Propagation of elastic waves in one-dimensional composites

Materials" Science and Engineering, A 122 ( 1989) 71 -75

71

Propagation of Elastic Waves in One-dimensional Composites A. ALIPPI Department of Energetics, University of Rome L Rome (Italy) and lstituto di Acustica "0. M. Corbino ", Consiglio Nazionale delle Ricerche, Via Cassia 1216, 00189 Rome (Italy)

(Received May 30, 1989)

Abstract

Consideration has been given to piezoelectric composite plates formed by a series of alternating strips of piezoelectric material and epoxy. By the use of a model that takes into consideration the transmittivity and reflectivity coefficient of plate modes at the boundary between different media, dispersion velocity curves of such plates have been deduced and compared with the experimental frequen O' response. Symmetry conditions and the fact that they give rise to stop band and passband frequency edges are discussed.

1. Introduction

Analyses of propagating elastic waves in a composite material have been previously reported [1, 2] (for a general reference on wave propagation in periodic structures, see ref. 3). The principal problem is defining the frequency response of the material structure or, in the case of infinitely extended periodic composites, the dispersion relation of the waves. Testing of composites and tailoring of acoustic transducers are fields of interest [4] where knowledge of the propagation of acoustic waves is preliminary to any further investigation. Here, the study of elastic propagation in onedimensional composites is presented through a model that makes use of the scattering matrixes at the material interfaces. This permits determination of the field at each point of the structure, and therefore the effective velocity within the structure and the overall transmission function of a finite sample of composite. Results are presented relative to 1-3 composite plates of piezoelectric materials for propagation of Lamb modes of zero order, in the simplifying assumption that the frequencies of the waves are low enough that no higher orders can be excited. Frequency 0921-5093/89/$3.50

response measurements have been performed and compared with the theoretical model. 2. Theoretical model

We assume that an acoustic perturbation is propagating in a one-dimensional structure made up of several segments of two materials 1 and 2 which alternate along one direction and where no variation occurs in the normal directions. For each medium, the segment or layer thickness, the acoustic velocity, the mass density and the acoustic impedance are d, c,/9, Z = p c respectively. A number n of acoustic modes or, alternatively, different kinds of wave, are assumed to propagate in each layer. Amplitudes will be labelled a + for positive progressive waves and a- for negative progressive waves. Quantities in medium 1 have no primes, and quantities in medium 2 have primes. By the use of matrix notation, we describe the acoustic amplitudes at the left-hand end of each segment as column vectors [a x] and [a '-+] of dimensions n and n' respectively for the two media. At the limiting surface of each segment the boundary conditions will couple each mode propagating in one medium with all the modes transmitted into the second medium, as well as with all the modes reflected into the same medium, in a way that can be best described through the use of the transparency and reflectivity matrixes It] = [t'] and [r] = -[r']. The term ti/ is the ratio of the transmitted mode j in medium 1 to the incident mode i in medium 2 in the case where the media are infinitely extended in one direction so that no stationary conditions are established by the finite dimensions of the segments. The converse definition applies to the term t/i. Similarly, rkt is the ratio of the reflected mode l to the incident mode k in medium 1, and conversely for rlk' in medium 2. Matrixes [r] and © Elsevier Sequoia/Printed in The Netherlands

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Jr'] are square matrixes (n x n and n' × n' respectively), while matrixes [t] and [t'] are rectangular matrixes (n x n' and n' x n respectively). We shall further introduce propagator vectors [S] and [S'] as row vectors, defined by the single components S ± =exp(-Y-ikd) S '± = exp(~ik'd')

(1)

where the wavevectors k = ~o/c, k' = e)/c', with w the angular frequency, should be identified for each propagating mode. Multiplication of the column vectors [a -+] at the left-hand end of one section by the propagators [S ±] shifts the acoustic fields from the values that they have at the lefthand end of a segment to those at the right-hand end. With the notation just introduced and according to the scheme indicated in Fig. 1, we may write for the amplitudes of the acoustic waves propagating in structured materials a set of coupled equations that derive from the continuity conditions that are to be satisfied at each segment interface [a +] = [r] [a-] + It'] IS'+ ] [a '+ ]

(2)

[S'- ][a'- ]=[t][a -] +[r'][S'+ ][a '÷ ]

In the above equations, for simplicity, no indexing has been used in order to identify each specific segment of the structure where continuity conditions are written. Obviously, this should be correctly done if the whole system of equations has to be written. Equations (2) can be ordered in such a way as to consider the amplitudes with primes as functions of the amplitudes without primes or, equivalently, the amplitudes in one medium as directly given by the amplitudes in the left-hand adjacent medium: [a '+] = [Ll[a +] +[M][a-]

(3)

[a' +] =[P][a +] +[Q][a -] d" a"

>~

d ~ ab

*_t

-,. t b

o~__r, I

d a

-r-

~- r

a~

Fig. 1. Definition of p r o p a g a t i o n p a r a m e t e r s in a threem e d i u m structure.

where [L] =[t'][S'-] [M] ={t'][r]{S'-] [P] =[r'][t'][ S'+]

(4)

[Q] =([t] -[r'] [t'] [r])[S' +] At the next interface, the equations will be conversely written by replacing the quantities with primes by the quantities without primes, and vice versa. In this way, eqns. (2) represent a system of 2 ( n ' N ' + nN) recursive equations, with N' and N the number of segments of materials with and without primes respectively, with an equal number of unknown quantities, represented by the amplitudes [a ±] and [a '~] on each of the N + N ' segments, proportional to the incident wave. In order to compute the field in the structure, or to determine its dispersion relation, we may start by solving the equations on the very last boundary, where the overall output field [a+]last can be taken as a datum and [ a - ] l a s t is equal to zero. The equations are then decoupled from one another, and the system can then be solved in a recursive way as a system of decoupled equations, up to those [a +]first and [a-]first that describe the field at the first segment, which finally give the input field as a function of the output field. Before presenting experimental data relative to particular composite structures, we would like to discuss ideal cases that are of relevance to general considerations. The easiest example to be considered is that in which only one mode of the elastic perturbation is allowed in each segment, as it is the case of normally incident bulk waves in different isotropic media; in this case, of course, the matrix notation can be replaced by scalar notation. If the two media alternate with constant segment widths d and d' along the propagation direction, we may compute what the overall transparency T = [(alast+/afirst+)12 of the structure will be for different values of the transmittivity coefficient t between the two media as a function of the acoustic path lengths (or kd and k'd' products), for given value of the number N* = N + N' of elements. Figure 2 represents the transparency function T for N* = 5, 11 and 31, equal values of the acoustical paths k d = k ' d ', and a transmittivity value t of 0.99 between the two media. A noticeable drop in the transmission function is evident for a range of frequencies that would form a stop band in the case of infinitely extended

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~

0

N~ N*=31

N*=11

(a

0

.,'t:)2

R

(..d (d / c ÷ d'/c~)

o

Fig. 2. Transparency function T vs. frequency in a twomedium periodic structure, for different total numbers of

elements.

50

100

S

0

Ib

T ,5

Fig. 4. Transparency function T vs. number N* of segments for (a) w d / c = 6 3 ( ~ r / 1 8 0 ) and (b) ~od/c=64(~/180), with equal segment widths d = d'. 1

o

t~ 12

;'T,

(d/c ÷ d'/~

Fig. 3. Transparency function T vs. frequency in a twomedium periodic structure for different values of the transmittivity coefficient t between the two media.

structure. Indeed, the high value of t, wh!ch corresponds to media of very similar acoustic characteristics, is chosen in order to show clearly how the stop band features of a finite structure depend on the number of elements considered. Obviously, the stop band conditions are satisfied better where there is a greater number N* of elements contributing to produce Bragg scattering conditions. Correspondingly, the discrimination between passband and stop band becomes sharper, corresponding to an improvement in the filtering qualities of the structure. The number of side lobes in the frequency response curve is related to the number of segments. Figure 3, however, represents the transmission coefficient T of a five-element structure, for different values of the transmittivity coefficient t=0.75, 0.91 and 0.99 between the two media. The transmission function approaches zero for

frequencies within the stop band at decreasing values of t and, correspondingly, the stop band becomes larger. This is easy to understand, since the increased mismatch between different elements of the structure itself limits the transmission coefficient. Finally, we would like to show how the transmission T is changed by varying the number N* of elements at two different frequencies chosen outside the stop band but very close to its edge. Figure 4 represents function T vs. N* for equal acoustical paths kd = k'd' for two different frequency values: in Fig. 4(a), ~od/c=63(:~/180) and, in Fig. 4(b), ~od/c = 64(~/180), with the stop band edge ~od/c =64.5(7r/180). The transmission function then fluctuates between 0 and 1 with a periodicity which decreases as the frequency approaches the stop band edge frequency. It is worth considering the stop band frequency value as that for which the periodicity would go to zero. Outside the stop band, in fact, there is always a series of numbers N * of elements for which the structure is transparent to the acoustic wave. The limiting case N * is well known in the literature [4-6], as it gives rise to the definition of the

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dispersion relation of the waves obtained by setting the periodicity condition or Bloch condition. Indeed, we can make use of eqn. (2) at one interface between two adjacent elements for defining the field in one element as a function of the field in the adjacent element; successively iterating the process a second time, the field in one element (with a double prime) can be defined in terms of the field in the next equivalent element (with no prime). This gives [a "+] = [L*][a +] +[M *][a-]

(5)

[a"-] =[P *][a +] +[Q *][a-] where

[L*] = [L'] [L] + [a'] [P] [M *] =[M'][Q] +[L'][M]

(6)

[a" +] = exp(i¢)[a +]

with real ¢. The result for the scalar case of one single mode of propagation is the well-known dispersion relation [3, 5]

with k the effective wavenumber, and D = d + d'. Without discussing the well-known results just given, it is interesting to find the periodicity condition (or the dispersion equation) for the case when elements of three different media are alternating in a one-dimensional structure with widths d, d' and d" and wave velocities c, c' and c". This is

[P *] =[P'][L] +[Q'][P] cos(kD) =

[Q *] =[P'][M] +[Q'][Q] and [L'], [M'], [P'] and [Q'] are obtained with the correct permutations of the corresponding quantities with no primes. The Bloch condition is applied by setting

(7)

[a"-1 =exp( - i~b)[a -]

1{ ( dd, coso)

+7+75)

+ r' r" cos~

d d/

-~+7]c

+rr" COS (.o( - - -d- + ~ +d'7 d") c

-=O

I ----L I

I

I

i

I

I

~3--~

+rr

I

I

I

I

I

I

I

I

I

I

I

I

--i

~-0

I --H-~--~J_

t8

_!

!

!-

I

I

I -

-I

I I

14

-

I i

~'~ •

•

0

t

--

i

-I

; - -i-

0.7.

~ .4

I

i

I

I

I -;-

I

I

I

0

-iI

--

__1

I

-i-

-

-I I

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-i-

- !I

-I-

I

I 13

( d d d'/!

cosa~ - c

c~l-7)I

(9)

_1 .... 4-----3-:---,-i-i-

A

N "T" =.a

,

c

I

I.

/

N

~ / ~ I

e .£

I

e .~

C __

where D = d + d' + d", r, r' and r" are the reflectivity coefficient between different pairs of materials and t, t' and t" are the corresponding transmittivity coefficients. An example of an infinitely one-dimensional composite with three alternating constituent materials is given in Fig. 5, where the dispersion relation is represented according to the common practice of folding the curve at values of kD that are multiples of n. The reported case has been computed for three different materials: lead zirconate titanate (PZT) ceramic (quantities with no primes), nickel (quantities with single primes) and polymer (quantities with double primes), with thickness ratios did '= 1 and did "= 5. The velocity curves in bulk materials for longitudinal waves are drawn as straight lines (curve C, ceramic; curve N, nickel; curve P, polymer).

!

k (-n/d) Fig. 5. Dispersion curve for longitudinal bulk waves propagating in a periodic three-medium structure: ceramic PZT, nickel and polymer.

3. Experimental results The experiments were performed on piezoelectric composite plates, where the reinforcement

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consists of parallel thin rods of piezoelectric PZT ceramic sliced out of a plate with poling axis normal to its surfaces, and the matrix is epoxy resin moulded in between the ceramic elements. The surfaces of the composite plate thus obtained are then polished and fully metallized and the electrical impedance of the structure is measured vs. frequency with an impedance meter. In the frequency range below the threshold condition of excitation of the first thickness mode of the plate, only the symmetrical Lamb mode s0 can propagate in the plate with a real wavenumber. Thus the assumptions for a scalar theory approximation are fulfilled and the resonance conditions of the plate can be compared with the transmission function T of the theoretical model. Figure 6 represents the real admittance frequency spectra of composite plates made of different numbers N* of elements, represented in the insets at the upper right-hand end of each

d,. o:9 mr. A

'

J,

~

m

uJ (J Z

spectrum (dark for the ceramic and white for the epoxy) (from ref. 7). The resonant condition of one single element at frequency f = 1.75 MHz develops for increasing number of elements into a passband, which is well defined by the stop band edge frequencies with more and more elements. Within the stop band, a number of transparency peaks is marked with arrows at the frequency positions where the theoretical model predicts maximum values. Owing to the experimental conditions used, only symmetrical modes (solid arrows) can be excited and they correspond to the higher peaks, while antisymmetric modes (broken arrows) do not appear in the spectra. The increasing relative magnitudes of band edge frequencies with an increasing number of elements is evidence of the k symmetry selection rule, which for an infinite structure would give resonances only at values of the wavevectors k = 2 n n / D , with n an integer. In conclusion, a model has been presented for the calculation of wave fields propagating in a one-dimensional composite structure. This permits the boundary equations for each interface of the structure to be decoupled. A matrix representation has been used which reduces to a scalar representation when no coupling occurs between different propagating modes at the interfaces. The theoretical model has been tested vs. experimental results obtained in a composite piezoelectric plate measured with an impedance meter. The Lamb wave conditions of propagation are such that, within a certain frequency range, results can be directly interpreted with the prediction of the theoretical scalar model.

l.-r~

References

w

!~

1.2

,,~

I

I

1

[

I

1.4

1.6

1,8

2.0

2.2

2.4

FREQUENCY (M Hz}

Fig. 6. Measured admittance function vs. frequency for composite plates with varying number of elements.

1 S. Lees and C. L. Dadidson, IEEE Trans. Sonics UItrason., 24 (1977) 222. 2 B.A. Auld, in A. Alippi and W. G. Mayer (eds.), Ultrasonic Methods in Evaluation of lnhomogeneous Materials, in NA TO Adv. Study Inst. Set., 126 ( 1987) 227-240. 3 L. Brillouin, Wave Propagation in Periodic Structures, McGraw-Hill, New York, 1946. 4 T. R. Gururaja, W. A. Schulze, L. E. Cross, R. E. Newnham, B. A. Auld and Y. J. Wang, IEEE Trans. Sonics Ultrason., 32 (1985) 481. 5 S. M. Rytov, Sov. Phys.--Acoust., 2 (1956) 68. 6 B. Jusserand, E Alexandre, J. Dubard and J, Paquet, Phys. Rev. B, 33 (1986) 2897. 7 A. Alippi, F. Craciun and E. Molinari, J. Appl. Phys., in the press.