Relaxation modulus function

For a blend system consisting of two monodisperse components whose molecular weights are both greater than Mg and far apart from each other (i.e. M2 > Ml > Me), the blending law can be basically described by the following relaxation-modulus function (not covering the time region of the... 

Solve them to show that the stress relaxation modulus function G(t) is given Iqr... 

The formulations of (47) and (55) have been criticized by Leonov [L9, Lll], among others, as not being tested for consistency with the second law of thermodynamics. For Newtonian fluids, such testing requires a positive shear viscosity. For a linear viscoelastic material, one may show that the relaxation modulus function must be always positive to satisfy the second law. The requirements for Eqs. (47) and (55) are not so clear. Leonov has sought to develop nonlinear viscoelastic rheological models based on thermodynamic arguments. 

The inelastic liquid is recovered with the relaxation modulus function being set to... 

The most often used relaxation modulus function is the multi-mode Maxwell memory function ... 

For the alternative generalization of the Zener solid shown below, derive the differential equations relating strain, stress, and time. Solve them to show that the stress relaxation modulus function G(r) is given by... 

In this study we used the iteration method with regularization and the relaxation modulus function was used to determine the linear rheological parameters of the materials. For the use of the regularization equation, the experimental window (( max iiim) was divided into several equal intervals on a logarithmic scale. It was assumed that the middle of every logarithmic interval is equal to a relaxation time X . Matlab software was used to programme the methods. 

Equation (9) has an empirical origin but a theoretical foundation can be proposed as follows. Indeed, quite a common assumption in many approaches of nonlinear viscoelasticity consists in considering time-strain separability (or factorability). Such an assumption readily means that the nonlinear relaxation modulus function G(t, y) can be separated into a time-dependent and a strain-dependent contributions, so that ... 

Kontou and Spathis [44] carried out an investigation into the relationship between long-term viscoelasticity and viscoplastic responses of two types of ethylene-vinyl acetate metallocene-catalysed linear low-density polyethylene using DSC, DMTA and tensile testing. A relaxation modulus function with respect to time was obtained from values of relaxation spectra and treated as a material property. This relaxation modulus function was used to describe the corresponding tensile data and a constitutive analysis, which accounts for the viscoelastic path at small strains and the viscoplastic path at high strains, was employed to predict the tensile behaviour of the ethylene polymers (see also [45 9]). 

What form does the relaxation modulus function have Figure 3.1.3 indicates that we should try an exponential decay... 

In practice, the conversion of an experimental data set, G, G, o, into a relaxation modulus function is usually carried out by representing G(t) in terms of a generalized Maxwell model. Thus, the oscillatory shear data are transformed into a set [Gj, Tj], i.e., a discrete relaxation spectrum. This transformation is based on the discrete form of Eqs. 4.40a, b ... 

Making use of equations presented earlier in this section, one can show that if the discrete spectrum is an accurate representation of the entire relaxation modulus function, the moments defined above can be used to calculate combinations of linear properties as follows ... 

Once they have been shaped into a suitable object, rubber compounds are vulcanized in such a manner that the full development of their mechanical properties is achieved, without creep phenomena that are normally exhibited by all polymer materials when on their rubbery plateau. In other terms, vulcanization extends toward infinity the rubbery plateau of the relaxation modulus function G t). Furthermore, reinforcing fillers somewhat increase the magnitude of the modulus at a given time. The combination of vulcanization and reinforcing effects induces quite complex changes in material functions of polymers, as easily demonstrated through purposely simple calculations. 

Let us consider, for instance, the relaxation modulus function G(T) of a pure SBR, as reported by Nielsen (Figure 5.25). The effect of CB loading (at constant temperature) can be approached by rewriting the Guth, Gold and Simha equation as follows ... 

Several functions are used to characterize tire response of a material to an applied strain or stress [4T]. The tensile relaxation modulus E(t) describes tire response to tire application of a constant tensile strain l/e -. 

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. 

The transition strongly affects the molecular mobility, which leads to large changes in rheology. For a direct observation of the relaxation pattern, one may, for instance, impose a small step shear strain y0 on samples near LST while measuring the shear stress response T12(t) as a function of time. The result is the shear stress relaxation function G(t) = T12(t)/ < >, also called relaxation modulus. Since the concept of a relaxation modulus applies to liquids as well as to solids, it is well suited for describing the LST. 

Fig. 2. Relaxation modulus G(t) of a set of polydimethylsiloxane samples with increasing extent of crosslinking plotted against time of crosslinking. The linear PDMS chains (Mn 10 000, polydis-persity index 2) were endlinked with a four-functional silane crosslinker catalyzed by a platinum compound. Samples with different extent of reaction were prepared by poisoning the reaction at different times. The actual extent of reaction was not determined. Two of the samples are clearly before the gel point (LST) and two beyond. The third sample is very close to the gel point. Data of Chambon and Winter [5] evaluated by Baumgartel and Winter [8]...

Adolf and Martin [15] postulated, since the near critical gels are self-similar, that a change in the extent of cure results in a mere change in scale, but the functional form of the relaxation modulus remains the same. They accounted for this change in scale by redefinition of time and by a suitable redefinition of the equilibrium modulus. The data were rescaled as G /Ge(p) and G"/Ge(p) over (oimax(p). The result is a set of master curves, one for the sol (Fig. 23a) and one for the gel (Fig. 23 b). 

Here m is the usual small-strain tensile stress-relaxation modulus as described and observed in linear viscoelastic response [i.e., the same E(l) as that discussed up to this point in the chapter). The nonlinearity function describes the shape of the isochronal stress-strain curve. It is a simple function of A, which, however, depends on the type of deformation. Thus for uniaxial extension,... 

Figure 22 Stress-relaxation modulus as a function of Crystallinity at temperatures above I f Numbers on the curves are rough values of the degree of Crystallinity.

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Figure 5. Shear relaxation modulus of NR as a function of crosslink density at 25°C.

FIGURE 14.10 Logarithm of the relaxation modulus as a function of temperature for three polymer samples. Sample (a) is (largely) crystalline vinyl pol5uner sample (b) is an amorphous vinyl polymer that contains light cross-linking and sample (c) is an amorphous vinyl pol5uner. The Tg for the amorphous polymer is about 100°C and the for the crystalline polymer is about 180°C. 

The shear stress relaxation modulus of the fluid, G(t), is a monotonically decreasing function of time, with G(oo) = 0. If the fluid initially at rest is given a small shear deformation y0 at t=0, the shear stress at later times becomes simply ... 

For our present purposes, the network theories suffer from an additional defect. They supply no information on the form of the memory function. The memory function must be obtained for each system by rheological experiments, and there is no way at present to predict how it should vary with the molecular structure of the polymer. For example, M(t) can be obtained from the stress relaxation modulus G(t) ... 

Fig. 5. Reduced relaxation modulus for the linear array as a function of molecular...

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 ... 

A series of curves drawn for the same molecular weights of Fig. 5 is presented in Fig. 6. Notice that the modulus function falls off much more rapidly in the three-dimensional case, reflecting the sharper distribution of relaxation times. (The broken line indicates the slope-1/2 associated with the linear chain behavior.)... 

In the previous two sections the way in which the relaxation times can be calculated has been shown. As soon as these relaxation times axe known, the construction of the relaxation modulus in shear should not be too difficult a task. From this modulus Lodge s memory function can be derived with the aid of eq. (2.6). Other rheological properties can then be calculated using Lodge s equations. 

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