DTIC. Naval Research Laboratory. Influence of Shear Modulus on the Behavior of Acoustically Transparent Materials ELECTE J, JUN 3 0 IM

Transcription

1 ID INFLUENCE OF SHEAR MODULUS ON THE BEHAVIOR OF I ACOUSTICALLY TRANSPARENT MATERIALS(U) NAVAL RESEARCH LAB MASHINGTON DC P S DUBDELDAY 36 APR 66 NRL-NR-5745 UNCLASSIFIED F/O 1711 ML MoEs"n

2 L3, 6 m 46

3 Naval Research Laboratory Washington, DC NRL Memorandum Report 5745 April 30, 1986 N Influence of Shear Modulus on the Behavior of Acoustically Transparent Materials Cn - P.S. DUBBELDAY * Materials Section Transducer Branch Underwater Sound Reference Detachment Naval Research Laboratory P. 0. Box Orlando, Florida DTIC ELECTE J, JUN 3 0 IM Cm) U- Approved for public release; distribution unlimited

4 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE REPORT DOCUMENTATION PAGE la. REPORT SECURITY CLASSIFICATION lb. RESTRICTIVE MARKINGS UNCLASSIFIED None 2a. SECURITY CLASSIFICATION AUTHORITY 3 DISTRIBUTION IAVAILABILITY OF REPORT Nennp Approved for public release; distribution 2b. DECLASSIFICATION IDOWNGRADING SCHEDULE None is unlimited 4. PERFORMING ORGANIZATION REPORT NUMBER(S) 5. MONITORING ORGANIZATION REPORT NUMBER(S) NRL Memorandum Report 5745 None 6a. NAME OF PERFORMING ORGANIZATION 6b OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION (If applicable) Underwater Sound Ref. Det. Naval Research Laboratory Code 5975 None 6c. ADDRESS (City, State, and ZIP Code) 7b. ADDRESS (City; State, and ZIP Code) P.O. Box Orlando, FL None 8a. NAME OF FUNDING/SPONSORING ORGANIZATION 8b. OFFICE SYMBOL (if applicable) 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER ONR/NRL Non-Special Focus Prgm Code 5900 Various 8c. ADDRESS (City, State, and ZIP Code) 10. SOURCE OF FUNDING NUMBERS P.O. Box PROGRAM PROJECT TASK WORK UNIT Orlando, Orlando,_FL_ _ FL ELEMENT NO. NO NO RR011- ACCESSION NO N TITLE (Include Security Classification) Influence of Shear Modulus on the Behavior of Acoustically Transparent Materials 12 PERSONAL AUTHOR(S) Dubbelday, Pieter S. 13a. TYPE OF REPORT 113b. TIME COVERED/ 114. DATE OF REPORT (Year, Month, Day) 15. PAGE COUNT Interim IFROM_1LjSL to 15L April SUPPLEMENTARY NOTATION 17 COSATI CODES 18 SUBJECT TERVIS (Continue on reverse if necessary and identify by block number) FIELD GROUP SUB-GROUP rho-c material X reflection of acoustic waves, acoustic transparency 19 ABSTRACT (Continue on reverse if necessary and identify by block number) Materials are under construction that match the density rho and dilatational sound spced c as closely as possible to the values of seawater, while maintaining sufficient rigidity to serve for structural purposes. The demand for rigidity implies a larger shear modulus than is typical for the usual rho-c elastomers. Matching of density and sound speed results in transparency for fluids only; the finite shear modulus in a solid admits the presence of a shear wave, which causes deviation from ideal rho-c behavior. In this report the effect is analyzed of a finite shear modulus on the reflection of plane waves by an infinite plate of the rho-c material. Examples are given of the reflection coefficient as a function of incidence angle for various combinations of density and sound speed close to ideal, and various ratios of plate thickness to dilatational wavelength. The effect of a finite loss factor in the shear modulus is shown. 20 DISTRIBUTION/ AVAILABILITY OF ABSTRACT 21 ABSTRACT SECURITY CLASSIFICATION ]UNCLASSIFIED/UNLIMITED 0' SAME AS RPT 0-OTIC USERS UNCLASSIFIED 22a NAME OF RESPONSIBLE INDIVIDUAL 22b TELEPHONE (Include Area Code) 22c. OFFICE SYMBOL Dr. Pieter S. Dubbelday 305, Code 5975 DD FORM 1473,84 MAR 83 APR edition may be used until exhausted SECURITY CLASSIFICATION OF THIS PAGE All other editions are obsolete i UNCI,AS; I F I ED ~~~~~~~~~~~~~~~...'.%,.:...,.,.." _..... _., -.-:......_...,'.,-

5 SECURITY CLASSIFICATION OF THI1S PAGE (This page blank) 07. SECURITY CLASSIFICATION OF

6 DUBBELDAY CONTENTS Abst ---- act AbTRDCTI i ANALYSIS RESULTS AND) DISCUSSION DTIC SJUN A1I

7 NRL EMO REPORT 5745 INFLUENCE OF SHEAR MODULUS ON TEE BEHAVIOR OF ACOUSTICALLY TRANSPAENT MATERIALS INTRODUCTION Several publications [1-4] have appeared in recent years that report on the development of an acoustically transparent material based on a fluoroepoxy filled with microballoons. Such a material can be designed to have a density and dilatational sound speed that closely match those of the acoustic medium, usually seawater, but with sufficient rigidity to serve as a material for structural members or backing plates of acoustical systems. The matching of density and dilitational sound speed leads to perfect acoustic transparency for reflection of sound at the interface of fluids only. A solid has a finite shear modulus; and for other than perpendicular incidence, a shear wave will be present in addition to the dilatational wave. This shear wave causes a deviation from ideal transparency. In this study an analysis is given of the influence of the shear modulus on the reflection and transmission of a plane sound wave by an infinite plate with strictly or approximately matching density and sound speed. It shows the limits on plate thickness and frequency imposed by the requirement of negligible reflection. The effect of a finite loss factor in the shear modulus is also investigated. ANALYSIS The analysis of the reflection and transmission of a plane sound wave by a plate is given in Ref. 5. The main points of the derivation are repeated here. In the general case (Fig. 1) a fluid with density p 0 is present at one side of the plate (thickness h = 2d) where the incident wave arrives, with MUnu riipt &pproved 7 April 1986 ;-.;i///... /.;-'.. ",., i '; "",,- " "';-' ---.,:i: /."-1-..., -.m.-,.,. --.._,

8 DUBBELDAY WAVE REFLECTED WV FLUID DENSITY Po d -- -PLATE -- DENSITY P$ ~~FLUID p let ~ DENSITY P TRANSMITTED WAVE Fig. 1 - Geometry of fluid-loaded plate angle 0 to the normal and wavenumber k o. sound speed c' is at the other boundary of the plate. forms an angle 0' with the normal. Another fluid with density p' and The transmitted wave The propagation of waves in the solid, excited by the incoming wave, is treated by exact elasticity theory. The plane-crested wave in the solid with wavenumber k is a combination of a dilatational wave, represented by a scalar potential 0 with amplitudes A and B for the symmetric and antisymmetric contributions, respectively, and shear wave represented by the 5-component of a vector potential # with amplitudes D 5 and Ca. Other field variables are the pressure amplitudes of the incoming wave Pi, of the transmitted wave P, and of the reflected wave Pr* At the boundaries of the plate with the two fluids, one imposes conditions of continuous normal stress and velocity and zero tangential stress. The matrix of the coefficients of the seven field variables in the six boundary conditions (after manipulation of the rows) is 0 i 2 shown in Table 1. Here P 0 = P. exp (ikod cos 0), P = Pr exp (-ikod cos 0),

9 NRL MEMIO REPORT 5745 Ckq 7SM - - U - a A "a Iq - 4. gqe Ue r bo eqia 0- to V M I 002 eq (.d eq 4 U -6. 3

10 DUBBELDAY and P' =P exp (-ik'd cos 0'). The wavenumbers ko, k, and k' are related through the coincidence condition k = k sin e = k' sin 0'. Also q, 2 =2 and = k 2 -k 2 where kd and k are the wavenumbers of the dilatational wave and shear wave in the solid. The ratios of the field variables may be found from the matrix as ratios of the corresponding subdeterminants. One finds for the case where the same fluid is present on both faces of the plate that the reflection and transmission coefficients R and T are given by pt T i = p o exp (2ikod cos 0), (2) where Po R_. _ A + D-" ~ -A+D(3) A + B- C +D' A + B -C + D (4) A 5 (q'd) 2 (kd) cos q'd sin q'd cos s'd sin s'd, (5) B il] k d q'd A cos q'd cos s'd cos 0, (6) 4

11 NRL MEMO REPORT 5745 C i [POkod q' d A. sin q' d sin s' d cos 0, (7) (8 D (kod) ha A. (cos 2 0 )/(k s d) 4 and A 8 =11',22 - m 12 m 2 1, (9) A =m m -m m (10) s ' in terms of the elements m.. of the matrix in Table 1. It is of interest to formulate the condition for zero transmission through the plate, which is equivalent to total reflection if there are no losses in the plate. From Eq. (4) one sees that this is equivalent to B + C = 0. Inserting the expressions for A and A from Eqs. (9) and (10), one may reduce a this to [(kd) 2 - (s'd)2] cos s'd sin s'd + 4(kd) 2 q'd s'd cos q'd sin q'd = 0. (1I) A computer program was written to evaluate the reflection coefficient R and transmission coefficient T according to Eqs. (1) and (2). Input to the program is the ratio of density of the fluid to that of solid po/ps, sound speed in the fluid relative to the dilatational speed in the solid co/cd, the ratio of shear to bulk modulus in the solid G/K, the loss tangent in the shear modulus ' 7 ' and the thickness of the plate relative to the wavelength of dilatational waves in the plate material h/x. The bulk modulus K is assumed lossless. If one introduces the following symbols -7

12 w-r r * *..- + ',r DUBBELDAY. = (po/ps) q'd (ksd) 2 cos q'd cos s'd a' (Po/Ps) q'd (ksd) 2 sin q'd sin s'd d = A kd/(kd) 2 d' =A 1 k 0 d/(k d) 2 (12) one may write for the expressions A, B, C, D, A = 1/2 aa' B = 1/2 i ad' cos 0 C = 1/2 i a'd cos e D = 1/2 dd' cos 2 0 (13) The reflection factor P/P 0 may be expressed as. (d'/a' (d/a) cos (14) Po (1 -i d cos 0)(1 d (14)co A 'structural response function' 0 was introduced in Ref. 6 by setting P +P i_ oo (15)-, 0 Here one finds a similar structural response function from P +P B-C+2D P -P 2A + B- C 0 6

13 NRL MEMO REPORT 5745 as.1,~ (d'/a') - (d/a) - 2i(d/a)(d'/a') cos i(d'/a') cos 0 - i(d/a) cos ( The structural response function is related to the effective impedance Z of the plate surface, defined as the ratio of the total pressure P + P 0 at the side of the incoming wave to the surface particle speed perpendicular to the plate. One finds P0c0 P 0+ P =oco 0 ifp cos a P -P 0 (17) One may define effective surface impedances for antisymmetric and symmetric waves separately. First the pressure at the side of the incoming wave is split in a part contributing to antisymmetric waves and a part contributing to symmetric waves, namely P +P= (P+P - P ') +1 (P+ pp'). (18) Similarly the displacement perpendicular to the plate, w(d), is written as w(d) = 1 [w(d) + w(-d)] + 1 [w(d) - w(-d)]. (19) The two impedances are then defined by P + P-P' a w(d) + w(-d) (20) and ~ rs

14 -~~~ ~~~ DUBBELDAY P +P+P, zs w(d) - wc-d)" (21) In terms of the symbols introduced before, one finds Z = -id/a and Z idl/al. It would be instructive to express Z in terms of Z and Z in the form of a an equivalent circuit diagram. Thus far this has not been successfully accomplished. RESULTS AND DISCUSSION Figure 2 shows the reflection coefficient as a function of incidence angle 0 for various values of relative density and sound speed. The three figures arranged along a column have equal values of po/ps, namely 0.995, 1.000, and The three figures arranged in a row have equal value of co/cd: 0.995, 1.000, and The four curves marked 1, 2, 3, 4 vary in value of the thickness h (=2d) relative to the wave length X of dilatational waves in the material. The four curves correspond to h/x = 0.50, 0.25, 0.10, The value of the shear modulus G relative to the bulk modulus K was found from measurements [7] to be In Fig. 2 the loss tangent in G is assumed zero. A striking feature in Fig. 2 is the appearance of a spike in the h/x 0.5 curve indicating 100% reflection. This may be explained by considering Fig. 3, which is the locus of points for which the relative wave speed 7= c/c s is such that zero transmission occurs, as a function of the dimensionless frequency ksd. The expression for this was given in Eq. (11). According to the coincidence relation c 0 = c sin 0, the points for 7 when the angle 0 is varied lie on a vertical line starting at the point with abscissa k d and 5 ordinate c 0 /c s. If this line intersects the characteristic curve of Fig. 3, total reflection will occur. In the example depicted in Fig. 3 this occurs for 7 = 18.9" and 0 = 9.16". This is relatively rare for near-pc materials. A small amount of loss in the shear modulus reduces the height of the spike to a negligible size. 8

15 NRL HUO REPORT 5745 bp "I I i" NN ik NN 11 I I I II o4-1 I:-, -..-,...-.:.....I' I,, I.. I,,/",, 0~~~ OD0 D P S C -."

16 DUBBELDAY * C.D C E.C KSD Fig. - V&vespeed 7 = c/c, as a function of dimensionless frequency k d for which zero tr~asmisuion occurs. Arrows indicate asyuptotes.. The vertical line in the locus of points with h/x = 6.5 (k d = 4.719) and 0 varying from 96 to Mo. 8 A global viewing of the nine sets of curves in Fig. 2 shows that at the intermediate range of 0, say from 30 to 60", there is little difference in the appearance of the curves for varying density and sound speed. As to the small angle range, it appears that the relevant parameter is the product pc; the groups that have similar appearance are a; b,d; c,e,g; f,h; and i. This corresponds to the fact that for fluids the behavior near 0 = 0* is determined by the product pc. Near 0 = 90" one sees that there are three groups distinuished by the presence or asence of zero reflection. For c there is an angle where the presence reflection is zero; for co/cd = /cd the zero reflection occurs at 90"; ad for c 1.05 there is no angle near 90 at which the reflection coefficient is zero. In Figure 4 examples are given where both the density and the sound speed differ by +5% or -5% compared with ideal pc. It appears that too small a density and sound speed are more deleterious than too large a density and sound speed, judging by the reflection coefficient. 10

17 NRL MEMO REPORT 5745 'I. a r r/ Fig. 4 Reflection coefficient i i as a function of co / incidence ngle ) /P c/c 0 / d,.9 s.all/ SI C.2 curve 1I /~. K. curve 2 h/)x = curve 3 h/x = 6.1 curve I -/ W.65 0 t ' 7E - THE7A In Figure 5 the effect of a finite loss tangent in C is shown, namely =0.01, which is about the value measured for the shear modulus of the rigid pc material. Three combinations of density and sound speed were chosen. One aces that the effect is quite small. %S

18 DUBBELDAY E0.10.o S0.05 C L [ I-- C.10 ~ ' CD' Fig. 5-0 Reflection coefficient as a function of incidenc angle / :. " ap /P* /C 1.9 )p /p,c /c = =_o 0 a o o/=, d 0.1 "),p " c/a"p cuve 1 - = 6.6 curv e - = w.w C VC THETA U 1 2,-

19 1. NRL MEMO REPORT 5745 "- REFERENCES 1. C.M. Thompson and J.F. Griffith, 'Development of an acoustically transparent plastic for underwater applications,' J. Acoust. Soc. Am. Suppl. 1, 70, S74 (1981). 2. C.M. Thompson and R.Y. Ting, 'A study of some epoxy polymers for underwater acoustic use,' in Organic Coatings and Applied Polymer Science Proceedings, Vol. 46, Am. Chemical Society, R.E. Montgomery, F.J. Weber, D.F. White, and C.M. Thompson, 'On the development of acoustically transparent structural plastics,' J. Acoust. Soc. Am., 71, (1982). 4. C.M. Thompson and J.R. Griffith, 'Rigid, acoustically transparent plastic based on fluoroepoxy,' to be presented at the Acoust. Soc. of Am. Mtg., Cleveland, OH, May P.S. Dubbelday, 'Application of a new complex root-finding technique to the dispersion relations for elastic waves in a fluid-loaded plate,' SIAM J. Appl. Math, 43, (1983). 6. P.S. Dubbelday and A.J. Rudgers, 'An analysis of effective shear modulus for flexural and extensional waves and its application to reflection of sound by a plate,' J. Acoust. Soc. Am., 70, (1981). 7. P.S. Dubbelday and K.W. Rittenmyer, Influence of shear modulus on the behavior of acoustically transparent materials,' to be presented at the Acoust. Soc. of Am. Mtg., Cleveland, OH, May 86. r 13 I.. ',- "a " " ",. - % " -,... a ". a -a - -1a.' 2...',." " ".'. " '.".,,,

20 * ~ 2.:.:.. *. a a V.t.r V.r *2. ~t<...*. :. -*. -- e S p 4~. -D...