Viscous material Kelvin model

The static tests considered in Chapter 8 treat the rubber as being essentially an elastic, or rather high elastic, material whereas it is in fact viscoelastic and, hence, its response to dynamic stressing is a combination of an elastic response and a viscous response and energy is lost in each cycle. This behaviour can be conveniently envisaged by a simple empirical model of a spring and dashpot in parallel (Voigt-Kelvin model). 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

The gel layer is treated as a Kelvin-Voight viscoelastic material where the gel shear is acting in parallel. This viscoelastic material stress has both an elastic and viscous component acting in parallel. This viscoelastic material model was substituted into the momentum equation and the resulting gel layer equation of motion is... 

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),... 

However, various combinations of eiastic and viscous elements have been used to approximate the material behavior of polymer melts. Some models are combinations of springs and dashpots to represent the elastic and viscous responses, respectively. The most common ones being the Maxwell model for a polymer melt and the Kelvin or Voight model for a solid. One model that represents shear thinning behavior, normal stresses in shear flow and elastic behavior of certain polymer melts is the K-BKZ model [28-29]. 

As discussed briefly in the next section, polymers have a unique response to mechanical loads and are properly treated as materials which in some instances behave as elastic solids and in some instances as viscous fluids. As such their properties (mechanical, electrical, optical, etc.) are time dependent and cannot be treated mathematically by the laws of either solids or fluids. The study of such materials began long before the macromolecu-lar nature of polymers was understood. Indeed, as will be evident in later chapters on viscoelasticity, James Clerk Maxwell (1831-79), a Scottish physicist and the first professor of experimental physics at Cambridge, developed one of the very first mathematical models to explain such peculiar behavior. Lord Kelvin (Sir William Thomson, (1824-1907)), another Scottish physicist, also developed a similar mathematical model. Undoubtedly, each had observed the creep and/or relaxation behavior of natural materials such as pitch, tar, bread dough, etc. and was intrigued to explain such behavior. Of course, these observations were only a minor portion of their overall contributions to the physics of matter. 

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