Small-Amplitude Oscillatory Motion

The next experiment we consider is that studied by Fraenkel (29) and by Kirkwood and Auer (39). In this experiment a fluid is placed between two flat plates perpendicular to the y-direction, one of which is fixed and the other is oscillating in the x-direction with very small amplitude. The gap between the plates is sufficiently small that the velocity profile is almost linear the velocity gradient is varying sinusoidally with time, so that fc(t) = PAe kV , where k° is a complex quantity which specifies the amplitude and phase of the oscillating velocity gradient, and o is the 

Here ryx and xfj are complex quantities specifying the phase and amplitudes of the oscillating stresses, and dp a real quantity, is the displacement of the oscillatory normal stress from the time axis. 

We now define the frequency-dependent material functions, tf, 8, and 0 thus  

Similar definitions can be made for p and p1 related to the oscillating secondary normal stress difference. The quantities 0 and p1 are real, whereas tf, 8, and (t are complex. It is customary to write these complex quantities thus12  

For this small-amplitude oscillatory flow, we have to solve Eq. (5.1) with K(t) = 0ls K°eimt. A form for ip may be postulated  

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Example 3.3 Small Amplitude Oscillatory Motion of a Linear Viscoelastic Body... 

Small-Amplitude Oscillatory Motion ( Dynamic Testing )... 

This expression when substituted into Eq. (7.25) will give for small-amplitude oscillatory motion exactly the expression for >7 in Eq. (7.17). Linear viscoelasticity theory cannot, however, predict the normal stress behavior. 

In the limit as Km tends to zero, the expressions in Eqs. (8.16)—(8.18) simplify properly to the formulas obtained previously for small-amplitude oscillatory motion. 

In small-amplitude oscillatory motion, the rigid dumbbells give a non-zero high-frequency asymptote for t] whereas the elastic dumbbells do not. 

The corresponding complete numerical solutions may be seat in Fig. 11. For small-amplitude oscillatory motion the dynamic properties are ... 

Another is the Small amplitude oscillatory motion, usually tangential, between two solid surfaces in contact." Note Here the term fretting refers only to the nature of the motion without reference to wear, corrosion or other damage that may follow. The term fretting is often used to denote fretting corrosion and other forms of fretting wear. 

In contrast to simple elastic solids and viscous liquids, the situation with polymeric fluids is somewhat more complicated. Polymer melts (and most adhesives are composed of polymers) display elements of both Newtonian fluid behavior and elastic solid behavior, depending on the temperature and the rate at which deformation takes place. One therefore characterizes polymers as viscoelastic materials. Furthermore, if either the total strain or the rate of strain is low, the behavior may be described as one of linear or infinitesimal viscoelasticity. In such a case, the stress-deformation relationship (the constitutive equation) involves not just a single time-independent constant but a set of constants called the relaxation spectrum,(2) and this, too, may be determined from a single stress relaxation experiment, or an experiment involving small-amplitude oscillatory motion. 

Fretting corrosion is a combined wear and corrosion process in which material is removed from contacting surfaces when motion between the surfaces is restricted to very small amplitude oscillations (often, the relative movement is barely discernible ranging from a few tens of nanometers to a few tens of micrometers). Fretting occurs where low-amplitude oscillatory motion in the tangential direction takes place between contacting surfaces, which are nominally at rest. (Waterhouse) It is necessary that the load be sufficient to produce a distortion of surfaces. This is a common occurrence, since most machinery is subjected to vibration, both in transit and in operation. Figure 6.41 shows a typical example. Pressed-on wheels can often fret at the shaft/wheel hole interface.31... 

Bates 1984 Fredrickson and Larson 1987 Fredrickson andFIelfand 1988). The relaxation of these fluctuations involves collective motion of many molecules, and thus it is slower than the relaxation time of individual molecules. In small-amplitude oscillatory shearing, the fluctuation waveform is deformed, producing a slowly relaxing stress. Presumably, this accounts for (a) the anomalous contribution to G and (b) a similar, but smaller, contribution to G" (Rosedale and Bates 1990 Jin and Lodge 1997). (Similar anomalies are observed in polymer blends.) An asymmetric version of this PEP-PEE polymer that forms cylindrical domains shows an even larger low-frequency anomaly (Almdal et al. 1992). 

Having discussed steady-state shear flow in 6 and small-amplitude oscillatory shearing motion in 7, we now consider a superposition of these two types of flow this type of superposed flow has only recently been studied. 

In practice, this model is oversimplified since the exciting wake shedding is by no means harmonic and is itself coupled with the shape oscillations and since Eq. (7-30) is strictly valid only for small oscillations and stationary fluid particles. However, this simple model provides a conceptual basis to explain certain features of the oscillatory motion. For example, the period of oscillation, after an initial transient (El), becomes quite regular while the amplitude is highly irregular (E3, S4, S5). Beats have also been observed in drop oscillations (D4). If /w and are of equal magnitude, one would expect resonance to occur, and this is one proposed mechanism for breakage of drops and bubbles (Chapter 12). 

One definition of fretting is "Wear phenomena occurring between two contacting surfaces having oscillatory relative motion of small amplitude." Note Fretting is a term frequently used to include fretting corrosion. This usage is not recommended. 

Because the largest pressure amplitudes are generated for wavelengths less than the contact width, the maximum variation in stress occurs near the surface. The overall effect of the oscillatory motion on the maximum stress is, therefore, small. 

To develop the tube theory of polymer motion, we consider the response of the melt to a step deformation. This is an idealized deformation that is so rapid that during the step no polymer relaxation can occur, and the polymer is forced to deform affinely, that is, to the same degree as the macroscopic sample is deformed. The total deformation, though rapid, is small, so that the chains deform only slightly this is called a small amplitude step strain. Because the deformation is very small, the distribution of chain configurations remains nearly Gaussian, and linear viscoelastic behavior is expected. In Chapter 4 we saw that the assumption of linear behavior makes it possible to use the response to a small step strain experiment to calculate the response to oscillatory shear or any other prescribed deformation. 

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