Viscous response

Since the strain is the same in both elements in the Voigt model, the applied stress (subscript 0) must equal the sum of the opposing forces arising from the elastic and viscous response of the model ... 

Next suppose we consider the effect of a periodically oscillating stress on a Voigt element of modulus G and viscosity 77. Remember from the last section that for a Voigt element the appUed stress equals the sum of the elastic and viscous responses of the model. Therefore, for a stress which varies periodically, Eq. (3.64) becomes... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

Over the years there have been many attempts to simulate the behaviour of viscoelastic materials. This has been aimed at (i) facilitating analysis of the behaviour of plastic products, (ii) assisting with extrapolation and interpolation of experimental data and (iii) reducing the need for extensive, time-consuming creep tests. The most successful of the mathematical models have been based on spring and dashpot elements to represent, respectively, the elastic and viscous responses of plastic materials. Although there are no discrete molecular structures which behave like the individual elements of the models, nevertheless... 

In Chapter 2 when the Maxwell and Kelvin models were analysed, it was found that the time constant for the deformations was given by the ratio of viscosity to modulus. This ratio is sometimes referred to as the Relaxation or Natural time and is used to give an indication of whether the elastic or the viscous response dominates the flow of the melt. 

The inelastic wave shows rise times that vary quite substantially. Recognizing that the rise time is a direct indication of the balance between the viscous response of the sample and the driving force, Grady [81G01] has analyzed and compared the effective viscosity of a range of materials. These viscosities are manifestations of the dynamic deformation controlled by the shock-induced defects, heterogeneities, and their motions. 

When a plastic material is subjected to an external force, a part of the work done is elastically stored and the rest is irreversibly (or viscously) dissipated hence a viscoelastic material exists. The relative magnitudes of such elastic and viscous responses depend, among other things, on how fast the body is being deformed. It can be seen via tensile stress-strain curves that the faster the material is deformed, the greater will be the stress developed since less of the work done can be dissipated in the shorter time. 

Most pigmented systems are considered viscoelastic. At low shear rates and slow deformation, these systems are largely viscous. As the rate of deformation or shear rate increases, however, the viscous response cannot keep up, and the elasticity of the material increases. There is a certain amount of emphasis on viscoelastic behavior in connection with pigment dispersion as well as ink transportation and transformation processes in high-speed printing machines (see below). Under periodic strain, a viscoelastic material will behave as an elastic solid if the time scale of the experiment approaches the time required for the system to respond, i.e., the relaxation time. Elastic response can be visualized as a failure of the material to flow quickly enough to keep up with extremely short and fast stress/strain periods. 

Polymer rheology can respond nonllnearly to shear rates, as shown in Fig. 3.4. As discussed above, a Newtonian material has a linear relationship between shear stress and shear rate, and the slope of the response Is the shear viscosity. Many polymers at very low shear rates approach a Newtonian response. As the shear rate is increased most commercial polymers have a decrease in the rate of stress increase. That is, the extension of the shear stress function tends to have a lower local slope as the shear rate is increased. This Is an example of a pseudoplastic material, also known as a shear-thinning material. Pseudoplastic materials show a decrease in shear viscosity as the shear rate increases. Dilatant materials Increase in shear viscosity as the shear rate increases. Finally, a Bingham plastic requires an initial shear stress, to, before it will flow, and then it reacts to shear rate in the same manner as a Newtonian polymer. It thus appears as an elastic material until it begins to flow and then responds like a viscous fluid. All of these viscous responses may be observed when dealing with commercial and experimental polymers. 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

G" = —sin(<5) viscous response - out of phase with imposed strain (3.48)... 

Schreiber and co-workers have noted very persistent history effects in linear polyethylenes (69). Fractions which have been crystallized from dilute solution required times of the order of hours in the melt state at 190° C in order to attain a constant die swell behavior upon subsequent extrusion. The viscosity on the other hand reached its ultimate value almost immediately. The authors concluded from this result that different types of molecular interactions were responsible for elastic and viscous response. However, other less specific explanations might also suffice, since apparent viscosity might be relatively intensitive to the presence of incompletely healed domain surfaces, while die swell, requiring a coordinated motion of the entire extrudate, might be affected by planes of weakness. It would... 

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). 

A linear output is associated with a liquidlike or viscous response. The retardation response of a Newtonian liquid is a straight line through the origin, where the slope of the line is inversely proportional to the viscosity. A stress relaxation response for a Newtonian liquid is a narrow spike, as energy is dissipated very quickly. Typically the recoil part of a creep curve will be almost horizontal if there is little elasticity in the sample. 

With the above information, it becomes possible to combine viscous characteristics with elastic characteristics to describe the viscoelasticity of polymeric materials.86-90 The two simplest ways of combining these features are shown in Figure 2.49, where a spring having a modulus G models the elastic response. The viscous response is modelled by what is called a dashpot. It consists of a piston moving in a cylinder containing a viscous fluid of viscosity r. If a downward force is applied to the cylinder, more fluid flows into it, whereas an upward force causes some of the fluid to flow out. The flow is retarded because of the high viscosity and this element thus models the retarded movement and flow of polymer chains. 

Figure 7.1. Definitions of elasticity and viscoelasticity. If a weight is placed on a rubber band at time zero and the initial length is /0, then instantaneously the weight causes a length increase that is followed by a continual slow length increase until the rubber band reaches some new equilibrium length l0 + AL. The elastic response is the length increase that occurs instantaneously, and the viscous response is the time-dependent length component.

To fully understand the behavior of biological materials we need to address the issue of viscoelasticity. When a weight is placed on viscoelastic material, there is an instantaneous elastic response and a time-dependent viscous response (see Figure 7.1). For polymers the elastic response reflects the change in macromoleular conformation, which is usually time independent if no bonds are broken. The viscous response is the flow of macromolecules by each other similar to what happens during the flow of fluids in a tube. Fluid flow is a time-dependent process. Polymers exhibit viscoelastic behavior because they have both a time-independent response and a time-dependent response. 

Newtonian shear flow of polymer melts is a stable process. This means that small disturbances in the flow conditions, caused by external effects, are readily suppressed. As the rate of shear increases, however, the elastic response of the melt becomes more pronounced relative to the viscous response. In other words, components of the stress tensor in directions different from the direction of the shear stress become more important. As a result, small disturbances are not so readily compensated and may even be magnified. 

It is important to realize that this type of behavior is not just a simple addition of linear elastic and viscous responses. An ideal elastic solid would display an instantaneous elastic response to an applied (non-destructive) stress (top of Figure 13-74). The strain would then stay constant until the stress was removed. On the other hand, if we place a Newtonian viscous fluid between two plates and apply a shear stress, then the strain increases continuously and linearly with time (bottom of Figure 13-74). After the stress is removed the plates stay where they are, there is no elastic force to restore them to their original position, as all the energy imparted to the liquid has been dissipated in flow. 

FIGURE 13-74 Schematic diagrams of (A) purely elastic response and (B) purely viscous response. 

This equation is derived by integrating Eq.( 11-29) with boundary condition)/ = 0, T = To at r = 0. Although the model has some elastic character the viscous response dominates at all but short times. For this reason, the element is known as a Maxwell fluid. 

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