Definition of Third-Order Material Constants

This choice makes kd k a differential increment of internal energy per unit actual volume and EkcIDk the corresponding increment of internal energy per unit reference volume. It is also worthwhile to note that - because of the properly l kdxk = ElcIXl, implied by (6.6) - for electrostatic situations the fields may be represented by the potentials (x) = P(X) with equipotential surfaces that can be identified with material surfaces in the two configurations and we have as a consequence of S = —gradcp also 

Among the thermal quantities the absolute temperature is clearly unaffected while entropy is quite naturally taken per unit reference volume without further complications. For convenience of notation we choose the temperature of the reference state to be o = 273.15 Since only temperature deviations from this reference value are of relevance for the subsequent discussion we identify with the variable = - o the temperature in degrees Celsius. 

For the following discussion we assume in addition all components of the electric field vector E, all stresses tAs, and all strains Vab to be zero in the reference state. However, we allow for the presence of spontaneous polarization P. 

To illustrate the notation and meaning of the third-order coefficients let us discuss the electric Enthalpy H2 (6.16) and the electric Gibbs function G2 (6.17). With Ek and Vab as independent variables they are important for experiments involving elastic waves arrd vibrations. 

The general strategy follows ideas by Bragger (1964), Tiersten (1971), Nelson (1979), and Adam et al. (1988) which is to start from the idea of stieking to the symbols of the second-order quantities arrd just augment them by additional indices according to the variable irrvolved in the additional derivative. The procediue is most easily demonstrated with the third-order stiffnesses 

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