Viscoelastic Model Approach

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. To model this material behavior, viscoelasticity utilizes spring constants ( i and E2), dashpots (viscosity, v), and St. Venant sliders (a slider to account for nonrecoverable deformation) elements. The properties of these elements may be selected to cover a wide range of elastic and time-dependent viscous behavior. Viscoelastic models can be divided into both the number of elements employed and whether the elements are in a series or parallel arrangements. These elements may be linear or nonlinear and are combined as necessary for the model to describe the behavior of the sediment imder study. These models describe short-term behavior reasonably well, but tend to not yield reliable predictions of deformation for extended time periods. 

Viscoelastic models to simulate soil creep. Equations to the right of model govern dashport behavior, (a) Murayama and Shibata (1964) (b) Schifhnan (1959) (c) Christensen and Wu (1964) (d) Singh (1966). (From Hirst, T.J., The influence of compositional factors on the stress-slrain-time behavior of soils, PhD Thesis, University of Cahfomia, 

This short summary has shown that viscoelastic models are capable, for engineering purposes, of describing the time-deformation relationship of soils over a given time interval and a range of applied stresses. In general, the method does not offer a correct mathematical description of soil behavior over all times and all stresses. 

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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... 

It is appropriate at this time to introduce viscoelastic flow analysis. Fiber spinning is one of the few processes that can be analyzed using analytical viscoelastic models. Here, we follow the approach developed by Denn and Fisher [4], Neglecting inertia, we can start with the momentum balance by modifying eqn. (6.79) as,... 

X. L. Luo, A Control Volume Approach for Integral Viscoelastic Models and Its Application to Contraction Flow of Polymer Melts, J. Non-Newt. Fluid Mech., 64, 173-189 (1996). 

This assumption of a linear relationship between stress and strain appears to be good for small loads and deformations and allows for the formulation of linear viscoelastic models. There are also non-linear models, but that is an advanced topic that we won t discuss. There are two approaches we can take here. The first is to develop simple mathematical models that are capable of describing the structure of the data (so-called phenomenological models). We will spend some time on these as they provide considerable insight into viscoelastic behavior. Then there are physical theories that attempt to start with simple assumptions concerning the molecules and their interactions and... 

When plastics are unloaded, the creep strain is recoverable. This contrasts with metals, where creep strains are permanent. The Voigt linear viscoelastic model predicts that creep strains are 100% recoverable. The fractional recovered strain is defined as 1 — e/cmax, where e is the strain during recovery and Cmax is the strain at the end of the creep period. It exceeds 0.8 when the recovery time is equal to the creep time. Figure 7.9 shows that recovery is quicker for low Cmax and short creep times, i.e. when the creep approaches linear viscoelastic behaviour. 

To capture the onset of extrudate distortions which can be associated with melt flow instabilities in the die, several modelling approaches have been followed [4]. Two common hypotheses are forwarded and centre around the so-called constitutive and slip instability issues. The constitutive approach starts with the premise that, on the basis of some viscoelastic theory, the shear stress becomes a many-valued function of shear rate. As a consequence of this noiunonotone function, a melt flow instability and the associated distorted extrudate will develop. For many commercial polymers, the nonmonotone function could be considered as the sum total of many nonmonotone functions, each associated with a specific molecular weight fraction. The associated experimental apparent shear stress-apparent-shear rate curve could then become monotone, i.e. as in Figure 1(b), as is the case for PP. It should be noted that for viscoelastic materials, no direct linear relation exists between a constitutive shear stress-shear rate function and the experimental pressure (apparent shear stress)-flow rate (apparent shear rate) curve. 

Kumar B, Das A, Alagirusamy R. An approach to determine pressure profile generated by compression bandage using quasi-linear viscoelastic model. J Biomech Eng Trans Asme 2012 134(9). 

The blown film process is known to be difficult to operate, and a variety of instabilities have been observed on experimental and production film lines. We showed in the previous chapter (Figure 10.10) that even a simple viscoelastic model of film blowing can lead to multiple steady states that have very different bubble shapes for the same operating parameters. The dynamical response, both experimental and from blown film models, is even richer. The dynamics of solidification are undoubtedly an important factor in the transient response of the process, but the operating space exhibits a variety of response modes even with the conventional approach of fixing the location of solidification and requiring that the rate of change of the bubble radius vanish at that point. 

The previous sections give a brief review of some elementary concepts of solid mechanics which are often used to determine basic properties of most engineering materials. However, these approaches are sometimes not adequate and more advanced concepts from the theory of elasticity or the theory of plasticity are needed. Herein, a brief discussion is given of some of the more exact modeling approaches for linear elastic materials. Even these methods need to be modified for viscoelastic materials but this section will only give some of the basic elasticity concepts. 

They presented a theoretical approach to predict the behavior of silicone rubber under uniaxial stress. The model is based on the concept of the classical Maxwell treatment of viscoelasticity and stress relaxation behavior, and the Hookean spring component was replaced by an ideal elastomer component. From the test data, the substitution permits the new model estimation of the cross-link density of the silicone elastomer and allows a stress level to be predicted as a complex function of extension, cross-link density, absolute temperature, and relaxation time. Tock and co-workersh" ] found quite good agreementbetweenthe experimental behavior based on the new viscoelastic model. By using dynamic mechanical analysis (DMA), the authors would have been able to obtain similar information on the silicone elastomer. 

The approaches above can describe only steady state, time-independent viscosities. In Chapter 4 we will show that for time-dependent viscoelastic models, like Maxwell s, extensional thickening arises naturally. 

Increasing the number of interconnected spring and dashpot elements in building viscoelastic models will increase the degrees of freedom in fitting the models to experimental data. Generalized models based on an infinite number of single elements will match the continuum mechanics approach of solid- and fluid dynamics. 

A purely viscous non-Newtonian approach was followed by Han and Park (1975b). They used the power-law model and the energy equation, assuming that the effects of crystallization were insignificant. The agreement of this model with experimental data in terms of the bubble radius and thickness as a function of the axial distance for LDPE and HDPE was reported to be reasonable. In terms of viscoelastic models, Luo and Tanner (1985) considered the Leonov model, and Cain and Denn (1988) considered the upper convected Maxwell and Marrucci models in nonisothermal cases of film blowing. In some of the cases analyzed, multiple steady-state solutions were present (see also Problem 9C.2). 

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