N5WC/WOL/TR 76-20

IIT

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WIEOAK LABORATORY

METHODS FOR SOLVING THE VISCOELASTICITY EQUATIONS FOR CYLINDER SHRPROBLEMS

BYVG.C. Gaunaurd 22 MARCH 19768

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41REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM

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r NSWC/WOL43- 6-a/

FMet~hods For Solving The ViscoelasticityEquations Xor Cylinder and Sphere Prob~lems. 6 EPRIGOG EOTNNE

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9PEFRIGOGNZTONAEADADDRESS iC. PROGRAM ELEMENT. PROJECT,.TASK

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Viscoelastic ity Equatious Sound-Absorption4Acoustic Scattering Cylinder-Sphere

Viscoslastic ModelsKelvin-Maxwell

ASSTRACT K6fCWAlNu G*A eeeee 1111" It 08C9OOp*IoIf 7be emb.

This report considers techniques used to solve the Xavier field equations ofvic--oelasticity in the Kelvin-Voigt or Maxwell models, for clrlindrical or spher-ical geometries. Introducing scealar and vector potenitials into the viscoelas-

ticity equations fon~ulation, %ltimately yields ý6elegraph-type'perisld3.Cferentia.l equations governing those potentials. For harmonic tinie-depeindencthese reduce to sealar and vector !ielinholtz' s equations with complex propagatioccastants. These cons'tants are shown tc be rela~ted to the viscoelastic ateri N

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~costatsin a more or less complicated fashion depending on the viscoelasticmodel used. The stresses, strains and displacements are then found from thesepotentials for a dozen cases of' interest in those two coordinate systems. Theformulation resembles that of electrodynamics in a Coulomb gauge.

Ac above information is vital to set-up and solve various kinds ofboundary-valua~-problems of dynamic viscoelasticity which appear when studyingcase .z ol- acoustic scattering from sound-absorbing structures, problems we arenow addressing. The analysis is summarized in two large Tab t-up in acorveuiently accessible form. Remarks on "~complex-moduli examined in a

rinal section under the light of the viscoelasticity "Correspo dence Theorem"and a list of reco~enda~tions and conclusions is given at the d.

le aYUNCL&SSIFIED .

.1.,7

NSWC/WOL/TR-76-20

INSWC/WOL/TR-76-20 22 March 1976

METHODS FOR SOLVING THE VISCOELASTICITY EQUATIONS FOR CYLINDERAND SPHREBEPROBLEMvS i w erer~,tovsolsi oes n ~oe

dynamic viscoelasticityintogoeretovsolscmdladadzndifferent situations of interest. This information is needed to study acousticscattering from sound-absorbing structures of cylindrical and spherical shapes,which are cases presently under study by the author. This report sets up mostof' the theoretical viscoelasticity foundations needed to deal with the othersound scattering problems under study.

This work is continuing and it was done as part of' an NS1WC project entitled"Acoustical Properties of Ordnance Materials", Task No. MAT-03L-OOO/ZROO-O0l-OlO,Problem 127, which deals with acoustic scattering from objects covered withviscoelastic materials. This is a progress report describing work done duringI7Y T6.

J. R. DIXONBy direct ion

NSWC/WOL/TR-76-20

CONTENTS

Page

I. INTRODUCTION -- The basic viscoelastic models. 3

II. SOLUTION OF THE FIELD EQUATIONS 7

1. Definition of Plane Strain 132. Constitutive Relations 133. Strain-Displacement Relations 13

4. Displacement-Independent Potential Fi,%tions 145. Stress-Displacement Relations 146. Strain-Independent Potential Relations 147. Field Equations 148. Stress-Independent Potential Relations 159. Helmholtz's Equations For The Scalar and Vector Potentials 15

10. Helmholtz's Equations For The Independent Scalar Potentials 1611. Solenoidal Solution of The Vector Helmholtz Equation 1612. Remarks 16

III. VISCOELASTIC MODELS AND COMPLEX MODULI 18

I. The Correspondence Theorem 21

2. Conclusions 22

IV. BIBLIOGRAPHY 25

TABLES

1. Equations of Linear Dynamic Viscoelasticity in Cylindrical and

Spherical Coordinates For G.eneral Arbitrary Time-Dependence in

Various Cases of" Interest.2. Equations o" Linear Dynamic Viscoelasticity in Cylindrical a.nd

Spherical Coordinates For Harmonic Tme-Dependence in Vario.as,(e, six) Cases of Interest. 11

2

NSWC/WOL/TR-76-20

I. INTRODUCTION The basic viscoelastic models.

The deformation of a viscoelastic solid under any kind of external loads isusually studied by means of viscoelastic models. Two basic models commonly usedare associated with the names of Kelvin&-Voigtz and Maxwell 3 . In the Kelvin-,Voigt model, the elastic and viscous properties of each material point (or par-ticle)of the body, which can be respectively represented by a spring and a dash-pot, are assumed connected in parallel. In the Maxwell model they are assumedconnected in series. We releat that this description applies at each point inthe viscoelastic solid and it is as if the body contained a continuous distri-bution of damped oscillators, (viz, elementary mass-spring-dashpot systems) oneat each one of its material points. As if it were not clear enough already, theentire "Kelvin-Voigt solid' can not be replaced by ONE spring connected in paral-

A lel with ONE dashpot. This simplistic view of a deformable solid as a singleparticle, may be useful in some other elementary context such as that used whenone treats a body as a particle, but it is of no use in viscoelasticity. Other-wise viscoelasticity could not be viewed as a field-theory capable of describingstress and displacement fields at each point in a body, since by that oversimpli-fication, the body has been reduced to a particle. We emphasize this rathertrivial, but quickly forgotten point, because it is common to find authors whotry to use, say, the "Kelvin model" as a single spring in parallel with a singledashpot, only to find that the model is "no-good" and that they must go to "more-degrees-of-freedom systems", such as three "Kelvin-models" in series, or othersimilar configurations, to obtain meaningful results. It is clear that this ap-proach does not give tho Kelvin model, as it truly is, even a charce to %rork".These authors have replaced the continuous field-equations of viscoelasticity?,by a set of three ordinery second-order differential equations of the simple

William Thomson (Lord Kelvin), b. at Belf'..+, +' ; d, near Glasgow, 1907.Professor at Glascov Univ, 1846-1889. Buried at Westminster Abbey, London(near Newton's tomb).

a Woldemur Voit, b. at Leiptig 1850; d. at Gattingen 1919. Professor ofMechanics at GOttingen Univ, German-y.

3 James C. Maxwell, b. Edinburgh, 1831; d. Cambridge, 1879. Professor ofPhysics at King's College and at Cambridge Univ, 1860-1879. Founder of theCavendish Lab at Cambridge.

S ,ie, a set of three scalar partial differential equations hopelessly coupledand non-linear, governing the displacement-field in the body, which can belinearized for small deformations and small deformation-gradients, and the"linear" theory of viscoelasticity then results. These equations are calledthe Navier equations of viscoelast."ity.

3

NSWC/WOL/TR-76-20

damped-oscillator type. This replacement is, in no way, equivalent to solving thefield-equations of Navier.

The basic point of the continuum field-theory approach to viscoelasticity isthat the model assumptions, that the spring and dashpot are connected in seriesor in parallel at each point in the body, are immediately reflected in the fact

that the field-equations resulting from either one of those models turn out to bedifferent. To fix the ideas we now give the linearized form of the field-equationsfor both these models.

i) Kelvin-Voigt Model (parallel):

1 + i 1 + 1 i2

it I. I_+

ii) Maxwell Model (series):

1+ • a2

' - 1 ;~3i28 at 2 e2-y j•e(C( 8

a a tC: at.. C at C8

Here 0 is the material density • , X u are the elastic Lamg constants,u is the vector-displacement field, and e\ ,e' (or a, 8 in the tsxvell model)are the viscosity coefficients. As given agove', in differential operator form,these equations hold in any coordinate system. These are the Navier equations one

must solve in the viscoelastic body. It is possible to derive eqs. (i) and (ii)starting from the basic idea that the spring and dashpot at each material point

S'are connected either in parallel or in series respectively. Once the displacement

field components are found by solving (i) or (ii) Vith suitable boundary conditions,one can then find the stresses from the=.

In the absence of viscosity (ie, X 0, u, 0 for the Kelvin model orCL 0, 8 = 0 for the MAwell model) both field eýuations (i) and (ii) given abovereduce to,

iii) The Pield equations of linear dynamic elasticity:

22.t1 - )Ca at2

!i4

.7• .m.m m m m•m• e~•mm . ..

NSWC/WOL/TR-76-20

since now there are only springs and no dashpots at each material point. The

scalar components of this vector equation are the Navier equations of elasticity

in the absence ofbody-forces as found by Naviers for the elastic body. Equations

Wi) and (ii) are also called the Navier equations (of viscoelasticity) by exten-

sion, since Navier never worked with viscoelastic solids.

In most materials of interest, the "constants" X and the "coefficients"

Y '~ • are really not constants, but frequency-dependent parameters. Thus, few

materials are completely describable by the Kelvin or the Maxwell models in thecontinuum sense of eqs. Ui) and (ii). The material behavior of viscoelastic sub-

11 i stances tends to follow one or the other model in different regions of the para-meters involved, say, frequency among others. This means that in general, thesemodels of field-equations (i) and ci) are quite "good". Rubbers at low frequen-cies are known to be well described by the Kelvin model (i). Pulse-tube measure-ments exploit this factual observation. Since these models are not perfect, re-

searchers in this field have proposed more complicated models.

It is not hard to see that various Maxwell elements in series at each point-lI in the body have the properties of a single Maxwell element with equivalent spring

n

and dashpot constants given by 1/keq - l/ki and 1A

respectively. Various Kelvin elements in parallel at each point in the body have

the properties of a single Kelvin element Vith ke- ka eq4 ni

On the other hand, Kelvin elements in series, or Maxwell elements in parallel,have more complicated properties. In their desire to generalize the basic models

-i . (i) and (ii), researchers have invented the so called "standard viscoelastic=odel". It consists of a Maxwell element in parallel with a Kelvin element ateach pcint in the body. The linearized field equations which result in this modelwhen two springs and two dashpots are connected as described above must be verycomplicated and rare, sinte I can not find one single reference to them. 1 wasable to derive the particular subcase which results when the two dashpots aredescribed by one single viscous constant n and both springs by the snne twoelastic constants Xe aud ve" The resulting field equations in this still very

2e

• C. L. Navier, (1827) M- oire sur les lois de !'equilibre et du mouvement deocorps solides ilastiques. X5m. Acad. Sci. Paris 7.

• !i•,

NSWC/WOL/TR-76-20

general case are,

iv) 1e4 - v2 • + t

(X. + + t

where tl is the Kronecker delta equal to one (zero) for t = t (or t # t2).t t 1 2 1 21 2

Further t (or t ) equal n/e.1 For tI = 0 and t 2 n/Ue (here n i.'), eqs.1 2 e' e v

(iv) reduce to the Kelvin model equations (i). When t = t = n/h (which amountsto setting 28 = i/nand -3a = 1/n), equations (iv) reAuce io the Aaxwell model

:- equations ai) with 3a + 28 = 0. The quantities tI and t are the retardation orrelaxation times of the Kelvin and Maxwell models respeciively. The constitutive.(e, stress-strain) relations of this viscoelastic model are also given in ref (6).I know of no viscoelastic boundary-value-problem that has ever been analyticallysolved using this model, which many agree is more realistic than (i) or (ii),since it can describe wider material behaviors and a wider variety of materials.Since the "standard viscoelastic model" is so hard to handle, we must realisticallyconclude that all analytical viscoelastic problems we are bound to see solved inthe near future will be based on either the Kelvin or the Maxwell models of fieldequations (i) or (ii) respectively.

Occasionally we find a reference in the literature which contains a verycomplicated network such as a doten "Maxwell elements" in parallel. Such a com-plicated network immediately implips that this is not a continuum field-theoryapproach, but rather that the body has been replaced by twelve coupled damped-Soscillators. Work of this nature happens to be mostly experimental, chemical, andcontaining little mathematical analysis. These complicated networks are basicallyintended as pictorial descriptionu without much physico-mathematical discussionof the respon3e of the twelve coupled-oscillators, which, per se, is far from

I If being a trivial problem. Sometimes one comes across an entire textbook dealingwith various aspects of viscoelasticity without a single reference to the contin-u.uum approach, or the field-equations for the viscoelastic models. This old fash-ioned tendenc.y is out-dated today. Mechanics of deformable media has become more

"* highly mathematited now-a-days, than ever before.

6

NSWC/WOL/TR-76-20

II. SOLUTYION OF ThI FIELD EQUATIONS. (Tables 1 & 2)

We will now present a way to solve the field-equations of linear dynamicviscoelasticity (i) and (ii). The technique we will follow is comm~on in otherfield-theories (ie, electro-(iynsmics) but we believe it is novel in viscoelast city.It consists of introducing scalar and vector potentials 0 and such that u

+ x 4'and * ' 0. By splitting the displacement field u into irrotation-al and solenoidal parts in this fashion, it turns out that the field equations ofeach model (i) or (ii) are automatically satisfied provided that the scalar andvector potentials satisfy certain scalar and vector te~.'-.&ý!ph-type equations.The vector ~otentale. has three scalar components, but sinee the solenoidal gaugecondition ' 4 =0 must be satisfied, only two of the three scalar functionsare independent. Those two ('and X where 4'is not [r j) together with the scalarpotential ý, form the three independent scalar potentials that can be used t3solve the problem. If these three independent scalar potentials satisfy threescalar telegraph-type equations then it can be shown that the f4,eld equations areautomatically satisfied.

It is then possible to express all the stress and displacement components interms of these independent potentials, which are determined first by solving thetelegraph-type equations they must satisfy. In this f shion we cani determine allthe stresses and displacements in the bod~y needed to completely solve the problem.

lo discuss these sclutions for the Kelvin-Voig-t or Ifaxwell solils we haveconstructed two charts. (Tables 1 mid 2). Table I deals with these viscoelasticmodels Cortheger~eral case of arbitrary tice-dopendence. Table 2 covers the im-portant case of harmonic time-dependence of the form exp (-iwt). The coordinatesystems covered in those tables are the cylindrical and the spheri.,al. '"hecYlindr.ca system is studiel in general in colunins B and C for the Kelvin orXu~ell models respectively. It is also studiel in the (:-independent) paestrain, subcaie, whichn is applicable to Infinitely long cylinders, in cclums D andE, 'for the Kelvin and ý ~vell models, respectively. The spherical system isalsopresented in fUll Seneri-lity in col=m F for the Kelvin zoiel. It is &lso given.in column G for the axi-s~.y'metric case without atinuthal dependence 0, u.gain fcrthe Xelv~in m-odel. The Maxwell model is not covered for spherical coordinates inthese tab'es. Mhe basic resul.t. of these tables, particularly -*able 2, for har-monic ti-e-dependence, is that if we introduce independent scalar potentialswhich satisfy the Holiholtz's equations with cozplex propagation cor.st~nts givernin ilter, (0, ten the field eq%;ations given in items (7) are automatically satis-fied for each of the cases considered. ýFurthermmore, the ditplacettent and stress-

iel d co~cveents are found f"rom the p.,,terntials by the relations in items (4) sand()respelctivrely. Nctice that the cetiplex prepa~atior. coustants river. i.- item

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~UN T ION5 ' T-1.H INDICATVD

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(A r) e- r Ito no

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3oct ~vi>'-A:C:;'11/12

N5WC /;~LT-76-2o

in a more or less complicated fashion depending on the model being used. Forexamp.le, the relation is more complicated for the Maxwell than for the Kelvinmodel, in any coordinate system. These tables provide us with the methodologyand information needed to solve boundary-value-problems of viscoelasto-dynamicsfor spherical or cylindrical geometries, in the Kelvin-Voigt or Maxwell models.The approach and material covered in these tables seems to be novel as applied toviscoelasticity. The reader familiar with electro-magnetic theory will quicklynotice the analogy of the presentatioq in 4 this table to that of electrodyn~mic+

theory in the less familiar Coulomb (V• = 0), rather than Lorentz (ie, V+l

c 0) gauge.

To verify that all the entries in the tables check with each other, we willillustrate the use of the formulas by analyzing one example. We arbitrarilyselect column D, which dee.-l with the Kelvin-Voigt model in cylindrical coordi-nates for the subcase of .ane-strain.

1. Definition oý. ?lanc-Strain

In plane-str- In, the strain tensor is two-dimensional and all strain compo-nents with a sub.xdex z (axial coordinate) vanish. The stress-tensor is not, andthere is a nonvanishing normal stress T . Another equivalent way to define thissituation is by stating that the axial afsplacement u vanishes, and that the

zother twc displactment components u and ue are z-independent. Plane-strain is ao~~~her~~ tweipa~etcmoet

particularly useful approximation for bodies which are very long in the z-direc-tion.

2. Constitutive Relations

These are the relations between the stress and strain field components,There are three normal and one shear-itre.•" components related to two normal andone shear-strain components. Stress and strain are independent concepts. Onlywhen these two notions are linked through constitutive relations, is that a visco-elastic theory is formed. The first of these four relations is,

T rr X ( e - iWXv ")A+ 2(Ue iWPv) E rr

where A is the trace of the strain terzsor (ie, e + -) and it is called the

dilatation. The other three relations are shown in Table 2.

3. Strain-Displacement Relations

Since the elastic or viscous constants do not appear in these relations,they are the same as in elasticity, namely,

r= .r , Oue + Ur, 2 e 1 Ur + u ur r r ae r r ar r

all other strs.in-components vanishing by definition of plane-strain.

13

A L• ..A

!tSWC/WOL/TR-76-20

4. Displacement Inderendent Potential Relation

These relations are intimately linked to the solenoidal solution of thevector Hcilmholtz equation in plane polar coordinates which is discussed below initem (11). The relations can be written as a single vector equation as follows,

uvq + N X Z

z z

where 4 and ý_ are the two independent scalar potentials 2needed in this plane-strain formulatia. It is shown in item (11) that where is a

solution of the scalar Helmholtz's equation ( V2 + <2 ) X w=r0.

5. Stress -- Displiacement Relations

If the strain-displacement relations (3) are substituted intc the constitu-tive relations (2), the result is the Lvress-displacement relations. The firstsuch relation is,

S•ur- [;, -iWX] A + 2 [ iw"v]•rr a Pe v ;

where the dilatation A is now expressed in terms of the displacement componentsas follows,

r +1 I + rar r ae r

6. Strain•- Independent Potential Relations

If the displacement-independent potential relations (4) are substituted intothe strain-displacemez• relations (3), the result is the strain-potential relation.The first of these three is,

%£ •"!i •~+ Z • •.

rr 2 2

and analogously for the other two. Here 4 and Z are the two independent

scalar potenti.ls. lot! that , 2is the axial component of the vector potentiali. Further, K_ = X -,' X where X is ano(ther independent scalar poten-tial vhich can uLso be-used here. (See item (11) below)

7. Field Epuaticns

These are the :Navier equations given initially it. (i). For harmonic time-dependence they can be written as follows,

..................... •. +....................

NSWC/WOL/TRh-76-20

2e v 2

wher quatiie dilatation; t. For plane-

stanin cylindricals, th uniis u ad Atk nthe followingsimplified forms,

2u 2 - O +uaVr 2 2 C 2 Vu, u 2 r0 Cr r r r

grad A Tr r r a6

and also we have,

u u C + ur r

2'2nd 2+ + 2. (r,6).

r 1a 2- r ar 232r r

8.Stress -Independent Potential Relations

If the strain-potential relations (6) are substituted into the constitutiverelati±ons (2), we obtain the stress-potential relations. One form of the first ofthese relations is,

1822

2The second form can be foune by setting, K~ X~. The formulas for theother three stresses present in this case can be found in Table 2.

9. Helmho).tt's Equations For The Scalar And Vector Potentials

Subitituting u grad 0 + c~.r1 into the Navier equations written in~the alte,1..w' ve form,

"'Pe Uj) ~.) U# x ('x U) + U* K. u~ 0

where X* X iwx U,1 U1 ie - and K w w vnta' id

15

1~4

NSWC/WOL/TR-T6-20

x+2p*) ~ I 2 + K 2 + P*~ X +V-(~~ +K 2 0 where

K 2 2-X+_ iw +p - and K 2 is as we defiL-ed it above in

item (7). It is obvious that the above exprdssion is satisfied if,

V + K 1 and V ~+ K 2 0 provided that

4 I =0 (ie, "Coulomb'" gauge).

10. Helmholtz's Equations For The Independent Scalar Potentials

Since in this case, = P then, V V2 and the result is,Z z z z

(V2 + K12) 0, (V2 + 2 )J 0

II11. Solenoida~l Solution of The Vector Hielmholtz Equation

2 2It is not hard to show that the solenoidal solution of ( + 2 ) 4 0

22ist five, tye abv reltio is0 ,adý K2 x. whrhe(, se ar schea

th clrfunction X~ (r,O). Note that, indeed V 0. Note that in theabsence of viscosity (ie, XA 0, p 0) the complex propagation constants K1

and Kbecome real. Thus viscous damnping is accounted for in these models bycomplix propagation constants, related to the material "constants" as shown abovein items (T) and (9).

12. Remarks

a) In cylindrical coordinates the plane-stress results are derivable fromthe plane-strain results by replacing the elastic constant A in -,he plane-strainresults, by the fictitious "constant" Ae defined to be, A~ = e 2XU I(Ae + 211e)In spherical.] coordinates there is no way to define the plane-streas subcase. Inplane-stress, the stress tensor is two-dimensional, which means t.iat all stress-components with a subizrdex z, vanish. The strain-tensor is not two-dimensionaland there is an e nonvanishing strain. This case is ideally suited for bodieswhich are very thIA in one direction, the z-di'rection.

b) Plane-strain (or plane-stress) with a~xial-.sy=.etry about the z-directioncan be obtained from the results in the eleven items above by merely settingu0 and L (any variableJ 0. There is only radial dependence in this situation.

16

P`'

NSWC/WOL/TR-76-20

c) All the above applies to the Kelvin-Voigt model in cylindrical (actually,plane polar) coordinates, for the plane-strain subcase. This is Column D of Table2. There are twelve cases covered in Tables 1 and 2 and this one was intended asan example to show how the various entries are derived from the others, how theycheck with each other, and how one could proceed to drive any other case. We haveused the information in these tables to rigorously set up and solve acoustic scat-tering problems where the scattering bodies are elastic cylinders and spheres withtheir outer surface coated with layers of viscoelastically absorbing materials.'One reason to have written this report is to gather these fundamental viscoelastic-ity relations, which are not available elsewhere to this degree of detail andapproach, in one document that we could refer to in future work as -he place wherethe theoretical background is derived and presented, in the form in which ie willuse it.

d) Some authors have stated that in order to solve problems involving absorp-tive bodies one must deal with Helmholtz's equation with a complex propagationconstant K. This is indeed correct but there is much more to it than Just that.Just with a complex K we would not know how the real and imaginary parts of K arerelated to the elastic and viscous material constants of the solid in the variousviscoelastic models that one could use. In fact, we would not know this relation-ship in any model. Furthermore, we would not know how to relate the stresses anddisplacements in the body to the solution of that Helmholtz equaticn with a com-plex K. Hence, although the idea is correct, in practice one really needs to de-rive all the detailed information contained in these tables, and that is why we

i i developed them. Careful examination of these tables shows that the problem isreally harder than anticipated, since we must solve not one but several (ie, three)Helmholtz's equations with various (ie, two) complex propagation cc.nstants whichare different, and then go through various other sets of equations (ie, 4 and 8)to obtain the displacements and stresses from the solutions of those Helmholtz'sequations. We finally point out that this procedure yields different results ineach one of the various models and cases presented there.

e) Column A, Table 1 shows some general formulas valid, for all coordinatesystems. Note, however, that since the solution of the vector telegraph equationvaries with the coordinate system, not many general entries can be filled in thatcoluzmn.

' G. C. aunaurd, Scund Scattering frcm an Elastic Cylinder Covered With aViaccel ast4.c -oatiag, JASA 58, SIC', 1075. Also, Proc. of "I., U.S.-Fed.R epublic of Germany, Hydroacoustics Sympcsium, %Ufllch, Germany, Vol 1, Fart "•pp 4-11. May 19T5 M

17

::•"2•" • - " ~~~~~~~. .. .. .. ............ '............. ' I i... ....: I"•i: • •.... :... :: •]

NSWC/WOL/TR-76-20

:i+KX+i~t 22or also, C(x,t) = A e- where, K2 = - (3)

2(1+iWP)

A Here C is real and the - signs in front of Kx describe waves travelling to theright Er left of some origin. (if one were interested in time-dependence exp(-iwt)we would replace i by -i in all these results.)

Now let us consider the standard one-dimensional wave equation,

2 2

axc C at'

The solution of eq. (h) for harmonic time-dependence exp(iwt) can analogouslybe written as,

+ 2O(x,t) = A ei[+ KX + wt] where K2 (5)

C2+KX + iWt 2

or also, C(x,t) = A e- where K =c2

eqaIt is obvious that2 solutiuns (2) and (5) and also (3) and (6) can be madeSequal- by setting C = C (U + iwP). It is clear that we can solve the telegraph

pequation with a real propagation speed C by ignoring the damping term, ie, bysolving the wave-equation with a suitably chosen complex propagation speed. Com-plex speeds are artificially produced by fictitious complex elastic constants thatone can introduce for this ptrpose.

Let us now look at the case of shear waves in a solid. Define a complexshear modulus u* ul' + ip" = i (U1 U i+), where 6 = L"/L'.

The equations of linear viscoelasticity are the same of those of elasticity.ie, no viscosity) if X and ue in the elasticity equations are replaced by thee e

quantities, X* U'e + iWU L' e iL v (C)e[ .v: e v

,.OTE: This is true only in the Kelvin-Voigt viscoelastic model with assumed time-dependence of the form exp(iwt).

Clearly ue = w' and wv Lit"/w. Thus, for shear waves,Su~~e V• 8

V 11,+ 19

Li N - = and C C 'li ' (1)Le (Ajll, W 1 p4

19

NSWC/WOL/TR-76-20

Hence, for harmonic shear waves in the Kelvin mudel, by solving the wave-equation, with a complex shear modulus, we have the solution of the telegraphequation. This can also be done in the Kelvin model for longitudinal (or dilats-tional) waves.

It turns out that the Kelvin model with harmonic time-dependence assumed, isthe only case where the viscosity coefficientsX , . of the viscoelastic solid are

linearly proportional to the imaginary parts X", I of the "complex elastic con-

stants" X*, 0*, the proportionality factors being I/w. It is also the only casewhere the elastic constants X , .i of the solid are just the real parts X', U' ofthe "complex elastic constantS.'' e

To show this, let us now consider the Maxwell model, where the springs anddashpots at each material point are now connected in series. It can be quicklyshown that the field equations and constitutive relations of the Maxwell solid:1 are the same as those of elasticity, provided that the elastic constants Xe) 1 eof the elasticity equations are replaced by the operators,

X (2 •+ 2e ap2 e ~ e Tt

28 t 3a + 28 + j' 2B +8

where a, 8 are the viscosity coefficients of the Maxwell model (which are analo-gous to X_, W of the Kelvin model). For time-dependence of th.e form exp(-iwt)we can caYl t~ese operators by the name "complex elastic constants" X*, u* ie,

X ( +28 -iw)iw) -W e e Ue (10)

It is obvious that if there is no viscosity (ie, a = 0,$ = 0) these quantities)•, •' would reduce to X , • respectively. It is also clear that a and 8 arenow not proportional to t~e imaginary parts of A* and u* respectively. Further-more, X and ie are not the real parts of X* and i anymore. Thus the "trick" ofe ethe complex elastic constants does not work here at all. The trick" also failsfor the standard viscoelastic model, or any more complicated model which containsat least one la•xwell element. In fact it fails even when there is no Maxwellelement in the model provided there is more than one Kelvin element. Thus, onlyfor one single Kelvin element will it "work."

For the reasons given above we believe that the most clear, systematic andnatural way to handle viscoelastic problems of this sort is to measure the (real)elastic constants of the material independently from the (real) viscous coeffi-cients and then use those numerical values in the field-equations when analyti-cally solving them. Of course, we must keep the damping terms in the field equa-tions. The final result will be ,omplex, but this fact is due to the presence ofthe damping terms rather than because of any "complex rooduli" we wanrt to ficti-tiously introduce because of our insistance on ignoring the dampirg terns instead.We have seer. that the complex --oduli "trick" becomes exceedingly difficult, in f,.ctimpossible, in any case other than the Kelvin-Voigt model with harmonic ti-me-depead-ence.

20

r -•

NSWC/WOL/TR-76-20

1. The Correspondence Theorem

A very useful theorem that permits us to relate solutions of elasticproblemsto those of viscoelasticity (in any model) is the Correspondence Theorem.' Thistheorem has a conceptually very simple statement, but it is very difficult toapply it to actual cases. The theorem basically says that if we want the solutionto a viscoelasticity problem (static or dynamic, and in any model) we should firstsolve the "corresponding" elastic problem without viscosity. Then we Laplace-transform the solution. Then we replace the elastic constants appearing in thatLaplace-transformed solution, by certain "memory functions" (which vary with theviscoelastic model being used). The result of this replacement is the Laplacetransform of the solution of the "corresponding" viscoelastic problem. Invertingit, we have the solution of the viscoelastic problem we wanted to solve. We hererecall the Laplace Transform pair,

F(s) f(t) eJtdt , f(t) _ fc+i- F(s)e ds

The prescription given by the Correspondence Theorem is very straight forward.It turns out in practice that when the elastic constants are replaced by thosememory functions", the resulting Laplace-transformed solution that must now be

inverted is quite formidable in most cases of interest (ie, dynamic cases). Weshould point out that the memory functions depend on the Laplace transform variable

s in a more or less complicated manner depending on whatever viscoelastic model oneA uses. See equations (T) or (10) with iw replaced by s, for the Kelvin or the

Maxwell model respectively.

I don't really want to discuss the Correspondence Theorem or its applications.My point is that those "memory functions" mentioned in it, are precisely theequivalent or the analogue of those "complex-elastic constants" that some authorstry to introduce fictitiously in real-space rather than in the Laplace-space, be-fore inversion to the real time-domain. Hence, the way to use those "complexelastic constants" in a way that Irorks," is really in the light of the correspond-ence theorem. Unfortunately, that is quite a difficult task.

We have already pointed out that another method which "works" is to measurethe (real) elastic and viscous constants of the material, each set independentlyof the other, and then solve the field-equations with the damping terms included,with those numerical values measured for the constants. This method is alsogeneral and works for any viscoelastic model and it is, incidentally, the wayvisicous flow problems are attacked in Fluid Mechanics.

• See A. Cemral Eringen, "Continuum Mechanics," John Wiley and Sons, inc., 1967,Article 9.12, p. 368.

2).• ]•,1

¶. . . . . , . . , .. , . . . , . ,• . :, . . . . . H . , , , . ' "-, ' • : .• ' ' , . " "

NSWC/WOL/TR-T6-20

Observation of Table 1 and 2 shows why the telegraph-type equations we havediscussed here are so important in viscoelasto-dynamical problems. Note that thetelegraph-type equations for the potentials 4, i, X are of the type given here in4q. (1), for the Kelvin-Voigt model only. Note, for exmaple, that for the Maxwellmodel they are substantially different. (Table 1, eqs. C-10). The equation forSis, C

2C

[26 + 4 + V = - a + 4 + 2LI +-~ 2B (3a + 2a) (11)c d chd 2aT t2

which is of the same "type" but not quite of the form given here in eq. (1).

We can conceptually always introduce complex moduli (as in eqs. (7) or (10)or their analogues for the "standard" model) provided we work in Laplace space asprescribed by the Correspondence Theorem. If we introduce them in the real time-domain and then try to physically identify the real (or imaginary) parts of thesecomplex constants with the real elastic constants (or the viscous coefficients),then the process only "works" in the Kelvin-Voigt model. In this sense, "works"means that the equivalence of both approaches can be established.

One way to experimentally measure the absorption losses of material samplesis the impedance tube or pulse tube. It is customary for the literature on thistechnique to report measurements of quantities such as E' and E" or 1' and 1j" etc...This technique is a good source for the popularity of "complex moduli". We shouldnote that the lossy samples tested in this fashion are always characterized by the

jI Kelvin-Voigt viscoelastic model with harmonic time-dependence, an assumption thatmay not always be Justified.

i 2. Conclusions

We summarize our points as follows,

(a) The continuum (infinite number of degrees-of-freedom) approach to visco-elasticity as a field-theory is the only way to go today. Discrete approachesleave much to be desired and are simplistically unrealistic.

(b) It looks like for some time to come, we will be dealing with the Kelvin-

.Voigt and Maxwell models, since any other model presents too many analyticaldifficulties.

(c) A systematic way to set up viscoelastic boundar-y-value-problems andsolve the field equations in these two models, is presented here for cylinder andsphere problems. This approach is shown in Tables 1 aud 2 which are self-explanatory.

(d) The most general method available to solte static or dynamic problemsof viscoelasticity in any model is by means of the Correspondence Theorem. Here%we must work in Laplace-space and the inversions are hard. This approach istotally equivalent (at least for the Kelvin and Maxwell models) to our approachdescribed above in (3) and in Tables 1 and 2.

I A 7

NSWC/WOL/TR-76-20

(e) Another technique, which holds for a model, is also given here. Itconsists of solving those resulting "telegraph-type" equations for the scalarpotentials keeping in them the damping terms, but using real values for theelastic and viscous constants which are to be experimentally measured as it isdone in Fluid Mechanics.

(f) We claim this technique is more systematic and less confusing than toignore the damping terms introducing instead "complex elastic moduli". Thecomplex moduli "trick" only works for the Kelvin model with harmonic time-dependence anyway.

(g) It seems clearer to us when talking about elastic constants, say,Young's modulus E, to think of the slope of the stress-strain curve as found in atension test, than of that "complex Young's modulus" which contains the "loss" inits imaginary part, all because some want to solve wave equations rather thantelegraph equations, particularly in the only situation when one is no harder tosolve than the other.

(h) The determination of the solid's elastic constants should be kept sep-arate and independent from the determination of the viscosity coefficients. Iknow this can be done in some instances, but I am not aware of how plausible thisrecommendation can be in all instances.

(i) Finally it is worth stating that the pulse tube measurements implicitlydescribe the viscoelastic "losses" in the sample by the Kelvin-Voigt model, anobservation that escaped me (and perhaps others) until recently.

(1) Equation (10) can be used to express the complex shear and dilationalmoduli for the Maxwell model in terms of the viscosity coefficients and the elasticconstants of the model as follows,

124f Azi and

X* + 20 + 211 +

2 + -i 3a,' +'e 2+( e)]+

++

These rel&tions may be useful when trying to interpret "losses" in the lightof the Maxwell model if one day one wishes, or the need arises to do so. It isobvious that these relations are considerably more complicated than the analogousones for the Kelvin model, which are,

23

IA S....'•"

NSWC/WOL/TR-76-20

X +V2

N ~ and M=e e

(See Table 23, column A, item(9.

24

BIBLIOGRAPHY

I. A. Cemal Eringen, Mechanics of Continua, John Wiley & Sons, 19672. A. Cemall Eringen, Non-Linear Theory of Continuous Media, McGraw-Hill Book Co.,

New Yorit, 19623. Witold N~owa~cki, Dynamic of Elastic Systems, John Wiley & Sons Inc, 190634&. J. D. Ferry, Viscoelastic Properties of Polymers, John Wiley & Sons Inc,

New York, 19615. S. C. Hunter, "The Solution of' Boundary Value Problems in Linear Viscoelastic-

ity" in Mechanics and Chemistry of Solid Propellants, pp 257-295, Proc of IVSymp on Naval Structural Mech, Perhamon Press, N Y, !967

6. C. Truesdell and R. Toupin, "The Classical Field Theories" Handbuch der Physik,Vol 111/1) Springer-Verlag, Berlin, 1960

7. C. Truesdell and W. ~.ill, "The Non-Linear Field Theories of Mechanics",Randbuch de. Physik, Bd 111/3, Springer-Verlag, Berlin, 1965

8. M.. Junger and D. Feit, Sc-.4d, Structures and Thv~ir 'nteraction, MIT Press,Boston, 1971

9. D. R. Bland, The Theory of Linear Viscoelasticity, Pergamon Press, N Y 196010. L. Cremer, M. Heck)., E. E. Ujngar, Structure-Bor'ne Sound, Springer-Verlag,

N Y, 1973I 11. C. H. Wilcox, "Debye Fotentials", J. Math Mech 6, 167-201, 195"A212. Hi. Ii~nl, A. W. Maue, K. Westpfahl, "Theorie der Beugung", in Handbuch der

Physik, B4 25, Springer-Verlag, Berlin, 19611.3. W. P. Mason, ~hs~lAcoustics and the Proertes of Solids. D). Van Nostrand

Co, Inc, N Y 1.95816.* H. Ubera.1l, "Surf'ace Waves in Acoustics", in Ph~scal Acoustics Vol X,

Edited by W. P. Mason, Academic Press, K Y, 19713, pp 1-6* T~his articlecontains over one hundred up-to-date references on the subject of' acousticscatte,,Ing by elastic objects.

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