REPORT NO. NADC-75324-30 . /

ULTRASONIC SPECTRUM ANALYSIS FOR NDT OFLAYERED COMPOSITE MATERIALS

W. R. ScottAir Vehicle Technology DepartmentNAVAL AIR DEVELOPMENT CENTERWarminster, Pennsylvania 18974

31 DECEMBER 1975

FINAL REPORTAIRTASK NO. R02201001/DG204

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

Prepared forNAVAL AIR SYSTEMS COMMAND

Department of the NavyWashington, D.C. 20361

NADC-75324-30

NOTICES

REPORT NUMBERING SYSTEM - The numbering of technical project reports issued by the Naval AirDevelopment Center is arranged for specific identification purposes. Each number consits of tne Centeracronym, the calendar year in which the number wasassigned, the siquence number of the report withinthe spe-.ific calendar year, and the official 2-digit correspondence code of the Command Office or theFunctional Department responsible for the report. For example: Report No. NADC-7301F-40 indicatesthe fifteenth Center report for the year 1973, and prepared by the Crew Systems Department. Thenumerical codes are as follows:

CODE OFFICE OR DEPARTMENT

00 Commander, Naval Air Development Ceniter01 Technical Director, Naval Air Development Center02 Program and Financial Maagement Department03 Anti-Submarine Warfare Program Office04 Remote Sensors Program Office05 Ship and Air Systems Integration Program Office06 Tactical Air Warfare Office10 Naval Air Facility, Warminster20 Aero Electronic Technology Department30 Air Vehicle Technology Department40 Crew Systems Department50 Systems Analysis and Engineering Department60' Naval Navigation Laboratory81 Technical Support Department

- 85 Computer Department

PRODUCT ENDORSEMENT - The discussion or instructions concerning commercial products herein donot constitute an endorsement by the Government nor do they convey or imply the license or right to usesuch products.

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APPROVED BY: DATE: 31 December 1975P. M. STURMConnm.nder, USNDeputy Director, AVTD

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13; 1) 6;.-..~~ILLIAM RJ SCOTT /_9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

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IS. SUPPLEMENTARY NOTES

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Nondestructive TestingUltrasonicsSpectrum AnalysisLaminates

K. Composite Materials20 ASSTRACT (Corninu* an reverse elde if necessary a d Identify by block number)

The structural use of advanced composite materials has increased inrecent years and a number of problems have arisen associated with assuringthe integrity of composite structures.

Because ultrasonics has emer as an important tool for inspectingcomposite materials, it has beco ,iessary to understand more about the

interaction of acoustic waves with composites and what information can beI obtained by monitoring this interaction..,

DD JAN•73 1473 EDITION OF I NOV 65 S OBSOLETEN5S'N 0102-014- 6601 1/UCASFE

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UNCLASSIFIED- jjiTy CL/PSSIFICATION OF THIS PAGE(Whan Date Entered)

I -In this paper a simple model is presented which predicts the ultrasonicA frequency spectra for a broad class of layered composite materials having a

finite number of laminae. Experimental verification of this model isdemonstrated for arrays exhibiting both periodic ani aperiodic frequencyspectra. The relationship between the spectra of periodic arrays havingboth finite and infinite numbers of layers is discussed.

Important results relating to NDT which have emerged from thisstudy include methods for predicting the results of spectrum analysisstudies on layered materials and techniques for mapping small changesin the modulus and thickness of composite materials. Also discussedis the existence of forbidden frequency bands at which ultrasoundcannot be transmitted through thick layered composites.

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SECUAITY CLASSIFICATION OP" THIS I AGE(qWhen Oat. 6., .red)

NADC-75324-30

I N T R O D U C T I O N

A number of problems are associated with describing the propagationof ultrasonic waves In laminated materials. In general such materials areboth inhomogeneous and anisotropic; therefore, there exist a number of modesof wave propagation associated with different directions of propagation.Attempts to treat these materials as monolithic materials with effectiveelastic moduli provide reasonable descriptions of quasi-static behavior;'however, when one attempts to describe transient phenomena with such amodel, It Is found that no choice of effective parameters is adequate todescribe the high frequency behavior of propagating waves as influenced bthe periodic nature of the composite. Recently a number of investigators , 3

have approached the problem of waves propagating in a composite mediumutilizing Fluoquet's Theorem for describing wave propagation inInfinitely periodic structures. Most of this work on infinite laminateshas been carried out in order to predict wave speeds associated with theeffect of Impact on laminates. These results have been promising; however,

N I it is not clear to what extent they can be applied to finite periodicstructures.

The problem of wave propagation in finite laminates has beenconsiderd by a number of investigators in the field of nondestructivetesting,, 5 who have been investigating the effect of laminates on thepropagation of ultrasonic pulses, in order to develop nondestructive testingtechniques for probing the structure of these laminates.

M.

A number of investigators have observed periodic fluctuations inthe frequency transmission and reflection spectra associated with laminategstructures. This periodictty in the frequency spectrum has been explainedas resulting from interference of coherent reflections of ultrasonic pulsesfrom two separated interfaces within a sample.

It will be shown that this two interface model, while phenomeno-logically correct In some cases, fails in general to accurately describewave propaiation in laminates in which several interfaces exist. Amodified model will be derived here which is capable of dealir.q with thegeneral case of finite laminates and their ultrasonic frequency spectra.Also, the general interpretation of frequency spectra in laminates will bediscussed with emphasis on applications of spectrum analysis to nondestructivetesting of laminates and composites.

NADC-75324-30

T AB LE O F CO0N T EN TS

Page No.

I NTRODUCTION . . . . . . . . . . . ............ *.*I

LIST OF FIGURES. . . . . . . . . . . . . . . . . . . . . . . . . . 2- 4

A. Acoustic Waves Incident on a Boundary.* 5 -6

B. Wave Propagation in a Single Layer 6 7 ....

C. Attenuation in a Layer ................- 7 -8

E. Wave Propagation in Multi-Layered Media. . . . 8 -9

F. Calculations of Reflection and Transmission Spectra. . . 9

1. Single Laminae in an Ambient Medium . . . . . . 9

2. Mlti-Layered Periodic Media . . . . . . . . . 9

3. Infinite Periodic Media. . ..... 1

EXPERIMENT ........................ 10

RESULTS AND DISCUSSION . . . . .. . . . . . . . . . . . . .. . 1

A.Transmission and Reflection Through a Single Finite Medium 11

B. Velocity of Sound Measurement. ............. 1

C. Multilayer Media ............ .12

D~. Agreement between Theory and Experiment., 12

E. Frequency Spectra from Structural Composite Materials. *13 -14

F. Forbidden Frequency Bands in Composites.* 13 -14

G. Applications of Spectrum Analysis in Nondestructive Testing 14 -15 -16

SUMMARY ANDOCONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . 16

RECOO4ENL'ATIOWS. 9 9 17

REFERENCES e . v e 9 a e 9 9 e e e 36

2

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L I S T 0 F F I r, t1 R E S

Figure No. Title Page No.

1 ltrasonic Pulse Incident upon Medlur 18

of Finite Thickness

2 Ultrasenic Pulse Incident upon 19

- Layered Media

3a Transmission Spectrum for .3T" Thick 20

Steel Plate In Water

3b Reflection Spectrum for IA, Thick 21

Steel Plate in Water

3c Transmission Spectrum for Glass Plate 22

1.22mm Thick In Water

3d Transmission Snectrum for two l.7 mm 23

Glass Plates in Water

3e Transmission Spectrum for three 1.?7mm 24Thick Glass Plates in Water

3f Transmission Spectrum for four 1.22mm 25Thick Glass Plates in Water

3q Transmission Spectrum for five 1.gramW 26Thick Glass Plates In Water

4 Apparatus for easurIna Acoustic Peflectlon 27and Transmisslon r.oefficients as a runcttonof Frequency

5 Frequency Dependent Transmission of a .1n' 28

Steel Plate-

6 Transmission Spectrum from Lavered Array 29

of three ,lass Plates (Measured Fxnerimentallv)

:-=- 30

7 Ultrasonic rreouencv Snectra of rraohlte3Epoxy Panels

8 Ultrasonic Velocity vs Freauencv In franhtte 31

Epoxy

9C-scan of Btoron Aluminum InpcTlmns 32

3

NADC-75324-30

L I S T O F FI GU R ES

FiueNo. Title Page No.

O~a Ultrasodic Frequency Spectra for 33

Boron Aluminum

-- 34

lOb Ultra'qonic Frequency Spectrum forBoron Aluminum

11C-scan for Graphite Epoxy PanelShowing Natural and SyntheticFlaws

f-

NADC-75324-30

THEORY

A. Acoustic Waves Incident on a Boundary

It is a well known result 7 that In the absence of dispersion thetransmission and reflection amplitude coefficients T and R of acousticplane waves incident normally from a medium with acoustic Impedance ZIonto a medium of acoustic Ifpedance Z2 are given by:

22-_ (l) 12 = 2 -ZI (lb) T12 =2Z2

z2 +Z1 z2 + Z22 1

where the frequency independent Z for each medium is given by the productof the density p and the speed of sound C.

When more than two media are involved, interference effects producefrequency dependent transmission and reflection coefficientb for waves

M traveling through media of finite thickness.

In general, for a given frequency, steady state solutions exist whichcan satisfy the boundary conditions for wave propagation through arbitrarylaminar media; these solutions may be obtained through the solution of 2n-ll:near simultaneous equations. The result of solving these equations isto generate steady state frequency dependent transmission and reflectioncoefficients. T(w) and R (w).

Because of the linear nature of the qoverning equations, implicitsolutions to transient problems can be constructed from these transmissionand reflection coefficients using a Fourier integral technique. Forexample if a number of layered media are impacted by a finite wave trainof amplitude A(t) and Fourier transform A(w), the transmitted frequencyand time dependent wave forms will be given by:: f -i~

AT (t) fe - T(w) A(w) dw

AT (w) T (w) A(-)

and the spectrum Atl t2 (w) of any time slice (t2 - t) of that transmittedwave train Is given by:

~-l

Atlt 2 (w) - (211) 2 AT(W) eiWt dt' t2 >t) > t=°

where t - o Is the time of arrival of the wave train at the first interface.

5

NADC-75324-30

Both A (w) (the spectrum of the total transmitted wave form)and A ( T (the spectrum of the gated wave form), can be

tit 2experimentally determined from ultrasonic measurements.

In this paper we will develop expressions for the quantitiesT(c) and R (w) by directly applying the principle of superpositionto determine the response of a layered medium to a transient wave train.

This approach will generate a summable infinite series forR (w) and T (a w); In addition to expressions for At t (G ) fromfinite partial sums. An Iteration scheme will be 1 2 developedfor calculating exact values of R (w ) and T (w ) for layered media.These values will be built up from the results for a single layer, andeach successive iteration will add an additional layer to the medium.Thus in computing T ( w ) and R ( w ) for n layered media, results forn-1, n-2, n-3, etc. layers will be simultaneously generated.

B. Wave Propagation in a Single Layer

Consider the geometry of Figure 1. Le' waves incident from theleft upon medium 2 have transmission and reflection coefficients T12 andR 2 and those incident from the right T21 and R21. The coefficients fort boundary between medium I and 3 are anlogously defined.

A portion of the pulse A(t) ir-'?A-nt from the left onto medium 2will be transmitted with amplitude T1- and, i-on emerging in medium 3,will produce a disturbance of the form

T2 3 T12 A(t -d )

In addition to the portion of the wave transmitted into medium 2,there will be a number of pulses emerging which have undergone multiplereflections within this medium. The total of all of these waves isgiven by:

Ad T T A (t- d ) + T12 R23 R2 1 T2 3 A (t -3d )......13, 12 23 12R3 2

N(2) A 13 (t) - T12 T23 j (R) R )A(Mt-(2N+)d)

232 c

where c - c2

Similarly the total wave reflected back into medium 1 is given by:

(3) R1 (t) d R12 A(t) + T12 T21 R21 (R23 R2 1) A(t 2NdC

N-1

6

NIADC-75324-30

Taking the Fourier transformis of expressions (2) and (3) andutilizing the Identity

ff(t -,&t) e itdt -e iAAt ff(t) e iad

we have:-icadc~r) -2iw~d/c1

(4) A' () - T A (w R 1 e ~)13 1223 .-o 23 21

-2iwd/c -2^iwd/c N*(5) RI() (F ( 1 + T1 T R & ~ (R R e A A(w))

21 23

where A' 3 M. is the Fourier transf~rm of A 3 (t), and R' 1W) is

the Fourier transform of R' (t), etc.

The sum appearing In expressions 4 and 5 can be readily evaluated forN either finite or Infinite allowing the definition of generalizedfrequency dependent reflection and transmission coefficients r(W) and t(w)for the layered medium. For 5 infinite we have the expression

(6) t(.) A3 (M) T1 2 T e -iwd/c

13 12 21 3

A w' -2cd/cI - 2 R21 e

C. Attenuation in a Layer

The use of complex notation allows attenuation effects to be introducedinto equations (6) and (7). This is done most easily by allowing d to be

d d (1-1 a)

where d is a real distance and Ct becomes a linear attenuation coefficient,corresponding to a wave whose amplitude is exponentially damped withdistance.

D. Generalized Reflection and Transmission Coefficients

Clearly, for the case of layered media of finite thickness a singleref lection coef ficient does not exi st for the time depedent revresentationOf the wave A (t); however, we can define op~erators R. em i Vhich wihenoperating on A(t) generate the expressions for the time dependent reflectedand transmitted waves respectively. The appropriate definition. for these"prations is:

7

NADC-75324-30

(8) i A(t) -- 1z/2 f 1/

(27) jft(w) exp (-imt) (211) A(t') exp(it') dt')dw

(9) T A (t); 9 (t (w) 61-(A (t)) -/ f- ° (1/12 -1/2

-(21) (t(w) exp (-iwt) (211) A(t') exp (iwt)dt')dw- whe 6 and are the Fourier transform and its inverse. The use

of this operator notation permits a layer to be represented as a boundaryor virtual Interface having these operators as their reflection andtransmission coefficients (see Figure 2).

E. Wave Propagation in Multi-Layered Media

Consider the layered media in Figure 2. This is essentially thesame as Figure 1 except that the boundary between media 2 and 3 is nowI replaced by a boundary medium with transmission and reflection operators

"23and T23

Rewriting expressions (2) and (3) in the form of a series andrealacine T std. R by operators we have,° 23 23"

(10) A' ( T) R R2 A(t-(2l) d/c13 12 23 23 Ac

00 N

(11) R' -R A(t) + T T R (R ( A(t-21td/c)1 12 12 21 23 N-O 23 21

it is readily verified by recourse to their definitions that thetransmission and reflection operators commute and that the Fourier transform

of a number of operators acting on a function is just the product of thecorresponding frequency reflec ion and transmission coefficients timesthe Fourier transforms of the function.

T R Aft)- R T A(t)N N N N

and F(R T A(t))- r (w) t (w) "(A(t))

Applying these results tc take the transform of expression (10) and (11)we have,(12) T' (W) A' (w) A (w) -T t) (r (w) R )exp (-i(2N+l)d/)

13 12 21

(13 R' (o) R (w) + T T r Wo) z (w) R ) exp (-21Nwd/c)12 12 21 21

4 ft

NADC-75324-30

Again summing the series we have,

-' - '-iwd/c

(12a) T(,) lT 12 t (w) e

R21 e

-2 iwd/c(13a) R (w) R12 + T12 T21 r(w) e

, - r1 R e -2iwd/c

Successive application of these formulae yields a simple iterative proce-" dure for tabulating transmission and reflection coefficients for media having an

arbitrary number of layers of varying thickness and acoustic parameters.

Normal Incidence reflection and transmission coefficients forseveral finite arrays have been calculated using formulae (la), (lb), (12a)and (12b) and the results of these calculations are discussed below.

F. Calculations of Reflection and Transmission Spectra

1. Single Laminae in an Ambient Medium

The values of the functions log o0 t(w) and log I r(t) for a0.30" steel plate in water are plotted In rgu es 3a and ib respectively.The values of these coefficients were computed directly from expressions(12a) and (13a) using a Fortran computer program "Layer" written for thispurpose. A computer controlled plotter was used to reproduce the plotsdirectly from computed values with a resolution of 100 points per Inch onthe frequency axis. This resolution, although acceptable for plottinglog 10 It(w) Iis inadequate to reveal the sharp resonances In the functionlog lj r() which is In fact periodic. The existence of these sharpper ically spaced resonances termed thickness resonances (period f- cf2d )can be exploited for highly accurate sound velocity measurements.

2. Multi-layered Periodic Media

In addition to reproducing the known analytical results for trans-mission and reflection from single layers, "LAYER" has been applied tocalculating loglI t(w)I and log 10 lr(! for finite periodic layered arrays.The res ,a ts nF £uch calculstions are seen In Figures 3c through 3q, whichplot log 10' Ie(, )Ifor stacks consisting of I to 5 glass plates 1.22mm thickspaced .lm apaL In an ambient medium of water. It is interesting to notethat when more thaa a single plate Is present log 10, It(w)I Is no longer asimply periodic function.

The shallow tightly spaced minima seen in Figures 3d through 3g- correspond roughly to the thickness reasonance of the entire array and they

Increase in -nunber as plates are added to the array. However, the broaderresonances remain constant in number but rapidly deepen as layers are added,producing frequency bands having increasingly high acoustic attenuation.

9

NADC-75324-30AA3

3. Infinite Periodic Media

It can be shown that in the limit of a periodic medium with aninfinite number of layers, there will be frequencies for which no travelingplane wave solutions exist Inside the medium. This is a limiting case ofthe behavior seen in the transmission curves in Figures 3c through 3g.Robinson and Leppelmeler (Journal Applied Mecbanics, March 1974, p. 89)have studied the case of wave propagation In a two component layeredperiodic medium and have given an expression for the wave number k'oflongitudinal traveling waves propagating in such a medium:

(14) cos ki ,,coo (w dl) cos (w d2) -1 (Pi Cl + P C2) siftw d )sin (w d2)F 2_ 2 2

C1 C2 P2 2 PC, C C2

where d and d2 are the layer thicknesses for media having sound velocitiesCi and I2 respectively and I- d1 + d . Whenever p1 C1 i C2 in expression14, there exist values of k fir which icos KlI>'I. This implies a complexvalue of k and frequency regions corresponding to such wave numbers arereferred to as forbidden bonds, because there exist no travelin2 wavespropagating at these frequencies. Using expression (14), the first fourforbidden frequency bands have been calculated for the layered glass arrayof Figures 3c - 3g and are indicated by arrows in Figure 3g. tt is seenthat these forbidden bands correspond to broad transmission minima which areseen to deepen rapidly as the number of layers In the finite array isincreased.

Experiment

Acoustic reflections and transmission coefficients were measured fora number of materials using the apparatus pictured schematically in Figure 4.Broadband longitudinal ultrasonic stress waves are,-projected onto a sample bya pulsed piezoelectric trandsucer (A). The projected pulse interacts withthe sample producing transmitted and reflected wave trains. The transmittedwave train is detected by a second transducer (B) placed behind the specimen,while the reflected wave is detected by the projecting transducer.

Either the transmitted or the reflected oulse mav be selected foramplification and gated electronically to obtain the desired portion or timeslice of the signal. This gated signal Is processed by a spectrum analyzer,which gencratec a Piiv of' -, - a'- 1...... .... the .ated Signal :m ......

frequency. (i.e. tAt.1t2 (W))

10

... ....- -Z: -- -: - -_-- -:

NADC-75324-30

R E S U L T S A N D D I S C U S S IO N

A. Transmission and Reflection Through a Single Finite Medium

Figure 5 shows the results of an experimental measurement ofthe frequency dependent transmission coefficient, t(w), for a O.30" steelplate. The projecting transducer in this measurement was a PanametricsVIP 15 MHz broadband device with a 1/4" diameter piezoelectric element; thereceiver was a 25 MHz Automation Industries transduzer with a 1/2" diameterelement. The distance betwwen the sample and the receiving transducer was2". The gate was set to receive all of the transmitted pulse for which theamplitude was greater than the ambient noise.

The upper reference curve in Figure 5 Is a plot of thelogarithm base 10 of the amplitude of the signal oroduced at the receivingtransducer vs. frequency(with no sample presenO. The lower curve is a plotof the logarithm of the transmitted amplitude as a function of frequency.The spectrum has a bandwith of I KHz filtered to remove most of the rapidoscillations associated with the pulse repetition rate of the pulserreceiver. Using this type of plot, the logarithm of the transmissioncoefficient can be measured as the difference in the ordinates of the twocurves.

B. Velocity of Sound Measurement

The theoretical values fr the transmission coefficient of a0.3" steel plate computed from the absolute value of expression (6) areplotted in Figure 3a. The function Is periodic with period

(05a) N - CN d

Frequencies for which (5a) is satisfied are often referred to as thicknessresonances. Knowing the thickness of the plate this formula can be used tomeasure the phase velocity of sound as a function of frequency from experi-mental data using the relationship.

(15b) CfN 2fd

For steel this is most accurately measured using reflectance minima forwhich the resonances are much sharper than for transmission peaks (see

] Figure 5). The precision of this technique is easily better than one part

N1 f ste ; h.wever, the var!atins in nuhlished values for C make itimpossible to ,nfirmn thase vajlues to better than 2 sionificant Figures.

E at

LL

NADC -75324-30

C. Multilayer Media

Figure 6 is a plot of the transmission spectrum from a layeredarray of three glass plates .00122M thick separated by .001 meters of water.The entire array was immersed In water during the measurement and the system

represents a somewhat Idealized laminate. Figure 3e Is a plot of thetransmission spectrum for this laminate as computed from formulas 12a and13a. The agreement between the theoretical and experimental spectraindicates that the linear acoustic approximation used is a reasonable model

-A for wave propagation In laminated structures. Interesting features of thefl complicated laminate spectrum are:

1. Transmission peaks do not recur with simple periodicityas they do with monolithic materials.

2. Although peaks approximately corresponding to the firsttwo thickness resonance modes for the entire array do appear, the thirdharmonic is absent and the next maximum appears after a large frequency gap.

A These results are noteworthy because It has often been IndicatedIn the nondestructive testing literature that reflections from a lamina-like

dJ structure within a specimen will produce a periodic spectrum. This isprobably true only when othe' reflections in the structure are small, are

_-A: of an essentially Incoherent nature or can be Isolated using a gatingtechnique.

0. Agreement between Theory and Experiment

As mentioned above, velocities of sound measured by the spectrumanalysis technique have been found to be in good agreement with publishedvalues. In addition, when published values for pand c are used in equations(1), (6) and (7) to compute spectra of laminates, we are able to computepositions and relative magnitudes of extremely detailed features of thesespectra. (See Figures 6 and 3e).

A somewhat more conclusive test of the agreement between theoryand experiment is the prediction and as' n btlute values of trans-mission and reflection coefficients. For ex t ? transmissioncoefficient of the steel layer in Figure 5 Is about dB above its prodictedvalue of -24dB and the maximum transmission coefficient, which should reachOdB varies + 2dB from this value.

The observed discrepancy between theory and experiment Isprobably due to near field effects of the transducer. This hypothesis hasbeen partajl,, conf!r.d by experlmnnts which have shown decreasing agree-ment between predicted and observed frequency spectra as frequency isIncreased. Since the near field of the transducer has a radius which isproportional to the transducer diameter divided by the square of the wave-length, the near field always extends further at hiah freauencies.Efforts to work .ntirely in the far field of the transducer have beenhampered by poor signal to noise ratios and by the physical constraints of

12

NADC-75324-30

the Immersion tank. Improved agreement between theory and experiment rayrequire the employment of analytical techniques for modeling near fieldtransducer effects.

Additional experimental error results whenever the bandwidthof the spectrum analyzer Is on the order of the spacing between spectralfeatures. This results In the smearing out of sharp spectral features suchas the transmission peaks In Figure 5, which should be much sharper andshould reach a higher maximum value.

E. Frequency Spectra from Structural Composite Materials

Figure 7 shows a plot of the frequency spectra from threedifferent graphite/epoxy laminates. For each of these spectra reflectanceminima exist which correspond roughly to thickness resonances. (I.e. f-Nc/2d).This suggests that to a first approximation, a graphite epoxy laminate may

z be modeled as a monolithic material. However, it is seen that as in thecase of the layered glass laminate the portion of the frequency spectrum

- corresponding to the thickness resonance of a single period of the laminatebehaves anomalously (i.e. resonance number 4 is missinq for the 4 yiv laminate.resonance number 18 is missing for the 18 ply laminate). Pursuing theimplications of modeling the composite as a monolithic material, formula(15b) may be applied to compute a dispersion relation c(f) for eachfrequency at which a thickness resonance occurs. Such a dispet.;Ion relationIs plotted in Figure 8 for an 18 ply graphite/epoxy laminate. Significantfeatures of this dispersion relation are,

(1) A general increase in velocity with increasing frequency,(a phenomenon expected in viscoelastic materials).

(2) Finite discontinuities in velocity at multiples of4 MHz.

F. Forbidden Frequency Bands in Composites

The existence of discontinuities in the dispersion relations

for composite materials results from the periodic structure of the compositeand may be predicted from equation (14) and the relation,

c-f/21k

It is found that finite discontinuities '.n velocity occurwhen frequencies of high attenuation referred to as fo.-bldden frequency bandsare reached. Alternately wp may say that these discontinuities occur wheneverthe wave number K is a multiple of 2fl/i where I ; Lhe thckness of one period

of the medium .(See Figure 7). Thus, although wave propagation does notcease in the forbidden bands for finite layered laminates, transmission

Is

strongly attenuated. In addition if experimentallv measured dispesion curvpsfor finite media are examined, it is found that regions of anomalous oisper-slon occur In the vicinity of the forbidden bands.

13

NADC-75324-30

It is interesting to note that the graphite/epoxy composites-- whose spectra are plotted in Figure 7 are not laid-up in a truely periodic

array but consists of 00, 900 and + 450 layers. Some indication of theeffects of ply arrangment can be seen in Figure 7 (Velocity dispersionfor an 18 ply composite). In addition to the discontinuities which occurnear 8 and 16 MHz corresponding to resonances of a single layer, anomaliesalso occur near 4 and 12 MHz which correspond to resonances of laminapairs. Strictly speaking one cannot exaplain such resonances in terms ofthe simple theory presented here, since for normal incidence longitudinalwaves identical angle ply laminae of different orientation should behaveidentically. It appears that for the general graphite/epoxy laminate adetailed model for stress wave propagation must include the effects of

- - waves being scattered In directions other than parallel to the incidentwave.

4 G. Applications of Spectrum Analysis in Nondestructive Testing

The reasonable success in modelling the ultrasonic frequencyspectrum of layered materials and the empirically confirmed capability ofpredicting qualitatively the frequency spectra of fiber reinforced compositessuggest a number of nondestructive testing applications.

The existence of thickness resonances in the reflection andtransmission spectra of composites, analogous to those present in monolithicmaterials, suggest that a type of generalized thickness gauging can beperformed using spectrum analysis. For example the peak position ofthickness resonances below the forbidden band may be given approximatelyby the expression,

(16) f anc/2d (for n less than the number of laminae)

peak

where d Is the thickness of the composite

n is an integer

and c is the phase velocity of sound in the composite.

Clearly, the position of the resonance peak is a function of the samplethickness and the velocity of sound, the latter being a function of theelastic constants (cI*) of the material and its density.

For longitudinal waves propagating normal to the surface of a compositewith hexagonal symmetry, the velocity of sound in the low frequency limit

!Z is given by,

c c33/P

14

-ADC-75324-30

Thus the resonance peak position In t e composite spectrum can be usedto monitor variations in density, thickr-ss and elastic constants. Anefficient reans of accomplishing this monitoring it-to use the amplitudeof one component of the frequency Spectrum at or near a resonance peak tomodulate the Intensity of a c-5can. In this manner a composite sample may

-A be Imaged using a particular frequency of ultrasound. For sharp resonancessmall changes In thickness or modulus will cause large changes in the ampli-tude of ultrasound being trAnsmitted at or near this frequency, producingcorresponding changes In c-scan Intensity.

An example of such a transmission c-scan is shown In Figure 9, whichIs a c-scan of two boron/aluminum 4 ply unidirectional tensile specimens.The nominal thickness of these specimens Is 0.041" and thetir ultrasonicthickness resonances for longitudinal waves occur at about 3.8 9Hz.(see Figures 10a and lib). The c-scan Intensity was modulated using theamplitude of the spectral component centered at 3.7 MHz. In this mannerslightly thicker specimens having peak transmission at !nwer frequenciesproduce darker c-scans than thinner ones. In fact, micrometer measurementsof the specimens showed that the darker specimen (11716 Figure 9) had anaverage thickness of 0.0419", while the lighter specimen had an averagethickness of 0.0402". Ultrasonic frequency spectra plotted for selectedpositions on each specimen show that the expected resonance shifts do infact occur, the frequency shift between the dark and light areas beingabout 0.2 MHz.

The agreement between these two results may be checked by differentiatingformula (16) which gives for the frequency shift,

pd/d)-feak 0.19 MHzpeak d

This.result Is well within the experimental variation in the thick-ness measurements.

In addition to the difference in average Intensity observed betweenthe two specimens In Figure 9 there Is a variation seen within each specimen.This gradation in shading appears as bands oriented parallel to the fiberdirection In the specimens. These bands also have frequency shiftsassociated with them, and the peak Intensity of the spectral thicknessresonances associated with the bands varies only slightly. This indicatesthat these bands are associated with thickness or modulus variations ratherthan delaminations.

The sensitivity of the resonances to delaminations can be seen inFigure II, which is a c-scan of a graphite/epoxy panel for which the intensity

I is modulated by the acoustic amplitude corresponding to a reflection thick-ness resonance (antiresonance). This amplitude is small excopt In thepresence of defects. Using this technique a number of natural andsynthetic defects were revealed without the necessity of accessing bothsides of the sample.

15

NADC-75324-30

A potential application of the analytical model presented here isthe prediction of the frequency spectra from diffusion bonded or adhesivelybonded laminates with arbitrary stacking. Because the model Is quiteaccurate for arrays of monolithic materials, the spectrum resulting froma particular condition In the laminate can be predicted without referenceto standard specimens.

S U MM ARY A ND CO0N CL US IO0N S

A model has been constructed for describing the interactions ofultrasonic plane waves with laminated structures. Using this modelmeasurable parameters such as the frequency dependent transmission andreflection coefficients can be accurately calculated for laminates

j consisting of layers of elastic monolithic materials.

Experiments and calculations for some simple finite periodic

laminates have demonstrated the following:

(1) In general ultrasonic frequency spectra of laminates including periodiclaminates are not simply periodic functions.

(2) If a finite periodic laminate is modeled using the model for anInfinite laminate, regions of high frequency attenuation will be localizedaccurately for laminates having a sufficient number of layers. However,this model cannot predict detailed transmission or reflection spectra.

(3) For wavelengths longer than f (where I Is the period of thelaminate) periodic laminates behave acoustically in much the same way asdispersive monolithic materials and their transmission and frequency spectraexhibit thickness reasonances, with spacing Af = c(f), where c(f) is the

2d

v._ e velocity in the composite. For wavelengths on the order of nl(n aninteger) the laminate exhibits anamalous dispersion and strong attenuation.

(4) Structural composites (e.g. graphite/epoxy, fiberglas, B/Al)have acoustic properties which are qualitatively similar to those describedabove for simple laminates.

The characteristics of laminate spectra reveal a number of veryImportant facts which should be considered when performing ultrasonic

Inspection of layered composite materials.

When Inspecting thick laminates ultrasonically, the existence of11] forbidden transmission bands and narrow thickness resonances dictatesthat ultrasonic frequencies be carefully selected. This is particularlyimportant if tuned transducers are being used. The location of forbid-fenfrequency bands will vary with material properties, thickness, lay-upand the number of plies In a laminate but will remain relatively constantIn thick laminates having the some periodicity.

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NADC-75324-30

Features of ultrasonic spectra can be used as sensitive monitors ofchanges In thickness and modulus both of which are linearly related tothickness resonance frequencies. Acoustic amplitudes of waves propagatingnear these resonant frequencies can be used to modulate c-scan intensities,thereby generating maps of a ame, which reveal small changes in eithermodulus or thickness.

Th model which has been constructed for ultrawic wave propagationIn laminates has proven to be quite accurate In predicting detailedfeatures of the ultrasonic frequency spectra of layered structures andshould be useful for describing wave propagation in any structureconsisting of monolithic laminae.

RECOM HENDAT IONS

I. The existing mode; for wave propagation In laminates should beapplied to more complicated laminates including laminates with defects andaperiodic arrays. This modeling will require the use of effective parameterswhen dealing with viscoelastic and fiber reinforced materials.

2. Efforts should be made to generalize the present model to casesInvolving scattering, shear wave propagation and propagation of waves atoblique incidence to laminate Interfaces.

3. Ultrasonic NDT techniques and devices should be developed whichexploit the resonant nature of laminated structures. These might includethe following:

a. Selection of frequencies for maximum penetration Into laminates.

b. The use of tuned transducers, filters and amplifiers for improvedsignal-to-noise ratio when Interrogating laminated composites near theirresonant frequencies.

c. The use of multi-frequency techniques for monitoring lay-upsand consistency in structural composites.

17

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R E F E R E N C E S

(1) J. E. Zimmer, Determination of the Elastc Constants of aFiber Composite Using Ultrasonic Velocity Measurements,McDonnell Doualas Paper 10181, June 1969

(2) C. WJ. Robinson, G. W. Leppelmeier, Experimental Verificationof Dispersion Relations for Layered Composites, Journalof Applied Mechanics March 1974, pp 89-91

(3) Subrata Mukherjee and Erastus H. Lee, Dispersion Relationsand Mode Shapes for Waves in Laminated Viscoelastic Composites

I7 by Finite Difference Methods, Stanford University

Department of Applied Mechanics Report No. 74-7,September 1974

(4) F. H. Chang, G. W. Yee, J. C. Couchman, Spectral AnalysisTechnique of Ultrasonic NDT of Advanced Composite Materials,

1 Nondestructive Testing, August 1974, pp 194-198

(5) J. L. Rose, P. A. Meyer Ultrasonic Procedures for PredictingAdhzz.=ve Bond Strength, Materials Evaluation, June 1973

(6) V. A. Simpson Jr, Time Frequency-Domain Formulation ofUltrasonic Frequency Analysis, Journal of AcousticalSociety of America Vol 56, No. 6 December 1974, pp 1776-1781I (7) L. E. Kinsler and A. R. Frey, Fundamentals of AcousticsWiley 1962, pp 128-132

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36

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