Viscoelasticity functions

The relaxation modulus (or any other viscoelastic function) thus obtained is a mean s of characterizing a material. In fact relaxation spectra have been found very useful in understanding molecular motions of plastics. Much of the relation between the molecular structure and the overall behavior of amorphous plastics is now known. 

Table 1. Phenomenological viscoelastic functions (independent of molecular models)...

The viscoelastic functions derivable from this model are presented in Table 3. It should be emphasized that these results are the same whether or not the basic responding unit of the linear lattice is the gaussian segment (RB model) or the (monomer level) torsional oscillator (DTO... 

Table 3. Viscoelastic functions of the linear array Name Form Reference...

Table 5. Viscoelastic functions of the cubic array in the continuum limit... 

A summary of analytic expressions obtained in this manner for all the viscoelastic functions is presented in Table 4 and 5 for the linear and cubic arrays. The well-known phenomenological analogy (8) between dynamic compliance and dielectric permittivity allows the formal use of Eqs. (T 5), (T 6), and (T 11), (T 12) for the dielectric constant, e (co), and loss, e"(co), of the linear and cubic arrays, respectively (see Table 6). The derivations of these equations are elaborated in the next section and certain molecular weight trends are discussed. 

The reduced expressions of Table 4 form a set of universal viscoelastic functions. Given the polymer molecular weight, material constants [Jg, Je, etc.), and one extremal relaxation/retardation time, one should be able to predict, roughly, the nature of the system response (within the framework of the linear models) from Eqs. (T 1)—(T 6) and Fig. 2—5. 

Calculation of the viscoelastic functions proceeds as above where, for example, Eq. (T 7) is the reduced relaxation modulus for the cubic array. The incomplete gamma function of order 5/2 may be obtained in simpler form through a recurrence relation and ... 

Up to this point concern has been with viscoelastic functions. In view of the phenomenological analogy between the dielectric permittivity and... 

An extensive study by Koppelmann (1958) of viscoelastic functions through one of the secondary mechanisms in Poly(methyl methacrylate) is shown in Fig. 13.15. Dynamic storage moduli and tan <5 s near 25 °C are plotted vs. angular frequency. It shows first, that a secondary mechanism is present, secondly that E and G are not completely parallel, because the Poisson constant is not a real constant, but also dependent on frequency (the numbers in between both moduli are the actual, calculated Poisson constants) and thirdly that tan <5E is practically equal to tan <5G. 

The time-temperature equivalence principle can also be applied to other viscoelastic functions in a similar way. Again, this leads to shift factors that are identical with those obtained from stress relaxation ... 

Interrelations between different viscoelastic functions of the same material... 

FIG. 13.59 Accuracy of interrelationships between static and dynamic viscoelastic functions. 

As a result, the microscopic viscoelastic function g as a function of y0 for the idealized model is given as... 

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... 

The same shift factor, a-j-, must superpose all of the viscoelastic functions. One must first perform the time-temperature shift on one of the viscoelastic functions and determine the values of the WLF constants. The same constants must then be applied to the other viscoelastic functions to determine their consistency in shifting the data. This process may need to be repeated several times in order to determine the best set of... 

The nomograph is created by plotting the viscoelastic function vs. reduced frequency. Reduced frequency is defined as where... 

The reference temperature (T ) is chosen differently depending on whether an empirical data fit or a "universal" WLF fit are carried out. If the data are fit empirically, T, should be taken as the value that gives the best fit of the data. The values of constants Cj and C2 are then calculated after superposition. This is done by shifting the curves and must be performed prior to using the nomograph program. The empirically determined constants may then be substituted in the WLF equation and a nomograph of the desired viscoelastic functions plotted. 

As can be seen by inspection, the time-temperature superposition for these WLF values appears quite good. The same WLF equation was used to shift the viscoelastic functions E and E". 

One may use the linear viscoelastic data as a pure rheological characterization, and relate the viscoelastic parameters to some processing or final properties of the material inder study. Furthermore, linear viscoelasticity and nonlinear viscoelasticity are not different fields that would be disconnected in most cases, a linear viscoelastic function (relaxation fimction, memory function or distribution of relaxation times) is used as the kernel of non linear constitutive equations, either of the differential or integral form. That means that if we could define a general nonlinear constitutive equation that would work for all flexible chains, the knowledge of a single linear viscoelastic function would lead to all rheological properties. 

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

All viscoelastic functions may be expressed in terms of a single reptation parameter (for example the plateau modulus or tube diameter) and the monomeric friction coefficient (or mobility factor in our terminology), in agreement with the above phenomenological presentation. 

Viscoelastic function in the whole time frequency domain Thus the relaxation modulus may be calculated from a very limited number of... 

The relaxation modulus is the core of most of the viscoelastic descriptions and the above expression can be checked from experimental viscoelastic functions such as the complex shear modulus G (co) for instance. In addition to the molecTilar weight distribution function P(M), one has to know a few additional parameters related to the chemical species the monomeric relaxation time x,... 

Rheological properties of food materials over a wide range of phase behavior can be expressed in terms of viscous (viscometric), elastic and viscoelastic functions which relate some components of flie stress tensor to specific components of the strain or shear rate response. In terms of fluid and solid phases, viscometric... 

For a specific food, magnitudes of G and G are influenced by frequency, temperature, and strain. For strain values within the linear range of deformation, G and G" are independent of strain. The loss tangent, is the ratio of the energy dissipated to that stored per cycle of deformation. These viscoelastic functions have been found to play important roles in the rheology of structured polysaccharides. One can also employ notation using complex variables and define a complex modulus G (o) ... 

The appropriate viscoelastic functions are the dynamic rheological properties (storage modulus G and the loss modulus G", and the dynamic viscosities f and T ") extrapolated to infinite dilution and are called the intrinsic dynamic rheological properties ... 

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