Nonlinear viscoelastic phenomena

Doi molecular theory adds a probability density function of molecular orientation to model rigid rodlike polymer molecules. This model is capable of describing the local molecular orientation distribution and nonlinear viscoelastic phenomena. Doi theory successfully predicts director tumbling in the linear regime and two sign changes in the first normal stress difference,as will be discussed later. However, because this theory assumes a uniform spatial structure, it is unable to describe textured LCPs. 

The classical viscoelastic properties are the dynamic shear moduli, written in the frequency domain as the storage modulus G ( y) and the loss modulus G a>), the shear stress relaxation function G t), and the shear-dependent viscosity j (k). Optical flow birefringence and analogous methods determine related solution properties. Nonlinear viscoelastic phenomena are treated briefly in Chapter 14. 

At some point, the constraints of time and space insist that the discussion be curtailed, so we here present a taxonomy of nonlinear viscoelastic phenomena, without the considerable quantitative analyses seen in prior chapters. The objective is to represent the range of observed phenomena and provide references that give entries into the literature. No effort has been made to give a thorough collection of published results. If Lord Rayleigh s critique - science is divided between quantitative measurement and stamp collecting - is invoked on this chapter, the stamps are indeed beautiful, but are likely to be more thoroughly quantitatively examined when readers become aware of their existence. 

The rubberlike liquid model is able to predict, qualitatively, certain nonlinear viscoelastic phenomena. In particular, some effects arising from the finite orientation of chain segments are predicted, for example a nonzero first normal stress difference. However, it fails to describe many other nonlinear effects. For example, it predicts that the viscosity is constant with shear rate and the second normal stress difference is zero. In fact, all its predictions for the shear stress in simple shear are the same as those of the Boltzmann superposition principle. We can gain some insight into the origins of nonlinearity by examining the features of the rubberlike liquid model that limit its predictive ability. 

Differential models obtained by replacing the ordinary time derivative in Eq. 10.21 by one that can describe large, rapid deformations are able to describe some nonlinear viscoelastic phenomena, but only qualitatively. To improve on such models, it is necessary to introduce additional nonlinearity into the equation. In the popular Phan-Thien/Tanner model, the Gordon-Schowalter convected derivative is used, and nonlinearity is introduced by multiplying the stress term by a function of the trace of the stress tensor. The Giesekus and Leonov models are other examples of nonlinear differential models. All of the models mentioned above are described in the monograph by Larson [7j. 

This section considers the behavior of polymeric liquids in steady, simple shear flows - the shear-rate dependence of viscosity and the development of differences in normal stress. Also considered in this section is an elastic-recoil phenomenon, called die swell, that is important in melt processing. These properties belong to the realm of nonlinear viscoelastic behavior. In contrast to linear viscoelasticity, neither strain nor strain rate is always small, Boltzmann superposition no longer applies, and, as illustrated in Fig. 3.16, the chains are displaced significantly from their equilibrium conformations. The large-scale organization of the chains (i.e. the physical structure of the liquid, so to speak) is altered by the flow. The effects of finite strain appear, much as they do when a polymer network is deformed appreciably. 

The example is an impressive demonstration both of a physical phenomenon that is driven by nonlinear viscoelasticity and of the use of numerical simulation. It is probably not the dominant mechanism for interface movement in multilayer systems, however, where viscosity and normal stress jumps across the interface are more important than the rather weak effect of the nonzero second normal stress difference. 

Viscoelasticity is a phenomenon observed in most of the polymers since they possess elastic and viscous characteristics when deformed. The properties such as creep, stress relaxation, mechanical damping, vibration absorption and hysteresis are included in viscoelasticity. If a material shows linear variation of strain upon the application of stress on it, its behavior is said to be linear viscoelastic. Elastomers and soft biological tissues undergo large deformations and exhibit time dependent stress strain behavior and are nonlinear viscoelastic materials. The non-linear viscoelastic properties of solid polymers are often based on creep and stress-... 

In the following several time dependent failure laws will be considered that can be used by the design engineer to make estimates of the probable time for rupture failure in uniaxial tensile tests. The section will conclude with a brief discussion of how to apply these approaches to more complicated structures. While the following methods have been developed primarily for a creep to rupture phenomenon, they can potentially be used for creep to yield as well or even possibly as a means of determining the demarcation between linear and nonlinear viscoelastic regimes. Some of the examples included are applied in this manner. 

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). 

Although it is a powerful means of investigating molecular structure and of basic characterization, and provides a general indication of the influence of M on flow behavior, the restrictions imposed by linear viscoelasticity make it inapplicable to a wide variety of practical problems. Nonlinearity is often associated with the phenomenon of shear thinning , that is, a reduction of the viscosity with shear rate in steady flow, characteristic of many polymer melts at intermediate shear rates [15]. This contrasts with the Newtonian behavior implied by the Boltzman superposition principle for steady flow [Eq. (54), in a liquid, G(s) must vanish as s coj. 

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