Simple Constitutive Equations

SIMPLE CONSTITUTIVE EQUATIONS 1.2.1 Linear and Non-linear Behaviour... 

H. Giesekus, A Simple Constitutive Equation for Polymer Fluids Based on the Concept of Deformation-dependent Tensorial Mobility, J. Non-Newt. Fluid Mech., 11, 60-109 (1982). 

Leonov AI (1992) Analysis of simple constitutive equations for viscoelastic liquids. J Non-Newton Fluid Mech 42(3) 323-350... 

S.A.Khan, R.G.Larson, Comparison of simple constitutive equations for polymer melts in shear and biaxial and uniaxial extensions, J. Rheol. 21 (1987), 207-234. 

Giesekus, H., A simple constitutive equation for polymer fluids based on the concept of deformation dependent tensorial mobility, J. Non-Newtonian Fluid Mech., 11, 69-109 (1982). 

In the special case of step strain, one can solve the set of equations (7.246)-(7.250) rigorously and obtain the results given in eqn (7.122). In the general case, the solution of the equation needs numerical calculation. It turns out that the difference between eqns (7.194) and (7.246) is not large for the usual flow history discussed in Section 7.6. For such flows, the simple constitutive equation will be useful. 

Accurate description of flow of the polymer melt through the die requires knowledge of the viscoelastic behavior of the polymer melt. The polymer melt can no longer be considered a purely viscous fluid because elastic effects in the die region can be significant. Unfortunately, there are no simple constitutive equations that adequately describe the flow behavior of polymer melt over a wide range of flow conditions. Thus, a simple die flow analysis is generally very approximate, while more accurate die flow analyses tend to be quite complicated. 

Interface stability in co-extrusion has been the subject of extensive analysis. There is an elastic driving force for encapsulation caused by the second normal stress difference (56), but this is probably not an important mechanism in most coprocessing instabilities. Linear growth of interfacial disturbances followed by dramatic breaking wave patterns is observed experimentally. Interfacial instabilities in creeping multilayer flows have been studied for several simple constitutive equations (57-59). Instability modes can be traced to differences in viscosity and normal stresses across the interface, and relative layer thickness is important. 

This familiar equation is more usually represented as a consequence of the molecular theories of a rubber network. Here we see that it follows from purely phenomenological considerations as a simple constitutive equation for the finite deformation of an isotropic, incompressible solid. Materials that obey this relationship are sometimes called neo-Hookean. 

As mentioned in Sections 1 and 3, a rheological constitutive equation relates the components of the three-dimensional extra-stress (matrix) to the components of the strain (matrix) or the rate of strain (matrix) in any given flow field. Extensional viscosity data of the kind shown in Figure 5 should therefore be explainable on the basis of a proper constitutive equation. Unfortunately, there is no simple constitutive equation that accurately predicts the behavior of a polymer melt in all the commonly encountered flow situations. Some equations do a remarkably good job... 

The linear viscoelastic phenomena described in the preceding chapter are all interrelated. From a single quite simple constitutive equation, equation 7 of Chapter 1, it is possible to derive exact relations for calculating any one of the viscoelastic functions in shear from any other provided the latter is known over a sufficiently wide range of time or frequency. The relations for other types of linear deformation (bulk, simple extension, etc.) are analogous. Procedures for such calculations are summarized in this chapter, together with a few remarks about relations among nonlinear phenomena. 

Considering that On is also fitted by an Eyring equation (3) as demonstrated by Nitta et al. (Nitta Takayanagi, 2006), we have the following simple constitutive equation ... 

Apply simple constitutive equations (especially for indexers)... 

The primary purposes of this chapter were first to introduce some representative phenomenological constitutive equations used to describe the viscoelasticity of flexible homogeneous polymeric liquids, and then to show how such constitutive equations may be used to describe relatively simple flow problems (e.g., steady-state shear flow and steady-state elongational flow). For such purposes, in this chapter we have presented only relatively simple constitutive equations. Owing to the space limitations here we have not presented other more complicated constitutive equations, which have been dealt with in the monographs by Bird et al. (1987) and Larson (1988). 

Likhtman, A. E., Graham, R. S., McLeish, T. C. B. How to get simple constitutive equations for polymer melts from molecular theory. Proc. 6th Eur. Cong. RheoL (2002), pp. 259-260... 

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