Experimental Determination of Viscoelastic Properties

There are a great number of techniques for the experimental determination of viscoelastic functions. The techniques most frequently found in the literature are devoted to measuring the relaxation modulus, the creep compliance function, and the components of the complex modulus in either shear, elongational, or flexural mode (1-4). Although the relaxation modulus and creep compliance functions are defined in the time domain, whereas the complex viscoelastic functions are given in the frequency domain, it is possible, in principle, by using Fourier transform, to pass from the time domain to the frequency domain, or vice versa, as discussed earlier. 

Each of these techniques relates the response to the perturbation field with the material function under study through an auxiliary method of analysis. Under some rigorous boundary conditions, this analysis should give the exact solution to the field equations. However, to overcome in practice the inherent technical difficulties, it is necessary to introduce some approximations that can affect the equations just as the boundary conditions do. Some experimental methods are better conceived than others, and, as a general rule, a simple method is always more desirable than a more complex one because it allows these technical difficulties to be eliminated or simplified. On the other hand, most modern experimental equipment incorporates the method of analysis as part of the software, and consequently it is not possible to discern in detail the process of calculation of the physical function under study. 

When the inertial forces can be neglected and the deformations are infinitesimal, the relationships between stress and strain can be assimilated into the relationships between force and displacement through a coefficient directly related to the geometry of the system, which, somewhat inadequately, is called a form factor 

In the analysis of viscoelastic systems by dynamic methods, a linear onedimensional system of second order in terms of force and displacement is customarily used. This approach also requires the introduction of the geometric form factors with the purpose of solving the real problem in a simple way (Fig. 7.1.) 

As shown, a detailed study of the same problem reveals that on some occasions the factors in question depend on the physical properties of the viscoelastic system. Hence, it is important to analyze carefully the hypothesis on which such a reduction of the problem is based, in order to be able to calibrate the quality of both the approximations achieved and the results obtained. Such a critical analysis must be implemented rigorously in the experiments. The previous considerations become more important the nearer one works to the limit stipulated by the technical specifications of 

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An analogous representation for the viscoelastic fluids is possible, see Fig. 1.33. For the experimental determinations of the rheological material properties the... 

The time-temperature superposition principle, t-T, has been a cornerstone of viscoelastometry. It has been invariably used to determine the viscoelastic properties of materials over the required 10 to 15 decades of reduced frequency, COaj, [Ferry, 1980]. Measuring the rheological properties at several levels of temperature, T, over the experimentally accessible frequency range (usually two to four decades wide), then using the t-T shifting, made it possible to constmct the complete isothermal function. 

In non-Newtonian fluids K a also depends on their physical and rheoiogical properties. The contribution of the latter has been normally expressed in terms of the apparent viscosity, and there is general agreement that this dependence is of the form Kj a 0(11 ) % where z can take values between 0.4 to 0.7. In the case of viscoelastic materials, inclusion of the fluid rheology is less straightforward. Several authors have tried to include the effect of elasticity via the Deborah number, which for stirred tanks is defined as the product of a characteristic time of the fluid and impeller speed. However, determination of the former is not an easy task because it is not always possible to characterize experimentally the viscoelastic properties of the fluid. Determination of the characteristic time of the fluid from experimental shear viscosity vs. shear rate curves [29] and from interpolation of published experimental data on viscoelastic properties [30] has been tried in the past. However, values thus obtained are not necessarily representative of the actual behavior of the liquid. At present, inclusion of the Deborah number in dimensional or dimensionless correlations has not been completely successful. 

We now devote several chapters to the experimental determination of the viscoelastic properties whose general features have been surveyed in Chapter 2 and whose interrelations have been summarized in Chapters 3 and 4. 

The ronaining sections of this chapter treat dynamic or oscillatory measurements of viscoelastic properties. Here, in simple shear, the experimental determination is usually a complex ratio of force to displacement (or torque to angular displacement) measured at a surface in contact with the sample or else a force measured... 

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

When the experimentalist set an ambitious objective to evaluate micromechanical properties quantitatively, he will predictably encounter a few fundamental problems. At first, the continuum description which is usually used in contact mechanics might be not applicable for contact areas as small as 1 -10 nm [116,117]. Secondly, since most of the polymers demonstrate a combination of elastic and viscous behaviour, an appropriate model is required to derive the contact area and the stress field upon indentation a viscoelastic and adhesive sample [116,120]. In this case, the duration of the contact and the scanning rate are not unimportant parameters. Moreover, bending of the cantilever results in a complicated motion of the tip including compression, shear and friction effects [131,132]. Third, plastic or inelastic deformation has to be taken into account in data interpretation. Concerning experimental conditions, the most important is to perform a set of calibrations procedures which includes the (x,y,z) calibration of the piezoelectric transducers, the determination of the spring constants of the cantilever, and the evaluation of the tip shape. The experimentalist has to eliminate surface contamination s and be certain about the chemical composition of the tip and the sample. 

The four variables in dynamic oscillatory tests are strain amplitude (or stress amplitude in the case of controlled stress dynamic rheometers), frequency, temperature and time (Gunasekaran and Ak, 2002). Dynamic oscillatory tests can thus take the form of a strain (or stress) amplitude sweep (frequency and temperature held constant), a frequency sweep (strain or stress amplitude and temperature held constant), a temperature sweep (strain or stress amplitude and frequency held constant), or a time sweep (strain or stress amplitude, temperature and frequency held constant). A strain or stress amplitude sweep is normally carried out first to determine the limit of linear viscoelastic behavior. In processing data from both static and dynamic tests it is always necessary to check that measurements were made in the linear region. This is done by calculating viscoelastic properties from the experimental data and determining whether or not they are independent of the magnitude of applied stresses and strains. 

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