Elastic constants measurement techniques

The elastic constants of liquid crystalline polymers can be measured in terms of the Frederiks transitions under the presence of a magnetic or electric field. Raleigh light scattering is also a method for measuring the elastic constants. Those techniques successfully applied to small molecular mass liquid crystals may not be applicable to liquid crystalline polymers. This is why very few experimental data of elastic constants are available for liquid crystalline polymers. 

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... 

A variety of experimental techniques are available for the investigation of the electron-lattice interaction. For static phenomena such as thermal expansion and magnetostriction one can use dilatometric and X-ray techniques. For dynamic effects such as elastic constant measurements, ultrasonic propagation and phonon dispersion the methods of sound velocity and attenuation measurements, and inelastic neutron or light scattering are available. In addition high-pressure work can give valuable information for some quantities. 

CsCl-structure materials. Perhaps the best studied materials, with respect to magnetoelastic and quadrupolar interactions, are the CsCl-structure materials (TmCd, TmZn etc.). The Grenoble group (Morin, Schmidt and collaborators), in particular, has done a variety of investigations on these simple cubic compounds. In addition to elastic constant measurements, as discussed before, they have applied other techniques to illustrate the electron-phonon coupling mechanisms ... 

The calculated binding energies in oxides (rows from 11 to 13 of Table 9.1) also agree weU with experimental data. The first-principle simulation correctly predicts the greatest cohesive energy for CaO and similar values for MgO and SrO. Precise measurements of the bulk moduli for oxides is difficult. The reported values are in agreement with elastic constants measured by ultrasonic techniques. 

J-g -K ) for alumina and 55.8 J-mof -K ( = 0.453 J-g -K" ) for zirconia. These values are in satisfactory agreement with literature values [Munro 1997, NIST 2002, Salmang Scholze 1982]. With this input information at hand. Equations (40) and (41) (the latter in connection with approximate values for the shear and bulk moduli, cf Table 8 below for the definite values) can now be used to obtain estimates for the differences that have to be expected at room temperature between adiabatic elastic constants (measured via dynamic techniques) and isothermal elastic constants (measured via static techniques). For alumina and zirconia... 

Pd4oCu4oP2o, Pd5oCu3oP2o, and Pd6oCu2oP20 alloys were measured by resonant ultrasound spectroscopy (RUS). In this technique, the spectrum of mechanical resonances for a parallelepiped sample is measured and compared with a theoretical spectrum calculated for a given set of elastic constants. The true set of elastic constants is calculated by a recursive regression method that matches the two spectra [28,29]. 

Data from ref [31] the elastic constants were measured by the pulse-echo technique j3 was determined from low-temperature specific heat data. 

The discrepancy between the first determination of the elastic constants and the following ones is not surprising, due to the very indirect way utilised in [33]. The difference between the Brillouin spectroscopy results on the one hand and those obtained with resonance ultrasonic spectroscopy on the other one is mostly due to the difference in C33. Although the authors claim that the most precise measurements of elastic moduli are obtained using ultrasonic techniques, the samples should have a... 

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... 

The chosen technique was X-ray diffraction that is widely used for non-destructive surface measurement of applied and residual stresses. Stress analysis relies on the determination of the lattice strain using the interplanar spacing as a gauge by measuring the peak shifts in a fixed O-direction for different /-tilt of the sample [3]. Stresses are calculated from measured strains using diffraction elastic constants which were calculated theoretically. As the Co phase takes up a certain amount of W and transforms after cooling into a solid solution with a variable W content, the measurements were limited to the WC phase. 

Several experiments were carried out to investigate the elastic constants of nematic polymers. They were essentially in agreement with the theory. But the available data are insufficient for checking theoretical predictions. Systematic and careful experiments are required to investigate the relationship of elastic constants to the flexibility and molecular length. We will introduce some measurement techniques and experimental data for elastic constants. 

Experimental shock-wave methods, or Brillouin scattering techniques used to measure elastic constants at either ambient, or non-ambient conditions. 

These standing waves are fundamental in vibration problems and are known as natural vibrations, or vibrational modes. Other types of progressive waves could be analyzed in a similar fashion but this is left to other texts. Resonant vibrations can be used as a technique to measure elastic constants and this will be discussed in the next section. 

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. 

The engineering elastic constants of isotropic materials can be measured from static loading. Name two other general measurement techniques. 

Equation (102) shows that MAQO can provide important information about the electronic parameters (extremal Fermi surface cross-sectional area, effective masses, electronic relaxation times) and about the electron-phonon interaction (strain derivatives of the cross-sectional area for different symmetry strains). With the help of this technique, combined with de Haas-van Alphen susceptibility measurements, one can put the deformation potential interaction and the temperature dependence of the elastic constants, discussed above in sect. 3.2, on a solid basis. In the following we discuss some compounds. 

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