AD-A149 216 ESTIMATE OF SOUND ABSORPTION DUE TO MAGNETOELASTIC iINTERACTIONS(U) NAVAL SURFACE WEAPONS CENTER SILVERSPRING MD K P SCHARNHORST 22 MAR 84 NSWIC/TR-84-226

UNCLASSIFIED F/G 2/ NL

EhhohEEmhhEmhEmoshhhEEEEmhhEI

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MWCROCOPY RESOLUTMA TEST CHART~m o Wa aw *D or3 W AM7% A

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A 1C TR 84-2U

SIMATE OF SOUND ABSORPTION DUETO MAGNETOELASTIC INTER9ACTIONS

B Y KtIRT P. SCHARNORST

RESARC AD TECHNOLOGY DEPARTMENT

22 MARCH 1934

1TI

VOEC 4 198

A

NAVAL SURFACE WEAPONS CENTER1.1.1 ~~Ddtm.Vrgnia 2244 o Sllw Sprin, Maryland 20910

.%o I -_

UNCLASSIFIED89CUmFT CLASSIFICATION Of THIS PAGE (fthn Da Eniwed) -_ -,-

PAGE READ INSTRUCTIONSREPORT DOC ENTATION BEFORE COMPLETING FORM

I. EPORT NUMBER " J 0. 3. RECIPIEisT'S CATALOG NUMBER

ISWC TR 84-2264. TITLE (and uae) S. TYPE OF REPORT & PERIOO COVERED

ZSTIMAZ' OF SOUND ABSORPTION DUE TONMUTOELASTIC INTERACTIONS Final

S. PERFORMING ONG. REPORT NUMBER

G. AUTNONf) . CNUMSR(4)Kurt P. Scharnhorst IR 4R 020

9. PEN1FOMING ORGANIZATION NAME AN ADORES$ 10. POGN LEMEN? PROJECT. TASC

AREA & VONK UNIT NUMUERS ?" "

Naval Surface Weapons CenterWhite OakSilver Spring, D 2091011. CONTROLLING OFFICE NAME AND ADDRESS 1 111119. N-tf

I. NUMBER OF PAGES5014. MONITORING AGENCY NAME 4 AOORESS(II diEffrent rom Conerolln Office) 10. SECURITY CLASS. fel ski* report)

UNCLASSIFIEDli6. OEC.ASSIFICATION/ OOWNGRAOINGSCHE, LE

16. DISAJOUTION STATEMeNT (of Wea ,ap*")

Approved for public release, distribution is unlimited.

17. DISTRIUUTION STATEMENT (of Mle uatjact enire I a.& 30, If ifferent hem Repe)

I. SUPLEM,TARY NOTES

Is. Tr wonos eCae arn. ever.. Sde If . ecooar OWm Identify by block a,,b,)

Sound AbsorptionMagnetola ticityEddy Currents

2&._A TRACT (CeNt0u.. 401 00We.. 414 It 4S SOa~e suE ma" 0ib bloc kirknmber)

Estimates of sound absorption by means of magnetoelastic induction ofconduction currents are given. Two examples are analyzed: (a) Induction ofeddy currents by means of magnetoelastic coupling between a viscoelastic absorb-er and a conducting ground plane, and (b) Induction of eddy currents in anelectrically conducting viscoelastic material in a uniform magnetic field. Weconclude that significant absorption occurs only if the average magnetic field _-7energy density and the mechanical kinetic energy density given by the product,--

DO , o' 1473 50,,1oR OF , NOV,,,,S OSSOLETE LSSF "':5/N I@1O"2143 s"u0,L01. o601s so~~ UNCLASSIFIEDS/N 0 102 - F 0 14- 6601 SECURITY CLASSIFICATION OF THIS PAGE (Whoa 0... Entered)

UNCLASSIFIED -* UUWUV CLASWIFICATION OF T1416 PAGSL (Wham Does Boo*

20. Abstract (Cont.)

of the mass density and the square of the speed of sound in the material are ofcomparable magnitude.

-. 7.

14-4M a112. LP. 04 14601 .UNCLASSIFIED

SECURITY CLASSIFICATION OF THIS PAGg(Uh., Does Entemd) -

5 4-

NSWC TR 84-226

FOREWORD

The present work arose out of a need to consider sound absorption processesin viscoolastic materials other than naturally occurring dissipation bymolecular relaxation. The latter becomes especially weak at low frequencies andshould be supplemented by alternative mechanisms. of several possibleelectromechanical and magnetomechanical dissipation processes we choose toinvestigate absorption due to mechanically induced eddy currents.

The work was carried out in the Nonmetallic Materials Branch of theMaterials Division of the Research and Technology Department under an IR grant.

Approved by:

Materials Division

ji ity Codes

NSWC TR 84-226

CONTENTS

chapter Page

ACOUSTIC REFLECTION FROM A UNIFORMLY MAGNETIZED VISCOELASTIC

PLATE ON AC ONDUCTINNGGROUNDPPLNEE ... .. ..... .. . ..

2 ACOUSTIC ENERGY ABSORPTION IN AN ELECTRICALLY CONDUCTING,VISCOELASTIC MATERIAL IN A UNIFORM MAGNETIC FIELD .. ....... 29

DISTRI BUTION . . . . .. .. .. .. .. .. .. .. .. .. .(1

ift*~

F + t.

+'..

NSWC TR 84-226

ILLUSTRATIONS

Figlare Page

I CROSS SECTION OF A MAGNETOVISCOELASTIC SLAB IN A VISCOELASTICMATRIX WITH EQUIVALENT SURFACE CURRENTS ............ 2

2 ACOUSTIC REFLECTION FROM VISCOELASTIC AND MAGNETOVISCOELASTICPLATES; 802 AND R (Bo = 0 gauss, a - 1.0 mho/u,Pc 2 /p0c0

2 - 0.1*(1.-iO.1), p/p0 - 1.0) . . . . . . . . . .. 203 ACOUSTIC REFLECTION FROM VISCOBLASTIC AND MAGNETOVISCOELASTIC

PLATES; 002 AND R (Bo - 15gauss, a - 107 mho/m,PC2 /pc 2 - 0.5*O.-iO.1), p - 1.0) . . . . . . . .... ,21

4 ACOUSTIC REFLECTION FROM VISCOLASTIC AND MAGNETOVISCOELASTICPLATES; 602 AN B O asa - 107dilu

2/pc 2 - 0.1*(.i.), p/po - 1.0),----22

5 ACOUSTIC REFLECTION FROM VISCOELASTIC AND MAGNETOVISCOELASTICPLATES; go2 AND R (Bo - 5 x 105 gauss, a _ 107 o/ho/m,

oPCo~o .o...-i.) .o .... .. .... 2o3i~

6 ACOUSTIC REFLECTION FROM VISCOELASTIC AND MAGNETOVISCOELASTICPLATES; 4o2 AND R (Bo - 106 gauss, a - 104 uuha/m,oc2/p0oc

2 - 0.1*(I.-iO.I), /p0o - 1.0) . . . . . . . . . .. 247 ACOUSTIC REFLECTION FROM VISCOELASTIC AND MAGNETOVISCOELASTIC

PLATES; 402 AND R (Bo" 106 gauss, a - 1o7 mho/r,p,.2 pC 2 = o.1*(l.-i0.1), P/po - 5.0) . . . . . . . . . . 25

8 ACOUSTIC REFLECTION FROM VISCOELASTIC AND MAGNETOVISCOELASTICPLATES; 602 AND R (Bo 10 gauss, a - 107 o/,P2 C o2- a .0*(l.-iO.1), p/po - 1.0) . . . . . ...... 26

9 ACOUSTIC REFLECTION FROM VISCOELASTIC AND MAGNETOVISCOELASTICB2 R 7' n0

PLATES; So AND R (Bo - 10 gauss, a - 10 mho/,P12/p.co2 - 0.1*(1.-i0.1), p/Po - 1.0) ........... 2710 DISPERSION RELATION FOR SOUND WAVES IN A VISCOEL ASTIC

ELECTRICAL CONDUCTOR ( - 10 gauss, a 104 mho/m,c2 /p0oc0 2 - 1.0*(1.-iO.1), P/po - 5.0) .......... 34

11 DISPERSION RELATION FOR SOUND WAVES IN A VISCOELASTIC-ELECTRICAL CONDUCTOR (Bo a 10 gauss, a -104 mho/u,

1c2/oCo2 - .0*(I.i0.), p/po - 1.0) .3......... 3512 DISPERSION RELATION FOR SOUND WAVES IN A VISCOELASTIC.

ELECTRICAL CONDUCTOR ( o a gauss, a _ 104 ho/ ,-2/poC2 - 0.1*(1.-iO.1), V/po - 1.0) . ...... . 36

13 DISPERSION RELATION FOR SOUND WAVES IN A VISCOELASTICELECTRICAL CONDUCTOR (Bo - 106 gauss, a = 104 mho/m,Pc2/pc 0

2 0.1*(l.-i0.1), p/po 1 1.0) . . . . . . . . . 37

iv

11~ ~ ~~ :~ DIPRSO REAINFO ON ASI A VSCEASI

NSWC TR 84-226

ILLUSTRATIONS (Cont.)

Figure Page

14 DISPERSION RELATION FOR SOUND WAVES IN A VISCOELASTICELECTRICAL CONDUCTOR (Bo - 5 x 105 gauss, a 104 mho/m,pc2/ 0oc0 2 - 0.1*(1.-iO.1), p/po - 1.0). ............ 38

15 DISPERSION RELATION FOR SOUND WAVES IN A VISCOELASTICELECTRICAL CONDUCTOR (Bo m 106 gauss, q = 10

5 mho/m,Pc2 /.c0

2 - 0.1*(I.-i0.1), p/p0o = 1.0) ........... 3916 DISPERSION RELATION FOR SOUND WAVES IN A VISCOELASTIC

ELECTRICAL CONDUCTOR (Bo - 5 x 105 gauss, a 105 mho/m,

oc2/ 0oCo2 - 0.1*(1.-iO.l), p/po = 1.0) ............. 40

17 DISPERSION RELATION FOR SOUND WAVES IN A VISCOELASTICELECTRICAL CONDUCTOR (Bo - 10

6 gauss, a - 105 mho/m,Pc2 1poC02 - O.2*(1.-i0.1), pipo - 1.0)............ 41

18 DISPERSION RELATION FOR SOUND WAVES IN A VISCOELASTIC'4 ELECTRICAL CONDUCTOR (Bo - 10

6 gauss, a = 105 mho/m,Pc2/poco2 " 0.05*(1.-iO.1), p/po 1.0) ............... .. 42

I. ,,

- .°

V/jvi ~1

~,.u. vv;* % **: %V% * .-.:-!

NSWC TR 84-226

CHAPTER I

ACOUSTIC REFLCTZON FROM A UNIFORMLY MAGNETIZED VISCOELASTIC PLATEON A CONDUCTING GROUND PLANE

The purpose of this report is to assess the performance of an acousticattenuator in the form of a viscoelastic sheet attached to an electricallyconducting backing, the sheet consisting of either alternating magnetized andunmagnetized strips of material or a distribution of magnetized disks in anunmgnetized lattice. In the present paper we consider the situation where themagnetized regions, assumed to be electrically nonconducting, are wellseparated, thus focussing attention on absorption due to induced eddy currentsin the backing or ground plane while neglecting interactions between magnetizedregions. We also treat the magnetized regions as if they were infinite inextent as far as the acoustic response is concerned, thus neglecting edgeeffects due to differences in the elastic responses of magnetized andunmagnetized regions. These elastic responses may be different not only becauseof the magnetic moment but also because of basic differences in the elasticmoduli and densities of the materials.

The two simplest magnetic structures in this category are uniformlymagnetized, either in the plane of the sheet or perpendicular to it. In thepresent paper we focus on uniform, perpendicular magnetization.

In order to come up with a readily calculable problem, we consider onlyone-dimensional geometries. We realize at the outset however, that finiteunifor ly magnetized plates do not generate uniform magnetic fields. Wetherefore have to make a number of simplifying assumptions to reduce themagnetoelastic interaction problem to a one-dimensional one. Consequently, theresulting theoretical developments will be highly approximate. In addition, wehave to make certain mathematical approximations. Nevertheless it is hoped thatthe results will convey at least the order of magnitude of the expectedattenuation.

Consider a uniformly magnetized plate with the magnetization vectorperpendicular to a conducting ground plane and bonded laterally to an -"

u=tgnetized viscoelastic material, as shown in Figure 1. Let the shape of theplate be either a circular disk or a long, rectilinear slab. We will argue thatwe may replace the magnetization by an equivalent surface current distribution,as shown in Figure 1. We may then solve for time varying fields above and belowthe ground plane as functions of time varying surface currents, 1o(z,t) (ampere-turn/cm), where the time variation of I is ultimately due to acousticvibrations in the plate.

- *** P % 'j*q% jW* *.

I i"-. '.

NSWC TA 84-226

A'AB

,C9

1011

2

*r -.V.-*

NSWC TR 84-226

We assume that the local magnetic dipole moment is frozen into theviscoelastic material. lence,it does not change because of the relativedisplacemet of two neighboring points of the material. The only change in thempmtization envisioned here is that due to changes in the local density ofdAwle moments which occurs as a consequence of such local relative displace-ments of the material. The magnetization, No, is the magnetic moment per unit ,-Volume: "\

voiNo

dVNO= d (1) ::

In line with our goal of considering a one dimensional problem, we assumethat the motion of the plate in response to a normally incident acoustic wave isstrictly in the a-direction as shown in Figure 1. This makes No and 10'f nctions of z only. Furthermore, in this case V x M. W 0 and since themost general source function for the vector potential, so, is:

& -- YMJ dV + da (2)

V S

we find that it is a function of the surface current 10 = Mo x I only - even%ben a compressional wave exists in the slab. Throughout the motion of the slabwe a tbherefore calculate the vector potential on the basis of the currentsource distribution shown in Figure 1. The source potential becomes:

1%

= f I(f°'t) da (3)

S

where I (zo,t) is a function of the local displacement u(zo,t). Thuscoupling between the magnetic and the viscoelastic problem occurs because of thepresence of u (Sot) in ao.

The dependence of I on u may be calculated as follows. Suppose we aregiven a distribution f(s) on a < s < b. Consider the new distribution ht(z') ona't

.-

NSWC TR 84-226

we have

h[g(z,t) f(z) (5).

In the present problem we are working with the transformation:

z'(t) - z + u(zt) a g(z't) (6)

and the distribution f(z) - Io * constant on the interval 0 < z < L, whichexists when no acoustic disturbance is present in the plate. Identifyingh(a'(t)) with I(z,t) at any given time, we find:

Itz'lt]=0z ) (7) ""-

Hence,in the presence of a compressional wave, the current at z' z + u(z,t) ismodified by ou(z,t)/Iz at z. For the long wavelength limit, which is the one weare interested in here 3u/z is small and we therefore disregard this dependenceand replace Ulz') by 10.

In order to solve for the acoustic response, we need the magnetic volumeforce exerted by the total magnetic field at any point inside the slab. We aregoing to asume that this force may be calculated in a certain average sensefrom the same equivalent surface current, I., that is being used to calculatethe magnetic field. Suppose we have solved the magnetodynamic boundary valueproblm for the current lo(z,t) and have found the vector potential at z > 0,A(x,t). Let A(za) be the Fourier transform. The resulting Lorentz force onan equivalent current loop will generally not be normal to the plane of theslab. But in keeping with our aim of simplifying this problem to the pointwhere only vertical forces and motions remain, we consider only the y-componentof the total magnetic field. We also consider only the case of a rectilinearstrip of material of width 2A and thickness L, since this geometry is ofprimary interest to us. The case of the circular disk can be analyzed inexactly the same way. We give the corresponding vector potential below. Hencethe Fourier transform of the Lorentz force per unit height and unit length ofslab on e.g.,a right hand surface current element of strength I(z,() at z is:

1(z,) - I(s,) X By(zw)

SA(z ,w) ,--- -I(z,u) X Az (8)

a:

4 **I.~ * -. A2....% * *..**-...- .t%~iSo o°

. .... ~al- -I CE _2 • . ..... . -. . ; --LT -_VIL 11u. - 7 w.- w -;1 r --.... .. .. . ...

NSWC TR 84-226

In order to obtain an expression for the total magnetic volume force (force/unitvolume) normal to the plane of this current, we first add the force on the lefthand current filament. This force is in the same direction, even though thefield is in the opposite direction, because the current is reversed. Dividingby the width of the strip, 2A, we obtain the average magnetic volume force or,more accurately, we obtain a simple analytical quantity which we will use inplace of the true average force inside the material:

1 aA(z 0)" UW-) -- (z,&,) x ZA (9)

From this force we have to substract the equilibrium magnetic force which existswhen no acoustic field is present because the displacement, u(z,t) in thesolution of the acoustic problem represents the displacement from positioncoordinates reached not only under the influence of static, mechanical forces r .but also under the influence of magnetic forces due to the static magnetization,No . Hence the resultant volume force is:

8 YA(zw) -,%(z)

P(z,M)-- X - x 3z

1.0 X S zw o(z):' '.3'.Az, 3 L (1a)s J-

This is the Fourier transform of the local magnetic force due to all sources;internal, due to other magnetic moments, as well as external, due to inducededdy currents in the ground plane. The latter force will enter the dynamicequations explicitly, whereas the magnetic force due to other internal momentswill be taken care of by means of a phenomenological modification of Hooke'sLaw. However, for the moment we will not distinguish between these varioussources and calculate the total vector potential resulting from all time varyinginternal (equivalent surface) and external (eddy) currents. The part of theresultant potential that is due to internal sources will be discarded later.

Let the applied vector potential for unit stationary surface current atz - zo, be ao(z,zo). This yields the slab's static magnetization which isunmodified by the presence of the ground plane if we assume that the magneticpermeability of the ground plane is that of free space, which we do. Then theapplied potential due to a suface current element to 10 at z = z + u(zo,t) attime t is ao(zz o + U(Zot)). Expanding this to first order with respect to Zo,or more accurately, taking the Frechet differential with respect to u(zo,t) atz * zo, we write:

ao(z,zo + U(zo,t)) ao(z,zo) + u(zo,t) u

- ao(z,z o) + &ao(zz o ) (II)

5-.-

V -\.' ~ V'~~: 4 ~*%%**~.*.~o** *.

16

NSUC TR 84-226

Nov let a and a be the Fourier transforms of the solutions for the totalvector potentials corresponding to a. and &so, respectively. Then:

Ao$z) - (z,:4o) dzo (12)

and:

A(zuv) - [a(zzo) + LA&U,zoaa)J dzo (13)

Hence, the total local magnetic pressure is:

L

00 /

P*(z10 S -.j az (0 w Z0Zod

I a f u(:0u") a 0 dz0 (14).

where we recall that the vector potential in Equation (14) is understood to beonly that part of the total potential that is due to induced eddy currents inthe ground plane.

This press ure has to be accounted for in the dynamic equations. WhenPa 0 these are:

Hooke's Law:

2 32u OP ~ 1aPic) azat at(1a

and:

Newton's 2nd Law:

ai~ (15b)

where P is the mechanical pressure, p the mass density and [c) the speed ofsound in the absence of magnetic forces. in order to obtain the correspondinguegnetoviscoelastodynamic equations we simply add the magnetic volume force on

6

NSWC TR 84-226 S

the righthand side of Equation (15b) and modify Hooke's Law implicitly bychanging the wave speed in the material from [c) to c-c(o). In other words,the internal magnetic interactions change the elastic properties of the material,conceivably even its viscoelastic properties, by moving the equilibrium state ofthe dynamic response to a different point on a nonlinear response characteristic.To account for this change, we simply modify Hooke's Law in the way indicatedabove. The resulting equations are: --

2lu Iai

pc2 - - .. (16a)z t St .. a-

7 - + Pm (16b)

We regard c as an emperically determined quantity. It's dependence on MO J.,

should be kept in mind throughout the remainder of this paper. When the changein performance of the magnetic slab is investigated as a function of theinteraction with the ground plane, via Pm, it must be kept in mind that asPm (specifically Io) changes, so does c(Mo), unless we assume that thelongitudinal modulus of the material in which the magnetic moments are embeddedis adjusted to conpensate for the effects of internal magnetic interactions.Conversely, if we change c while keeping P. fixed, we will ultimately . .encounter incompatible combinations of the two and may obtain unphysical results.

Since u(z,t) appears under the integral sign on the right hand side of . WEquation (L6b) this is an integro-partial differential equation. It can besolved readily by means of perturbation techniques and we indicate below howthis is done. A quicker way is by resorting to the following approximation. Wereplace u(zo,t) in Pm by its average value on o < zo L:

L-u(zo,) f u(zo,e) dzo i u(zo,w) a a(w) (17)

0

and obtain:

P(Z,) 20 - a(z,L,w) a(zO,

=04 [by(z,L,w) - by(z,O,w)] (18)

.. '

7

W . O.o

oO A

NSWC TR 84-226

In this form the equations may be integratedreadily. We are still takingaccount of the coupling between the acoustic and electromagnetic fields, but aresacrificing the point by point amplitude and phase relationship that existsbetween the local displacement and the y-component of the magnetic field. Inview of other approximations that were made earlier, this seems to be areasonable step at this point. The solution for u(zopt) will be a function ofa and later on we will have to solve for a using Equation (17). Sincea(*) will occur linearly on both sides of this equation, this step istrivial.

For the reflection coefficient we need the steady state solution ofEquation (16) with Pm as defined by Equation (18). Like a(w), the magneticfields in Equ~tion (18) remain unspecified for the moment. Introducing the timedependence Clot and setting v = au/st we obtain:

2 dv m i.-

PC . (19a)dz

dP"''- - + + P (19b)

Substituting Equation (19a) into Equation (19b) one finds:

d2v 2 ikd 2 k ; =- i PR ; kM (20)

This inhoogeneous, second order, ordinary differential equation must now besolved subjict to certain boundary conditions. Introducing the incident andreflected waves:

P-- k iko(z-L) , 3 ik"(z-L)

V 0 - 1 [Alp iko(zL) - 0iko(z-L]

and requiring that both v and P be continuous across the boundaries, we find:

z = L ; pocov(L) B E1 - Al = PoCovo (21)

and:

k dvs*= L ; - k =nA I +EBIn= Po (22) "

~ ~ .'. ... .. o

F ri --I V- N7 %r . %- , TI i - 7 - "- 0

NSWC TR 84-226

Defining the known input admittance of the backing as G(O)=-v(O)/P(O), one has:

sOdv (23'X 0 G (O) L - v (0) =0 (23) .;

dz

Combining Equations (21) and (22) we obtain the boundary condition on v at zaL:

M2ikA dv ioCok_z = L ; (24)

pc dz PC/

and from Equation (23) at z=O we have:

d ik (25)dz P.C G(O)J

Suppose the solution of Equation (20) has been found. The final step is tocalculate the reflection coefficient, B2 - IBl/AuI 2 . Using Equations (21) and(22) one finds that:

oCov (L) I - B vk1 I -pco G(L) (26)PI~l' - ik (l oo (L 26 ''

where, by definition, G(L) is the input admittance of the magnetic layer.Equation (26) will therefore yield the answer to the problem once v(z) has beenfound.

The most compact expression for v(z) is obtained with the aid of theGreen's function, g, of this problem. This function must satisfy mixedhomogeneous boundary conditions according to Equations (24) and (25), i.e., forg we must have:

'I°'

= 0 ; DI(g 1 EVg4 a V() g 0 (27)

.;.

ikpoco

D2(nL 00S (28)

Referring to the general solution of this type of problem in three dimensions asgiven in standard texts, the solution of the present problem may be written:

9

o~r'~ .* % % ' .*. . * ~~. 4.

NSWC TR 84-226

r1)., m dv(z'w) dg(z,:°'e) zo - L6.-v(zdw) - z w dZ j ,, 0 "19zo 0 :

L" (" Pm(z°,w) g(zpz°9#A) dz° ; 0 < z < L (29)

where g is the solution of:

dg2 (z,zom) 2.(30)

+ k SZzo, - - 6(z-zo ) (30)

In Equation (29) the usual surface integral has been reduced to functionevaluations at the two boundary points and the volume integral has become a lineintegral. Using the boundary conditions, Equations (24), (25), (27), and (28),one finds:

v(z,") "1 + #2 (31)

where:

'1 " g(z,L,w) (32)

and:

I2 - 'L P,(zO, ) g(zzOw) dzO (33)The function *1 therefore yields the reflection coefficient of thenonmagnetic layers, whereas #2 yields the effects due to magnetization.

Had we not used the average displacement, a(m) in Equation (18), wewould now have a double integral on the right hand side of Equation (33) with vappearing in the integrand. Equation (31) could,therefore,be used as thestarting equation for a perturbation expansion of the solution. For instance, -with the aid of Equations (14), (31), (32), and (33) one finds that to firstorder, the solution for v is:

10

NSWC TR 84-226

V (a*, ) g(z,L,l) - g(zo,L,w)

aa(z' ,ZoW)

3z azo g(z,z',w) dz' dzo (34)

with remainder:

0 a(t',tW)

Av(zw*) -\cIfv(toa aW t

*g(5,tW) g(zz',w) dz' dz dt dt(3

" followsais

petrbto series is::?-

~2- '~)~~r)(36)

The ratio of the magnetic* field2eeg density, Polo~ and themechanical kinetic energy density PC in Equation (36) is a small quaotityin all real magnetoelastic materials. Furthermore, it may be assumed thatlIki is of order unity or larger at all frequencies. Hence, X. is a smallnumber and the expansion converges rapidly. On the other hand, the smallness ofthis parameter also tells us that the effects we are going to observe will beweak since the solution with finite Pm will differ little from that for anonmagnetic slab.

In order to find v explicitly, we need g. By definition:

yl(z) y2 (zo) z zo

g (37)~

a Y2(z) Ylz) z zo

where yj. and Y2 are two linerarly independent solutions of:

LIZ +k~y 0(38)hteo

~..r~r~d '~ % ~ * % ~ ~ * % * % % * ~ * * *4 . - ..-%*

-7.. - - A - .

NSWC TR 84-226

and £o,(yly2) is the Wronskian:

jj dY2 dyl)£o(ylsY2) XYI dz .- Id~ (39) y

Evidently, & will satisfy its boundary conditions if yl and Y2 are chosensuch that:

zin 0 DI(yl) 0(40)

zinL ;2Y2

We use the two linearly independent functions:

yl sin(kz -1 (41)

and

Y2 - inlk(z-L) -021 (42)

with Wroskian:

£0(Yj,Y2) -- k sin(kL - 1 -2 (3

Applying the boundary conditions yields:

tanelS -ipc G(O) (44)

and:

tanIS 2J a OLc (45)

for the two integration constants in Equation (31) and the problem is solved.

12

NSWC TR 84-2267..:

For g(z~z,99W) one finds:

[sin[kz- 11 sin[k(L-zO) - 2 ) z zo ;:

Removing the singularity in the slope of the Green's function from the integralin Equation (33) leads to a more explicit form of the solution:

zik [Y(z) Y2(zo) - Y2(z) yl(zo)] Pm(zo)

v(z) - c1 yl(z) - c h dzo (47)

where:

ik f Pm(zo) Y2(zo) dzo=l c 2 + -- :( 4 8 )PC1

0

and:

2ikAl Y2(L)-c2 (49)

Observing that the Wronskian is independent of z, we note that: ......

I LA 0 (Y 1 , Y 2 ) = - ' 2 - 2 Y l ) - Y I ' - i l''':L jL -L "...:

=-Y 2(L) D2[Y(L)]

Substituting into c2, we find that:

2ikA1 (50)2 -;c" D2[y1(L)]

'13

,.:.,,'.

,,q:13 ""* .. ~.~'

NSWC TR 84-226

Before writing down the reflection coefficient, ye calculate a(m).Writing Equation (31) in the form:

v(z) = Alfl(z) + a(*) f2(z) (51)

where:

fl(z) * 2ik S(z,L,a) (52)

and:

Lik f Putzo,*) ?ii ef2(z) - g-(zzo,) dzo (53)

0

and using Equation (17) one finds:

a(*) (54)

where:

e~u) -I - '2

-" / fl dz (55)

and:

4 " ' f2 dz (56)

For the purpose of estimating the performance difference between magnetized

and unmagnetized states we write B in the form B - go + A, where B0 is thereflection coefficient in the absence of magnetization and AB is the partdepending o!. A measure of the effectiveness of the magnetization for twootherwise similar materials is given by:

14• %;-

NSWC TR 84-226

Rt [181j2 -IB.121 2 Real [a So*] + 1AS12

We therefore want to calculate both 2 and So . With the aid of Equations (24),(26), (51) and (54) we find:

1 (fl(L) (1 - #2) + 41 f2(L)- C ° (L) + - (2 + P0c fI(L)] (1 - 2

) + P0 0% 1 f 2(L) (58)

Setting *2 - 0 P. we obtain:

0 -P 0o -0 f 1(L)-- c G(L) p(L)o (09)

so 0 o 2 + oCo fl(L)

where G(L)o, is the input admittance of a nonmagnetic layer. An explicitexpression for this function may be obtained by observing that according toEquations (46) and (52):

2i sin (kL - 0I) sin(82) (60)PC I sin (kL- 0 1 - 02)"

Substituting in Equation (59) and using Equations (44) and (45), we obtain thewell-known expresssion:

G( - tan (UL)I - ipc G(o) tan (kL)

With the-aid of Equations (58) and (59) we now find the desired expressions forB, 0 , and A:

- + o Co G(L)] , GIL' (61)

6r 1where: "--

L11 ocoso GL (62)

+o ., C G(L)o16)

15

I*.

NSWC TR 84-226

from Equation (59) and:

q f2(L) (63)

Hence from Equations (59), (61) and (63):

f2 (64)

[T-- 2"]

Combining Equations (59) and (63) we also obtain:

L ~_ J Pc° *1 f2(L) "so 2(1 + Po o fI(L)) (I - 2) (65)

At this point we have an explicit expression for R, the change of thereflection coefficient due to the magnetization M.o, in terms of the Fouriertransform of the vector potential in Equation 14. Hence, as a final step wecalculate the quantity "a(Zzou)/(*) - &a(zzoPW)/8zo from the gradient ofthe applied potential in Equation (11), i.e., from hao(Zzo,) - a(")$aol(&zo,9l*)IzoU0 O. Because of our approximation of u( 0 ,N) it is notnecessary to perform the differentiation on z., and we calculate the resultantpotential a(z,zo,w) from the applied potential ao(z,zo)e-igt. Clearly, if wedid not want to use the average of u in the dynamic equations we could obtain theintegrand in the expression for the magnetic pressure by simply differentiatingthe expression for a(z,zo,w), as calculated below, with respect to zo, andmultiplying by u(zo,*). Conceptually the potential a(z,zo,w) is the result ofapplying a bifilar current at z-zo with unit amplitude and frequency w. -Physically we are dealing with the motion of a constant current at z - z+ +U(zo,t). This current is changing its flux linkage with the ground plane andhence is inducing eddy currents. If the motion Su(zo, )/8t were uniformwe could have derived the form of hao(Z,Zo,t) from the classical theory ofelectromagnetism in moving media. For if the applied flux is denoted by * wehave for any arbitrary current filament in the plane of the boundary with areaelement da and line element ds:

itjLa dsm JEo ds

- V X(v X bo ) * da - (v X bo ) * ds , (66)

16

0, ..

-::::q.:

NSWC TR 84-226

i" e":

-- =- X bo(z,zo) (67)

or:

Aa0 - - u X bo - -u X (boz k + boy j) "iu boy

t 3ao t 3ao0';:Iu A . u (68)

since 80, is a function of (z-zo ) and the subscript zero on the electricfield Eo, the magnetic field bo refers to "applied" quantities. Equation (68)was obtained previously, Equation (11), in spite of the fact that in the presentcase motion was assumed to be uniform. This agreement is evidently due to theapproximation we made in the derivation of Equation (11). Had we gone to secondorder, we would have picked up a term of the form Iu(zo,t)1 2 a2ao/3zo2Iu=Owhich would have introduced the curvature of the potential as well as the firstharmonic, with frequency 2w.

The differential equations for a(zzo,W) are:

z > 0; V2a = -iaolO4(z - zo) [6(y - A) - 6(y + A)] (69)

and:

da ."-.

z 0; V2a moo -idaoa (70)

where the time dependence eiIt has been cancelled on both sides. TheFourier transform of the source potential is logarithmic and may be representedby the following integral:

Volo k1Z -of sin(ka) sin(ky) dk -..,

= -- • (71)0

Solving the boundary value problem by requiring the continuity of a(z,zow)and the parallel component of the magnetic field, i.e., of the gradientsa(,£oU)/I£ at x - o leads to the Fourier transform of the totalpotential a(z,zow). When z > o:

17

2" -. 2!2"2 "

NSWC TR 84-226

a(z,z0.w) -a(Zzo) + - e -~~ 0

*in(kA) sin(ky) (-.-~d 9 (2

where q (k2 - iumouo)lI2. in passing, we note that the correspondingsolution for the case of a circular disk of radius bl is:

a~zz0,.) - 01&bf {kI~Z-oI , -k(z+zo)}

*JI(kbl) Jl(kr) dk (73)

where JI(x) is the cylindrical Bessel function of order one which is regularat X - 0.

Note that the absolute magnitudes of the integrands decrease exponentiallywith increasing kc. The jiportant contributions to the values of the integralstherefore occur in the neighborhood of kc a 0. The intergals are readilyevaluated if in the neighborhood of kc - 0 the expression ((k-q)/(k+q)) isapproximated by an exponential function with an exponent linear in kc. Wetherefore write:

f(k) - -explg(k)J (74)

and expand g(k) -log [f(Ic)] in a Taylor series near k-O to first order in k.We obtain:

g(O) -logif(0)] 0

and:

1 dfj -2 j - -2dk f (k) dk ~ 0 k ~ I~h

Ulk-0~~~-. *5**ia:OkO =v

NSWC TR 84-226

Hence:

2

f(k) 'exp[-k(l + i)6] ,62 - 2 (75)

and 6 is the electromagnetic "skin depth." Substituting this into a(zZo,w)and integrating one finds:

- AOIO [(z -zo0 2(y )2 *a(z,zo,%) -log(z , -)2 +

(z + zo 2 + (y +-

-- ( 7 6 )

Evidently, the part of a(zzoiw) in Equation (76) that is independent of6 is due to internal sources. The only reason for carrying it to this pointin the calculation was that it was needed as the source-term for the eddycurrent problem. Its effect on the dynamic response of the plate has alreadybeen accounted for in Hooke's Law, Equation (16a). We will therefore discard itnow.

Substituting the remaining term in Equation (76) into Equation (18) andsetting y - + A (y - -A was accounted for when the volume force wascalculated) one finds:

20°2alo z + L + x I

Pm(z') (z + L + ) 2 + (2A) 2 z + L + x

(z + x) l (77)(z + x) 2 + (2A)2 Z x (1 +

where ,aoIo2 =B2 =(gauss)/4w is the magnetic field energy density per cm3

in cgs units.

The expression for Pm(zw) may now be substituted into Equation (61) toyield the reflection coefficient for a normally incident pressure wave and intoR, Equation (57), to yield the change in performance.

We have evaluated R for several different combinations of p, c, B and6, see Figures 2 through 9. The plate was assumed to be 10 cm thick and 20 cmwide, i.e., both L and A in Figure 1 were 10 cm. The XFIL integrationroutine of the NSWC/WOL computer library was used to calculate the variousintegrals in Equation (64).

19 'a

NSWC TR 84-226

cu

* *0

-p-h

I d

>

QLIO

LL g'I-jo

..

00001 00910 00200 00210- 0090- O0*t-

(iN3383d 001+ 01 O1-)8 (T 01 0) OSI.L38

20

NSWC TR 84-226

cu

la

UQ.

- - 4N* a zh.C, I ,

It qP~

II U-

I.

ujl~

Son UA

w IL

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0 le.

NSWC TR 84-226

cu

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NSWC TR 84-226

cu

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zl

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NSWC TR 64-226

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24u

NSWC TR 84-226

cu

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to

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NSWC TR 84-226

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26

NSWC TR 84-226

cu

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27

NSWC TR 84-226

As indicated in the text we expected to have to use rather large internalfields in order to obtain observable effects. The results bear this out; unlesswe apply at least several hundred thousand gauss, unless we use an Mo thatproduces a 3 of this order of magnitude, there are no discernible differences inthe responses of magnetic and nonmagnetic plates. We find that the energy ratioin Rquation (36) is indeed the controlling parameter in this situation. Theconductivity of the substrate is of relatively minor importance.

In closing we might add that although we believe the order of magnitude ofthe effect was calculated correctly, we could have done better had we used theperturbation expansion of the solution in our numerical calculations. Inretrospect we feel that the phase information that was suppressed when the localdisplacement in the magnetic volume force was averaged, might have had asignificant influence on the details of the frequency dependence of a2. Onthe other hand, in view of the many conceptual simplifications we have made inthe course of solving this problem, this refinement would have been academic.

7. -

II&

oo.oI

I .,.I b ..I-

I -,

28.

~ ~.*~: ~ ~-.': -2."

NSIC TR 84-226

CHAPTER 2 .-. %

ACOUSTIC ENERGY ABSORPTION IN AN ELECTRICALLY CONDUCTING, VISCOELASTICMATERIAL IN A UNIFORM MAGNETIC FIELD

In Chapter 1 the sources of the time varying magnetic field on the one handand the electromagnetic absorber on the other were spatially separate. If themagnetoviscoelastic material was made electrically conducting both functionscould be combined in the same volume. Two apparently different mechanisms wouldnow give rise to conduction currents. In addition to the mechanism discussed inChapter 1, which we think of as a 3B/3t effect, there would now be the (v XB) effect, which is due to convective transport of charge. Both effects are dueto the relative motion of "sources" and "observers." In the first case the"sources" move, hence aB/at at the observer was picked up as (v X B) inChapter 1, where v referred to the motion of the source (the magnetic moment).In the second case the "observer" moves, i.e., v refers to the motion of theobserver (the mobile charge carrier). Note that in the former case relativemotion of source and observer is detectable only because B is nonuniform.

The method of analysis used in Chapter I is too approximate for a situationin which the magnetic material discussed in that section simply becomesconducting. The direction of the magnetic field inside the magnetized slabwould also be wrong, since it is roughly in the same direction as the localdisplacement and therefore eliminates the (v X B) effect., If on the other handthe uniform magnetization is assumed to be parallel to the ground plane insteadof perpendicular to it, then the internal induction will also be roughly uniformand parallel to the ground plane and effects due to finite conductivity aremaximized. These effects may then be analyzed readily if at the same time weneglect the magnetic field variations that were the source of dissipation inChapter 1, namely those due to local displacements of magnetic moments under theinfluence of dilatational waves and focus only on convective charge transportarising from the translational motion of the molecules; i.e., on the (v X B) .... -effect. The constant magnetization Mo may then be replaced by the constantmagnetic induction, Bo, where Bo - Po~o, or equivalently we mayconsider an unmagnetized material in a constant parallel field with a wavetravelling at right angles to the field direction. In fact, since we reallyonly need the attenuation constant, or the imaginary part of the propagationvector, k, in order to estimate changes in power dissipation, we will only solvefor a wave in an infinite medium rather than for reflections from a layer.Henceowe are looking for the complex dispersion relation k(w,o,B).

Under the action of a pressure wave in the material we think of the mobilecharge as being "carried along" by local displacements of molecules, i.e., thereis no charge separation. The (v X 3) electric field creates a current, J, and

2

29 ,- .

.a.,: : __ .. *.-..:::.:.w'- . °- * *

NSWC TR 84-226

this current locally exerts a Lorentz force, J X B on the material. As inChapter I the equations of motion are:

+_J X H (78)

and:

OC2 Ov ap (79)

As in the theory of magnetohydrodynamics, the equation for the magneticfield is obtained from:

V X = B (80)at

V X B = la (81)

and:

a a[E + (v X B)) (82)

in mks units, where the last equation is stricly true only for a uniformvelocity, v. Eliminating J and E from Equations (80), (81), and (82) yields:

- v2 + V X (v X B) - . (83)"°t

for the magnetic induction. We assume that B consists of a strong, stationarybackground field, B9, and a relatively weak response field, b; B - Bo + b.Furthermore, if Bo is assumed to point in the x-direction and v in the -s-direction, then J is in the y-direction and the response field, b is parallelto Bo.

Assuming. the time dependence e-it in Equations (78), (79), and (83)yields:

iM(V X i) X - pc 2 AV- pw2V (84)

30

4.,

NSWC TR 84-226

and:

L V2B + V X (v X B) - -iwb (85)

Taking account of the fact that the only coordinate dependence is on z, we find:

-iw d(B2) d2v + '286)

2iwc 2 d " d-z2 -C2

and:

d(vB) d2B i'ib(7dz z2 + iob (87)

Using the fact that Bo, is constant, Ib/B9 l

NSWC Ti 84-226

where:

x Mc

and:

y

The expression in the curly bracket of Equation (92) therefore modifies the

propagation constant ko - w/c which holds in the absence of a background

field or when oao. Note that the parameter x may also be written:

2 (93)2 2ko26

where:

6 ;2(94)

is the electromagnetic "skin depth," or:

Il2x S x 10-2 (95)

where Ao is the wavelength in the material in the absence of magnetic effects.

The two parameters, y and 6 are seen to be the controlling variables of this

problem, as they were in the reflectivity problem of section (A). Let:

4., Henry+P o 4w7'- lt

-_. .

a - 102 Mh!.o

* - 2w x 102 Hz

c = 1.5 x l03 -7

sec

32

-. . • ,,-v . .

- -- . . .. . " " . .. ., ,, " " '*,*' " " " " % • . • ." " " " ,.." ." ." " " P,...r Y_,o=

"ar'.'

NSWC TR 84-226

Bo 103 gauss = 10-1 Weber

-103

Then:

x .45

and:

y 3.54 x I0-6

Hence,with these orders of magnitudes for the parameters, the ratio of magneto-static to kinetic energy density is very small; y-o and from Equation (92) itfolows that the dispersion relation is the same as in the case of o - o.

The induction inside the material has to rise to the order of magnitude ofseveral hundred thousand gauss before the dispersion relation, and hence theabsorption mechanisms are significantly altered. Note also from Equation (88)

that:

b - L " (96)

Since vo in this type of situation is of the order of magnitude of 10- 5 to

10-6 u/sec, the assumption that ib/Boi

p~I p 7

NSWC Tr' 84-226

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NSWC TR 84-226

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NSWC TR 84-226

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NSWC TR 84-226

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NSWC TR 84-226'

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