Certain asymptotic relations for the dynamic moduli in superposed oscillatory shear

ht. J. Engng Sci.. 1972. Vol. IO. pp. 263-272.

Pergamon Press.

Printed in Great Britain

CERTAIN ASYMPTOTIC RELATIONS FOR THE DYNAMIC MODULI IN SUPERPOSED OSCILLATORY SHEAR B. BERNSTEIN, R. R. HUILGOL Illinois Institute of Technology, Chicago, Illinois 60616, U.S.A. and R. I. TANNER Brown University, Providence, Rhode island 029 12, U.S.A. Abstract- We consider the incompressible BKZ fluid subjected to small in-line and transverse oscillations superposed on steady shearing Sow. Investigations are made of the asymptotic behavior of the storage and loss moduli at ultrasonic frequencies. Asymptotic formulae for both shear and normal stress modufi are obtained. We also point out the relations~ps between the uhrasonic storage moduli and acceleration waves. 1. INTRODUCTION RECENTLY, Tanner and Williams[l], and Bernstein and Huilgol[2] have considered the ultrasonic dynamic viscosities of viscoelastic fluids in superposed oscillatory shearing motions. In such flows the velocity field is assumed to be given, in a fixed Cartesian co-ordinate frame, by

jl=:i=(),

A= Ky-ysinwt,

(1.1)

or by i=

KY,

3=

i=--ayoinof_ h

0,

(1.2)

The flow in equation (1.1) involves in-line oscillations while that in (1.2) concerns transverse oscillations. In both flows, K is known as the rate of steady shear, a is the amplitude and o is the frequency of oscillation of the superposed velocity field, and h is the gap width. In what follows, a is assumed to be small enough so that all linearizations in a are permissible. Corresponding to the flow in equation (1. l), write the shear stress as T(Xy)

(1.3)

=T(K)+~COSwtG;(K,W)-~SinotC;‘(K,W)

where 7(K) is the steady shearing stress; while for the motion (1.2), T(yz) =acoswtGI(~,Of-~sin&G~(~,~).

(1.4)

In (1.3) and (1.4), G ’ and G ’ represent the storage and loss moduli respectively. Let the normal stress differences for the motion (1.1) be written asi T(~~)-_T(yy)=N~(K)+~COS~~N~i(K,Of-~Sinof~~~(K,o),

lThe normal stress differences with frequencies here.

2w are proportional 263

(1.5)

to a2 and hence are not included

264

B. BERNSTEIN, T(yy)-T(zz)

R. R. HUILGOL

and R. I. TANNER

=N2(K)+~coswtN;~(K,W)-~sinwtN;:(K,W),

(1.6)

where N1 ( K) and N, (K) are respectively the first and second normal stress differences of the steady shearing flow, with N:$, Ni: and Nii, Nii being the corresponding dynamic moduli. For the flow in (1.2), the dynamic normal stress differences are not linear in a, and hence, to the first order in a, these are equal to the steady flow differences. That this statement is true can be verified by using the nearly viscometric flow theory of Pipkin and Owen[3], especially their parity relations. Hence we do not consider Ni,, etc., here. Previously, it has been shown [ 1,21 that for incompressible BKZ fluids [4], (1.7)i and that [2]

(1.8) where qi, ,ql are the dynamic viscosities for the flow (1.2) and (1.1) respectively and q’ is the dynamic viscosity of the linear theory. It was proved in [2] that the above two relations do not hold for all fluids. Here we shall develop general asymptotic formulae for the dynamic moduli G ;(K, w), G~(K,o), G~(K,w) and G~(K,w) as well as for the components Nii(~,u), NI:(K,w), N&(K, w) and Nli(~, 0). We shall report these limits for the incompressible BKZ fluid only. For this fluid [4], the constitutive equation is given by T+pl=

2

aCIC-‘(t-s)-aCTC alp 811 f

t

(r-s)

(1.9)

where U = U (I,, I*, s) is the stored energy function, C,( f - S), 0 5 s < 03,is the right relative Cauchy-Green tensor, C ;’ ( t - s ) its inverse, and I, = rrc;‘(t-s),

(1.10)

12 = rrC,(t--s).

(1.11)

In simple shearing flow, the shear stress function and N2(~) are given by [5]

T(K)

and the normal stress differences

N1(~)

7(K) =2/0m(~+$jKSdS,

(1.12)

(1.13) Whis result makes precise the relation (29a) of [ 11.

Dynamic moduli in superposed oscillatory shear

265

(1.14) (1.15) 2. DYNAMIC

SHEAR

MODULI

From [2], the following relations can be recorded for the BKZ fluid:

G;(K,@)=

l;.f( K&S)(1-cosws)dS,

GI)(K,

f%f(~~,

w) =

G~(K,w)

G:(K,

s)

sin os ds,

~(Ks,s)(~-COSOS)&,

=fr

W) = JQwg(lcs, s)

sin 0s ds,

(2.1) (2.2) (2.3) (2.4

where f(KS, s) = 2 (“‘I al+

g(KS,s)=

~)+4~~~~(!$+2~+~), ar

2(~+~),

t2.6)

and the right sides are evaluated at II = Zz= 3 + Consider the following integral If

h(KS,

(2.5)

K’S

s)e’““dS

2.

(2.7)

as a representative

of the above relations. We shall now obtain an asymptotic expansion for this integral. To this end, we examine

for some absolutely integrable function 4(s). Under the assumptions that 4’(s) is absolutely integrable and htit 4(s) = 0, we may apply integration by parts to (2.8) and derive

Furthermore, under the assumptions that !~IJ4’(s) = 0 and that #‘(s) is absolutely integrable, one may substitute (b’(s)for #J(S)in (2.9) and obtain (2.10) which may be used

UESVd.10No.3-D

in conjunction with

(2.9) to yield

B. BERNSTEIN,

266

R. R. HUILGOL

fom $(s)e’Ugds =:+(a)

and R. I. TANNER

+(;f;)’ @I’(O)+(i)’

fO=$“(s)eiosds.

(2.11)

Now suppose that for some integer n, we have: I”.lim$P(s) =0 , k= I, 2, . . , n - 1, and Z” . #(*j(s) is absolutely integrable for k=O,;T:., n; then a repeated application of the above process of integration by parts yields

Ioz

$(s)efWsds = b#(O) +(i)2

4’(O) +* *+(~j~~(~-l)(0)

+(b)* 1% #(“‘(s)eiwsds. 0

(2.12)

Since, according to the Riemann-Lebesgue Lemma [61, (2.13)

Ail Jr #(n)(s)e~* ds = 0, we see that (2.12) yields lim w* om#(s)eti~ds-~$(0)-(~)2#‘(0)-.. WC? [I

+-(-$*@k-l)(O)]

=O,

k= 1,2,..,n,

(2.14)

i.e. the asymptotic expansion of the integral (2.8) is given by

f

:

m. ‘+(~j~~(~-~)~O),

#(s)e&“ds - ~~(O~+(~j2#~(0)

(2.15)

where (2.15) is valid for any n for which 1Oand 2Ohold. We now define hCk’(KS, s) by k”“( KS, S) =$h(KS,S),

(2.16)

k=0,1,2,...n;

so that (2.15) yields

I

cc

0

hfKS,

s)eio8 dS

-~h(O.O)+(~)zh”‘(O,O)+.~ *+ (J” ~(~-l)(O,O~,

(2.17)

i.e. 00

lim ok llJ-*m

[S

h(~s,s) eiwsds-

(~;)h(O,O)-(~;)Zh”‘(O,O)-.

.

.-(~)kh~~-“(o,o)]=o,

0

k=O,l,....,

n-l,

(2.18)

which is valid for any k for which ~z’~‘(K.s, s) is absolutely integrable, k = 0, I, . . , n; and for which ~~mhh(k)(~~r~)=O, k=O,l,..,n-1. A function with a monotonic

Dynamic module in superposedoscillatoryshear

267

exponentially bounded approach to zero as s * 00 will ensure these conditions. Whenever equations (2.17) or (2.18) are applied, it will be tacitly assumed that these conditions hold. Now, in [2], it was shown that (2.19)

~&~[wG~(K,w)]=~~[~G~(K,w)]=-G(O),

where G (7) is the derivative of the shear relaxation modulus G(T) of linear viscoelasticity. The argument given in [2] to establish (2.19) could be thought of as essentially an application of (2.18) to (2.2) and (2.4) with k = 1. We are now in a position to obtain many other relations of this type. Two such relations of interest involve

where ‘Im’ implies the imaginary part of, and

obtained from (2.1) and (2.3) respectively.

According

1

to (2.20), (2.18) and (2.5) now,

=f(KS,S)Slg+=o,

(2.22)

where (2.22), follows since the terms in (2.18) involving even powers of o are all real. Similarly, we obtain

1

cop& (G;

(K, w))

=g(KS,S)Sl,=o=o,

1

= 0,

(2.23)

=-$(,(KS,S)S)lszo. ~“-$G;(w)) 1

It appears then that the first non-vanishing term in the asymptotic expansion of (2.20) or (2.21) may not be lower than cubic order. Our next task is to evaluate these coefficients. It appears, as can be seen by examining (1.12) (1.15), (2.5) and (2.6) and I, =

26X

B. BERNSTEIN,

R. R. HUILGOL

and R. I. TANNER

I2 = 3 + K’s ‘, that f( KS, s) and g (KS. s ) are even functions of their first variable, i.e. f(y, s) and g(y, s) are even functions of y. Suppose we write the Taylor expansion off(y, s ) about y = 0, for fixed s, to obtain f(Y,

Sincef(y,

s) =.6(s)

+hb)Y+"fz(~)y2+o(Y2).

s) is even in y, we have that&(s) f(Y.

where, of course,&(s)

f(Ks,

= 0, and are left with

3) =.6(s) +h(S)Y2fO(Y2).

=’ (a’/ay”)f(y, s)s

(2.24)

(2.25)

s) Iy=,,.Thus (2.26)

=fo(s)s+f2(s)K2s3+0(s3).

Hence

-$(f(

KS,

s)s)

Iszo

=

-2&(o).

(2.27)

-4 (s),

(2.28)

Now, it was shown in [2] that

f(0, s) = g(O, s) = and from (2.25), it follows that fo(s) = -G(s). gives hrir

[

“3&(G;(~,w))]

This together with (2.27) and (2.22),

=2e(0).

(2.29)

1

(2.30)

An exactly similar argument will give w3&

(G;)K,w))

=

2G (0).

In summary, then, we have shown that

=ili w3$(G;( W3&(GI(KIO)) 1 [

K,W))]=~~~[03~(G'(~))]=2e(0).

(2.3 1); We now make the following important observation on the first non-vanishing coefficients in the asymptotic expansions of these quantities. The observation is that these limits are independent of the rate of shear K of the base motion. We wish to point out that our calculations do not shed any light on whether the limits ~&%G;(K,

W) =

2

tThis result makes precise the relation (29b) of [ 11.

I~~(Ks,

S)

ds

(2.32)

Dynamic moduli in superposed oscillatory shear

269

or the limit (2.33) is related to $5 G ’ (co). The experimental results of Booij[7,8], Osaki, et al.]9], Kataoka and Ue~[lO] as well as those of Walters and .Iones[ll] show that in some instances G~(K, W)-+ G’(o), while in others the curves of G~(K, o) vs. cobecome parallel (as w increases) for different values of K. It is not clear whether these disagreements in the published experimental data are due to the property of the log-log plot or due to lack of accuracy at the high frequency range. Similarly the data of Simmons [ 121 and Tanner and Williams [ 11 on Gi (K, o) are not very conclusive either. So we are unable to say whether the experimentaldataonG~(~,~) (OrG~(K,~))vs.~tendtoG’(w)aswbecomesve~l~ge. While we cannot find any relation between these storage moduli and G ’ (o), we point out in Section 4 the relations between the storage moduli and speeds of acceleration waves.

3. DYNAMIC

NORMAL

STRESS

DIFFERENCES

For the normal stress differences, one can show that N;, (6 0) = J; ~(K~,~)(l-COS~~)d~, Nfi(K,

0)

=

j;

U (KS,

S)

sin OSds,

(3.1) (3.2)

N;<(K,W) = j-r U(KS,S)(l-cosCUS)ds,

(3.3)

NI,;(K,W) = J; v ( KS,

sin OSds,

(3.4)

(~+2~+~),

(3.5)

S)

where I((KS,

S) =

-V(KS,S)

=

4 (~+~~

KS+4K3S3

4~KS+4K3S~

(3.6)

2

and the right sides are evaluated at I1 = i2 = 3 -I-~5s~. Employing (1.13), we have the result of Bernstein [ 133:

limN;;t’G w) _ dN,(K) 0e.o

w

dK

*

(3.7)

Similarly, by using (I. 14), one obtains lim

%i(K, 0) -;:-dNztK)

W-+0

w

dK

*

(3.8)

B. BERNSTEIN,

170

R. R. HUILGOL

and R. 1. TANNER

We now turn to ultrasonic limits. Bernstein [ 131 showed that (3 -9) which may be verified by referring to (1.12) and employing the Riemann-Lebesgue Lemma on the cosine component of the integrand in (3.1). Applying (2.18) to (3.2) we read off:

= 4K$

(g(

KS,S))18=o=

4K&0).

(3.10)

Similarly, follow the results: (3.11) s=o,r1=12=3

It is obvious, now, that in principle one may generate other asymptotic limits by evaluating the higher order coefficients in (2.17) or (2.18). As far as the authors are aware, there exist only one set of dynamic normal stress difference measurements due to Booij[8, 141. He measured Nii(~, o) and N;\ (K, o) for o ranging from 1 to 10 rad./sec. So we are unable to verify our predictions (either as w + 0 or w * 00)from his data. 4. CONCLUDING

REMARKS

In Sections 2-3, we derived some asymptotic formulae for the various moduli. It was noted in section 2 that our calculations do not connect Gi (K, co) and GI (K, w) with G ’ (co) of infinitesimal viscoelasticity as w ---, w. However we wish to point out that one may prove the following results for a fluid of density p. If an acceleration wave with an ~plitude vector a moves in the direction of a unit vector n, when the fluid is undergoing steady simple shearing ahead of the wave, it can be shown that the wave speed pU 2 is equal to the limit &i$ G i (K, w) or I& G i (K, w) as follows: Direction of a X

Direction of n

Valid for

Y

all simple fluids,

$m G; (K, a)

2

3’

all simple fluids,

$m G; (K, a)

X

z

BKZ fluids.

~od~li l&Gj(K,m)

271

Dynamic moduli in superposed oscillatory shear

The first two results are obtained by using the general results of Huilgol[ 151 on the acoustic tensor of the incompressible simple fluid (see Appendix for details). They are consistent with Coleman and Gut-tin’s [ 161 theorem of equivalence establishing that ultrasonic shear wave speeds and acceleration wave speeds are identical. The last one is valid for BKZ fluids only and was derived by Sadd [ 173.7 We wish to suggest that the above set of experiments offer easier ways of measuring the acceleration wave speeds than those currently practised. Finally, as far as the authors are aware, experimental results are not available for N[ and N:’ at high frequencies. It would be interesting to see if these observations, when available, lend further confirmation of the BKZ theory. REFERENCES 111 R. I. TANNER and G. WILLIAMS, Rheol. Acfa (in press). I21 B. BERNSTEIN and R. R. HUILGOL, Trans. Sec. Rheol. 15.73 1 (1971). [31 A. C. PIPKIN and D. R. OWEN, Physics Fluids 10,836 (1967). [41 B. BERNSTEIN, E. A. KEARSLEY and L. J. ZAPAS, Bur. Stand. J. Res. 68B, 103 (1964). Mech. 2,329 (1966). PI B. BERNSTEIN,Acra Fourier series and Boundary Value Problems, Second Edition p. 87. McGraw[61 R. V. CHURCHILL, Hill (1963). 171 H. C. BOOIJ, Rheol. Acra 5,215 (1966). Ml H. C. BOOIJ, Ph.D. Thesis, Univ. of Leiden (1970). PI K. OSAKI, M. TAMURA, M. KURATA and T. KOTAKA, J. phys. Chem. 69,4183 (1965). 1101 T. KATAOKA and S. UEDA, J. Polym. Sci. (Part A-2) 7,475 (1969). 1111 K. WALTERS and T. E. R. JONES, Proceedings ofthe 5th International Congress Rheology, Edited by S. ONOGI, Vol. 4, p. 337, University Park Press 1970. 1121 J. M. SIMMONS, Rheol. Acta 7, 184 (1968). [131 B. BERNSTEIN, 1nt.J. non-linear Mech. 4,183 (1969). [141 H. C. BOOIJ, Rheol. Acta 7,202 (1968). 1151 R. R. HUILGOL, to be published. [I61 B. D. COLEMAN and M. E. GURTIN, Arch. ration. Mech. Analysis 19,239 (1965). [171 M. SADD, Private communication. [ISI B. D. COLEMAN, M. E. GURTIN and I. HERRERA, Arch. ration. Mech. Analysis 19, 1 (1965). 1191 B. D. COLEMAN and M. E. GURTIN, J. Fluid Mech. 33,165 (1968). (Received

19July

1971)

APPENDIX By using arguments [ 181 which are fairly standard in the theory of acceleration waves in materials with memory, Huilgol[ 151 has shown that the acoustic tensor for the propagation of waves in an incompressible simple fluid is given by cz

Qij(n) = 2wb6 zPpmkr

(Al)

a=0

In (Al), Bz(. /g(s)) is the tirst variationofthe

functional

TE = T+pl=

s@O(C,(t-sl),

(A2)

which is the constitutive equation of the simple fluid, in the direction g(s), 0 5 s < ~0.Thus w(. [g(s)) is a linear functional of g(s) and in (A I), ( Ct (t - s) )J1is the strain history at the wave front. If one normalizes %p(. ) in (A2) so that tr Se0 (C,(r-s)) ?It was indeed Mr. Sadd who showed

US

= 0,

(A3)

that the above three relationships are valid for the BKZ fluid.

B. BERNSTEIN,

272

R. R. HUILGOL

and R. I. TANNER

and further if the wave moves into a region undergoing steady simple shearing, then the linear functionals 6X,,kl appearing in (Al) are identical to the functionals 69 ,,kl of nearly viscometric flows [3]. Using the velocity field (1.1) in (A 1) one may calculate the speed of propagation plJ 2,for a wave propagating in ydirection with an amplitude in the x-direction, to be given by [ 151

pU*= 21:

{~1Zz2(~,S)~S-~,212(~,S)}dS.

(A4)

Incidentally, we note that for all simple fluids

which is equation (7.6) of [3]. That (A4) and (A5) are such that the integrands must differ by s can also be established[l5] by computing the variations -6 850 (-KSIS) and-d Se0 (-~~11) ofColemanandGurtin[l9]. Comparing equation (33) of [2] with equations (2.1) and (2.2) here, we note that for the simple fluid GI(K, W) = 2

K,S)KS-~IC~~~~(K,S)}(~-COSWS)~S. IOU ic~1222(

Hence (A4), (A6) and the Riemann-Lebesque

Lemma imply that

which verifies the first assertion in the table in section 4. One can establish the other claims in the table in a similar manner. Resume-Nous considerons le fluide incompressible BKZ comme &ant soumis a de petites oscillations longitudinales et transverses superposees a un Ccoulement constant de cisaillement. Des recherches sont faites sur le comportement asymptotique des modules d’accumulation et de perte aux frequences des ultrasons. Des formules asymptotiques pour les deux modules de contraintes normale et de cisaillement sont obtenues. Nous faisons Cgalement ressortir les relations entre les modules d’accumulation des ultra-sons et les ondes d’acctleration. Zusammenfassung- Wir untersuchen die inkompressible BKZ Fltissigkeit, die kieinen Reihen- und Querschwingungen unterwotfen ist welche einer stetigen Scherstriimung iiberlagert sind. Nachforschungen iiber das asymptotische Verhalten der Lagenmd Verlustkennzahlen bei Ultraschahfrequenzen werden gemacht. Asymptotische Formeln fiir Kennzahlen fur Scherspannung und normale Spannung werden erhahen. Wir weisen such auf die Beziehung zwischen Ultraschalllagerkennzeichen und Beschleunigungsweilen hin. Sommario - Si tratta l’argomento de1 fluid0 BKZ incomprimibile sottoposto a piccole oscillazioni in linea e trasversali sovrapposte a un flusso di tranciatura uniforme. Si studia il comportamento asintottico dei moduli di accumulo e di perdita a frequenze ultrasonore. Si ricavano le formule asintottiche sia nei riguardi de1 module di tranciatura the di quell0 di sollecitazione normale. Si fa anche presente i rapporti the esistono fra i moduli d’accumulo ultrasonoro e le onde di accelerazione. AB~T~~IcT- RaccMoTpeHa HecXoiMaeMas mmOCTb BKZ, IxonBepXeHHaa K He60JIbmnM JInneiHbiM n IIOnepOYHbIM KOJI&HHfiM, HKJIOlKCHHbIM Ha CTBUHOHBPHOM CABWrOBOM IIOTOKC. kiCCJIe~OBaH0 aCAMIITOTIIYeCKOe nOJl,‘WHbI

IIOBCj.(eHHC

aCHMUTOTWIeCKHe

HtUI&%ElCKeHEDl. KpoMe TOIBOJIH yCKO~HEUI.

MOJQ’JIeB ~OpMyJlbI

Mb1 IIOKaECM

HaKOIIJICHliR AJIK MOAYJICB 3BBHCHMOCTW

II

IIOTepH

Ha

W HUIp%KeH&iK MOnyJIeB

YJlbTpa3BYKOBbIX

‘iBCTOTBX.

IlpEi CABHr-2 H HOpMWIbHOrO

)‘JIbTpa3ByKOBOI-0

HaKOIUIeHRR

OT