Four-parameter model, deformation

The four-parameter model provides a crude quahtative representation of the phenomena generally observed with viscoelastie materials instantaneous elastie strain, retarded elastic strain, viscous flow, instantaneous elastie reeovery, retarded elastie reeovery, and plastic deformation (permanent set). Also, the model parameters ean be assoeiated with various molecular mechanisms responsible for the viscoelastic behavior of linear amorphous polymers under creep conditions. The analogies to the moleeular mechanism can be made as follows. 

Macromolecular materials usually possess entropy elasticity together with viscous and energy-elastic components. Such behavior was only partly comprehensible by use of the models discussed up to now. It can be described very satisfactorily, however, by a four-parameter model in which a Hooke body, a Kelvin body, and a Newton body are combined (see the lowest figure in Figure 11-11). With this model, the deformation must again be added, i.e., with Equations (11-49), (11-52), and (11-57),... 

At aU technically relevant temperatures, polymers deform by creep. To describe the time-dependence of plastic deformation, we again exploit equation (8.3). In contrast to the viscoelastic deformation, there is no restoring force in viscoplasticity. Equation (8.3) is thus used to describe the dashpot element connected in series in the four-parameter model from figure 8.7(b). 

The four-parameter model provides at least a qualitative representation of all the phenomena generally observed in the creep of viscoelastic materials instantaneous elastic strain, retarded elastic strain, steady-state viscous flow, instantaneous elastic recovery, retarded elastic recovery, and permanent set. It also describes at least qualitatively the behavior of viscoelastic materials in other types of deformation. Of equal importance is the fact that the model parameters can be identified with the various molecular response mechanisms in polymers, and can, therefore, be used to predict the influences that changes in molecular structure will have on mechanical response. The following analogies may be drawn. 

Consider, for example, the creep response of the four-parameter model (Fig. 18.8). For this model, a logical choice for A would be the time constant for its Voigt-Kelvin component, Jja/Gz- For De> 1 (t - Ac), the Voigt-Kelvin element and dashpot 1 will be essentially immobile, and the response will be due almost entirely to spring 1, that is, almost purely elastic. For De 0 (t, > A ), the instantaneous and retarded elastic response mechanisms have long since reached equilibrium, so the only remaining response will be the purely viscous flow of dashpot 1, and the deformation due to viscous flow will completely overshadow that due to the elastic response mechanisms (imagine the creep... 

In terms of the four-parameter model, below Tg, only spring 1 is operative, and the material is almost completely elastic (low damping). In the vicinity of Tg, the viscosity of dashpot 2 drops to the point where it can deform and dissipate energy, giving the damping peak. At higher temperatures, its viscosity drops to... 

FIG. 17 Force-deformation relationship and description of Equation 16, shown in a four-parameter compression model (adapted from Yan and Barbosa-Canovas, 1997). 

Figure 9. Four-parameter or Burger model for the deformation behaviour of polymers.

Figure 12 represents all steps of craze formation in crystalline polymers in a single model. It is based on Hornbogen s model for a crack tip in a polymer crystal, under the utilization of individual block drawings by Schultz for the fine scale nature of plastic deformation in semicrystalline thermoplastics. The classification into four regions A to D (after ) helps to describe and imderstand the influence of molecular parameters on craze strength and craze breakdown. 

The Bird-Carreau model is an integral model which involves taking an integral over the entire deformation history of the material (Bistany and Kokini, 1983). This model can describe non-Newtonian viscosity, shear rate-dependent normal stresses, frequency-dependent complex viscosity, stress relaxation after large deformation shear flow, recoil, and hysteresis loops (Bird and Carreau, 1968). The model parameters are determined by a nonlinear least squares method in fitting four material functions (aj, 2, Ai, and A2). 

In this paper we have reviewed the structures appearing at onset of electro-convection in nematic liquid crystals. The influence of the relevant material parameters (ca and ao) and the role of the initial director alignment were explored. Our calculations using a linear stability analysis of the standard model of electroconvection (performed for zero frequency) revealed that four different scenarios characterized by different ranges of the wavenumber q can be identified (1) the Qf= 0 mode (a homogeneous deformation known as the Freedericksz transition) predicted and observed in cases C, D, E and H, which is... 

Most authors who have studied the consolidation process of solids in compression have used the basic model of a porous medium with point contacts which yields a general form of the mass and momentum balances. This must be supplemented by a model describing filtration and deformation properties. Probably the best model to date is that of Kos, which uses two parameters to define the characteristic behaviour of suspensions. His model can potentially be applied to four processes sedimentation, thickening, cake filtration and expression. 

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