Generalized linear viscoelastic, fluid ,

Filbey equation (7). For cases of small deformation and deformation gradients, the general linear viscoelastic model can be used for unsteady motion of a viscoelastic fluid. Such a model has a memory function and a relaxation modulus. Bird and co-workers (6, 7) gave details of the available models. 

The most important limitation to the generalized Newtonian fluid is that memory effects (viscoelasticity) in the material are not predicted. In this sense, a step forward is the generalized linear viscoelastic flnid. In integral form, it can be written as follows ... 

These expressions are general for any viscoelastic fluid in the linear... 

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. 

In flow situations where the elastic properties play a role, viscoelastic fluid models are generally needed. Such models may be linear (e.g., Voigt, Maxwell) or nonlinear (e.g., Oldroyd). In general they are quite complex and will not be treated in this chapter. For further details, interested readers are referred to the textbooks by Bird et al. [6] and Barnes et al. [25],... 

Figure 13.3 Generalized response to oscillating shear for a fluid showing linear viscoelasticity. The shear stress r results from the imposed shear strain y, and is offset by S X can be resolved into the in-phase contribudon r and the out-of-pha contribution r"...

The term rheology dates back to 1929 (Tanner and Walters 1998) and is used to describe the mechanical response of materials. Polymeric materials generally show a more complex response than classical Newtonian fluids or linear viscoelastic bodies. Nevertheless, the kinematics and the conservation laws are the same for all bodies. The presentation here is condensed one may consult other books for amplification (Bird et al. 1987a Huilgol and Phan-Thien 1997 Tanner 2000). We begin with kinematics. 

The creep and creep recovery behavior of a four-parameter fluid is shown in Fig. 5.4 and is recognized as the response of a thermoplastic type polymer as given earlier in Fig. 3.13. The three stages of instantaneous elasticity, delayed elasticity and flow represents the most general type behavior possible for a linear viscoelastic material. Note Some texts do not include the flow term as a viscoelastic component, preferring instead... 

A few of these interrelationships can be derived from postulates that are general enough to be valid for most polymeric fluids. For example, the steady state shear viscosity and first normal stress coefficient at low shear rate can be derived from low frequency linear viscoelastic measurements ... 

This analytical solution review is tractable only for very limited assumptions, such as homogeneity and linearly elastic behavior (not to mention excluding variations that are time- or temperature-dependent). The first deviation that must be examined is the elastic linearity assumption for polishing pads. Polymers, in general, show behavior that lies between that of an elastic solid and a viscous fluid. The term viscoelastic has been applied to this behavior. 

General Regimes of Response. The nonlinear viscoelastic response of polymers, of course, follows some of the same classifications as does the linear response. Hence, the behavior above the glass temperature and into the terminal zone is fluid behavior, and often follows time-temperature superposition. The phenomenology of polymer melts and solutions is commonly described by... 

General Regimes of Response. The nonlinear viscoelastic response of polymers, of course, follows some of the same classifications as does the linear response. Hence, the behavior above the glass temperature and into the terminal zone is fluid behavior, and often follows time-temperature superposition. The phenomenology of polymer melts and solutions is commonly described by constitutive laws that relate the stress and strain histories to each other (59,69). A brief description of the K-BKZ model (70-72) is provided as it seems to capture most of the behaviors of polymer melts and solutions subjected to large deformations or high deformation rates. At the same time the nonlinear form of the reptation... 

In the previous chapter, it was pointed out how the chemical structure of polymers influences their mechanical properties. The glass transition temperature and the rate of cooling from the melt determine whether the polymer will be a hard, stiff material, or a soft flexible material. These qualitative differences can be quantifled by measuring standard mechanical properties. The student, generally, is introduced to the subject of mechanical behavior through the study of linear elastic metals that exhibit solid behavior at normal operating temperatures and conditions. However, this study of polymers reveals that polymers exhibit fluid as well as solid behavior and are viscoelastic and viscoplastic at room temperature. The mechanical properties of polymers are strain rate sensitive and highly temperature dependent. 

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