Continuous viscoelastic models

The continuous viscoelastic models discussed in this section make use of the theory of rubber elasticity and of the time-temperature superposition principle. 

The continuous chain model includes a description of the yielding phenomenon that occurs in the tensile curve of polymer fibres between a strain of 0.005 and 0.025 [ 1 ]. Up to the yield point the fibre extension is practically elastic. For larger strains, the extension is composed of an elastic, viscoelastic and plastic contribution. The yield of the tensile curve is explained by a simple yield mechanism based on Schmid s law for shear deformation of the domains. This law states that, for an anisotropic material, plastic deformation starts at a critical value of the resolved shear stress, ry =/g, along a slip plane. It has been... 

In a further development of the continuous chain model it has been shown that the viscoelastic and plastic behaviour, as manifested by the yielding phenomenon, creep and stress relaxation, can be satisfactorily described by the Eyring reduced time (ERT) model [10]. Creep in polymer fibres is brought about by the time-dependent shear deformation, resulting in a mutual displacement of adjacent chains [7-10]. As will be shown in Sect. 4, this process can be described by activated shear transitions with a distribution of activation energies. The ERT model will be used to derive the relationship that describes the strength of a polymer fibre as a function of the time and the temperature. 

The total contribution of the shear strain to the fibre strain is the sum of the purely or immediate elastic contribution involving the change in angle, A0e=0o- , occurring immediately upon loading of the fibre at f=0, and the time-dependent or viscoelastic and plastic contribution A0(f)=0(f)-0o [7-10]. According to the continuous chain model for the extension of polymer fibres, the time-dependent shear strain during creep can be written as... 

The phase angle changes with frequency and this is shown in Figure 4.7. As the frequency increases the sample becomes more elastic. Thus the phase difference between the stress and the strain reduces. There is an important feature that we can obtain from the dynamic response of a viscoelastic model and that is the dynamic viscosity. In oscillatory flow there is an analogue to the viscosity measured in continuous shear flow. We can illustrate this by considering the relationship between the stress and the strain. This defines the complex modulus ... 

To make MCT calculations simpler, the solvent dynamic quantities are sometimes modeled in such a way that the self-consistency is avoided. This is where the viscoelastic models (VEMs) play an important role. The VEM is usually expressed as a Mori continued fraction where the frequencies are de-... 

In theory, each mechanism of mobility corresponds to the proper simplified solvable dynamic models of the polymer chain. They describe continuous viscoelastic or discrete rotameric mechanisms of internal mobfl-... 

The problem will be solved for the case where the viscoelastic half-plane is characterized by a discrete spectrum model (Sect. 1.6). The more general continuous spectrum model is discussed by Golden (1977). The proportionality assumption (Sect. 1.9) will be adopted for the material so that a unique Poisson s ratio exists. Therefore, from (1.6.25, 28, 29), (3.5.20) and (3.5.22), we have... 

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) ... 

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. 

Equations of continuity and motion in a flow model are intrinsically connected and their solution should be described simultaneously. Solution of the energy and viscoelastic constitutive equations can be considered independently. 

Solution of the flow equations has been based on the application of the implicit 0 time-stepping/continuous penalty scheme (Chapter 4, Section 5) at a separate step from the constitutive equation. The constitutive model used in this example has been the Phan-Thien/Tanner equation for viscoelastic fluids given as Equation (1.27) in Chapter 1. Details of the finite element solution of this equation are published elsewhere and not repeated here (Hou and Nassehi, 2001). The predicted normal stress profiles along the line AB (see Figure 5.12) at five successive time steps are. shown in Figure 5.13. The predicted pattern is expected to be repeated throughout the entire domain. 

Creep modeling A stress-strain diagram is a significant source of data for a material. In metals, for example, most of the needed data for mechanical property considerations are obtained from a stress-strain diagram. In plastic, however, the viscoelasticity causes an initial deformation at a specific load and temperature and is followed by a continuous increase in strain under identical test conditions until the product is either dimensionally out of tolerance or fails in rupture as a result of excessive deformation. This type of an occurrence can be explained with the aid of the Maxwell model shown in Fig. 2-24. 

For a viscoelastic solid the situation is more complex because the solid component will never flow. As the strain is applied with time the stress will increase continually with time. The sample will show no plateau viscosity, although there may be a low shear viscous contribution. This applies to both a single Maxwell model and one with a spectrum of processes ... 

An important and sometimes overlooked feature of all linear viscoelastic liquids that follow a Maxwell response is that they exhibit anti-thixo-tropic behaviour. That is if a constant shear rate is applied to a material that behaves as a Maxwell model the viscosity increases with time up to a constant value. We have seen in the previous examples that as the shear rate is applied the stress progressively increases to a maximum value. The approach we should adopt is to use the Boltzmann Superposition Principle. Initially we apply a continuous shear rate until a steady state... 

There is a wealth of microstructural models used for describing nonlinear viscoelastic responses. Many of these relate the rheological properties to the interparticle forces and the bulk of these consider the action of continuous shear rate or stress. We will begin with a consideration of the simplest form of potential, a hard rigid sphere. 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

The presence of a low-viscosity interfacial layer makes the determination of the boundary condition even more difficult because the location of a slip plane becomes blurred. Transitional layers have been discussed in the previous section, but this is an approximate picture, since it stiU requires the definition of boundary conditions between the interfacial layers. A more accurate picture, at least from a mesoscopic standpoint, would include a continuous gradient of material properties, in the form of a viscoelastic transition from the sohd surface to the purely viscous liquid. Due to limitations of time and space, models of transitional gradient layers will be left for a future article. 

The non-isothermal viscoelastic cell model was used to study foam growth in the continuous extrusion of low density foam sheet. Surface escape of blowing agent was successfully incorporated to describe the foaming efficiency. Reasonable agreement was obtained with experimental data for HCFC-22 blown LDPE foam in the sub-centimetre thickness domain. 11 refs. 

In equation (6.33), the stresses in the moving viscoelastic liquid (6.31) are added to the stresses in the continuum of Brownian particles. When the equations of motion are formulated, we have to take into account the presence of the two interacting and interpenetrating continuous media formed by the viscoelastic liquid carrier and the interacting Brownian particles that model the macromolecules. However, the contribution of the carrier in the case of a concentrated solution is slight, and we shall ignore it henceforth. 

A plastic material is defined as one that does not undergo a permanent deformation until a certain yield stress has been exceeded. A perfectly plastic body showing no elasticity would have the stress-strain behavior depicted in Figure 8-15. Under influence of a small stress, no deformation occurs when the stress is increased, the material will suddenly start to flow at applied stress a(t (the yield stress). The material will then continue to flow at the same stress until this is removed the material retains its total deformation. In reality, few bodies are perfectly plastic rather, they are plasto-elastic or plasto-viscoelastic. The mechanical model used to represent a plastic body, also called a St. Venant body, is a friction element. The... 

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