Isothermal elastic constants

The calculated isothermal elastic tensor for yS-HMX is compared in Table 8 to the one reported by Zaug (isentropic conditions). Uncertainties in the calculated elastic coefficients represent one standard deviation in values predicted from five contiguous two nanosecond simulation sequences from the overall ten nanosecond simulation. As mentioned above, Zaug s experiments sufficed to determine uniquely five of the thirteen elastic constants (modulo the... 

Further properties include the isothermal bulk modulus (Kt), the thermal expansion coefficient (/ ) and the constant pressure heat capacity (Cp). The isothermal bulk modulus is calculated first (or from corrected elastic constants as discussed in Section 6.2) and is usually defined as the Reuss bulk modulus... 

The elastic constants described above neglect the effect of temperature. The temperature component depends on whether adiabatic elastic constants (at constant entropy) or isothermal elastic constants (at constant temperature) are required. 

Table 3.4 Comparison of uncorrected and pressure and temperature corrected isothermal elastic constants of Mg2Si04 at 5 GPa and 1000 K.

However, elastic constants can show observable effects. In the limits given above the electrons experience a quasi-static strain induced by the sound wave. The resulting energy shift d k)e is the net shift with electron relaxation processes already included, as discussed in sect. 3.1. One can calculate the isothermal elastic constants using the free-energy density of band electrons, (Ziman 1960) ... 

Relations between the isothermal elastic constant c, the thermal expansion (3 and the specific heat C can be obtained ... 

The free energy of a crystal in a magnetically ordered phase or in an applied magnetic field may be presented only as an expansion in powers of displacement tensor components Uafi (see eqs. 92 and 96) the contributions of finite deformations and electron-rotation interactions to isothermal elastic constants. ... 

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). 

The influence of the crystal field is evident also from the elastic properties. Similarly, as for the magnetic susceptibility, the isothermal elastic constants Cj, can be calculated (Mullen et al., 1974) as a strain susceptibility due to the crystal field... 

A rough estimate of the difference between isothermal and adiabatic elastic constants (e.g. E and E, respectively) can be made as follows Since it is known that the isothermal and adiabatic shear moduli are equal [Silhavy 1997], i.e. 

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... 

Table 3. Comparison between adiabatic and isothermal elastic constants (ICF GPa) at room temperature...

Isothermal elasticity, eQi measured under such conditions that the temperature remains constant. 

Figure 5.2 Temperature dependence of the isothermal elastic stiffness constants of aluminium [10].

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