Third‐Order Padé Thermoelastic Constants of Solid Rocks

Abstract

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.

Ionospheric–Thermospheric Responses to Geomagnetic Storms from Multi-Instrument Space Weather Data

“…Various authors have shown latitudinal and longitudinal ionosphere responses due to PPEF [60][61][62][63][64]. EEJ variations at different longitudes are due to the underlining effects of local winds, which are responsible for EEJ driving [65][66][67][68][69]. In addition, longitudinal differences in EEJ are caused by the different nature of the propagating diurnal tides, the meridional winds, and the dynamics of the migratory tides [70][71][72][73][74].…”

A Neural Network Trained by Multi-Tracker Optimization Algorithm Applied to Energy Performance Estimation of Residential Buildings

“…As is known, the research related to the building sector covers a wide range of subjects from construction safety [47,48] and material analysis [49,50] to energy systems. In this work, the focus was on energy performance analysis that was successfully carried out using sophisticated machine learning.…”

Prediction of Rock Tensile Strength Using Soft Computing and Statistical Methods

“…The mechanical properties of foundation materials affect the stability of civil and mining structures [36,37]. Mineralogical and physical properties have a significant effect on the mechanical properties and stability of mines [38].…”