Linear viscoelastic range

The relaxation phenomenon which has been discussed so far is within the linear viscoelastic range. Under large deformation, the global relaxation time has to include the contribution from the external work Aw done on the lattice site and takes the form (20)... 

Figure H3.1.5 Determination of the linear viscoelastic range with respect to strain for a food shortening. The linear range appears to exist for strains of <0.02%.

Figure H3.1.6 Dependence of the extent of the linear viscoelastic range of a food shortening on the frequency of the applied sinusoidal shear stress. As the stress frequency decreases, the range of linear behavior also decreases.

There is strong interest to analytically describe the fzme-dependence of polymer creep in order to extrapolate the deformation behaviour into otherwise inaccessible time-ranges. Several empirical and thermo-dynamical models have been proposed, such as the Andrade or Findley Potential equation [47,48] or the classical linear and non-linear visco-elastic theories ([36,37,49-51]). In the linear viscoelastic range Findley [48] and Schapery [49] successfully represent the (primary) creep compliance D(t) by a potential equation ... 

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 function F(t — t ) is related, as with the temporary network model of Green and Tobolsky (48) discussed earlier, to the survival probability of a tube segment for a time interval (f — t ) of the strain history (58,59). Finally, this Doi-Edwards model (Eq. 3.4-5) is for monodispersed polymers, and is capable of moderate predictive success in the non linear viscoelastic range. However, it is not capable of predicting strain hardening in elongational flows (Figs. 3.6 and 3.7). 

Time-temperature superposition was first suggested by H. Leaderman who discovered that creep data can be shifted on the horizontal time scale in order to extrapolate beyond the experimentally measured time frame (9-10). The procedure was shown to be valid for any of the viscoelastic functions measured within the linear viscoelastic range of the polymer. The time-temperature superposition procedure was first explicitly applied to experimental data by... 

For deformation within the linear viscoelastic range. Equation 3.76 expresses the generated stress (ao) in terms of an elastic or storage modulus G and a viscous or loss modulus G". 

While the dynamic experiments described above are to be conducted in the linear viscoelastic range, another experiment can be conducted in which the results obtained in the non-linear range are useful. With a controlled-stress rheometer, one can conduct an experiment in which the stress is increased continuously at a constant oscillatory frequency, say 1 Hz. Results obtained in such an experiment are shown schematically in Figure 3-40. As the stress is increased continuously, initially, G and G" remain relatively constant until at a critical value of stress, Oc, the magnitude of G decreases sharply and that of G" also decreases not as sharply after a slight inerease. One may also use the value of the applied stress at which the curves of G and G" intersect... 

Creep-compliance studies conducted in the linear viscoelastic range also provide valuable information on the viscoelastic behavior of foods (Sherman, 1970 Rao, 1992). The existence of linear viscoelastic range may also be determined from torque-sweep dynamic rheological experiments. The creep-compliance curves obtained at all values of applied stresses in linear viscoelastic range should superimpose on each other. In a creep experiment, an undeformed sample is suddenly subjected to a constant shearing stress, Oc. As shown in Figure 3 1, the strain (y) will increase with time and approach a steady state where the strain rate is constant. The data are analyzed in terms of creep-compliance, defined by the relation ... 

In addition, other measurement techniques in the linear viscoelastic range, such as stress relaxation, as well as static tests that determine the modulus are also useful to characterize gels. For food applications, tests that deal with failure, such as the dynamic stress/strain sweep to detect the critical properties at structure failure, the torsional gelometer, and the vane yield stress test that encompasses both small and large strains are very useful. 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

Cubic liquid crystalline systems have been described as clear, stiff gelsJ As such, they show shear thinning after an apparent yield stress has been exceeded. The viscoelastic properties are also typical for the gel character a broad linear viscoelastic range and a frequency-independent elastic component, which is considerably higher than the viscous component, are observed. ... 

FIG. 6. The diagram of a sequence of step strains. The resulted stress can be superposed if all strain is within the linear viscoelastic range (see Eqs. (8-10) after Dealy and Wiss-brun, 1990]. 

In the linear viscoelastic range, various other material functions relate to one another (Ferry, 1980)... 

Time-aging time superposition was possible at all strains up to the yield strain and at all ternperamres smdied. The aging time shift rate, [t, was semi to vary systematically with both temperamre and strain. In the linear viscoelastic range, the shift rate remmned constant. At increasing strains the shift rate was seen to decrease, though this effect was less evident at higher ternperamres. The shift rate increased with increasing temperamre up to approximately 110 °C to 120 T, after which it rtqiidly decreased to zero at Tg. 

Consider a material subjected to an oscillating load of small amplitude that is in the linear viscoelastic range. The angular frequency of the sinusoidal oscillation is sinusoidal stress a will produce a sinusoidal strain b, and vice versa. However, because of the viscous component of the deformation, there will be a phase shift between stress and strain. The pertinent quantities can be represented as follows ... 

DMA experiments are performed under conditions of very small strain so that the material response is in the linear viscoelastic range. This means that the magnitude of stress and strain are linearly related and the deformation behavior is completely described by the complex modulus function, which is a function of time only. The theory applies both for the case of a tensile deformation or simple extension and for shear. In the latter case the comparable modulus is with components G ico) and G" co). As a first-order approximation, E = 3G. The theory is developed assuming deformation under isothermal conditions, and temperature does not appear (nor is implicit) as a variable. 

The viscoelastic analysis for DMA requires that the sample be in the linear viscoelastic range. In practice, this means that the strain/stress behavior is independent of the strain/stress level. Unmodified polymers, such as PMMA and PC, which are amorphous, are not likely to exhibit strain-dependent behavior as long as the strain amplitude is kept below about 0.3%. However, certain filled materials, especially carbon black or sUica-filled rubbers, may... 

In principle, the shear viscosity function in the linear viscoelastic range reduces to... 

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