMPARISONWITH EXPERIMENT

Then for the theoretical analysis the following approach was used. As it is well known [11] the taking into account the attenuation of the mechanical oscillations and acoustic waves in the theoretical analysis may be conducted by use of the concept of viscosity. The viscosity is characterized by the viscosity coefficient �ijkl, which is the tensor of fourth rank and has the same symmetry as the corresponding modules of elasticity cijkl. In this case for the harmonic oscillation on the frequency f the elastic constant becomes complex in the constitutive equation for the mechanical stress Tij and the imaginary part of the elastic

( )

2k k

ij ijkl ijkl mij ml l

kijkl ijkl mij m

l

u uT c e Ex t x

uc i e Ex

η

ωη

∂ ∂= + − =∂ ∂ ∂

∂+ −∂

constant is equal to the product of the angular frequency and viscosity coefficient. Here uk is the mechanical displacement, tis the time, xl is the spatial coordinate, emij is the piezoelectric constant, Em is the electric field intensity, i is the imaginary unit, �=2�f is the angular frequency. It is obviously that the variational problem based on the Rayleigh-Ritz method becomes more complicated. But the performed experiments have shown that the samples of ceramics on the kHz frequencies are characterized by high values of Q-factor. In this case for the theoretical analysis one can use the perturbation method and solve the problem by step to step method. At first by using the known material constants of the sample one can determine the resonant frequencies and shape of oscillations by neglecting the viscosity. Then after choosing the certain value of the resonant frequency and using the known values of viscosity one can introduce the complex elastic modules and repeat the procedure of the calculation. At that the coefficients in the expansion of the sought functions and the basic functions may be considered as real values. The complexity appears only in matrix equations due to the complexity of the elastic modules and frequency. Thus by using the calculation program for analysis of the samples without viscosity one can find the complex values of the resonant frequencies. At that the real part has the previous physical meaning and the imaginary part gives the decrement of the attenuation of the oscillations in time at any point of the sample. But in our case the coefficients of the viscosity turned out to be unknown. Therefore we made the attempt to estimate these coefficients by using the experimental data. At first we introduced in the calculation the values of the imaginary parts of the elastic modules which are proportional to the corresponding real parts. We used the algorithm of calculation based on the representation of the sought functions as linear combinations of Legendre’s polynomial with considering the symmetry of the sample and material [8]. Then we found the

b

a

c

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fractional change of the resonant frequency and attenuation on one period of oscillations as functions of the imaginary part of the elastic constant c33. These dependencies for the aforementioned materials are presented in Figs. 2 and 3, respectively. One can see that with increase of the imaginary part of elastic constant (and respectively viscosity) the resonant frequency insignificantly increases. This fact has the clear physical meaning because as it is well known with increase in the viscosity of the material the phase velocity of acoustic wave increases [11]. As for the attenuation on one oscillation period as it has been predicted [11] it increases with the increase of the viscosity coefficient and respectively

1.E-12

1.E-08

1.E-04

1.E+00

1.E+05 1.E+07 1.E+09 1.E+11

1.E-12

1.E-08

1.E-04

1.E+00

1.E+05 1.E+07 1.E+09 1.E+11

1.E-12

1.E-08

1.E-04

1.E+00

1.E+05 1.E+07 1.E+09 1.E+11

Fig. 2. The dependencies of the fractional change of the resonant frequency on the imaginary part of the elastic constant Im �33 for the piezoceramic samples of PZ27 (a), PMN34.5%PT (b) and PZ26 (c).

imaginary part of elastic constant in accordance with the linear law (Fig. 3). Fig. 4 shows the dependence of the Q-factor of the mechanical oscillations on the imaginary part of constant �33. One can see that with the increase in the imaginary part of elastic constant Q-factor decreases in accordance with linear law and all dependencies are practically the same for all materials under study. These dependencies allowed to estimate the corresponding values of the imaginary parts of c33 and corresponding values of the viscosity coefficients by using the experimental values of the Q-factor presented in the table 1. This data is presented in the table 2 for the aforementioned materials. This table also shows the attenuation of the

1.E-05

1.E-03

1.E-01

1.E+01

1.E+05 1.E+07 1.E+09 1.E+11

1.E-05

1.E-03

1.E-01

1.E+01

1.E+05 1.E+07 1.E+09 1.E+11

1.E-05

1.E-03

1.E-01

1.E+01

1.E+05 1.E+07 1.E+09 1.E+11

Fig. 3. The dependencies of the attenuation of oscillations on one period on the imaginary part of the elastic constant Im �33 for the

piezoceramic samples of PZ27 (a), PMN34.5%PT (b) and PZ26 (c)

(f-f 0)

/f 0(%

)

Im C33 (N/m2)

a

(f-f 0)

/f 0(%

)

Im C33 (N/m2)

b

(f-f 0)

/f 0(%

)

Im C33 (N/m2)

c

Atte

nuat

ion

(dB

/per

iod)

Im C33 (N/m2)

c

Atte

nuat

ion

(dB

/per

iod)

Im C33 (N/m2)

b

Atte

nuat

ion

(dB

/per

iod)

Im C33 (N/m2)

a

1740 2012 IEEE International Ultrasonics Symposium Proceedings

amplitude on one period of oscillations and the fractional difference between the theoretical and experimental values of the resonant frequency (�f/f0). One can see that these values do not exceed the usual spread of the values of the material constants (±3%) due to the technological peculiarities of their obtaining and processes of the aging materials.

1.E+00

1.E+02

1.E+04

1.E+06

1.E+05 1.E+07 1.E+09 1.E+11a

1.E+00

1.E+02

1.E+04

1.E+06

1.E+05 1.E+07 1.E+09 1.E+11b

1.E+00

1.E+02

1.E+04

1.E+06

1.E+05 1.E+07 1.E+09 1.E+11c

TABLE 2

Material Im c33, Pa �33, Pa s �f/f0,% Atten., dB/period

PZ27 1.13×109 616 -2.5 0.27 PMN34.5%PT 2.5×108 315 0.17 0.04

PZ26 6.0×107 104 0.9 0.01

III. CONCLUSIONThus the possibility of the estimation of the viscosity

coefficients of solid by the variational method was shown for the piezoelectric samples of the cube form. This estimation was carried out by the comparison of the experimental value of the Q-factor for the lower frequency resonance with the theoretical dependence of the Q-factor on viscosity coefficient. The experimental determination of the Q-factor was performed by using the frequency dependence of the sample admittance measured with the help of meter of the LCR parameters. It should be noted that in the paper we made the not exactly suggestion that the imaginary parts of the elastic constants are proportional to their real ones. This allowed to reduce five independent components of the viscosity tensor to one and significantly to simplify the problem. However for more exact determination of all components of the viscosity tensor one can use the method described in [8] by including in unknown material constants all independent components of the viscosity tensor.

ACKNOWLEDGMENT

The work is financially supported by the Grants RFBR 10-02-01313a and 12-02-01057a.

REFERENCES[1] E.E.P. Nisse, “Variational method for electroelastic vibration analysis”,

IEEE Trans. on Ultrason., Ferroel., and Freq. Contr., vol. SU-14, no.14, pp. 153-160, 1967.

[2] R. Holland, “Resonant properties of piezoelectric ceramics rectangular parallelepipeds”, J. Acoust. Soc. Amer., vol.43, pp.988-997, 1967.

[3] R. Holland R. and E.E.P. Nisse, “Variational evaluation of admittances of multielectroded three –dimensional piezoelectric structures”, IEEE Trans. on Ultrason., Ferroel., and Freq. Contr., vol. SU-15, no.2, pp. 119-132, 1967.

[4] H.H. Demarest Jr., “Cube resonance method to determine the elastic constants of solids”, J. Acoust. Soc. Amer., vol.49, no. 3, pp.768-775, 1971.

[5] I. Ohno, “Free vibration of a rectangular parallelepiped crystal and its application to determination of elastic constants of orthorombic crystals”, J. Phys. Earth, vol.24, pp. 355-379, 1976.

[6] I. Ohno, “Rectangular parallelepiped resonance method for piezoelectric crystals and elastic constants of alpha-quartz”, Phys. Chem. Minerals, vol.17, pp. 371-378, 1990.

[7] H.H. Demarest Jr, “Cube resonance method to determine the elastic constants of solids”, J. Acoust. Soc. Amer., vol.49, pp.2154-2161,1971.

[8] T. Delaunay, E. Le Clezio, M. Guennou, H. Dammark, M.P. Thi, and G. Feuillard, “Full tensorial characterization of PZN-12%PT single crystal by resonant ultrasound spectroscopy”, IEEE Trans. on Ultrason., Ferroel., and Freq. Contr., vol. 55, no. 2, pp. 476-488, 2008.

[9] L.D. Landau and E.M. Lifshitz. Theoretical Physics, vol. 5, Statistical Physics, Moscow: Nauka, 1964, 570P.

[10] G.A. Maximov, “Generalized variational principle for dissipative hydrodynamics: shear viscosity from angular momentum relaxation in the hydrodynamical description of continuum mechanics” in book “Hydrodynamics –Advanced Topics”, ISBN 978-953-307-596-9, edited by Harry Edmar Schulz, Andre Luiz Andrade Simoes, and Raquel Jahara Lobosco, Publisher In Tech, December, 2011, pp. 35-50.

[11] D. Royer and E. Dieulesaint. Elastic waves in solids I. Free and guided propagation. Berlin,: Springer, 2000, 374P.

Fig. 4. The dependencies of Q – factor on the imaginary part of the elastic constant Im �33 for the piezoceramic samples of

PZ27 (a), PMN34.5%PT (b), and PZ26 (c).

Q- f

acto

r

Im C33 (N/m2)

Q-f

acto

rQ

-fac

tor

1741 2012 IEEE International Ultrasonics Symposium Proceedings