Viscoelastic master curves

Figure 2. Viscoelastic master curves of poly(styrene-b-a-meth-ylstyrene). Solid curve sample BPI broken curve sample BPll...

Figure 2. Illustrative Viscoelastic Master Curves Represented on Reduced Frequency Nomograph, Using Simulated Data.

Viscoelastic Master Curves. In order to evaluate whether a given material is suitable for a particular damping application we need to know its viscoelastic properties over a broad range of temperature and frequency. However, in most instances we can measure these quantities only over a limited range of temperature or frequency. The data are then extended to other temperatures and frequencies by using the time-temperature-superposition procedure (5, 6) to form viscoelastic master curves that correlate the data and extend its utility. 

Figure 2. Viscoelastic master curves represented on reduced temperature nomograph. Key solid symbols, modulus values and open symbols, loss tangent values. Insert at upper left shows the shift factor function, aT, used for data reduction.

Klti Kliippel, M. Evaluation of viscoelastic master curves of filled elastomers and applications to fracture mechanics. J. Phys. Condens. Matter 21 (2009) 035104 (10 pages). 

Fig. 9.3 Viscoelastic master curve of HEUR with Mw = 35K and 16 carbons in the end chain [2], The reference temperature is 5°C. The activation energy and the number of elastically effective chains can be found from the shift factors. (Reprinted with permission from Ref. [2].)...

The linear viscoelastic master curve of a material serves as an important fingerprint for its mechanical behavior and the fine features of these master curves correlate with the particular materials molecular details. For these reasons, master curves are widely generated in practice. Below, we illustrate an example [38] of master curve generation where the original data were taken under dynamic testing. Fig. 2 shows the data. Fig. 3 shows the master curve obtained by means of the shift factor calculated from the data in Fig. 2. Finally, Fig. 4 shows a plot of the shift factor that is seen to display WLF-type behavior. 

Fig. 3. Linear viscoelastic master curves of PSA in relation to Dahlquist criterion for tack.

Although difficult, it is possible to measure stress vs. strain curves of PSAs. Examples of such work include that of Christenson et al. [.3J and Piau et al. [23J. One can do this at various elongation rates and temperatures and create a material response function. Of course, it is much easier to obtain rheological data at small strains than to obtain tensile stress-strain data. One can assume a shape of the stress vs. strain function (i.e. a constitutive relationship) and then use the small strain data to assign values to the parameters in such a function. In order for a predictive model of peel to be useful, one should be able to use readily obtained rheological parameters like those obtained from linear viscoelastic master curve measurements and predict peel force master curves. 

Therefore, with the exception of the Giesekus model, the parameters for all of these constitutive equations can be deduced from the relaxation time spectrum of the material which can be obtained from the small strain linear viscoelasticity measurements alone. There are various numerical methods in the literature which allow the determination of this spectrum from measured viscoelastic master curves, such as dynamic modulus, relaxation modulus, and creep compliance. 

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

It was pointed out in Section 26.2.1 that the friction coefficient is considerably larger on a rough track than on a smooth one when the log ajv values of the master curve are small, i.e., when the temperature is high, and the speed is low, i.e., when the viscoelastic losses are low. Moreover, the adhesion friction, which enables tangential stresses to be built up, is low. 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

Dynamic mechanical experiments yield both the elastic modulus of the material and its mechanical damping, or energy dissipation, characteristics. These properties can be determined as a function of frequency (time) and temperature. Application of the time-temperature equivalence principle [1-3] yields master curves like those in Fig. 23.2. The five regions described in the curve are typical of polymer viscoelastic behavior. 

Another often used representation of the viscoelastic flow behavior utilizes normal stress coefficients P/ = Ni/y. Figure 10 depicts flow curves of a family of PAA/water solutions differing in concentrations and therefore in their viscosities. Normalized by the zero-shear viscosity fiQ and by a constant shear rate /q shear stress value of to= 1 N/m they produce master curves for viscosity and the normal stress coefficient. The preparation... 

The free-volume concept was applied most widely in the theory of viscoelastic properties of polymers developed by Williams, Landel and Ferry (WLF theory), presented in detail in12. According to WLF theory, the changes in liquid viscosity with frequency and temperature from glass temperature T% to T may be plotted on a single master curve by using the reduction factor... 

The creep of a viscoelastic body or the stress relaxation of an elasacoviscous one is employed in the evaluation of T] and G. In such studies, the long-time behavior of a material at low temperatures resembles the short-time response at high temperatures. A means of superimposing data over a wide range of temperatures has resulted which permits the mechanical behavior of viscoelastic materials to be expressed as a master curve over a reduced time scale covering as much as twenty decades (powers of ten). 

We have seen in the previous sections that viscoelastic scaling, employing the scaling factor ac, produces master viscosity curves for polymer-gas solutions that are identical to the master curve for the pure polymer. This means that the effect of dissolved gas on the rheology of polymer melts can be described entirely by the variation of ac with gas content. We have not, of course, demonstrated that all polymer-gas systems follow this scaling beha-... 

All polymer-gas systems studied here exhibit ideal viscoelastic scaling, whereby viscosity measurements taken at different gas compositions can be unified to a master curve of reduced viscosity r (c, y)/a versus reduced... 

Stress relaxation master curve. For the poly-a-methylstyrene stress relaxation data in Fig. 1.33 [8], create a master creep curve at Tg (204°C). Identify the glassy, rubbery, viscous and viscoelastic regions of the master curve. Identify each region with a spring-dashpot diagram. Develop a plot of the shift factor, log (ax) versus T, used to create your master curve log (ot) is the horizontal distance that the curve at temperature T was slid to coincide with the master curve. What is the relaxation time of the polymer at the glass transition temperature ... 

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