Viscoelasticity models

A computer program was written to perform all the calculation. It is found that the three-element viscoelastic model provides reasonable estimation of the behavior of the polyvinyl chloride material during the impact... 

In the following section representative examples of the development of finite element schemes for most commonly used differential and integral viscoelastic models are described. 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

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... 

In viscoelastic models in addition to the described conditions, stresses at the inlet should be given. As already mentioned there is no universal method to define such conditions, however, the following options may be considered (Tanner, 2000) ... 

Simulation of the Couette flow of silicon rubber - viscoelastic model... 

The dry-processed, peel-apart system (Fig. 8b) used for negative surprint apphcations (39,44) is analogous to the peel-apart system described for the oveday proofing apphcation (see Fig. 7) except that the photopolymer layer does not contain added colorant. The same steps ate requited to produce the image. The peel-apart system rehes on the adhesion balance that results after each exposure and coversheet removal of the sequentially laminated layer. Each peel step is followed by the apphcation of the appropriate process-colored toners on a tacky adhesive to produce the image from the negative separations. The mechanism of the peel-apart process has been described in a viscoelastic model (45—51) and is shown in Figure 8c. 

Recovery is the strain response that occurs upon the removal of a stress or strain. The mechanics of the recovery process are illustrated in Fig. 2-34, using an idealized viscoelastic model. The extent of recovery is a function of the load s duration and time after load or strain release. In the example of recovery behavior shown in Fig. 2-34 for a polycarbonate at 23°C (73°F), samples were held under sustained stress for 1,000 hours, and then the stress was removed for the same amount of time. The creep and recovery strain measured for the duration of the test provided several significant points. 

Similar instability is caused by the electrostatic attraction due to the applied voltage [56]. Subsequently the hydrodynamic approach was extended to viscoelastic films apparently designed to imitate membranes (see Refs. 58-60, and references therein). A number of studies [58, 61-64] concluded that the SQM could be unstable in such models at small voltages with low associated thinning, consistent with the experimental results. However, as has been shown [60, 65-67], the viscoelastic models leading to instability of the SQM did not account for the elastic force normal to the membrane plane which opposes thickness... 

The viscoelastic effects on the morphology and dynamics of microphase separation of diblock copolymers was simulated by Huo et al. [ 126] based on Tanaka s viscoelastic model [127] in the presence and absence of additional thermal noise. Their results indicate that for

bulk modulus of both blocks, the area fraction of the A-rich phase remains constant during the microphase separation process. For each block randomly oriented lamellae are preferred. 

The above model assumes that both components are dynamically symmetric, that they have same viscosities and densities, and that the deformations of the phase matrix is much slower than the internal rheological time [164], However, for a large class of systems, such as polymer solutions, colloidal suspension, and so on, these assumptions are not valid. To describe the phase separation in dynamically asymmetric mixtures, the model should treat the motion of each component separately ( two-fluid models [98]). Let Vi (r, t) and v2(r, t) be the velocities of components 1 and 2, respectively. Then, the basic equations for a viscoelastic model are [164—166]... 

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 ... 

One feature of the Maxwell model is that it allows the complete relaxation of any applied strain, i.e. we do not observe any energy stored in the sample, and all the energy stored in the springs is dissipated in flow. Such a material is termed a viscoelastic fluid or viscoelastic liquid. However, it is feasible for a material to show an apparent yield stress at low shear rates or stresses (Section 6.2). We can think of this as an elastic response at low stresses or strains regardless of the application time (over all practical timescales). We can only obtain such a response by removing one of the dashpots from the viscoelastic model in Figure 4.8. When a... 

The application of finite strains and stresses leads to a very wide range of responses. We have seen in Chapters 4 and 5 well-developed linear viscoelastic models, which were particularly important in the area of colloids and polymers, where unifying features are readily achievable in a manner not available to atomic fluids or solids. In Chapter 1 we introduced the Peclet number ... 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

Patterson GK, Zakin JL (1968) Prediction of drag reduction with a viscoelastic modell AIChE J 14 434... 

If data is needed for the more sophisticated viscoelastic models now being introduced then results from forced vibration dynamic tests (Chapter 9) or stress relaxation tests (Chapter 10), as appropriate, would be used. 

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