Viscoelasticity solid polymers

Viscoelastic phenomena always involve the change of properties with time and, therefore, the measurements of viscoelastic properties of solid polymers may be called dynamic mechanical. Dynamic mechanical thermal analysis (DMTA) is a very useful tool for studying... 

In case of copper some rheological experiments carried out at a given polymer concentration and increasing amoimt of cations indicates that copper/pectin systems in the one-phase domain behave as a viscoelastic liquid rather than a viscoelastic solid referred to as true gel (G (co) = G, when to—>0 with Gg the equilibrium shear modulus)[35]. Despite the lack of experimental data the range in cation and polymer concentration in which true gels may be observed seemed very limited. These results corroborate the strength of the binding of copper by pectins evidenced by the properties of the phase separation curves. 

The macroscopic long-time behavior of dense polymer liquids exhibits drastic changes if permanent cross-links are introduced in the system [75-77], Due to the presence of junctions the flow properties are suppressed and the viscoelastic liquid is transformed into a viscoelastic solid. This is contrary to the short-time behavior, which appears very similar in non-cross-linked and crosslinked polymer systems. 

Here we describe the strain history with the Finger strain tensor C 1(t t ) as proposed by Lodge [55] in his rubber-like liquid theory. This equation was found to describe the stress in deforming polymer melts as long as the strains are small (second strain invariant below about 3 [56] ). The permanent contribution GcC 1 (r t0) has to be added for a linear viscoelastic solid only. C 1(t t0) is the strain between the stress free state t0 and the instantaneous state t. Other strain measures or a combination of strain tensors, as discussed in detail by Larson [57], might also be appropriate and will be considered in future studies. A combination of Finger C 1(t t ) and Cauchy C(t /. ) strain tensors is known to express the finite second normal stress difference in shear, for instance. 

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

When there is no volume change, as when an elastomer is stretched, Poisson s ratio is 0.5. This value decreases as the Tg of the polymer increases and approaches 0.3 for rigid solids such as PVC and ebonite. For simplicity, the polymers dealt with here will be considered to be isotropic viscoelastic solids with a Poisson s ratio of 0.5, and only deformations in tension and shear will be considered. Thus, a shear modulus (G) will usually be used in place of Young s modulus of elasticity E Equation 14.2) where E is about 2.6G at temperatures below Tg. 

Since polymers are viscoelastic solids, combinations of these models are used to demonstrate the deformation resulting from the application of stress to an isotropic solid polymer. Maxwell joined the two models in series to explain the mechanical properties of pitch and tar (Figure 14.2a). He assumed that the contributions of both the spring and dashpot to strain were additive and that the application of stress would cause an instantaneous elongation of the spring, followed by a slow response of the piston in the dashpot. Thus, the relaxation time (t), when the stress and elongation have reached equilibrium, is equal to rj/G. 

The effects of test rate and temperature described above are obviously interrelated. This is not an unexpected observation since epoxy polymers are viscoelastic solids and so it would be predicted that reducing the rate of testing, for example, would be... 

The models described so far provide a qualitative illustration of the viscoelastic behaviour of polymers. In that respect the Maxwell element is the most suited to represent fluid polymers the permanent flow predominates on the longer term, while the short-term response is elastic. The Kelvin-Voigt element, with an added spring and, if necessary, a dashpot, is better suited to describe the nature of a solid polymer. With later analysis of the creep of polymers, we shall, therefore, meet the Kelvin-Voigt model again in more detailed descriptions of the fluid state the Maxwell model is being used. 

So, polymer melts display elastic as well as viscous behavior. In other words they are viscoelastic. Do polymer solids display some viscous behavior Also, we ve used the word relaxation when we talk about time-dependent behavior, but what do we mean by this To find out we now need to explore the subject of viscoelasticity in more depth. 

Polymers can and do display both of these elements in their response to an applied stress (an elastic response and permanent deformation), but the defining feature of the creep curve of a viscoelastic solid is some sort of... 

When the inclusion is an air bubble in a viscoelastic solid the shear modulus of the air is zero, and also it is reasonable to neglect the bulk modulus of the air since it is several orders of magnitude less than the bulk modulus of the solid. The Kerner mean field model then predicts the following expression for the effective bulk modulus of the air-polymer composite,... 

Polymeric (and other) solids and liquids are intermediate in behavior between Hookean, elastic solids, and Newtonian, purely viscous fluids. They often exhibit elements of both types of response, depending on the time scale of the experiment. Application of stresses for relatively long times may cause some flow and permanent deformation in solid polymers while rapid shearing will induce elastic behavior in some macromolecular liquids. It is also frequently observed that the value of a measured modulus or viscosity is time dependent and reflects the manner in which the measuring experiment was performed. Tliese phenomena are examples of viscoelastic behavior. 

In real food polymers, a distinction can be made between a viscoelastic solid, which contains some cross-links that are permanent, and a viscoelastic liquid, where, under the influence of stress, the relative movement of whole molecules will be observed. As shown in Figure 8.9, in the case of a viscoelastic solid, after application of the stress, the strain will eventually reach a constant value, and upon removal of the stress, the strain will finally return to the remaining value of food primary energy, which was not entirely dissipated. For a viscoelastic liquid, a permanent deformation will remain after removal of the stress. In the stress relaxation area, the deformation value will decay to zero for a viscoelastic liquid, whereas for a solid, it will reach a constant, nonzero value. It can also be seen as either a decreased value to the zero or a constant, nonzero value, as pointed out by the dashed line. Both values characterize the rheology parameters of foods under certain conditions. One of the reasons for this is that the factors of time-dependent foods are not necessarily related to their elastic modulus. This can be explained by the series of small deformations without rupture, which are dependent in different ways and are based on the primary molecular microstructure of foods. The time required for the stress to relax to a definite fraction of its initial value is the relaxation time. 

Stress is related to strain through constitutive equations. Metals and ceramics typically possess a direct relationship between stress and strain the elastic modulus (2) Polymers, however, may exhibit complex viscoelastic behavior, possessing characteristics of both liquids and solids (4.). Their stress-strain behavior depends on temperature, degree of cure, and thermal history the behavior is made even more complicated in curing systems since material properties change from a low molecular weight liquid to a highly crosslinked solid polymer (2). ... 

A single weightless Hookean, or ideal, elastic spring with a modulus of G and a simple Newtonian (fluid) dash pot or shock absorber having a liquid with a viscosity are convenient to use as models illustrating the deformation of an elastic solid and an ideal liquid. Because polymers are often viscoelastic solids, combinations of these models are used to demonstrate deformations resulting from the application of stress to an Isotropic solid polymer. 

Modern processing equipment usually employs an Archimedes-type screw that conveys the solid polymer (powder or granules) and causes it to melt progressively through heat transfer from the barrel and heat generation by viscoelastic dissipation of the energy introduced by the screw rotation. 

Samples of bulk polymers respond to applied stresses in several ways some materials behave as elastic solids, some as viscous liquids, and stiU others exhibit viscous as well as elastic properties. The latter are called viscoelastic solids. If a constant force is applied to a viscoelastic sample, the extension of this solid may be divided into three parts (1) an instantaneous elastic deformation, (2) a delayed elasticity or creep, and (3) a viscous flow. This may be seen best by referring to Fig. 15-17 which illustrates an example of a tensile force Fo applied at zero time, maintained until... 

Our discussion of the viscoelastic properties of polymers is restricted to the linear viscoelastic behavior of solid polymers. The term linear refers to the mechanical response in whieh the ratio of the overall stress to strain is a function of time only and is independent of the magnitudes of the stress or strain (i.e., independent of stress or strain history). At the onset we concede that linear viscoelastie behavior is observed with polymers only under limited conditions involving homogeneous, isotropie, amorphous samples under small strains and at temperatures close to or above the Tg. In addition, test conditions must preclude those that ean result in specimen rupture. Nevertheless, the theory of linear viseoelastieity, in spite of its limited use in predicting service performance of polymeric articles, provides a useful reference point for many applications. 

Plots of compressive yield stress versus density on logarithmic scales, for polystyrene and HOPE foams, have slopes 1.5. If the lines are extrapolated to the density of the solid polymer, the yield stress is close to (tq, that measured for the solid. Consequently, the form of Eq. (8.20) is confirmed. Cell faces in HDPE foams behave in a non-linear viscoelastic manner when bent. The stress distribution, resembles that in a plastic hinge (Fig. 8.7b), so it is not surprising that the exponent in the yield stress-relative density relationship is the same as for polystyrene. We will return to these materials in the cycle helmet case study in Chapter 14. 

In chapter 7 the phenomenon of creep in a viscoelastic solid is considered. For an ideal linear viscoelastic medium the deformation under a constant stress eventually becomes constant provided that in equation (7.4) is zero. If the load is removed at any time, the ideal material recovers fully. For many polymers these conditions are approximately satisfied for low stresses, but the curves (b) and (c) in fig. 6.2 indicate a very different type of behaviour that may be observed for some polymers under suitable conditions. For stresses above a certain level, the polymer yields. After yielding the polymer either fractures or retains a permanent deformation on removal of the stress. 

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