More Accurate Constitutive Equations

As our second major topic, we present the simplest equations from each of the three important classes of constitutive equations, namely the differential equations from the retarded-motion expansion, the Maxwell-type differential equations, and the integral equations. Third and finally, we summarize the more accurate constitutive equations that we feel are the most promising for simply and realistically describing viscoelastic fluids and for modeling viscoelastic flows. More complete treatments of nonlinear constitutive equations are available elsewhere (Tanner, 1985 Bird et al., 1987 Larson, 1988 Joseph, 1990). Throughout this chapter, our examples are drawn from the literature on polymeric... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

The proposed mechanisms of models to explain the drag reduction phenomenon are based on either a molecular approach or fluid dynamical continuum considerations, but these models are mainly empirical or semi-empirical in nature. Models constructed from the equations of motion (or energy) and from the constitutive equations of the dilute polymer solutions are generally not suitable for use in engineering applications due to the difficulty of placing numerical values on all the parameters. In the absence of a more generally accurate model, semi-empirical ones remain the most useful for applications. 

In my ventures around the world, I have seen that Eq. (4.4) for VTWO constitutes good practice and a conservative approach. I also admit, however, that I believe a more accurate terminal velocity criterion may be derived, using and applying the equations given in this book. Lend-... 

Although this simplified version of Fourier s heat conduction law is well known to be an accurate constitutive model for many real gases, liquids, and solids, it is important to keep in mind that, in the absence of empirical data, it is no more than an educated guess, based on a series of assumptions about material behavior that one cannot guarantee ahead of time to be satisfied by any real material. This status is typical of all constitutive equations in continuum mechanics, except for the relatively few that have been derived by means of a molecular theory. 

The Chow equations, which constitute a large set that is too long and complex to reproduce here, are sometimes more accurate. Both of these sets of general-purpose equations (Halpin-Tsai and Chow) are applicable to many types of multiphase systems including composites, blends, immiscible block copolymers, and semicrystalline polymers. Their application to such systems requires the morphology to be described adequately and reasonable values to be available as input parameters for the relevant material properties of the individual phases. 

Master equations have been used to describe relaxation and kinetics of clusters. The first approaches were extremely approximate, and served primarily as proof-of-principle. ° Master equations had been used to describe relaxation in models of proteins somewhat earlier and continue to be used in that context. " More elaborate master-equation descriptions of cluster behavior have now appeared. These have focused on how accurate the rate coefficients must be in order that the master equation s solutions reproduce the results of molecular dynamics simulations and then on what constitutes a robust statistical sample of a large master equation system, again based on both agreement with molecular dynamics simulations and on the results of a full master equation.These are only indications now of how master equations may be used in the future as a way to describe and even control the behavior of clusters and nanoscale systems of great complexity. ... 

Finding and describing approximate solutions to the electronic Schrodinger equation has been a major preoccupation of quantum chemists since the birth of quantum mechanics. Except for the very simplest cases like H2, quantum chemists are faced with many-electron problems. Central to attempts at solving such problems, and central to this book, is the Hartree-Fock approximation. It has played an important role in elucidating modern chemistry. In addition, it usually constitutes the first step towards more accurate approximations. We are now in a position to consider some of the basic ideas which underlie this approximation. A detailed description of the Hartree-Fock method is given in Chapter 3. 

Accurate description of flow of the polymer melt through the die requires knowledge of the viscoelastic behavior of the polymer melt. The polymer melt can no longer be considered a purely viscous fluid because elastic effects in the die region can be significant. Unfortunately, there are no simple constitutive equations that adequately describe the flow behavior of polymer melt over a wide range of flow conditions. Thus, a simple die flow analysis is generally very approximate, while more accurate die flow analyses tend to be quite complicated. 

The Newtonian constitutive equation is the simplest equation we can use for viscous liquids. It (and the inviscid fluid, which has negligible viscosity) is the basis of all of fluid mechanics. When faced with a new liquid flow problem, we should try the Newtonian model first. Any other will be more difficult. In general, the Newtonian constitutive equation accurately describes the rheological behavior of low molecular weight liquids and even high polymers at very slow rates of deformation. However, as we saw in the introduction to this chapter (Figures 2.1.2 and 2.1.3) viscosity can be a strong function of the rate of deformation for polymeric liquids, emulsions, and concentrated suspensions. 

Factorization of the function concentrated polystyrene solution, time-strain factorability is not valid at short times after the imposition of the step shear. An accurate K-BKZ constitutive equation for shearing flows of this material will be much more complex than that for melt I. Furthermore, in strain histories in which a strain reversal takes place, such as constrained recoil (Wagner and Laun, 1978) or double-step strains with the second strain of sign opposite the first (Doi, 1980 Larson and Valesano, 1986), good agreement... 

Finally, more accurate differential and integral constitutive equations were presented, and their successes and failures in de-st bing experimental data, were discussed. No single nonlinear constitutive equation is best for all purposes, and thus one s choice of an appropriate constitutive equation must be guided by the problem at himd, the accuracy with which one wishes to solve the problem, and the effort one is willing to expend to solve it. Generally differential models of the Maxwell type are easier to implement numerically, and some are available in fluid mechanics codes. Also, some cmistitutive equations are better founded in molecular theory, as discussed in Chiqpter 11. 

In this paper, we have developed a true 3D numerical approach for the simulation of flow-induced residual stress. By solving the primitive variables Navier-Stoke equations and constitutive equation of viscoealstic behavior, we can correctly predict the first normal difference, the second normal difference and flow-induced residual Von Mises stress, which are essential to have accurate predicting physical properties of the finished parts, such as warpage behavior and optical properties. By the results of our numerical approach > we can expect to have more accurate analysis of warpage. 

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