Viscoelastic solid behavior

This last domain gives a specific viscoelastic solid behavior to the blend. As a consequence, we have particularly studied the behavior of this terminal region. 

Contact mechanics, in the classical sense, describes the behavior of solids in contact under the action of an external load. The first studies in the area of contact mechanics date back to the seminal publication "On the contact of elastic solids of Heinrich Hertz in 1882 [ 1 ]. The original Hertz theory was applied to frictionless non-adhering surfaces of perfectly elastic solids. Lee and Radok [2], Graham [3], and Yang [4] developed the theories of contact mechanics of viscoelastic solids. None of these treatments, however, accounted for the role of interfacial adhesive interactions. 

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. 

Viscolelastic materials can be divided into viscoelastic solids and viscoelastic, or simply elastic, liquids. All viscoelastic liquids are non-Newtonian, but not all non-Newtonian liquids are viscoelastic. Non-Newtonian liquids show nonlinear rheological behavior, and this may be time dependent (Barnes et al., 1989). 

For N20 dispersions in UP and VE resins a different behavior is observed. N20-VE resin dispersions containing 65% of resin show liquid behavior with low storage modulus and G < G". Addition of styrene increases the storage modulus with G > G". Hence, a viscoelastic solid is obtained. N20-UP resin dispersions with a resin content of 65% already show the properties of a viscoelastic solid ((7 > ( ). However, addition of styrene only slightly increases the storage modulus. 

Once more, the rheological behavior of many pharmaceutical and biomedical materials is more appropriately described by a number of Voigt units connected in series (18). The model illustrated in Figure 10.4 describes the rheological behavior of a viscoelastic solid as, in this case, the elastic contribution is sufficient to ensure that there is no unlimited, nonrecoverable viscous flow. However, if the spring in one of the units possesses zero elasticity (i.e., G = 0), then nonrecoverable viscous flow will be observed, and the material is better described as a viscoelastic liquid or, alternatively, an elastoviscous system. 

For a solid material, the typical difference in deformation behavior between a Hookean solid and a viscoelastic solid can be explained in terms of an applied constant load. 

For a Hookean solid, say a metal, the load will produce a deformation that stays constant over time. On the other hand, for a polymeric material the same load will produce an initial deformation, followed by a slow and constant deformation up to a certain value (creep). This is an illustration of a retardation process, where the final response of the material to the load is retarded. On the other hand, one can also visualize an experiment where a constant strain is imposed to both, a Hookean solid and a viscoelastic solid. Under these experimental conditions, a constant stress is developed in the first case, whereas in the second case, the stress is nonconstant it starts at an initial value and then decreases up to a zero value. This experimental behavior constitutes a relaxation process. 

As can be seen, the Maxwell-Weichert model possesses many relaxation times. For real materials we postulate the existence of a continuous spectrum of relaxation times (A,). A spectrum-skewed toward lower times would be characteristic of a viscoelastic fluid, whereas a spectrum skewed toward longer times would be characteristic of a viscoelastic solid. For a real system containing crosslinks the spectrum would be skewed heavily toward very long or infinite relaxation times. In generalizing, A may thus he allowed to range from zero to infinity. The concept that a continuous distribution of relaxation times should be required to represent the behavior of real systems would seem to follow naturally from the fact that real polymeric systems also exhibit distrihutions in conformational size, molecular weight, and distance between crosslinks. 

Rubber is a viscoelastic solid formed by crosslinking a polymer, which is initially a viscoelastic liquid. In spite of this difference there still are some common issues in understanding the physics of the glass temperature and the viscoelastic mechanisms in the softening dispersion (i.e., called the glass-rubber transition zone in Ferry (1980). A case in point can be taken by comparing the viscoelastic behavior of the neat epoxy resin Epon lOOlF (Plazek and... 

We recall that the combination of both liquid and solid behavior is termed viscoelasticity. We have already discussed the basic law for the "simple" liquid, Newton s law. For solids, Hooke s law defines the relationship between stress S and deformation 7, using a material parameter called the modulus of elasticity S = Gy. G represents a shear modulus, while E represents Young s modulus in tension (S = Ey). That is the behavior of so-called "ideal elastic solids." This may occur mostly in metals or rigid materials, while in the case of polymers, Hooke s correlation is foxmd only in the glassy state, below Tg. 

Since by definition the viscoelastic material in shear exhibits both the behavior governed by equation 6 and than represented by equation 8, the constitutive relation for the linear viscoelastic solid can be written as... 

Usually this criterion allows us to unambiguously classify a phase as either a solid or a fluid. Over a sufficiently long time period, however, detectable flow occurs in any material under shear stress of any magnitude. Thus, the distinction between solid and fluid actually depends on the time scale of observation. This fact is obvious when we observe the behavior of certain materials (such as Silly Putty, or a paste of water and cornstarch) that exhibit solidlike behavior over a short time period and fluid-like behavior over a longer period. Such materials, that resist deformation by a suddenly-applied shear stress but undergo flow over a longer time period, are called viscoelastic solids. 

In contrast to simple elastic solids and viscous liquids, the situation with polymeric fluids is somewhat more complicated. Polymer melts (and most adhesives are composed of polymers) display elements of both Newtonian fluid behavior and elastic solid behavior, depending on the temperature and the rate at which deformation takes place. One therefore characterizes polymers as viscoelastic materials. Furthermore, if either the total strain or the rate of strain is low, the behavior may be described as one of linear or infinitesimal viscoelasticity. In such a case, the stress-deformation relationship (the constitutive equation) involves not just a single time-independent constant but a set of constants called the relaxation spectrum,(2) and this, too, may be determined from a single stress relaxation experiment, or an experiment involving small-amplitude oscillatory motion. 

Once the cubic nature of the blue phase was established, attempts to measure the elastic constants using more sensitive techniques appeared shortly thereafter [25], [96], [97], with those of Kleiman et al. [25] being the most extensive. The latter experiments are very delicate, since the blue phase lattice is both soft (small elastic constants) and weak (small elastic limit). Torsional oscillators configured as cup viscometers were used and the shear distortion was kept to less than 0.02%. Figure 7.12 shows results for both the shear elasticity G and the viscosity rj. These data are taken at various frequencies and must be extrapolated to 0 Hz to obtain the static properties. In the helical phase the extrapolation is somewhat dependent on the model nevertheless, the authors claim that G becomes nearly zero in the helical phase and about 710 dyn cm in BPI. (This figure should be compared to 10" dyn cm 2 in a metal ) However, since BPI also possesses viscosity, its behavior is that of a viscoelastic solid. 

The sharp dichotomy between viscoelastic solids (ordinarily, cross-linked polymer systems) and viscoelastic liquids (ordinarily, uncross-linked polymer systems) is apparent in all the time-dependent and frequency-dependent viscoelastic functions which describe their mechanical behavior in small deformations. Examples of these functions for various types of each of the two classes will be surveyed in the next chapter. 

For both viscoelastic liquids and viscoelastic solids, combinations of large shear rates or large static deformations, respectively, with small time-dependent deformations result in a variety of characteristic behavior, some examples of which will be given in subsequent chapters. 

For a viscoelastic solid with linear viscoelasticity corresponding to a model with springs and dashpots such as Fig. 1 -9 or 1 -10, G" should also be directly proportional to CO at very low frequencies. Such behavior is not observed for the examples on the right side of Fig. 2-4 experiments have never been carried to sufficiently low frequencies to test this prediction. 

Finally, it is worth mentioning another approach used to describe nonlinear viscoelastic solids nonlinear differential viscoelasticity [49, 178, 179]. This theory has been successfully applied to model finite amplitude waves propagation [180-182]. It is the generalization to the three-dimensional nonlinear case of the rheological element composed by a dashpot in series with a spring. Thus in the simplest case, the stress depends upon the current values of strain and strain rate rally. In this sense, it can account for the nonlinear short-term response and the creep behavior, but it fails to reproduce the long-term material response (e.g., relaxation tests). The so-called Mooney-Rivlin viscoelastic material [183] and the incompressible version of the model proposed by Landau and Lifshitz [184] belraig to this class. 

To further illustrate the point of a liquid with both elastic and viscous behavior, the flow of a rheological liquid is shown in Fig. 1.2. Here a polymer liquid is in a clear horizontal (to avoid gravity effects) tube and a dark reference mark has been inserted that moves with the fluid. The liquid is unpressurized in frame 1 but a constant pressure has been applied in frames 2 through 5 where motion can be seen to have taken place as time progresses. In frame 6 the pressure has been removed and in frames 7 and 8 the liquid can be seen to partially recover. No recovery would take place if this were an ordinary viscous liquid. This is known as an elastic after effect and a similar effect or creep recovery is observed in viscoelastic solids and/or all polymers provided the correct temperature is chosen. 

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