THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA VOLUME 36, NUMBER 6 JUNE 1964

Validity of Thin-Plate Theory in Dynamic *

M. A. B•or

300 Central Park West, New York, New York 10024

AND

F. V. POHLE

Adelphi University,Garden City, New York (Received 17 October 1963)

Resonancedamping for a vibrating plate is investigatedboth accordingto the exact equationsof dynamical viscoelasticityand the classicalthin-plate equationsderived in mechanicsof materials.The plate is assumed as isotropicand homogeneousand no shear-or rotatory-inertiacorrections have beenincluded in the thin- plate approximations.Two typesof materialsare investigatedthat correspondto real and complexvalues of the bulk modulus.For eachcase, the complexshear modulus is t•(lq-ig) and valuesof g up to 0.10 were used in the calculations.The two theories are in excellent agreementin a range of wavelengthsas low as about ten times the thickness.It is found that thin-plate theory evaluatesthe dampingmore accurately than it doesthe static rigidity.

LIST OF MAIN SYMBOLS complexratio of load to dynamicdisplace- coefficientsin Eq. (3) ment, exact theory 61 ratio of static load to static , 62 E(xq-2tz)/p] '• plate theory ½ = e•,•q-eyu Rp complexratio of load to dynamicdisplace- exx =Ou/Ox ment, plate theory exy = (1/2) (Ou/Oy+Ov/Ox) R/R -1/(eq-iG), amplificationfactor at reso- eyy =Ov/Oy nance, exact theory E Young'smodulus Rp/l•p = 1/(epq-iGp), amplificationfactor at reso- = 4u(X+u)/(X+ 2u) nance, plate theory g dampingconstant in shearmodulus t time G measureof the damping,exact theory u = Oqv/Ox-Of/Oy, horizontal displacement, Gp measureof the damping,plate theory Fig. (1). h thicknessof the plate v = 0 qv/Oyq-Of/Ox,vertical displacement, i Fig. (1). K bulkmodulus= Xq-(2/3)u U, V displacementamplitudes of u, v, respectively l wavenumber,Eq.(8) x,y horizontaland vertical coordinates, Fig. (1). wavelength(œ/h= r/'•) 1-- (co•'/c•'l•') q•/q, •q-iG(exact theory) 1- (co•'/c•.•'l•') qrp/qsp •pq-iGp(plate theory) Eq. (24) R ratioof static load to staticdisplacement, =lh/2 exacttheory 0 =ty * Presented at the Sixty-Fifth Meeting of the Acoustical frequency Societyof America,New York, 15-18 May 1963. =co/c,1 1110

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= 2l•e•y which had been developedearlier by Alfrey 7 and was = 2lzeyyq-Xe restricted to the static analysis of isotropic massdensity incompressiblemedia. solutionsof wave equations,Eqs. (5) The generalization to compressibleisotropic and = E/J2 (1q- v)•, shearmodulus anisotropicmedia was derived in a paper by Biot in = vEil(l+ v)(1-- 2•)•, Lam•'s constant 1954.1In two companionpapers by the sameauthor in - lz(1 q- ig), complexshear modulus 1955,5,6 the term correspondenceprinciple (or corre- = (2•/3)[3v/(1- 2r)-ig•, complexLam•'s spondencerule) was introducedin order to emphasize constant for material 1. the far-reachingconsequences of this development.It Poisson's ratio was pointed out that by this principle a vast body of Laplace's operator, in rectangular co- resultsin the theory of becomesimmediately ordinates applicable to viscoelasticity, including problems of exp(x) = e x vibrations and wave propagation in materials with isotropicor anisotropicproperties. INTRODUCTION Applications to the dynamic theory of viscoelastic plateswere alsodeveloped in somedetail in the quoted papers,*.• along with somegeneral theoremsfor visco- IBRATINGouslylarge amplitudesstructural elementsat resonant mayfrequencies. reach danger-In elastic continua derived by variational procedures.This the past, the resonantconditions have frequentlybeen general formulation and its extension and generali- avoidedin the designstages, but in many present-day zationsmake possiblethe solutionof a large number of problemsthis is no longerpossible unless heavy weight technically important problemsinvolving plates and penalties are acceptable.It is therefore necessaryto shells.We considerhere an application to problemsof introduce damping into the design of the structural viscoelasticdissipation in plates. componentsto reducethe peak stressesand amplitudes Practical engineeringneeds require the use of simpli- and to minimizethe problemof fatigue. fied theoriessuch as the well-knownthin-plate approxi- Among the many sourcesof structural damping,that mation that is derived in texts on the mechanics of due to viscoelasticdissipation (material damping) is materials. It is not clear how accuratelytheories of this consideredin this report. The presenceof even a small type can predict the energy dissipationand resonant amount of damping can have a decisiveeffect on the amplitude due to the viscoelasticnature of the material. resonant amplitude and stress levels, and the actual The elementarytheories of the type indicated usually reduction is very sensitiveto the precise amount of predict displacementsquite accurately, but stresses dampingthat is present. must be interpreted carefully for designpurposes; the General studies of three-dimensional viscoelastic dissipativeeffects are an even more sensitivereflection media have been developedin recentyears basedupon of the displacementpatterns and, for a proper evalua- new methodsand conceptsin linear irreversiblethermo- tion of the effectivenessof simplified theory, it is dynamics.•-4 In the context of thesegeneral theories, a necessaryto work out fully a problem by the exact correspondenceprinciple and an associatedvariational equationsof dynamic viscoelasticity.The correspond- principle were developed•,5,6 that are of extreme ence principle•,*,• can then be used to determine the generality and are applicable to problems including damping accordingto the exact theory and the results dynamics and anisotropy. They provide a complete can be comparedwith thosederived from the thin-plate generalizationof the so-called viscoelasticanalogy, theory.In the presentapplication, no shearor rotatory inertia correctionshave been made to the thin-plate 1 M. A. Blot, "Theory of Stress-StrainRelations in Anisotropic equations. Viscoelasticityand Relaxation Phenomena,"J. Appl. Phys. 25, To carry out this program, a simple but typical 1385-1391 (1954). example is taken that is capable of an exact solution 2 M. A. Biot, "Variational Principles in Irreversible Thermo- dynamicswith Applicationsto Viscoelasticity,"Phys. Rev. 97, by the three-dimensionalequations of elasticity; the 1463-1469 (1955). elasticmoduli can then be replacedby their complex a M. A. Biot, " of ViscoelasticPlates Derived from equivalents for the viscoelastic material and the Thermodynamics,"Phys. Rev. 98, 1869-1870(1955). 4 M. A. Biot, "Linear Thermodynamicsand the Mechanicsof dissipationcan be calculated.An approximatenumerical Solids,"Cornell Aeron. Lab. Rept. No. SA-987-S-6(June 1958); procedure has been devised for the evaluation of presentedas a generallecture at the 3rd U.S. National Congressof resonanceamplitudes, which eliminates handling of AppliedMechanics, Brown University, June 1958,and published in Proc. Natl. Congr.Appl. Mech., 3rd U.S. (1958). complexfunctions. The thin-plate theory can also be 5 M. A. Biot, "Dynamicsof ViscoelasticAnisotropic Media," in appliedto the sameproblem, and a precisecomparison Proc.Midwestern Conf. Solid Mech., 4th, PurdueUniv., Purdue can then be made to determine the validity of the Engr. Expt. Sta. Publ. No. 129 (Sept. 1955). 0 M. A. Biot, "Variational and LagrangianMethods in Visco- thin-plate approximations.The procedureto be used elasticity,"in Proceedingsof theMadrid CoIIoquiumon Deforma- tionand Flow of Solids,September 1955 (Springer-Verlag,Berlin, 7T. Alfrey, "Non-HomogeneousStress in Viscoelasticity," 1956). Quart. Appl. Math. 2, 113-119 (1944).

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subsequentlyin this paper begins with the exact

elasticity equations and shows how the thin-plate Fro. 1. Notations approximationarises as the result of geometricassump- for the plate. tions as to the "thinness"of the plate, and dynamical approximationsthat involve characteristictimes of the applied sinusoidal loads and the velocities of the elasticwaves in the plate. in time. The deformation is antisymmetric with The object of such comparisonsis to arrive at the respectto the x axis and representsa of the simplesttechnologically useful governing equations for use in more realistic situations. plate. The displacementsare functionsof (x,y,t), and at y-q- (h/2) it is requiredthat The general problem of energy-dissipationmecha- nisms in structures has been consideredby Lazan,8,ø u= q- U sin(/x) exp(icot), with a discussionof various engineeringmeasures of (2) v= V cos(/x)exp(icot). the dissipation.The measurementof the damping has been considered by Plunkett•ø; vibrating-reed tests The surfacestresses (q,r) and the surfaceamplitudes have been describedby Bland and Leen and by Horio (U, V) can be relatedlinearly as and Onogi12 that determine the complex moduli of (r/lt.t) = a11U-4-a12V, given materials. (al•= a•l) (3) Demerla has presentedan extensivebibliography in (q/lt•)= a•l S-4-a•2V, the field of material damping up to 1955; a selected bibliographyTM to 1959 is also available. where u is the shearmodulus. The first problemis the determinationof the a•i coefficientsin Eqs. (3). I. SOLUTION BY EXACT EQUATIONS OF It is well-known that the displacements can be DYNAMIC VISCOELASTICITY written in the form

The plate, Fig. (1), is assumedto be isotropicand u= (O•/Ox)- (O•/Oy) and v= (O•/Oy)q-(O•/Ox), (4) homogeneous;-- m < x< m, --h/2_

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plate the spanis representedby œ/2. A value of 0.3 for t correspondsto a spanof about 5 times the thickness; tions of the plate see Fig. 2. We show that this particularvalue plays for œ= 10h (•,=0.3).. an importantpart as a limiting casefor the validity of DrawnFro.to2.scale. Deforma- the thin-plate theory. j ,Z'=Ioh j The functionsf• and f2 satisfy the ordinary differ- ential equations Thus, the relationship between the normal-load com- ponentq and the displacementcomponent V is known, ('=d/dO) (9) Eq. (15). For a purely elastic plate, resonancetakes place if q=0, r-0. The characteristicequation is derived by where •2=1-(•o2/c•212) and 1•22=1--(•o2/c2212).The introducingthis conditionin Eqs. (13), or, equivalently, appropriate solutionsthat have the correct evenness by putting q=0 in (15). This yields the well-known and oddness conditions for the vertical coordinates are form f•=A•cosh(#•O) and f•=A•sinh(#•O). (10) tanh(/g2•)/tanh (/•) = 4/•/•/(1 q-/g•2)•. (16) Thus, the functions (•,•k) are now known in terms of This characteristicequation is a relationshipbetween (O,x,t) to within arbitrary constants(A•,A2). If the frequencyand wavelengththat may be representedon a results,Eqs. (7), and (10), are usedin Eq. (4), u and v plot usingthe nondimensionalvariables •=co/c•l and •. are knownand by (2) U and V can be written as The physicalsignificance of • becomesevident whenit is recalledthat •o/l is the phasevelocity alongx. U= A •x sinh(fi•t)- A 2 sinh (11) In the presentanalysis, we assumethat if the medium V=.42 cosh(•)+.4 2• cosh(•). is elastic--that is if • and X are real--Eq. (16) is satisfied.In other words, the frequency is chosenas a The stressesare given by functionof the wavelengthin sucha way that the plate (1/l•)tryy= 2eyy-•-(X/u)e, (1/t•)•xj= 2exy, (12) is at resonance.Moreover, the resonant condition is assumedto be definedby the lowest-frequencybranch where of Eq. (16). This branch correspondsto bending e= e•x+ eyj= (au/ax)+ (av/ay) vibrations that degenerateinto Rayleigh waves for and wavelengthssmaller than the plate thickness. e•v= «[(au/ay)+ (av/ax)•. Substitutionof Eqs. (11) into Eq. (3) leads to the A. Complex Moduli load-displacementrelations in the form To computethe dampingat resonance,the moduli t• (q/lu)= 2A• sinh(/•7)+A •(1+/• •) sinh(/•7), and X are replacedby complexquantities representing (13) the viscoelasticproperties of the medium. The moduli (r/lu)= --A •(1+/• 2) cosh(/•,)- 2A2/•2 cosh (•,). are functionsof the imaginary frequencyp=ico. The The coefficientsa•j in Eqs. (3)can now be found by values adopted here are those correspondingto the eliminationof A 1, .4• from amongEqs. (11), (13). The resonantfrequency. We considertwo typesof materials' results are onewith zerobulk damping,the otherwith equalvalues of bulk and sheardamping. Two valuesu= } and u= « Daii=/•2(1--/•i2), of Poisson'sratio will be assumed. The examples treated below are intended as extreme cases in order to Da12=Da2•= 21•11•2 tanh(/g•)-- (lq-B••') tanh(/g2•), (14)bracket a range of propertiesof commonmaterials.

D= tanh(/•)--/•8• tanh(/•). 1. Material I

If onlynormal surface forces are present on the plate, This material is purely elastic for volume changes then r=0, q•0, and and viscoelastic for shear deformation. The shear modulusu is replacedby #=u(lq-ig). For a material [•11•22• •1221 with bulk elasticity,it was shown• that X is replacedby (q/ltz)=[.• .aV. (15) •,= K-- (2/3)# (17) Direct substitutionsof the a•i from Eq. (14) into (15) where K is a real quantity representingthe bulk show that modulus.This may also be written as •= (2/3)tz[(3v)/(1--2v)--ig•, (18) al• /g2(1-/g• 2) wherevis Poisson'sratio for the purely elasticmedium.

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2. Material II write q= RV,, For a material of the second type, we choose a :• (25) particularcase where # andX containthe sam•Complex q= R V,.. factor; that is, Hence, #= t•(lq-ig), •= X(1-+-ig). (19) Vd V, = R/R = 1/ (• nt- iG) (26)

The bulk viscoelasticityis then representedby the representsthe amplificationfactor at resonance. complexmodulus At resonance,R vanishesfor a purely elasticmedium.

2 The quantity G is a measureof the dampingand is of the K= (X-+-ate) (1-+- ig). (20) order g for small g. The real part • is of the order g2. General properties of materials of this type were Sinceg is sufficientlysmall, only the linear termsare re- discussedin an earlier paper.6 The term "homogeneous tained in the analysisand Gig is then independentof spectrum" was used to indicate this property. The g. The validity of this approximationis established term uniform spectrumseems preferable and is usedin in the numericalanalysis of the problem. this paper. ,,.•, II. THIN-PLATE APPROXIMATION Representativevalues of g for structural materials are in the range 0.01 to 0.10, with values 0.25 being •:The equation ofmotion of a thinplate is typical of materialsused in solidpropellant grains. Eh a O4V O2V •+oh•= applied load. (27) 3. AmplificationFactor at Resonance 12(1-- v• Ox4 OF This factor is the ratio of the dynamic deflectionat resonance to the static under the same load If V(x,t) in Eq. (27) is assumedas V cos(/x)exp(icot) at zero frequency. and the appliedload as 2qvcos(/x)exp(cot), then Eq. We first evaluatethe plate amplitudeunder dynamic (27) becomes

conditions.By the correspondence_ principle, we now 2qv= --ohw2Vq-[Eha/12(1--•,•)•14V, (28) replacet• and X by # and X in Eq. (15). The resultis or, with '•=lh/2 and E/(1--•,2=4t•(Xq-t•)/(Xq-2t•), as q/l• = (alla22-a122)V/all, (21)

or

q=RV with R= (ltffd11)(d11622--6122).(22) (q•,/l•)=L3(X+2•/-• IV. (29) Theload_q is nowa complexquantity •. The complex In the static case(co= 0), the result is quantitiesfil, fi2and the &j alsoreplace fil, fi2and respectively. (4/3)(Xq-• •,•aV. (30) Under static conditions,co= 0 and the static load q8is (q,v/lt•)= \X-+-2u/ given by the limiting value of Eq. (15) as fil and:fi2 approach unity. Hence, With q•v denoting the complexload at resonancefor the viscoelasticthin-plate theory, we set q•v/q•v 2/t•(X-+-t•) =•vq-iGv, which is the analog of the expression qs=RV with R= (tanM,-sech2•). (23) (•q-iG) from the exact theory. By the samereasoning X-+-2t• usedin obtainingEq. (26), we find the amplification factor at resonancefor the thin plate to be What we are interestedin here is the ratio q/q, at resonance. We write this ratio as Pr•,/V,• 1/(ev+iOv. (31) qr/q•=l•/R= eq-iG. (24) In the presentcase, we find that The complexload qr is that given by Eq. (21) at resonance;that is, assumingthat if we put g=0 Eq. (16) is satisfied.As pointedout above,this relatesthe frequencyto the wavelengthby a curve that is chosen to be the lowest-frequencybranch of the characteristic equation(16). x[1+ (4/9)[ (1 - 2u)/(1 - u)•22j (32) The physicalsignificance of the ratio eq-iGis obtained or by comparingthe static deflectionV• to the dynamical 1--•+ •2• (33) deflectionV, under the same load q at resonance,We Gv/g=(2/3)(•--7 '/

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for g2<

+--3 1-•1 2•{•22(1+}•12)2--4}•14-]V. (34) A. Material I Material I is purely elastic for volume changesand Since •5•2= 1- (w•c•F), •2•= 1- (w•c•F), and c•• viscoelasticfor shear deformation (K-K). We have = (t•/o), c22=(X-t-2t•)/•, we may write Eq. (34) as the following data'

14hal• v=-}, O

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where IV. PRESENTATION AND DISCUSSION OF RESULTS The ratio C/Cv gives,.a comparisonof the amplifi- F(ig)=.41•fi•.tanh(/•23')--(1+/•22) 2tanh(/•.3'),. (37) cationfactor at resonancefor the exactand ihin-plate Since/51•':=1-122/ (1+ig), (dt•2/dg)g=o= (122/2151)i and/52 theories.This rati ø as a functionof 3'-lh/2 is shownin Fig,= (3) for materialsi and 2 (v=« andi). In the

may be expanded_ in termsof g; the sameprocedure can calculationsmade to obtain the curves,the quantities be usedfor tt2and for all combinationsthat appearin f(ig). The quantities/51and/52 are, of course,real and Gig and Gv/g were computedfor small g. Sincethe technologicalrange of interestis for valuesof g to 0.10, known sincef• is known. Consistentapplication of this the resultsare accuratefor purposesof comparisonand expansionprocedure leads to the value of f'(0), which for direct usein applications. will involveonly the real quantities/52,/52,3' and hyper- bolicfunctions of real arguments. Figure (3) showsthat in the range0< 3'< 0.3 thereis almostno detectableerror in the casev= «; for the error is about 4% at •,= 0.30. At values of 3,=0.60, B. Material II the errorsare (V= «) 1% and (v=¬) 9% for Material For a material of the secondkind, which exhibits and 3% for Material II. In each case,the thin-plate bulk •= (Xq-]u) (1q- ig) and •,= X(1-big), theory under-estimatesslightly the dampingthat is fi= u(1q-ig); thismaterial has a uniformspectrum. For present. a given value of v, the resonant (f•--3') relationship For completeness,the resonanceamplitudes Vr and is the same as for the first material. However, lvrvthemselves should be compared.This requiresan =l--f•2/(lq-ig) and l•.•'=l--fF/3(lq-ig) if v=•, and evaluation of the static deflection. We write this change requires only slight modificationsin the previouswork; in fact, only two numericalcoefficients of F in Eq. (37)must be changed and all othernumerical work is the same. Whenv= «, X= o• and/•22=1--122,/%.•'= 1;/•22= 1--fF/ (1-t-/g),t]•. •'= 1 andthese results are the same as before in the incompressiblecase. Thus, the resultsfor both material are the sameif v= «. From Eqs. (23), we find the static-deflectionratio An exact evaluation for the elasticity solution is involved but straightforward;a linearizedcalculation (V•v/V,)= 3/(23'a) (tanh3'- 3'sech23') is againa simplematter. A complete(nonlinear) calcu- = 1- (4/5)'r•'+ (51/105)-r%. lation in this caseshowed precisely the samebehavior upon g as in the thin-plate.approximation, which Thisratio is plottedin Fig. (4). justifiesthe useof the linearizedcalculations throughout For the static deflection(at zero frequency),the the analysis. thin-platetheory underestimatesthe deflection;for 3'= 0.30,the erroris about7%. The resonantamplitude resultsfrom the combinedeffect of the dampingand the static-deflectioncorrections, which act in opposite directions.The latter correctionis appreciablylarger and the validityof the thin-platetheory is appreciably better for the evaluationof the dampingthan for the staticrigidity. For wavelengthslarger than ten times the thickness,the error of the thin-plate theory de- creasesrapidly.

1.0

0.8

0.6 V__sp vs 0.4

I a '7'=œh/a 0.2 i _J = ,o '; 0 Fro. 3. Ratio of dampingfactor G/Gv as a functionof q,=lh/2. Material I: Elastic for volume changesand viscoelasticfor shear deformation. Material II' Uniform spectrum (exhibits bulk viscosity).Corresponding values of wavelength-to-thicknessratio Fro. 4. Comparisonof staticdeflections by exact(V,) andthin- œ/h are also shown. plate (V,•) theories.

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V. CONCLUDING REMARKS overestimate the resonant-amplitudefactor. Com- In evaluatingthe resonanceamplitude of a homoge- parison of the thin-plate and exact theoriesfor static neous isotropic viscoelastic plate, the thin-plate deflectionsshows an oppositebehavior; i.e., the thin- approximationis foundto holdin the range0 < • < 0.30, plate theory underestimatesthe deflections.The error or for wavelengthslarger than about ten times the of the thin-platetheory for the dampingis smallerthan thickness.The present results, Fig. (3), shoW:'that for the static deflection

G/G• is closeto unity in this range, for both cases ACKNOWLEDGMENT •=¬ and «. The incompressiblecase (u=«) provides better agreementthan u=l, althoughthe latter is in This work was prepared at the Adelphi Research errorby lessthan 4% in the statedrange. In eachcase, Center; the support of the Aeronautical Systems the thin-plate theory slightly underestimatesthe Division, U.S. Air Force,under contract is gratefully damping;the thin-plate theory will thereforeslightly acknowledged.

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