Extended temperature dependence of elastic constants in cubic crystals

Telichko, Arsenii; Sorokin, B. P.

doi:10.1016/j.ultras.2015.02.020

Cited by 5 publications

(5 citation statements)

References 29 publications

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“…Nonlinear thermoelasticity theories generally ignore humidity, salinity, grain geometry, and microstructures. The theories are based upon the assumption that the strain energy in a material is not simply a function of the square of the strain, but includes higher‐order terms as well, resulting in the high‐order thermoelastic constants (e.g., Garber & Granato, 1975; Shrivastava, 1980; Sorokin et al., 1999; Telichko & Sorokin, 2015; Yang, Fu, Fu, et al., 2019) in addition to the standard elastic stiffnesses. Like these thermoelastic constants, the Padé coefficients presented in this study also belong to macroscopic constants and are not correlated directly to microstructures and heterogeneities.…”

Section: Experimental Samples and Resultsmentioning

confidence: 99%

“…The quantitative analysis of the correlation between Padé coefficients and rock properties, however, such as mineral proportions, lithological diversities, grain scales, and microcracks, will be the object of further studies. Measuring the higher‐order elastic constants in crystals and solid rocks at high temperatures and pressures is challenging (e.g., Hiki et al., 1967; Telichko & Sorokin, 2015; Winkler & Liu, 1996; Winkler & McGowan, 2004). Hence, we obtain these elastic constants of porous rocks by least squares fitting the experimental measurements.…”

Section: Discussionmentioning

confidence: 99%

“…Based on the assumption of finite static strain for temperature changes and small-amplitude acoustic waves, the fourth-order Taylor thermoelastic constants are introduced to develop nonlinear thermoelastic models to account for the second-order (Telichko & Sorokin, 2015) and third-order nonlinear temperature dependence of elastic wave velocities for cubic crystals and isotropic materials, respectively. For example, the third-order nonlinear thermoelastic model for isotropic materials is defined as…”

Section: Thermoelastic Models For Theoretical Predictionmentioning

confidence: 99%

“…Equation 46 accounts for the linear temperature dependence of elastic wave velocities in isotropic materials. Based on the assumption of finite static strain for temperature changes and small‐amplitude acoustic waves, the fourth‐order Taylor thermoelastic constants are introduced to develop nonlinear thermoelastic models to account for the second‐order (Telichko & Sorokin, 2015) and third‐order (Yang, Fu, Fu, et al., 2019) nonlinear temperature dependence of elastic wave velocities for cubic crystals and isotropic materials, respectively. For example, the third‐order nonlinear thermoelastic model for isotropic materials is defined as

\left\{\begin{array}{l}\rho {V}_{p}^{2}=\mathit{\lambda }+2\mu +{\gamma }_{1}^{P}{\epsilon}+{\gamma }_{2}^{P}{{\epsilon}}^{2}+{\gamma }_{3}^{P}{{\epsilon}}^{3},\hfill \\ \rho {V}_{s}^{2}=\mu +{\gamma }_{1}^{S}{\epsilon}+{\gamma }_{2}^{S}{{\epsilon}}^{2}+{\gamma }_{3}^{S}{{\epsilon}}^{3},\hfill \end{array}\right.

where the synthetical third‐ and fourth‐order Taylor thermoelastic constants (

{\gamma }_{1}^{P}

{\gamma }_{1}^{S}

) and (

{\gamma }_{2}^{P}

{\gamma }_{3}^{P}

{\gamma }_{2}^{S}

{\gamma }_{3}^{S}

) for elastic P‐ and S ‐waves, respectively, are written as

({, \begin{matrix} γ_{1}^{P} = 2 italic... \end{matrix})

…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…The temperature‐associated elastic constants in most crystals are usually weak and linear at low temperatures but gain prominence with increasing temperatures (e.g., Diederich & Trivisonno, 1966; Meeks & Arnold, 1970). Such a nonlinear temperature dependence at higher temperatures results in higher‐order thermoelastic constants (e.g., Sorokin et al., 1999; Telichko & Sorokin, 2015) and consequently determining more effective elastic constants. These theoretical crystal models have been extended to single‐crystal rocks (garnet and rock salt) to investigate the temperature‐associated behavior of elastic constants (e.g., Bonczar et al., 1977; Overton & Swim, 1951; Soga, 1967).…”

Section: Introductionmentioning

confidence: 99%

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Third‐Order Padé Thermoelastic Constants of Solid Rocks

Yang

et al. 2022

JGR Solid Earth

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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.

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Section: Experimental Samples and Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Thermoelastic Models For Theoretical Predictionmentioning

confidence: 99%

\left\{\begin{array}{l}\rho {V}_{p}^{2}=\mathit{\lambda }+2\mu +{\gamma }_{1}^{P}{\epsilon}+{\gamma }_{2}^{P}{{\epsilon}}^{2}+{\gamma }_{3}^{P}{{\epsilon}}^{3},\hfill \\ \rho {V}_{s}^{2}=\mu +{\gamma }_{1}^{S}{\epsilon}+{\gamma }_{2}^{S}{{\epsilon}}^{2}+{\gamma }_{3}^{S}{{\epsilon}}^{3},\hfill \end{array}\right.

where the synthetical third‐ and fourth‐order Taylor thermoelastic constants (

{\gamma }_{1}^{P}

{\gamma }_{1}^{S}

) and (

{\gamma }_{2}^{P}

{\gamma }_{3}^{P}

{\gamma }_{2}^{S}

{\gamma }_{3}^{S}

) for elastic P‐ and S ‐waves, respectively, are written as

({, \begin{matrix} γ_{1}^{P} = 2 italic... \end{matrix})

…”

Section: Theoretical Backgroundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Third‐Order Padé Thermoelastic Constants of Solid Rocks

Yang

et al. 2022

JGR Solid Earth

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Low temperature acoustic characterization of PMN single crystal

Smirnova

Сотников

Ktitorov

et al. 2017

Journal of Applied Physics

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This acoustic study of the relaxor single crystal PbMg1/3Nb2/3O3 is focused on the temperature evolution of longitudinal and shear ultrasonic wave velocities as well as on their attenuation within the temperature range of 4.2–300 K. Temperature dependences of all three independent elastic constants for the cubic structure (C11, C44, and C12) along with the bulk modulus B, the Cauchy ratio, and the degree of lattice anisotropy were derived from the velocity data set. Important parameters such as the Debye temperature, the Grüneisen parameter H, and the Poisson ratio were determined at cryogenic temperatures as well. Deep minima were found for C11 and C44 around the temperature Tm where the dielectric constant reaches a maximum, followed by a drastic increase of C11 and C44 at decreasing temperature and ultimately by a saturation at temperatures below 25 K. The behavior of these constants at T < Tm corresponds well to relations resulting from the Einstein-oscillator model. A changeover of C12(T) dependence from increasing at T < Tm to unexpected monotonic decreasing behavior upon cooling from inflection temperature Ti ≈ 220 K followed by a saturation at T < 25 K was found. Furthermore, the behavior of C12 (T) and interrelated parameters such as B and H was analyzed on the basis of the concept of phonon pressure due to the thermal expansion. It is essential that the electrostriction contribution was taken into account, as well.

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On three-stage temperature dependence of elastic wave velocities for rocks

Yang

et al. 2021

Journal of Geophysics and Engineering

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For most rocks, the typical temperature behavior of elastic wave velocities generally features a three-stage nonlinear characteristic that could be expressed by a reverse S-shape curve with two inflexion points. The mechanism regulating the slow-to-fast transition of elastic constants remains elusive. The physics of critical points seems related to the multimineral composition of rocks with differentiated thermodynamic properties. Based on laboratory experiments for several rocks with different levels of heterogeneity in compositions, we conduct theoretical and empirical simulations by nonlinear thermoelasticity methods and a S-shape model, respectively. The classical theory of linear thermoelasticity based on the Taylor expansion of strain energy functions has been widely used for crystals, but suffers from a deficiency in describing thermal-associated velocity variations for rocks as a polycrystal mixture. Current nonlinear thermoelasticity theory describes the third-order temperature dependence of velocity variations by incorporating the fourth-order elastic constants. It improves the description of temperature-induced three-stage velocity variations in rocks, but involves with some divergences around two inflexion points, especially at high temperatures. The S-shape model for empirical simulations demonstrates a more accurate depiction of thermal-associated three-stage variations of P-wave velocities. We investigate the physics of the parameters ${a_1}$ and ${b_1}$ in the S-shape model. These fitting parameters are closely related to thermophysical properties by being proportional to the specific heat and thermal conductivity of rocks. We discuss the mechanism that regulates the slow-to-fast transition in the three-stage nonlinear behavior for various rocks.

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Extended temperature dependence of elastic constants in cubic crystals

Cited by 5 publications

References 29 publications

Third‐Order Padé Thermoelastic Constants of Solid Rocks

Third‐Order Padé Thermoelastic Constants of Solid Rocks

Low temperature acoustic characterization of PMN single crystal

On three-stage temperature dependence of elastic wave velocities for rocks

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