A N EXPLICIT CONSTITUTIVE EQUATION BASED ON AN INTEGRATED STRAIN HISTORY D. C .
BOGUE
Department of Chemical and Metallurgical Engineering, University of Tennessee, Knoxville, Tenn.
An explicit constitutive equation is developed by empirically modifying a second-order integral theory in such a way as to predict reasonable shear-dependent viscosities and normal stress functions. The model involves material constants GI, Gf.. with units of stress (elastic moduli), constants hJ2 with units of time (relaxation itimes), and two dimensionless constants. A form expressed in terms of a continuous relaxation spectrum is also presented. By reducing the general statement of the theory one is able to obtain
.
...
explicit results for ( 1 ) general smooth, near-viscometric flows expressible in terms of Rivlin-Ericksen tensors, (2) steady simple shearing flow, yielding shear-dependent viscosity and normal stress functions, (3) classical linear viscoelasticity including step and sinusoidal changes of strain, (4) recoil following cessation of steady shear, and (5) stress buildup (overshoot) following start of steady shear. Bingham plastic behavior results G2) truncations of the series for the viscosity and normal stress funcas a special case of 2. Two-element (GI, tions are fitted to experimental data for polyisobutylene solutions taken from the literature. The elastic moduli and relaxation times so obtained are reasonable when compared with relaxation spectrum data obtained independently.
K
1957 Green and Rivlin (70) formulated a general nonlinear
I theory for viscoelastic materials in the form of stress equal to a sum of integrals (more generally, a functional) involving the strain history and a memory function. Much of the recent work in continuum mechanics, in particular that of Coleman and Noll, is a generalizatllon of this basic concept. Of particular interest is Coleman and Noll’s simple fluid theory, which is the least general iiheory that incorporates the notions one normally associates with fluids (5) : The material flowsi.e., it can sustain a steady shearing motion indefinitely under an applied shearing stress-and it has no intrinsic preferred state. This latter notion does not preclude the development of orientation effects under shear, and thus the alignment of polymer molecules is a mechanism within the scope of simple fluid theory. For the purposes of the present work the starting point is the constitutive equation for a n incompressible fluid in the second-order theory of finite viscoelasticity (7) :
Jo Jo where J i j ( s ) = g k c bXk/bxi d X L / b x j , deformation history tensor; m ( J ) , b(s,, s b j , G(s,, $0) = unspecified decay functions; s, ,s, sb = backward-running time indices. The deformation history tensor, J i j ( s ) ,is a measure of the strain difference between the present state (at time t ) and the state a t some earlier time ( t - s). Index s (and also indices s, and sb) are backward-running times, s = 0 being now (time t ) and s = m being a n infinitely long time ago. x i are the coordinates of some point at time t and X iare the coordinates a t the earlier time, t - s. Basically the tensor, J i j ( s ) ,is a function of both the present time ( t ) and time in the past (s). To emphasize this double functionality the symbol Jij,(s) is employed by some authors (6). An understanding of Jij(s) is best obtained by observing how it is written down in the examples which follow.
The theory given in Equation 1 is “finite” in the sense that it is not limited to small deformations; it is “second-order” in the sense that it weights history to only some second-order extent-or, more precisely stated, in the sense that the motion is slow enough that a second-order memory is adequate. A mathematical definition of second-order involves consideration of the general functional relating stress to past strain: If the error in a second-degree polynomial approximation of the functional approaches zero faster than the second power of a certain scalar whi,ch is a measure of the “magnitude” of the deformation (the Hilbert-space norm), the theory is said to be second-order (7). This measure of the ’.magnitude” is a mathematically convenient but physically arbitrary quantity. Of more immediate concern is the fact that Equation 1 is the least general simple fluid theory with the following properties : I t reduces to classical linear viscoelasticity, and it predicts stress relaxation phenomena. It has the serious disadvantage, however, of predicting a constant (Sewtonian) viscosity. Clearly, then, some extension or modification of second-order theory is needed for fitting experimental data. A systematic third-order theory, however, is too general to be useful in the sense that one must introduce a triple integral with additional unspecified functions into Equation 1. Also it will yield a n (odd) polynomial relationship between stress and shear rate which is not convenient for fitting data except in limited ranges. One is thus led to consider an adjustment of Equation 1 or to different theory. For comparison consider the mechanistic continuum theory of Pao (72, 73). From his molecular theory Pao extracts the notion of following a fluid element in a rotating coordinate system and proceeds to develop a continuum theory independent of the molecular arguments. The qualitative idea of his theory can be stated by \vriting Stress
=
J7
d(strain)
(exponential decay function of time) dt‘
The time derivative is one which rotates with the fluid element. I t is similar to the Jaumann derivative [see Equation 1 ( 2 ) ] but differs from it in that it tracks the fluid rotation for a finite rather than an infinitesimal time period. VOL. 5
NO. 2
MAY 1966
253
Pao’s theory is very attractive in that there is a mechanistic argument behind it and in that it is easily related to molecular theory. The difficulties are the mathematical complexity (one is faced with simultaneous equations even in simple shear) and the ambiguity as to how one is to treat the time derivative in flow fields other than simple shear. But the qualitative notion is useful: The stress depends on the history of strain. past strains weighted with respect to how long ago they occurred and to how much rotation (shear) has taken place since they occurred. The Coleman-Koll type of formulation states that past strains are to be weighted only with respect to how long ago they occurred. Shear-dependent behavior enters through the high-order cross terms in the functional. I t is the proposition of the present work that the past strains be weighted with respect to both how long ago they occurred and also how much “working over” (expressed in terms of the second invariant) the fluid has received since they occurred. This dependence is introduced in an exponential manner. because the equation can then be treated analytically and it gives experimentally reasonable results. For contrast the proposed form is
where XE is an effective time constant, dependent on the material and on the deformation rate history. In essence what has been done is to preserve the form but not the rigor of the Coleman-Sol1 second-order theory and to adapt it empirically using qualitative concepts from the Pao theory. I t would be erroneous, however, to suggest that the concepts from Pao theory have been used directly: The Pao theory depends mechanistically on the rotation of fluid elements to motivate a shear-dependent fluid memory, whereas the proposed theory does not suggest any mechanism for this effect. The details are presented below. The theory presented there is similar to that in an earlier report (7), but with a change in the form of the decay functions.
The general siatement of the proposed theory is
where l / X E , = l/X, constant” (reciprocal) ;
+
uK,,(s),
an “effective
Gz
. . ., hl,
A? .
. .,a
time
= material constants
1
T o avoid the cumbersome notation ~ I I ~ ~the l ~positive z ~ , square root is understood throughout the discussion. The double integral term is needed only for special problems (notably, for the second normal stress difference, 7 2 2 - 733) and thus is presented separately as follows: Double integral term =
254
l&EC FUNDAMENTALS
where dij = ui(
= - d - s13
+ 1 A . J 2 ) s 2+ -
13
,
, , ,
.
(4)
+ ujli,deformation rate or first Rivlin-Ericksen
Rivlin-Ericksen tensor. In the present discussion, terms higher than those shown in Equation 4 are not used. These higher order terms are in fact identically zero for a wide class of important f l o ~ s :known as viscometric flows, which include such geometries as pipe flow, axial and/or rotating flow between concentric cylinders, etc. ( 5 ) . One must be in the appropriate coordinate system for the higher order terms to vanish. Thus if one tries to describe concentric cylinder flow using Cartesian coordinates, the higher order terms will occur. In the present discussion, then, one assumes that the flow is viscometric or “near-viscometric” and that one is in an appropriate coordinate system for the flow. Furthermore i t is assumed that the history is changing slowly enough so that one may write
K&)
Is,’( I I ~ ( ~ ) I ~ / z
= -
dt =
I I I ~ ( ~ = o)ll/z
(5)
The final equation needed is the continuity equation for incompressible fluids : ut]* = 0
(6)
(2)
(l/s) l l I I d ( t ) 1 1 / 2 dc, an average deformation rate; GI,
J 13, (s)
Kow Equations 4, 5, and 6 are to be substituted in Equations 2 and 3 and the results simplified. Certain high-order terms resulting from the double integral term art: to be neglected. The lowest order neglected term is
Statement and Application of Theory
double integral term
+
where 1 / A ’ R n = l / X n u’R,, ( ) ; u’ = material constant (u’ > > u ) ; (~(1,.sb) = unspecified function but of same magnitude as the decay function of first term. The factor 2 has been arbitrarily introduced before the time constants in order to give reasonable results for 7 2 2 - 7 3 3 . General Smooth, Near-Viscometric Flows: Rivlin-Ericksen Type Equations. For smooth flows one can write out the deformation history tensor as a series in the first and higher derivatives of the deformation rate ( 7 7) :
In viscometric flows the first term in the equation for ~~2 is of order K , whereas the above term is of order K3. Thus the above term is of vanishingly small order for small K. At high K it is also of small order, provided a’ >> a. There is little information to evaluate the validity of the assumed inequality u’ >> a, since u’ manifests itself only in the second normal stress difference (Equation 11) and a t high shear rates. Terms of order d . . A ( 2 ) ,, .(as above), of order A ( ? ) .. A ( 2 ) .. . and of higher order are, however, neglected in the present discussion. The combination of Equations 2 through 6 yields finally: 711
where
=
+ Tdij + TAij(” +
In order for b, to be a hydrostatic pressure in the absence of flow and Equation 7 to be internally consistent, one must have the following definition for p:
These results can be varied by introducing the time constant into Equation 3 in a different manner. One could also use the original theory (Equations 2 and 3) directly by writing out the kinematical description as follows :
Xi Equations 7 and 8 provide the viscoelastic extension of the Newtonian assumption. They can be combined with the force balance equation tcl provide the viscoelastic analog of the Navier-Stokes equation (see 2). Simple Shearing Flow. Consider now simple steady shearing flow in which the only velocity gradient is K = bvl/bx2 [see Figure 1 , a (2) 1. For this flow the Rivlin-Ericksen tensors
=
XI
x:!=
XZ
- KSX~
and J12
=
- K s , Jll= K2s2,other Jii
=
0
>>
One obtains Equations 9 and 12, provided a’ a. The form of Equations 9 and 12 predicts Bingham plastic behavior as a special case. Thus for a two-element model with XI X p and aKh2 1, one has
>>
