The torsion pendulum

The simplest device consists of a specimen of circular cross-section suspended vertically with its upper end rigidly elamped [8]. Its lower end supports a dise, or 

For an elastic rod the equation of motion is 19 + rO = 0, where T is the torsional rigidity of the rod, which is related to the shear modulus G through 

The high sensitivity to sample dimensions implies that inter-specimen comparisons can be subject to large uncertainties. 

A Torsion wire B Inertia disc C Mirror D Specimen 

When the vibrations are damped the amplitude decreases with time, but with light damping there is only a small effect on the period 

For linear viscoelastic solids, the torsional modulus is complex, and may be written as G = Gi + 1G2. When the damping is small, it is justified to replace Gi for G in Equation (6.1), hence 

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Glass-transition temperatures are commonly determined by differential scanning calorimetry or dynamic mechanical analysis. Many reported values have been measured by dilatometric methods however, methods based on the torsional pendulum, strain gauge, and refractivity also give results which are ia good agreement. Vicat temperature and britde poiat yield only approximate transition temperature values but are useful because of the simplicity of measurement. The reported T values for a large number of polymers may be found ia References 5, 6, 12, and 13. 

Free- Vibration Methods. Free-vibration instmments subject a specimen to a displacement and allow it to vibrate freely. The oscillations are monitored for frequency and damping characteristics as they disappear. The displacement is repeated again and again as the specimen is heated or cooled. The results are used to calculate storage and loss modulus data. The torsional pendulum and torsional braid analy2er (TBA) are examples of free-vibration instmments. 

The Metravib Micromecanalyser is an inverted torsional pendulum, but unlike the torsional pendulums described eadier, it can be operated as a forced-vibration instmment. It is fully computerized and automatically determines G, and tan 5 as a function of temperature at low frequencies (10 1 Hz). Stress relaxation and creep measurements are also possible. The temperature range is —170 to 400°C. The Micromecanalyser probably has been used more for the characterization of glasses and metals than for polymers, but has proved useful for determining glassy-state relaxations and microstmctures of polymer blends (285) and latex films (286). 

Torsion property As noted, the shear modulus is usually obtained by using pendulum and oscillatory rheometer techniques. The torsional pendulum (ASTM D 2236 Dynamic Mechanical Properties of Plastics by Means of a Torsional Pendulum Test Procedure) is a popular test, since it is applicable to virtually all plastics and uses a simple specimen readily fabricated by all commercial processes or easily cut from fabricated products. 

One such case arises in the theory of clocks. As is known, a dock is a mechanism consisting of two parts a torsional pendulum with a small damping, and an escapement mechanism replenishing the energy lost by damping in the torsional pendulum. 

In the phase plane the trajectory of the torsional pendulum between... 

Dynamic shear moduli are conveniently determined with automated equipment, for instance, with the torsion pendulum. However, moduli derived from dynamic tests are often higher than the results from static tests for lack of relaxation. Examples are shown in Table 3.3. Young s moduli of the polymers A, B, C, D, derived from tensile tests (frequency 0.01 Hz) are compared with shear moduli S determined with the torsion pendulum (frequency > 1 Hz). For rubberlike materials is 3S/E = 1, according to Eq. 

ISO 537 International Standard Plastics — Testing with the torsion pendulum... 

This second group of tests is designed to measure the mechanical response of a substance to applied vibrational loads or strains. Both temperature and frequency can be varied, and thus contribute to the information that these tests can provide. There are a number of such tests, of which the major ones are probably the torsion pendulum and dynamic mechanical thermal analysis (DMTA). The underlying principles of these dynamic tests have been covered earlier. Such tests are used as relatively rapid methods of characterisation and evaluation of viscoelastic polymers, including the measurement of T, the study of the curing characteristics of thermosets, and the study of polymer blends and their compatibility. They can be used in essentially non-destructive modes and, unlike the majority of measurements made in non-dynamic tests, they yield data on continuous properties of polymeric materials, rather than discontinuous ones, as are any of the types of strength which are measured routinely. 

We may now express the damping term of the torsion pendulum experiment in terms of the dissipation factor by the simple equation (an approximation which holds for most cases) ... 

Before considering particular test methods, it is useful to survey the principles and terms used in dynamic testing. There are basically two classes of dynamic motion, free vibration in which the test piece is set into oscillation and the amplitude allowed to decay due to damping in the system, and forced vibration in which the oscillation is maintained by external means. These are illustrated in Figure 9.1 together with a subdivision of forced vibration in which the test piece is subjected to a series of half-cycles. The two classes could be sub-divided in a number of ways, for example forced vibration machines may operate at resonance or away from resonance. Wave propagation (e.g. ultrasonics) is a form of forced vibration method and rebound resilience is a simple unforced method consisting of one half-cycle. The most common type of free vibration apparatus is the torsion pendulum. 

The torsional pendulum principle, via a braid sample, was used to measure the variation of shear modulus with temperature, and the loss angle, and hence to derive the glass transition temperature. This gave for the gelled but uncured material Tg 80°C, and for the material which had been kept at 70°C for 120 h, and could therefore be regarded as fully crosslinked, Tg -60°C. 

The problems due to anisotropy may essentially be divided into two categories. First there is the problem that often the particular solution used to calculate a modulus itself assumes isotropy, e.g. for the torsion pendulum few authors use the full expressions when considering anisotropic materials. These problems may, in general, be overcome by finding solutions for anisotropic materials. 

We thank W. Geymayer for the electron microscopic plates, G. Schreyer and J. Hennig for carrying out the torsion pendulum measurements, W. Wunderlich for determining molecular weights, and H. Knoll for technical assistance. 

One of most popular techniques for dynamic mechanical analysis is the torsion pendulum method. In a modification of this method designed to follow curing processes, a torsion bar is manufactured from a braid of fibers impregnated with the composition to be studied this is the so-called torsional braid analysis (TBA) method.61 62,148 The forced harmonic oscillation method has been also used and has proven to be valuable. This method employs various types of rheogoniometers and vibroreometers,1 9,150 which measure the absolute values of the viscoelastic properties of the system under study these properties can be measured at any stage of the process. The use of computers further contributes to improvements in dynamic mechanical analysis methods for rheokinetic measurements. As will be seen below, new possibilities are opened up by applying computer methods to results of dynamic measurements. 

The torsion pendulum is detailed in ISO 4663, where one end of the sample is fixed and the other can rotate freely. Frequencies of vibration of between 0.1... 

Various types of vibration experiments can be carried out to measure E and E2 at a certain frequency. An example is the torsion pendulum, in which the sample, connected to an auxiliary mass, is brought into a free torsional oscillation. From the frequency of the pendulum (around 1 to 10 sec) E is calculated, from the rate of damping tan 8 and E2. Other types of dynamic mechanical measurements can be carried out at higher frequencies, such as bending vibrations with or without extra mass, wave propagation, etc. By combining a number of these different techniques, a time scale ranging from 10 to 10"8 sec can be covered. 

The loss factor, tan 8, can be measured with the aid of dynamic-mechanical experiments (such as the torsion pendulum). The deformation in such a test varies as indicated in Figure 7.13 the damping follows from the logarithmic decrement , A, it can be easily shown that... 

It should be noted that a TL, l would not be expected in this polystyrene) braid sample because of its high molecular weight (approximately 100,000) (46). Also, this analysis of the composite structure of the braid supports Nielsen s (47) suggestion that rL, L arises solely from the residual elasticity of the glass. The torsional pendulum, braid, and TL, l will be discussed in more detail elsewhere (16,17). 

All of the Tg data reported in Table II were obtained with the resilience method a few selected results were doubly checked with the values obtained from the torsion-pendulum data. Note the identical value of the Tg for MS/AN/MMA measured by the resilience method and that obtained with the torsion-pendulum data on Figure 12. 

Details of the torsion pendulum analyses were given previously (5). Pneumatic impact strengths were determined by a well-known method (7). 

The torsion pendulum determinations were made at low frequency ( lHz) while the duration of the impact strength measurements is on the order of 1 msec. To correct for this frequency difference, we used the WLF equation (9) and found that the 10°C (1 Hz) tan 8 information should approximately correspond to the impact measurements taken at room temperature. In Figure 1, we plot the impact strengths vs. tan 8 (10°C). A single linear correlation does not exist, but the tan 8 values predict that the S sample would have a higher impact strength than the B/S sample and that the B sample, at the other extreme, would have a lower impact strength than the S/B5 specimen. Most models would favor... 

One can determine the shear modulus of a fiber from a torque per unit area versus twist curve. In practice, a simple apparatus called a torsion penduliun is used more commonly. An experimental setup to measure the shear modulus of small fibers is shown in Fig. 9.8 (Mehta, 1996). The torsional pendulum, placed in a vacuum oven, allows the measurement of shear modulus as a func-... 

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