Simple Nonlinear Constitutive Equations

In the first place, the averaged model equations are highly nonlinear and require sophisticated numerical analysis for solution. For example, the attempt to obtain numerical solutions for motions of polymeric liquids, based upon simple continuum, constitutive equations, is still not entirely successful after more than 10 years of intensive effort by a number of research groups worldwide [27]. It is possible, and one may certainly hope, that model equations derived from a sound description of the underlying microscale physics will behave better mathematically and be easier to solve, but one should not underestimate the difficulty of obtaining numerical solutions in the absence of a clear qualitative understanding of the behavior of the materials. 

Analytical solutions (e.g., obtained by eigenfunction expansion, Fourier transform, similarity transform, perturbation methods, and the solution of ordinary differential equations for one-dimensional problems) to the conservation equations are of great interest, of course, but they can be obtained only under restricted conditions. When the equations can be rendered linear (e.g., when transport of the conserved quantities of interest is dominated by diffusion rather than convection) analytical solutions are often possible, provided the geometry of the domain and the boundary conditions are not too complicated. When the equations are nonlinear, analytical solutions are sometimes possible, again provided the boundary conditions and geometry are relatively simple. Even when the problem is dominated by diffusive transport and the geometry and boundary conditions are simple, nonlinear constitutive behavior can eliminate the possibility of analytical solution. 

From a numerical viev point, rapid progress has been made in the last few years in studies generally devoted to the entry flow problem, together with the use of more and more realistic constitutive equations for the fluids. Consequently, more complexity was involved for the munerical problem, in relation to the nonlinearity induced by the rheological model in the governing equations. The use of nonlinear constitutive models required approximate methods for solving the equations, such as finite element techniques, even for isothermal and steady-state conditions related to a simple flow geometry. 

Equations (3-32)-(3-34) are equivalent to the so-called Oldroyd-B equation. The Oldroyd-B equation is a simple, but qualitatively useful, constitutive equation for dilute solutions of macromolecules (see Section 3.6.2). Refinements to the simple elastic dumbbell model, such as the effects of the nonlinearity of the force-extension relationship at high extensions, are discussed in Section 3.6.2.2.I. 

A mathematical expression relating forces and deformation motions in a material is known as a constitutive equation. However, the establishment of constitutive equations can be a rather difficult task in most cases. For example, the dependence of both the viscosity and the memory effects of polymer melts and concentrated solutions on the shear rate renders it difficult to establish constitute equations, even in the cases of simple geometries. A rigorous treatment of the flow of these materials requires the use of fluid mechanics theories related to the nonlinear behavior of complex materials. However, in this chapter we aim only to emphasize important qualitative aspects of the flow of polymer melts and solutions that, conventionally interpreted, may explain the nonlinear behavior of polymers for some types of flows. Numerous books are available in which the reader will find rigorous approaches, and the corresponding references, to the subject matter discussed here (1-16). 

Many constitutive equations have been proposed in addition to those indicated above, which are special cases of fluids with memory. Most of these expressions arise from the generalization of linear viscoelasticity equations to nonlinear processes whenever they obey the material indifference principle. However, these generalizations are not unique, because there are many equations that reduce to the same linear equation. It should be noted that a determined choice among the possible generalizations may be suitable for certain types of fluids or special kinds of deformations. In any case, the use of relatively simple expressions is justified by the fact that they can predict, at least qualitatively, the behavior of complex fluids. 

Now we turn to the less complicated case of constitutive equations (3.129)— simple fluid with (nonlinear) viscosity and heat conduction in which an independence of the density gradient h was assumed from the start. By inspection of the results of... 

Other Constitutive Modei Descriptions. The above work describes a relatively simple way to think of nonlinear viscoelasticity, viz, as a sort of time-dependent elasticity. In solid polymers, it is important to consider compressibility issues that do not exist for the viscoelastic fluids discussed earlier. In this penultimate section of the article, other approaches to nonlinear viscoelasticity are discussed, hopefully not abandoning all simplicity. The development of nonlinear viscoelastic constitutive equations is a very sophisticated field that we will not even attempt to survey completely. One reason is that the most general constitutive equations that are of the multiple integral forms are cumbersome to use in practical applications. Also, the experimental task required to obtain the material parameters for the general constitutive models is fairly daunting. In addition, computationally, these can be difficult to handle, or are very CPU-time intensive. In the next sections, a class of single-integral nonlinear constitutive laws that are referred to as reduced time or material clock-type models is disscused. Where there has been some evaluation of the models, these are examined as well. 

The linear viscoelastic phenomena described in the preceding chapter are all interrelated. From a single quite simple constitutive equation, equation 7 of Chapter 1, it is possible to derive exact relations for calculating any one of the viscoelastic functions in shear from any other provided the latter is known over a sufficiently wide range of time or frequency. The relations for other types of linear deformation (bulk, simple extension, etc.) are analogous. Procedures for such calculations are summarized in this chapter, together with a few remarks about relations among nonlinear phenomena. 

Behavior in simple extension is, of course, a relatively elementary aspect of nonlinear viscoelasticity. Other types of deformation, and combinations of deformations, can provide additional information which must be described by appropriate constitutive equations and eventually interpreted on a molecular basis. Some investigations of time-dependent properties in pure-shear - and biaxial extension of thin, flat specimens - of soft rubbery polymers have been reported. 

In Chapter I we saw that nonlinear normal stresses can arise in simple shear (eq 1.5.15) and in torsion (Example 1.7.1) of an elastic solid. In this chapter our immediate goal is to develop constitutive equations that can predict normal stresses and other nonlinear phenomena in flowing viscoelastic liquids. The eventual goal of this effort is to use these equations to predict and control viscoelastic... 

To accomplish these goals systematically, a brief discussion of constitutive linearity and non-linearity is introduced in Section 2. In Section 3 the "stress softening" or "Mullins Effect" [2-8] is analyzed from a simple mechanical failure model of the composite material s microstructure and is seen to contain this type of nonlinear behavior. Also the p order Lebesgue norms [13,14] of the strain history are presented as being excellent memory measures of the strain history to use in the constitutive equations. In Section 4 the development is extended to include the general three-dimensional constitutive equation for isotropic materials. 

An argument to resolve the discrepancy between the failure envelopes obtained for different modes of straining is indicated by the work of Blatz, Sharda, and Tschoegl [42]. These authors have proposed a generalized strain energy function as constitutive equation of multiaxial deformation. They incorporated more of the nonlinear behavior in the constitutive relation between the strain energy density and the strain. They were then able to describe simultaneously by four material constants the stress-strain curves of natural rubber and of styrene-butadiene rubber in simple tension, simple compression or equibiaxial tension, pure shear, and simple shear. 

A few additional comments about when and under what conditions one must use a nonlinear viscoelastic constitutive equation are discussed here. At this time it seems that whenever the flow is unsteady in either a Lagrangian DvIDt 0) or a Eulerian (9v/9r 0) sense, then viscoelastic effects become important. In the former case one finds flows of this nature whenever inhomogeneous shear-free flows arise (e.g., flow through a contraction) and in the latter case in the startup of flows. However, even in simple flows, such as in capillaries or slit dies, viscoelastic effects can be important, especially if the residence time of the fluid in the die is less than the longest relaxation time of the fluid. Then factors such as stress overshoot could lead to an apparent viscosity that is higher than the steady-state viscosity. In line with these ideas one defines a dimensionless group referred to as the Deborah number ... 

Hooke s law, the direct proportionality between stress and strain in tension or shear, is often assumed such that the constitutive equations for a purely elastic solid are o = fjs for unidirectional extension and x = qy in simple shear flow. The latter expression is recognized from Chapter 7 as the constitutive relationship for a Newtonian fluid and, in analogy to Hooke s law for elastic solids, is sometimes termed Newton s law of viscosity. For cross-linked, amorphous polymers above 7, a nonlinear relationship can be derived theoretically. For such materials v = 0.5. When v is not 0.5, it is an indication that voids are forming in the sample or that crystallization is taking place. In either case, neither the theoretical equation nor Hooke s law generally applies. Before turning to one of the simplest mathematical models of viscoelasticity, it is important to recall that the constitutive equations of a purely viscous fluid are a = fj for elongational flow and x = qy for shear flow. 

A model consisting of the codeformational MaxweU constitutive equation coupled to a kinetic equation for breaking and re-formation of micelles is presented to reproduce most of the nonlinear viscoelastic properties of wormlike micelles. This simple model is also able to predict shear banding in steady shear and pipe flows as well as the long transients and oscillations that accompany this phenomenon. Even though the model requires six parameters, all of them can be evaluated from single and independent rheological experiments, and then they can be used to predict other flow situations. The predictions of our model are compared with experimental data for aqueous micellar solutions of cetyltrimethylammonium tosilate (CTAT). 

An exact derivation and implementation of the constitutive equations needed for this multimode model is not given here the interested reader is refeued to References 13 and 67. What is important to note is that the method just desctihed allows the determination of the relaxation spectrum from a simple set of uniaxial tensile or compression experiments at different strain rates and does not require lengthy relaxation experiments. The nonlinearity in stress is the same as used in the flow stress itself... 

Although there have been numerous numerical simulations of EHL contacts lubricated with non-Newtonian liquids, few have assumed for the lubricant shear response the ordinary shear-thinning that can be observed in rheometers. One of the authors has developed a numerical scheme [12] for the calculation of central film thickness that can utilize any generalized Newtonian fluid model for the nonlinear shear response and any general pressure-viscosity response. It is an extremely simple Grubin style inlet zone analysis that, instead of utilizing the Reynolds equation, integrates the constitutive relation across the thickness of the film. The pressure boundary condition that we use for the Hertz boundary has been criticized [13] however, the film thickness result is... 

---