Elastic constants measurements

In the perfectly elastic, perfectly plastic models, the high pressure compressibility can be approximated from static high pressure experiments or from high-order elastic constant measurements. Based on an estimate of strength, the stress-volume relation under uniaxial strain conditions appropriate for shock compression can be constructed. Inversely, and more typically, strength corrections can be applied to shock data to remove the shear strength component. The stress-volume relation is composed of the isotropic (hydrostatic) stress to which a component of shear stress appropriate to the... 

The authors thank A. Migliori for facilitating his ultrasonic spectroscopy apparatus for the elastic constant measurements. This work was supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science. 

Elastic behavior, ceramics, 5 613-615 Elastic constants, measured values of,... 

Porosity is known to decrease the elastic moduli of ceramic materials (8-9). To estimate the magnitude of this effect for the current elastic constant measurements, we have used theories for the elastic modulus decrement due to spherical pores by Mackenzie (10) and Ledbetter and Datta (1 ) to calculate the "zero-porosity"... 

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. 

Between 1 and 2 K, a lattice contribution of 0.449 7 J/(mol K) was added based on a Debye temperature of 163 K determined by Rosen (1967) from elastic constant measurements. With this addition, heat capacity measurements of Lounasmaa (1964a) (0.40-3.8 K) were used to derive a different interpretation of the large magnetic contribution as 133.5722 2 extended representation of the low-temperature heat capacity to 3.8 K ... 

The heat capacity measurements of Lounasmaa (1962a) (0.4. 0 K) were analyzed in terms of nuclear, electronic, and combined lattice-magnetic terms. Further to these measurements, a number of new determinations have helped to improve the fit. The measurements of Lounasmaa and Sundstrom (1967) showed that the discrepant Run III of Lounasmaa (1962a) was incorrect and therefore rejected. Rosen (1967) determined the Debye temperature from elastic constant measurements to be 169 K at 4.2 K, equivalent to a lattice contribution to the heat capacity as 0.403 mJ/(mol K" ). With the values of the nuclear and lattice contributions fixed, the heat capacity values of Lounasmaa (1962a) were used to derive the electronic... 

At the time of the heat capacity measurements of Lounasmaa (1964c) (0.4. 0 K), the lattice and magnetic contributimis could not be separated but later Rosen (1968a) determined the limiting Debye temperature from elastic constants measurements to be 118 K, equivalent to a lattice contribution to the heat capacity of 1.18 mJ/(mol K ) so that the magnetic contribution could then be separately calculated from a reassessment of the measurements of Lounasmaa (1964c) over the temperature range of 0.50-1.25 K. When combined with the nuclear terms as determined in Part 11.11, results in a revised heat capacity equation valid to 1.25 K ... 

While the value of 6d obtained by Wells et al. (1976) agrees with a value of 177 K obtained by Rosen (1968b) from elastic constant measurements, the measurements of neither Wells et al. (1976) nor Dceda et al. (1985) agree with a value of 185.6 K obtained by Palmer (1978) also from elastic constant measurements. In view of the very high purity of the samples used by Ikeda et al, the selection of these values and the inclusion of the nuclear contribution obtained from Part 13.12 lead to the heat capacity up to 4 K being represented by... 

Additional contributions to the total heat capacity with the nuclear contribution omitted were fitted to the equation C°p T) = yT+BT, where y is the electronic coefficient and B is a combination of the lattice and magnetic contributions to the heat capacity. From elastic constant measurements on single crystals, Rosen et al. (1974) determined a limiting Debye temperature of 191.5 K, equivalent to a lattice contribution of 0.28 mJ/(mol K" ), while similar measurements by Palmer (1978) determined 186.8 K, equivalent to 0.30 mJ/(mol K" ). Averaging these values to 0.29 mJ/(mol K ), then the magnetic contribution to the heat capacity is given by the subtraction is U=B-0.29mJ/(molK ) (Table 143). 

The Debye temperature selected by Pecharsky et al. (1993) is notably lower than the experimental values obtained from elastic constant measurements at 192 K determined by Rosen (1968b) and 187.8 K determined by Palmer (1978). The selected values are basically those determined by Pecharsky et al. (1996) but adjusted so that the first term in the nuclear contribution is restored to the value of 19.45 mJ/(raol K) obtained from the NMR measurements of Sano et al. (1972) rather than the value of 18.0 mJ/(mol K) determined by Pecharsky et al. (1996) during the fitting process. The total heat capacity up to 6 K can then be represented by... 

The measurements of Bucher et al. (1970) were carried out using pure alpha phase and indicate that the metal used by Lounasmaa (1963) must have also been very rich in this phase. Both values of are in excellent agreement with the value of 117.5 K obtained by Rosen (1971) from elastic constant measurements. 

Elastic constants measured as a function of temperature are available for most of the lanthanides in polycrystalline form (Rosen, 1967, 1968) and for Tb, Dy, Ho and Er single crystals (Palmer, 1970 Palmer and Lee, 1973 and du Plessis, 1976). For a summary of the elastic properties of the lanthanides reference can be made to Taylor and Darby (1972, section 2.4) and to ch. 8, section 9. If a suitable lattice dynamical model were devised, we should be able to calculate Cl from first principles. This was done for Gd, Dy and Er metals (Sundstrom, 1968), but at the time of these calculations, elastic constants were available only for polycrystalline samples at a few fixed temperatures. Nevertheless the results obtained did indicate that Lounasmaa s (1964a) interpolation idea was reasonable. With the elastic constant data available today it should be possible to calculate Cl for the entire region of interest, although this appears not to have attracted much attention, presumably because the uncertainty involved in separating off the contributions in experimental heat capacity results makes comparison with theory unrewarding as far as Cl is concerned. 

At or near absolute zero the elastic constants determine the thermal spectrum of lattice vibrations and, consequently, the Debye temperature. Since elastic constants can be determined with high precision, it is useful to calculate the Debye temperature from acoustic wave velocities, i.e. from elastic constants measured near absolute zero. In the Debye theory the characteristic temperature at absolute zero is given (Anderson, 1963) by... 

In comparing elastic constants measured acoustically with those obtained in a static (very low frequency) test, note that acoustic values are measured under adiabatic conditions, while static values are isothermal. The two t5q>es of bulk modulus measurements are related by the standard thermodynamic relation... 

Thus ATeff is comparable with the elastic constants measured independently. 

SO that the critical regime extends over a fraction of a degree. It may easily switch from observable to non-observable with a slight change of the roughly estimated numerical constant that lead to Eq. (40). Furthermore, the crossover regime may be different from one observable to another. Recent estimates of the Ginzburg criterion, for instance, indicate that elastic constants measurements are one hundred times more sensitive to fluctuations than heat capacity ones [37]. Experiments tend to confirm this point a mean field behavior of the heat capacity has been reported [36] whereas fluctuations seem to be important in some tilt, susceptibility or bulk modulus measurements [38, 39]. 

P. E. Cladis, Phys Lett. 1974, 48A, 179 and unpublished. This is a qualitative result. While it may not be possible to make elastic constant measurements for and deep enough into the N-SmA critical regime to establish quantitative certainty about critical exponents, there should be no problem for Xi which does not diverge at this transition. 

It is also possible to use turbidity measurements to evaluate elastic constants (Fig. 5, though elastic constants measured in this way for alkylcyanobiphenyls (nCBs) [25] and mixtures [26] are consistently higher than those determined by Frederiks transition studies. Hakemi [27] measured j, 22 and 33 for 8CB (octylcyano biphenyl) close to the nematic-smectic A phase transition and deduced a value of the critical exponent from the divergence of 22 and 33. 

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... 

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