Linear viscoelastic model, description

Note 7 There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in Definition 5.2. (See, for example, [11].) Such definitions have certain advantages regarding their mathematical generality. However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models). 

The question of the detailed limits of validity of the reptation model thus remains a pending question. What appears puzzling is the fact that, on one hand, the reptation model and the Doi - Edwards description of the linear viscoelasticity work so well both qualitatively and quantitatively for some experiments, while, on the other hand, they seem unable to account for all the existing data. This may suggest that the reptation model does not contain the whole story of linear polymer dynamics, and that one needs to learn more on other possibly competing processes. 

Table 7 gives a summary of qualitative performances and problems encountered for simple shear and uniaxial elongational flows, using the Wagner and the Phan Thien Tanner equations or more simple models as special cases of the former. Additional information may also be found in papers by Tanner [46, 64]. All equations presented hereafter can be cast in the form of a linear Maxwell model in the small strain limit and therefore are suitable for the description of results of the linear viscoelasticity in the terminal zone of polymer melts. 

Though it is desirable to confirm the experimental results, the comparison of the theoretical formulas (148) with the experimental results (150) indicates that, insofar as the influence of the ambient macromolecules on the dynamics of a chosen macromolecule is concerned, the ambient macromolecules are equivalent to a certain relaxing medium. The reptation effect is due to terms of order higher than the first in the equation of motion of the macromolecule, and it is actually the first-order terms that dominate the linear viscoelastic phenomena. Attempts to describe viscoelasticity without the leading linear terms lead to a distorted picture, so that one begins to understand the lack of success of the reptation model in the description of the viscoelasticity of polymers. Reptation has to be included when one considers the non-linear effects in viscoelasticity. 

We have shown (Section 5.2.7) tliat the standard linear solid, a three-component spring and dashpot model, provides to a first approximation a description of linear viscoelastic behaviour. Eyring and his colleagues [52] assumed that the deformation of a polymer was a thermally activated rate process involving the motion of segments of chain molecules over potential barriers, and modified the standard linear solid so that the movement of the dashpot was governed by the activated process. The model, which now represents non-linear viscoelastic behaviour, is useful because its parameters include an activation energy and... 

For the generalized linear Maxwell model (GLM) and the upper convected model (UCM), the only material parameters needed are contained in the relaxation time spectrum of the material which can be obtained from simple linear viscoelastic measurements. For the Giesekus model, one needs in addition the mobility factors which Christensen and McKinley obtained by fitting the stress-strain curves of the adhesive. The advantage of the Giesekus model was that it provided them with a better description of the stress-strain curves. This, of course, is to be expected since those curves were used to deduce the parameters of the model. 

The purpose of this chapter has been to give a description of some of the most useful contact mechanics expressions as they relate to studies of adhesion. The primary assumption regarding the properties of the materials themselves is that a linear constitutive model is obeyed throughout the strained region, with the possible exception of a relatively small cohesive zone at the contact edge. Many of the results obtained for simple linear elastic behavior are analytic. Linear viscoelasticity can be handled as well, although in this case numerical approaches... 

It may be shown that any combination of linear elements must be linear, so any models based on these linear elements, no matter how complex, can represent only linear response. Just how realistic is linear response Its most conspicuous shortcoming is that it permits only Newtonian behavior (constant viscosity) in equlibrium viscous flow. For most polymers at strains greater than a few percent or so (or rates of strain greater than 0.1 s" -), linear response is not a good quantitative description. Moreover, even within the limit of linear viscoelasticity, a fairly large number of linear elements (springs and dashpots) are usually... 

The mathematical representation of the elastic behavior of oriented heterogeneous solids can be somewhat improved through a more appropriate choice of the boundary conditions such as proposed by Hashin and Shtrikman [66] and Stern-stein and Lederle [86]. In the case of lamellar polymers the formalisms developed for reinforced materials are quite useful [87—88]. An extensive review on the experimental characterization of the anisotropic and non-linear viscoelastic behavior of solid polymers and of their model interpretation had been given by Hadley and Ward [89]. New descriptions of polymer structure and deformation derive from the concept of paracrystalline domains particularly proposed by Hosemann [9,90] and Bonart [90], from a thermodynamic treatment of defect concentrations in bundles of chains according to the kink and meander model of Pechhold [10—11], and from the continuum mechanical analysis developed by Anthony and Kroner [14g, 99]. 

The current version of the tube model combines all the above mechanisms to describe the linear viscoelastic data of linear and star-branched chains. For example, the Milner-McLeish (MM) model, " adopting the DTD molecular picture for the CR mechanism, considerably well describes the data, as shown with the solid curves in Figures 9 and 10. Equally good description can be obtained for some other models combining those mechanisms in a way different from that in the MM modd. - - ... 

These predictions of the Zimm model are compared with experimental data on dilute polystyrene solutions in two -solvents in Fig. 8.7. The Zimm model gives an excellent description of the viscoelasticity of dilute solutions of linear polymers. 

Qearly it is a crude simplification to characterize the effect of the environment by a single parameter a. A more appropriate description would be to assign a viscoelastic character to the environment. However, we proceed here using the simplest possible model. As we shall show later various experimental results indicate that this model is adequate for linear polymers with narrow distribution of molecular weight. Limitations of the model will be discussed later. 

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