Elastic modulus adiabatic

Pandit and King (1982) and Bathe et al. (1984) presented measurements using transducer techniques, which are somewhat different from the accepted values of Kiefte et al. (1985). The reason for the discrepancy of the sonic velocity values from those in Table 2.8 and above is not fully understood. It should be noted that compressional velocity values can vary significantly depending on the hydrate composition and occupancy. This has been demonstrated by lattice-dynamics calculations, which showed that the adiabatic elastic moduli of methane hydrate is larger than that of a hypothetical empty hydrate lattice (Shpakov et al., 1998). 

Polycrystalline, adiabatic, elastic moduli for the Ba2YCu307 x superconducting ceramics were evaluated from measurements of the ultrasonic wave speeds in these materials. The ultrasonic technique used for these measurements is described in detail elsewhere (5,) specifics of the current procedure are briefly described here. Ultrasonic elastic moduli, C, are determined from the general relation ... 

Since heat does not have a chance to flow during the period of a high-frequency ultrasonic wave, these are adiabatic elastic moduli. Results are tabulated in Table III for both samples. 

This expression clearly has parallels with Equation 5 if we interpret adiabatic compressibility, k, as the reciprocal of the elastic modulus, M. This simple equation is the key to an important class of food measurements because k is related to changes in the tertiary structure of proteins, opening up the possibility of rapid characterization and the detection of denaturation (Apenten et al., 2000). For mixtures of materials, Wood (1941) invoked the idea of an effective medium to represent a mixture consisting of particles suspended in a continuum with compressibility and density given as ... 

Most seismological constraints on mantle composition are derived by comparison of values of seismic wave velocities inferred for particular regions within the Earth to the values measured in the laboratory for particular minerals or mineral assemblages, with such comparisons being made under comparable regimes of pressure (P) and temperature (T). The primary parameters of interest, then, are the compressional (or P-) wave velocities (Vp) and the shear (or S-) wave velocities (Ej). These wave velocities are simply related to the density (p) and to the two isotropic elastic moduli, the adiabatic bulk modulus (Ks)... 

The sign in the right hand part of this expression should be chosen so that Svp (T) > at r < Ti (adiabatic regime) and the opposite inequality should take place at r > Ti (isothermal regime for the Jahn-Teller system s contribution to the elastic modulus). 

The velocity of the wave is related to the density (p) and the elastic constant (Cm) of the medium through which it is propagating (in the equation shown below). The elastic constant is unique to the mode of propagation and to the material. For example, in liquids Cm is the adiabatic bulk modulus (B) ... 

The adiabatic and isothermic bulk elasticity modules for water only slightly differ. The adiabatic modulus for water is 2.2-lO Pa. The bulk elasticity modulus for gas can be obtained from the equation of state. For ideal gas the isothermic modulus is approximately equivalent to Pa, and the adiabatic modulus is equivalent to yPA, where Pa is the pressure inside the cell, y = 1.4 the adiabatic constant. The isothermic modulus can be used in the case of infinitely slow processes. In the case of pressure oscillations at sound frequencies the adiabatic modulus should be used. Gas is much more compressible than liquids. The bulk elasticity modulus for gas is four orders of magnitude smaller than for water. Therefore, even small amounts of gas much smaller than the volume of the solution (Voas Va), can mimic small values of the effective cell elasticity modulus Eq. (6), i.e. a strong decrease of the cell resistance to pressure variations. It is extremely important to avoid the presence of any small amounts of gas in the solution because it can lead to uncontrolled changes of the effective cell elasticity modulus. 

Elastic modulus values are classified into two groups one is the static modulus, and the other is the dynamic modulus. The former is called the isothermal modulus and is obtained from the linear relationship between load and displacement of a specimen. The latter is called the adiabatic modulus and is determined from the resonance frequency or the velocity of an ultrasonic wave (USW) in a specimen. The difference between them is caused by thermal expansion, which results from the adiabatic behavior of the specimen during the propagation of an ultrasonic wave pulse in the latter. Some difficulties cannot be avoided in the determination of the isothermal modulus. For example, a relatively large specimen is needed for the static measurement of a small strain. Thus, the elastic modulus is usually determined from the velocity of an ultrasonic wave in a single crystal of a material, for which it is difficult to prepare a large specimen. 

Often the compressibility ( t ) is used rather than the bulk modulus (K). These quantities are simple reciprocals for isotropic materials, i.e., x = Ultrasonic methods for measuring elastic properties are inherently adiabatic processes. Compressibility, on the other hand, is frequently determined by hydrostatic pressurization techniques which are isothermal in nature. Conversion of isothermal values ( t) to adiabatic values ( s) is accomplished via the following relation ... 

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... 

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). 

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