Stress Oscillation

Fig. 28. Photograph of the alternating transparent and stress-whitened zones in poIy(ethylene terephthalate) drawn under stress oscillations...

The multiple stress oscillations are produced by the tumbling orbits of the director as it flips end over end, or winds up, in the shearing flow at high Er. These stress oscillations... 

Note that the director rotation rate is proportional to the shear rate y. Hence, the period of the stress oscillations is equal to the period P for rotation of the director through an angle of 7T, given by Eq. (10-6) ... 

The damping of the stress oscillations presumably arises from a gradual loss of spatial coherence in the phase of the tumbling orbit across the sample. In a plate-and-plate rheometer, the strain is linearly dependent on the radial distance from the axis of rotation. As a result, the gap-averaged director orientation varies as a function of radial position in the sample. When this source of inhomogeneity in the tumbling orbit accounted for by integrating the torque contributions predicted by Eq. (10-31) over... 

Figure 11.18 Predictions of the tumbling parameter A as a function of reduced concentration C/C2 from the Smoluchowski equation for hard rods with the Onsager potential. The exact result from the spherical-harmonic expansion is shown, compared to approximate results from an analytic formula and from the perturbation expansion of Kuzuu and Doi. The open circles (O) are estimates from the periods of shear stress oscillations in transient shearing flows for PEG solutions (see Walker et al. 1995), and the closed circle ( ) is from a direct conoscopic measurement of Muller et al. (1994).

More generally, the linear response of a viscoelastic material always has the stress oscillate at the same frequency as the applied strain, but the stress leads the strain by a phase angle 8. 

Comparing the stress distributions in the coarse and the fine TBC microstructures generated with the higher-order analysis, we observe virtually no difference in the pure ceramic region. In the functionally graded regions, differences are expected due to differences in the microstructural scales. However, one common feature exhibited by the stress distributions in the two mierostructures is the common envelope within which the stress oscillations in the... 

In a typical case of dynamic mechanical analysis, a small stress oscillates periodically in a sinusoidal mode with amplitude cr and frequency co, and the small strain e follows the modulation with a certain phase lag The sinusoidal stress is the imposed stimulation, and in a complex form. 

They are almost completely elastic until the upper yield strength Ren (UYS) is reached. At this stress, plastic deformation sets in rather suddenly, which is localised in so-called Liiders bands or flow lines. While the stress oscillates, these lines extend until they cover the whole specimen. The lowest stress occurring during this process is called lower yield strength i eL (lys). Why this localised plastic deformation occurs, will be explained in section 6.4.3. After the specimen has plastified completely, it behaves identical to a metal without apparent yield point. 

In a viscoelastic system (such as the case with a flocculated suspension), the stress oscillates with the same frequency, but out-of-phase from the strain. From measurement of the time shift between strain and stress amplitudes (At) one can obtain the phase angle shift 3,... 

In dynamic (oscillator) measurements, a sinusoidal strain, with frequency v in Hz or CO in rad s (concentric cylinder) or plate (of a cone and plate) and the stress is measured simultaneously on the bob or the cone, which are connected to a torque bar. The angular displacement of the cup or the plate is measured using a transducer. For a viscoelastic system, such as the case with a cosmetic emulsion, the stress oscillates with the same frequency as the strain, but out-of-phase [11). Figure 12.4 illustrates the stress and strain sine waves for a viscoelastic system. 

In dynamic (oscillatory) measurements, one applies a sinusoidal strain or stress (with amplitudes yo or < o and frequency co in rad s ) and the stress or strain is measured simultaneously. For a viscoelastic system, the stress oscillates with the same frequency as the strain, but out of phase. From the time shift of stress and strain, one can calculate the phase angle shift <5. This allows one to obtain the various viscoelastic parameters G (the complex modulus), G (the storage modulus, i.e. the elastic component of the complex modulus) and G" (the loss modulus or the viscous component of the complex modulus). These viscoelastic parameters are measured as a function of strain amplitude (at constant frequency) to obtain the linear viscoelastic region, whereby G, G and G" are independent of the applied strain until a critical strain above which G and G begin to decrease with further increase of strain, whereas G" shows an increase. Below y the structure of the system is not broken down, whereas above y the structure begins to break. From G and one can obtain the cohesive energy density of the structure... 

Fig. 2.32 Polyisoprene extrusion effect of repeated capillary passes at 150 sec on lower stress plateau-stress oscillation region of flow curves [51]. Numbers in graph are the number of extrusion passes.

The mechanisms of the distortion of IR absorption bands of stressed polymers have been discussed especially by Gubanov [7—9], Kosobukin [13], Vettegren and Novak [15] and Wool [36]. There is a general agreement that the distorted IR absorption band profile D(i ) can be related to a large number of separate oscillators with strongly overlapping absorption bands whose maxima are shifted by different amounts. The possible causes of the frequency shift of an individual stressed oscillator have been considered to be the quasielastic deformation of the harmonic oscillator (re-... 

As discussed previously molecular stresses are a function of molecular strains which depend on sample morphology and chain orientation. Any real sample, therefore, contains a variety of differently stressed oscillators. The profile of the deformed band, D(p), representing such a system of oscillators can be expressed by a convolution integral ... 

Here F (n) is the distribution of stressed oscillators, U v) the shape of the normalized undeformed band, and n an integration variable. The distribution F (p) can be obtained by deconvolution and transformed into a molecular stress distribution by Eq. (8.1). 

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