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An integral constitutive law for viscoelastic fluids based on the concept of evolving natural configurations: stability analysis
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International Journal of Non-Linear Mechanics 39 (2004) 1275 – 1287
An integral constitutive law for viscoelastic $uids based on the concept of evolving natural con(gurations: stability analysis Liviu-Iulian Palade∗ Ecole Sup erieure de Plasturgie, B.P. 807, 01108 Oyonnax, Cedex, France Received 24 January 2003; received in revised form 18 August 2003; accepted 18 August 2003
Abstract A general conceptual framework has recently been developed within the context of large deformations for the study of materials (solids and $uids) that either undergo microstructural changes or for which stress states are related to the presence of several relaxation mechanisms. The corner stone of this new approach is that all materials exhibit an in(nity of stress-free natural con(gurations that are evolving in a thermodynamically admissible process. In this paper, that is exploratory in nature, we study the behavior of a non-linear integral viscoelastic constitutive equation (CE) for a $uid assumed to posses an in(nity of evolving natural con(gurations. The CE stability pattern with respect to small perturbations about the rest state is also addressed. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Natural con(guration framework; Stability
1. Introduction Polymer $uid $ows have received a lot of attention due to both wide spread industrial applications and because of challenging fundamental physical aspects [1–6]. Consequently, there is a large amount of experimental studies that have been published together with a certain number of constitutive equations (CE) based on continuum and/or statistical mechanics concepts intended to model the $ow behavior. Of particular interest is the $ow of viscoelastic liquids that undergo microstructural changes, such as $ow-induced crystallization (FIC) [7], polymerization and/or crosslinking during reactive injection molding, for which there is still a need on one hand for a deeper and more complete understanding of the physical phenomena and on the other hand for sound and trustworthy models. In order to deal with such a complex task, many authors have chosen the way of generalizing existing concepts. For example, McHugh used the Poisson bracket formalism developed by Beris and Edwards [8] to formulate a model that accounts for polymer FIC [9–11]. Another innovative approach has been pioneered by Rajagopal and his co-workers [12–14] and is based on the newly introduced concept of evolving natural con(gurations. Brie$y, the material response is elastic from these natural con(gurations and the rate of dissipation (which is maximized) determines how they evolve. Moreover, the maximization of the dissipation rate does not imply that the material reaches instantaneously ∗
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the natural con(guration, it only means it reaches it at the highest admissible rate. The viscoelastic response is determined by the stored energy related to the elastic response from the (stress free) natural con(gurations and the rate of dissipation due to viscous eEects. This approach has led to Oldroyd-type diEerential CEs and has been used very successfully in modelling FIC [15–18]. We point out here that while viscoelastic CEs involving intermediary (i.e. more than two) con(gurations have already been considered in the past [19–21], the thermodynamic framework set up by Rajagopal is new. In this work—which is exploratory in nature—we shall apply the Rajagopal’s ideas of natural con(gurations (in(nitely many) framework to generalize an integral non-linear CE that has the following form: t T(t) = −pI + S(t); S(t) = m(t − t )h(IIC )Ct (t ) dt ; (1) −∞
where Ct (t ) = FtT (t )Ft (t ) is the right Cauchy–Green tensor. The memory function, which can be obtained from either reptation theories [22] or from linear viscoelastic measurements, is given by [23] m(t) =
N Gn n=1
n
e−t= n ;
(2)
while for the damping function we choose the general form [23] ; h(IIC ) = ( − 3) + IIC
(3)
where IIC = (1=2)[(trC)2 − tr(C2 )] and ¿ 0 is a parameter. We point out here that authoritative molecular theories for polymer melts result in this type of integral CEs. For instance, Wagner’s model [4,23–25] which is a generalization of the de Gennes, Doi and Edwards theory [22], leads to a CE of the following form: t u ⊗ u dt ; S(t) = m(t ˙ − t )5f2 (4) |u |2 −∞ where in the above f = f(u ) is a molecular stress function that depends on the vector u = Eu, E being a suitable strain gradient tensor that acts on a unitary vector u of isotropic statistical distribution and whose end is located on a sphere of radius 1. Hence, there is a strong interest in studying this type of models within diEerent conceptual frameworks. This paper is organized as follows. We (rst present the generalization of the CE given by Eqs. (1), then we deal with the stability properties of its linearized form (i.e. the behavior with respect to small amplitude perturbations about the rest state).
2. The CE Let B(t), Bkp (t) and B(t ), where t; t ∈ R, be a current con(guration, the associated natural con(guration and a deformed con(guration at time t , respectively. All con(gurations are thought as diEerentiable manifolds. From Fig. 1 we observe that Ft (t ); Gt (t ) = Fk−1 p (t)
(5)
where the strain gradients are given by Ft (t ) = grad x(t) ⊗ (t );
Gt (t ) = grad (t) ⊗ (t );
Fkp (t) = grad x(t) ⊗ (t):
(6)
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Fig. 1. The con(gurations B(t1 ); : : : ; B(tn ); : : :, the associated natural con(gurations Bkp (t1 ) ; : : : ; Bkp (tn ) ; : : : and a deformed con(guration B(t ) are shown together with the corresponding strain gradient tensors.
Next, the left Cauchy–Green tensor Bkp (t) = Fkp (t) FkTp (t) is the solution of the following thermodynamic evolution equation that was derived when a generalization of the Maxwell model has been worked out in this new framework (for details see Ref. [14]): Bkp (t)
+ Bkp (t) = I:
(7)
Instead of deriving the evolution equation for Bkp (t) from a thermodynamic standpoint, we shall assume that ), the same evolution equation (7) holds for the model considered here, where =2Gg (T )=0 and =3=tr(Bk−1 p (t) 9 with Gg (T ) the temperature-dependent glassy modulus (whose value is ¿ 10 Pa), 0 the zero shear viscosity, and ( ) is Oldroyd’s upper convected derivative [2,6] D( ) ( ) = (8) − L( ) − ( )LT : Dt Such an assumption is similar in spirit to prescribing evolution equations for the plastic strain in theories for viscoelastic bodies. Next, since the $uid is assumed isotropic, one can pick a basis such that Fkp (t) = Vkp (t) , with Vkp (t) the left stretch tensor arising in the polar decomposition theorem. With this assumption, from Eq. (7) one obtains the strain gradient tensor Fkp (t) = (Bkp (t) )1=2 and further on the right Cauchy–Green tensor Ct (t ) by using the Eq. (5), that is F−1 Ft (t ): Ct (t ) = Gt (t )T Gt (t ) = FtT (t )Fk−T p (t) kp (t) Therefore, the CE to be studied in this work is given by the following set of equations: t S(t) = m(t − t )h(IIC )Ct (t ) dt ; −∞
h(IIC ) =
; ( − 3) + IIC
¿ 0;
(9)
(10) (11)
Ct (t ) = GtT (t )Gt (t );
(12)
Ft (t ); Gt (t ) = Fk−1 p (t)
(13)
Bkp (t)
(14)
+ Bkp (t) = I:
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It is important to note that Eq. (14) plays a crucial role in determining the stress tensor S(t) in (10) as Ct (t ) is determined by Fkp (t) as indicated in (9). 3. The stability of the rest state Hadamard stability and thermodynamics are related to each other, as it can be observed from previous studies [26–28]. In this section, we examine the stability of the rest state of the material as given by Eq. (14) using the linearized theory. In doing this, we assume the material contained in a suMciently smooth bounded volume has been set in motion at time t = 0 by an imposed (rst-order disturbance in the pressure, motion, strain and stress (elds, i.e. ˆ Fˆ kp (t) ; T)e ˆ i(k·x−ct) + o(2 ) (p; f(x); Fkp (t) ; T(t)) = (p0 ; f0 (x); F0; kp (t) ; T0 (t)) + (p; ˆ f; = (p0 ; f0 (x); F0; kp (t) ; T0 ) + (pˆ 1 ; fˆ 1 ; Fˆ 1; kp ; Tˆ 1 ) + o(2 );
(15)
where ¿ 0 is a small amplitude disturbance parameter, f(x) = f(x((t ); t)) is a prescribed (given) motion, (p0 ; f0 ; F0; kp (t) ; T0 ) are the zeroth-order terms each of which—but f—may depend only on t, while ˆ Fˆ kp (t) ; T) ˆ are the real (rst-order amplitudes of the corresponding disturbances, k(t) ∼ (k1 ; k2 ; k3 ) the (p; ˆ f; wave vector and c the wave velocity. Moreover, the asymptotic expansion (15) holds true if for each of its terms a limit of the following type does exist: A(x; t) − A0 (t) − A1 (x; t) lim = 0; (16) →0+ 2 where A may be either a scalar, or a vector or tensor (eld. If A is a tensor, i.e. A, then one can use A = tr 1=2 (AT A). Although other norms may be de(ned, since tensors are linear and continuous operators with domains and ranges subsets of (nite-dimensional vector spaces, all norms are equivalent. Hence, we assume that relationship (16) must hold irrespective of the norm used. The type of -order disturbance chosen here (see Eq. (15)) is diEerent from the one used by Kwon and Leonov [29–31] who have chosen perturbations with asymptotically large frequency and wave vector. Indeed, these former authors being interested in studying perturbations relevant for rapid $ows, have used the following disturbance instead: ( ) = ( )0 + (∧ )e
i(k·x−!t) 2
+ o(2 )
(17)
and have shown that some continuum mechanics models are Hadamard ill-posed with respect to this particular perturbation. Before proceeding further, we make the following remarks: • Any quantity A(x; t) that has asymptotically been expanded in 15 may be re-written equivalently as ˆ ik·x e−ict + o(2 ) A(x; t) = A0 (t) + Ae −t ˜ = A0 (t) + A(x)e + o(2 );
∈ C;
(18)
where A˜ = A˜ Re + iA˜ Im is generally a complex function. However, the particular case when A˜ is a real function—that is of interest for certain applications—may be considered. Moreover, in order to simplify notations (and to some extent calculations) we shall also make use, instead of (15), of Eq. (19) −t ˜ F˜ kp (t) ; T)(x)e ˜ (p; f(x); Fkp (t) ; T(t)) = (p0 ; f0 (x); F0; kp (t) ; T0 (t)) + (p; ˜ f; + o(2 )
= (p0 ; f0 (x); F0; kp (t) ; T0 ) + (p1 ; f1 ; F1; kp ; T1 ) + o(2 ):
(19)
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• We now remark that in many previously published studies on Hadamard stability of viscoelastic CEs the velocity (eld is assumed to be a problem-independent variable (see for example [31–33]). We shall, however, show in the next section that if the perturbed f(x; t; t ; ) motion (eld is known (as it is assumed in Eq. (15)) then the perturbed velocity (eld v(t; x; ) can be calculated from it. Incompressibility restrictions will also be addressed. Next, in order to perform stability calculations we need the following equations: the evolution law for the Fkp (t) tensor, the CE in linearized form, momentum balance, compatibility restrictions. We shall deal with each of them subsequently. 3.1. Perturbed velocity 6eld Let the motion x = f(x(); t) in perturbed form be given by −t ˜ f(x(); t; ) = f0 (x(); t) + f(x)e + o(2 ):
(20)
From (20) we get that Ft (t; x; ) = grad ⊗ f(x(); t) −t ˜ = grad ⊗ f0 (x(; t)) + (grad ⊗ f(x))e + o(2 ) −t ˜ = F0; t (t) + F(x)e + o(2 ):
(21)
Since incompressibility requires det[F0; t (t)] = 1, then necessarily det[Ft (t; x; )] = 1 as well. Hence, the determinant of the rhs of Eq. (21) must be equal to 1. Next, we can use the well-known relationship L0 (t) = grad x ⊗ v0 (t; x) = [(d=dt)F0; t (t)]F0;−1t (t) written in perturbed form d L(t; x; ) = Ft (t; x; ) Ft−1 (22) (t; x; ) dt to obtain v(t; x; ). To do so, assume the perturbed velocity (eld be given by ˜ x)e−t + o(2 ); v(t; x; ) = vQ0 (t) + v(t;
(23)
where vQ0 and v˜ are vector (elds to be further calculated. The velocity gradient L(t; x; ) can now be expressed as L(t; x; ) = grad x ⊗ v(t; x; ) ˜ x)]e−t + o(2 ) = grad x ⊗ vQ0 (t) + [grad x ⊗ v(t; ˜ x)e−t + o(2 ): = L0 (t) + L(t;
(24)
Next, from (21) and (22) we get that L(t; x; ) =
d −1 −t ˜ + o(2 ): [F0; t (t)]F0;−1t (t) − [I + L0 (t)]F(x)F 0; t (t)e dt
(25)
Now, from the rhs of Eqs. (24) and (25) after equating the corresponding zeroth- and (rst-order terms in we successively obtain the following results: L0 (t) = grad x ⊗ vQ0 =
d [F0; t (t)]F0;−1t (t) ⇒ vQ0 = v0 ; dt
(26)
where v0 is a given velocity (eld, and −1 ˜ x) = grad x ⊗ v(t; ˜ ˜ x) = −[I + L0 (t)]F(x)F L(t; 0; t (t);
(27)
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where F0;−1t (t) = F0; t (t ). Eq. (27) is a (rst-order PDE whose general solution provides the v˜ (eld components. However, we shall solve it for a particular motion that is of interest for certain engineering applications. Let a (zeroth-order in ) mixed shear and extensional strain (eld x = f((t ); t) be given by
x1 (t) = e−( =2)(t−t ) [1 (t ) + s(t − t )2 (t )];
x2 (t) = e−( =2)(t−t ) 2 (t );
x3 (t) = e (t−t ) 3 (t );
(28)
to which corresponds the following velocity (eld v0 (t; x) ∼ (v0; 1 ; v0; 2 ; v0; 3 ): v0; 1 (t) = sx2 (t) − x1 (t); v0; 2 (t) = − x2 (t); v0; 3 (t) = x3 (t) (29) 2 2 with divx (v0 (x)) = 0 and where s ¿ 0; ¿ 0 are, respectively, the constant shear and extensional rates. From (28) one obtains that [F0; t (t )]lm = @l (t )=@xm (t) is given by ( =2)(t−t ) −s(t − t )e( =2)(t−t ) 0 e ; det[F0; t (t )] = 1; [F0; t (t )]lm = (30) 0 0 e( =2)(t−t ) 0
0
e− (t−t
)
while from (29) we get [L0 (t)]lm = @vl =@xm as − s 0 2 [L0 (t)]lm = 0 0 ; tr[L0 (t)] = 0: − 2 0 0
(31)
We now turn our attention to -order terms, and assume perturbations of the form ˜ f(x()) = (x1 (1 ); x2 (2 ); 0): This corresponds to perturbations that propagate uniquely along the x1 - and x2 -axis. We further get 0 0 F˜ 11 (x1 ) ˜ ˜ F(x) = grad ⊗ f˜ = F˜ 22 (x2 ) 0 0 ; det[F(x)] = 0; 0 0 0
(32)
(33)
where in the above F˜ mm (xm ) = @xm =@m ; m = 1; 2. Since the F˜ mm (xm ) components are real-valued functions, this is a particular case of the general framework that has been described in Eq. (19). Next, from Eqs. (27), (30), (31) and (33) we get
− s e( =2)(t−t ) F˜ 11 (x1 ) −s − s (t − t )e( =2)(t−t ) F˜ 11 (x1 ) 0 2 2
( =2)(t−t ) ˜ ˜ x) = (34) grad x ⊗ v(t; F 22 (x2 ) 0: 0 −s e 2 0 0 0 Integrating (34) provides the v˜ components as
v˜1 = − s e( =2)(t−t ) F˜ 11 (x1 ) d x1 − s(t − t )x2 F˜ 11 (x1 ) + C1 (t); 2
− s e( =2)(t−t ) F˜ 22 (x2 ) d x2 + C2 (t); v˜3 = C3 (t); v˜2 = 2
(35)
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where Cl (t); l = 1; 3 are arbitrary time-depending functions. Hence, Eqs. (23), (29) and (35) give the v(t; x; ) velocity (eld. We are now ready to deal with incompressibility restrictions, i.e. divx (v) = 0. From Eq. (23) above we see ˜ = 0 as well. Using (35) we easily get that that divx (v) F˜ 11 (x1 ) − s(t − t )x2 F˜ 11 (x1 ) + F˜ 22 (x2 ) = 0;
(36)
which can be interpreted either way: if F˜ 11 (x1 ) is given, then it provides F˜ 22 (x2 ), and if F˜ 22 (x2 ) is assumed known, it is an ODE for F˜ 11 (x1 ). Hence, the two perturbations F˜ 11 (x1 ); F˜ 22 (x2 ) are actually related to each other because of incompressibility. If the $ow is shear free, i.e. s = 0 (in which case grad x ⊗ v˜ ∈ Sym), then incompressibility triggers F˜ 11 = −F˜ 22 . 3.2. Evolution law (EL) for the Fkp (t) tensor For convenience let us recall here the EL (7) Bkp (t)
+ Bkp (t) = I;
= 3=tr(Bk−1 ): p (t)
(37)
The above EL must now be expressed in perturbed form, and we shall deal with it in the following. From (19) we get Bkp (t) (t; x; ) = [Fkp (t) FkTp (t) ](t; x; ) = F0; kp (t) F0;T kp (t) + [F1; kp (t) F0;T kp (t) + (F1; kp (t) F0;T kp (t) )T ] + o(4 ) = B0; kp (t) + Ekp (t) + o(4 );
Ekp (t) ∈ Sym;
(38)
where Ekp (t) is a short-hand notation for the second term on the rhs. From (38) we immediately get that Bk−1 (t; x; ) = B0;−1kp (t) − B0;−1kp (t) Ekp (t) B0;−1kp (t) + o(2 ) p (t)
(39)
and further on, since tr(B0;−1kp (t) Ekp (t) B0;−1kp (t) ) = tr(B0;−2kp (t) Ekp (t) ), that
(t; x; ) = 0 − tr(B0;−2kp (t) Ekp (t) ) + tr(o(2 )):
(40)
We now turn our attention to the upper-convected term. From Eq. (8) we observe that we must (rst express the material derivative operator in perturbed form. Hence D D @ + v1 · grad x + o(2 ): (41) = + v0 · grad x + v1 · grad x + o(2 ) = Dt @t Dt ; 0 Next, by making use of (23) and (41) back into (37), the perturbed upper convected term can now be written as D T T Ek (t) −[L1 (t; x)B0; kp (t) +(L1 (t; x)B0; kp (t) ) ]−[L0 Ekp (t) +(L0 Ekp (t) ) ] ; Bkp (t) (t; x; ) B0;kp (t) + Dt ; 0 p (42) ˜ −t . Moreover, since diEerentiation with respect to x of o(2 ) and higher-order relations is not where L1 = Le always permissible [34], the order relations were dropped when calculations have been carried out, hence the use of the sign in (42). Next, making use of (38), (40) and (42) back into (37) and after equating the corresponding zeroth- and (rst-order terms in , we get for the zeroth order B0;kp (t)
+ B0; kp (t) = 0 I;
0 = 3=tr(B0;−1kp (t) )
(43)
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and for the (rst-order D Ek (t) − [L1 (t; x)B0; kp (t) + (L1 (t; x)B0; kp (t) )T ] − [L0 Ekp (t) + (L0 Ekp (t) )T ] + Ekp (t) Dt ; 0 p = − tr(B0;−2kp (t) Ekp (t) )I:
(44)
Several remarks are to be made at this level. First, one can use the zeroth-order equation (43) to calculate the Fkp (t) tensor in the way outlined in Section 2. Indeed, from (31) and (43) we get after some algebra that 2s2 s 0 0 + 1 0 + ( + )2 ( + )2 s 0 0 (45) [B0; kp (t) ] = : 0 2 ( + ) + 0 0 0 − 2 Next, since Fkp (t) =(Bkp (t) )1=2 , the next step involves calculations of the B0; kp (t) eigenvalues and corresponding eigenvectors so that B0; kp (t) can be written in diagonal form. Since this triggers tedious algebraic calculations, we have used the Mathematica v4.0 software formal calculation package, and we shall give here only the (nal results. Speci(cally, to the following eigenvalues qm ; m = 1; 3:
0 0 2 2 + s2 q1 = ( + ) ; ( + ) + s s − ; q2 = − 2 ( + )3 q3 =
0 ( + )2 + s s + ( + )2 + s2 3 ( + )
do correspond the (unnormed) eigenvectors qm ; m = 1; 3 s − ( + )2 + s2 q1 = {0; 0; 1}; q2 = ; 1; 0 ; +
(46)
q3 =
s+
( + )2 + s2 ; 1; 0 : +
(47)
As usually, the above {qm } vectors, upon normalization, generate an orthogonal matrix denoted [Q] that is used to get the diagonalized [B0;?kp (t) ] = [Q][B0; kp (t) ][Q]T matrix (in the {qm } basis) as 0 0 0 − 2 0 2 0 [( + ) 0 3 ( + ) : (48) [B0;?kp (t) ] = 2 + s2 )] ( + ) + s(s − 0 2 0 0 [( + ) 3 ( + ) 2 2 + s(s + ( + ) + s )] The tensor F0;?kp (t) = V0;?kp (t) ∈ Sym = (B0;?kp (t) )1=2 can now be obtained. Next, [F0; kp (t) ] = [Q]T [F0;?kp (t) ][Q]. It has an awkward algebraic expression and will not be given here since it is not critical for the stability argument.
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Second, incompressibility triggers that det[B0; kp (t) ] = 1. Using (45) we get that 0 is a solution of the following algebraic equation: (1 + s2 )3 3
− 0 + 2 = 0; ( + )4 0
0 = 2 :
(49)
For given ; ; s values, 0 can be found numerically. The second remark is that by working out the -order term, as given by Eq. (44), one gets the F1; kp (t) = F˜ kp (t) (x; t)e−t tensor. Indeed, by making use of explicit forms of Ekp (t) ; B0; kp (t) ; D=Dt; 0 operators, (44) can be re-written as ˜ 0; kp (t) + (LB ˜ 0; kp (t) )T ] ( − )(Akp (t) + AkTp (t) ) + v0 · grad x (Akp (t) + AkTp (t) ) − [LB − L0 (Akp (t) + AkTp (t) ) − (Akp (t) + AkTp (t) )L0T = −2 tr(F0;2 kp (t) Akp (t) )I;
(50)
where Akp (t) = F˜ kp (t) F0; kp (t) . This equation can (rst be solved componentwise for Akp (t) , and F˜ kp (t) =Akp (t) F0;−1kp (t) is obtainable right after. This step certainly involves lengthy algebra. However, one may be rather interested in particular solutions. For example, if it is further assumed that F˜ kp (t) ∈ Sym, since F0; kp (t) has been chosen so that ∈ Sym, then Akp (t) ∈ Sym as well. Hence, (50) gets the simpler form v0 · grad x (Akp (t) ) + ( − )Akp (t) − (L0 Akp (t) + Akp (t) L0T ) + tr(F0;2 kp (t) Akp (t) )I =
1 ˜ ˜ 0; kp (t) )T ]: [LB0; kp (t) + (LB 2
(51)
This is a linear, constant coeMcient (independent of x), (rst-order PDE in [Akp (t) ]mn ; m; n = 1; 3. From Ref. [35, Chapter 12], we observe that it has unique solution under certain restrictions (i.e. the sum of squared coeMcients be strictly positive) and for appropriate boundary conditions like the following (Dirichlet) ones: Akp (t) (x)|x∈@Bkp (t) = A0 (x) ∈ Sym;
(52)
where A0 (x) is a prescribed (known) value on the boundary. 3.3. Linearized form of the CE CE (10) can be expressed in condensed form as t S(t) = g(Gt (t )) dt ; −∞
(53)
where g = m(t − t )( =(( − 3) + IIC ))Ct (t ), Ct (t ) = GtT (t )Gt (t ). It is this function g that we shall linearize. Ft (t ) tensor. We recall that To do so, we (rst proceed by calculating the perturbed form of the Gt (t ) = Fk−1 p (t) Eq. (21) gives the explicit form of Ft (t; x; ). Since Ft (t ; x; ) = Ft−1 (t; x; ), we easily get from (21) that −t ˜ Ft (t ; x; ) = F0; t (t ) − F0; t (t )F(x)F + o(2 ): 0; t (t )e
(54)
However, the perturbed form of Fkp (t) =F0; kp (t) +F˜ kp e−t +o(2 ) has been obtained in the precedent section. Hence Fk−1 (t; x; ) = F0;−1kp (t) − F0;−1kp (t) F˜ kp (t) F0;−1kp (t) e−t + o(2 ): p (t)
(55)
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Now, using (54) and (55) above we can explicitate Gt (t ; x; ) as Gt (t ; x; ) = Fk−1 (t; x; )Ft (t ; x; ) p (t) ˜ 0; t (t )]e−t + o(2 ) = F0;−1kp (t) F0; t (t ) − [F0;−1kp (t) F˜ kp (t) F0;−1kp (t) F0; t (t ) + F0;−1kp (t) F0; t (t )FF ˜ −t + o(2 ): = G0; t (t ) + Ge
(56)
The linearization of the function g comes about from an asymptotic expansion limited to (rst-order in @ g(G0; t (t ) + G1; t (t )) g(Gt (t ; x; )) g(G0; t (t )) + @ =0
g(G0; t (t )) + DG0; t (t ) g(G0; t (t ))[G1; t (t )];
(57)
where the weak (directional) derivative is to be evaluated at G1; t (t ). Using the linearity of the weak derivative, i.e. Dg(G0; t (t ))[G1; t (t )] = m(t − t ){(Dh(IIC0; t (t )[G1; t (t )])C0; t (t ) + h(IIC0; t (t ) )(D(C0; t (t ))[G1; t (t )])}
(58)
and its properties (see [36] for details), we get, after some algebra, the following linearization for the extra stress part of the CE: t t ˜ − tr[C0; t (t )]tr[GT (t )G]) ˜ 2 (tr[C0; t (t )G0;T t (t )G] 0; t g(G0; t (t ) dt + m(t − t ) C0; t (t ) 2 ( − 3 + II ) C0 −∞ −∞ + (GT (t )G˜ + G˜ T G0; t (t )) dt e−t + o(2 ): (59) − 3 + IIC0 0; t 3.4. Momentum balance equation The momentum balance law (with ' = 1 for simplicity) is given by Dv = divx (T) Dt
(60)
and must now be expressed in perturbed form. From Eqs. (1), (19), (23) and (41) we get for 0 order grad x (p0 (x)) = −(v0 · grad x )v0 :
(61)
Using Eq. (29) we can easily see that the rhs vector (eld in (61) is not solenoidal, i.e. divx [(v0 · grad x )v0 ] = 3 2 =2. The problem of (nding a solution for PDE (61) can be handled in the following way: by taking the divergence of both members of (61) one gets a Poisson PDE for p0 (x) S3 p0 (x) = −
3 2 ; 2
p0 ∈ C2 (B(t)):
(62)
As it is known [37], its solution can be written as p0 = p0; h + p0; p ;
(63)
where p0; h is the solution of the corresponding Laplace (homogeneous) problem, and p0; p is a particular solution. In order to solve the Laplace problem one needs to make further assumptions regarding the p0 (x) behavior on a boundary. For instance, we might be led to solve an interior Dirichlet problem (IDP) like the
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following one: S3 p0; h (x) = 0; p0 (x)|x∈@B(t)⊂R3 = h(x) ∈ C0 (B(t));
h = given;
(64)
whose explicit solution (depending on the domain geometry) may be found using, for example, the Fourier– Bernoulli method. Of course other boundary conditions may be set up, depending on speci(c physical arguments invoqued. Next, p0; p can be expressed using the Newtonian potential, as dX1 dX2 dX3 1 3 2 p0; p (x) = ; (65) 3 B(t)⊂R 4( 2 x − X x;X∈B(t) where x ∼ (xl ), X ∼ (Xm ); l; m = 1; 3. We have thus obtained the solution p0 (x). Morever, it can be shown that it is well posed [37], in the sense that if h0 − h∗0 ¡ ;
∗ where h∗0 = p0; h |x∈@B(t) ;
h0 := max |h0 (x)|; x∈@B(t)
(66)
then ∗ p0; h − p0; h ¡
(67)
as well. We now turn our attention to the -order terms. From Eqs. (23), (41), (59) and (60) one gets after some algebra ˜ x)) − (v˜ · grad )v0 + v˜ − @v˜ − (v0 · grad )v˜ = r(t; x): ˜ = divx (S(t; (68) grad x p(x) x x @t Using Eqs. (29) and (35) the rhs term r(t; x) can be calculated. It obviously involves messy algebra and, since we are interested here in qualitative stability features, we shall not deal with the explicit form of the p(x) ˜ solution of (68) above. It can, however, be obtained using arguments similar to those invoqued for the 0 -order term solution. 3.5. Compatibility restrictions By taking (rst the divergence, then the gradient of both members of the momentum balance equation (60), we get Dv ; (69) grad x ⊗ grad x (p) = grad x ⊗ divx (S) − Dt from which we get that and
grad x ⊗ grad x (p) ∈ Sym
(70)
Dv ∈ Sym: grad x ⊗ divx (S) − Dt
(71)
As an aside we observe that (70) and (71) are fully equivalent with the following conditions curlx [grad x (p)] = 0; Dv = 0; curlx divx (S) − Dt
(72) (73)
respectively. Next, if (70) is true, then one must get that grad x ⊗ grad x [p0 (x) + p(t; ˜ x)e−t ] ∈ Sym;
(74)
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from which we (rst get that grad x ⊗grad x (p0 (x)) must ∈ Sym. Written otherwise, it gives (@2 p0 =@xm @x n )em ⊗ en ∈ Sym. If this is to happen, then necessarily @2 p0 =@xm @x n = @2 p0 =@x n @xm , and this last relationship holds true since p0 ∈ C2 (B(t)). Using a similar argument we get that grad x ⊗ grad x (p) ˜ ∈ Sym as well. Further on, from Eq. (59) one easily sees that the (rst term in (71) is always ∈ Sym, independent of the EL solution. The second term, grad x ⊗ Dv=Dt, can be explicitated using Eqs. (29), (35) and (41) as @v˜ ˜ ∈ Sym grad x ⊗ v˜ · grad x (v0 ) − v˜ + (75) + v0 · grad x (v) @t or in condensed form as grad x ⊗ w ∈ Sym. Hence, the oE-diagonal terms of grad x ⊗ w must be equal to each other. After some lengthy algebra, one sees that always @w1 =@x3 = @w3 =@x1 , @w2 =@x3 = @w3 =@x2 , but @w1 =@x2 = @w1 =@x2 only for shear free $ows, i.e. when s = 0. Stated otherwise, this means that for mixed (shear and extensional) $ows one must consider a perturbed motion whose components depend on both x1 ; x2 . Such a generalization, of course, will trigger even more tedious algebraic manipulations; however, the nature (i.e. their linearity) of the boundary value problems will not be aEected. 4. Discussion and conclusions The problem of (nding Hadamard stable and thermodynamically admissible CEs is not an easy issue (see [38–41] and references cited therein). However, signi(cant progress has been achieved. For example, Leonov [42] built up a thermodynamic framework that resulted in thermodynamically stable diEerential models. Recently, Rajagopal and his co-workers have accomplished a major break-through by providing a more general thermodynamic framework that produces both diEerential and integral models for viscoelastic $uids [41,43]. There is growing interest in integral models applicable to polymer melt $ow behavior, since molecular theories (mostly based on reptation concepts) result in this type of CEs. This suggests the possibility that integral CEs based on rational continuum mechanics may be linked to physical theories that take into account the polymer microstructure (chain topology and diEusion behavior, etc.). Like our previous work [33], this paper—which is exploratory in nature—has been motivated by the exceptional elongational $ow behavior of polylactides [44,45]. Indeed, in these experimental studies here mentioned a signi(cant increase of the elongational viscosity (of about two decades in magnitude) has been reported, and sample polydispersity (among others) has been invoked to possibly explain such behavior. Moreover, attempts to (t the experimental data based on a truncated K-BKZ law while using physically meaningful model parameters have failed. We have thus been led to search for new CEs that take into account the polymer morphology and could model its non-linear extensional $ow. Such CEs must not only be thermodynamically admissible but also Hadamard stable. In this work, we have addressed the Hadamard stability pattern of a model K-BKZ CE that is expressed in the conceptual framework set up by Rajagopal, that is to say of materials exhibiting an in(nity of natural con(gurations. In doing this, we have assumed that the right Cauchy–Green tensor originates from a thermodynamic evolution law derived by him and Srinivasa in Ref. [14]. Next, we inquired whether such CE is Hadamard stable or not with respect to -order perturbations about the rest state. We have found that the linearized CE is indeed Hadamard stable with respect to this perturbations, provided that the pressure (eld terms are the solutions of certain second-order boundary value problems. The Hadamard stability pattern of the initial non-linear CE has not been addressed in these preliminary studies. This and other aspects, like model features, comparison with experimental data and connection with polymer molecular morphology will be addressed in future work. Acknowledgements The author would like to express his deep gratitude to Prof. K.R. Rajagopal, Texas A& M University, College Station, for many enlighting discussions on continuum mechanics and thermodynamics that helped to
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signi(cantly improve this paper presentation. He also thanks Dr. Y. BWereaux, Ecole SupWerieure de Plasturgie, Oyonnax, for friendly computer assistance. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
G. Astarita, G. Marrucci, Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, Maidenhead, 1974. R.R. Huilgol, Continuum Mechanics of Viscoelastic Liquids, Hindustan Publishing, New Delhi, 1976. W.R. Schowalter, Mechanics of Non-Newtonian Fluids, Pergamon Press, Oxford, 1978. R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Wiley, New York, 1987. C.W. Macosko, Rheology: Principles, Measurements and Applications, Wiley, New York, 1994. R.R. Huilgol, N. Phan-Thien, Fluid Mechanics of Viscoelasticity, Elsevier, Amsterdam, 1997. A.J. McHugh, Flow-induced crystallization in polymers, in: J. Lyngaae-Jorgensen, K. Sondergaard (Eds.), Rheo-Physics of Multiphase Polymeric Systems, TECHNOMIC, Lancaster, PA, 1995. A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems, Oxford University Press, New York, 1994. A.K. Doufas, I.S. Dairanieh, A.J. McHugh, J. Rheol. 43 (1999) 85. A.K. Doufas, A.J. McHugh, J. Rheol. 45 (2001) 855. A.K. Doufas, A.J. McHugh, J. Rheol. 45 (2001) 1085. A.S. Wineman, K.R. Rajagopal, Arch. Mech. 42 (1990) 53. K.R. Rajagopal, A.S. Wineman, Int. J. Plasticity 8 (1992) 385. K.R. Rajagopal, A.R. Srinivasa, J. Non-Newtonian Fluid Mech. 88 (2000) 207. K.R. Rajagopal, A. Wineman, ARI: Interdisciplinary J. Phys. Eng. Sci. 51 (1998) 160. I.J. Rao, K.R. Rajagopal, Interfaces Free Boundaries 2 (2000) 73. I.J. Rao, K.R. Rajagopal, Int. J. Solids Struct. 38 (2001) 1149. K. Kannan, I.J. Rao, K.R. Rajagopal, J. Rheol. 46 (2002) 977. F. SidoroE, J. MWech. 14 (1975) 545. F. SidoroE, J. MWech. 14 (1975) 571. F. SidoroE, J. MWech. 15 (1976) 85. M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, Clarendon, 1986. R.G. Larson, Constitutive Equations for Polymer Melts and Solutions, Butterworths, Boston, 1988. M.H. Wagner, H. Bastian, P. Hachmann, J. Meissner, S. Kurzbeck, H. MYunstedt, F. Langouche, Rheol. Acta 39 (2000) 97. M.H. Wagner, P. Rubio, H. Bastian, J. Rheol. 45 (2001) 1387. R.L. Fosdick, K.R. Rajagopal, Proc. R. Soc. London A 339 (1980) 351. R.R. Huilgol, N. Phan-Thien, Int. J. Eng. Sci. 24 (1986) 161. A.I. Leonov, J. Non-Newtonian Fluid Mech. 42 (1992) 323. Y. Kwon, A.I. Leonov, J. Non-Newtonian Fluid Mech. 47 (1993) 77. Y. Kwon, A.I. Leonov, J. Non-Newtonian Fluid Mech. 58 (1995) 25. A.I. Leonov, Constitutive equations for viscoelastic liquids: formulation, analysis, and comparison with data, in: D.A. Siginer, D.D. De Kee, R.P. Chhabra (Eds.), Advances in the Flow and Rheology of Non-Newtonian Fluids, Elsevier, Amsterdam, 1999. L.I. Palade, P. AttanWe, R.R. Huilgol, B. Mena, Int. J. Engng. Sci. 37 (1999) 315. L.I. Palade, J.R. Walton, A. Farina, Int. J. Non-Linear Mech. 39 (2004) 379. J. Kevorkian, J.D. Cole, Multiple Scale and Singular Perturbation Methods, in: Applied Mathematical Sciences, Vol. 114, Springer, New York, 1996. H. Levine, Partial DiEerential Equations, in: S.-T. Yau (Ed.), AMS/IP Studies in Advanced Mathematics, Vol. 6, Series, Providence, RI, 1997. M.E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, New York, 1981. I. Stakgold, Boundary Value Problems of Mathematical Physics, in: SIAM Classics in Applied Mathematics, Vol. 29, SIAM, Philadelphia, 2000. I.M. Rutkevich, J. Appl. Math. Mech. (Russian) 36 (1972) 306. F. Dupret, J.M. Marchal, J. Non-Newtonian Fluid Mech. 20 (1986) 143. D.D. Joseph, Fluid Dynamics of Viscoelastic Liquids, in: Applied Mathematical Sciences, Vol. 84, Springer, New York, 1990. J.E. Dunn, K.R. Rajagopal, Int. J. Eng. Sci. 33 (1995) 689. A.I. Leonov, Rheol. Acta 15 (1976) 85. I.J. Rao, K.R. Rajagopal, Int. J. Non-Linear Mech., submitted. L.I. Palade, H.J. Lehermeier, J.R. Dorgan, Macromolecules 34 (2001) 1384. J.R. Dorgan, H.J. Lehermeier, L.I. Palade, J. Cicero, Macromol. Symp. 175 (2001) 55.
International Journal of Non-Linear Mechanics 39 (2004) 1275 – 1287
An integral constitutive law for viscoelastic $uids based on the concept of evolving natural con(gurations: stability analysis Liviu-Iulian Palade∗ Ecole Sup erieure de Plasturgie, B.P. 807, 01108 Oyonnax, Cedex, France Received 24 January 2003; received in revised form 18 August 2003; accepted 18 August 2003
Abstract A general conceptual framework has recently been developed within the context of large deformations for the study of materials (solids and $uids) that either undergo microstructural changes or for which stress states are related to the presence of several relaxation mechanisms. The corner stone of this new approach is that all materials exhibit an in(nity of stress-free natural con(gurations that are evolving in a thermodynamically admissible process. In this paper, that is exploratory in nature, we study the behavior of a non-linear integral viscoelastic constitutive equation (CE) for a $uid assumed to posses an in(nity of evolving natural con(gurations. The CE stability pattern with respect to small perturbations about the rest state is also addressed. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Natural con(guration framework; Stability
1. Introduction Polymer $uid $ows have received a lot of attention due to both wide spread industrial applications and because of challenging fundamental physical aspects [1–6]. Consequently, there is a large amount of experimental studies that have been published together with a certain number of constitutive equations (CE) based on continuum and/or statistical mechanics concepts intended to model the $ow behavior. Of particular interest is the $ow of viscoelastic liquids that undergo microstructural changes, such as $ow-induced crystallization (FIC) [7], polymerization and/or crosslinking during reactive injection molding, for which there is still a need on one hand for a deeper and more complete understanding of the physical phenomena and on the other hand for sound and trustworthy models. In order to deal with such a complex task, many authors have chosen the way of generalizing existing concepts. For example, McHugh used the Poisson bracket formalism developed by Beris and Edwards [8] to formulate a model that accounts for polymer FIC [9–11]. Another innovative approach has been pioneered by Rajagopal and his co-workers [12–14] and is based on the newly introduced concept of evolving natural con(gurations. Brie$y, the material response is elastic from these natural con(gurations and the rate of dissipation (which is maximized) determines how they evolve. Moreover, the maximization of the dissipation rate does not imply that the material reaches instantaneously ∗
Tel.: +33-474-81-9308; fax: +33-474-81-9301. E-mail address: [email protected] (L.-I. Palade).
0020-7462/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2003.08.006
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the natural con(guration, it only means it reaches it at the highest admissible rate. The viscoelastic response is determined by the stored energy related to the elastic response from the (stress free) natural con(gurations and the rate of dissipation due to viscous eEects. This approach has led to Oldroyd-type diEerential CEs and has been used very successfully in modelling FIC [15–18]. We point out here that while viscoelastic CEs involving intermediary (i.e. more than two) con(gurations have already been considered in the past [19–21], the thermodynamic framework set up by Rajagopal is new. In this work—which is exploratory in nature—we shall apply the Rajagopal’s ideas of natural con(gurations (in(nitely many) framework to generalize an integral non-linear CE that has the following form: t T(t) = −pI + S(t); S(t) = m(t − t )h(IIC )Ct (t ) dt ; (1) −∞
where Ct (t ) = FtT (t )Ft (t ) is the right Cauchy–Green tensor. The memory function, which can be obtained from either reptation theories [22] or from linear viscoelastic measurements, is given by [23] m(t) =
N Gn n=1
n
e−t= n ;
(2)
while for the damping function we choose the general form [23] ; h(IIC ) = ( − 3) + IIC
(3)
where IIC = (1=2)[(trC)2 − tr(C2 )] and ¿ 0 is a parameter. We point out here that authoritative molecular theories for polymer melts result in this type of integral CEs. For instance, Wagner’s model [4,23–25] which is a generalization of the de Gennes, Doi and Edwards theory [22], leads to a CE of the following form: t u ⊗ u dt ; S(t) = m(t ˙ − t )5f2 (4) |u |2 −∞ where in the above f = f(u ) is a molecular stress function that depends on the vector u = Eu, E being a suitable strain gradient tensor that acts on a unitary vector u of isotropic statistical distribution and whose end is located on a sphere of radius 1. Hence, there is a strong interest in studying this type of models within diEerent conceptual frameworks. This paper is organized as follows. We (rst present the generalization of the CE given by Eqs. (1), then we deal with the stability properties of its linearized form (i.e. the behavior with respect to small amplitude perturbations about the rest state).
2. The CE Let B(t), Bkp (t) and B(t ), where t; t ∈ R, be a current con(guration, the associated natural con(guration and a deformed con(guration at time t , respectively. All con(gurations are thought as diEerentiable manifolds. From Fig. 1 we observe that Ft (t ); Gt (t ) = Fk−1 p (t)
(5)
where the strain gradients are given by Ft (t ) = grad x(t) ⊗ (t );
Gt (t ) = grad (t) ⊗ (t );
Fkp (t) = grad x(t) ⊗ (t):
(6)
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Fig. 1. The con(gurations B(t1 ); : : : ; B(tn ); : : :, the associated natural con(gurations Bkp (t1 ) ; : : : ; Bkp (tn ) ; : : : and a deformed con(guration B(t ) are shown together with the corresponding strain gradient tensors.
Next, the left Cauchy–Green tensor Bkp (t) = Fkp (t) FkTp (t) is the solution of the following thermodynamic evolution equation that was derived when a generalization of the Maxwell model has been worked out in this new framework (for details see Ref. [14]): Bkp (t)
+ Bkp (t) = I:
(7)
Instead of deriving the evolution equation for Bkp (t) from a thermodynamic standpoint, we shall assume that ), the same evolution equation (7) holds for the model considered here, where =2Gg (T )=0 and =3=tr(Bk−1 p (t) 9 with Gg (T ) the temperature-dependent glassy modulus (whose value is ¿ 10 Pa), 0 the zero shear viscosity, and ( ) is Oldroyd’s upper convected derivative [2,6] D( ) ( ) = (8) − L( ) − ( )LT : Dt Such an assumption is similar in spirit to prescribing evolution equations for the plastic strain in theories for viscoelastic bodies. Next, since the $uid is assumed isotropic, one can pick a basis such that Fkp (t) = Vkp (t) , with Vkp (t) the left stretch tensor arising in the polar decomposition theorem. With this assumption, from Eq. (7) one obtains the strain gradient tensor Fkp (t) = (Bkp (t) )1=2 and further on the right Cauchy–Green tensor Ct (t ) by using the Eq. (5), that is F−1 Ft (t ): Ct (t ) = Gt (t )T Gt (t ) = FtT (t )Fk−T p (t) kp (t) Therefore, the CE to be studied in this work is given by the following set of equations: t S(t) = m(t − t )h(IIC )Ct (t ) dt ; −∞
h(IIC ) =
; ( − 3) + IIC
¿ 0;
(9)
(10) (11)
Ct (t ) = GtT (t )Gt (t );
(12)
Ft (t ); Gt (t ) = Fk−1 p (t)
(13)
Bkp (t)
(14)
+ Bkp (t) = I:
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It is important to note that Eq. (14) plays a crucial role in determining the stress tensor S(t) in (10) as Ct (t ) is determined by Fkp (t) as indicated in (9). 3. The stability of the rest state Hadamard stability and thermodynamics are related to each other, as it can be observed from previous studies [26–28]. In this section, we examine the stability of the rest state of the material as given by Eq. (14) using the linearized theory. In doing this, we assume the material contained in a suMciently smooth bounded volume has been set in motion at time t = 0 by an imposed (rst-order disturbance in the pressure, motion, strain and stress (elds, i.e. ˆ Fˆ kp (t) ; T)e ˆ i(k·x−ct) + o(2 ) (p; f(x); Fkp (t) ; T(t)) = (p0 ; f0 (x); F0; kp (t) ; T0 (t)) + (p; ˆ f; = (p0 ; f0 (x); F0; kp (t) ; T0 ) + (pˆ 1 ; fˆ 1 ; Fˆ 1; kp ; Tˆ 1 ) + o(2 );
(15)
where ¿ 0 is a small amplitude disturbance parameter, f(x) = f(x((t ); t)) is a prescribed (given) motion, (p0 ; f0 ; F0; kp (t) ; T0 ) are the zeroth-order terms each of which—but f—may depend only on t, while ˆ Fˆ kp (t) ; T) ˆ are the real (rst-order amplitudes of the corresponding disturbances, k(t) ∼ (k1 ; k2 ; k3 ) the (p; ˆ f; wave vector and c the wave velocity. Moreover, the asymptotic expansion (15) holds true if for each of its terms a limit of the following type does exist: A(x; t) − A0 (t) − A1 (x; t) lim = 0; (16) →0+ 2 where A may be either a scalar, or a vector or tensor (eld. If A is a tensor, i.e. A, then one can use A = tr 1=2 (AT A). Although other norms may be de(ned, since tensors are linear and continuous operators with domains and ranges subsets of (nite-dimensional vector spaces, all norms are equivalent. Hence, we assume that relationship (16) must hold irrespective of the norm used. The type of -order disturbance chosen here (see Eq. (15)) is diEerent from the one used by Kwon and Leonov [29–31] who have chosen perturbations with asymptotically large frequency and wave vector. Indeed, these former authors being interested in studying perturbations relevant for rapid $ows, have used the following disturbance instead: ( ) = ( )0 + (∧ )e
i(k·x−!t) 2
+ o(2 )
(17)
and have shown that some continuum mechanics models are Hadamard ill-posed with respect to this particular perturbation. Before proceeding further, we make the following remarks: • Any quantity A(x; t) that has asymptotically been expanded in 15 may be re-written equivalently as ˆ ik·x e−ict + o(2 ) A(x; t) = A0 (t) + Ae −t ˜ = A0 (t) + A(x)e + o(2 );
∈ C;
(18)
where A˜ = A˜ Re + iA˜ Im is generally a complex function. However, the particular case when A˜ is a real function—that is of interest for certain applications—may be considered. Moreover, in order to simplify notations (and to some extent calculations) we shall also make use, instead of (15), of Eq. (19) −t ˜ F˜ kp (t) ; T)(x)e ˜ (p; f(x); Fkp (t) ; T(t)) = (p0 ; f0 (x); F0; kp (t) ; T0 (t)) + (p; ˜ f; + o(2 )
= (p0 ; f0 (x); F0; kp (t) ; T0 ) + (p1 ; f1 ; F1; kp ; T1 ) + o(2 ):
(19)
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• We now remark that in many previously published studies on Hadamard stability of viscoelastic CEs the velocity (eld is assumed to be a problem-independent variable (see for example [31–33]). We shall, however, show in the next section that if the perturbed f(x; t; t ; ) motion (eld is known (as it is assumed in Eq. (15)) then the perturbed velocity (eld v(t; x; ) can be calculated from it. Incompressibility restrictions will also be addressed. Next, in order to perform stability calculations we need the following equations: the evolution law for the Fkp (t) tensor, the CE in linearized form, momentum balance, compatibility restrictions. We shall deal with each of them subsequently. 3.1. Perturbed velocity 6eld Let the motion x = f(x(); t) in perturbed form be given by −t ˜ f(x(); t; ) = f0 (x(); t) + f(x)e + o(2 ):
(20)
From (20) we get that Ft (t; x; ) = grad ⊗ f(x(); t) −t ˜ = grad ⊗ f0 (x(; t)) + (grad ⊗ f(x))e + o(2 ) −t ˜ = F0; t (t) + F(x)e + o(2 ):
(21)
Since incompressibility requires det[F0; t (t)] = 1, then necessarily det[Ft (t; x; )] = 1 as well. Hence, the determinant of the rhs of Eq. (21) must be equal to 1. Next, we can use the well-known relationship L0 (t) = grad x ⊗ v0 (t; x) = [(d=dt)F0; t (t)]F0;−1t (t) written in perturbed form d L(t; x; ) = Ft (t; x; ) Ft−1 (22) (t; x; ) dt to obtain v(t; x; ). To do so, assume the perturbed velocity (eld be given by ˜ x)e−t + o(2 ); v(t; x; ) = vQ0 (t) + v(t;
(23)
where vQ0 and v˜ are vector (elds to be further calculated. The velocity gradient L(t; x; ) can now be expressed as L(t; x; ) = grad x ⊗ v(t; x; ) ˜ x)]e−t + o(2 ) = grad x ⊗ vQ0 (t) + [grad x ⊗ v(t; ˜ x)e−t + o(2 ): = L0 (t) + L(t;
(24)
Next, from (21) and (22) we get that L(t; x; ) =
d −1 −t ˜ + o(2 ): [F0; t (t)]F0;−1t (t) − [I + L0 (t)]F(x)F 0; t (t)e dt
(25)
Now, from the rhs of Eqs. (24) and (25) after equating the corresponding zeroth- and (rst-order terms in we successively obtain the following results: L0 (t) = grad x ⊗ vQ0 =
d [F0; t (t)]F0;−1t (t) ⇒ vQ0 = v0 ; dt
(26)
where v0 is a given velocity (eld, and −1 ˜ x) = grad x ⊗ v(t; ˜ ˜ x) = −[I + L0 (t)]F(x)F L(t; 0; t (t);
(27)
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where F0;−1t (t) = F0; t (t ). Eq. (27) is a (rst-order PDE whose general solution provides the v˜ (eld components. However, we shall solve it for a particular motion that is of interest for certain engineering applications. Let a (zeroth-order in ) mixed shear and extensional strain (eld x = f((t ); t) be given by
x1 (t) = e−( =2)(t−t ) [1 (t ) + s(t − t )2 (t )];
x2 (t) = e−( =2)(t−t ) 2 (t );
x3 (t) = e (t−t ) 3 (t );
(28)
to which corresponds the following velocity (eld v0 (t; x) ∼ (v0; 1 ; v0; 2 ; v0; 3 ): v0; 1 (t) = sx2 (t) − x1 (t); v0; 2 (t) = − x2 (t); v0; 3 (t) = x3 (t) (29) 2 2 with divx (v0 (x)) = 0 and where s ¿ 0; ¿ 0 are, respectively, the constant shear and extensional rates. From (28) one obtains that [F0; t (t )]lm = @l (t )=@xm (t) is given by ( =2)(t−t ) −s(t − t )e( =2)(t−t ) 0 e ; det[F0; t (t )] = 1; [F0; t (t )]lm = (30) 0 0 e( =2)(t−t ) 0
0
e− (t−t
)
while from (29) we get [L0 (t)]lm = @vl =@xm as − s 0 2 [L0 (t)]lm = 0 0 ; tr[L0 (t)] = 0: − 2 0 0
(31)
We now turn our attention to -order terms, and assume perturbations of the form ˜ f(x()) = (x1 (1 ); x2 (2 ); 0): This corresponds to perturbations that propagate uniquely along the x1 - and x2 -axis. We further get 0 0 F˜ 11 (x1 ) ˜ ˜ F(x) = grad ⊗ f˜ = F˜ 22 (x2 ) 0 0 ; det[F(x)] = 0; 0 0 0
(32)
(33)
where in the above F˜ mm (xm ) = @xm =@m ; m = 1; 2. Since the F˜ mm (xm ) components are real-valued functions, this is a particular case of the general framework that has been described in Eq. (19). Next, from Eqs. (27), (30), (31) and (33) we get
− s e( =2)(t−t ) F˜ 11 (x1 ) −s − s (t − t )e( =2)(t−t ) F˜ 11 (x1 ) 0 2 2
( =2)(t−t ) ˜ ˜ x) = (34) grad x ⊗ v(t; F 22 (x2 ) 0: 0 −s e 2 0 0 0 Integrating (34) provides the v˜ components as
v˜1 = − s e( =2)(t−t ) F˜ 11 (x1 ) d x1 − s(t − t )x2 F˜ 11 (x1 ) + C1 (t); 2
− s e( =2)(t−t ) F˜ 22 (x2 ) d x2 + C2 (t); v˜3 = C3 (t); v˜2 = 2
(35)
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where Cl (t); l = 1; 3 are arbitrary time-depending functions. Hence, Eqs. (23), (29) and (35) give the v(t; x; ) velocity (eld. We are now ready to deal with incompressibility restrictions, i.e. divx (v) = 0. From Eq. (23) above we see ˜ = 0 as well. Using (35) we easily get that that divx (v) F˜ 11 (x1 ) − s(t − t )x2 F˜ 11 (x1 ) + F˜ 22 (x2 ) = 0;
(36)
which can be interpreted either way: if F˜ 11 (x1 ) is given, then it provides F˜ 22 (x2 ), and if F˜ 22 (x2 ) is assumed known, it is an ODE for F˜ 11 (x1 ). Hence, the two perturbations F˜ 11 (x1 ); F˜ 22 (x2 ) are actually related to each other because of incompressibility. If the $ow is shear free, i.e. s = 0 (in which case grad x ⊗ v˜ ∈ Sym), then incompressibility triggers F˜ 11 = −F˜ 22 . 3.2. Evolution law (EL) for the Fkp (t) tensor For convenience let us recall here the EL (7) Bkp (t)
+ Bkp (t) = I;
= 3=tr(Bk−1 ): p (t)
(37)
The above EL must now be expressed in perturbed form, and we shall deal with it in the following. From (19) we get Bkp (t) (t; x; ) = [Fkp (t) FkTp (t) ](t; x; ) = F0; kp (t) F0;T kp (t) + [F1; kp (t) F0;T kp (t) + (F1; kp (t) F0;T kp (t) )T ] + o(4 ) = B0; kp (t) + Ekp (t) + o(4 );
Ekp (t) ∈ Sym;
(38)
where Ekp (t) is a short-hand notation for the second term on the rhs. From (38) we immediately get that Bk−1 (t; x; ) = B0;−1kp (t) − B0;−1kp (t) Ekp (t) B0;−1kp (t) + o(2 ) p (t)
(39)
and further on, since tr(B0;−1kp (t) Ekp (t) B0;−1kp (t) ) = tr(B0;−2kp (t) Ekp (t) ), that
(t; x; ) = 0 − tr(B0;−2kp (t) Ekp (t) ) + tr(o(2 )):
(40)
We now turn our attention to the upper-convected term. From Eq. (8) we observe that we must (rst express the material derivative operator in perturbed form. Hence D D @ + v1 · grad x + o(2 ): (41) = + v0 · grad x + v1 · grad x + o(2 ) = Dt @t Dt ; 0 Next, by making use of (23) and (41) back into (37), the perturbed upper convected term can now be written as D T T Ek (t) −[L1 (t; x)B0; kp (t) +(L1 (t; x)B0; kp (t) ) ]−[L0 Ekp (t) +(L0 Ekp (t) ) ] ; Bkp (t) (t; x; ) B0;kp (t) + Dt ; 0 p (42) ˜ −t . Moreover, since diEerentiation with respect to x of o(2 ) and higher-order relations is not where L1 = Le always permissible [34], the order relations were dropped when calculations have been carried out, hence the use of the sign in (42). Next, making use of (38), (40) and (42) back into (37) and after equating the corresponding zeroth- and (rst-order terms in , we get for the zeroth order B0;kp (t)
+ B0; kp (t) = 0 I;
0 = 3=tr(B0;−1kp (t) )
(43)
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and for the (rst-order D Ek (t) − [L1 (t; x)B0; kp (t) + (L1 (t; x)B0; kp (t) )T ] − [L0 Ekp (t) + (L0 Ekp (t) )T ] + Ekp (t) Dt ; 0 p = − tr(B0;−2kp (t) Ekp (t) )I:
(44)
Several remarks are to be made at this level. First, one can use the zeroth-order equation (43) to calculate the Fkp (t) tensor in the way outlined in Section 2. Indeed, from (31) and (43) we get after some algebra that 2s2 s 0 0 + 1 0 + ( + )2 ( + )2 s 0 0 (45) [B0; kp (t) ] = : 0 2 ( + ) + 0 0 0 − 2 Next, since Fkp (t) =(Bkp (t) )1=2 , the next step involves calculations of the B0; kp (t) eigenvalues and corresponding eigenvectors so that B0; kp (t) can be written in diagonal form. Since this triggers tedious algebraic calculations, we have used the Mathematica v4.0 software formal calculation package, and we shall give here only the (nal results. Speci(cally, to the following eigenvalues qm ; m = 1; 3:
0 0 2 2 + s2 q1 = ( + ) ; ( + ) + s s − ; q2 = − 2 ( + )3 q3 =
0 ( + )2 + s s + ( + )2 + s2 3 ( + )
do correspond the (unnormed) eigenvectors qm ; m = 1; 3 s − ( + )2 + s2 q1 = {0; 0; 1}; q2 = ; 1; 0 ; +
(46)
q3 =
s+
( + )2 + s2 ; 1; 0 : +
(47)
As usually, the above {qm } vectors, upon normalization, generate an orthogonal matrix denoted [Q] that is used to get the diagonalized [B0;?kp (t) ] = [Q][B0; kp (t) ][Q]T matrix (in the {qm } basis) as 0 0 0 − 2 0 2 0 [( + ) 0 3 ( + ) : (48) [B0;?kp (t) ] = 2 + s2 )] ( + ) + s(s − 0 2 0 0 [( + ) 3 ( + ) 2 2 + s(s + ( + ) + s )] The tensor F0;?kp (t) = V0;?kp (t) ∈ Sym = (B0;?kp (t) )1=2 can now be obtained. Next, [F0; kp (t) ] = [Q]T [F0;?kp (t) ][Q]. It has an awkward algebraic expression and will not be given here since it is not critical for the stability argument.
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Second, incompressibility triggers that det[B0; kp (t) ] = 1. Using (45) we get that 0 is a solution of the following algebraic equation: (1 + s2 )3 3
− 0 + 2 = 0; ( + )4 0
0 = 2 :
(49)
For given ; ; s values, 0 can be found numerically. The second remark is that by working out the -order term, as given by Eq. (44), one gets the F1; kp (t) = F˜ kp (t) (x; t)e−t tensor. Indeed, by making use of explicit forms of Ekp (t) ; B0; kp (t) ; D=Dt; 0 operators, (44) can be re-written as ˜ 0; kp (t) + (LB ˜ 0; kp (t) )T ] ( − )(Akp (t) + AkTp (t) ) + v0 · grad x (Akp (t) + AkTp (t) ) − [LB − L0 (Akp (t) + AkTp (t) ) − (Akp (t) + AkTp (t) )L0T = −2 tr(F0;2 kp (t) Akp (t) )I;
(50)
where Akp (t) = F˜ kp (t) F0; kp (t) . This equation can (rst be solved componentwise for Akp (t) , and F˜ kp (t) =Akp (t) F0;−1kp (t) is obtainable right after. This step certainly involves lengthy algebra. However, one may be rather interested in particular solutions. For example, if it is further assumed that F˜ kp (t) ∈ Sym, since F0; kp (t) has been chosen so that ∈ Sym, then Akp (t) ∈ Sym as well. Hence, (50) gets the simpler form v0 · grad x (Akp (t) ) + ( − )Akp (t) − (L0 Akp (t) + Akp (t) L0T ) + tr(F0;2 kp (t) Akp (t) )I =
1 ˜ ˜ 0; kp (t) )T ]: [LB0; kp (t) + (LB 2
(51)
This is a linear, constant coeMcient (independent of x), (rst-order PDE in [Akp (t) ]mn ; m; n = 1; 3. From Ref. [35, Chapter 12], we observe that it has unique solution under certain restrictions (i.e. the sum of squared coeMcients be strictly positive) and for appropriate boundary conditions like the following (Dirichlet) ones: Akp (t) (x)|x∈@Bkp (t) = A0 (x) ∈ Sym;
(52)
where A0 (x) is a prescribed (known) value on the boundary. 3.3. Linearized form of the CE CE (10) can be expressed in condensed form as t S(t) = g(Gt (t )) dt ; −∞
(53)
where g = m(t − t )( =(( − 3) + IIC ))Ct (t ), Ct (t ) = GtT (t )Gt (t ). It is this function g that we shall linearize. Ft (t ) tensor. We recall that To do so, we (rst proceed by calculating the perturbed form of the Gt (t ) = Fk−1 p (t) Eq. (21) gives the explicit form of Ft (t; x; ). Since Ft (t ; x; ) = Ft−1 (t; x; ), we easily get from (21) that −t ˜ Ft (t ; x; ) = F0; t (t ) − F0; t (t )F(x)F + o(2 ): 0; t (t )e
(54)
However, the perturbed form of Fkp (t) =F0; kp (t) +F˜ kp e−t +o(2 ) has been obtained in the precedent section. Hence Fk−1 (t; x; ) = F0;−1kp (t) − F0;−1kp (t) F˜ kp (t) F0;−1kp (t) e−t + o(2 ): p (t)
(55)
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Now, using (54) and (55) above we can explicitate Gt (t ; x; ) as Gt (t ; x; ) = Fk−1 (t; x; )Ft (t ; x; ) p (t) ˜ 0; t (t )]e−t + o(2 ) = F0;−1kp (t) F0; t (t ) − [F0;−1kp (t) F˜ kp (t) F0;−1kp (t) F0; t (t ) + F0;−1kp (t) F0; t (t )FF ˜ −t + o(2 ): = G0; t (t ) + Ge
(56)
The linearization of the function g comes about from an asymptotic expansion limited to (rst-order in @ g(G0; t (t ) + G1; t (t )) g(Gt (t ; x; )) g(G0; t (t )) + @ =0
g(G0; t (t )) + DG0; t (t ) g(G0; t (t ))[G1; t (t )];
(57)
where the weak (directional) derivative is to be evaluated at G1; t (t ). Using the linearity of the weak derivative, i.e. Dg(G0; t (t ))[G1; t (t )] = m(t − t ){(Dh(IIC0; t (t )[G1; t (t )])C0; t (t ) + h(IIC0; t (t ) )(D(C0; t (t ))[G1; t (t )])}
(58)
and its properties (see [36] for details), we get, after some algebra, the following linearization for the extra stress part of the CE: t t ˜ − tr[C0; t (t )]tr[GT (t )G]) ˜ 2 (tr[C0; t (t )G0;T t (t )G] 0; t g(G0; t (t ) dt + m(t − t ) C0; t (t ) 2 ( − 3 + II ) C0 −∞ −∞ + (GT (t )G˜ + G˜ T G0; t (t )) dt e−t + o(2 ): (59) − 3 + IIC0 0; t 3.4. Momentum balance equation The momentum balance law (with ' = 1 for simplicity) is given by Dv = divx (T) Dt
(60)
and must now be expressed in perturbed form. From Eqs. (1), (19), (23) and (41) we get for 0 order grad x (p0 (x)) = −(v0 · grad x )v0 :
(61)
Using Eq. (29) we can easily see that the rhs vector (eld in (61) is not solenoidal, i.e. divx [(v0 · grad x )v0 ] = 3 2 =2. The problem of (nding a solution for PDE (61) can be handled in the following way: by taking the divergence of both members of (61) one gets a Poisson PDE for p0 (x) S3 p0 (x) = −
3 2 ; 2
p0 ∈ C2 (B(t)):
(62)
As it is known [37], its solution can be written as p0 = p0; h + p0; p ;
(63)
where p0; h is the solution of the corresponding Laplace (homogeneous) problem, and p0; p is a particular solution. In order to solve the Laplace problem one needs to make further assumptions regarding the p0 (x) behavior on a boundary. For instance, we might be led to solve an interior Dirichlet problem (IDP) like the
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following one: S3 p0; h (x) = 0; p0 (x)|x∈@B(t)⊂R3 = h(x) ∈ C0 (B(t));
h = given;
(64)
whose explicit solution (depending on the domain geometry) may be found using, for example, the Fourier– Bernoulli method. Of course other boundary conditions may be set up, depending on speci(c physical arguments invoqued. Next, p0; p can be expressed using the Newtonian potential, as dX1 dX2 dX3 1 3 2 p0; p (x) = ; (65) 3 B(t)⊂R 4( 2 x − X x;X∈B(t) where x ∼ (xl ), X ∼ (Xm ); l; m = 1; 3. We have thus obtained the solution p0 (x). Morever, it can be shown that it is well posed [37], in the sense that if h0 − h∗0 ¡ ;
∗ where h∗0 = p0; h |x∈@B(t) ;
h0 := max |h0 (x)|; x∈@B(t)
(66)
then ∗ p0; h − p0; h ¡
(67)
as well. We now turn our attention to the -order terms. From Eqs. (23), (41), (59) and (60) one gets after some algebra ˜ x)) − (v˜ · grad )v0 + v˜ − @v˜ − (v0 · grad )v˜ = r(t; x): ˜ = divx (S(t; (68) grad x p(x) x x @t Using Eqs. (29) and (35) the rhs term r(t; x) can be calculated. It obviously involves messy algebra and, since we are interested here in qualitative stability features, we shall not deal with the explicit form of the p(x) ˜ solution of (68) above. It can, however, be obtained using arguments similar to those invoqued for the 0 -order term solution. 3.5. Compatibility restrictions By taking (rst the divergence, then the gradient of both members of the momentum balance equation (60), we get Dv ; (69) grad x ⊗ grad x (p) = grad x ⊗ divx (S) − Dt from which we get that and
grad x ⊗ grad x (p) ∈ Sym
(70)
Dv ∈ Sym: grad x ⊗ divx (S) − Dt
(71)
As an aside we observe that (70) and (71) are fully equivalent with the following conditions curlx [grad x (p)] = 0; Dv = 0; curlx divx (S) − Dt
(72) (73)
respectively. Next, if (70) is true, then one must get that grad x ⊗ grad x [p0 (x) + p(t; ˜ x)e−t ] ∈ Sym;
(74)
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from which we (rst get that grad x ⊗grad x (p0 (x)) must ∈ Sym. Written otherwise, it gives (@2 p0 =@xm @x n )em ⊗ en ∈ Sym. If this is to happen, then necessarily @2 p0 =@xm @x n = @2 p0 =@x n @xm , and this last relationship holds true since p0 ∈ C2 (B(t)). Using a similar argument we get that grad x ⊗ grad x (p) ˜ ∈ Sym as well. Further on, from Eq. (59) one easily sees that the (rst term in (71) is always ∈ Sym, independent of the EL solution. The second term, grad x ⊗ Dv=Dt, can be explicitated using Eqs. (29), (35) and (41) as @v˜ ˜ ∈ Sym grad x ⊗ v˜ · grad x (v0 ) − v˜ + (75) + v0 · grad x (v) @t or in condensed form as grad x ⊗ w ∈ Sym. Hence, the oE-diagonal terms of grad x ⊗ w must be equal to each other. After some lengthy algebra, one sees that always @w1 =@x3 = @w3 =@x1 , @w2 =@x3 = @w3 =@x2 , but @w1 =@x2 = @w1 =@x2 only for shear free $ows, i.e. when s = 0. Stated otherwise, this means that for mixed (shear and extensional) $ows one must consider a perturbed motion whose components depend on both x1 ; x2 . Such a generalization, of course, will trigger even more tedious algebraic manipulations; however, the nature (i.e. their linearity) of the boundary value problems will not be aEected. 4. Discussion and conclusions The problem of (nding Hadamard stable and thermodynamically admissible CEs is not an easy issue (see [38–41] and references cited therein). However, signi(cant progress has been achieved. For example, Leonov [42] built up a thermodynamic framework that resulted in thermodynamically stable diEerential models. Recently, Rajagopal and his co-workers have accomplished a major break-through by providing a more general thermodynamic framework that produces both diEerential and integral models for viscoelastic $uids [41,43]. There is growing interest in integral models applicable to polymer melt $ow behavior, since molecular theories (mostly based on reptation concepts) result in this type of CEs. This suggests the possibility that integral CEs based on rational continuum mechanics may be linked to physical theories that take into account the polymer microstructure (chain topology and diEusion behavior, etc.). Like our previous work [33], this paper—which is exploratory in nature—has been motivated by the exceptional elongational $ow behavior of polylactides [44,45]. Indeed, in these experimental studies here mentioned a signi(cant increase of the elongational viscosity (of about two decades in magnitude) has been reported, and sample polydispersity (among others) has been invoked to possibly explain such behavior. Moreover, attempts to (t the experimental data based on a truncated K-BKZ law while using physically meaningful model parameters have failed. We have thus been led to search for new CEs that take into account the polymer morphology and could model its non-linear extensional $ow. Such CEs must not only be thermodynamically admissible but also Hadamard stable. In this work, we have addressed the Hadamard stability pattern of a model K-BKZ CE that is expressed in the conceptual framework set up by Rajagopal, that is to say of materials exhibiting an in(nity of natural con(gurations. In doing this, we have assumed that the right Cauchy–Green tensor originates from a thermodynamic evolution law derived by him and Srinivasa in Ref. [14]. Next, we inquired whether such CE is Hadamard stable or not with respect to -order perturbations about the rest state. We have found that the linearized CE is indeed Hadamard stable with respect to this perturbations, provided that the pressure (eld terms are the solutions of certain second-order boundary value problems. The Hadamard stability pattern of the initial non-linear CE has not been addressed in these preliminary studies. This and other aspects, like model features, comparison with experimental data and connection with polymer molecular morphology will be addressed in future work. Acknowledgements The author would like to express his deep gratitude to Prof. K.R. Rajagopal, Texas A& M University, College Station, for many enlighting discussions on continuum mechanics and thermodynamics that helped to
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signi(cantly improve this paper presentation. He also thanks Dr. Y. BWereaux, Ecole SupWerieure de Plasturgie, Oyonnax, for friendly computer assistance. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]
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