Viscoelasticity general models

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Accordingly, given the necessity from equilibrium coil dimensions that bt> 1, the shear rate and frequency departures predicted by FENE dumbbells are displaced from each other. Moreover, the displacement increases with chain length. This is a clearly inconsistent with experimental behavior at all levels of concentration, including infinite dilution. Thus, finite extensibility must fail as a general model for the onset of nonlinear viscoelastic behavior in flexible polymer systems. It could, of course, become important in some situations, such as in elongational and shear flows at very high rates of deformation. 

The model of a viscoelastic body with one relaxation time used above has one principal disadvantage it does not describe the viscous flow of the reactants before the gel-point at t < t. Thus it is important to use a more general model of a viscoelastic medium to interpret the results obtained. The model must allow for flow and may be constructed by combining viscous and viscoelastic elements the former has viscosity rp and the latter has a relaxation modulus of elasticity Gp and viscosity rp,... 

In this general approach to viscoelasticity, appropriate models are constructed for the interpretation of the stress-strain-time behavior of a polymer. Then, values of Young s modulus G of the elastic elements and the viscosities i] of the viscous elements are used to characterize and predict the general behavior of the material. 

Viscoelasticity has already been introduced in Chapter 1, based on linear viscoelasticity. However, in polymer processing large deformations are imposed on the material, requiring the use of non-linear viscoelastic models. There are two types of general non-linear viscoelastic flow models the differential type and the integral type. 

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],... 

Fig. 19 Mechanical-viscoelastic model of Lin and Chen (1999) with two Maxwell models to describe SME in segmented PUs. (a) General model, (b) Change of the model in the shape-memory cycle, (c) Shape-memory behavior for two PU samples. Solid lines indicate the recoverable ration curves of the model. Taken from ref. [36], Copyright 1999. Reprinted with permission of John WUey Sons, Inc.

The viscoelastic BKZ model can be used to predict a sudden Jump in strain timewise or the appearance of spatially discontinuous strain (necking). The discussion here has involved the general nature of the material function which appears in the BKZ constitutive equations. Without being more specific about this function, the relatively simple analysis here can deal with how the inception of a... 

Viscoelastic creep When a plastic material is subjected to a constant stress, it undergoes a time-dependent increase in strain. This behavior is called creep. It is a plastic for which at long times of applied stress, such as in creep, a steady flow is eventually achieved. Thus in a generalized Maxwell model, all the dashpot viscosities must have finite values and in generalized models must have zero stiffness. 

The shear thinning behavior, as generally observed with polymer systems, is a typical nonlinear viscoelastic effect, so that by combining the Carreau-Yasuda and the Arrhenius equations a general model for the shear viscosity function can be written as follows ... 

The authors of Ref [36] offered to use for pol5Tners stress-strain (o-e) curves description the following general model of viscoelastic body, based on the fractional order derivatives, which has the appearance ... 

Abstract Phase separation in isotropic condensed matter has so far been believed to be classified into solid and fluid models. When there is a large difference in the characteristic rheological time between the components of a mixture, however, we need a model of phase separation, which we call viscoelastic model . This model is likely a general model that can describe all types of isotropic phase separation including solid and fluid model as special cases. We point out that this dynamic asymmetry between the components is quite common in complex fluids, one of whose components has large internal degrees of freedom. We also demonstrate that viscoelastic phase separation in such dynamically asymmetric mixtures can be characterized by the order-parameter switching phenomena. The primary order parameter switches from the... 

Here, we focus our attention on phase separation in complex fluids that are characterized by the large internal degrees of freedom. In all conventional theories of critical phenomena and phase separation, the same dynamics for the two components of a binary mixture, which we call dynamic symmetry between the components, has been implicitly assumed [1, 2]. However, this assumption is not always valid especially in complex fluids. Recently, we have found [3,4] that in mixtures having intrinsic dynamic asymmetry between its components (e.g. a polymer solution composed of long chain-like molecules and simple liquid molecules and a mixture composed of components whose glass-transition temperatures are quite different), critical concentration fluctuation is not necessarily only the slow mode of the system and, thus, we have to consider the interplay between critical dynamics and the slow dynamics of material itself In addition to a solid and a fluid model, we probably need a third general model for phase separation in condensed matter, which we call viscoelastic model . 

An example of creep deflection in a tensile bar for an epoxy at different temperatures is shown in Fig 5.12. It will be noticed that the creep response for a temperature of 155° C still has a positive slope after seven hours. Without knowing the type of material, one might expect the response to be that of a viscoelastic fluid. The creep response for 165° C and 170° C clearly have reached a limit and has the character of a thermoset. Because of the nature of the response, the epoxy could be best characterized by a viscoelastic fluid model such as the four-parameter fluid for both the 155° C and 160° C data. On the other hand, the epoxy could best be characterized by a viscoelastic solid model such as the three-parameter solid for temperatures above 160° C. To characterize the material over all time and temperature ranges would require a generalized model with a large number of elements. Methods to accomplish this will be discussed in subsequent sections. 

The proportionality constant between modulus and viscosity is known as the relaxation time, r,. For creep experiments, the proportionality constant is known as the retardation time. For the generalized models, there is a spectrum of relaxation or retardation times to account for the various viscoelastic processes occurring at different time scales in the material. 

It should be noted that (3.7.23) is essentially as general as an arbitrary discrete or continuous spectrum model, in the present context. This is because all quantities must be linear in the viscoelastic functions, so that the results for a more general model are simply sums of terms of the form that will now be derived. Explicit results can be obtained for the problem with friction, in terms of Whittaker functions. However, these will not be introduced in the present work. We refer to Golden (1979a, 1986a) for further details. In the frictionless case (from (3.7.11) we see that d,o = 0) ... 

Thus in all our discussions for simple shear we must realize that the models can be applied to other types of deformation. In the next section we develop a general model for linear viscoelastic behavior in only one dimension. Then we extend it to three dimensions, and in Section 3.3 we examine its behavior for different deformation histories stress relaxation, creep, and sinusoidal oscillation. 

Housiadas, K. and Tsamopoulos,). (2000) Unsteady extrusion of a viscoelastic aimular film — I. General model and its numerical solution. /. Non-Neivtonian Fluid Mech., 88, 229-259. 

Increasing the number of interconnected spring and dashpot elements in building viscoelastic models will increase the degrees of freedom in fitting the models to experimental data. Generalized models based on an infinite number of single elements will match the continuum mechanics approach of solid- and fluid dynamics. 

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11. 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

As already discussed, in general, polymer flow models consist of the equations of continuity, motion, constitutive and energy. The constitutive equation in generalized Newtonian models is incorporated into the equation of motion and only in the modelling of viscoelastic flows is a separate scheme for its solution reqixired. 

In general, the utilization of integral models requires more elaborate algorithms than the differential viscoelastic equations. Furthermore, models based on the differential constitutive equations can be more readily applied under general concUtions. 

As mentioned in Chapter 1, in general, the solution of the integral viscoelastic models should be based on Lagrangian frameworks. In certain types of flow... 

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