Phenomenological Mechanical Models

In this section, elementary mechanical models that can describe some aspects of viscoelastic polymeric behavior are presented. Although these simple models cannot represent the behavior of real polymers over their complete history of use, they are very helpful to gain physical understanding of the phenomena of creep, relaxation and other test procedures and to better understand the relationship between stress and strain for a viscoelastic material. Undoubtedly, the first models were developed on the basis of observations and not just as a mathematical exercise. Generalized mechanical models are presented later in Chapter 5. 

The simplest mechanical models for viscoelastic behavior consist of two elements a spring for elastic behavior and a damper for viscous behavior. First it is convenient to introduce the model of a linear spring to represent a Hookean bar under uniaxial tension where the spring constant is the modulus of elasticity. As indicated in Fig. 3.19 the spring constant can be replaced by Young s modulus if the stress replaces P/A and strain replaces 6/L. 

Consider a semi-infinite fluid as shown in Fig. 3.20. If a flat plate at the top of the fluid is moved with a velocity, V = du/dt, and if the fluid is Newtonian the shear deformation varies linearly from top to bottom assuming a no slip boundary condition between the fluid and the plate as well as between the fluid and the container. 

The strain on a differential element of the fluid is given by du/dy. The Newtonian law of viscosity for the shear process shown in Fig. 3.20 may thus be expressed as, 

Spring and damper elements can be combined in a variety of arrangements to produce a simulated viscoelastic response. Early models due to Maxwell and Kelvin combine a linear spring in series or in parallel with a Newtonian damper as shown in Fig. 3.21. Other basic arrangements include the three-parameter solid and the four-parameter fluid as shown in Fig. 3.22. 

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In this chapter we describe the common forms of viscoelastic behaviour and discuss the phenomena in terms of the deformation characteristics of elastic solids and viscous fluids. The discussion is confined to linear viscoelasticity, for which the Boltzmann superposition principle enables the response to multistep loading processes to be determined from simpler creep and relaxation experiments. Phenomenological mechanical models are considered and used to derive retardation and relaxation spectra, which describe the time-scale of the response to an applied deformation. Finally we show that in alternating strain experiments the presence of the viscous component leads to a phase difference between stress and strain. 

Theoretical models of the film viscosity lead to values about 10 times smaller than those often observed [113, 114]. It may be that the experimental phenomenology is not that supposed in derivations such as those of Eqs. rV-20 and IV-22. Alternatively, it may be that virtually all of the measured surface viscosity is developed in the substrate through its interactions with the film (note Fig. IV-3). Recent hydrodynamic calculations of shape transitions in lipid domains by Stone and McConnell indicate that the transition rate depends only on the subphase viscosity [115]. Brownian motion of lipid monolayer domains also follow a fluid mechanical model wherein the mobility is independent of film viscosity but depends on the viscosity of the subphase [116]. This contrasts with the supposition that there is little coupling between the monolayer and the subphase [117] complete explanation of the film viscosity remains unresolved. 

However, a distinction should be made in that Eq. (12) is purely phenomenological and does not require any transport mechanism model while the Nermst-Planck equation used in the previous finely-porous membrane model requires a specific pore model. Another difference is that the salt concentration in Eq. (12) is that in the membrane while the quantity appearing in the Nernst-Planck equation refers to the salt concentration in the membrane pores. 

Because of the wide application of these resins to diverse industries and their very different kinetic models and mechanisms of cross-linking and reactions, phenomenological kinetic models for epoxy, vinyl ester, and phenolic resins are presented in the next three subsections. 

Time constants are related to the relaxation times and can be found in equations based on mechanical models (phenomenological approaches), in constitutive equations (empirical or semiempirical) for viscoelastic fluids that are based on either molecular theories or continuum mechanics. Equations based on mechanical models are covered in later sections, particularly in the treatment of creep-compliance studies while the Bird-Leider relationship is an example of an empirical relationship for viscoelastic fluids. 

Viscoelastic stress analysis of two component systems shows that a broadening of the dispersion zone is to be expected 166,167), even if the disperse phase (filler) is purely elastic 166) and it is not necessary to ascribe different molecular properties to the continuous phase. The simplest way to visualize this mechanical interaction is by the use of phenomenological mechanically equivalent models. The model of Takayanagi (/68) is illustrated in Fig. 16. The elastic solution for this model is easily derived from elementary considerations. By the correspondence principle of viscoelastic stress analysis 169), the viscoelastic solution is obtained simply by substituting complex moduli in place of purely elastic moduli... 

Such being the case, further inferences about the nature of the wear process follow. A disrupted fluid film allows localized contacts at the rubbing surfaces, and it is the mechanistic processes at these contacts that determine the course of lubricated wear. When the wear process is abrasive, it is most likely influenced directly by fluid film thickness and surface roughness, whereas processes such as adhesion, transfer, oxidation, additive reaction and the like are responsive to surface conditions at the contacts as well as to the number of contacts. These are the aspects of lubricated wear that are emphasized in this chapter, from the viewpoint of phenomenology, mechanisms and modeling. 

On a modest level of detail, kinetic studies aim at determining overall phenomenological rate laws. These may serve to discriminate between different mechanistic models. However, to it prove a compound reaction mechanism, it is necessary to determine the rate constant of each elementary step individually. Many kinetic experiments are devoted to the investigations of the temperature dependence of reaction rates. In addition to the obvious practical aspects, the temperature dependence of rate constants is also of great theoretical importance. Many statistical theories of chemical reactions are based on thermal equilibrium assumptions. Non-equilibrium effects are not only important for theories going beyond the classical transition-state picture. Eventually they might even be exploited to control chemical reactions [24]. This has led to the increased importance of energy or even quantum-state-resolved kinetic studies, which can be directly compared with detailed quantum-mechanical models of chemical reaction dynamics [25,26]. 

Attempts to introduce dispersed phase normal stresses on a purely phenomenological basis [32,43] or by means of simple mechanical models [51,52,56] are heuristic by their very nature and in no way help to solve the problem. Therefore, the results of the corresponding stability studies, and especially the studies treating... 

There are many different expressions proposed to model the elastic part They find their origins either in some statistical mechanical model like the Gaussian chains, the Flory [32] or the James and Guth models [33], or in some phenomenological laws tending to reproduce experimental observations [34]. In the... 

The tension at the surface of a liquid is one of the more striking manifestations of the forces that act between molecules, and attempts to explain it in these terms go back to the eighteenth century. The early attempts were in terms of crude mechanical models of a liquid, which we describe in Chapter 1. These pre-thermodynamic theories were aban> doned in the nineteenth century to be replaced by more phenomenological or quasi-thermodynamic methods. We introduce these in the second and thM chapters, and use them extensively in the last part of the book, since they are, when handled ri dy, still powerful methods of treating complicated problems such as those of multi-phase equilibria and critical phenomena. 

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