Elastic constants of orthotropic composite materials using plate resonance frequencies, classical lamination theory and an optimized three-mode rayleigh formulation

Compositrcs Engineering. Vol. 3, No 5, pp. 395-407, 1993. Printed in Real Britain.

0961-9526/93 s6.00+ .w 0 1993 Per@mon Press Ltd

ELASTIC CONSTANTS OF ORTHOTROPIC COMPOSITE MATERIALS USING PLATE RESONANCE FREQUENCIES, CLASSICAL LAMINATION THEORY AND AN OPTIMIZED THREE-MODE RAYLEIGH FORMULATION E. 0. AYORINDE and R. F. GIESON Advanced Composites Research Laboratory, Department of Mechanical Engineering, Wayne State University, Detroit, MI 48202, U.S.A. (Received 21 February 1992; revised version accepted 9 September 1992) Abstract-This paper shows how the four independent elastic constants (longitudinal and transverse Young’s moduli, in-plane shear modulus and major Poisson’s ratio) of an orthotropic material may be extracted from the modal resonance data of a freely-supported rectangular thin plate made from the material, using the classical lamination theory and an optimized three-mode Rayleigh formulation with a suitably formed least-squares objective function. Results are obtained for orthotropy ratios of unity (aluminum), about three (unidirectional E-glass/vinylester) and about 13 (graphite/epoxy), and plate aspect ratios of unity and about two. The results suggest that the method proposed has potential for rapid characterization of elastic properties of advanced composites.

1. INTRODUCTION

Compositematerialshavesteadilygainedimportancein aerospace,automotiveand many other applicationsbecauseof desirablepropertiessuchashigh strength-weightratio, high stiffness-weightratio and excellentcorrosion resistance.Thesematerialshavetherefore attractedincreasingresearchattention.Whatevertheir application,it is vital that methods be developedfor reliably establishingtheir elastic properties. The most widely used compositematerialsconsistof polymerresinsreinforcedwith high-strength,high-stiffness fibers. Although suchmaterialscanbe anisotropic,experiencehasshownthat many thin, laminated compositeplate, shell and panel structuresin automobile,aircraft and space applicationsareessentiallyorthotropic, and, indeed,some(e.g. chopped-fiber,randomly reinforcedtypes)arealmostisotropic. The elasticbehaviorof orthotropic compositesmay be adequatelydescribedby four elasticconstants-the longitudinal Young’smodulus,E, , the transverseYoung’s modulus, Ey , the in-plane shearmodulus, Grl, and the major Poisson’sratio, VW. Presently,thereare a number of standardsthat stipulatemethodsof testing for the elastic constantsof fiber composites.Notable among theseare the standardsfor high moduluscompositesdrawn up by the ASTM (1987).Someindustrial standardsalso exist. For example,the three major United Statesautomotive manufacturershavedeveloped specialstandardsfor automotivecomposites(ACC, 1990).All thesemethodsarebasedon static tests and have the major drawback of involving many samplesand specialtest fixtures. Thesemethodsareconsequentlyslow and expensive.It would be impractical to use any of thesestandardmethodsto continuously monitor the quality of composite componentson a production line. A major objectiveof this work is to developa reasonablyfast one-shotmethod for measuringall four elasticconstantsof a thin orthotropic material. Vibrating beam tests canonly providetwo of theseconstants,hencethe useof a plate. The platewith all edges free hasbeenchosenbecauseof the inability to exactly realizein practiceother standard boundaryconditionssuchas simply supported,clampedand pinned.The disadvantageof the free edgecondition is that thereis no known tractable, closed-formsolution for the plate vibration problem, and approximatemethodshaveto be used. The literature on the vibration of compositebeams,plates,panelsand shellsis rich (Bert, 1991a,b; Gibson, 1990).Apart from somepartial useof the full three-dimensonal 395

E. 0. AYOIUNDE

396

and R. F. GIBSON

elasticity solution, five main classesof approximatedisplacementtheorieshave been utilized. These are the classical lamination theories (CLT), including single-layeror homogeneousorthotropy cases(Hearmon, 1959;Cawley and Adams, 1978;Crawley, 1979;Leissaand Martin, 1990;Graesseret al., 1991,etc.), first-ordersheardeformation theory (FOST) (Yang et al., 1966; Sun and Whitney, 1973), higher-order shear deformationtheories(HOST) (Noor and Burton, 1989;Librescu, 1991),individual-layer (also called “ply-by-ply” or “discrete-layer”) theory (ILT) (Srinivas, 1973;Cho et al., 1991),and combinationtheories(Noor andBurton, 1990)whichutilize two or moreof the above. In addition, mixed-elasticity methods (Chandrashekaraand Chander, 1986), which simultaneouslyutilize displacementand stressformulations, havealso beenused. Whereasthis work primarily addressesthe “inverse” problem of extractingelastic constantsfrom frequencyinformation, the “forward” problem of estimatingfrequencies from the plate vibration model and known elastic constantsshould first be examined, since its solution determinesthe feasibility and relative easeof solution of the inverse problem. The partial differential equation governing the free transversevibration of a symmetricallylaminatedthin plate at equilibrium in the x-y planemay be written as: Dll

a4w 4 ax + +

4D26

4D16

a4w a4w + 2(D12 + 2D& ax3ay ax2ay2 -

a4w axay3+

4 D22

2

-+ph$=O, tyw

wherex andy are the orthogonalplane coordinates,the plate deflection w = w(x,y, t), the DiiS are the standardbendingstiffnessesof the classicallamination theory (Jones, 19’75),p is the plate density, or mass per unit volume, h is the plate thicknessand t representstime. The deflectionamplitude W may be approximatelyrepresentedby the series:

whereA, are undeterminedconstantsand (+(x) and +j(Y) are admissiblecharacteristic normal-mode beam functions in the appropriate directions. It may be alternatively representedby: w(x,Y)

= i:

Biyi(X,Y)*

(3)

i=l

where Bi are undeterminedconstantsand Yi(x, y) are admissible characteristicplate functions. The energymethod, in various forms, is the most widely usedapproximateanalysis procedure.Accordingly, the expressionsfor the maximum potential and kinetic energies of a harmonically vibrating plate may be respectivelyrenderedas:

(4)

and Tmax=+phiu2

= ‘W’dydx ss0 0

wherew is the frequencyof vibration, and a andb arethe platesidedimensionsalongthe x- and y-axes,respectively.

Elastic constants of orthotropic

composite materials

397

For specially orthotropic plates, eqns (1) and (4) become simpler, as Oi6 = Dza = 0, Dl, = D,, Dzz = Dy , D,, = vwD,, and D,, = Dw . In all of these equations, D, = EXh3/12(1 - v,v,,),

D,, = E,,D,/E,,

Dxy = h3GV/12,

E, and Ey are Young’s moduli in the x- and y-directions, respectively, G, is the shear modulus associated with the x-y plane, and vV and vyXare the major and minor Poisson’s ratios, respectively. The next step in the forward problem is to substitute the assumed deflection expression of eqn (3) into the energy expressions of eqns (4) and (5) and equate the maxima of the potential and kinetic energies according to the Rayleigh method. This yields the frequency, o. Different deflection functions have been used in eqns (4) and (5), including beam characteristic functions (Young, 1950; Warburton, 1954; Hearmon, 1959; Dickinson, 1978), simply supported plate functions (Dickinson and Li, 1982), beam characteristic orthogonal polynomials (Bhat, 1985), plate characteristic orthogonal polynomials (Lam et al., 1989) and simplified (nonorthogonal) polynomials (Kim et al., 1990). Dickinson and Li (1982) proved that once a free edge is involved, there is no clear advantage of accuracy in using either the beam or plate characteristic function representation. Accordingly, a beam representation should be preferable in such cases because it is simpler. Bhat (1985), using a Gram-Schmidt process, constructed an orthogonal polynomial beam representation which proved superior to the normal-mode type for lower modes and free edge conditions. An alternative, nonorthogonal set which required somewhat more computation, but much simpler function generation and integral evaluation procedures, was later proposed by Kim et al. (1990). Lam et al. (1989) also used two-dimensional orthogonal polynomials for the vibration of rectangular composite plates. From a study of the relative ease, accuracy and required computational effort involved in these approaches, it was concluded that a characteristic normal-mode beam representation should suffice for basic work and an improvement could later be sought with the use of nonorthogonal simple polynomials such as those proposed by Kim et al. (1990). Different energy methods have also been applied to the plate vibration problem, including the Rayleigh (Warburton, 1954; Dickinson, 1978) and Rayleigh-Ritz (Young, 1950; Warburton, 1954; Dickinson and Li, 1982; DeWilde et al., 1984) methods and Hamilton’s variational principle (Srinivas, 1973). The most popular approach has been to take a sufficient number of terms in the series representation of the deflection. The easiest and most direct approach is the basic Rayleigh method which has been confirmed to be reasonably adequate for lower plate modes as long as in-plane forces are low and no edge is free. The Rayleigh-Ritz method with a sufficient number of terms yields much better results, but at a significant cost of computational effort and time. In this respect, some workers (Leissa, 1973; Dickinson and Li, 1982; Deobald and Gibson, 1988) have used 36 terms (i.e. six in each direction), and others (Leissa and Narita, 1989) used 144 terms. The improved Rayleigh method introduced by Kim and Dickinson (1985) consists of applying the principle of minimum potential energy to obtain an optimized three-term deflection representation. The resulting expression is given in the following equation and may be qualitatively conceived as an optimal combination of the mode of interest (i.e. the resonance mode whose experimental data is being fitted to the frequency equation) and the next two higher modes, W-G Y) = e%W(Pj(Y)

- C4cmAY)

- a?t(x)goi(Y)1.

(6)

Here the mode shape of interest is the (i,j)th, A is a constant, @i(x) and #am are the appropriate ith andjth beam mode shapes, m = i + 1, n = j + 1 for dissimilar beam end conditions, and m = i + 2, n = j + 2 for the case where the conditions are the same at each end. Constants c and dare obtained by optimizing the frequency using the principle of minimum potential energy (i.e. a&,=/& = 0 and aV,,/ild = 0). Since this approach leads to a much simpler frequency expression, it is particularly adaptable to the solution of the inverse problem. The present work is based on this approach. Although, as will be seen later, results obtained by using this three-term expression for Ware quite reasonable,

398

E. 0. AYORINDE

and R. F. GIRSON

it is envisaged that one possible future improvement of our procedure would be to use a six-term expression for increased accuracy. In contrast to the abundance of works on the vibration of laminated plates as partly referenced above, the use of dynamic test methods and the solution to the inverse problem to determine the four elastic constants has been the subject of a relatively smaller research effort. Zelenev and Electrova (1973) and Wolf and Carne (1979) worked on isotropic materials. Rutkowski et al. (1981) outlined a method based on an impulse modal test and finite-element computations. Their frequencies were obtained from the eigensolution of a system of equations. This way, the number of frequencies obtained depends on the system size, and there is no direct way of assigning computed frequencies to specific modes. When an analytical frequency expression is used, as in this work, the analytical modal indices are explicitly input, and therefore known. DeWilde et al. (1984), McIntyre and Woodhouse (1988) and Deobald and Gibson (1988) later addressed the problem for orthotropic materials in independent attempts. The work of DeWilde et al. was based on Galerkin’s method while that of Deobald and Gibson was based on application of the Rayleigh-Ritz method to the vibration of a plate. The Deobald and Gibson method has been unable to consistently obtain reliable values for the Poisson’s ratio for orthotropic materials while the DeWilde method is more time-consuming and requires the preparation and test of various beam and plate specimens. Both methods have been applied only to unidirectional composite materials. McIntyre and Woodhouse used a power series displacement representation with Rayleigh’s method. They extract approximate mode shapes from a 36 x 36 eigenmatrix. With their method, for a particular material, it is necessary to try various plate aspect ratios until one obtains two that give modal responses of the cross and ring types, respectively. Experimental data from these are then used in their plate vibration model. They appear to acknowledge the analytical and experimental difficulties of their method for orthotropic materials. Fallstrom and Molin (1991a, b) have more recently applied the essence of the McIntyre and Woodhouse method, but replaced the Chladni powder trace procedure with a real-time, laser-based, TVholography system, and used the finite-element method for computation of the elastic constants. Their method is based on the use of some guessed functions to approximately represent the displacement of the plate in each of three particular modes of vibration (the l-l, cross and ring modes). It is considered that their approach is also significantly more costly both experimentally and computationally than that being proposed in this paper. 2. THEORETICAL

For specially orthotropic

ANALYSIS

rectangular plates, eqn (4) becomes:

and eqn (5) remains the same. By minimizing the potential energy in eqn (7) with respect to the constants c and d, respectively, in the deflection function of eqn (6) Rim and Dickinson (1985) obtained: c = (CdEu

- EjiF)/(Ci,Cq/

- F*)

d = (CittEji

- EcF)/(Ci,Cw

- F2)

03)

where C, = (DJH)Gf(d/a2)

= (D,/H)Gj(a2/b2)

+ 2[H,Hj

Eu = Hi(Kj + Lj)[2(O,/H)

- l] + 4(O,/H)JiMj

Eji = Hj(Ki + L,)[2(Dv/H)

- 11 + 4(0,/H)JJM,

F=

-(KiKj

+ LiLj)[2(O,/H’)

- l] + 4(O,/w)M,Mj

+ 2(O,/H)(J,Ji

- HiH,)] (9)

Elasticconstantsof orthotropic compositematerials

399

and the integrals Gi, Hi, Ji, Ki, Li, and Mi, which are basedon normal-modebeam characteristicfunctions, are givenby:

(W

and H = vxyD,, + 20,.

(11)

Whenthe maximum valuesof the kinetic andpotentialenergies[eqns(5) and (7)] are equatedaccordingto Rayleigh’smethod, the frequencyequationis obtainedas m2pw2

lt4H

= Cu + C2C, + d2Cw. - 2cEu - 2dEji + 2cdF . 1 + c2 + d2

(12)

For our purposes,it is betterto rearrangeeqn (12)as pha2bz a=

H(Co + C2C, + d2Cd - 2cEg - 2dEji + 2cdF) . ~‘(1 + c2 + d2)

(13)

The reasonfor this preferenceis that in the latter form, the left hand side (LHS) is completelydeterminedby the massdensityand geometryof a givenplate, and, unlike the right hand side(RHS), is totally independentof modal frequenciesand indicesas well as elasticmaterial constants.The LHS may thereforebetakenasa benchmarkagainstwhich changing RHS values may be compared as they are determined from different experimentalmodal parametersand trial valuesof the elasticconstants.In this regard, eqn (13) may be essentiallywritten as:

fL =fR*

(14)

If we know the resonancefrequencycuexactly and the four elasticconstants,then eqn (14)shouldbe exactlysatisfied.However,sincethis is not the case,the equationcan only be approximatelysatisfiedfor eachfrequency.Thus, for any measuredfrequency, we may sayfR - fL is equalto a residud, and define a dimensionlessresidual,S, as

This dimensionlessform is preferablehereandessentialin othercases(aswill be seenlater) whenthe objectivefunction involvesadditional contributions.In particular, for onemode set (i.e. four modal frequenciesand their correspondingmodal shapeindices,from which we canextractestimatesof the four unknownelasticconstants),we have:

f‘-lEdi fL

i = 1,2,3,4.

400

E. 0.

AYORINDE

and R. F.

GIBSON

Since6i may be positiveor negative,its squareis chosenas a measureof how accurately the frequencyeqn(12)is satisfiedfor the frequencyvalue. Thus, a goodmeasureof how well the four frequenciesand the trial valuesof the elasticconstantstogethersatisfythe frequencyequation,shouldbe indicatedby the sum of squaresof the four appropriateSi values.The residual-on-frequency equation satisfaction,R, , is thereforethe objective function that needsto be minimized. It is givenby: 4 R, = c S;,

(17)

i=l

wherethe subscriptQ denotes“equation”. The resultingsetof frequencyequationscan be solvedin many ways. Direct solution of the resultingmatrix equationusinginversion is possible, and was tried, using the Newton-Raphsonmethod and its variants but significantvariationsin the magnitudesof the elementstendto makesomeof the matrices ill-conditioned, henceincapableof yielding acceptableresults. Solutions by the finiteelement and finite-difference methods are also possible but would be much more computationally costly, although such methods could be consideredfor full threedimensionalanalysis.Even with the regressionmethod usedhere, severaloptions are possible,and many weretried, althoughthe presentpaperis basedon the useof only the objective function of eqn (17). More generalobjective functions could be formed in variousways. It is well known that experimentalmodal data are subjectto error. If we assume that one of the known plate parametersp, a, b, h (i.e. massdensity, major and minor dimensions,and overall thickness)is in fact unknown, and then utilize a set of experimental modal data and trial values of the elastic constantsto solve the frequency equationsfor the parameter, its computed value would, in general,be found to be different from its actualvalue. Thus, in addition to the degreeof accuracywith which a particular set of data (modal and elastic constants)satisfiesthe governingequations, similar measuresof how well it predictseachof the four known plateparameterscouldbe usedas contributingobjectivefunctions. The implication is that a completelycorrectset of modal and elasticdata would perfectlysatisfythe equationsand would alsoaccurately “predict” thoseparameters.The dimensionlessresidualcontributedby the estimateof an individual parameterwould thereforebe:

8,=

computedvalueof parameterK -1. actualvalueof parameterK >

(18)

Thus, an expandedobjectivefunction may be definedas: objectivefunction = R, + R, + R, + Rb + Rh

(19)

wherea typical residualR, is givenby: R,=

id:.

(20)

i=l

In eqn (20),i is summedoverthe four modesthat arealwaysrequiredto extractthe four elasticconstants,andK representsany of Q, p, a, b and h asdetailedin eqn(19).It is also conceivablethat, owingto somepeculiarbehaviorof an advancedcompositematerial,the proper effectsof the various contributionsin eqn (19) would not be uniform. A more generalobjectivefunction would thereforeinclude weightingfunctions C,, as objectivefunction = CQRQ + C,R, + C&R, + CbRb + ChRh.

(21)

The determinationof appropriateweightscould be donesemiempiricallyusingdatafrom known cases.From the large number of options tried, it was found that for some optimization schemes,in certaincircumstances,theweightingpatternsignificantlyaffects the convergencespeedand accuracy of computations. However, only the objective function of eqn (17) was usedto obtain the resultsin this paper. This is equivalentto settingCo = 1 and C, = C, = C, = C, = 0 in eqn (21).

Elasticconstants of orthotropic composite materials

401

3. EXPERIMENTAL ANDCOMPUTATIONAL PROCEDURES Experimentalvibration data for 254mm-squarealuminum andgraphite/epoxyplates of thicknesses3.16 and 1.483mm, respectively,from the work of Deobaldand Gibson (1988)were utilized in this work. In addition, a unidirectional E-glass/vinylester(FRP 470)plate (Sullivan, 1991),179.4x 165.9x 3.251mm, density 1590kg rns3and a 12-ply unidirectional graphite/epoxy plate 255.6x 126.2x 1.7526mm, density 1540kg m-‘, weretestedin the arrangementof Fig. 1, and their modal frequenciesand profiles were established.It shouldbenotedthat the plateboundaryconditionsin Fig. 1 arenot exactly the sameasthosein the work of Deobaldand Gibson(1988),wherethe plate wasplaced on a soft cotton pad to simulate free support. In the present work, the plate was suspendedby nylon filaments in an inverted “Y” arrangementwhich has extremelylow vertical impedance.However,the free-edgecondition holds approximatelyin both cases. The new experimentalapparatusis the subject of another paper in preparation. The modal frequencieswere obtainedby impacting the plate with the impulse hammer and observingthe spectrumanalyzerdisplay. Impacting was done at many points in turn to avoid missingany resonancein the rangeof interest(lower end of frequencyspectrum), and using the zoom feature of the analyzerto ensurehigh accuracyin obtaining each particular frequency. The mode shapes correspondingto these frequencieswere determinedby usingthe hammerto locatethe nodal linesresponsiblefor the subtraction of a particular mode. Using the convention of numbers of half waves along (or alternatively nodal lines perpendicularto) each principal direction gave the required modal shapeindices.The modal shapeindicesand correspondingfrequencieswerethen utilized as inputs to the computerprogram. In the program, a searchwas carriedout in the four-spaceof the parametersEx, Ey , GW, and vV . The residualof eqn (17)wasused as the objectivefunction to be minimized. The flow chart of Fig. 2 indicatesthe logical structureof the computerprogram. Sinceit wasdiscoveredfrom extensiveinvestigation that the Poisson’sratio seemedto belargelyresponsiblefor the high modal densityof the four-parameterobjective function, an alternative of optimizing that function in the E,-E,-G, spacefor specific valuesof Poisson’sratio was also employed.The results presentedare from this latter approach.The Poissonratio was searchedup to the upper theoreticalminimum (Jones,1975)given by (EJE,) 1’2.Typical computation time for a cycleon a 386personalcomputerusing QuickBasicwas about 6 min.

Force Transducer PCB 208 A02

Miniature Accelerometer

HP 9om Model 332

Fig.1. Blockdiagram

of

modalanalysis equipment.

402

E. 0. AYORINDE and R. F. GIBSON Select

value

of

v,,

Of

E,,

1 Select

trial

values

EVV

GM

I

Is

Values

Residual

of

< Last

constants

Residual

exhausted

?

?

Fig. 2. Flow chart for determination of elastic moduli. 4. RESULTS

The resultsare summarizedin Tables l(a-d), and typical variations of the residual againstthe Poisson’sratio aredepictedin Figs 3(a-d). The first proper(i.e. turning point type) minimum, along the Poissonratio axis is taken for each case,for reasonslater explainedin more detail. A weighted average, with the inverseof the residual as the weightingfactor, is taken overthe five modesets.In the rare caseof absenceof a proper minimum for a particular mode set, the results for that mode set are simply omitted altogether. 5. DISCUSSION

The methoddescribedabovehasbeensuccessfullyappliedto find with goodaccuracy the four elasticconstantsof materialswith orthotropyratios betweenunity (i.e. isotropic) and about 13, and plate aspectratios betweenone (i.e. square)and about two. The resultsin Tablesl(a-d) showthat the useof experimentaldatafrom evena single mode set (four resonances)generallygives a rather reasonableset of estimatesfor the elasticconstants,especiallythe elasticmoduli. As expected,the Poisson’sratio provesto be the most variableof the four constants. It also appearsto contribute significantly to the deviation of the residualfrom its ideal value of zero, as may be observedfrom suchentriesas row 5 of Table l(a), rows 1 and 5 of Table l(b), row 5 of Table l(c), and row 2 of Table l(d). There are some cases, although much fewer, in which the Poisson’sratio is practically correct and yet the residualis high. In suchcasesit may be seenthat the other predictedelastic constants deviatemore from their correctvaluesand are thereforeresponsiblefor the situation. One possibleway of decidingthe final result is simply to admit the completeset of predictedelasticconstantsfor which theresidualis minimum. The resultsin Tablesl(a-d) showthat this would also yield resultsthat in many casescomparefavorably with those

Elasticconstantsof orthotropic compositematerials

403

Table 1. (a) Elasticconstantsof aluminum from modal data of squareplate [mode/frequency (2,2: 156.73, (2.3: 411.7). (3,2: 411.7). (1.4: 744.9).(4, 1: 744.9).(4, 1: 744.9)]. (b) Elasticconstantsof graphite/epoxyfrom modal data of squareplate [mode/frequency(2,2: 49.37). (3, 1: 210.5), (2,4: 222.4), (3,2: 231.6), (3,3: 295.211. (c) Elastic constants of unidirectional E-glass/vinyl ester (FRP 470) from modal data of square plate [mode/frquency (2,2: 154.4), (1,3: 264.4), (2,3: 417.6), (3,2: 516.8), (1,4: 745.5)]. (d) Elasticconstantsof graphite/epoxy from modal data of rectangularplate [mode/frequency(3,1: 263.2).(3,2: 356.0). (2.3: 380.8), (3.3: 635.2). (4, 1: 732.8); plate aspectratio 2.031 Mode set*

(ii%)

(G?a)

(G%a)

Residualx lo*

69.1 72.9 76.7 77.2 60.9 73.0 72.4

69.1 72.6 76.5 76.5 58.2 72.7 72.4

25.2 24.1 25.4 25.4 24.9 24.2 28.0

0.35 0.30 0.18 0.18 0.48 0.29 0.33

11.347 0.143 3.700 2.800 10.342

122.4 126.1 122.0 125.0 123.5 124.5 127.9

10.05 10.89 9.81 11.69 10.06 10.50 10.27

6.29 5.97 6.54 6.35 6.29 6.26 7.3

0.53 0.15 0.25 0’ 0.35 0.22 0.22

23.263 2.388 5.033 6.373 14.613

21.6 22.6 22.1 21.5 12.8 21.8 24.0

6.87 7.38 7.29 7.29 7.38 7.30 6.87

3.06 3.06 3.20 3.20 3.23 3.09 2.89

0.473 0.350 0.320 0.38 0.22 0.358 0.325

8.973 1.683 22.089 19.993 19.358

138.4 141.4 141.2 141.2 141.7 140.3 127.9

10.31 10.91 10.40 12.45 10.91 10.80 10.27

6.62 6.09 6.67 6.09 6.14 6.4 7.3

0.275 0.475 0.15 0.26 0.23 0.26 0.22

1.362 5.450 2.246 3.799 4.063

Table l(a)

1 2 3 4 5 Weightedaverage Actual’ Tabie l(b)

1 2 3 4 5 Weightedaverage Actual’ Toble l(c)

1 1

L

3 4 5 Weightedaverage Actual’ Table I(d)

1 2 3 4 5 Weightedaverage Actual’

‘Frequenciesare in Hertz. *A mode set is a group of four successive modes chosenin cyclicalorder from the list of five experimental modesusedfor the particular case.Thus, mode set 1 here comprises(2,2), (2,3), (3,2) and (1,4). ‘From Deobald and Gibson (1988). ‘No proper minimmn identified. ‘From Sullivan(1991).

of the averagingmethod used.However, there are somecases,such as in Table l(b, d) whereit yieldssignificantlyworsePoisson’sratio valuesthan the averagingmethod.Since the relativecorrectnessof the resultsfrom eachmodesetis alreadyrecognizedby utilizing the inversesof the residualsas weightingfactors, the authorsbelievethat the weighted averageshould be the preferredanswer. All the elasticconstantspredictedfrom the rectangulargraphite/epoxyplate (aspect ratio 2.03) are observedto be slightly higher than those from the squareplate. The longitudinal elasticmodulus, E,, is about 7% higher, while the Poisson’sratio is about 18% higher. More extensiveinvestigationof the effect of the plate aspectratio upon the method being proposedis under way, but the presentresultsare still consideredgood, evenat this aspectratio. It is also observedthat the shearmodulus, G, , is generallymore under-predictedthan the other constantsfor the aluminum and graphite/epoxy, and over-predictedfor the vinylester(PRP 470). However,Deobaldand Gibson(1988)notedthat the “actual” elasticconstantsthey quoted from the metals handbook for aluminum should be taken as typical valuesas

E. 0.

lo-= 0.0

AYORINDE

and R. F.

I 0.5

1.0 ?olssows

GIFSON

I 1.5

am0

Fig. 3. (a) Equation residual from modal data of square aluminum plate [modes (2,2), (2,3), (3,2), (1,4)]; (b) Equation residualfrom modal data of squaregraphite/epoxyplate [modes(3, l), (2,3), (3.2). (1,4)]; (c) Equation residual from modal data of square unidirectional E-glass/ vinylester(FRP 470) plate [modes (1,3), (2,3), (3,2), (1,4)].

Elasticconstantsof orthotropic compositematerials

I 0.5

lo4 0.0

I 1.0

, 1.5

I 2.0

t 2.5

I 5.0

405

5.5

Fig. 3. (d) Equation residual from modal data of rectangular graphite/epoxy plate (aspect ratio 2.03) [modes(2.2), (3, l), (2,4), (3,2)].

they do not satisfy the standardrelationshipamongelasticconstants G = E/2(1 + v).

(22)

All the quoted “actual” elasticconstantsare static valuesexceptgraphite/epoxymoduli obtainedfrom beamvibration experimentsasquotedby DeobaldandGibson (1988).The facts that the elasticconstantsactually vary with frequency,and that we are usingonly a three-termapproximationfor displacementmay belargelyresponsiblefor thesmall errors in the computed values of these constants,as discussedin the precedingparagraphs. Unavoidableexperimentalerrorsmay accountfor the restof the errors.Possibleimprovementsto accuracyarebeing investigatedin a paperunderpreparationwheremore terms are usedin the expressionfor the admissibleplate deflection. In Figs 3(a-d), the behaviorof the residualwith respectto the Poisson’sratio seems to manifest certain patterns.Unimodal behavioris shown in someof the 20 mode sets considered.This was true of nearly all the aluminum mode set results, and may be generallytrue of isotropic cases.In most cases,however,two, and sometimesmore, minima were manifested. Generally, the first minimum, correspondingto the lowest minimum Poissonratio, alsoinvariably correlatedbestwith the correctvaluesof all four elastic constantsand was lower than the others. In one casethe only two minima were about the same,and in three casesthe secondminimum was somewhatlower than the first. However, in such cases,the four elastic constantscorrespondingto the second minimum deviate significantly from the correct values. Support damping, use of a truncateddisplacementexpression,and neglectof transverseshearmay be responsiblefor the slightly nonidealminima behavior observedin thosefew cases.However,theseand other points are being addressedin on-going work. It would then seemthat the first turning-point minimum always yields the best Poisson’s ratio. The upper theoretical maximum Poisson’sratio, basedon an assumptionof ideal two-dimensionalgeometryis given by (Jones,1975): VI2 = (E&!?2)“2. (23) If three-dimensionalbehavioris recognized,the boundson the Poissonratio values become(Jones,1975;Lempriere, 1968): +v12(9

+ [ 1 - ti2gy2[

< v21 < -Iy32v11@

1 - qg3)]1’2gy’2~

- [ 1 - v:1($)]“2[

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E. 0. AYO~NDEand R. F. hSON

One of the factors that complicate the responseof square plates is diagonal symmetry. If the material is isotropic, the squaregeometricalshapecreatessymmetric boundary conditions about each diagonal. Essentially, this means that some modes cannot be adequatelydescribedsimply as (m, n) but as (m, n f n, m). This issuewas addressedfor membranes(Strutt, 1945)and plates (Warburton, 1954; Leissa, 1973; Kim and Dickinson, 1985).The salientconclusionsare that the fully, simply supported, isotropic, square plate is not similarly affected, and an alternative formulation for configurationsmanifestingsuchbehaviorwasgiven.This point wasprovedin the course of this work by utilizing the modal dataof Deobaldand Gibson(1988)for aluminum, for which the useof diagonallysymmetricmodeswith the main formulation, asexpected,did not yield correctresults.Insteadof a generalapproachof utilizing two different formulations to coverboth typesof modal responses,the additional symmetrydata were simply excludedin this work. One sure,practicalway of eliminating their occurrenceis simply to avoid usingsquaregeometryfor isotropic cases.Conclusively,a simplemethodhasbeen introducedfor extractingthe elasticconstantsof orthotropicmaterialsfrom experimental plate vibration data. The method is adaptableto a wide variaty of formulations of the elastic-dynamicbehaviorof laminatedplates. Acknowledgements-The authors gratefully acknowledgethe support of a grant from The Ford Motor Company. We are alsograteful to Andy Wen, graduate assistant,who carefullyconductedthe experhnentson the glass/vinylesterand the rectangulargraphite/epoxy plates. REFERENCES AmericanSocietyfor Testingand Materials(1987).ASTM Standards and Literature References for Composite Materials, ASTM, Philadelphia,PA. Automotive CompositesConsortium (1990).Test Procedures for Automotive Structural Composite Materials. ACC, Troy, MI. Bert, C. W. (1991a).Researchon dynamicbehaviorof compositeand sandwichplates-V: Part I. Shock Vibr. Dig. 23(9), 3-14. Bert, C. W. (1991b).Researchon dynamicbehaviorof compositeand sandwichplates-V: Part II. Shock Vibr. Dig. 23(10), 9-21. Bhat, R. B. (1985). Natural frequenciesof rectangularplates using characteristicorthogonal polynomialsin Rayleigh-Ritzmethod. J. Sound Vib. 102(4),493-499. Caldersmith, G. and Rossing,T. D. (1984).Determination of modal coupling in vibrating rectangularplates. Appl. Acoust. 17, 33-44.

Cawley,P. and Adams, R. D. (1978).The predictedand experimentalnatural modesof free-free CFRP plates. J. Compos. Mater. 12, 336-347. Chandrashekara.S. and Chander, R. (1989). Structural vibrations and acoustics.Proc. 1989 ASME Design Technical Conferences-12th Biennial Conf. on Mechanical Vibration and Noise, DE-Vol. 18-3 (Editedby T. S. Sankaret al.), pp. 171-175.ASME, New York. Cho, K. N., Bert, C. W. and Striz, A. G. (1991).Freevibrations of laminated rectangularplatesanalyzedby higher order individual-layertheory. J. Sound Vib. W(3), 429-442. Crawley,E. F. (1979).The natural modesof graphite/epoxycantileverplatesand shells.J. Compos. Mater. 13, 195-205. Deobald, L. R. and GibsonR. F. (1988).Determination of elasticconstantsof orthotropic platesby a modal analysis/Rayleigh-Ritztechnique.J. Sound Vib. 124(2),269-283. DeWilde,W. P., Narmon, B., Sol, H. and Roovers,M. (1984).Determination of the material constantsof an anisotropiclamina by free vibration analysis.Proc. 2nd Modal Analysis Conf., Vol. I, pp. 44-49, Orlando, FL. Dickinson, S. M. (1978). The buckling and frequency of flexural vibration of rectangular isotropic and orthotropic platesusing Rayleigh’smethod. J. Sound Vib. 61(l), l-8. Dickinson,S. M. and Li. E. K. H. (1982).On the useof simplysupportedplate functions in the Rayleigh-Ritz method applied to the flexural vibration of rectangularplates.J. Sound Vib. 88(2), 292-297. Fallstrom, K. E. and Molin, N. E. (1987). A nondestructivemethod to determine material properties in orthotropic plates. Polym. Compos 8, 103-108. Fallstrom, K.-E. and Molin, N.-E. (1991a). A nondestructivemethod to determine material properties in anisotropicplates.Polym. Compos. 12, 293-305. Fallstrom. K.-E. and Molin, N.-E. (1991b). Determining material properties in anisotropic plates using Rayleigh’smethod. Polym. Compos. 12, 306-314. Gibson,R. F. (1990).Dynamicmechanicalpropertiesof advancedcompositematerialsand structures:a review of recentresearch.Shock Vib. Dig. 22(8), 3-12. Graesser,D. L., Zabinsky,Z. B.. Tuttle, M. E. and Kim, G. I. (1991).Designinglaminated compositesusing random searchtechniques.Compos. Struct. 18, 311-325. Hearmon, R. F. S. (1959).The frequencyof flexural vibration of rectangularorthotropic plateswith clamped or supportededges. J. Appl. Mech. M(l), 537-540. Jones,D. (1975).Mechanics of Composite Materiab. Hemisphere,New York.

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