Adiabatic elastic constants

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. 

The full definition of the adiabatic elastic constants is given by... 

Fig. 48. Schematic temperature variations of adiabatic longitudinal elastic constants in Ce HF compounds. In the various regimes c T) is determined by anhatmonic effects (r= fl ), magnetoelastic coupling (r A = CEF gap) and Gruneisen parameter coupling (F< T ). c, Cg are extrapolations from the anharmonic and magnetoelastic regimes, respectively.

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) ... 

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). 

Cijkimn are the third-order adiabatic elastic constants (TOEC),... 

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...

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