Torsion, experiments

Assuming these conditions apply to a torsion experiment, it also becomes possible to define two shear moduli, and G2. The first of these represents that part of the stress that is in phase with the strain, divided by the strain. 

The problem of definition of modulus applies to all tests. However there is a second problem which applies to those tests where the state of stress (or strain) is not uniform across the material cross-section during the test (i.e. to all beam tests and all torsion tests - except those for thin walled cylinders). In the derivation of the equations to determine moduli it is assumed that the relation between stress and strain is the same everywhere, this is no longer true for a non-linear material. In the beam test one half of the beam is in tension and one half in compression with maximum strains on the surfaces, so that there will be different relations between stress and strain depending on the distance from the neutral plane. For the torsion experiments the strain is zero at the centre of the specimen and increases toward the outside, thus there will be different torque-shear modulus relations for each thin cylindrical shell. Unless the precise variation of all the elastic constants with strain is known it will not be possible to obtain reliable values from beam tests or torsion tests (except for thin walled cylinders). 

Again, reliable creep modulus data have to be available in order to apply the deflection equations. Tables 25.2 and 25.3 (see also Fig. 25.3) give the expressions for the deflections and torsional deformations of bars. By means of these equations the modulus of engineering materials may be determined from deflection and torsion experiments. The reader is also referred to, e.g. Ferry (1980), McCrum et al. (1997), Whorlow (1992) and Te Nijenhuis (1980, 2007). 

Note that /3 and /4 are stress components in the plane of isotropy and, therefore, have the same Weibull parameters. The parameters i and /3i would be obtained from uniaxial tensile experiments along the material orientation direction, dt. The parameters a2 and /Efe would be obtained from torsional experiments of thin-walled tubular specimens where the shear stress is applied across the material orientation direction. The final two parameters, a3 and /33, would be obtained from uniaxial tensile experiments transverse to the material orientation direction. 

The NMR data presented above reveal a dynamic heterogeneity of filled PDMS in the frequency range from about 10 kHz to 100 MHz. To determine whether the heterogeneity remains at lower fi equencies, dynamic mechanical measurements are performed. The results for cured, unfilled silicon rubber are compared with those for filled samples containing different fraction of hydrophilic Aerosil (380 m g ). For a more straightforward analysis of the mechanical experiments, a random poly(dimethyl/methyl-phenyl) siloxane copolymer containing approximately 90 mol% dimethyl- and 10 mol% methylphenyl-siloxane units has been used for sample preparation. This copolymer is fully amorphous over the whole temperature range. The results of torsion experiments at a frequency of 1.6 Hz are shown as a function of temperature in Fig. 7. 

Usually, creep deformation of ice single crystals is associated to a steady-state creep regime, with a stress exponent equal to 2 when basal glide is activated . In the torsion experiments performed, the steady-state creep was not reached, but one would expect it to be achieved for larger strain when the immobilisation of the basal dislocations in the pile-ups is balanced by the dislocation multiplication induced by the double cross-slip mechanism. 

VsGa Cylindrical Layer (Fig. 5). The results in this case were quite similar to those for NbaSn layers. Torsion experiments were performed first. The broadening of the curve-system in high fields was again observed. Moreover, a marked temperature dependence and hysteresis was found. The hysteresis did not vanish completely after annealing at 250°C. Similar results were obtained in the pull tests however, the critical currents did not decrease to the same extent. 

Although the temperatures used in the volume recovery experiments were not identical to those used in the torsion experiments, one can reatUly see that the volume requires a much longer time to reach equilibrium. From Figure 12, it is tqiparent that the volume takes more than 10 s to equilibrate at 138 "C, whereas... 

Stresses should be small enough for linear behaviour to hold to a good approximation, a situation which is rarely in doubt in torsion experiments, but which may not hold for tensile strains greater than about 0particular cases. The linear range of each experimental material must therefore be investigated. 

Table 11.1 Pairs of craze-initiation parameters a- and s in tension-torsion experiments on PS (minerai-oii-free PS...

In eq. (11.15) the concentrated local mean normal stress ai and deviatoric shear stress Si are in the plastic enclaves that result in craze initiation (the time taken for the formation of 10 crazes per cm in 10 s in the tension torsion experiments is taken as the mean period r of craze initiation), which needs to be stated in terms of global stresses a and s through the use of the average stress eoneentrations and of surface grooves. How this is done is discussed below in Section 11.6 comparing the model predictions with the results of the tension torsion experiments. 

The comparison starts with determination of how the applied stresses 022 and an in the tension-torsion experiments are concentrated by the set of surface grooves of semielliptical shape. This is done using specific expressions provided by Neuber (1946). 

Another important aspect of finite elasticity theory is the ability to measure the strain energy fimction derivatives Wi = dW/dli and W2 = BW/9I2. Penn and Kearsley (94) showed how this is done nsing data from torsional experiments. An interesting aspect about torsion in finite deformations is that in order to maintain the cylinder at a constant length, it is necessary to apply normal forces at the ends of the cylinder. If the cylinder is left imrestrained, it will lengthen in an effect referred to as the Poynting (95) effect, first observed early in the last century in experiments with metal wires. When a cylinder of length L is twisted by an amount... 

Thus, from torque and normal forces in torsional experiments, the VL function derivative can be obtained. Typical data for natural rubber are presented in Figures 30, 31, 32. The figures illustrate the sequence that would be used to obtain the VL ftmction. First, obtain torque and normal force data (Fig. 30) and use equations 47 and 48 to obtain the strain energy function derivatives Wi and W2 (some typical results shown in Fig. 31). Finally, data of the sort shown in Figure 31 are used to obtain the VL fimction derivative w X). Figure 32 shows such data obtained from torsional measurements on natural rubber samples cross-linked to different extents (102). 

Torsional Experiments. The geometry and equations for torsion of an elastic cylinder are presented above. For the viscoelastic K-BKZ material, the equations look similar. For isochronal values of the strain potential function, one can define what looks like a time-dependent strain-energy ftinction Wi(Ii, I2, t) ... 

Fig. 56. Values of the time and strain-dependent strain energy function derivatives dW/dli and 9W/9/2 for a glassy PMMA determined from torque and normal force measurements in single-step stress relaxation torsional experiments. After McKenna (114).

Fig. 57. Values of the time and strain-dependent strain energy function derivatives Wi = 9W/9/1 and W2 = 9W/9/2 for a glassy polycarbonate determined from torque and normal force measurements in single-step stress relaxation torsional experiments, (a) >/ 0.017 0.033 A 0.050 v 0.067 0.083 O 0.10. (b) y A 0.017 0.033 o 0.050 T 0.067 v 0.083 0.10. After Pesce and McKenna (146).

There are three independent shear moduli Gi = 1/ 44, G2 = I/S55 and G3 = l/see corresponding to shear in the 23, 13 and 12 planes respectively. For a sheet of general dimensions, torsion experiments where the sheet is twisted about the 1, 2 or 3 axis will involve a combination of shear compliances. This will be discussed in greater detail later, when methods of obtaining the elastic constants are described. 

Material Parameter Characterization Elastic Modulus The first step in the process is to characterize the material model that describes the behavior of axes, flexible gear and rigid gear and also to quantify the variability in the material. Through physical measurements, tension/com-pression and torsional experiments, the material density, elastic and shear moduli were obtained calibrating to represent the dynamic behavior of them at room temperature. Using the data obtained from these samples and from these tests, a probabilistic description of the randomness of them was obtained and subsequently used in more complex system. More specifically, for these components, the elastic modulus is treated as random variable. 

Fig. 19. Double-torsion experiment direct loading for small torsions.

Fig. 20. Double-torsion experiment use of pulley and beam for large torsional deformations. Torque M = PL. (Taken from K. Cho and A.N. Gent, Intemati J. Fracture, 28, 239 (1985), published by Kluwer Academic Publishers.)...

Using much the same methodology, shear stress relaxation data can be corrected for instrument compliance effects. In a stress relaxation experiment, usually there is a commanded step in strain that is imposed on the sample. The resulting torque is then measured. If the instrument is compliant, not only does the sample twist (for torsion experiments) but the machine also twists. These results in a lesser strain applied to the sample. Referring to reference [6], we have the following ... 

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