Higher-order elastic constants

It is consistent with the approximations of small strain theory made in Section A.7 to neglect the higher-order terms, and to consider the elastic moduli to be constant. Stated in another way, it would be inconsistent with the use of the small deformation strain tensor to consider the stress relation to be nonlinear. The previous theory has included such nonlinearity because the theory will later be generalized to large deformations, where variable moduli are the rule. 

Within the elastic regime, the conservation relations for shock profiles can be directly applied to the loading pulse, and for most solids, positive curvature to the stress volume will lead to the increase in shock speed required to propagate a shock. The resulting stress-volume relations determined for elastic solids can be used to determine higher-order elastic constants. The division between the elastic and elastic-plastic regimes is ideally marked by the Hugoniot elastic limit of the solid. 

Table 2.2. Higher-order elastic constants (after Davison and Graham [79D01]). ...

The second- and higher-order elastic constant studies in single crystals with large Hugoniot limits have provided an examination of elastic behavior... 

The observed elastic shear constant C44 of aluminum is 2,8 x 10" crg/cm, whereas the electrostatic contribution is 14.8 x 10" erg/cm (Harrison, 1966a, p, 179 the value there was based upon an effective charge 7.9 percent larger tlian the 3,0 appropriate here). The band-structure energy is an estimate of the difference, —12.0 x 10" crg/cm. Even if we sum all terms, the result is approximate because of the neglect ofterms of higher order than two and the use of an approximate pscudopoteiitial. We obtain the effect of the nearest lattice wave numbers here. 

It should be noted that despite the similarity between the different estimates, there is still no perfect agreement with all experimental data. The total cross section, the low energy virial coefficient, the elastic constant and the Debye-temperature of the solid are examples of such deviations. Furthermore the question of the three-body forces is not solved completely. The fact that contributions of higher-order non-additive multipole forces were found to be... 

Finally, in this section, possible frequencies for the excitation of coherent vibrations should be considered. Originally special reference was made to membrane oscillations as their high polarization will then yield coherent electric oscillations (cf. Ref. 7, with earlier literature). Since the thickness of a membrane is about 10" cm and its elastic constant is equivalent to a velocity of sound of about lO cm/s, a frequency of the order of 10 Hz was expected, corresponding to electromagnetic waves in the millimeter region. When based on proteins or DNA, however, both higher and lower frequencies may be expected. 

The electrical resistivities of the dense Kondo systems CeNiln, CePdln, and CePtln have been measured under hydrostatic pressures up to 19 kbar (Kurisu et al., 1990). The Kondo temperature of CeNiln and CePtln shifts linearly with pressure to higher temperatures at rates of 2.3 and 1.5 K/kbar, respectively. For CePdIn, the pressures were not high enough to reach the CePtln or CeNiln state. Measurements of the elastic properties of CePdln reveal that all elastic constants exhibit softening at low temperatures due to the crystal electric field effect and the antiferromagnetic ordering (Suzuki et al., 1990). 

Generally speaking, the ability of shell model potentials to reproduce the elastic properties of ionic materials is much more limited, as compared to structures, with errors typically being an order of magnitude larger. This is a consequence of the fact that the perturbation of a structure about its equilibrium form is much more sensitive to higher order polarizabilities than the minimum itself, where any errors can be readily subsumed into the parameterization. A classic example is the failure of the dipolar shell model to reproduce the Cauchy violation in the elastic constants of simple cubic oxides, such as MgO (Catlow et al. 1976). 

In addition to the adiabatic or isothermal difference, acoustically determined elastic constants of polymers differ from static values because polymer moduli are frequency-dependent. The deformation produced by a given stress depends on how long the stress is applied. During the short period of a sound wave, not as much strain occurs as in a typical static measurement, and the acoustic modulus is higher than the static modulus. This effect is small for the bulk modulus (on the order of 20%), but can be significant for the shear and Young s modulus (a factor of 10 or more) (5,6). 

Elastic Constants. For the polymers listed in Table 2 for which both longitudinal and shear sound speeds are given, the elastic constants have been calculated at room temperature, ambient pressure, and a frequency of 2 MHz these are listed in Table 7. The moduli values are approximately 1 order of magnitude lower than those for metals. The range of Poisson s ratio values is somewhat higher than that for metals. A review of elastic properties of polymers is given by Hartmann (130). 

Landolt-Bomstein Tables, Neue Serie Vol. III/29a (Editor D.F. Nelson Springer Verlag. Every AG, McCurdy AK (1992) Low Frequency Properties of Dielectric Crystals. Second and Higher Order Elastic Constants. 

Brillouin spectroscopy enables the elastic constants of pol3oners to be determined at frequencies of several gigahertz, i.e. three orders of magnitude higher than those pertaining to ultrasonic measurements, which are known as h3qiersonic frequencies. 

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