Linear viscoelasticity relationship

In practice this relationship is only approximately correct because most plastics are not linearly viscoelastic, nor do they obey completely the power law expressed by equation (2.62). However this does not detract from the considerable value of this simple relationship in expressing the approximate solution to a complex problem. For the purposes of engineering design the expression provides results which are sufficiently accurate for most purposes. In addition. 

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. 

Whilst obtaining this is the ultimate goal for many rheologists, in practice it is not possible to develop such an expression. However, our mechanical analogues do allow us to develop linear constitutive equations which allow us to relate the phenomena of linear viscoelastic measurements. For a spring the relationship is straightforward. When any form of shear strain is placed on the sample the shear stress responds instantly and is proportional to the strain. The constant of proportionality is the shear modulus... 

Firstly, it helps to provide a cross-check on whether the response of the material is linear or can be treated as such. Sometimes a material is so fragile that it is not possible to apply a low enough strain or stress to obtain a linear response. However, it is also possible to find non-linear responses with a stress/strain relationship that will allow satisfactory application of some of the basic features of linear viscoelasticity. Comparison between the transformed data and the experiment will indicate the validity of the application of linear models. 

You will notice that this is the expression for a Maxwell model (see Equation 4.25). From Equations (4.121) to (4.125) we have applied a Fourier transform and confirmed that a Maxwell model fits at least this portion of the theory of linear viscoelasticity. The simple expression for the relationship between J (co) and G (co) allows an interesting comparison to be performed. Suppose we take our equations for a Maxwell model and apply Equation (4.108) to transform the response to an oscillating strain into the response for an oscillating stress. This requires careful use of simple algebra to give... 

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 mean times t and tw will be called the number-average and weight-average relaxation times of the terminal region, and tw/t can be regarded as a measure of the breadth of the terminal relaxation time distribution. It should be emphasized that these relationships are merely consequences of linear viscoelastic behavior and depend in no way on assumptions about molecular behavior. The observed relationships between properties such as rj0, J°, and G and molecular parameters provides the primary evidence for judging molecular theories of the long relaxation times in concentrated systems. 

The Eyring analysis does not explicity take chain structures into account, so its molecular picture is not obviously applicable to polymer systems. It also does not appear to predict normal stress differences in shear flow. Consequently, the mechanism of shear-rate dependence and the physical interpretation of the characteristic time t0 are unclear, as are their relationships to molecular structure and to cooperative configurational relaxation as reflected by the linear viscoelastic behavior. At the present time it is uncertain whether the agreement with experiment is simply fortuitous, or whether it signifies some kind of underlying unity in the shear rate dependence of concentrated systems of identical particles, regardless of their structure and the mechanism of interaction. 

Hooke s Law, which states that a proportional relationship exists between stress and strain, usually holds for a viscoelastic material at a small strain. This phenomenon is called linear viscoelasticity (LVE). Within the LVE region, the viscoelastic parameters G and G" remain constant when the amplitude of the applied deformation is changed. Consequently, parameters measured within the LVE region are considered material characteristics at the observation time (frequency). 

Stresses in viscoelastic materials "remember" deformation prehistory and so are not an unambiguous function of instantaneous deformations however, they may be expressed by a functional. For a linear viscoelastic material, the relationship between stresses and deformations... 

Butter and milk fat exhibit viscoelastic behavior at small stresses (Chwiej, 1969 Pijanowski et al., 1969 Shama and Sherman, 1970 Sherman 1976 Shukla and Rizvi, 1995). To probe this behavior, a very small stress or deformation is applied to a sample and the relationships between stress, strain and time are monitored. Viscoelastic testing is performed in the linear viscoelastic region (LVR) where a linear relationship between stress and strain exists and where the sample remains intact. Depending on the material, this region lies at a strain of less than 1.0% (Mulder and Walstra, 1974) or even less than 0.1% (Rohm and Weidinger, 1993). Figure 7.10 shows the small deformation test results for milk fat at 5°C. 

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). 

Static and dynamic linear viscoelastic measurements are used to gain insights into the relationships between cheese structure and rheological behavior. Non-linear viscoelastic measurements have been used to a relatively small degree to measure the response of cheese to large deformations. 

This chapter discusses current research on the use of sulfur in recycled asphaltic concrete pavements. In addition, it describes the results of laboratory tests and theoretical predictions using the latest linear viscoelastic layered pavement analysis methods (15,16) to compare the performance of various sulfur-asphalt concrete pavements with conventional asphalt concrete pavements in a variety of climates. The relationship between pavement distress and performance used in the computer program was established at the AASHTO road test (17). Finally, the results of domestic field tests of sulfur-asphalt pavements are presented along with a discussion of future trends for the utilization of sulfur in the construction of highway pavement materials. 

This assumption of a linear relationship between stress and strain appears to be good for small loads and deformations and allows for the formulation of linear viscoelastic models. There are also non-linear models, but that is an advanced topic that we won t discuss. There are two approaches we can take here. The first is to develop simple mathematical models that are capable of describing the structure of the data (so-called phenomenological models). We will spend some time on these as they provide considerable insight into viscoelastic behavior. Then there are physical theories that attempt to start with simple assumptions concerning the molecules and their interactions and... 

One can express linear viscoelasticity using the relaxation spectrum H X), that is, using the relaxation time X. The relationship between the relaxation modulus and the spectra is ... 

In this book, we review the most basic distinctions and similarities among the rheological (or flow) properties of various complex fluids. We focus especially on their linear viscoelastic behavior, as measured by the frequency-dependent storage and loss moduli G and G" (see Section 1.3.1.4), and on the flow curve— that is, the relationship between the "shear viscosity q and the shear rate y. The storage and loss moduli reveal the mechanical properties of the material at rest, while the flow curve shows how the material changes in response to continuous deformation. A measurement of G and G" is often the most useful way of mechanically characterizing a complex material, while the flow curve q(y ) shows how readily the material can be processed, or shaped into a useful product. The... 

The generalized stress-strain relationships in linear viscoelasticity can be obtained directly from the generalized Hooke s law, described by Eqs. (4.85) and (4.118), by using the so-called correspondence principle. This principle establishes that if an elastic solution to a stress analysis is known, the corresponding viscoelastic (complex plane) solution can be obtained by substituting for the elastic quantities the -multiplied Laplace transforms (8 p. 509). The appUcation of this principle to Eq. (4.85) gives... 

Linear viscoelasticity theory predicts that one component of a complex viscoelastic function can be obtained from the other one by means of the Kronig-Kramers relations (10-12). For example, the substitution of G t) — Ge given by Eq. (6.8b) into Eq. (6.3) leads to the relationship... 

It is clear that neither the additional torque nor its derivative is a linear function of the torsion angle. Consequently the corresponding viscoelastic relationships will not be either. 

The solution of a problem in linear viscoelasticity requires the determination of the stress, strain, and displacement histories as a function of the space coordinates. The uniqueness of the solution was proved originally by Volterra (11). The analysis carried out in this chapter refers exclusively to isotropic materials under isothermal conditions. As a rule, it is not possible to give a closed solution to a viscoelastic problem without previous knowledge of the material functions. The experimental determination of such functions and the relationships among them have been studied in a specific way in separate chapters, and therefore the reader s knowledge of them is assumed. At the same time, the methods of analysis carried out in this chapter and in Chapter 17 will allow us to optimize the calculation of the material functions. 

Chapters 5 and 6 discuss how the mechanical characteristics of a material (solid, liquid, or viscoelastic) can be defined by comparing the mean relaxation time and the time scale of both creep and relaxation experiments, in which the transient creep compliance function and the transient relaxation modulus for viscoelastic materials can be determined. These chapters explain how the Boltzmann superposition principle can be applied to predict the evolution of either the deformation or the stress for continuous and discontinuous mechanical histories in linear viscoelasticity. Mathematical relationships between transient compliance functions and transient relaxation moduli are obtained, and interrelations between viscoelastic functions in the time and frequency domains are given. 

It is remarkable that several predictions about linear viscoelastic properties in the terminal region can be obtained without special assumptions about the stress- orientation relationship. Thus, for a simple shear strain y which is small enough to evoke only a linear response, the shear stress after the rapid equilibratin process can be written... 

It is now easy to understand why the viscoelastic behavior of filled vulcanizates at large strains is so complex and why it is so difficult to investigate in definitive manner. None of the processes responsible for stress-softening occur instantaneously (16,182), nor can they generally be expected to obey the time-temperature relationships of linear viscoelasticity. Since time-temperature superposition is no longer applicable,... 

When we progress from the foregoing qualitative discussion of structure-property relationships to the quantitative specification of mechanical properties, we enter a jungle that has been only partially explored. The most convenient point of departure into this large and complex subject is provided by the topic of "linear viscoelasticity." Linear viscoelasticity represents a relatively simple extension of classical (small-strain) theory of elasticity. In situations where linear viscoelasticity applies, the mechanical properties can be determined from a few experiments and can be specified in any of several equivalent formulations (11). 

Overall, the regime of linear viscoelasticity is characterized by reasonable success in establishing structure-property relationships. The properties themselves are unambiguously and simply specifiable. The relevant structural features are largely recognizable aspects of molecular structure. Molecular theories exist that provide a bridge between the molecular structure and the macroscopic viscoelastic properties. 

Linear viscoelasticity is the simplest type of viscoelastic behavior observed in polymeric liquids and solids. This behavior is observed when the deformation is very small or at the initial stage of a large deformation. The relationship between stress and strain may be defined in terms of the relaxation modulus, a scalar quantity. This is defined in Equation 22.7 for a sudden shear deformation ... 

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