———. ——. . . . ..—

ELASTIC WAVES FROM A DISTRIBUTED SURFACESOURCE IN A UNIDIRECTIONAL COMPOSITE LAMINATE

Shyh-Shiuh 1, ihJet Propulsion Laboratory

California Institute of 1 ethnology

Ajit K. MalDepartment of Mechanical, Aerospace and Nuclear Engineering

University of California, Los Angeles

ABSTRACTThe response of a unidirectional composite plate of infirritc lateral dimensions to localized dynamic surface

sourws is investigated through theoretical modeling and laboratory tests. In the theoretical sitnulations, the materialofthc plate is assumed to be dissipative and transversely isotropic with its symmetry axis parallel to the fibers. Thesource is assumed to hsrve an arbitrary spatial and time dependence. The associated clastodynamic boundary valueproblem is solved by means of an integral transform technique followed by numerical evaluation of the inversionintc~als. The laboratory tests are carried out on unidirectional graphite epoxy plates of thicknesses ranging from1-25 mm and large Iateml dimensions (> 30 cm? excited by means of broadband transducers attached to its surface.The calculated surface response of the plate at different distances and directions from the source is shown to agreevery well with the recorded response in the ultrasonic range.

INTRODUCTIONIt is well known that Iaminatcd tibcr reinforced composites often suffer significant internal damage when they

arc subjected to Iocalizcd dynamic surface loads. The damage may involve frbcr breakage and debonding as wellas delamination bctwccn the individual laminae. Such damage has been observed to occur even at relatively lowimpact speeds resulting in a severe loss in the load carrying capacity of the larninac. Although the damage is clearlycaused by the stresses which develop within the material, the prccisc nature of these stresses and their relationshipto the degree and mode ofthc damage are not clearly understood at present. This is particularly true in the dynamiccase where the stresses are caused by waves whose propagation characteristics are strongly influenced by theinherent an isotropy and heterogeneity of the composite material.

Dynamic response of plates has been studied theoretically by many authors through past decades. The linearelastic solutions of the isotropic or anisotropic plates have been investigated by, c.g, Weaver and Pao (1982),Vasudevan and Mal (1985); Xu and Mal (1987), I,iu et al. (1991a, b), Mrrl and I,ih (1992, 1995). For lowfrequency response quasi-static and thin plate theories have been used (SCC e.g., Chow(1971 ), Moon(l 973); Sunand l’rm (1984); 1,al (1984)). While nurncrous analytical investigation of wave propagation in plate have beenexecuted, concurrent experimental studies of such wave processes in the laboratory have been far Icss prevalent.Most clXort in this direction were devoted to a cmnpwiwn of pred Icted phase and group velocities of surface waves.

lor wrvcfonn analysis, Medick (1960) used a 220 Switl rifle bullet impacting perpendicular to an aluminum plateto generate flexure waves Rwerrtly, German et al. (1989) used a lead break on the surface of a plate to generatedwavcy however, there are no available results for comparison between the measured and calculated time historyfor wave propagation in a cclmposite under dynamic surface loads.

In this paper a clas.sieal integral transform technique coupled with the matrix method developed by Mal and Lih(1992) is used for the theoretical simulation of the surface source problem, The integrals involved in the spatialinverse transform arc evaluated by means of a previously developed adaptive integration scheme. Themcrr..urcmcnts are earned out by means of an ultrasonic system called the Fracture Wave 1 letcctor by Digital WaveCorp. Numerical and cxperirncntal rrmr[ts for the response of a unidirectional composite Iarninate due to a quasi-sine pulsed load are compared.

THEORY

Material ModelingComposite materials arc known to bc strongly an isotropic as well as dissipative. In tibcr-reinforced composites

dissipation is caused by the anisotropy of fiber orientations, and the dissipation of the waves is caused by theviscxrclastic nature of the resin and by scattering from the fibers and other inhomogencities. Both effects can bcmodeled in the frequency domain by assuming that the stiffness constants, C,j, are eornplcx and frequcncy-dependcnt, A possible form of Ci, that can model the essential featu res of the dissipation caused by these factors,has been given in Mal, Bar-Coher~, and Lih (1 992) and will be used here. A brief description of the model ispresented for completeness; the details can be found in the cited paper.

We recall that the linear constitutive equation for a transversely isotropic elastic solid with its symmetry axisalong the xl-axis (Figure 1) can bc expressed in the form

II~11

%2

033

023

031

~12

c11 C12 C12 0 0 0

%2 C22 C 2 30 0 0

’312 C23 %2 0 0 0

0 0 0 C4400

0 0 0 0 C550

o 0000C5:

Ul,l

U2.2

U3,3

‘2,3 +

UI,3 +

UI,2 +

‘3,2

U3,1

u?>],

(1)

where Cli = (ct2 - cJ12, oij is the Cauchy’s stress tensor, w is the displacement vector and c1l, CIV czv C23, CSS arethe five independent real stiffness constants of the material. We introduce five additional constants al, al, al, adand as related to Cij and the density of the material, p, through

= c221p, a2 = c, / p , a 3 = (c12 + c55)@al(2)

a4 = (C22 - c*3y2p z cddip, a5 = c55iP

It is WCII known that the quantities da,, <az, J% J% and Jas r~present the velocities of five indcPcndent butkwaves that can be transmitted along certain spccitic directions in the transversely isotropic solid.

IA Cll, C,2, C21, C+,, C55 denote the urmplex and frequency-dependent stiffness constants of the fiber-reinforcedcomposite and let the complex constants Al,A~, A3, A, be A5 are defmcd through,

,4, = c22/p, A2 = c),/p, ,43 = (c,* + c55yp,

A4 = c44@, A5 = ~55ip(3)

f(x,,+,lj~ ._..–— ———–-3

FIGURE 1. THE SURF’ACE LOAD PROB1.EM FOR A UNIDIREC1”IONAL COMPOSITE

Wc assume that the two sets of constants ,4i and a, arc related by the equations

(4)

whcrcp is the damping factor (NM et al,1992). For many isotropic solids,p is independent of frequency in a broadfrequency range. This is generally true if the wavelengths are long compared to the rnicrc,structural dimensions(e.g., grain size) of the solid. At higher frequencies the damping, factor incrcascs with frequency duc to wavescattering. We assume that p is constant below a certain frequency roO, beyond which it becomes frequencydcpcndcnt and that it can be expressed in the form

where J== u/2z is the frequency in cycles, ~O = ~/2rt, II(f) is the IIeaviside step function, and~ ,Oa ~ J areconstants which determine the degree of decay in the amplitude of the waves with propagation distance.

‘fhc first tcnn in the right hand side of (5) rcpmsents dissipation duc to internal friction and other thermodynamiceffects, while the second term represents the attenuation caused by wave scattering by the fibers and otherinhomogcneitics in the m[ltcnd, rnoreovcr, wave attenuation due to scattering becomes more and more prominentasJ-yO increases.

The nmtcrial model has been used to calculate the reflected field from graphite/epoxy plates of differentthicknesses immersed in watcl- and the results have been compared with measured data in a pitch-catch typearrangement in an ultrasonic test bed (Mal et al. (1992)). ‘fhc predicted results have been found to be inremarkable agreement with the measured data. We now use this model to calculate the elastodynamic fieldproduced by a conccntratod SUI face load on a graphitefcpoxy Iami nate.

The Surface Load ProblemThe formal solution for a concentrated load problem can be found in (Mal and I,ih (1992)). The general

procedures arc dcscribcd briefly as follows. Consider a unidirectional composite Ian)inatc with thickness II andwith fiber directions parallel to -xl axis, The applied load fly, J /) is assumed to bc acting at the top surface(Yigurc 1). In abscncc of body folces, the governing equations of the problcm bccomc

‘w = Pu(,,, (6)

The boundary conditions can be expressed as

o,5(xl, X2 , o, f) = - @,> .+> O

0,3(X,, X2, 11, 2) = o(7)

where i = 1, 2, 3. In order to solve the problem, wc introduce lourier time transforms of all time-dependentvariablesJthrough

j(X1, X2 , 6)) = ( fix,, +> t) ~! ““ dt

(8)

( 9 )

and denote the Fourier time transform of the displacement and stress components u,(x, I), o,Jx, t) by ti~x, w), ~,{x,~)), Then ~I1(X, w), d,j(x, O) arc sohrt ions of the system

(lo)

6,3(X1, X2 , o, 0) = -j.(x,> Xy ~) (11)

~,@l> X2> 11, 0) = 0, (i ‘ 1, 2, 3) (12)

wherc~<xl, X2, O) is the lJourier time transform of~(~, ~, t), ‘l’he Cauchy’s equations of motion @,quation 10)must be supplcmcntcd by the constitutive Equation (1) and the solution must satisfy the outgoing wave (orradiation) condition at large lateral distances from the load In order to obtain a formal solution of theboundary-value problem in the frequency domain, we introduce the double spatial Fourier transforms of rl~x, u),r$(x, o), and~{xl, xl, o) through

(13)

(14)

(15)

To solve the problem, a six-dimensional “stress-displacement vector” {S) in the transformed domain is

introduced,

{S(X3) ) = {tl,(x3) X,3(X3S

lhcn {S} can be expressed in a partitioned matrix product form as

where

Klr, :Cz,,R ‘(x3 ) = diag[ e c ~! %’3]

(16)

(17)

(18)

In the above, c,’ are complex constants related to downgoing and upgoirrg, waves within the laminate, [Q,j] are3X3 rnatriccs rmd <arc the “vertical” wave nunbcrs of the three possible waves in the composite with “horizontal”wave numbers {,, ~~ ‘1’hc expressions for [Qb] and (i are given in the Mal and 1,ih (1992). Then the six constantscan be determined from the boundary conditions. The solutions fhr the top and bottom displacement can then beexpressed as

m)) = KL] - Q#Q2M#l[QM - Q*FQ,l’QwwDJ (19)

([(II)] : [Q,, - Q, ZEQ2;Q2,][A’][Q2, - Q2JQ;Q21W’ ~)<o)) (20)

where {X(O)} is the stress vector on the top surface of the plate. We assume that load is normal to the surface ofthe laminate, and that the applied load can be separated into a time dependent function and spatially distributedfunction p(xl, .V2). Denote the Fourier time transform ofj(t) byflm), and the spatial double Fourier transform ofP(x,, X2) by l’(C1, Q. ‘f’hcn {W)) bccon~es

{qO)) = -flco) (0, o, P($,, (2)1 (21)

(22)

In this paper t wo specific type of are ccmcerned onc is a unit concentrated load (Figure 2(a)), where P(xj, XJ= 6(xl)b(xJ so that T’({l, Q = 1. lhc other is a unit load uniformly distributed in a circular region with radius a(h’igure 2(b)) with,

Ht,> (2) ‘ J“ ~’ne ‘<t@sO ‘tlsino) dr ~o

= y~ J,(&J), (f * o )- ..—

na 2, (( = o)

(23)

where { = i(~l~ + Lzz) and J, (x) is the BCSSC1 function of the first kind of order 1. A recently developed adaptive

(a) f(ij

FIGURE 2. (a) A POINT LOAD (b) A DISTRIBUTED LOA[) IN A CIRCULAR AREA

ntnncrical integration schcrnc (Ma] and I,ih, 1992) has been used to evaluate the wrwcnurnbcr integral and theresulting spectra arc inverted by fa st l~ourier transfornr (IJH’).

THE EXPERIMENTThe expcrirncntal setup is shown in Figure 3. The source is a single pulse gcncratcd by a Stanford DS345

function generator and runplificd by a Kitcc A300 lU~ gatcd pulse anrplificx. Identical broadband transducers(l Jigital Waves, Mcdcl 131000,5 Ml V.) were used as transnriltcr ancl receiver. A Fracture Wave Detector (DigitalWaves, 1;4000) with four signal conditioning rnodulcs was used for data acquisition. I’hc rnodrdcs arc integratedwith triggers, tnggcr threshold, echo delay controller, tiltcrs, and AI) converteis which can acquire data in at rsrtcs

Stanford 1>S3451 timtion Gerwator

- $ _ _ _ _

[

L.- -.-.::.-.:1

v[ “1

t “- ‘“--.

1( ‘-- ----!

Signal ConditioningModules

Ull I

——0 0 0 00 0 0 0

[10 00 () Fracture

0 0 0 0 g:tor

● *

‘1

PC---m—- —-4/ .—e

Prearrrp

1. ?-. . ..1(50 L ‘---- ‘---””- ‘-” –-”””--[ ..-1’---“)

Rikc AWrO GatcdN‘ Psdsc Amplifier

(......a~k~

—--.—=- —>~___Z* .

C o m p o s i t e 1 .amir~te — -s

FIGURE 3. THE EXPERIMENTAL SETUP

B1OOO, High k’iddii y Transducer

FL PA 2040 G Low Noise PrmnDlifdcr

L“k +/- 0,5 v output

c;:+c~ActIvc Bessel k alter ActIvc Bessel kilter

I $p_4+-.—. ——

Signal Conclitioniug Module

‘:!!JK’’&,,tcrtnterfacc

FIGURE 4. F4000 FRACTURE WAVE DET[:CTOR BLOCK DIAGRAM

from 3.715 to 25 MIIZ, The block diagram of the Fracture Wave 1 lctector is shown in Figure 4. As depicted inFigure 3, the transduc.m were conncctcd to wideband preamplifiers then to signal conditioning modules to recordand digitize the signals at a rate of 12.5 MHZ. The data for each signal were transferred tc) a personal computerfor rrnalysis. “Ihc measurements were made for filed points at 0°,45° and 90° to the fibers,

A [O]lt 12 x 12 crnz unidirectional graphite/epoxy plate was used in the cxperirnc.nt. The material used wasA.%1/3502. ‘fhc plate was made in UC] Ak Composite Manufacturing Lab, The laminate thickness was 3.175 mm,with stiffness constants Cll = 155.01, qz = 6,44, qz == 15.6, q] = 7.89, 95 =- 5,00 (G]’a), and density p =- 1.56g/cm3. The damping cocflicients were assigned the values with pO ~~ 0.005, and a, = 0.

RESULTSThe calculations and the mcasurcrncnts wele carried out for a variety of spccirncns and sources to station

distances, IIcrc we present results for a caw in which the distance of the field point is much larger then the sourcedimensions.‘fhcresultfor wavcpmpagalioll irlalunlinum plate is first presentd for comparison, ‘I’hcrnalcria lconstantsfor

allirnil]url] wI; =72.48 GPa, p=2691 GI'a, andticdanlpirlg conskr(ts used arepO~ 0.005, a0 ‘O. lhcsourccfunction is plotlcd in Figure 5. It shou]d be noted that the source function generated from the function generatoris originally a 2MI KI Iz single sine pulse; however, through gating and amplifying together with the transformationthrough transducers, the original source function is altered

Figure 6 shows the comparison between measured and calculated time histories of normal displacement Uj ona3.175mmt hickahrminumpl atcat 50mmfromthesourcc. ltcanbc secnthat there iscxccllenta greebctwecnthccalculated andthcrncasured wavcforrns. Fi~;ure 7showsthe cort~pafison bctwecn n)casured and calculatedtime histories of the normal displacement UJ at 30 mm from source on a 3,175 mm thick graphite/epoxy laminate\titi~~lahnal wr~Smts nlentiorlcd above forwave propagation along ttle fiber. Although theagrccment is not asexcellent asthatwith ahrrninum, itisvcry good,. Figurc8is thesarnc case as Figure 7forpropagation 450tothcfiber, although thcreis sotl~cdiNerencc forthefirst arrivals, themain pulses arcingc~od agreement, F’igure9isthesamcas Figurc7for wave propagation 90°tothefibcr. Theresult again shows good agrccmeotb etweenthcmeasured andcalculatcd results. Notcthat thedifferences atthcend (}fthctime histc}V iscattsed byrcflcctionsfrom the boundaries of the plate.

I

}

Source

..— —

+

—-=--%/Lti

—. ,--,--- T -T-T -, —T-. ,.--,-– --r., . . ~.. ,-T , ..T_m.. .–T ~ , ~ _,1 1

0 . 0 2 0 . 0 4 0 . 0

‘Time (psec)

FIGURE 5. TIME HISTORY OF SOURCE

,.of..-._+.\–2.0-: -- --I T r r 1 rl-~~”’] “r-rT-TTl-rr ~l~-’”l’T r r- I f i“ l“”]

0 , 0 20$0 40.0 60.0

Time (psec)

FIGURE 6. COMPARISON BETWEEN MEASURED (DASHED LINE) ANDCALCULATED (SOLID LINE) 7 IME HISTORIES OF NORMALDISPLACEMENT U, ON A 3.175 MM THICK ALUMINUM PLATE AT 50 MMFROM SOURCE

—

1.o--{ 0°

‘1‘-l. o-~- T-l–l-[ ‘T r-l- i ‘f-”[-r-r-l”

~.l...T.T ,_ T .. ,._(..,- , -, , , ., ~_T 1

0.0 20.0 40.0 6 0 . 0

Time (~sec)

FIGURE 7. COMPARISON BETWEEN MEASURED (DASHE[J LINE) ANDCALCUIATE[) (SOLID LINE) TIME HISI ORIES OF THE NORMALDISPLACEMENT U,, ON A 3.175 MM THICK GRAPt{lTE/EPOXYLAMINATE PLATE AT 30 MM FROM SOURCE FOR WAVEPROPAGATION ALONG 0° TO THE FIBER

o.4-4 5 °

–0.4 --1-1-1-[ I l~il-rl-l[-lrrlrr~ll’ir-” rrrl

0.0 20. O 40.() 60.0

l’ime (psec)

FIGURE 8. SAME A IN FIGURE 7 FOR WAVE PROPAGATION 45° TOTHE FIBER

I

1.o- 1 90°

( ) .o- .—–.-..–-...-.–.-m..

,,

: .,,:

--1.o- - , - , , , ,1 ,,-1, -,-~, ~-,-, , ~-r,~,, ,, [Ill” 10 . 0 20.0 40.0 60.0

‘1’ime (pscc)

FIGURE 9. SAME AS IN FIGURE 7 FOR WAVE PRC)PAC+ATION 90° TOTHE FIE)ER.

CONCLUDING REMARKSAn investigation comprised of theoretical analysis and experit nents was performed to study the response of

composite laminates to dynamic surface loads. It shows good agreement of the main pulses between the measureddata and the calculated results; however the agreement between calculated and measured time history is excellentfor the isotropic plate, but it is reduced somewhat for the composite laminate. lmprovcrnents in the model andexperiment are nccdcd to obtain better results. The quantitative features of a wave propagating in a compositelaminate arc better understood. ‘Ihc understanding developed in this work will bc uscfrrl for the prediction of thedamage produced in composite Ianlinates subjected to low speed ilnpacts and for the ultrasonic NDE of structuralcomposite components.

ACKNOWLEDGEMENTSThis research was supported by the AFOSR under grant 1 ~49620-93-l -0320 monitored by Dr. Walter

Jones.

REFERENCESChow, 1. S,, 1971, “On the Propagation of Ilcxural Waves on ar, Orthotropic 1,aminrrtcd Plate and Its Response

to an hnpulsc 1.oad,” J. Conrposi[e A4aferial.r, Vol. 5, pp. 306-319.German, M., and Presser, W., 1990, “Al? Source Orientation by Plate Wave Analysis,” Journal of Accoustic

Emission, Vol. 9, pp. 283-288.I,al, K. M., 1984, “Coefficients of Restitution for low Velocity Transverse Impact on Thin Graphite-Epoxy

1 ,aminatcs, “ Composite 7echnologyReview, Vol. 6, pp. 112-117.I,ih, S, and Mal, A. K., 1995, “on the Accuracy of Approximate Plate Theories for Wave Field Calculations in

Theories for Wave Field Calculations in Composite 1 aminates,” Wave Mo[ion (in press),1,iu, G. R., lani, J., Ohyostri, 1. and Watanabe, K., 1991a, “Tral, sicnt Waves in Anisotropic I.aminated Plates,”

Part 1 &2. Jourml of Vibration and Acoustics, Vol. 113, pp. 230-239.I,iu, G. IL, Tani, J., ohyoshi, 1. and Watanabe, K., 1991 b, “Characteristic Wave Surfaces in Anisotropic

1.arninated Plates, “ Jourttal of Vibration atld Acoustics, Vol. 113, pp. 279-285.Mal, A, K., 1988,” Wave Propagation in Layered Composite 1,aminates Under Periodic Surface I.oads,” Wave

J

Motion, Vol. 10, pp. 257-266Mal, A, K., Ilar-Cohen, Y. and I,ih, S., 1992, “Wave Attenuation in Fiber-Reinforced Composites,”

International Symposium on M’II: Mechanics andMechanistns ofMaterialDamping, pp.245-261.Mal, A. K. and Lib, S., 1992, “ElastOdynamic Response of a Unidirectional Composite I aminate to Concentrated

Surface I,oads, Parts I, 11 “ Journal o~dppliedMechanics,, Vol. 5’), pp. 878-892.Medick M., J., 1961, “ on the Classic Plate Theory and Wave Propagation”, Jotirwal of Applied Mechanics,

Vo]. 28, pp. 223-228.Sun, C. T. and Tan, T. M., 1984, “Wave Propagation in a Graphite/Epoxy laminate,” Journal of Asfronaafica/

Science, Vol. 32, pp. 269-284.