Superposition, flow curves

Whilst the flow curves of materials have received widespread consideration, with the development of many models, the same cannot be said of the temporal changes seen with constant shear rate or stress. Moreover we could argue that after the apparent complexity of linear viscoeleastic systems the non-linear models developed above are very poor cousins. However, it is possible to introduce a little more phenomenological rigour by starting with the Boltzmann superposition integral given in Chapter 4, Equation (4.60). This represents the stress at time t for an applied strain history ... 

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

It has been experimentally demonstrated that the principles of temperature and concentration superposition of flow curves are applicable to melted keroplasts. In other words, in the fixed matrix FC compositions with different filler content, tp and T measured at different temperatures, can be interpolated through a flat and parallel displacement along the coordinate axes (in double logarithmic coordinates). 

A. The Phenomenology of Multiple M dianlsm Rehixation and Separation of the Mechanisms AH of the primary experimental data were obtained by the technique of stress relaxation. Since the primary curves (before shifting) show no unusual features, they will not be reproduced here. The characteristics of multiple relaxation do become evident, however, when an attempt is made to construct a master curve (54), It is observed that while in the glassy and transition regions the behavior is completely normal, in the flow region superposition is impossible. This manifests itself in a number... 

Figure 1.2 illustrates, using the same sample, how the rate of cooling through Tg and storage at room temperature bring into evidence the presence of the enthalphy relaxation effect as a superposition on the heat flow curve shift. Figure 1.2 also shows the extent of the Tg(onset)-value differences due to the presence of these endothermic peaks. It will be clear that a standardised Tg-value determination procedure is necessary to obtain reproducible results ... 

In multiphase polymeric systems the zero-shear Newtonian coefficient of shear viscosity, tjo, is difficult to determine, but the constant stress viscosity, tj , provides an acceptable alternative. The equivalence is easy to note considering superposition of flow curves - if the flow curves superpose at any stress then they also superpose within the Newtonian plateau. 

FIGURE 5.6 Heat balance in a CSTR (a) heat generated by reaction (b) heat removed by flow and transfer to the environment (c) superposition of generation and removal curves. The intersection points are steady states, (d) Superposition of alternative heat removal curves that give only one steady state. 

Fig. 5.1. Creep curve of a polymer sample under tension (schematic). The elongation ALz induced by a constant force applied at zero time is set up by a superposition of an instantaneous elastic response dashed /me), a retarded anelastic part dash-dot line) and viscous flow dotted line). An irreversible elongation is retained after an unloading and the completion of the recovery process...

Extensional flows yield information about rheological behavior that cannot be inferred from shear flow data. The test most widely used is start-up of steady, uniaxial extension. It is common practice to compare the transient tensile stress with the response predicted by the Boltzmann superposition principle using the linear relaxation spectrum a nonlinear response should approach this curve at short times or low strain rates. A transient response that rises significantly above this curve is said to reflect strain-hardening behavior, while a material whose stress falls... 

It is important to understand the creep behavior shown in Figure 16.4 is not a simple superposition of linear elastic and viscous responses. Figure 16.5 shows the typical strain-time curves of ideal elastic material, ideal viscous material, and viscoelastic polymer fibers under constant stress. The ideal elastic material deforms instantaneously as the stress is applied and the stain remains constant with time. The removal of the stress causes the ideal elastic material to return to its original dimension. For the ideal viscous material, the strain increases linearly with time as long as the stress is applied. The removal of the stress does not return the ideal viscous material to the original dimension. This is because the eneigy introduced by the woik of the external stress is dissipated in the flow, leading to a permanent deformation. Both the ideal elastic and viscous responses contribute to the creep-recovery curve of the viscoelastic polymer fibers. However, the creep-recovery curve of viscoelastic polymer fibers is not a simple superposition of these two ideal behaviors. In addition to the ideal responses, the creep-recovery curve of the polymer fibers also includes retarded elastic response, in which... 

Figure 22. Typical power-time curve for the action of cadmium intoxication on a freshwater snail Planorbis comeus). At the arrow, contaminated water is introduced to the sorption chamber at a higher flow rate to guarantee for a quick and effective water exchange. The bar near the ordinate indicates the standard deviation for a superposition of 3 individual traces [182].

Time-temperature superposition (tTs) was carried out for these multi-temperature multi-frequency tests based on Williams-Landel-Ferry (WLF) relationship. It considers the equivalency of time and temperature in the context of free volume theory for an activated flow process in viscoelastic materials such as PET. It has been found the tTs holds for the whole temperature/frequenQr range. The master curve generated from tTs is shown in Figure 13 for the 10 wt. % bamboo-PET composite at the 25.0 C reference temperature. The time-temperature superposition shift factor follows Arrhenius temperature dqrendence according to the expression ... 

The mobility of a material is influenced not only by temperature but also by stress, as was already discussed above when we dealt with plastic flow phenomena. Figure 10 (bottom) shows that, similar to high temperatures, stress also increases the yield stress. Instead of TTS, now time-stress superposition (TSS), employing Eyring stress aaivation, is applied to shift the data onto a master curve that coincides with the master curve obtained by temperature annealing. ... 

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