ON THE VISCOUS MODELS FOR WAVE PROPAGATION IN SOLIDLOADED WITH VISCOUS LIQUID

M.-P. Chang* T.T.WiT*

Institute of Applied MechanicsNational Taiwan University

Taipei, Taiwan 10617, R.O.C.

ABSTRACT

Recently, in the fields of biosensing and nondestructive of materials, there are increasing interests onthe investigations of the surface wave propagation in fluid loaded layered medium. Several differentmodels for the elastic coefficients of viscous liquids are usually adopted in the investigations. Thepurpose of this paper is to study the variations of choosing different viscous liquid models on thedispersion and attenuation of waves in liquid loaded solids. In the paper, a derivation of the elasticcoefficients of a viscous liquid based on the Stokes' assumption is given first. Then, for thehypothetical solid assumption of a viscous liquid, the associated wave equations and expressions of thestress components for different viscous liquid models utilized in the literatures are given. Finally,dispersion and attenuation of waves in a viscous liquid loaded Al half space and a SiC plate immersedin a viscous liquid are calculated and utilized to discuss the differences among these four differentmodels.

Keywords : Viscous liquid, Surface wave, Lamb wave.

1. INTRODUCTION

Recently, the development of micro-acoustic wavesensor in biosensing [1~3] created the needs for furtherinvestigations of the surface wave propagation in fluidloaded layered medium, especially, when the loadedliquid is viscous. On the other hand, in the field ofnondestructive evaluation using elastic waves, severalinvestigations on the viscosity-induced attenuation havealso been reported in recent years [4~8]. Although thefrequency ranges used are different, however, thetheoretical background of the theories is very similar.

The propagation of leaky Rayleigh waves under theinfluence of viscous damping and heat conduction inboundary layers was studied by Qi [4]. Later, Wu andZhu [5] neglected the heat conduction effect andproposed an alternative approach for studyingattenuated leaky Rayleigh waves due to viscousdamping alone. In a subsequent paper, Zhu and Wu [6]employed the same approach to study Lamb wavepropagation in a plate bordered with a viscous fluidlayer. Nagy and Nayfeh [7] investigated the viscosity-induced attenuation of longitudinal guided waves inrods loaded with fluid layer. A discrepancy betweentheir results and the works of Qi [4] and Wu and Zhu [5]for the case of flat interface was reported. In a

subsequent paper, on including the viscous effect on thelongitudinal wave in fluid, Nayfeh and Nagy [8] deriveda formal solution and examined the effects of fluidviscosity on the Lamb wave as well as leaky Rayleighwave.

The purpose of this paper is to study the effects ofchoosing different viscous liquid models on thedispersion and attenuation of waves in liquid loadedsolids. Firstly, the elastic coefficients of the viscousliquid are derived based on the Stokes' assumption.The evolution of these coefficients from the constitutiverelation of fluids is given. Secondly, for thehypothetical solid assumption, the associated waveequations and expressions for the stress components fordifferent viscous liquid models utilized in the literaturesare given. Finally, dispersion and attenuation of wavesin a viscous liquid loaded Al half space and a SiC plateimmersed in a viscous liquid are calculated and utilizedto compare the differences among these four differentmodels.

2. WAVES IN VISCOUS LIQUID

For a plane harmonic wave propagating in anisotropic Newtonian fluid, the constitutive relations canbe expressed as [9]

* Graduate student ** Professor, corresponding author

The Chinese Journal of Mechanics, Vol. 15, No. 3, September 1999 103

dux—OXX

dx2 dxx

dux du2 du3— + - — + -—OXX OX2 OX3

du2

—-ox3

du3

—-ox2

(4)

With Eq. (4), the elastic constants of the liquid in Voigtform read

%22 = -

T13 =

T 3 3 = -

du2

&7dux du3

dx3 dxi

du3

dux

dx2 dx3

dux du2 du3

— + — + —OXX OX2 OX3

du2 du3

ox3 ox2(1)

where % is the stress tensor, —p is the hydrostaticpressure, \xL is the coefficient of viscosity, XL is thesecond coefficient of viscosity, co is the circularfrequency, ux, u2, u3 are the wave displacements alongthe xl9 x2, x3 directions, respectively. In order tocompare with the constitutive relation of a hypotheticalsolid, the velocity gradients usually adopted in the fluidmechanics have been replaced by the displacementgradients.

In the Stokes hypothesis, the second coefficient ofviscosity XL is related to the coefficient of viscosity \iL

as [10]

(2)

In addition, the hydrostatic pressure can be expressed as

dxx dx2 dx3

(3)

where K is the bulk modulus and is equal to K = p Q2,CL is the longitudinal wave velocity of the liquid.With the Stokes hypothesis, on substituting Eqs. (2), (3)into Eq.(l), the constitutive relation of the liquid can berewritten as

2_

3

du\ du2 du3

dxx dx2 dx3

(dux du2T— + —

\ox2 oxx

2K i

3

dux du2 du3

—L + —£. + —1dxx dx2 dx3

du2— -dx2

dux

~dx~3

du3— -dxx

%33=\ K--l®\lL

dux du2 du3

dxx dx2 dx3

du3

dx3

K-\m\xL

0

0

0

0

0

i®\xL_

(5)

We note that the elastic constants in Eq. (5) arederived from the constitutive relations of isotropicNewtonian fluid based on the Stokes' assumption. Inparticular, the Stokes model show that Cu = K +

0

0

0

0

0

0

K-fiG)\XL

0

0

0

0

0

0

0

0

0

0

0

0

0

3. THE CONCEPT OF HYPOTHETICALELASTIC SOLIDS

In the literatures, Wu and Zhu [5] have utilized theviscous liquid model of Lamb to solve the solid-viscousliquid interaction problems. Recently, Nayfeh andNagy [8] pointed out that the model utilized by Wu andZhu for a viscous liquid has the deficiency ofincorporating the attenuation of longitudinal wave.Instead, they suggested possible models to improve theabove deficiency, one is modeling the viscous liquid(with the viscous coefficient denoted as |iL) as ahypothetical solid whose shear rigidity equals i co \iL.The other one is the use of the Stokes model as shownin the previous section. In this section, we are going toderive and compare the field equations and constitutiveequations of the hypothetical solids with differentchoice of the elastic constants.

3.1 The Governing Equations

For a plane harmonic wave propagating inhypothetical isotropic elastic solid, the waves can bedivided into the in-plane (xx-x3 plane) and the anti-planemotion. For the in-plane motion, the equation ofmotion can be obtained as

82ux d2u3

dxx dxt dx3

d2u3 d2ux

dx3 dxx dx3

dx2 dxx dx3

}d2u3 d2u3 d2ux

dx2 dx2 dxx dx3

= pu3

(6)

where isotropic assumption of Cu = C13 + 2 C55 has been

104 The Chinese Journal of Mechanics, Vol. 15, No. 3, September 1999

used. The anti-plane motion reads

C55 V2u2 = pu2 (7)

where, V2 = (a2 / dxx2 + d21 dx2). Equation (6) can be

decoupled by introducing the scalar potential (|> and thevector potential \j/ as [11]

dxx dx3 dx3

On substituting Eq. (8) into Eq. (6), the decoupled waveequations become

CnV2<|> = p<j> , C55V2i|/ = p y , C55V

2u2=pu2 (9)

While the constitutive relations become

T13 = C55 2dx3 d2x3 d2x{

T -C dUl

"23 - <-55 "TOX3

d2xx xi dx3 d2x3 dx\ dx3

3.2 Viscous Liquid Model I

One intuitive way of modeling the viscous elasticliquid is to replace the shear modulus of a solid by thefrequency dependent shear modulus i®\xL. Thisassumption leads to the set of elastic constants for anisotropic viscous liquid as

, Ci3=K , C55=i(D\lL (11)

On substituting Eq. (11) into Eq. (9), the correspondingwave equations become

^ L _ £L V2\i/ = 0, — - ̂ - \V2u2=0 (12)dt { p j 8t { p )

where kL = -y] pco2 / (K + H (o\xL) is the logitudinal wave

number of the viscous liquid. The stress componentsfor model I can then be obtained by substituting Eq. (11)into Eq. (10) as

T1 3 =

T 2 3 = / CO \XL

= K

x\ dx3 d2x3

du2

dx3

d2x3 dx3

(13)

3.3 Viscous Liquid Model II

For some instance, the attenuation of the longitudinalwave in the viscous liquid is neglected, and thence, theother choice of the elastic constants of the viscousliquid is

C U = K , Cl3=K-2i®\xL, (14)

In this model, the forms of the wave equations aresimilar to that of model I (Eq. (12)), except for the wavenumber of the longitudinal wave is kL = co / CL. Onsubstituting Eq. (14) into Eq. (10), the expressions forthe stress components read

T13 =

x23 = i co \xL

x\ dx3 d x3

du2

dx3

x33 = K V2(|) - 2 /X\

(15)

We note that the stress component x13, x23 are thesame as those of model I, while T33 is different.

3.4 Viscous Liquid Model III (Stokes Model)

As shown in Section 2, on adopting the Stokesassumption, the elastic constants of an isotropic viscouselastic liquid are

4 2Cl3=K--i®\iL ,

(16)

With the elastic constants in the above equation, thewave equations can be obtained as those shown in Eq.(12), except for now, the wave number kL is equal to^p©2 / (K + jiG>\xL). In a similar way, the stresscomponents are obtained as

T13 = i co \iL 252\|/ 52i|/

8x1 dx3 d2x3 d2xx

t 2 3 = I CO \iL •

i=KV2(|> +

du2

l (01

d2xx

(17)

3.4 Viscous Liquid Model IV

On neglecting the heat conduction effect, Wu andZhu [5] proposed an approach for studying attenuatedleaky Rayleigh waves due to viscous damping. In asubsequent paper, Zhu and Wu [6] employed the same

The Chinese Journal of Mechanics, Vol. 15, No. 3, September 1999 105

approach to study Lamb wave propagation in a platebordered with a viscous fluid layer. The wave equa-tions they employed are exactly the same as model II.That is the attenuation of the longitudinal wave isneglected. For the stress components, however, the ex-pressions they utilized are the same as those of model I.

In Ref. [8], Nayfeh and Nagy mentioned that Ref. [5]adopted model II for the viscous liquid elastic constants.However, we find that the viscous liquid model theyused is indeed model IV.

4. SURFACE WAVE IN VISCOUSLIQUID LOADED SOLID

The conventional way to study the propagation ofsurface waves in an isotropic layered half space can befound in Ref. [12]. In the conventional approach, theelement number of the determinant, which resultingfrom satisfying the appropriate interface and boundaryconditions, increases rapidly with the number of layersoverlaying the half space. However, it is known thatthe above mentioned problem can be overcome byadopting the sextic formalism [13]. In the sexticformalism, the equation of motion and the constitutiveequation are combined and arranged to form a first-order matrix differential equation. The displacementand the traction acting across the planes normal to thelayering surfaces are grouped into a six-dimensionalvector. In each layer, the solution of the matrix ODEforms a transfer matrix that can be utilized to map thevariables from one surface to the next layering surface.With this formulation, the size of the matrixencountered in the computation is independent of thenumber of layers.

The formulation of the sextic formalism for waves inanisotropic layered media can be found in the Ref. [13],however, a concise summary of the formalism isrecently reported by Wu and Wu [14]. In addition, inRef. [14], formulations of the wave velocity, tractionfields as well as the impedances of the viscous liquidare given. We note that in the sextic formalism, thetraction field is expressed in terms of the impedance andthe velocity field of the medium, and then, on em-ploying the boundary conditions or interface conditions,the dispersion of waves in a layered structure can beobtained in a straightforward way.

From the derivations shown in section 3, we foundthat the major differences among these models are twofolds. The first one is in the wave number kL of thelongitudinal wave. kL in Model I and III are complex,and this means that these two models allow thelongitudinal waves to be attenuated when propagating inthe viscous liquid. On the other hand, model II and IV,kL are real, and this represents that the attenuation of thelongitudinal wave in the viscous liquid is not included.The second difference among the four models is theexpression for the stress component x33. It is notedthat the construction of the local impedance and globalimpedance of a medium is dependent on the expressionof the stress component.

5. COMPARISONS OF THE VISCOUSLIQUID MODELS

A computer program modified from the one used inRef. [14], which incorporated the four different viscousliquid models, was written. A viscous liquid loaded Alhalf space and a SiC plate immersed in a viscous liquidare utilized to compare the differences among these fourdifferent models.

For the case of viscous liquid loaded Al half space,the elastic constants of Al are p = 2698 kg/m3, A, = 5.84x 1010 N/m2, \x = 2.59 x 1010 N/m2. The density, longi-tudinal wave velocity of the viscous liquid are assumedp = 1000 kg/m3 and Q = 1500m/s. Shown in Fig. 1are the phase velocity dispersions of the Rayleighsurface waves. The upper group of the curvesrepresents those for the case of \xL = 0.1 N-s/m2, and thelower curves are for the case of \LL = 1.0 N-s/m2. Fromthe results, we note that the dispersion of the Rayleighwave velocity preserves the similar trend for the fourmodels, that is, the Rayleigh wave velocity decreaseswith the increasing of the frequency. In addition, it isnoted that Rayleigh wave velocity decreases with theincrease of the viscosity, say for example, in this casefor frequency up to 150MHz. Figure 2 shows theattenuation of the Rayleigh waves in the viscous liquidloaded Al half space. From the results, one finds thatfor low viscosity and low frequency, the attenuationresults from the four different models are negligible.On examining the group of curves for \xL = 1.0 N-s/m2,we note that the differences of the curves generatedfrom model I and III are always close with each other.The attenuation calculated based on model II is thesmallest. This may due to the assumption of neg-lecting the attenuation of the longitudinal wave in theviscous liquid.

2900-

•gj 2 8 8 0 -

t-§ 2860 -

2840-

2820-

^=0.1 N-s/m2

=1.0 N-s/m2

Model #1

— — Model #2

Model #3

- r Model #4

I l I ' I40 80 120

Frequency (MHz)160

Fig. 1 The phase velocity dispersions of the Rayleighsurface waves in a viscous loaded Al half spacewith different viscous models

106 The Chinese Journal of Mechanics, Vol. 15, No. 3, September 1999

100000 -a

10000 -

1000-

100-

Model#l

— — Model #2

Model #3

Model #4

r "̂ i ' I40 80 120

Frequency (MHz)160

Fig. 2 The attenuation of the Rayleigh surface wavesin a viscous loaded Al half space with differentviscous models

l .(

0.5<Uo.:o.eO.fo.*0.:

0.2

0.1

o.c-0.]

-o.:-0.:

=10"' N s / m 2

#

| i L = 1 0 ' 3 N s / m 2

Model #1

Model #2

Model #3

Model #4

Fig. 3

i ' i • I • I • I l i 1 i " • I

0 2 4 6 8 10 12 14 16 18 20Frequency (MHz)

The attenuation of leaky Lamb wave in a lmmthick SiC plate immersed in viscous liquid withdifferent viscous models

Shown in Fig. 3 are the results for the case ofattenuation of leaky Lamb wave in a lmm thick SiCplate immersed in a viscous liquid. The elasticconstants of the viscous liquid are the same with thoseused in Ref. [8] as p = 3100 kg/m3, Cn = 42.6 x1010N/m2, C55 = 16.75 x 1010 N/m2. The density andthe bulk modulus of the liquid are p = 1000 kg/m3 and K= 2.25 x 109 N/m2. Four different magnitudes of theviscosity are considered, i.e., \iL = 10~3, 10~2, 10"4,1 .ON-s/m2. As noted in Ref. [8], to reveal the behaviorof the attenuation curves, the attenuation wasnormalized to the square root of viscosity accordingly.We note that the curves calculated based on the modelIII (Stokes model) are the same as those shown in Fig. 8of Ref. [8]. From the results, we demonstrated furtherthat for low frequency and low viscosity, the differenceon attenuation caused by choosing different viscositymodel is negligible. As the viscosity increases, thedifferences of the wave attenuation among the fourmodels are increased.

6. CONCLUSION

In this paper, we studied the effects of using differentviscous liquid models on the dispersion and attenuationof waves in liquid loaded solids. The elasticcoefficients of the viscous liquid are derived based onthe Stokes' assumption. Then, for the hypotheticalsolid assumption, the associated wave equations andexpressions of the stress components for differentviscous liquid models utilized in the literatures aregiven. Finally, dispersion and attenuation of waves ina viscous liquid loaded Al half space and a SiC plateimmersed in a viscous liquid are utilized to compare the

The Chinese Journal of Mechanics, Vol. 15, No. 3, September 1999

differences among these four different models. Fromthe results, we demonstrated that for low frequency andlow viscosity, the difference on attenuation causedby choosing different viscosity model is negligible.However, as the viscosity increases, the differences ofthe wave attenuation among the four models areincreased accordingly.

ACKNOWLEDGMENT

The authors thank the financial support of thisresearch from the National Science Council of R.O.C.through the grant NSC 88-2218-E-002-020.

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(Manuscript received May 21, 1999,Accepted for publication July 20, 1999.)

108 The Chinese Journal of Mechanics, Vol. 15, No. 3, September 1999