Viscoelastic model, general

Linear viscoelastic models General linear x + k AxIAt — tjy ... 

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... 

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. 

Differential Viscoelastic Models. Differential models have traditionally been the choice for describing the viscoelastic behavior of polymers when simulating complex flow systems. Many differential viscoelastic models can be described by the general form... 

Integral viscoelastic models. Integral models with a memory function have been widely used to describe the viscoelastic behavior of polymers and to interpret their rheological measurements [37, 41, 43], In general one can write the single integral model as... 

For irrotational and small deformation flows, Equation 3.84 reduces to the general linear viscoelastic model ... 

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 modeling and control of movements in this chapter relates to external control of muscles via so-called functional electrical stimulation. Macroscopic viscoelastic models started from the observation that the process of electrical stimulation transforms the viscoelastic material from a compliant, fluent state into the stiff, viscous state. Levin and Wyman [35] proposed a three-element model— damped and undamped elastic element in series. Hill s work [36] demonstrated that the heat transfer depends upon the type of contraction (isometric, slow contracting, etc). The model includes the force generator, damping and elastic elements. Winters [37] generalized Hill s model in a simple enhancement of the original, which... 

This short summary has shown that viscoelastic models are capable, for engineering purposes, of describing the time-deformation relationship of soils over a given time interval and a range of applied stresses. In general, the method does not offer a correct mathematical description of soil behavior over all times and all stresses. 

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 linear viscoelastic model assumes that the stress at the ciurent time depends not only on the current strain, but on the past strains as well. It also assumes a linear superposition. Its general form reads... 

A related issue is that the modulus is a viscoelastic property, as evidenced by the temperature/strain-rate dependence, and that for most poljnners (at least those without a large beta transition near the alpha transition) time-temperature superposition of, for example, the shear relaxation modulus is valid (80). Further, G Sell and McKenna (81) have shown that the 5neld stress vs strain rate also seems to obey time-temperature superposition. Hence there is a correlation between the viscoelastic properties and the yield response of pol5uners, though one that is not generally stated explicitly. We note that some of the models mentioned previously, such as those of Caruthers group (41,42), Tervoort and co-workers (40), and Knauss and Emri (35), are (nonlinear) viscoelastic models that have yield arising due to the nonlinear response induced by the material clock (see Viscoelasticity). 

For the generalized viscoelastic model given in Problem 4.12, show that real and imaginary parts of the complex dynamic modulus G oi) are given by... 

All the codes listed provide a wide range of inelastic models, which adequately cover such liquids as dispersions and emulsions, but the kinds of hquids foimd in the polymer, personal-product, detergent, pharmaceutical and general chemicals industries might need viscoelastic models, and these presently are only provided by the POLYFLOW code. 

A general misbehavior of differential and integral viscoelastic models has been highlighted (Table 5). 

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 . 

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