Viscoelasticity, linear limit

We can also calculate other viscoelastic properties in the limit of low shear rate (linear viscoelastic limit) near the LST. The above simple spectrum can be integrated to obtain the zero shear viscosity 0, the first normal stress coefficient if/1 at vanishing shear rate, and the equilibrium compliance J... 

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. 

The constant Tr is called the Trouton ratio10 and has a value of 3 in this experiment with an incompressible fluid in the linear viscoelastic limit. The elongational behaviour of fluids is probably the most significant of the non-shear parameters, because many complex fluids in practical applications are forced to extend and deform. Studying this parameter is an area of great interest for theoreticians and experimentalists. 

Dynamic measurements, at strain amplitudes within the linear viscoelastic limit, were made to establish the properties of the essentially undisturbed samples (Elliott and Ganz, 1977). Oscillatory experiments are a powerful tool to study the effects of aging, the amount and type of ingredients, and additives such as food gums on the rheological properties and quality of salad dressings (Munoz and Sherman, 1990). 

Brueller, O.S. and Schmidt, H.H., On the linear viscoelastic limit of polymers - Exemplified on poly (methyl methacrylate) . Polymer Engineering and Science 19, 1979, p, 883—887. 

The origin of Deborah s number is indicated in the frontispiece to this text. In Figure 4.1.2 we take the characteristic flow time to be the inverse of the typical deformation rate while in oscillatory flows we use the amplitude of the oscillatory strain times its frequency (yaO)). Die elastic, Newtonian, and linear viscoelastic limits illustrated in Figure 4.1.2 have already been discussed in Ch ters 1, 2, and 3, respectively. Second-order fluids, to be covered shortly, reside in a fringe of the regime of nonlinear viscoelasticity that lies just across the border from the Newtonian domain. 

To perform constant rate squeezing rather than constant stress requires programming the gap to close at an exponentially decreasing rate, eq. 7.2.4. Soskey and Winter (1985) have done this. They were able to get the linear viscoelastic limit, but for e > 1 they found it difficult to determine whether they had strain hardening or simply loss of lubrication. Isayev and Azari (1986) did the simpler constant velocity squeezing experiments. They calculated a biaxial viscosity from their force versus time curves using a differential constitutive model and found behavior very similar to Figure 7.3.5 for a polybutadiene gum (r <, 10 Pa-s). 

Khan and Larson (1991) have shown that step planar squeezing gives the correct linear viscoelastic limit at small strains. Lubrication is lost at strains similar to equibiaxial, < l.S. Further work is needed with this geometry. 

In the linear viscoelastic limit, the DEMG model reduces to the original reptation model of Doi and Edwards, with only the slowest relaxation mode retained. Since the DEMG given by Eqs. 11.38 through 11.40 does not contain reptative constraint release, we have set the relaxation... 

This modification of the convected Maxwell model contains one constant G (an elastic modulus) and the non-Newtonian viscosity function rj y). It describes the shear-rate dependence of the viscosity perfectly and the first normal stress coefficient rather well. In steady elongational flow it gives an infinite elongational viscosity, and does not simplify properly in the linear viscoelastic limit. Nonetheless it has been found to be useful in exploratory flow calculations aimed at assessing the interaction of shear thinning and memory. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

The theories of elastic and viscoelastic materials can be obtained as particular cases of the theory of materials with memory. This theory enables the description of many important mechanical phenomena, such as elastic instability and phenomena accompanying wave propagation. The applicability of the methods of the third approach is, on the other hand, limited to linear problems. It does not seem likely that further generalization to nonlinear problems is possible within the framework of the assumptions of this approach. The results obtained concern problems of linear viscoelasticity. 

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. 

For positive exponent values, the symbol m with m > 0 is used. The spectrum has the same format as in Eq. 8-1, H X) = H0(X/X0)m, however, the positive exponent results in a completely different behavior. One important difference is that the upper limit of the spectrum, 2U, has to be finite in order to avoid divergence of the linear viscoelastic material functions. This prevents the use of approximate solutions of the above type, Eqs. 8-2 to 8-4. 

The response of simple fluids to certain classes of deformation history can be analyzed. That is, a limited number of material functions can be identified which contain all the information necessary to describe the behavior of a substance in any member of that class of deformations. Examples are the viscometric or steady shear flows which require, at most, three independent functions of the shear rate (79), and linear viscoelastic behavior (80,81) which requires only a single function, in this case a relaxation function. The functions themselves must be determined experimentally for each substance. 

The viscoelastic response of polymer melts, that is, Eq. 3.1-19 or 3.1-20, become nonlinear beyond a level of strain y0, specific to their macromolecular structure and the temperature used. Beyond this strain limit of linear viscoelastic response, if, if, and rj become functions of the applied strain. In other words, although the applied deformations are cyclic, large amplitudes take the macromolecular, coiled, and entangled structure far away from equilibrium. In the linear viscoelastic range, on the other hand, the frequency (and temperature) dependence of if, rf, and rj is indicative of the specific macromolecular structure, responding to only small perturbations away from equilibrium. Thus, these dynamic rheological properties, as well as the commonly used dynamic moduli... 

The linear viscoelastic models (LVE), which are widely used to describe the dynamic rheological response of polymer melts below the strain limit of the linear viscoelastic response of polymers. The results obtained are characteristic of and depend on the macromolecular structure. These are widely used as rheology-based structure characterization tools. 

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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