Constitutive Tube model-based

Heinrich, G., Kaliske, M., 1997. Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity. CompuL Theor. Polym. Sci. 7 (3-4), 227-241. Heinrich, G., Straube, E., Helmis, G., 1988. Rubber Elasticity of Polymer Networks Theories Polymer Physics. Springer, Berlin/Heidelberg, pp. 33-87. 

Heinrich, G. and Kaliske, M. (1997) Theoretical and numerical formulation of a molecular based constitutive tube-model of mbber elasticity. Comput. Theo. Polym. Sci., 7, 227. 

Tube models capable of describing the essential features of nonlinear behavior are described in Chapter 11, which also introduces constitutive equations based on tube models. Such equations are of practical importance, as they aim to predict the way a melt behaves during industrial forming operations. 

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 Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties.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. 

A more detailed discussion of constitutive equations based on tube models is given in Chapter 11. 

The DE Constitutive Equations. The DE model (52-56) made a major breakthrough in polymer viscoelasticity in that it provided an important new molecular physics based constitutive relation (between the stress and the applied deformation history). This section outlines the DE approach that built on the reptation-tube model developed above and gave a nonlinear constitutive equation, which in one simplified form gives the K-BKZ equation (70,71). The model also inspired a significant amount of experimental work. One should begin by... 

Wagner et al. (63-66) have recently developed another family of reptation-based molecular theory constitutive equations, named molecular stress function (MSF) models, which are quite successful in closely accounting for all the start-up rheological functions in both shear and extensional flows (see Fig. 3.7). It is noteworthy that the latest MSF model (66) is capable of very good predictions for monodispersed, polydispersed and branched polymers. In their model, the reptation tube diameter is allowed not only to stretch, but also to reduce from its original value. The molecular stress function/(f), which is the ratio of the reduction to the original diameter and the MSF constitutive equation, is related to the Doi-Edwards reptation model integral-form equation as follows ... 

Constitutive equations can now be derived from the tube-based theories. Since the parameters in these equations are related to basic molecular parameters, these models should be truly predictive, but their ability to make quantitative predictions is limited because of the simplifying assumptions necessary to derive them. These equations and their strengths and weaknesses are presented in detail in Chapter 11. 

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