Measurement of elastic constants

The measurement of elastic constants is a very different undertaking for the two situations of a sheet and a fibre. The experimental methods employed for these two cases will therefore be discussed separately. 

In order to describe the elastic behavior of a body, the values of the elastic constants are needed. Thus, it is important to understand the experimental techniques that are used to measure these constants. The most obvious approach is to apply a stress and measure the resulting strain held (static loading). For such approaches, strain gages are often used to measure the strain. These gages are usually electrical resistors that are calibrated such that changes in resistance can be converted to strain. Newly developed optical techniques, such as laser exten-someters, are allowing strains to be measured without specimen contact. 

As a result of the above difficulties, alternative techniques were sought and have become established. One approach (sound velocity) involves passing ultrasonic waves through the material and determining their velocities. The other approach (dynamic resonance) uses the natural vibration of the material. The elastic constants are determined from specimen geometry and the resonant frequency. For these last two techniques, experimental accuracies of 0.1% are not uncommon. Both of these latter techniques can also be used on anisotropic materials but the current discussion will emphasize isotropic materials. 

The velocities v, and Vj of the longitudinal and transverse waves are given in Eqs. (2.83) and (2.84), respectively. Thus, from the measurement of these velocities and density, one can readily calculate B and /t for an isotropic body. The values of E and v are then obtained from the equations in Appendix 3. A 

In the dynamic resonance experimental technique, a body is forced to vibrate and the constants are determined from the resonant frequencies. The types of vibration utilized are usually the longitudinal, flexural or torsional modes. The first two allow E to be determined and the last gives the shear modulus. It is usually easier to excite flexural waves than longitudinal ones, thus the use of flexural and torsional waves will be emphasized in this discussion. To use the dynamic resonance approach, the solution to the differential equations of motion must be known and this has been accomplished for several specimen shapes. In particular, it is common to use specimens of rectangular or circular cross-section, as solutions are readily available. Vibrations in the fundamental mode usually give the largest amplitude and are, therefore, the easiest to detect. 

For torsional waves, the relationship between p and the fundamental resonant frequency/is given by 

The elastic moduli AT,-,/= 1,2,3) of low molecular mass nematics are typically of the order of several piconewton. The splay and bend constants 1 and 33 are often of com- 

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Thompson et al. described a series of ultrasonic techniques used for in-.situ measurements of elastic constants on thick-walled submersible vessels [149]. The elastic constants can provide information about fabrication errors such as wavy fibers and fiber disbonds. Elastic constant measurements can be performed using Rayleigh or Lamb wave modes, or by using angle beam techniques, It was shown that the effect of the ani.sotropy increases... 

In a series of related publications, Hennig has reported the measurements of elastic constants for oriented polymers which are either amorphous or of low crystallinity. In his earliest work." Hennig showed that in polyvinyl chloride and polymethylmethacrylate the relationship 3/Eo = S33 + 2S11, where is the modulus of the isotropic polymer, holds to a good approximation. Results for the anisotropy of the linear compressibility y in polyvinyl chloride, polymethylmethacrylate, polystyrene and polycarbonate were also reported. In this experiment Hennig measured the linear compressibility parallel to the draw direction 7ii, and that in the plane perpendicular to the draw direction Vi. For uniaxially oriented polymers yn = 2Si3 + S33 = S i -I-Si2-I-S 3. It was... 

The experimental work was performed to provide both material property Input to the design model and experimental verlflcatiori of model predictions. To this end small scale and scaled up joints were fabricated to determine their mechanical behavior. Experiments included measurements of elastic constants and of stress-strain curves as a function of temperature, tensile strength, shear strength, and fracture toughness. Some experiments were also performed on specimens fabricated entirely of the interlayer materials. 

Figure 5.11 Schematic diagram showing layout of ultrasonic apparatus used for measurement of elastic constants. (From Dyer et al., J. Phys. D (1992))...

Finally, let us briefly discuss the methods of the experimental measurement of elastic constants. Here we will give only general formulas and will not go deeply into the relevant electrooptical phenomena, which will be described in more detail in the corresponding chapters. 

Figure 6.12 Schematic diagram showing layout of ultrasonic apparatus used for measurement of elastic constants.

The measurement of elastic constants is of particular interest near the nematic-smee-tic A (Tn-sitia) transition. The bend and twist elastic constants contain terms that diverge at TiM.smA according to a power law in reduced temperature [329] KjjCritical exponents have been measured for the twist [77-79, 81-87, 171] and bend [77-79,81,82,85,88-91,93-97,168,171, 172] constants. X-ray diffraction measurements made by Bradshaw et al. [98] indicate a correlation between short-range smecticlike order and the TTjg/A i, ratio, even for non-smectogenic substances. 

The most frequently studied homologu-ous series of nematic liquid crystals with regard to the measurement of elastic constants is the n-alkylcyanobiphenyls (n-CB, n=5 to 8) [20, 30, 41, 63, 142-180]. In particular, data have been reported for... 

Measurement of Elastic Constants at Low Temperature by Means of Ultrasonic Waves Data for Silicon and Germanium Single Crystals, and for Fused Silica... 

Although TiAl is obtained over a large range of chemical composition from about 49 to 56 at% at room temperature (Massalski et al., 1990), TiAl single crystals have been prepared, not at the stoichiometric composition, but at an Al-rich composition. Thus a single crystal of Ti-S6 at% Al was used for measurement of elastic constants (Tanaka et at., 1996a He et al., 1995). 

B1 Measurement of Elastic Constants using Electric or Ms netk Field-Induced... 

B3.1 Measurement of elastic constants using the torsion pendulum... 

Measurements of elastic constants by classical methods and ultrasonic pulse methods are described in [3.23,24], respectively. Elastic constants can also be measured by Brillouin scattering or from inelastic neutron scattering techniques [1.35]. 

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