Linear Viscoelasticity - Fundamentals

The treatment of linear viscoelasticity presented in this chapter is sufficient for a full understanding of the models described in subsequent chapters. However, readers wishing to delve more deeply into this subject may wish to consult the monographs by Ferry [1] and Tschoegl [2]. Ferry treats the rheological properties of polymers, while Tschoegl s book is a comp endium of empirical models and relationships between various linear material functions. 

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At sufficiently low strain, most polymer materials exhibit a linear viscoelastic response and, once the appropriate strain amplitude has been determined through a preliminary strain sweep test, valid frequency sweep tests can be performed. Filled mbber compounds however hardly exhibit a linear viscoelastic response when submitted to harmonic strains and the current practice consists in testing such materials at the lowest permitted strain for satisfactory reproducibility an approach that obviously provides apparent material properties, at best. From a fundamental point of view, for instance in terms of material sciences, such measurements have a limited meaning because theoretical relationships that relate material structure to properties have so far been established only in the linear viscoelastic domain. Nevertheless, experience proves that apparent test results can be well reproducible and related to a number of other viscoelastic effects, including certain processing phenomena. 

The resulting stress was measured, and a discrete Fourier transform was performed to obtain the elastic and viscous moduli. The experimental variables in FTMS are the fundamental frequency, f, and the strain amplitudes, Yi, at each frequency, i. Each of the other frequencies are harmonics (integer multiples) of the fundamental frequency. The fundamental frequency was set at 1 rad/s, while the harmonics were chosen to be 2, 5, 10, 25, 40, 50, and 60 rad/s. The summation of the strain amplitudes at each frequency was below the linear viscoelastic limit of the NOA 61 sample. 

A general description of the fundamental relationships governing the dynamic response of linear viscoelastic materials may be found in several sources (28, 37, 93). In general, sinusoidally applied strains (stresses) result in sinusoidal stresses (strains) that are out of phase. Measurements may be made under uniaxial, shear, or dilational loading conditions, and the resultant complex moduli or compliance and loss-phase angle are computed. Rotating radius vectors are usually taken to represent the... 

The derivation of fundamental linear viscoelastic properties from experimental data obtained in static and dynamic tests, and the relationships between these properties, are described by Barnes etal. (1989), Gunasekaran and Ak (2002) and Rao (1992). In the linear viscoelastic region, the moduli and viscosity coefficients from creep, stress relaxation and dynamic tests are interconvertible mathematically, and independent of the imposed stress or strain (Harnett, 1989). 

Frequency sweep studies in which G and G" are determined as a function of frequency (o)) at a fixed temperature. When properly conducted, frequency sweep tests provide data over a wide range of frequencies. However, if fundamental parameters are required, each test must be restricted to linear viscoelastic behavior. Figure 3-31... 

A fundamental difficulty in the study of the linear viscoelastic behavior of filled rubbers is the secondary aggregation of filler particles, which greatly influences the behavior at small strains, where the response is linear. The effect of this aggregation is overcome at large strains, but now non-linearity and a number of other complications become problems. 

It is common to compare the behavior of polymers with the behavior of metals and to use similar types of experiments to evaluate their performance under mechanical deformation. It is, therefore, important to highlight any qualitatively significant differences between their behavior and the fundamental physical reasons for these differences. In metals, creep is neither linearly viscoelastic nor recoverable, since (unlike polymer chains) metals do not have entanglements. Furthermore, creep is significant only at very high temperatures in metals. 

In Chapter 4, we studied the fundamental importance of the relaxation modulus G t) in linear viscoelasticity. Here, we shall show how the theoretical form of G t) in the Doi-Edwards model is derived in terms of molecular structural and dynamic parameters. In the Doi-Edwards theory the study of G t) includes the nonlinear region. However, we shall postpone full discussion of G t) in the nonlinear region until Chapter 12. 

While the Choi and Schowalter [113] theory is fundamental in understanding the rheological behavior of Newtonian emulsions under steady-state flow, the Palierne equation [126], Eq. (2.23), and its numerous modifleations is the preferred model for the dynamic behavior of viscoelastic liquids under small oscillatory deformation. Thus, the linear viscoelastic behavior of such blends as PS with PMMA, PDMS with PEG, and PS with PEMA (poly(ethyl methacrylate))at <0.15 followed Palierne s equation [129]. From the single model parameter, R = R/vu, the extracted interfacial tension coefficient was in good agreement with the value measured directly. However, the theory (developed for dilute emulsions) fails at concentrations above the percolation limit, 0 > (p rc 0.19 0.09. 

The viscoelasticity can be categorized as either linear or nonlinear, but only the linear viscoelasticity can be described theoretically with uncomplicated mathematics. The fundamental viscoelastic parameters of a linear viscoelastic system do not depend on the magnitude of the stress or strain. Therefore, the linear viscoelastic regime is always used for studying the mechanical properties of viscoelastic blended materials. One of the accepted techniques for investigating the viscoelastic behaviours of natural rubber blended materials is the... 

Chen and Xu [60] found that a primary ionic crosslink network was formed at the initial stage of curing. At this time, the continuous covalent crosslink network of the NR matrix was not formed the whole crosslink network is dominated by ionic crosslinks. As shown in Chen and Xu s studies [60], the initial curing time is about in the first 1 min. The ionic crosslinks, with some physical crosslinks and primary covalent crosslink points, construct the crosslink backbone in this period. Although there are very small amounts of covalent crosslink points, they can not constitute a fundamental covalent network. Because of the formation of poly-ZDMA nanoparticles, some physical adsorption points also exist in this period. The non-linear viscoelasticity of the NR/ZDMA composites is quite different at this time. 

We finally observe that delayed response phenomena akin to creep and relaxation occur in other areas of Mechanics and Physics, and are attributable to the same fundamental cause, namely (usually internal) frictional losses. The mathematical techniques used for analyzing such phenomena are similar to those used in analyzing the properties of the viscoelastic functions. Such a close analogy exists between certain phenomena in the theory of Dielectrics and Linear Viscoelasticity, as emphasized by Gross (1953). 

I. Viscoelastic Solutions in Terms of Elastic Solutions. The fundamental result is the Classical Correspondence Principle. It is based on the observation that the time Fourier transform (FT) of the governing equations of Linear Viscoelasticity may be obtained by replacing elastic constants by corresponding complex moduli in the FT of the elastic field equations. It follows that, whenever those regions over which different types of boundary conditions are specified do not vary with time, viscoelastic solutions may be generated in terms of elastic solutions that satisfy the same boundary conditions. In practical terms this method is largely restricted to the non-inertial case, since then a wide variety of elastic solutions are available and transform inversion is possible. 

It is interesting, from a fundamental point of view, to characterize the viscoelastic behavior of a fluid, which reflects the forces between the particles and hence the structure of the fluid. This characterization is done dynamically by applying on the fluid a sinusoidal stress with frequency N and low amphtude, so as not to break the structure of the fluid (linear viscoelastic mode). The response is a deformation of the same frequency. Elastic systems have a response in phase with the stress. In a Newtonian system, the shear stress is proportional to the strain rate and the sinusoidal strain response is dephased by 90° compared to the sinusoidal stress applied. The phase angle 5 (0 < 8 < 90°) is therefore a characteristic of viscoelastic behavior. We use the formahsm of complex numbers to write ... 

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