ACOUSTIC ATTENUATION IN SOLIDS - … · acoustic field, the emitted wave is completely reflected and the average acoustic potential on surface S T of the transducer functioning now

Attenuation and the Nonlinear Behavior of Solids

M. A. Breazeale

National Center for Physical AcousticsUniversity of Mississippi

University, MS 38677USA

We have measured nonlinearity of single crystals, samples of a pseudocrystalline material, amorphous solids, a composite,piezoelectric crystals and polarizable materials. As a result, we observe that the nonlinearity parameter (the ratio of thecoefficient of the nonlinear term to that of the linear term in the wave equation) of single crystals typically is between 3 and 15,about the same as that for liquids. The extremely high nonlinearity parameters measured in other samples often depend onfactors other than the elastic nonlinearity. In samples with high attenuation, care must be taken in the measurement. Theseobservations, and others, are clarifying our understanding of the nonlinear behavior of materials, and since much of theattenuation resulting from nonlinearity can be controlled by using small amplitudes, we can measure a new and useful acousticalparameter.

INTRODUCTION

Nonlinear behavior of solids can lead to attenuation, butattenuation is not a good reason for investigating nonlinearbehavior—yet. Even at very small amplitudes thenonlinearity of solids can be observed. Our measurementof nonlinearity of solids began in 1965 [1]. Since then wehave presented results for single crystals of copper, fusedsilica, germanium, silicon, CsCdF3 and. KZnF3, quartz,LiNbO3, KMnF3, NaCl, Gallium Arsenide, and PZTceramics. Recently we measured the nonlinear behavior ofa graphite-epoxy composite. We have presented a reviewof the nonlinear properties of diamond lattice solids [2], areview of solid state nonlinearity [3], and a review of thetechniques for measurement of the nonlinear properties ofsolids [4]. All of these publications show the effect oftemperature change, and together they summarize ourexperience with measurement of the nonlinear properties ofsolids.

THEORY

In our experiments the effect of nonlinearity is found to bemuch larger than that of attenuation. Thus, it is appropriateto ignore attenuation terms in the differential equationdescribing the process. If the attenuation is small enough,the appropriate nonlinear differential equation for a puremode direction is

�o�2U�t2 �K2

�2U�a2 � 3K2 �K3� �

�U�a

�2U�a2 �... (1)

This equation has a solution of the form

U � A sin� �

�a Ak� �2

8cos2� � ... (2)

where the nonlinearity parameter

� �

3K 2 � K 3

K2

(3)

appears, and � � ka ��t . This form emphasizes thefact that both the fundamental amplitude A and the second

harmonic amplitude �a Ak� �2

8 must be measured.

The quantity �, the ratio of the coefficient of the nonlinearterm to the coefficient of the linear term, is a measure ofthe nonlinearity of the solid. This is the quantity we seek tomeasure. Expressions for K2 and K3 for cubic crystals aregiven in Table l, where we also have given values for K2and K3 that would correspond to propagation in liquids orgases.

EXPERIMENT

Measurement of the nonlinearity of solids can beaccomplished by use of a capacitive detector, which iscapable of making absolute measurements. Typically, wemeasure at a fundamental frequency of 30 MHz, althoughwe have successfully covered the range between 5 and 40MHz. Since the amplitudes are between 0.01 and lAngstrom, the capacitive gap mustbe quite small in order that large fields (and hence greatsensitivity) can exist in the capacitive detector withoutdangerously high voltages. Typically, we work with a gapspacing of 8 microns. The smallest amplitude measured todate is 0.0001 Angstrom. Although the capacitive detectoris capable of measurement at cryogenic temperatures,variations with temperature we have measured indicate thatthis may not be necessary.

Table 1. Values of K2 and K3 for different substancesSubstance K2 K3

CubicCrystal [100] C11 C111 [110] 1/2[C11+C12+2C44] 1/4[C111+3C112+12C166] [111] 1/3[C11+2C12+4C44] 1/9[C111+6C112+12C144

+24C166+2C123+16C456]Fluids Liquids �oco

2 -�oco2(B/A+5)

Gases �oco2 -�oco

2(�+4)

RESULTS

Some of our measured results are presented in Figures 1,2, and 3. It is to be observed that the nonlinearityparameter of cubic crystals is not strongly temperaturedependent. In addition to these data we recently havemeasured a graphite-epoxy composite and single crystals oflithium niobate. In the graphite-epoxy composite we findthat the nonlinearity parameter along the fibers is of theorder of 5, while that perpendicular to the fibers is closer to10. In both cases there is a frequency dependence. In thelithium niobate we find a small nonlinearity parameter—1.5 and the complicating factor of an electromagneticevanescent wave along the piezoelectric direction.

Figure 1. Measured [100] Nonlinearity Parameters

Figure 2.Measured [110] Nonlinearity Parameters

Figure 3. Measured [111] Nonlinearity Parameters

REFERENCES

1. M. A. Breazeale and Joseph Ford, Ultrasonic studies ofthe nonlinear behavior of solids, J. Appl. Phys. 36, 3486-3490 (1965).2. M. A. Breazeale, Ultrasonic studies of the nonlinearproperties of diamond lattice solids at low temperatures,Acta Physica Slovaca 37, 203-214 (1987).3. M. A. Breazeale and Jacob Philip, Determination ofthird-order elastic constants from harmonic generationmeasurements, in Physical Acoustics edited by W. P.Mason and R. N. Thurston, Vol. XVII, Academic Press,New York, pp. 1-60 (1984).4. M. A. Breazeale, Third order elastic constants of cubiccrystals, in Handbook of Elastic Properties of Solids,Liquids, and Gases edited by M. Levy, H. E. Bass, and R.R. Stern, Vol. I, Academic Press, San Diego, pp. 489-510(2001).

A Pulsed Technique for the Measurement of the UltrasonicAttenuation

H. Djelouah (*), N. Bouaoua, A. Alia

Faculté des Sciences – Physique –Université des Sciences et de la Technologie Houari BoumedienneB.P. 32 El Allia, Bab Ezzouar, 16111 Alger, Algeria E-Mail : (*) [email protected]

In viscous fluid media the attenuation α of the acoustic waves is proportional to the square of the frequency f ( )α β= f 2 . If a rigid targetof small surface is placed in the acoustic field, the emitted wave is completely reflected and the detected pressure consists of three impulsesarriving at different instants. The first corresponds to the direct or plane wave. Here, we take the advantageous of the transient mode to selectthis impulse which is time resolved from the two others in order to deduce the β coefficient without using diffraction correcting terms.

INTRODUCTION

In the method that will be described in this papaer,the measurement of the attenuation of ultrasonic wavesin viscous media is done by exploiting the radiatedultrasonic plane wave, reflected by a small size targetplaced near the transmitter, before the diffraction (edgewave) appears.

THEORY

In viscous fluid media the attenuation of the acousticwaves is proportional to the square of the frequency f:

α β= f 2 and the equation of propagation for theacoustic potential is [1]:

01 22

2

22 =∇

∂∂−

∂∂+∇− φβφφ

ttC (1)

When the acoustic wave is radiated by a plane pistonembedded in a rigid baffle and for small values of βcoefficient, the acoustic potential φ, at a point M, isgiven by [1]:

),()(),( tMtvtM iφφ ⊗= (2)

where

dStMS RCt

e tC

CtR

∫∫−−

=βπ

βφ2

3

22

2][

)2(),( (3)

represents the impulse response for the potential andv(t) the velocity of the surface source, dS being anelementary surface centered around the point M0

belonging to the source and R MM= 0

If the point M is on the axis of a circular transducer ofradius a, the impulse response is :

−−

−+=tC

CtzErftCCtazErfCtzi 22

22

222),(

ββπφ

(4)

If a rigid target of small surface SC is placed in theacoustic field, the emitted wave is completely reflected andthe average acoustic potential on surface ST of thetransducer functioning now as a receiver is given by:

ii

T

C

zSStvt φφφ ⊗

∂∂⊗= )()( (5)

For a weak attenuation, the detected average pressure canbe written [1]:

ii

T

C

ztv

SSCp φφρ ⊗

∂∂⊗

∂∂= (6)

Figure 1 represents the detected pressure. This pressureconsists of three impulses arriving at instants centeredrespectively on

tz

Ct

z a zc

and tz a

c02

12 2

2 22 2

= =+ +

=+

, .

t0 corresponds to a wave coming from the projection ofthe target on the surface of the transducer (direct wave orplane wave). t1 represents the simultaneous arrival of twowaves corresponding to a propagation from the center ofthe transducer towards the edge of the transducer andreciprocally. Finally t2 corresponds to the arrival of a waveissuing from the transducer edge, reflected by the targettowards the transducer edge.

Figure 1 : Simulation of the detected pressure in the caseof a circular transducer with a=10mm, and a circular targetwith a radius of 0.4mm; C=1.5mm/µs, z= 20mm, β=0.001mm-1.MHz -2.

Because of the dispersion these arrival timescorrespond to mean values since they are preceded byprecursors [1 ]. For two different positions z1 and z2, letus select the first impulse which corresponds to thedirect wave and calculate the attenuation coefficientdefined by

−=

)f;()f;(Log

)(21α

11

22

12 zPzP

zz (7)

where P2(z2; f) and P1 (z1; f) are respectively thespectrum of the first impulses detected when the targetis at a distance z2 and z1 on the transducer axis. It hasbeen checked that the graph of α versus f 2 is a linewhose slope is equal to β .

PRINCIPLE OF THE METHOD

An ultrasonic broad band transducer functioning intransmitting-receiving mode is used. The resultsobtained with a transducer with a diameter equal to 20cm and a nominal frequency equal to 2.25 Mhz, andwith a target which radius is 0.4mm are represented byFigure 2.

Figure 2 Echos from the basis of a rod made of duraluminwith a radius of 0.4mm, immersed in glycerine(C=1.92mm/µs)

Figure 3 : Plot of α versus f2. ( ___ ) : obtained from Equation(7) ; (- - -): α=β f2.

The positioning of the target on the axis of the transduceris carried out by seeking the position for which edge wavesare of maximum amplitude.

The signal processing consists in reading the signalsstored at two different positions, then to use the relation (7)to plot α versus f2 . The measurement of the slope of thelinear interpolation of these results allows to calculate β.Figure 2 represents the results obtained with glycerine at atemperature T=16.8°C. The obtained value β=0.0032 mm-

1.MHz -2 , is in accordance with the results obtained by otherauthors [2 ].

The principal advantage of this method is that themeasurements are carried out in the field close to a planetransducer, thus avoiding the encountered problems ofalignment when the measurements are made in the far field.Nevertheless, if some precautions are not taken, the resultscan be particularly disappointing; indeed it is essential touse a wide-band transmitter with a large radius, and asmall-size plane target in order to separate in time thevarious impulses.

CONCLUSION

The attenuation coefficient of the acoustic waves in theviscous fluids can be measured by the proposed method byavoiding any corrective diffraction term. This method canbe used advantageously by replacing the reflectors by abroad band hydrophone of small size placed in the fieldnear to the transmitter. In this case, the signal pressure isonly made up of two pulses [ 3].

REFERENCES1. M.Deville et al., Proceedings of IEEE Ultrasonics Symposium, 1985,683-687.

2. Kinsler et al., Fundamental of acoustics, Wiley & Sons.

3. H.Djelouah, et al.,Ultrasonics,27,80-85 (1989).

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

frequency2 (MHz2)

alph

a (1

/mm

)

0 2 4 6 8 10-0.5

-0.3

-0.1

0.1

0.3

0.5

volts

time (µs)

t 0 t1 t2

20 24 28-150

-50

50

150

time (µs)

pres

sure

(Pa)

t 0 t 1 t 2

PRECISION IMPROVEMENT OF ATTENUATIONDETERMINATION IN SOLIDS BY MEANS OF

STATISTICAL FILTRATION

I.Rugina, C.Rugina

Institute of Solid Mechanics of the Romanian Academy, C.Mille 15, Bucharest, Romania

Abstract: The paper presents a method to improve the attenuation determination by means of so-called “statistical filtering”.This kind of filtration gives good results in the determination of attenuation introduced by different solid materials in thepropagation of ultrasonic waves. The method uses different types of correlation theory functions to improve the ratio betweenthe signal containing the needed information on the attenuation, and the noise or perturbation signal. In his way, one canobtain a substantial diminution of the errors in the determination of the needed characteristics obtained by the signal analysis,especially in ultrasonic systems.

INTRODUCTION

To improve the ultrasonic system characteristicdetermination [2], [3], some classic signal filteringmethods are usually used. These methods cannot beso easy used in all cases. If the signal represents aquasi-stationary or a non-stationary process, othermethods can be used. Even if the signal represents astationary process, some good results can be alsoobtained, in some cases by other methods,especially by means of so-called “statisticalfiltration”. This method uses different types ofcorrelation theory functions [1], to improve theratio between the signal containing the neededinformation, and the noise or perturbation signal. Inthis way, one can obtain a substantial diminution ofthe errors in the determination of the neededcharacteristics obtained by the signal analysis[4],[5].

In the high power ultrasonic systems, thestatistical filtration can be sometimes used with agood efficiency. We will show the possibility to usethis method in the case of the attenuationdetermination. This determination is generallyperformed by means of a sine signal, decreasing intime, which represents the variation of the vibrationamplitude at the end of a resonator constituted bythe analyzed dissipative solid material. Itslogarithmic decrement, given by two consecutivemaximum values of the sine signal, determines theneeded attenuation value. Unfortunately, somerandom perturbations given by the internal noise ofelectronic measuring system or by randomvibrations of the ultrasonic system componentsunder the control, are almost already superposed onthe decreasing sine signal Using the correlationfunction of this signal, one can improve the ratiobetween the signal containing the neededinformation on the attenuation, and the noise orperturbation signal.

ATTENUATIONDETERMINATION BY MEANS OFAUTOCORRELATION FUNCTION

If we consider an attenuated sine signal, havingsuperposed some random perturbations, we candescribe it in the following form:

( ) ( )ttsinetf t ε+ω= α− (1)

whereα - the attenuation coefficientω - angular frequencyε(t) - a small random perturbationThe attenuation coefficient α, can be obtained

by means of two consecutive maximum values ofeq.(1), at the moments tn and tn+1, for ε(t)= 0. Wewill have:

n1n

1nn

tt

)]t(fln[)]t(fln[

−−

=α+

+ (2)

But the random perturbation ε(t), affects thecorrect values f(tn) and f(tn+1), giving some errorswhich can be reduced by means of correlationfunction.

The autocorelation function R(τ) of f(t), can bewell approximated by:

( ) ( )[ ] ( ) ( ) ( )[ ]dtttsinettsineT

1R

T

0

tt∫ τ+ε+τ+ω⋅ε+ω=τ τ+α−α− (3)

By neglecting some terms, having very smallvalues, we can write:

( ) ( )∫ τ+ω⋅ω≅τ α−ατ− T

0

t dttsintsineT1

eR (4)

After some routine calculation, we find:

( ) ( )[ ]ϕ

Ι⋅ϕ+ωτ≅τ ατ−

cosTsineR 2 (5)

with: 2

1

I

Iarctg=ϕ (6)

and:

( ) ( )[ ]TcosTsinTsin224

eI 2

22

T2

1 ωω+ωαωα−ωα+ωα

=α−

(7)

( ) ( )[ ]T2cosT2sin4

eI

22

T2

2 ωω+ωα−ωα+ω

=α−

(8)

We can see that eq. (1) and eq. (5) represent thesame exponential function, which modulate a sinefunction having the same period, given by the sameangular frequency ω. So, their logarithmicdecrement α, will be almost the same, but the factorI2/Tcosϕ, depending also from α, can introducesome small errors.

The most important fact is that the errors, givenby the random perturbations superposed on thesignal, can be considerably reduced. Fig.1a showsan attenuated sine signal, with some randomperturbations, superposed, expressed by eq. (1).Fig.1b shows its autocorrelation function. One cansee that the random perturbations are pretty wellfiltrated by autocorrelation function, so its errorcomponent can be almost eliminated.

0 0.1 0.2 0.3 0.4

0

1

a) unfiltered signal

0 0.1 0.2 0.3 0.4

0

0.01

b) filtered signal by autocorrelation function

Fig.1

This method can be successfully used toimprove the precision in attenuation determinationobtained for ultrasonic system components. In thiscase the logarithmic decrement given by twoconsecutive maximum values, of the modulatedsine function, having small differences betweenthem, can be easily affected even by small randomperturbation, and so the statistical filtration bymeans of autocorrelation function can improveconsiderably the precision in attenuationdetermination.

REMARKS AND CONCLUSIONS

The filtration of the signals, containing theneeded informations about the attenuation, bymeans of correlation functions can eliminate animportant part of errors given by randomperturbations of electronic measuring system or byrandom vibrations of the ultrasonic systemcomponents, which are under the control.

The results obtained for the attenuationcoefficient α, given by eq.(2), in the case of somecomponents of a high power ultrasonic transducer,constituted from aluminum, confirmed that wecould obtain a precision improvement from 4% to1.2%. The precision improvement was appreciatedby the reduction of the variance, for the α value,obtained for different two pairs of the moments tn

and tn+1, corresponding of two consecutivemaximum values of the analyzed signal.

Even if the statistical filtering of the signal canintroduce itself some small errors, given by somemodifications of the signal characteristics impliedin the values under the control, however in thiscase, we can considerably reduce the total error,and so we can obtain an important improvement ofthe precision in the needed determination.

ACKNOWLEDGEMENTSThe support for this work by The National

Agency for Science, Technology and Innovation(ANSTI) Bucharest, Grant nr.5208/1999-2001-A6,is gratefully acknowledged.

REFERENCES

1. J.S. Bendat, A.G. Piersol, “Engineering Applicationsof Correlation and Spectral Analysis”, JohnWiley&Sons, 1980.

2. R. Truell, C. Elbaum, B. Chick. Ultrasonic Methodsin Solid State Physics, Academic Press, New Yorkand London, 1969.

3. A. Puskar, “The use of high-intensity ultrasonics”,Elsevier Co., Amsterdam, 1982.

4. I. Rugina, V. Chiroiu, C. Rugina, C. Chiroiu,“Precision improvement of ultrasonic systemcharacteristics determination by correlationmethods”, The Annual Symposium of the Instituteof Solid Mechanics, SISOM ’99,Bucuresti 1999, pp.81-86.

5. I. Rugina, C. Rugina, "Dynamic behaviour analysisof the ultrasonic systems by means of statisticalmethods", The Annual Symposium of the Instituteof Solid Mechanics SISOM‘2000, Bucharest,November 26-27, 2000,pp. 21-26.

The Green Function Method for Propagation of DampedAcoustic Waves in Isotropic Media

E.L. Albuquerquea, P.W. Maurizb and L.S. Lucenaa

a Departamento de Física, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazilb Departamento de Ciências Exatas, Centro Federal de Educação Tecnológica do Maranhão, 65025-001,São Luís-MA, Brazil, and Departamento de Física, Universidade Federal do Ceará, Fortaleza-CE, Brazil

A Green function technique is employed to investigate the propagation of damped acoustic wave in isotropic media. Thecalculations are based on the linear response function approach, which is very convenient to deal with this kind of problem.Both the displacement and the displacement gradient Green functions are determined. All deformations in the media aresupposed to be negligible, so the motions considered here are purely acoustic waves. The damping term � is included in aphenomenological way into the wavevector expression. In order to examine the acoustic wave excitation, we consider a semi-infinite isotropic media occupying the region z<0, with an interface parallel to the xy-plane and vacuum outside. By using thefluctuation-dissipation theorem, the power spectrum of the acoustic waves is also derived with interesting properties.

Recently there is a revival interest to investigate thepropagation of acoustic waves in elastic medium. Forinstance, fundamental representations for the acoustic,isotropic and anisotropic elastic cases were recentlydeveloped based on an integral representation for thewavefield at a receiver point [1]. These representations canbe recast as modeling formulas for reflection from atransparent interface by exploiting the Kirchoffapproximation, which express the unknown scattered fieldand its normal derivative in terms of the incident field [2].The result is called the Kirchoff-Helmholtz integral.

As an extension of these previous works, Schleicher et al[3] used another mathematical model, based on ageometrical ray approximation (GRA) Green functionformalism, within a framework in which the GRA isexpressed by the group velocities and a relativegeometrical-spreading factor of the acoustic wave.

It is the aim of this work to treat the same problem butconsidering a Green function formalism based on thefrequency distributions of the acoustic wave spectra. Wefirst investigate the simple case of propagation of theacoustic waves in isotropic media. In forthcomingpublications, we intend to consider more sophisticatedgeometries, as well as anisotropic effects.

The frequency distribution of acoustic waves is mainlydetermined by the power spectra of the thermally-inducedfluctuations in the degrees of freedom of the many scattererfound in the medium [4]. These power spectra, orcorrelation functions, are most conveniently calculated bythe use of Green functions within the linear responsefunction theory [5]. Taking into account the imaginary partof these Green functions, the required power spectra areobtained via the fluctuation-dissipation theorem [6].

We consider a semi-infinite isotropic media occupyingthe region z < 0, with an interface parallel to the xy-planeand vacuum outside. The equation of motion for the

propagation of an acoustic wave in such an elastic mediumcan be written as [7]:

j

iji

rS

tu

�

��

�

�

2

2

�

where � is the density of the medium, ui is the i-component of the displacement vector, and Sij is the stresstensor, given by Sij = Cijkl skl. Also, Cijkl is the 4th-orderelastic tensor, and skl the strain tensor defined by skl =(1/2)(�uk/�rl+�ul/�rk). Here, ijkl can be any Cartesian axis,i.e. x, y, or z.

The three simultaneous equations for the components uihave solutions for the vibrational frequency distinguishedby a branch label � (related with the one longitudinal andthe two transverse modes polarization of the acousticwaves), whose corresponding normal-mode coordinatesare U

�q = ��q

kuk. It is not difficult to show that thistransformation separates the equation of motion (1) intothree uncoupled harmonic-oscillator equations given by:

022

2

���

�

qqq U

t

U��

�

�

It is important to note that in general the three acousticvelocities �

�q = ��q/q depend on the direction but not the

magnitude of the wavevector. For isotropic medium, in theabsence of damping, this relation is simply � = �/q.However, the acoustic waves in isotropic media can sufferspatial and/or temporal damping. It is sufficient, for thepresent calculation, to introduce the dampingphenomenologically. With Cartesian superscripts removed,the stress-strain relation is replaced by S = Cs + C� (ds/dt).The second term on the right introduces a relaxationtime � into the strain caused by a time-varying stress.Its insertion into the equation of motion (1) produceswavevector versus frequency relation of the form q2 =�

2/v2(1-i��). The phenomenology can be made more

(1)

(2)

realistic by the assumption of branch and wavevectordependent relaxation times.

We now determine the Green functions by a classicallinear response method [5,6]. The expressions to bederived are valid for transverse as well as longitudinalmodes, provided the appropriated velocity is replaced. Inthis sense, the displacement Green functions are obtainedby calculating the effect of a fictitious external appliedpoint force

)'()exp( zztiFz �� ��

which is parallel to the z-axis and applied to a point z' inthe medium. Hook’s law gives the interaction energybetween the force and the z-component of thedisplacement, i.e.:

)exp()( tiFzuE zzint ����

This applied force produces displacement in both x andz-directions, whose magnitudes are determined by theinsertion of (3) into the right-hand-side of (1), i.e.:

)'()exp(2

2

2

zztiFrr

uCtu

zlk

iijkl

i���

��

��

�

����

The particular solution of (5) is:

)]'exp()(

)'exp()[2/()(2

2

zziqqq

zziqqiFzu

LTx

LLzz

��

�� ��

where qL,T={[�2/�L,T2(1 – i��)]2 – qx

2}1/2, qx being thecommon wavevector x-component, and the subscripts Land T mean longitudinal and transverse modes,respectively.

The homogeneous (or complementary function) solutionof (6) can be given by:

)exp()exp()( ziqBziqAzu TLz ����

where A and B are constants to be found through theusual boundary conditions, i.e: the continuity of the z-component of the displacement uz and the stress Szz at z =0.

The Green functions are obtained from (7) by applicationof the linear response theory. In view of the standard form(4) of the interaction energy, the displacement Greenfunction is simply equal to

zzzz Fzuzuzu /)()();( �������

�

where �� ... ��� is the Zubarev's form [8] to express the

Fourier transformed Green function of the argumentsshown. The displacement gradient Green function is givenby

)();()2()();(��

�������������

���� zzuzzuzzzzzuzzzu

Taking into account the fluctuation-dissipation theorem[6]:

� ��

�

�� ������)0();0(Im)0( 2

zzBz uuTku

where kB is the Boltzmann’s constant, we have, after a bitof algebra:

� � ��

�

�

��

�

�

�� 222224

23

232

24Re)0(

xTTLxT

LxT

xT

Bz

qqqq

qqq

Tku

���

��

����

In (10) and (11), Im and Re means the imaginary andreal part of the arguments shown.

Fig.1 shows the acoustic power spectrum, as describedby the dimensionless term inside the bracket in (11). Wehave considered the ratio vL/vT = 2. As we can infer, thespectrum has three parts: for vLqx < �, both qL and qT arereal, and the spectrum is continuous. For vTqx < � < vLqx, qTis still real, while qL is imaginary. Therefore, the spectrumis also continuous in the range 1 to 2 shown in the figure.For � < vTqx, both qL and qT are imaginary, and thecontribution of a surface wave (of Rayleigh type) will bethe dominant one (not shown here).

FIGURE 1. Power spectrum for acoustic wavespropagating in an isotropic, as a function of the dimensionlesterm �/vTqx.

The main application of our results are related with theseismic reflection data [9], since they enable us to find aspecifically chosen reflection term without the necessity tocalculate other parameters that might be considered noisein a real situation.

ACKNOWLEDGMENT

We thank the Brazilian Research Agencies CAPES-PROCAD and CT-Petro for partial financial support.

REFERENCES1. N. Bleistein, Mathematical Methods for Wave Phenomena, Academic

Press Inc., Orlando, 1984.2. M. Tygel, J. Schleicher and P. Hubral, J. Seismic Expl. 3, 203 (1994).3. J. Schleicher, M. Tygel, B. Ursin and N. Bleistein, Wave Motion, to be

published.4. G.W. Farnell, in Physical Acoustics Vol. VI, W.P. Mason editor,

Academic Press, NY, 1969.5. W. Hayes and R. Loudon, Scattering of Light by Crystals,Wiley, NY,

1978.6. L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon, Oxford,

1968.7. B.A. Auld, Acoustic Field and Waves in Solids, Vols. 1 and 2, Wiley,

NY, 1973.8. D.N. Zubarev, Soviet Phys. Uspekhi 3, 320 (1960).9. Hubral, J. Schleichert and M. Tygel, Geophysics 61, 742

(1996).

(3)

(4)

(5)

(6)

(7)

(8)

(11)

(9)

(10)

New Mechanism of Ultrasonic Attenuation in SolidInsulator with Frozen-in Magnetization

V.V. Sokolov

Physics Department, Moscow State Academy Instrumental Engineering and Information Science, 20 Stromynka St., Moscow 107846 Russia

The theory of attenuation of ultrasound in magnetic cubic crystals based on general thory of magnetoelasticity for nonconductingmagnetic media with frozen – in magnetization is presented.

GENERAL EQUQTIONS OFMAGNETOELASTICITY

Recently, we have developed general theory ofmagnetoelasticity for perfect solid insulator with thefrozen magnetization and its was applied to describethe ultrasonic wave propagation. Fairly goodquantitative agreement between theory andexperiment was demonstrated [1]. Up to now thefield dependence of ultrasonic attenuation in anonconducting magnetic media has not beensatisfactory explained. The arm of the present work isdeveloped the theory of attenuation in such mediabased on next general equations of magnetoelasticity:

;0����

����

�

�

�

�

�

i

i

xq

tdtd

��

��

eqiii

kk

i HHt

qx

mdt

dm ���

�

���

�

; (1)

� �� � ;; ,

,2

2

�

�

����

Tieqi

j

ji

eqin

i

kki

nkn

i

mfHxm

H

Hmxq

qf

xdtqd

����

�

��

�

�

��

�

���

�

����

�

�

�

�

�

�

��

� �;42

j

j

xm

�

����

�� .

ii x

H�

����

Here we took into account the relaxation time � ofthe magnetic field strength to its thermodynamic-equilibrium value eq

iH . The system of equations isclosed by setting a specific form of the free energydensity f, which depends on the invariants of thetensor composed of the spatial derivatives qi,j of thedisplacement vector qi for individual points of thesolid, on the temperature T, and on the components ofthe specific magnetization vector mi,. The latter twoequations of system (1) are the Maxwellmagnetostatic equations, where � is the scalarpotential of the magnetic field. Here we assume thattemperature of solid is constant. The specific featureof system (1) is the equation for the magnetizationthat expresses the condition of the magnetizationfreezing in the material of the solid. As result thesystem of magnetoelastic equations (1) makes itpossible to study the behavior of solids withoutrestricting oneself to the case of magnetic saturation.

By considering a cubic crystal the appropriate freeenergy expression is found to be

� �

� � � �

� � � �

� �.,

22

2

)(2

222222

2

2221

22244

1222211

zxyzyx

zxzyzzyxyyx

zzzyyyxxxzxyzxy

zzxxzzyyyyxxzzyyxx

mmmmmmK

mxmmmmm

mmmc

cc

f

��

���

������

�������

����

�������

����������

where c11, etc. are the coefficients of elasticity, �1 and�2 are the magnetoelastic coupling constants, �xx, etc.are the components of the strains.

ATTENUATION

As an elastic wave propagates through a solid theelastic motion reacts on the magnetization throughthe magnetoelastic coupling. The relaxation of themagnetic field strength to its thermodynamic-equili-brium value lead to the damping of the elastic waves.As a result the ultrasonic attenuation becomes depen-dent on the direction and value of magnetization.This mechanism is effective if the frequency of wavefar from magnetic resonance.For example, let us consider a propagation oftransverse wave along [110] with the displacementalong [001]. We assume that the cubic crystal understudy is placed in a homogeneous, stationary,constant external magnetic field . The magnetic fieldis applied parallel to the [001]. Then, in anunperturbed state, the magnetization vector has onlyone nonzero component mz = m0. Using the explicitform of the functional dependence of the free energy(2), we solved equations (1) linearized near theunperturbed state. The solutions determine thevelocity of the transverse wave

� � � �,

2/1

22220

22

22220

���

�

���

��

��

��

�

����� a

abacb

abcc

tt

��

��

��

and the attenuation of this wave

� �� �,

2 22220

2

abaccb

t ��

�

��

��

�

where ;2

4 20

20

��

Kma � b= .;

0

44203

0

20

22

��

� cc

mt �

The dependence of relative attenuation on field atconstant frequency �=100 MHz and �=2�10-6 s isshown in Fig. The parameters used in thiscalculations were those characteristic of nickel.According [2] the most striking feature of theattenuation of ultrasound in nickel single crystals is asharp peak of the attenuation of the longitudinal andof the transversal modes and also the magneticcontribution to attenuation vanishes between 0.58 and0.82 Bs (Bs is saturation induction).

0 4 8

0

2

B, kG

�

The limits of applicability of our theory aredetermined by the following conditions: 1��� ,where � is the frequency of the elastic disturbanceand � is the relaxation time of the magnetic fieldstrength relaxing to its thermodynamic-equilibriumvalue, and �<<�0, where �0 is the Larmor fre-quency.

REFERENCES1. V.V. Sokolov and V.V. Tolmachov,

Acoustical Physics 46 474-480(2000).2. A. Zielinski, G. Dietz, C. Becker, D. Lenz

and K. Lucke, J. Magn. Magn. Mat. 82 33-42(1989).

Surface Roughness of Semiconductor Materials and Effecton Surface Acoustic Wave Propagation

C.M. Flannery and H. von Kiedrowski

Paul Drude Institute for Solid State Electronics, Hausvogteiplatz 5-7, D-10117 Berlin, Germany

The effect of surface roughness on adhesion and tribological properties of films and interfaces is known to be of keyimportance. Therefore it is of the utmost importance to be able to measure this quantity and to predict the perturbing effectsdifferent roughness levels may cause. Roughness is known to affect the propagation of surface acoustic waves on a material butthere is little useful quantitative data on the topic. This work investigates the dispersive effect of roughness on surface acousticwavepackets (30-200 MHz frequency range) for different degrees of nanometer roughness on silicon (001) and (111) surfaces,We show that the dispersion effect is significant, and that although available theory agrees qualitatively with the results, thetheory is not adequate to predict the real SAW dispersion. These experimental results have considerable implications for designof SAW devices, for accuracy of Brillouin spectroscopy measurements and for possible applications to non destructive testing.Previously unknown dispersive effects on anisotropic crystal surfaces are also demonstrated.

INTRODUCTION

The surface state determines the useful properties ofmany materials, in particular the tribological, elasticand adhesion properties. Every surface possesses aroughness and this can have a critical effect on materialperformance and any devices in which it is employed.Propagating Surface Acoustic Waves (SAW) are verysensitive to the surface state and are affected by thelevel of roughness. Many techniques rely on surfaceacoustic waves to achieve their goals but little attentionis paid to the possible perturbing effects that roughnessmay have: the important fields of SAW devicesrequires ever more accurate knowledge of acousticparameters to design high quality devices operating atGHz frequencies; Brillouin Spectroscopy regularlyrelies on measurement of SAW velocity dispersion athigh frequency to obtain elastic constants; Non-Destructive testing (NDT) techniques frequentlyinspect structures of very large roughness. But in noneof these areas is attention given to the roughnessperturbation which at high frequencies or largeroughnesses may deleteriously affect results. In thiswork we show experimentally that the effect issignificant and should not be ignored.

SAW Theory .

It is well known that SAWs propagating on a roughsurface will become dispersed: with increasingfrequency the velocity will slow and scattering willincrease, attenuating the wave. Here we concentrate on

the more important velocity dispersion effect. Despitecopious theory�1,2,3�, there exist very fewexperimental studies of SAW interaction withroughness, the few published works�5,6� were done atlow frequencies (a few MHz) on isotropic materialsand results were not convincing. Here we measuredispersion at high frequencies (100s MHz) and onrelevant crystal materials with submicron roughnesslevels. The theory of SAW interaction with roughnessis extremely complex and rather difficult tounderstand. However the result of several papers is, inthe experimentally-realistic long wavelength limit theSAW phase velocity dependence on frequency is:

vv a( )� �

�

0

2

2�� �

Where ���� is SAW velocity at frequency �, �� isvelocity at zero frequency, � is rms roughness and a istransverse correlation length of the roughness. � is aconstant which is a function of the elastic properties ofthe material. All authors assumed a random Gaussianroughness distribution described by � and a. Theimportant feature to note is that the velocity dispersionis linear with frequency and depends on the square ofthe roughness. On anisotropic materials, where elasticparameters vary depending on direction, � and � willalso be functions of SAW propagation direction.

EXPERIMENTAL

We prepared a range of Silicon (001) wafers andpolished/roughened them to various levels ofroughness varying from �=10-250 nm., a was in the15-30 µm range for all samples. SAWs were generatedby light from a pulsed nitrogen laser (0.5 ns duration)and focused into a line shape on the samples. The heatenergy cause a rapid expansion of the absorbing sourcearea, giving rise to wideband SAWS propagatingacross the substrate. These were detected at differentpropagation distances by a piezoelectric foil withknife-edge detector. Frequency-dependent phasevelocity dispersion curves were obtained via Fouriertransforms of SAWS of 10 mm path length difference

RESULTS AND DISCUSSION

Fig. 1(a) shows obtained dispersions on the samples ofdifferent roughnesses and with linear fits to each curve.It is clear that the slope (rate of dispersion) increaseswith increasing roughness and that dispersion is linearwith frequency — the first verification of thepredictions of the theory and the first measurements onanisotropic materials. Fig. 1(b) plots the dispersions asa function of roughness; the result is somewhat

perplexing, the dispersion increases with roughness butthe behavior is not dependent on the square of theroughness, contradicting the theory. The dispersions ofthe Pseudo-SAW mode on the ��110�� direction werelarger than those for the Rayleigh mode the ��100��direction. This is also surprising because thepenetration depth of the Pseudo SAW is longer andshould be less affected by roughness, no theory fordispersion of Pseudo SAW modes exists so this is acompletely new result. The Rayleigh SAW dispersionis almost constant over the detectable range ofpropagation angles on (001) Si. However it ispredicted�3� that the dispersion can vary by 40% onthe Si(111) surface, on which the Rayleigh SAW isdetectable at all propagation angles. To this end wemeasured SAW dispersion on a �=300 nm Si (111)wafer between the velocity extremes of �1 -1 0� and �2-1 1� angles. We found that the dispersion indeed doesvary on this cut, the minimum being along �1 -1 0� andincreasing with angle to a maximum about 30% higheralong �2 -1 -1�. The overall dispersion was lower on Si (111)than on (001), also predicted by ref. �3�.

CONCLUSIONS

We have been the first to characterise SAW dispersiondue to roughness at 100 MHz frequencies, and onanisotropic materials. The predictions of the theoryhave been qualitatively verified but quantitatively donot agree. We feel that some assumptions of the theory,that roughness is Gaussian and that SAW wavelength roughness length may not be valid. Thesedispersions are significant, and, especially at GHzfrequencies, should be taken into account for SAWdevice design and Brillouin spectroscopy (which isknown to underestimate constants); for NDT, resultsmay also allow remote determination of roughness bymeasuring SAW dispersion.

REFERENCES

1. A.G. Eguiliz and A.A. Maradudin, Phys. Rev. B, 28, 728-747, (1983).

2. V. V. Krylov, Prog. Surf. Sci., 32, 39-110, (1989).

3. A.V. Shchegrov, J. Appl. Phys, 85, 1565, (1995).

4. M. De Billy, G. Quentin and E. Baron, J. Appl. Phys.,61, 2140-2145 (1987)

5. V.V. Krylov and Z.A. Smirnova, Sov. Phys. Acoust., 36,583585,1990.

0 50 100 150 200 2504960

4980

5000

5020

5040

5060

5080

Frequency (MHz)

250 nm

130 nm

80 nm

40 nm

10 nm

Vel

0 50 100 150 200 250

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4 [100] RSAW [110] PSAW

slop

e �v/

V o at 1

00 M

Hz

(%)

� roughness (nm)

FIGURE 1. (a) Measured SAW dispersions on (001)Si. (b) Normalised slope versus roughness.

Variable Forgetting Factor Recursive Total Least SquareMethod for Ultrasound Attenuation Coefficient Estimation

J.-il Songa, J.-seok Lim b, Y.-G. Pyeon b and K.-M. Sunga

a School of Electrical Engineering, Seoul National University, Seoul, 151-742 Koreab Department of Electronics Engineering, Sejong University, Seoul, 143-747 Korea

This paper deals with the application of nonstationary spectral analysis for attenuation coefficient estimation on thebackscattered ultrasonic signal from biological tissues. Ultrasound attenuation caused by scattering and absorption shifts thefrequency of the instantaneous spectrum moderately or highly with time. This time dependent frequency-shift property makesthe signal nonstationary. Therefore, the conventional methods such as block AR estimation or recursive AR with fixedforgetting factor do not work well. In order to improve the performances of the conventional estimators, both in weakly and instrongly attenuating media, we propose a new recursive total recursive least square autoregressive algorithm with a variable-forgetting factor. The variable-forgetting factor can handle adaptively the nonstationarity of the backscattered ultrasound.

INTRODUCTION

The measurement of ultrasound attenuation inbiological tissues has received much interest in thefield of ultrasonic tissue characterization. As anultrasound pulse propagates through soft tissue, itexperiences an attenuation-dependent frequency-shift.Several techniques based on either a time domain or afrequency domain processing, can be used todetermine the frequency shift of backscattered signals.Among the techniques, spectral methods are preferredbecause these not only analyze the properties of tissuesversus depth but also correct the diffraction effect [1].However, they have a problem that the performance ofthe estimator will be degraded as the nonstationarity ofsignal increases.

To improve the performances in such nonstationaryenvironment, we propose a new recursive total leastsquare regressive AR algorithm with variableforgetting factor (VFF-RTLS-AR). This method canupdates the forgetting factor as well as the ARparameters at each time according to the signalnonstationarity. Therefore, this algorithm canadaptively cope with the nonstationarity in echoes andprovide more accurate estimation results in wide rangeof attenuation.

ATTENUATION ESTIMATION

The reflected nonstationary signal x(n) is digitizedwith a sampling frequency fs . This numerical signal ismodeled as the output of a linear filter driven by white

Gaussian noise u(n) with zero mean and variance �s2

[3]. It is given by

��

����

p

ii nuinxnanx

1)()()()( (1)

where ai(n) are the AR parameters at time n and p isthe order of the AR model. In practice, the order of theAR model must be chosen. In our study, it appears thata second-order AR process (AR2) can efficientlyestimate the maximum energy frequency or thecentroid frequency by experiments.

From the previous researches, it is known thatattenuation coefficient � can be derived from severalparticular frequencies such as maximum energyfrequency [1]. A simplified relationship between theattenuation coefficient and the maximum energyfrequency is shown below.

dttdf

CcmMHzdB)(max

)/( ��� C : positive constant (2)

Therefore, AR parameters must be estimatedprecisely in order to carry out the accurate attenuationestimation.

RECURSIVE TOTAL LEAST SQUAREAR PARAMETER ESTIMATION WITH

VARIABLE FORGETTING FACTOR

In general total least square (TLS) methodestimates better than ordinary least square (OLS)

method[2]. This paper proposes VFFR-RTLS-ARmethod in which newly derived recursive total leastsquare (RTLS) is combined with variable forgettingfactor. The variable forgetting factor properly adjustsitself to the nonstationarity and it is determined as theminimizing arguments of cost function.

Table 1 summarizes the VFF-RTLS-AR algorithmfrom the nonstationary AR model to the maximumenergy frequency estimation. Therefore, using eq. (2),we can obtain the ultrasound attenuation with theestimated frequency.

Table 1. Summery of the VFF-RTLS algorithm

Initialize 0P , 0� , 0S

Tnnnn

Tnn

n PP

K���

�

11

1

��

�

�

�

1�� nnne ��

� �11

1�

�

�� nnnnn

n PKPP ��

1�� nnn P ��

�

���

���

������ ]Re[ *

11 nnnnn e

� � � �

n

nnn

n

nnnnnn

n

PKK

KISKIS

��

���

���

������

1

11

11 ��

�� nnnnn PS ���

���

����

����

����

��

� �

)(ˆ11

4)(ˆ

cos2

)(ˆ2

11max na

nafnf s

�

where Tpn nana )](ˆ,),(ˆ[ 1 ��� ,

Tn nxpnnx )](),(,),1([ ���� ��

SIMULATION AND RESULTS

The performance of the proposed method is testedon uncorrelated computer simulations. Different mediawith attenuations ranging from 1 to 5 dB/cmMHz areconsidered. The sampling frequency is 400 MHz andthe transducer center frequency is 45 MHz. Eachsimulation contains 300 times averages of 1000samples, with a speed of ultrasound of 1530 m/s.

FIGURE 1. Simulation Results of fmax vs. time with(�,�) (OLS:dotted, VFF-RTLS:dash, True value:solid)

CONCLUSIONS

As shown in FIGURE 1., the proposed algorithmworks well in respect to the estimation error.Therefore, this VFF-AR method can be used tononstartionary spectral estimation problems withoutany pre-process.

REFERENCES

1. T. Baldeweck, P. Laugier, A. Herment, and G.Berger, IEEE Trans. Ultrason., Ferroelect., Freq.Conlr., 42, 99-110 (1995)

2. G.H.Golub and C.F. Van Loan, SIAM Journal ofNumerical Analysis, 17, 883-893 (1980)

��

�����

��

�

�

�

�����

���

��

�����

��

�

�

�

�����

���

� �����

��

��

�

�

�

�����

New possibilities in HBAR spectroscopy

G. Mansfeld, S. Alekseev, and I. Kotelyansky

Institute of Radioengineering and Electronics, RAS, 11, Mokhovaya st., 103907 Moscow, Russia

The new modified version of HBAR spectroscopy based on the measurements of only parallel resonance bandwidth at eachresonance frequency has been developed and used for the study of bulk acoustic wave attenuation in langatate(La3Ga5.5Ta0,5O14) and the frequency dependences of the acoustic wave attenuation coefficients in tungsten, aluminum andplatinum films.

INTRODUCTION

The typical high overtone BAW composite resonatorstructure is schematically shown in Fig.1. It consists ofa rather thick plate made in our case of the materialunder investigation with flat parallel faces, electrodelayers (aluminum films), and a piezoelectric Zinc oxidefilm. The attenuation study in high overtone bulk acousticwave resonator (HBAR) spectroscopy is based on themeasurements of the difference between thepeculiarities of s11 (microwave reflection coefficient)phase frequency dependences near the position of theindividual resonant peaks.This approach proved to bethe most fruitful for the measurements of parallelresonances bandwidth. In practice sometimes seriesresonances can influence the accuracy of bandwidthmeasurements. Due to motion properties andadditional parasitic inductances, capacitances andresistance these phase data are sometimes distorted. Itresults in serious experimental error. A convenient way of the evaluation of the acousticlosses is suggested. After the measurement of thefrequency dependence of the real and the imagine partsof electrical impedance of the resonator it is possible tofind on the frequency points between the positions ofthe resonant peaks the additional electric impedancedue to resonator series capacitance and parasiticreactance and resistance. These data can berecalculated to resonant point and subtracted from thecomplex impedance measured on the resonance.Remaining impedance will represent the data forparallel LCR circuit alone. The losses can be obtainedfrom both impedance modulus and phase frequencydependences. The existence of series and parallel resonances canbe understood from the equivalent circuit diagram. The equivalent circuit diagram of high overtoneBAW resonators (Fig.1b) follows from the expression

for input electric impedance Ze of the piezoelectriclayers with electrodes loaded acoustically by the layerunder investigation

a)

� 2

Z2 Z1Zt

Ze

Z 0

t2 t1

� 1q

d l

b

piezo-electric p l a t e

Mei n v e s t i g a t e d

l a y e r

b)

Z e

R 0C 0

L 1

C 1

R 1

L n

C n

R n

FIGURE 1. a) The schematic diagram of the BAWcomposite resonator under investigation; b) Equivalentcircuit diagram. R0, C0 - are cancelled in the modifiedHBAR spectroscopy.

SAMPLES

In the HBAR experiments with langatate the sampleswere prepared as thin (0.5…0.8 mm) properly oriented(similarly to bars) LGT plates with parallel and flatfaces with Al - ZnO -Al transducers deposited on oneface. HBAR experiments were carried out at thefrequencies 0.5 - 6.0 GHz. Shear waves were excitedusing ZnO films with inclined C- axis (25є).The faces of the samples were polished parallel (<8’’)and flat.

EXPERIMENTAL PROCEDURE

Microwave Network analyzer HP8753ES was usedfor reflection coefficient and impedance measurementsof the samples. Network analyzer was controlled by acomputer so if it was possible to obtain a very highfrequency resolution in a wide frequency band. Themeasurements procedure is as follows. First theexperimental frequency dependence of impedance wasmeasured. Then the independently measured resistancecurve is subtracted from the calculated dependenceReZe, and the independently measured dependence ofthe series reactance data are subtracted from ImZe .Then only parallel peaks remain and it is easy tomeasure Qn-factor from the bandwidth measured from

the level 2/2 of the maximum of the modulus orfrom the slope of the phase characteristics φn of Ze as:

���

����

�

�

��

�

�nnn fQ

21

.

Using the data obtained the attenuation constant instructure was calculated as:

��

���

���

sdB

QMHzf

n

nn

�

��

)(68.8

Roughness losses were evaluated and subtracted fromthese total losses together with the losses in ZnOtransducer.

THE RESULTS OF THE EXPERIMENTS

Typical frequency dependences of the attenuationcoefficient for longitudinal waves for various directionsof propagation in LGT are shown in Fig.2. Thesecurves represents f2-dependence typical for Akhiezerlosses. To obtain these curves the losses in ZnO film,diffraction losses and losses due to roughness weresubtracted from the measured dependence.

Attenuation constants for different directions of thepropagation formally recalculated to 1 GHz keeping inmind square frequency dependence of the attenuationconstant are listed in Table. A relatively high value ofthe attenuation constant was observed for rotated ± 45oY-cuts but the frequency dependence for thesedirections also was square

1 2 3 4 5 6 72

3

456789

10

20

30

4050

f, GHz

f2

dB/ s�

1

2

3

FIGURE 2. Typical frequency dependences of theattenuation coefficient for longitudinal waves: 1- X-direction, 2- Y-direction, 3- Z-direction.

TABLE

Direction of wavepropagation

Wavepolarization

AttenuationdB/µs GHz2

Y FS 1.1±0.1Y SS 0.90±0. 02Y L 0.62± 0.02Z L 0.60±0.02Y-45º L. 4.7±0.1X L 1.18±0.02

The value of Qf product for practically importantcase of slow shear BAW in Y - direction was found Qf˜ 3 1013. The modified acoustic HBAR spectroscopy methodwas developed under the grant ISTC 1030.

Harmonic Waves in Viscoelastic Media Modelledby Fractional Derivatives or Fractional Operators

of Two Different Orders

Yu. A. Rossikhin and M. V. Shitikova

Department of Theoretical Mechanics, Voronezh State University of Architecture and Civil Engineering, ul.Kirova 3-75, Voronezh 394018, Russia, e-mail: [email protected]

Harmonic waves are investigated in viscoelastic materials whose behaviour is described by the generalized Kelvin-Voigt, Zenermodels and other models containing fractional derivatives and other fractional operator of two different orders. For each of theabove mentioned models, the comparative analysis between the behaviour of its rheological and dynamic characteristics iscarried out. The vector diagrams and the mechanical loss tangent are used as the rheological characteristics, and the velocitiesand coefficients of attenuation of harmonic waves are considered as the dynamic characteristics. The analysis shows thatdepending on the relative magnitudes of the orders of the fractional derivatives and fractional operators entering into therheological model, some of the models considered may describe both the wave and diffusion processes occurring in mechanicalsystems, but others describe only wave or only diffusion phenomena. As this takes place, the behaviour character of the vectordiagrams is similar to that of the velocities of harmonic waves as functions of the frequency for the corresponding models.

THE SIMPLEST RHEOLOGICALMODELS WITH TWO FRACTIONAL

PARAMETERS

Consider the simplest rheological models involvingfractional derivatives or fractional operators of twodifferent orders, namely: the modified Zener model

),(0 ετετεστσ ββσ

αασ

ααε DDED ++=+ (1)

the modified Kelvin-Voigt model ),(0 ετετεσ ββ

σαα

σ DDE ++= (2)and the model with the fractional operator ],)1([ ετυεσ βαα

εε−

∞ +−= DE (3)where σ and ε are the stress and strain, respectively, tis the time, ετ and στ are the relaxation and creep

times, respectively, 0E and ∞E are the relaxed and

nonrelaxed moduli of elasticity, respectively, α and β(0 <α, β <1) are the orders of the fractional derivativesor operator, σαD or εαD is the Riemann-Liouvillefractional derivative defined as

∫∞−

′Γ′′−=

t

t

tttt

D ,)-(1

d)(dd

αα

ασσ

)1( α−Γ is the Gamma-function, 1−∞∆= EEευ , and

0EEE −=∆ ∞ . Each of the above mentioned rheological models inthe Fourier space can be written as ,)(* εωσ iE= (4)

where σ and ε are the Fourier transform of the stress

and the strain, respectively, )(* ωiE is the complex

modulus of elasticity, and ω is the frequency. The complex modulus is the basic value to determinethe dissipative characteristic of the each rheologicalmodel, namely, the tangent of the mechanical lossangle:

,)()(

tanωω

χE

E′′′

= (5)

where )()( * ωω iEE ℜ=′ and )()( * ωω iEE ℑ=′′ .

The relationships for the complex modulus )(* ωiEand the mechanical loss angle tangent χtan for

different rheological models involving two and moredifferent fractional parameters, among them for themodels (1)-(3), are presented in [1]. If one constructthe E ′ - dependence of E ′′ (so called vector diagram)for the models (1)-(3) (see, for example, [2]), then itsanalysis shows that for the model (1) at αβ = thecurves of the vector diagrams issue out of zero andarrive at unit, at αβ > these curves issue out of zeroand go to infinity, but at αβ < there exist such

domains of the values iωτ ),( σε=i within whichχtan becomes negative. In other words, the model (1)

at αβ = and αβ > describes the wave and diffusionprocesses, respectively, but at αβ < the given modellosses its physical meaning. For the model (2), thecurves of the vector diagrams at any magnitudes of

1,0 << βα issue out of zero and go to infinity, in so

doing χtan remains positive for the whole domain of

σωτ . As for the model (3), the curves of the vectordiagrams at any magnitudes of 1,0 << βα issue outof zero and arrive at unit, in so doing χtan remains

positive for the whole domain of the value εωτ . Inother words, at any magnitudes of 1,0 << βα themodels (2) and (3) describe only diffusion and waveprocesses, respectively.

HARMONIC WAVES IN FRACTIONALCALCULUS VISCOELASTIC MEDIA

Let us illustrate the peculiarities of the behaviour ofthe rheological models (1)-(3) by the example of adissipating harmonic wave. In this case the relationshipfor the displacement ),( txu at 0≥x has the form ],)(exp[),( xkxtiAtxu δω −−= (6)

where A is the amplitude, 1−= ck ω is wave number, cis the wave velocity, and δ is its damping coefficient. Substituting (6) into the equation of motion of theviscoelastic rod yields

χωωρ21

cos|)(| 2/1*2/1 −= iEk , (7a)

χωωρδ21

sin|)(| 2/1*2/1 −= iE , (7b)

where ρ is the density of the rod’s material. Considering formulas (4) and (5) in equations (7),one can determine the ω-dependence of the wavenumber k (the phase velocity c) and the dampingcoefficient δ for the rheological models (1)-(3).Theasymptotic expansions of the values c and δ for largefrequencies ω are of most interest for our investigation,since the main difference in the behaviour of wavecharacteristics for different rheological models isobserved at large frequencies. These expansions havethe following form:for the model (1) at β >α, γαβ == , and αβ < ,respectively,

−

+

−−

−

=

−−

−∞∞

−−

,2

)(cos)(

21

1

,2

cos)(21

21

12

,4

)(sec

220

0

2220

αβπωτττ

πγωτ

ωαβπττ

αβσ

α

σ

α

ε

γσ

αββ

σ

α

ε

c

E

Ec

c

c (8a)

−

−

−

=

−−+−−

−−∞−∞

−−−−

,2

)(sin

21

,2

sin21

2

1

,4

)(sin

12210

1

0

1

21

2210

αβπτωττ

πγτω

αβπωττ

δ

αβσ

αβα

σ

α

ε

γσ

γ

αββ

σ

α

ε

c

E

Ec

c

(8b)

for the model (2) at β >α and αβ < , respectively,

=

4sec

4sec

20

20

παω

πβω

α

β

c

cc , (9a)

=−−

−−

4sin

4sin

211

0

211

0

παω

πβω

δ α

β

c

c, (9b)

and for the model (3) at 0 < α, β < 1

−= −

∞ 2cos)(

21

1παβωτν αβ

εεcc , (10a)

2sin

21 11 παβτωνδ αβ

εαβ

ε−−−

∞= c , (10b)

where ρ/00 Ec = , and ρ/∞∞ = Ec .

The analysis of the formulas (8)-(10) shows that forlarge magnitudes of the frequency ω the behaviour ofthe velocities of harmonic waves for the models underconsideration is similar to the asymptotic behaviour ofthe vector diagrams at large ω: for the model (1) thevelocity increases to infinity at β >α and tends to ∞cwhen γαβ == ; for the models (2) and (3) the

velocities at any 0 < α, β < 1 either only increasewithout bounds (the model (2)) or only tend to ∞c (themodel (3)). When αβ < , the asymptotic magnitudes

of δ at large ω take on negative values, and the waveloses it physical meaning. As ω increases, the damping coefficient δ for allrheological models increases without any bounds forthe magnitudes of the fractional parameters α and βdifferent from unit, and remain restricted values onlywhen 1== αβ .

REFERENCES

1.Rossikhin, Yu. A., and Shitikova, M. V., Mech. Time- Dependent Materials 5, 1- 45 (2001).2. Rossikhin, Yu. A., and Shitikova, M. V., Appl. Mech. Rev. 50, 15-67 (1997).

Anomalies of Ultrasound Reflection from Boundary withStrong Dissipative Medium

D.A. Kostiuk, J.A. Kuzavko

Brest State Technical University, 224017, Moskovskaya-267, Brest, Belarus

The reflection of an ultrasound longitudinal wave from flat boundary of a solid body with strong dissipative medium or SDM(e.g. viscous liquid or material in its relaxation stage) is considered. It is found that reflection and transition factors aresubstantially depending on dissipative parameter of SDM and wave frequency. The experimental confirmation of earlierunknown phenomenon – the anomalous acoustic wave reflection factor change – was received.

The reflection of continuous and pulseacoustic signals from medium boundary wasinvestigated rather in detail. Nevertheless, the case ofacoustic wave reflection from medium with strongabsorption of ultrasound is unknown to us and isinteresting in scientific and practical plan. We considerreflection of an ultrasound longitudinal wave (LW)from flat boundary of a solid body with SDM (e.g.viscous liquid or material in its relaxation stage).

Let suppose that continuous harmoniclongitudinal wave (LW) is spread in a solid bodywithout attenuation. It is partially reflected at normalfall on boundary with a viscous liquid, and past LW ina liquid rather quickly fades.

The wave equation for LW in dissipative mediumlooks like:

txxxxxxx bucuu ,, ����� , (1)

where uх - component of longitudinal displacement inLW, с - elasticity module, ρ - density, b - dissipativelosses parameter, determined in factors of shift η andvolumetric ξ viscosity and thermal conductivity factorχ according to a ratio [2]:

)( 1134 ��

���� pv ccb ��� , (2)

in which cp and cv are thermal capacities ofmedium at constant pressure and volume accordingly.

Thus factor of absorption of sound α isunequivocally expressed through the parameter ofdissipative losses b according to expressionα=ω2/2ρSl

3, where ω=2πf - cyclic frequency of asound wave, Sl - speed of a longitudinal sound. Let'snote, that at b=0 these equation determines theacoustic oscillations in a solid body with theappropriate material constants.The decisions for falling, reflected and past waves aresearched in a standard kind [2]:

� �

� �

� �) ( exp

) (exp

) (exp

202

101

101

txkixuu

txkiuu

txkiuu

TT

RR

II

��

�

�

����

���

��

(3)

where k1=ω/Sl1, k2=ω/Sl2 - wave numbers, Sl1 and Sl2 -speed of a longitudinal sound in a solid body (1) andliquid (2), t - time.

The boundary conditions at x=0 are representingthe continuity of displacement and stress in anacoustic wave and will be written down as follows:

Txtx

Txx

Rxx

Ixx

Tx

Rx

Ix

ubucuuc

uuu

,2,2,,1 )(

,

���

��

(4)

That decisions (3) satisfy to the appropriate waveequations, and being substituted in (4), give thesystem of the linear equations to define the factors ofreflection and transition

(T=1+R). Reflection factor has thefollowing kind [3]:

� �

)1(22

)1(1

)1(22

)1(1

20202/12

20202/120

xxTixTx

xxTixTxRR

�����

�����

��

(5)

Here R0=(Z2-Z1)/(Z2+Z1) and T0=2Z2/(Z2+Z1) arereflection and transition factors of acoustic waveaccordingly (when ω�0), x=ω/ωc , Z1=ρSl1 andZ2=ρSl2,0 are acoustic impedances of solid and liquidmediums (without dissipation), ωc=ρ2S2

l2,0/b is someeffective frequency to characterize the dissipativemedium, Sl2,0 is sound velocity (when ω=0).

Starting from (5), a statement for a reflectedsignal phase can be followed:

IT uuT 0102�

� �

� � � � )21(2

)1(1)1()1(12

)1(1)1()1(

222

02/12200

22/120

2/122/1200

xxTxxRTxR

xxxTRtg R

��������

������

�(6)

Thus, accordingly to (5) and (6) at reflectionof an acoustic wave from dissipative medium itsamplitude and phase varies. The same can be saidabout the transited wave.

Proceeding from the given dependence Rωand using direct and inverse Furier transformationswith the help of the computer the signal reflected fromboundary of plexiglas - epoxy pitch was estimated formodeled acoustic pulses.

If the reflection occurs from less denseacoustic medium (Z2 < Z1), at ω<<ωc then there is aninversion of a signal (ΨR=π). In a vicinity ω~ωc theminimum of reflection factor is observed at the furtherincrease of a phase of the reflected signal concerning aphase of a signal, falling on boundary. Further atω>>ωc Rω�1 and ΨR

�2π. There is a completereflection of a signal. Otherwise at reflection frommore dense medium the inversion of a signal does notoccur (ω>>ωc , Rω�R0 and ΨR

�0). Similarly at ω ~ωc the minimum of reflection factor Rω is observed ata maximum of a phase ΨR

ω. Further at ω>>ωc Rω�1and ΨR

�0.To confirm the theoretically predicted above

phenomenon - dependence of the reflection factorfrom the dissipation of ultrasonic energy in reflectingmedium the following experiment was carried out. Thepulse generator feeds ultrasonic piezoceramicaltransducer (UPT) with resonance frequency of 3.5MHz. An acoustic pulse close to the theoreticallyconsidered form was radiated into the structure ofplexiglass - epoxy pitch. Radiated and reflected signalswere registered by oscilloscope.

The epoxy pitch was prepared accordingly tothe state standard (10 g of epoxy pitch to 1.2 g ofcuring agent). Let's note that acoustic impedances ofliquid and hard phase of epoxy pitch are differing nomore than 100%. During the mix hardening

temperature grew no more 10OС in comparison withroom, that practically did not influence on the acousticparameters of the mix. We suppose the reduction ofreflection factor, which changed in 2,5 times, to beexplained only by the theory advanced here, namelysharp change of energy dissipation in an epoxy layerwhile hardening. Also at hardening of the epoxy pitchthe duration of the reflected acoustic signal changedfrom τ=3 �s up to τ=2 �s, that will be coordinated toconclusions of the advanced here theory.

As a result of carried out theoretical andexperimental researches the earlier unknownphenomenon - anomalous change of reflection factorof an acoustic longitudinal wave from boundary of asolid body with strong dissipative medium isestablished. The unique opportunities on measurementof a spectrum of reflected acoustic signals in such orsimilar structure are interesting in development offunctional devices of solid-state electronics, and alsoin development of the express-method ofmeasurement of viscosity of liquids.

References

1. Kayno S. Acoustic waves. Devices, visualizationand analog processing of signals / M.: “Mir”.1990. 656 p.

2. Vinogradova M. B., Rudenko O. V., SuhorukovA. P. Theory of waves / M.: “Science”. 1990. 432p.

3. Kostiuk D. A., Kuzavko Yu. A. Anomalies ofreflection of acoustic pulses from boundary withstrong dissipative medium. / Proceedings ofinternational conference on neural networks andartificial intelligence. Brest, 12-15 October, 1999.p.p. 183-188.

Acoustic Attenuation at Elastic Phase Transitions in DeuteratedBetaine Phosphate/Betaine Phosphite Solid Solutions

V.Samulionis a, J.Banys a , G.Völkel b and A. Klöpperpieper c

aDepartment of Physics, Vilnius University, Sauletekio al. 9/3, 2040 Vilnius, LithuaniabUniversität Leipzig, Fakultät für Physik und Geowissenschaften, Linnéstr. 5, D - 04103 Leipzig, Germany

cFachbereich Physik der Universität des Saarlandes, 66123 Saarbrucken, Germany

The temperature dependencies of ultrasonic attenuation and velocity in deuterated betaine phosphate, betaine phosphite and theirsolid solutions are presented near antiferrodistortive (AFD) phase transition. Measurements have been performed at 10 MHzfrequency for longitudinal waves propagating along crystallographic X, Y and Z directions. It is shown, that the critical ultrasonicbehaviour is very anisotropic and strongy depends on composition of solid solution.

INTRODUCTION

The understanding of elastic AFD phase transitions(PT) is one of the fundamental problems in the condensedmater physics. Ultrasonic method directly probes elasticdegrees of freedom and enables to obtain informationabout the mechanism of such PT. Typical examples ofsystems exhibiting antiferodistorsive ordering are betaine[1-5] compounds: deuterated betaine phosphate (DBP),deuterated betaine phosphite (DBPI) and their solidsolutions DBP xDBPI1-x , which undergo the ferroelasticphase transitions in the temperature interval 356-366 K.The monoclinic crystal structure exhibits the chains ofbetaine molecules and PO 3 , PO4 groups directed alongthe b axis [6], therefore these crystals are highlyanisotropic. The NMR investigations of DBPI/DBPrevealed an order-disorder character of AFD PT [7]. Inthe AFD phase the order parameter was interpreted interms of the tilt of the betaine molecules out of the mirrorplane. The structural similarity of DBP and DBPI allowsto obtain the mixed compounds DBP x DBPI1-x in thewhole range of composition. Therefore the purpose of ourstudies was to investigate orientation dependencies ofultrasonic velocity and attenuation at the AFD phasetransition in solid solutions of DBP x DBPI 1-x.

RESULTS AND DISCUSSION

The typical v=f(T) dependencies for longitudinalwaves are shown in Fig.1 along Z direction (this directioncorresponds to the crystallographic c axis). The mostinteresting finding is that the PT temperature, whichcorresponds to the velocity anomaly, at first increasesalmost linearly with x , then reaches a maximum near theconcentration x =0.35, and then decreases down to To DBP

= 365 K. Velocity anomalies are sharp enough for pureDBP and DBPI, but in the middle of composition rangethey clearly become more wide. This is determined by theincrease of the order parameter relaxation time (prefactor

τ0 also has a maximum in the same region of x ( see [8] ).This phenomenon is seen more explicitly in α = f (T)dependencies, which are shown in Fig. 2. The width ofattenuation peaks is directly proportional to τ 0.

350 355 360 365 370 375

-8

-6

-4

-2

0

2

DBP contentsz - axis

0.3

0.40.5

0.15

10

∆ v

/ v ,

%

T , K

FIGURE 1 . The temperature dependencies of longitudinalultrasound relative velocity for Z - mode in mixed crystalswith different DBP content.

355 360 365 3700

1

2

3

x = 0.4x = 1 x = 0

α , a

.u

T , K

FIGURE 2. The temperature dependencies of longitudinalultrasound attenuation for Z - mode in mixed crystals withdifferent DBP content.

Increase of τ0 also reduces step of velocity at T0. So, wecan assume that the step of static value of ∆v/v for Z-mode is almost independent on x. In contrary, for X -mode the velocity step clearly depends on x (Fig. 3).

3 5 0 3 5 5 3 6 0 3 6 5 3 7 0 3 7 5

- 8

- 6

- 4

- 2

0

2D B P c o n t e n t s

1

0 . 5

0 . 3

0 . 4

0 . 1 5

0

x - a x i s

∆ v

/ v ,

%

T , K

FIGURE 3. The temperature dependencies of longitudinalultrasound relative velocity for X - mode in mixed crystalswith different DBP content.

The ∆v/v step at To at DBP concentrations till 0.4decreases from the value ∆v/v = 7.5 % in pure DBPI andthen almost saturates at the value ∆v/v = 3 % for pureDBP. The effect of the increase of τo also is clearly seen.

For longitudinal ultrasound along Y - axis the relativestep at PT in velocity is completely absent for phosphateconcentrations x < 0.15, then increases with increasingDBP concentration and reaches the maximum value ∆v/v= 7.5 % in pure DBP ( Fig. 4). The additional part ofultrasonic velocity for DBP concentrations x < 0.15 inAFD phase follows the temperature variation of square ofthe order parameter [9,10], which is determined by thedegree of order of betaine molecules [6]. In this range ofx there is no critical dip of velocity which clearly appearsin the compound DBP 0.3DBPI0.7. Therefore we can pointout that in the range 0.15<x<0.3 long range elastic fieldshows considerable variation. In the same compositionrange AFD PT temperature also has a maximum.

350 355 360 365 370 375

-8

-6

-4

-2

0

2

4

x=1

x=0.15

x=0.3

x=0.4

x=0.5

x=0

∆ v

/ v ,

%

T , K

FIGURE 4. The temperature dependencies of longitudinal relativeultrasound velocity for Y - mode in mixed crystals withdifferent DBP content.

The differences in ultrasonic anomalies at AFD PT forvarious solid solutions possibly can be determined bychange in the coupling anisotropy of betaine moleculewith PO4- and PO3- tetrahedra (i.e. the substitution of oneoxygen by proton in tertraedron leads to the reduction ofhydrogen bonds from two in DBPI to one in DBP).

Structurally it means that the step of a string alongscrew b axis is different for DBPI. We believe that rathersubtle structural changes are responsible for such unusualelastic behaviour, but can not at this moment point it outclearly. It must be noted that in the same compositionrange 0.15<x<0.3 the unusual additional peak of α havebeen observed in heating cycle below T o (Fig. 5). Furtherstructural investigations are needed to obtain moreprecise information about structural changes in thesesolid solutions (0.15<x<0.3).

ACKNOWLEDGMENT

These investigations have been supported by theLithuanian State Science and Studies Foundation.

REFERENCES

1. J. Albers. Ferroelectrics. 78 , 3-11 (1988).2. J. Albers, A. Klöpperpieper, H.J. Rother and K.H. Ehses.

Phys. Status Solidi(a). 74 , .553-557 (1982).3. W. Schildkamp and J. Z. Spilker. Kristallogr. 168, 159-171

(1984).4. M.I. Santos, J.M. Kiat, A.Almeida, M. Chaves, A. Klöpper-

pieper and J. Albers. Phys. Stat. Sol. 189 , 371-387 (1995).5. V. Samulionis, J. Banys, G. Völkel and A. Klöpperpieper.

Phys. Status Solidi. 168 , 535 - 541 (1998).6. J. Totz, H. Braeter and D. H. Michel. J. Phys.: Condens,

Matter. 11 , 1575-1588 (1999).7. P. Freude, J. Totz, D. Michel and A. Klöpperpieper.

Ferroelectrics. 208/209 , 93-103 (1998).8. V. Samulionis, J. Banys and G. Völkel. Journal of Alloys

and Compounds. 310 , 176-180 (2000).9. V. Valevichius, V. Samulionis and J. Banys. Journal of

Alloys and Compounds. 211/212 , 369-373 (1994).10. V. Samulionis, V. Valevicius, J. Banys and A. Brilingas.

Journal de Physicue IV. C6 , 405 -408 (1996).

330 335 340 345 350 355 360 365 3700,0

0,3

0,6

0,9 x - axis

DBPI0.85

DBP0.15

α ,

cm - 1

T , K

FIGURE 5. The temperature dependence of ultrasonicattenuation. Arows show direction of temperature variation .

Acoustic Properties of Metals Under Large Torsion Strains

O. Yu. Serdobolskaja

Acoustic Chair, Department of Physics, Moscow State University, 117234 Moscow, Russia

Sound velocity and attenuation in metal rods under great torsion strains are investigated experimentally for increasing andalternating loading. The results in plastic region are related with calculation using the hysteron model.

Acoustic methods are widely used in researches ofdestruction process kinetics. Dislocations densityincrease and cracks formation leads to the change ofsound velocity and attenuation. In the present workacoustic parameters of longitudinal waves areinvestigated in metal rods under torsion deformationup to the strength limit. Samples of aluminum andbronze are examined. The pulse acousticmeasurements (f = 5 MHz) were carried outsimultaneously with measurement of dependence oftorque M on rotation angle �. The behavior of acousticand mechanical parameters for one of aluminumsamples is presented on Figure 1.

FIGURE 1. Acoustic and mechanical properties of thealuminum sample (vl = 6.4 km/s, G = 32 Gpa) undertorsion stress.

Acoustic wave propagated along the spool formsamples, so that the uniform deformation took placeonly in free of clips thin part of sample. Acousticparameters presented in Figure 1, are averaged overcross section.In elastic area (����0.5) the attenuation does not vary;by the beginning of plastic deformation the attenuationincreases due to scattering on arising dislocations; atthe beginning of microcracks formation the essentialgrowth happens. The sound velocity in elastic areavaries weakly (only with account of higher orderselastic moduli), and then decreases to the region of

cracks formation, where it begins to grow again.Probably, it is connected to fastening of dislocationsand diminution of internal tension.The interesting effect is observed at the breakdown oflarge cracks. The sound amplitude sharply drops, thenduring 10-15 s a signal is reverted virtually to the initialvalue. Apparently, the edges of the formed crack "sticktogether" at the further torsion and again begin to pass asound well.The rotating strain enables to apply reversal loads to asample. At cyclical loads the hysteresis is observedboth for mechanical and acoustic parameters (Figures2,3).

FIGURE 2. Mechanical hysteresis of bronze sample.Solid curve is calculated using hysteron model.

The hysteresis of a sound velocity look like "butterfly"as it does not depend on the strain direction. Howeveralready at amplitudes of a strain used in our experimentaccumulation of fatigue defects in one cycle is large,therefore the final velocity is less than the initial one.

FIGURE 3. Sound velocity hysteresis of bronzesample. Solid curve is calculated using hysteronmodel.

For the description of hysteresis loop we used themodel of elastic hysterons, offered in [1] forlongitudinal strains in rocks. Regardless to the physicalnature of hysterons (cracks, dislocations, grains and soon) it is assumed that such hysterons are of identicallength and have a rectangular hysteresis loop. Eachhysteron is "opened" up to load magnitude Po and haslength �l; on an inverse paths it is "closed" at pressurePc , and its length becomes equal to zero. The commonextension in plastic area (neglecting elasticdeformation) is proportional to the sum of openhysterons, which, in turn, depends on the hysteronsdensity �(Po,Pc). From Preisah-Mayergoyz (P-M)diagram in coordinates Po and Pc for a definite statisticsof hysterons we can obtain the form of hysteresiscurves.In our model we consider torsion hysterons, where thetorque M and the angle of rotation � correspond topressure and expansion� The signs of stress and strainswas taken into account. The proposed model allows toreceive the residual deformation and the fatigue effect.Note that the rotation angle is proportional to thealgebraic sum of open hysterons; on the other hand, thesound velocity is proportional to their total number anddoes not depend on their sign.The results of calculations are presented in Figures 2, 3by solid curves. The P-M diagram for the relativehysteron density ��(Mo, Mc) is presented in Table 1.

Table 1. P-M diagram.Mo 4 3 2 1 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

Mc1 10 10 9 8 6 5 3 2 12 20 19 18 15 12 10 7 5 3 2 13 40 39 37 30 25 19 13 9 6 4 3 2 14 80 78 75 61 50 38 26 18 12 7 4 3 2 3

Here the torques are given in relative units. Thesame diagram is for "negative" hysterons withopposite sign of torques. The density on the matrixdiagonal (Mo = Mc) increases as N n1= 2�10n where nis the row number, the density in each row decreasesas exp(-km2) (m is counted from diagonal, k is the fitparameter). After each cycle with some stressamplitude Mmax open hysterons with |Mc| < Mmaxremain. Mechanical loop is closed because thehysterons of opposite sign compensate each other.For sound velocity the loops are not closed, and wehave the linear dependence of velocity on the cyclenumber in plastic region (Figure 4)

.

FIGURE 4. Relative change of sound velocity at themaximal stress of cycle loadig. Black and white pointscorrespond to opposite signs of deformation.

However near to the threshold of destruction at acracking and large density of interacting defects theconsidered model can not be applied.

REFERENCES

1. K. E.-A. Van Den Abeele, P. A. Johnson, R. A. Guyer, Ê.R. McCall. J. Acoust. Soc. Am., 101, No. 4, 1885-1898(1997)