Elastic constants pressure dependence

The simple theory of electronegativity fails in this discussion because it is based merely on electron transfer energies and determines only the approximate number of electrons transferred, and it does not consider the interactions which take place after transfer. Several calculations in the alkali halides of the cohesive energy (24), the elastic constants (24), the equilibrium spacing (24), and the NMR chemical shift 17, 18, 22) and its pressure dependence (15) have assumed complete ionicity. Because these calculations based on complete ionicity agree remarkably well with the experimental data, we are not surprised that the electronegativity concept of covalency fails completely for the alkali iodide isomer shifts. 

Instead of measuring the force-temperature dependence at constant volume and length, one can measure this dependence at constant pressure and length but in this case it is necessary to introduce the corresponding corrections. The corrections include such thermomechanical coefficients as iso-baric volumetric expansion coefficient, the thermal pressure coefficient or the pressure coefficient of elastic force at constant length 22,23,42). 

Figure 3. Spontaneous strains and elastic properties at the 422 < i> 222 transition in Te02. (a) Spontaneous strain data extracted from the lattice-parameter data of Worlton and Beyerlein (1975). The linear pressure dependence of (e - (filled circles) is consistent with second-order character for the transition. Other data are for non-symmetiy-breaking strains (e + 62) (open circles), 63 (crosses), (b) Variation of the symmetry-adapted elastic constant (Cn - Cu) at room temperature (after Peercy et al. 1975). The ratio of slopes above and below Po is 3 1 and deviates from 2 1 due to the contribution of the non-symmetry-breaking strains. (After Carpenter and Salje 1998).

Values for most of the coefficients in Equation (23) were extracted from experimental data. Values were assigned to 4 and ke arbitrarily in the absence of the relevant experimental observations. The bare elastic constants were given a linear pressure dependence based on the variations calculated by Karki et al. (1997a). When experimental data become available, a comparison between observed and predicted elastic constant variations will provide a stringent test for the model of this phase transition as represented by Equation (23). 

Elastic constants depend on pressure and temperature because of the anharmonicity of the interatomic potentials. From the dependence of bulk and shear moduli on hydrostatic and uniaxial pressure, third order elastic constants and Griineisen parameters may be determined. Griineisen parameter shows the effect of changing volume, V, on the phonon mode frequencies, co. 

C. W. Garland C.F. Yamell (1966). J. Chem. Phys., 44, 1112-1120. Temperature and pressure dependence of the elastic constants of ammonium bromide, (table 5). R.S. Seymour A.W. Pryor (1970). Acta Ciyst, B 26, 1487-1491. Neutron diffraction study of NH,Br andNH, . 

J.H. Gieske and G.R. Barsch, Pressure dependence of the elastic constants of single crystalline aluminum oxide, Phys. Status Solidi 29, 121-131 (1967). 

Material Constants, Elastic wave velocities have been obtained for oil shale by ultrasonic methods for various modes of propagation. Elastic constants can be inferred from these data if the oil shale is assumed to be a transversely isotropic solid (9). This is a reasonable approximation considering the bedded nature of the rock. Many of the properties of oil shale depend on the grade (kerogen content), which in turn is correlated with the density ( 10). The high pressure behavior of oil shale under shock loading has been studied in gas-gun impact experiments (11). 

To some extent the way the computation is carried out is governed by the objectives. Usually information about one or more of the following is sought pressure distribution in the lubricant film minimum thickness of the film shape of the film. Items of input into the problem are load, radii of curvature of the boundaries, material properties (such as viscosity and density of the fluid together with their pressure and temperature dependence, elastic constants of the solid boundary material), and speed. A set of assumed initial conditions is used to... 

In most unsaturated rocks and soils, elastic properties and plastic flow depend on the capillary pressure which is related to water saturation degree through water retention curve (Fredlund and Rahardjo 1993). In this work, for simplicity, we neglect the variation of elastic constants with capillary pressure. However, we intend to account for the influence of capillary pressure on plastic behaviour of argillites. Only a small number of triaxial compression tests with different water saturation degrees are available. We can only provide a first approximation of such a influence. We consider that the failure parameter A (see Equation 13) linearly increases with capillary pressure ... 

Radiation pressure is a steady constant pressure for continuous wave signals. For acoustic pulses, however, this pressure varies periodically at the pulsing frequency. Radiation pressure thus provides a mechanism for producing foree at frequencies other than the normal ultrasonic frequencies and potentially in the audible zone. For small particles and non-absorbing interfaces, radiation pressure has a direction and amplitude which depend on the elastic properties of the material in question. This extra force may also result in particle movement. 

The various procedures for obtaining rate constants from experimental data are next considered. In most of these, the ionic yields are deduced solely from their power absorption from the observing rf field. Expressions for the rate constant thus depend on the nature of the power absorption, according to whether (1) the ion is in free flight (zero collisions in the limit of zero pressure), (2) it is experiencing elastic, nonreactive collisions, or (3) it is undergoing chemically reactive collisions. The three different procedures which have been developed are now considered in turn. 

Next we note that there are two physieally different sources of temperature and pressure dependence of the elastic constants of polymers. One, in common with that exhibited by all inorganic crystals, arises from anharmonic effects in the interatomic or intermolecular interactions. The second is due to the temperature-assisted reversible shear and volumetric relaxations under stress that are particularly prominent in glassy polymers or in the amorphous components of semi-crystalline polymers. The latter are characterized by dynamic relaxation spectra incorporating specific features for different polymers that play a central role in their linear viscoelastic response, which we discuss in more detail in Chapter 5. 

That this form does indeed hold had been demonstrated for the cavitation of glassy polypropylene (Mott et al. 1993) in a computational study furnishing the validity of this extension of the universal binding-energy relation to symmetrical bulk response. The additional attraction of this expression is that it points out directly that application of a pressure produces symmetrical elastic compaction in an isotropic solid. However, more interestingly, one notes that, when dilatation is imposed, the bulk modulus monotonically decreases and eventually, at a dilatation of 1 /p, vanishes. This also leads to the observation that, if dilatation results from thermal expansion in response to a temperature increase, the bulk modulus also decreases. This simple observation represents the essence of the temperature dependence of all other elastic constants in anisotropic solids, beyond the mere effect on the bulk... 

Temperature and pressure dependencies of the elastic constants sometimes are required and are often informative. This chapter contains graphical presentations of the temperature dependence of polycrystalline constants and Q s where data were available. However, pressure dependencies have not been included because so little data exists. Readers who are concerned with pressure dependencies of elastic properties are referred to a paper by Guinan and Steinberg (1974) which lists estimated temperature and pressure derivatives of the polycrystalline constants for all the rare earth metals except promethium. When experimental data for the pressure dependence of rare earth metal elastic constants were presented, the writer has provided appropriate references in the pertinent section for the metal involved. 

During their investigation Palmer et al. (1974) examined the effect of a 2.5 Tesla magnetic field on the temperature dependence of the elastic constants of terbium, Monfort and Swenson (1965) and Stephens and Johnson (1969) obtained the pressure dependence of the compressibility. 

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