REFLECTION OF LAMB BENDING WAVES FROM THE

INTERFACE BETWEEN TWO BUTT-JOINED HALF STRIPS

I. P. Getman and O. N. Lisitskii UDC 539.3:534.2

The methods developed in [2, 4] are used to examine the problem of the incidence of an antisymmetric Lamb wave on the interface between two butt-joined half-strips with different elastic properties. The reflection of the energy of the incident wave across the interface is analyzed within a broad range of frequencies, and a study is made of the character of the stress-strain state in the neighborhood of the interface.

Let two half-strips occupy a region D: {--oo<x<oo; [zI~h . Here, the plane bound-

ary x = 0 is an interface separating materials having different properties. The elastic moduli and densities of the materials will be designated as A I, ~l, Pl, in the subregion x < 0 and as A2, P2, P2 in the subregion x > 0.

We introduce four-dimensional vector W with the components llYl = Ux, W2 = uz, •3 = Gxx,

W4 = ~xz- We will assume that the boundaries of the region D z = • are free of stresses

and that the vector W is continuous at the interface x = 0.

With the use of the vector W, we write the equations describing steady vibrations of a strip in the form

I dW "7" d"-'~ = TW, (i)

where T is the following differential operator:

0 ~+ 2~

T=

where ~ is the dimensionless frequency,

--0 0

-- pf~ 0

0 __p~2 ~+2~+ X+2-----~

= ~l~z.

1 0 ~+2~ 1 0

0 --0

0

Assuming in (i) that W= V(z) e ~vx and adopting homogeneous boundary conditions on the faces of the strip, we arrive at the following eigenvalue problem:

TV -- ?V = O, AV I~-+h = O;

A=] i'O (;~+2p$0 0 0 0 0l[ 1" (2)

As in known, real eigenvalues of problem (2) correspond to propagating uniform waves, while complex eigenvalues of the problem correspond to nonuniform waves which are localized in the neighborhood of the boundary x = 0 and decay with increasing distance from it.

The operator T is J-self-adjoint [6] in Hilbert space H with the scalar product (X,u

h 4

E xh~dz' i.e., (JT)* = Jr, where J is a unitary operator of the form

--h k=l

Rostov-on-Don. Translated from Prikladnaya Mekhanika, Vol. 27, No. 8, pp. 5~-59, August, 1991. Original article submitted December 4, 1989.

780 0038-5298/91/2708-0780512.50 �9 1992 Plenum Publishing Corporation

II 'if J = i 12 O' '

Here, O, and I are zero and unit operators.

The properties of the operator T lead to the conditions of generalized orthogonality of eigenfunctions V k [5]

~ It should be noted that the period-mean energy transported through the cross section

of the waveguide by a wave with the real wave number ys can be represented in the form [6]

P (W (?m)) = 4 t" (ou "-5 r,w - - $u - - 7w) dz = ..ih

f~ = "7-- (JV (W~), V (~'zh)) = P.~;

u = u.. (V,,3, rr = u~ (~'tO, o = ~.,~ (~,.,), " ~ = ~ = (%'t,3.

(4)

Here and below, the first index in the two-index notation corresponds to the number of the subregion (s = i, 2), while the second corresponds to the eigenvalue.

We will examine a problem in which the interface is struck by an antisymmetric Lamb wave of unit amplitude originating at x = -~.

W~s (x, z, 0 = V (z, ?,s) e'er"x-a~ �9

The components of the vector V(Ys are determined by the following relations [i]:

sh (z~z 2czzcz 2 sh cc,z ) us = iYs " ch a2h 2 , L-T~c~ ch~,h '

c, 2 ch ~ z ".Ys" ch =,z W s - - c h ~ h V~+ ~ ch=,h ;

o s = iu (L + 2ix ) u s + ~OWs; .r s = ~ (OUs + iu

~ = ~ Ore 2 Off" - - - ' T ' ~ = " ~ - % + 2~ "

(5)

To simplify the notation in Eqs. (5), we omit the index s = I, 2 corresponding to the number of the subregion.

With allowance for the radiation conditions in the first and second subregions, the wave field can be represented in the form

W, = V (Y,s) e"V'ff-m) + ~, A,.V (-- V,.) e i(-vI"*-a~ n

W~ = E B~nV (72,~) ellV~"*-~176 ; n

Im ~z~ ~ O; d~/d'~h > 0 (Ira "~zh = 0).

(6)

In Eqs. (6), Azn and Bzn are the unknown complex amplitudes of the reflected and trans, mitted waves.

781

. . . . . ~ , ~ f /

C ' / ! - -Y "J" !

f ~ ,

H r

Fig. 1

The condition of continuity of the vector W on the boundary x = 0 leads to the follow- ing functional equality (the multiplier exp (-i~t) is omitted)

v (viJ) + ~ A,.v (-- v,.) = V B,.v (v=.). n n

(7)

By using generalized orthogonality conditions (3), we reduce system (7) to an infinite sys- tem of linear algebraic equations

paj6~=~B2~bh ~(m=l,2 .... ). (8) k

We use B2k found from (8) to determine the amplitudes Aim from the formulas

AI~ = ~ B2hak~. (9) k

In Eqs. (8) and (9) , 6jm is the Kronecker symbol; the c o e f f i c i e n t s bkm and akm are found through the following scalar products of characteristic vector-functions:

=/(JV(v,,,), v(v,,.)), Irnv,,.=o bhm [(JV(v2,,), V(-- ?,m)), Im V,. ,~ 0 '

1 /(JV(y~h), V(--y~.,)), Imv,. ,= 0 ah"-- P~., [(JV(72k), V(?~m)), Imy~m=# : 0 '

=t--(JY(--7*m)' V(--Tzm)), I m ? , , . = O ; Pz" [ (Jr (-- ?zm), Y (?z.~)), Im ?z.~ :f= O.

System (8) was solved by the reduction method. Here, the order of the reduced system is determined by the degree of satisfaction of contact conditions (7). In order to satisfy the contact conditions for the displacements to within 1-2% of the maximum value of Ux(Z) in practical calculations, it is sufficient to account for four pairs of nonuniform waves in each half-strip (in addition to the propagating waves). In this case, outside a small region around the points of inflection z = ih, x = 0, the contact conditions for the stresses will be satisfied to within 5-6% of the maximum stress Oxx(Z) in the incident wave.

In the case of the presence of stress singularities occurring for certain combinations of contacting media, allowing for a large number of nonuniform waves does not lead to a deterioration in the stress contact conditions in the indicated neighborhood. This has to do with the fact that the stress series do not converge near the points of inflection; the singularity must be removed to improve convergence.

After the amplitudes were found from the algebraic system, we checked to see if the energy of the incident wave was equal tO the sum of the energies of the reflected and trans-

782

IA,31 i \

�9 / %.~,

Fig. 2

mitted waves. Besides being concerned with satisfaction of the contact conditions, the energy balance serves as an additional criterion of the reliability of the results.

The amplitudes A1n, B2n found from (8), (9) can be used to calculate the wave fields in both half-strips, as well as to perform an energy analysis of the reflection and transmis- sion of waves through the interface.

As is known, the propagation of bending waves in a layer has several distinctive fea- tures compared to the propagation of tension-compression waves. These waves are associated

.with a high level of dispersion at low frequencies, which is due to the more complex char- acter of their reflection and transmission through the interface.

Figure 1 shows dispersion curves along with the frequency dependence of the reflection coefficient in the incidence of the first antisymmetric wave on the interface. The solid lines show the dispersion curves of the left half-strip with the parameters %1 = 2.06"i011 N/m2, Pl = 1.53"1011 N/m2, 01 = 18.7"102 kg/m3, while the dashed lines show the same for the right half-strip with %2 = 0.59"1011 N/m2, ~2 = 0-33"i011N/m~, 02 = 2-7"102 kg/m~. By the reflection coefficient, we mean the ratio of the energies of the reflected waves to the

N

energy of the incident wave k r = p~l • E IA*mI2 P,m (N is the number of waves propagating

at the given frequency in the first half-strip).

The reflection coefficient increases monotonically in the frequency region 0 < ~ < i.I, when single travelling waves propagate from the left and right of the interface. Here, the coefficient reaches its maximum value at the frequency corresponding to the appearance of the second propagating wave in the reflected field. In this case, within the frequency range 0 < ~ < 0.2, the coefficient nearly coincides with the value k r = 10% obtained from the solution of the analogous problem in the applied theory of beam flexure.

The value of k r gradually decreases with the appearance of two propagating waves in the reflected and transmitted fields, with the value eventually reaching a minimum at the fre- quency ~*l = 3.16. A characteristic feature in this case is an increase in the amplitude of the first reflected nonuniform wave IAl31 in the neighborhood of this frequency. The corresponding complex wave number is ~ll 3 (the solid line in Fig. 2). The value of the maximum IA131 depends on the elastic properties of the contacting materials. Thus, if the properties of the material of the first half-strip remain unchanged and if as the material of the second half-strip we take a softer material with the moduli %1/~2 = 87.5; DI/~2 = I00; pz/p2 = 200 - such as ~',i - then at the frequency ~',i the first reflected nonpropagat-

ing mode is excited somewhat more readily (the dashed line in Fig. 2). In this case, the frequency ~''i is shifted relative to ~*l and is close to the boundary resonance frequency for the first half-strip [3].

As in the case of symmetric vibrations, such a frequency can be referred to as the boundary resonance frequency for flexural vibrations. However, it should be noted that non- uniform waves are excited to a somewhat lesser extent at the boundary resonance frequency when the vibrations are antisymmetric instead of symmetric [2]. (The difference for the given pair of materials is roughly fivefold.)

With an increase in frequency and the appearance of three or more travelling waves in the reflected and transmitted fields (~ > 3.33), the behavior of the reflection coefficient stabilizes in the neighborhood of the value k r = 49% - deviating from the latter by no more than 5-6%. The exception is the neighborhood of the higher-order boundary resonance frequen- cies ~'2 and ~,3. Here. k r has local minima which are less pronounced than the minimum at the frequency ~,i. (Five travelling waves propagate in the left and right half-strips at the frequency ~,3.) As at the frequency ~'i. nonuniform waves are also excited in the neighborhood of the interface at ~'2 and ~,3.

783

Yet one more difference that can be noted between the above-examined antisymmetric vibrations of a compound strip and its symmetric vibrations is the lower sensitivity of the reflection coefficient to the frequencies of thickness resonances of both half-strips. In order to more fully analyze the reflection and transmission of energy through the inter- face, it is best to examine the distribution of incident energy between the normal waves participating in the wave process. Here, we will concern ourselves with the main features of the modal distribution of the energy of the reflected and transmitted waves in different frequency ranges for the indicated pair of materials. Nearly throughout the investigated frequency range, all of the energy of the transmitted waves is transferred by the first nor- mal mode. This ceases to be the case only in the neighborhood of the boundary resonance frequency ~,i, where most of the energy of the transmitted waves is transferred in the sec- ond mode. The energy distribution between the reflected normal modes is considerably more complex and depends to a large extent on frequency. Thus, with the appearance of the second reflected mode, it becomes the more energy-intensive. An increase in frequency is accom- panied by a gradual reduction in the fraction of energy transferred by this wave. This frac- tion then increases somewhat in the neighborhood of ~,i, where much of the reflected energy is also transported by the second normal wave. In contrast to the transmitted waves, the first reflected wave is not always the most energetic wave in the reflected field. As a rule, the largest amount of reflected energy is transported by the waves with the highest (first) and lowest (last) wave numbers at the given frequency.

We studied a fairly broad range of frequencies, when up to eight travelling waves pro- pagated in both half-strips. At higher frequencies - when wavelength is considerably less than the thickness of the waveguide - the kinematic properties of the lowest Lamb waves (symmetric and antisymmetric) nearly coincide with the kinematic properties of a Rayleigh wave. Thus, the calculations performed here can serve as a basis for the solution of prob- lems concerning the incidence of a Rayleigh wave on the interface of two butt-joined half- strips.

LITERATURE CITED

I. I. A. Viktorov, Physical Principles of Ultrasonic Rayleigh and Lamb Waves in Engineering [in Russian], Nauka, Moscow (1966).

2. I. P. Getman and O. N. Lisitskii, "Reflection and transmission of sound waves through the interface between two butt-joined elastic half strips," Prikl. Mat. Mekh., 52, No. 6, 1044-1048 (1988).

3. V.T. Grinchenko and N. S. Gorodetskii, "Reflection of a Rayleigh wave from the free end of a waveguide," Prikl. Mekh., 20, No. 9, 12-16 (1984).

4. B. A. Kasatkin, "One class of diffraction problems for normal and surface waves," Akust. Zh., 28, No. i, 232-237 (1982).

5. A. G. Kostyuchenko and M. B. Obrazov, "Certain properties of roots of self-adjoint quadratitic bundles," Funkts. Anal. Priloz., 2, 28-40 (1975).

6. P. E. Krasnushkin, "Resonances in an infinite elastic cylinder and the transformation of complex waves into nondecaying waves," Dokl. Akad. Nauk SSSR, 266, No. 1 (1982)

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