Third‐Order Padé Thermoelastic Constants of Solid Rocks.

Classical third‐order thermoelastic constants are generally derived from the theory of small‐amplitude acoustic waves in isotropic materials during heat treatments. Investigating higher‐order thermoelastic constants for higher temperatures is challenging owing to the involvement of the number of unknown parameters. These Taylor‐type thermoelastic constants from the classical thermoelasticity theory are formulated based on the Taylor series of the Helmholtz free energy density for preheated crystals. However, these Taylor‐type thermoelastic models are limited even at low temperatures in characterizing the temperature‐dependent velocities of elastic waves in solid rocks as a polycrystal compound of different mineral lithologies. Thus, we propose using the Padé rational function to the total thermal strain energy function. The resulting Padé thermoelastic model gives a reasonable theoretical prediction for acoustic velocities of solid rocks at a higher temperature. We formulate the relationship between the third‐order Padé thermoelastic constants and the corresponding higher‐order Taylor thermoelastic constants with the same accuracy. Two additional Padé coefficients α1 ${\mathit{\alpha }}_{1}$ and α2 ${\mathit{\alpha }}_{2}$ can be calculated using the second‐, third‐, and fourth‐order Taylor thermoelastic constants associated with the Brugger's constants, which are consistent with those obtained by fitting the experimental data of polycrystalline material. The third‐order Padé thermoelastic model (with four constants) is validated by the fourth‐order Taylor thermoelastic prediction (with six constants) with ultrasonic measurements for polycrystals (olivine samples) and solid rocks (sandstone, granite, and shale). The results demonstrate that the third‐order Padé thermoelastic model can characterize thermally induced velocity changes more accurately than the conventional third‐order Taylor thermoelastic prediction (with four constants), especially for solid rocks at high temperatures. The Padé approximation could be considered a more accurate and universal model in describing thermally induced velocity changes for polycrystals and solid rocks. Plain Language Summary: Using the Taylor series for thermoelastic constants at higher temperatures is challenging owing to the involvement of the number of unknown parameters. Thus, we propose the third‐order Padé approximation to replace the Taylor series. Applications to laboratory measurements show that, with much less complexity, the Padé approximation presents a tendency of higher accuracy than the third‐order Taylor series and has similar precision with the higher‐order Taylor series at higher temperatures (up to 1500 K for polycrystals and 1000°C for solid rocks). Then we use such a model to predict the temperature‐dependent velocities of elastic waves. Finally, we analyze the physics of two additional Padé coefficients by associating them with the thermal expansion mismatch due to rock heterogeneities in lithology and with the thermally induced deformation of microcracks. Key Points: We apply the Padé rational function to the total thermal strain energy functionWe correlate the Padé and higher‐order Taylor thermoelastic constants through the equivalency of approximation accuraciesThe physics of the Padé coefficients is relevant to the thermal expansion mismatch and thermally induced deformation of microcracks [ABSTRACT FROM AUTHOR]