Stress measurement oscillations

The electromagnetic spectrum measures the absorption of radiation energy as a function of the frequency of the radiation. The loss spectrum measures the absorption of mechanical energy as a function of the frequency of the stress-strain oscillation. 

In the first of these techniques an approximation to uniform rate of shear throughout the sample is achieved by shearing a thin film of the liquid between concentric cylinders. The outer cylinder can be rotated (or oscillated) at a constant rate and the shear stress measured in terms of the deflection of the inner cylinder, which is suspended by a torsion wire (Figure 9.2) or the inner cylinder can be rotated (or oscillated) with the outer cylinder stationary and the resistance offered to the motor measured. 

In a rheomety experiment the two plates or cylinders are moved back and forth relative to one another in an oscillating fashion. The elastic storage modulus (G - The contribution of elastic, i.e. solid-like behaviour to the complex dynamic modulus) and elastic loss modulus (G" - The contribution of viscous, i.e. liquid-like behaviour to the complex modulus) which have units of Pascals are measured as a function of applied stress or oscillation frequency. For purely elastic materials the stress and strain are in phase and hence there is an immediate stress response to the applied strain. In contrast, for purely viscous materials, the strain follows stress by a 90 degree phase lag. For viscoelastic materials the behaviour is somewhere in between and the strain lag is not zero but less than 90 degrees. The complex dynamic modulus ( ) is used to describe the stress-strain relationship (equation 14.1 i is the imaginary number square root of-1). 

When the sin / method (Sec. 16-4) is used, some specimens that have been plastically deformed in the region examined yield values of df that vary with sin in an oscillatory manner, rather than linearly [16.33, 16.34, 16.26]. These oscillations in di are not fully understood. They must be caused by a system of micro-stresses more complex than pseudo-macrostress, because pseudo-macrostress, like true macrostress, yields a linear variation of di with sin ij/. When oscillations occur, the standard two-exposure method of stress measurement (Sec. 16.4) can be seriously in error.)... 

The tools are exactly the same as the ones described in the previous section, the main difference with flow measurements lies in the fact that we now apply a strain or stress that oscillates sinusoidally with time instead of being... 

Vertical oscillations are always present in rotating members. These can be particularly annoying with normal stress measurements on high viscosity materials. Adams and Lodge give an es-... 

Figure 16.8), where the cyclic load (stress or strain) can be applied, with the resulting strain or stress measured. For liquid samples, the geometries discussed in conjunction with rotational viscometry are often used with the drive system modified to produce sinusoidal rather than steady rotational deformation. Flexible samples such as libers, films, and rubber are preloaded in tension and oscillated about a positive tensile strain so that they do not go slack at the bottom of the sine wave. Such tests give dynamic tensile properties, E, E", etc., which are related to the corresponding shear properties by... 

We commented above that the elastic and viscous effects are out of phase with each other by some angle 5 in a viscoelastic material. Since both vary periodically with the same frequency, stress and strain oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the lag between the two waves. Another representation of this situation is shown in Fig. 3.14b, where stress and strain are represented by arrows of different lengths separated by an angle 5. Projections of either one onto the other can be expressed in terms of the sine and cosine of the phase angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and perpendicular to a. Thus we can say that 7 cos 5 is the strain component in phase with the stress and 7 sin 6 is the component out of phase with the stress. We have previously observed that the elastic response is in phase with the stress and the viscous response is out of phase. Hence the ratio of... 

Another resonant frequency instmment is the TA Instmments dynamic mechanical analy2er (DMA). A bar-like specimen is clamped between two pivoted arms and sinusoidally oscillated at its resonant frequency with an ampHtude selected by the operator. An amount of energy equal to that dissipated by the specimen is added on each cycle to maintain a constant ampHtude. The flexural modulus, E is calculated from the resonant frequency, and the makeup energy represents a damping function, which can be related to the loss modulus, E". A newer version of this instmment, the TA Instmments 983 DMA, can also make measurements at fixed frequencies as weU as creep and stress—relaxation measurements. 

A technique for performing dynamic mechanical measurements in which the sample is oscillated mechanically at a fixed frequency. Storage modulus and damping are calculated from the applied strain and the resultant stress and shift in phase angle. 

The degree of vulcanisation of a rubber compound is assessed technically by the indefinite terms of undercure, correct cure, optimum cure and overcure. It may be given precision by (a) measurement of stress-strain relationship of a range of cures, (b) measurement of the modulus at 100% elongation, (c) measurement of the volume swelling in benzene, or (d) by the use of instruments such as the oscillating disc rheometer and the moving die rheometer. 

As with the elastic solid we can see that as the stress is applied the strain increases up to a time t = t. Once the stress is removed we see partial recovery of the strain. Some of the strain has been dissipated in viscous flow. Laboratory measurements often show a high frequency oscillation at short times after a stress is applied or removed just as is observed with the stress relaxation experiment. We can replace a Kelvin model by a distribution of retardation times ... 

The hardness of a polymer can also be estimated from the modulus of elasticity E (high E modulus indicates high hardness). The advantage here is that every region of elasticity and every degree of hardness can be detected with a single kind of measurement (determination of stress-strain-behavior or torsional oscillation). 

The Weissenbeig Rheogoniometer (49) is a complex dynamic viscometer that can measure elastic behavior as well as viscosity. It was the first rheometer designed to measure both shear and normal stresses and can be used for complete characterization of viscoelastic materials. Its capabilities include measurement of steady-state rotational shear within a viscosity range of 10-1 —13 mPa-s at shear rates of 10-4 — 104 s-1, of normal forces (elastic effect) exhibited by the material being sheared, and of an oscillatory shear range of 5 x 10-6 to 50 Hz, from which the elastic modulus and dynamic viscosity can be determined. A unique feature is its ability to superimpose oscillation on steady shear to provide dynamic measurements under flow conditions all measurements can be made over a wide range of temperatures (—50 to 400°C). 

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