Torsion pendulum

To find convenient expressions for the storage modulus and the loss tangent for a viscoelastic material under free oscillation in torsion, it is necessary to return to the equation of motion given by 

As is well known, the solution of Eq. (7.1) is a damped sinusoidal wave that can be written as 

The logarithmic decrement A, as a measure of the damping, can be defined as the logarithm of the quotient of the amplitude of two successive waves. Then 

Normally the elastic support friction Sp can be disregarded, and the frequency of the wave can be written as 

The constant S can be calculated in a measurement carried out without the test sample. In fact, in many measurements the equipment is prepared in such a way that S = 0 that is, the purely elastic rigidity of the suspension thread, E, is negligible. Since the flexibility of suspension wires of cylindrical cross section depends on the fourth power of the diameter, this requirement is achieved by using a wire of very small diameter. In these conditions 

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This involves the determination of the damping of the oscillations of a torsion pendulum, disk, or ring such as illustrated in Fig. IV-8. Gaines [1] gives the equation... 

Torsional Braid Analyzf Torsional pendulum Tortoiseshell Torulaspora rosei Torula yeast... 

The melting point of commercial Teflon PEA is 305°C, ie, between those of PTEE and EEP. Second-order transitions are at —100, —30, and 90°C, as determined by a torsion pendulum (21). The crystallinity of the virgin resin is 65—75%. Specific gravity and crystallinity increase as the cooling rate is reduced. An ice-quenched sample with 48% crystallinity has a specific gravity of 2.123, whereas the press-cooled sample has a crystallinity of 58% and a specific gravity of 2.157. 

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. 

Softening and cure is examined with the help of a torsional pendulum modified with a braid (65), which supports thermosets such as phenoHcs and epoxies that change from a Hquid to a soHd on curing. Another method uses vibrating arms coupled to a scrim-supported sample to measure storage and loss moduH as a function of time and temperature. An isothermal analytical method for phenoHc resins provides data regarding rate constants and activation energies and allows prediction of cure characteristics under conditions of commercial use (47). 

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. 

Fig. 42. Torsion pendulum and typical damped sine wave output. P is the period of the motion and M2 are successive ampHtudes (241).

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). 

There are several other comparable rheological experimental methods involving linear viscoelastic behavior. Among them are creep tests (constant stress), dynamic mechanical fatigue tests (forced periodic oscillation), and torsion pendulum tests (free oscillation). Viscoelastic data obtained from any of these techniques must be consistent data from the others. 

The shear modulus of a material can be determined by a static torsion test or by a dynamic test employing a torsional pendulum or an oscillatory rheometer. The maximum short-term shear stress (strength) of a material can also be determined from a punch shear test. 

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. 

Table 3.3. Comparison of results from torsion pendulum and from tensile tests (quasistatic) at T = 500 K...

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

Robeson et al. studied the secondary loss transitions of a series of poly(arylene ether)s using a torsion pendulum.15 They found that the secondary loss transitions are closely related to the segmental motion of the aryl ether bonds. The secondary... 

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. 

Figure 7. Dynamic shear response (expressed as G, G , J" and tanS) of isotactic polypropyienes of varying crystallinity (torsion pendulum, 1Hz, ref 6). O 65%, O 56%, A 50%, achieved by annealing of quenched specimen ( 45%) + 68%, isothermally crystallized. Curves are calculated from parameterized equation (8).

The networks studied were prepared from reactions carried out at different initial dilutions. Aliquots of reaction mixtures were transferred to moulds, which were maintained at the reaction temperature under anhydrous conditions, and were allowed to proceed to complete reaction(32). Sol fractions were removed and shear moduli were determined in the dry and equilibrium-swollen states at known temperatures using uniaxial compression or a torsion pendulum at 1Hz. The procedures used have been described in detail elsewhere(26,32). The shear moduli(G) obtained were interpreted according to Gaussian theory(33 34 35) to give values of Mc, the effective molar mass between junction points, consistent with the affine behaviour expected at the small strains used(34,35). 

Gel Point and Tg. The variation of Tg with ac was investigated (18, 2J5,29) for dry networks formed from system 3 of Figure 9 at different initial dilutions of reaction mixtures. Measurements were carried out at 1Hz using a torsion pendulum(26). The results were shown in Figure 10. The two limiting values of Tg for this system correspond to networks with Me = > and Mc = Me-Thus, the horizontal broken line gives the minimum Tg, that of a linear MDI/POP polymer having a repeat unit of molar mass equal to Mq, and the maximum value of Tg at ac = 0.5 was obtained by extrapolation of (1/MC, Tg) data to 1/M . 

Figure 5 Schematic diagram of a torsion pendulum and a typical damped oscillation curve. Modified from L. E. Nielsen,...

ISO 4663 1986 Rubber - Determination of dynamic behaviour of vulcanizates at low frequencies - Torsion pendulum method... 

Note 3 A damping curve is usually obtained using a torsion pendulum, involving the measurement of decrease in the axial, torsional displacement of a specimen of uniform cross-section of known shape, with the torsional displacement initiated using a torsion bar of known moment of inertia. 

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