Rheologica Acta

Rheol. Acta 21,683

-

691 (1982)

On the equivalence of simplest non-linear rheological equations for viscoelastic polymer media A . I . Leonov Institute of Problems in Mechanics, Academy of Sciences of the U.S.S.R., Moscow Some equivalence conditions are formulated for non-linear models of polymer melts and solutions that are analogous to known conditions for threeconstant linear rheological equations. The resulting model is analysed in simple shear and elongational flows. The kinematics of finite elastoviscous strains is considered in an appendix.

Abstract:

Non-linear rheological model, viscoelastic polymer fluid, equivalence condition, simple shear flow, simple elongational flow, kinematics of finite strains

Key words:

1. Introduction A constitutive theory for polymeric liquids is formulated in [1, 2]. A simple version of this theory is analysed in shear and shows good agreement with experiments over a large range of shear rates [3]. However, some discrepancies exist with experiments at large strain rates. One way to improve the comparison between theory and experiments is to use a more complicated expression for the elastic potential W such as that considered in [5]. Another approach, to be explored here, is the use of a more complex evolution equation to determine the elastic deformation tensor c. To reduce the number of unknown constitutive functions in this theory we demand certain equivalence conditions with the previous theory that are analogous to those satisfied by similar linear rheological models. Below, the general scheme of forming such equations is formulated as well as a simple rheological relationship taking into account the equivalence. This model is analysed in simple shear and extension.

2. Formulating the problem We shall consider an isotropic, incompressible elastoviscous medium under isothermal conditions. (Compressible materials are described in [6] while 847

nonisothermal conditions are discussed in [1, 2, 6].) Deformation from an equilibrium state may be characterized by a positive definite tensor ~, the principal components of which for a pure elastic (equilibrium) homogeneous deformation are the ratios of the squares of lengths for an elementary body particle in the deformed and non-deformed states. The tensor may be defined by the element-wise unloading of the medium from stresses (the principle of "unloading in the small"). In the equilibrium state, such a medium is completely determined by the specific free e n e r g y f = VV'(i1 ,It2, T)/D0 where i a - tr(0), Ir2 ~ tr(~ -a) are the basic invariants of the tensor ~, P0 is the mass density, T denotes the Kelvin temperature and I~ is the elastic potential. In the nonequilibrium (elastoviscous) state, we can describe the medium by virtue of a specific free energy f dependent upon the same parameters used for the equilibrium state. As shown in [1, 2], the positive dissipation rate D in the isothermal case under consideration can be written in the form D = ~r:e -

PoJ'IT = a:e

-

#e:~e > O.

(1)

Here ~ is the symmetric Cauchy stress, e denotes the rate of deformation tensor, and ~e is the reversible stress tensor ~e ~ - - / ~

0f

+ 2P0~'--,

O0

(2)

684

Rheologica Acta, Vol. 21, No. 6 (1982) + rh~k) [0. e . 0 - ( 1 / 3 ) 8 tr(0 2. e)]

defined in terms of the isotropic pressure/3 and unit tensor t~. F o r isothermal d e f o r m a t i o n o f the m e d i u m , the sources o f nonequilibrium are the differences ~p = e - ee ;

~ p = o" -

ee,

+ /'r/~k)[0-1 • e" 0 -1 - ( 1 / 3 ) 8 tr(0 - 2 . e)]

where ~p and #p m a y be called the irreversible strain rate and stress since they vanish in the equilibrium state. By analogy with (3), we can write tbp = to - tbe where to is the complete vorticity tensor and tbp is the irreversible part. As in [1, 2], we assume here that this part vanishes identically. These kinematic relationships are further discussed in the appendix. Substituting (3) into (1) we obtain

We shall also consider the following constitutive relations trp

= Ml(c):e,

ep

: g3(c):

~7e

:

cr~t

--P¢~ + 2 c W 1 - 2 c - l W 2 ;

= e e. c + c - e e ; D = ep:e

-

#e:ep

>

(6)

+ ( l / 2 ) m t k o ) ( e . e . 6 -1 + e -1 . e . e ) .

(3)

Wk = OW/OI~,

det(c) = 1.

(7)

(4)

O.

This inequality indicates the sources o f the irreversibility in the m e d i u m . We now consider two sets o f possible constitutive relations for incompressible materials (tr(e) = 0). In the first we have

Here the symbol M e : e has the same c o n n o t a t i o n as (7) and the f o u r t h - o r d e r tensors M 1 and M 3 are positive definite. The second set o f equations has been used in [2, 3] to describe the rheological properties o f polymers with (8)

Ml:e = ml(Ii,12,T)e.

t~p : /17/1(0) : e - h7/2(0): O e,

We now assume the existence of the following limit transitions:

~p = /17/2(0): e - /17/3(0): Oe', t~e =

-'k 2Cl/~/r 1 -- 2 0 - 1 I ~ 2 ;

--/~

l~kmOl£V/Oik,

d - - ~: + 0 " t o -- t o ' 0 = ~ e - 0 + 0 " ~ ; det(O) = 1,

(5)

a) the linear case o f d e f o r m a t i o n when e --* 0, 0 ~ t~ + 2g, c ~ tJ + 2E where tr0?) = tr(e) = 0 (here and e are infinitesimal elastic strain tensors); b) the equilibrium state of elastic m e d i u m with finite recoverable strains when ~(/k ~ 0, M~ ~ 0. For the linear case, the f o r m u l a s (5) take the f o r m

where a superposed prime denotes the deviatoric or traceless part of a tensor and the o p e r a t o r (~) is called the J a u m a n n or co-rotational time derivative. The fourth order tensors hT/1 and hT/3 are positive definite [1, 2] which follows f r o m the dissipation inequality (4). Each o f the tensors hT/k has 10 constitutive scalars r?/! k ) ( ] 1 , I 2 , T) (k = 1, 2, 3; i = 1, 2 . . . 10) such that /~7/k: e denotes ~7/k: e = rhtk)e + [rh~k)tr(0 • e) + th~t) tr(0 - 1 . e)l • (0 - 1 / 3 i l t J ) + [rh(4k) t r ( 0 . e) + rh~*) tr(c - 1 . e)](0 -1 - 1 / 3 i 2 6 )

°+, o+~ #e = - p 6

o+,/" ~-

(5 a)

and f o r m u l a s (7) reduce to O'p,

+ rh~*)[0 -1 .e + e.C"-1 - ( 2 / 3 ) t ~ t r ( ~ - l . e ) ]

ep = g / O ;

=

g - ~/t~,

+ 2G~,

E = e'e,

+ rht6k) [~ • e + e • ~ - (2/3) 6tr(t~ • e)]

o+~

2r/2e 0 -- v l l / G 1 ,

Leonov, On the equivalence of simplest non-linear rheological equations for viscoelastic polymer media a e=p6

685

explicitly on temperature. We assume the decomposition

+ 2Gig,

= ee,

(7a) I~= Wo(T)~(il,i2);

where ~, G1 are elastic constants, 0,r/0,rh,v/2 are viscosities and 0, 0 are characteristic times for the materials. Note we can exclude the internal parameters g and E and obtain the functional relationship between the observed external quantities e and a ' . If both models describe the same material, then this relationship has the form

W=

Wo(T)w(Ii,I2).(13)

As we restrict our attention to isothermal conditions, the temperature T will occur in all relationships only as a parameter, and very often we shall ommit it from the number of arguments of different functions.

3. The equivalence of the two representations 0t

= 2~/°

+ 1 e,

(9)

where the material constants of both models are related by the parameter S such that

s-0-

0 0

r/o + 0

_

4_ qo

0 < S < 1.

; 0(1 - S)

(10)

In passing from (5 a) and (7 a) to (9), it can be shown that g = (1 - S)E,

(11)

To establish our objective, we have to show the existence of a certain transformation which relates (5) to (7). The result (11) for the linear models suggests that ~ and c are related. This relationship should be isotropic and include only numerical constants. We assume the relationship between ~ and c can only have the form ~ = f ( c ) where f ( c ) is an isotropic tensor function. Moreover, this function should reduce to (11) in the linear limit discussed earlier. Under these conditions, we can explicitly determine f ( c ) . Let ~ and c be two positive definite tensors satisfying conditions

= 6+2g+O(g2); e=6-'.'

which establishes a simple connection between the internal variables in the linear models (5 a) and (7 a). An immediate consequence of (10) and (11) is

+O(t:);

g = (1 - S)e;

ItZPI--l/tr(g:)--,0;

det(~)= 1,

Ilell=l~(t:)~0;

det(e)=l,

0
1.

(14)

Then the isotropic function ~ = f ( c ) has the form I ~ = (~tr(g 2) - r/°(1 - S) tr(g2) = Gltr(g2) = W, 0 which implies the elastic potential is the same for both models. Returning to the analysis of the nonlinear models (5) and (7), we seek to establish conditions under which they describe the same material. Of course, these conditions distinguish only a limited class of relations described by (5). In essence, the results follow from the quasilinearity of the formalism under suggestion. A necessary requirement is the relationship

¢v(i, ,i2, 73 = w ( z , , i2, r) ,

(12)

which implies the invariance of the free energy upon the representation of the rheological equations. Since i k, I k are kinematic measures, they do not depend

= f(c)

(15)

= c ~-s.

To prove (15), note that the tensors t~ and c have coinciding principal directions which implies

C =

liOOol !OOol I! c2

,

0

~=

C3

0

(cl) 0

C3

l

f(c2) 0

~2

f(c3)

where det(e) = q c 2 c 3 = 1 and det(~') = ~1~2~3 = f ( Q ) f ( c 2 ) f ( c 3 ) = 1. Since ~ and c are positive definite then ci > 0, ei > 0 andf(ci) > 0. Solving for c3 andf(c3) in terms of the independent variables cl, c2 a n d f ( q ),f(c2) we obtain the equation f ( q ) " f(c2) = [ f ( c ~ I . c [ 1)]- 1.

686

Rheologica Acta, Vol. 21, No. 6 (1982)

Differentiating this with respect to q and c2 gives f ' ( c 1 ~ ' c 2~) = " - ~ - ~ i Cf~)

1 . C2C2 '

f , (c i- 1 . C21) f ( c l ) f ' ( c 2 ) = f2(ci-1 • C2 a)

1 Cl~ '

f'(q)f(c2)

(ci - cj)(ci + cj) (e.)ij

Since ee = e - dp a n d e e = e - ep, then f r o m ( 1 8 ) w e obtain (~)i;

from which it follows thatf(c) = Z c m. Then from (14), we conclude thatA = 1 and m = 1 - S. In the remaining part o f this section, we derive the particular f o r m o f (5) obtained by substituing d for c in (7). The tensor 117/2 can be f o u n d in terms o f the principal values ci, and the tensors a e, AT/~ can be determined in terms o f a e, M~, respectively. Moreover, the existence o f a linear limit for (7) insures the existence o f an analogous limit for (5). Let c = ~x where x = (1 - S) -~ and consider the o r t h o g o n a l t r a n s f o r m a t i o n Q to a particular orthogonal coordinate system moving with the continuum. The coordinate directions coincide with the principal axes o f c such that

E* = Q . e .

QT =

09" = Q . o . Q T

+ th~2)~';2)] + [r~z)(~ Oue*a) (2

+ rht2)(Ea" e;'e*a)](ei

- ~-h)

+ [ff¢42)(~a' Ge*a)

+ rh~2)(E G'e*~)l(~F' - ~i2) 6~

+ (rh~2) + 2th~2)ei + 2rht2)e/-1 + m~2)~i + m(q2)E/-2+ rht~)e* + 6(~'i)

c3

+ ,~)eie; +

(21)

if')

a#)e;'e;'

+ (-~)rhtZd(~ff I + djdF1)]e~

~'3 e* = Q . e . Q

(i = j ) ,

(~$)u = [a~I ~) + ,~?)(el + eA + a42)(e~ - ' +

c'2

+ O.QT;

(19)

Now we determined .h7/2(c). Using the formulas (5)2 for ~p and (6) for hT/2:e we obtain

I!1° :1 0

(i:j),

(~)ij =

C2

0

(1-1)e*+l(e~)ii

:

I!°:]

c* = Q . c . Q T =

(18)

(i ¢ j ) .

(i c j )

(22)

where 6((~) =- ~r3:~; is an isotropic function• Now, comparing (19) with (21) we find

T,

1

6(e,) = ~(c,);

in this system. In terms o f the new variables, the evolution eqs. (5)4 and (7)4 reduce to

~ 2 ) = r~2) = r~2) = r~t2) = 0 ,

r~ 2) + 2 th~2)ci + 2 r ~ 2)(~/--1 + /~2)~i + F~/~2) 1~/-2 + F~t~ dci dt

2ei(~*)ii = 0

(e i - dj)co~ =

(ci

(i = j ) ;

+ cj)(e*)ij

=

(i 4= j ) ,

(16)

1 - 1/x,

(23)

where 0(c) --- M 3 : a ' . becomes

In tensor form, the first result

1

dci dt

2ci(ee*)ii = 0

6 = -- ~. K

(i = j ) ;

(24)

Using the Hamilton-Cayley identity

(Ci -- Cj)(.D~" = (C i + q ) ( e * ) i j

(i ¢ j ) ,

(17)

where no s u m m a t i o n is implied. Since c i = ~7 then

q-i,#+1:,in

(23)3 , we

1=o,

obtain the scalar equations

2rh~z) + I, rh~2) + rh~2) = 0;

(e*e)ii = 1 (e,)ii K

(i

2rh~z) + rh~2) + Z2r~2) = 0,

j) a # ) + ,~t~ - i~rn~ ~ - i , , ~ 2 ) = 1 - 1 / , ¢ ,

(25)

Leonov, On the equivalence of simplest non-linear rheological equations for viscoelastic polymer media Comparing (20) with (22) we find

r~:} + ,~2)(e~ + e~) + ,~a)(e;-, + e~ b + ,~t2}e~e~ + r~pe,-'ef ~ + mt~o)(e?l~,.-' + e £ ' )

d(7-cJ)(Ui + ~ I

=l-

(i--/:j).

\<+e,/

(26)

Note that all the rhl2) can be expressed in terms of rht2), rh~~) and ~t~ such that e 4 ~) = t - ~ / ~ + ~ p / ~

+ ~})

region of the plane ]1, i 2 for which these relationships make sense. Then from (27) and (29) it follows that the quantities ~2) depend on the invariants il, i 2. Therefore, the tensor M2 is completely determined. Another consequence of (29) is that the quantities r~ 2), rh~2), r~t2) tend to a constant value the limit 0 ~ & which is consistent with the linear limit discussed earlier. Let us note once more an important relationship. Any deviatoric isotropic tensor function ~,(c) = ~(~) has the property

f42: (u = (1 - 1/x)(u.

For the principal components of the tensor ~, this formula follows immediately from (19), (20) and (23).

;

~ I ~) = - ~-(m~} + l~m~)).

(27) The quantities rhl2), t ~ ~}, r ~ are determined by the set of equations

Now, we will relate the stress tensor for the models (5) and (7). Using (12) and (15), note that

ae+p6=

2c.

,~,~,~[~) + & r ~ , ~) + ro,~t~ 2~

c

= .o

-

(3O)

+ ~ 2 } / ~ _ ~t{¢;

,~2} = ,~t2} = ,~{p = ,~t2) = o ,

~{? = - ~ ( l , ~ p

7

+ o,

K

'

where

OW OW -2c.-Oc O~ 0W

- -

0~

-

1

(,~+p,~)

x

- ~-(~, - ~ ) ~ ;

ae = ~- 6e-

g-=-g\

#q

(el - e,) ~ y~ ; e~e+

/'

2 ]

=

-T

v12

La~(e~

2 [

~%2) = - T

-

Then, with the help o f (30) and (31) we can rewrite the relationship for #p in (7) in the f o r m

v31 e~)

1)12

#p = h T / l ( g , ) : e - (1 -

v23

e2(~3 - e , )

-

~(~3 - ~)

1 [ = -T

ci + c2 ~,

v,~¢~

L

V31 ^e,

+ e,

q - cl

q_ V23 42~

c3

+_C3 ]J

'

(29) where V~-- (C3 -- C1)(C3 -- C2)(C2 -- ~1)"

These quantities are symmetric functions of 0s, Oj. The formulas (29) make sense in the limit as ci ~ cj since (C i --

(32)

~2-el hT/1 = M l .

v° -

1/X)Oe.

We note that stress tensors for the two models are the same if

] v2~ [

v3-----2--~+

(31)

(28)

for i ~e j. The solution of this system is

'~

O~ Oc

which implies that

~o -

~}

687

~j)2 (X -- 1) 2 [1 "b

12~:~

O(C i --

Cj)],

(Ci --+ Cj)'

Also cl and c2 are unambiguous functions of ~ = ~ + c2 + ~-l~i-1 and i2 = ~1-1 + ~i-I + ~l~2in the "wedge-shaped"

(33)

We have shown that the t r a n s f o r m a t i o n (15) conserving the free energy, allows us to relate the models (5) and (7). Moreover, the inverse also holds true. If the tensor f'/2(~) is defined by (27) and (29), then the equality W = l~is satisfied automatically. Evidently, the rheological eq. (5) and (7) can be m a d e isomorphic like their linear counterparts. The fact that eqs. (7) belong to the class o f equivalent theological equations, provides an additional i m p o r t a n t relation between the tensoric material functions M1 (c), ep(C) and tre(C) in eqs. (7). To extract this relation let us consider the retardation process. In the case o f eqs. (7) it means tr = 0, while for the equivalent eqs. (5) it implies: ~p = 0, a = 0. Under the

688

Rheologica Acta, Vol. 21, No. 6 (1982) 4. S i m p l e s h e a r a n d s i m p l e e x t e n s i o n a l f l o w

latter conditions eqs. (5) take the form 37/1(6):e = /~2(6): t~" - tTe ; (34)

37/2(6):e = - / ~ 3 ( 6 ) : ~e = - ~ ( 6 ) .

Eqs. (34) show that under retardation conditions e is an isotropic function of 6. Then, by employing eq. (30) and eliminating e f r o m eqs. (34) the latter reduces to (k

-

1) A?/~(~): ~(~)

o

T

-1

,

o

0

0

Cll C12 ! 1 12 C22 '

C22

C=

,

0 ~7

=

--

-- C12 o |

C12 0

Cll 0

011

'

Where

11 = 12 = I--~ Cll + C22 + 1 .

1/311~) - b2(c -1 - 1/3/2t~)

where bi = b i ( T , !1,/2), and using eq. (6) and the expression for ae in (7)3 one can obtain two algebraic relations which connect the constitutive scalars mi', l~i and bi: 3b1(ml + mlo) + m E [ 2 b l ( I ~ - 3/2) + b 2 ( I l i 2 -

¢-1

0

(35)

= 0"e.

Now by using the representation of ¢ ( c ) [1, 2] ta(c) = bl(c-

e=

=

which, in turn, using eqs. (24), (31) and (33), results in (k - 1)M1 (c): ¢(c)

Simple shear is a well studied flow for which theory and experiments can be compared. In this case, the kinematic matrices have the f o r m

All constitutive scalars depend only on m!k) (/). The stresses in steady flow for are given in terms of the sum a = ae viscometric functions a12, a 1 -= all - 633 where

I, i.e. mSk) = the model (7) + a p and the 022, 02 = 0"22

9)] 0.f2 :

+ c

2(m2- m3 -

+ ms)

m3 [bl (/1/2 - 9) + 2b2(I 2 - 3h)]

+ 2i--(Cli + C22)(m 6 + m7) + ~-(1 + 2c22)

+ 2 m 6 ( 2 b l h + b2/2)

1 2 + c22 _ 2c22) ; • (m 8 + m9) + ~-m10(Cll

6mTb2 + m 8 [ b l ( 2 I 2 - 3/2)

+ b2(I112 - 3)] - m9(262/2 + b l h ) 6 W~/(k

-

0.~2 -- 2(W1 + W2)c12'

1), 0.~ = ~,c12(cll - c22)(m2 - m 3 - m 4

3b2(ml + ml0) - m 4 1 2 b l ( I 2 - 3/2) + b 2 ( I l i 2 - 9)1 + ms[bl(IlI2

+ m s + m s + m 9 - mlo ) ;

- 9) + 2b2(I22 - 3/1)1

0.~ = 2(cll -

6m6bl

+ 2m7(blI~

+ m 9 [ b 1 ( I l I 2 - 3) + 212 - 3h)b2] -

1).

c22)(W 1 +

W2),

0.~ = - ~ , c 1 2 [ ( m 2 - m3)(1 - c22) + (ms - m4)

- m 8 ( 2 b l h + bE/E)

= 6 W2/(k

-

+ 2b2/2)

(36)

These relations show that: (i) not all the scalars m i are independent within the frames o f the equivalence principle; (ii) only five material functions (ml + ml0, m6, m7, m8 and m9) are substantial and independent; (iii) the simplest case of rheological equations (8) is forbidden by the relations (36).

• (Cll

-

--

1)

+

m7 -

m 6 + m9Cll

m8c22 - ½ m10(cll - c22)] ;

0.[ = 2Wl(1 - c22) + 2W2(Cll - 1),

(37)

where the coefficients are understood to be associated with the tensor M 1 . Despite of some quantitative differences, the qualitative dependence of the viscometric functions upon the shear rate ), is similar to

Leonov, On the equivalence of simplest non-linear rheological equations for viscoelastic polymer media that exhibited by polymeric liquids• This dependence is shown in figure 1. We assume the particular constitutive functions

Using (38) and (39) in (7) we shall obtain the constitutive relations

rr = - p ~ + 2 p c +

w=uG-3)

(w~=.>o,

3 p ( K - 1)

Co

w2=o), (38)

C(I) = C 0 e x p [ - f l / 2 ( / l + / 2 - 6)]

689

• exp[fl(I 1 + 12 - 6)/2] [2I~ + 1112 - 3/2

9(I1 + 12)]l~211+ \11 + 212

~,{81

. [c-l.e + e.c

12\ /

l___326tr(c-l.e)]

+ 2 c-e.c-

tr(c 2 e) 3

ep = C o e x p [ - f l ( 6 + /2 - 6)/2]

[c c l+ ,I2 i,, t Fig. 1. The qualitative dependence of the viscometric functions tyl2, al and (72 on the shear rate ~, for the model (8)

b - c. e - e. c = - 2c. ep(C) ; tr(e) = 0 ; as used in [ 1 - 3]. Here C(I1,I2) is a characteristic relaxation time. The experiments discussed in [3, 4] show that 0 < fl 4 1. Under moderate elastic strains, it can be assumed t h a t / / = 0. In addition, considerations of simplicity and comparison with the data along with eqs. (36) suggest that all mk equal zero except

m

7

-

det(¢) = 1.

(40)

When c ~ 6E, 11 ~ I2 ~ 3, from (40) we obtain (7a) where

Co

-

1

O=SO,

40'

tlo-

2pO

1-S

,

1 ~ c - - - .

1-S

(41)

3u(~:- l) expIfl(/1 + /2 - 6)/21 Co

In steady shear, assuming fl = O, we find that

x . l/'2 e 1 1 = ~ / 2 ( l _ x ) 1/2' c22- ( l + x ) v : ;

• (211_+6 \ I 1 + 212

2F

: •

I1 +

')' I2

312-91a+212/

I~+I1/2-

C12

, (39)

--

_

_

,

l+x I1=12=I=

6p(xm

8

-

1)

co

•

[2

I 2+ILI

1 +)/2(1 +x),

exp[fl(/a + / 2 - 6)/21

2. 2-

(_/,+/2)1'

312- 9\11+ 2/2

.

+

( i + x; 1'2

1- s

8F2S 1 + ~ - ;1[~2(~ + x) + 21 •

(1 + x ) [ 2 ~

690

Rheologica Acta, Vol. 21, No. 6 (1982)

~-2_= ° ' 2 ( 1 - S) - _ ( 1 _ 2p

S) [

l/2 ] (1 + x ) l/z

A+1

4 F 2 S [ 2 ] / 2 - (1 + x) 1/21

+

2

+ 6/"S

6

d2

2+1

2

dr

2

exp[Bzl,

( 2 2 -- 2 - 1 )

exp[-flX)

= 6F,

(1 + x)3/2[~/2(1 + x) - 11 []/2(1 + x) + 2 ] '

(45) where

~12 ----

a12(1 - S) 2p

2/"(1 - S) 1 +x

2Z(2) -- 22 + 22 + 22 -1

6;

F-

0) ;

"c - t / O .

FS

+

+ 2 -2 -

]/2(1+x)-

1 4F3S

+

(1 + X)2[l/2(1 + X) - 1][1/2(1 + x) + 21 (42) where x - (1 + 4/'2) 1/2 and F - y0. These expressions have the asymptotic expansions

Similar systems have been studied in [1, 4]. This differs from that considered in [4] by the factor 2 / ( 2 + 1) in the second term o f (45). The recoverable strain a measured under retardation conditions (a : 0) is related to 2 by the formula a = 21- s which differs f r o m that given in [4]. W h e n fl = 0, the steady flow regime, defined by expressions (7 :

2 2 -- 2 - 1 ;

o1 = 2/"2[ 1 - S + 0 ( / " 2 ) ] ;

/ " _ _ _ +2 1 ( 2 2 _ 2 _ 1 ) ,

(46)

62

t7 2 =

-/"211 - 2 S + 0(/"2)] ;

(712 :

r[l + o(/"2)1,

always exists. Moreover, the following asymptotic expression hold true:

(0 < / " < +); a=3F[I+O(F)I;

2 = I + F + O ( F 2)

(F,~I),

~1 = (2 - S ) F I / z [ 1 + O ( F - 2 ) ] ; a = 6 E l l + O((6/")-1/2)]; b2 :

--(1

-- S / 2 )

;

+ O(F-l/z)

2 = ]/6/'[1 + O ( ( 6 F ) - l / 2 ) l ~ O"12

= S F 1/2 + O ( 1 ) ,

(F~> ~)

which are in qualitative agreement with data for polymeric fluids. In simple extension, the vorticity vanishes and

22

e=

~

1

,

2

0

C-l =

°

These formulas predict that the T r o u t o n viscosity r/r a / F increases with strain rate approaching twice its value at low rates. For fl 4= 0, the viscosity may more than double. At sufficiently large values of F, no steady flow exists for the medium under consideration. -

----

Appendix: The kinematics of finite strains for an elastoviscous medium

2-

.

(44)

0

After a very long calculation using (31), we obtain O" --

(47)

,,]-1

0

-

A

C

(F~> 1).

(43)

0"11 -- 0"22 (1 -- S) = (1 - S)(2 2 - 2 -1 ) 2p

Let ~ = {~i} and x = ~xi} denote the coordinates of a material point initially and at time t. Then the smooth, oneto-one mapping x = f(~, t)

(A 1)

determines the motion the continuum. We postulate that the elastoviscous medium may be characterized with a particular "unloaded" state r/ = {r/i}which can be defined as the "instantaneous" element-wise unloading of the medium from stresses. Thus, the existence of a smooth one-to-one depend-

Leonov, On the equivalence of simplest non-linear rheological equations for viscoelastic polymer media ence r/ = r/(~, t) is postulated which can be rewritten using (A1) as x

=

(A2)

O(q,t).

dx = F . d~.

(A3)

In the elastoviscous medium, we can also define the gradient tensors of recoverable F~ and irrecoverable Fp strains as F~.d*l;

dq=

Now, let us consider the relationship (A8) written in the form Fe" H

:

(Vx~)e)

(A 12)

" Fe" FT.

We can write F~ = Ve " Re where Ve is a symmetric tensor and R e is orthogonal. Substituting into (A12) gives

The deformation gradient F is defined as

dx=

691

(A4)

Fp.d~.

~. ve + v~.Re.R~.

Ve : ( V x v e ) - c ~ ,

(A13)

where ce = F 2. After transposition of this equality and addition with (A 13), we obtain

b~

=

(Vxve) • c~ + ce" (V~v,,) T.

(A14)

These definitions imply Combining this result with (A 10) gives (A5)

F = Fe'F p .

be -

We define the complete v, recoverable v e and irrecoverable vp velocities such that =

k ¢ =- Of

;

t) e =

k.

=

0¢

;

(A16)

which coincides with the evolution equations for c in (7) if we substitute into the latter coe = co and c = ce. The same results also apply to the model (5).

Vp = V -- V e .

(A6) Note that dv = F'd~

Coe . c e + Ce ' e 9 e = e e . c e + c e ' e e ,

Acknowledgement

The author is indebted to Mr. W. E. VanArsdale for revision of the English text.

= F ' F -1 . d x ,

which implies Vxv = F - F - ~ .

(A7)

Analogously, the recoverable strain rate Vxve is defined as ~7*Ve =/~'e" F e I ,

(A8)

where dve = F e . a t l = iZe. F p . d ¢ = Fe.F;

= k e . Fp" F - I " a x

1 "dx.

The irrecoverable strain rate Vxvp is defined as VxVp = VxV-

Vxve=Fe'iTp'F;

1 " F e a.

(A9)

Further, we can write Vxv = e + e) ;

V x V e = e e -[- (19e ;

VxVp

References 1. Leonov, A. I., Preprint of lecture in the International School "Problems of Heat and Mass Transfer in Rheology of Complex Media", A. V. Luikov's Institute of Heat and Mass Transfer of AN BSSR (Minsk, April 1 6 - 28, 1975). 2. Leonov, A. I., Rheol. Acta 15, 8 5 - 9 8 (1976). 3. Leonov, A. I., E. Ch. Lipkina, E. D. Paskhin, A. N. Prokunin, Rheol. Acta 15, 411 - 426 (1976). 4. Prokunin, A. N., Nonlinear Elastic Phenomena under Extension of Polymer Liquids. Experiment and Theory, Preprint No. 104, Institute of Problems in Mechanics of Academic of Sciences of the USSR, Moscow (1978). 5. Leonov, A. I., A. N. Prokunin, Rheol. Acta 19, 3 9 3 - 403 (1980). 6. Gorodtsov, V. A., A. I. Leonov, Prikladnaya Matem. i Mekh. USSR 32, 7 0 - 94 (1968). (Received January 26, 1981; in revised form April 8, 1982)

= e p + O.)p,

(A 10) where e, e e , e p are the symmetric parts of the complete, recoverable and irrecoverable strain rate and ( D , ( D e , t O p a r e the corresponding antisymmetric parts. From (A9) and (A10), we conclude e=

ee + e p ,

~=

toe + ¢oa.

(All)

Author's address: Prof. A. I. Leonov Academician II'yushin Street 1, ap. 54 USSR-125319 Moscow