Thermoelastic Properties of Solids based on Equation of State (original) (raw)
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Application of high pressure–high temperature equation of state for elastic properties of solids
Physica B-condensed Matter, 2002
The theory of high pressure-high temperature equation of state recently developed is used to investigate the elastic properties of solids under the effect of temperature as well as pressure. The calculated values of temperature dependence of bulk modulus of NaCl are found to present a better agreement with the experimental data as compared with earlier relation. The results obtained for second-order elastic constants are found to present a good agreement with experimental data. It is concluded that the present approach is very simple and far better compared with the theory of lattice dynamics and two-body central potential. It makes the situation very simple and straightforward as compared with earlier investigations. The results are reported for NaCl, KCl, CaF 2 , MgO, CaO, Mg 2 SiO 4 and Al 2 O 3 . r
Volume-based thermoelasticity: Thermal expansion coefficients and the Grüneisen ratio
Journal of Physics and Chemistry of Solids, 2012
In an extension of our current studies of volume-based thermodynamics and thermoelasticity (VBT), we here consider the parameters at ambient temperature of the dimensionless Gr + uneisen ratio (or Gr + uneisen parameter), g th , which is a standard descriptor of the thermophysical properties of solids:
Geochimica et Cosmochimica Acta, 2003
Grüneisen's parameters are central to studies of Earth's interior because these link elastic data to thermodynamic properties through the equation of state and can be measured using either microscopic or macroscopic techniques. The original derivation requires that the mode Grüneisen parameter (␥ i) of the longitudinal acoustic (LA) mode equals the thermodynamic parameter (␥ th) for monatomic solids. The success of the Debye model indicates that ␥ LA ϭ ␥ th is generally true. Available elasticity data for crystalline solids contain 30 reliable measurements, covering 10 structures, of the pressure derivatives of the bulk (K S) and the shear (G) moduli. For these phases, the measured values of ␥ th and ␥ LA agree well. Other solids in the database have disparate ␥ LA values, suggesting large experimental uncertainties within which ␥ LA ϭ ␥ th. This relationship allows inference of the pressure (P) derivative of the shear modulus (ѨG/ѨP ϭ G') from widely available measurements of ␥ th , the isothermal bulk modulus (K T), ѨK T /ѨP, and G. We predict G' as 1.55 for stishovite, 1.6 to 2.15 for MgSiO 3 ilmenite, 1.0 for ␥-Mg 1.2 Fe 0.8 SiO 4 , and 0 for FeS (troilite). Similarly, G' measured for MgSiO 3 perovskite suggests that K S ' ϭ 4, corroborating volume-pressure data. For many materials, pairs of G' and K S ' ϭ ѨK S /ѨP from independent elasticity studies of a given phase define a nearly linear trend, suggesting systematic errors. Non-hydrostatic conditions and/or pressure calibration likely cause the observed variance in K S ' and G'. The best values for pressure derivatives can be ascertained because the trend defined by measured pairs of G' with K S ' intersects the relationship of G' to K' defined by ␥ LA ϭ ␥ th at a steep angle. Our results for isostructural series show linear correlations of K S ' with K S and of G' with G. Values of K S ' are nearly 4 for high-pressure phases, which is consistent with the harmonic oscillator model, whereas G' has a wide range of Ϫ1 to 3. Hence, inference of a detailed mineralogy inside the Earth is best constrained by comparing seismic determinations of shear moduli to laboratory measurements.
A Simple, Potential-Free Model to Calculate Elastic Constants of Solids at High Temperature
American Journal of Condensed Matter Physics, 2012
A simple method for the determination of temperature dependent second order elastic constants (SOEC) of MgO, CaO, Mg 2 SiO 4 and Grossular garnet[Ca 3 Al 2 (SiO 4 ) 3 ] using a potential free model based on thermodynamical relationships, has been proposed. The equations developed here are based on the linear relationship between elastic constants at temperatures higher than the Debye temperature. The extrapolated data for elastic constants at very high temperatures obtained in the present study are useful to understand the thermoelastic properties of given solids. It is found that the calculated values of elastic constants, in general, decrease with temperature. The theoretical predictions incorporating the concept of Debye temperature, reported in this paper, are well supported by the available experimental data. The proposed emp irical relationship provides a method to estimate the thermoelastic properties of geophysical minerals and solids at high temperature range.
The dependence of the Anderson-Grüneisen parameter δT upon compression at extreme conditions
Journal of Physics and Chemistry of Solids, 1993
The parameter 6, is a dimensionless thermoelastic parameter important in thermodynamic studies involving high temperature at high pressure. We give a brief history of attempts to define 6, in terms of fundamental interatomic potentials. We show that ab initio calculations provide a way to find 6,(~, T) (where 9 = V/V,,), since it is possible to find the minimum in the isochoric Br vs Tdata arising from the ub initio analysis (Br is the isothermal bulk modulus). The method is demonstrated for MgO, where 6, is calculated over a wide q, Tfield. We find 6, decreases at high compression, but is independent of T.
Journal of Alloys and Compounds, 2004
The main objective of this study is to develop a rigorous thermodynamic apparatus for elucidating the temperature and pressure dependencies of thermal and elastic properties in an integrated manner. Towards this cause, an isobaric linear scaling relation or approximation connecting logarithm of adiabatic bulk modulus (ln B S) with enthalpy increment (H = H T − H 0) of the form ln(B S) = ln(B 0) + k S (H T − H 0), is invoked in this study. k S is a temperature independent thermoelastic parameter. It is found that this relation is obeyed by a number of solids irrespective of their bonding peculiarities, that includes metals, ceramics and minerals of geophysical and nuclear interest. A rigorous analysis of the thermodynamic implications of this scaling relation is presented in this paper. In particular, useful approximations for the temperature and pressure dependencies of bulk modulus, entropy and enthalpy are obtained. More importantly, the analysis brings out vividly the underlying physical basis for the apparent temperature independence of certain popular thermoelastic quantities like Grüneisen and Anderson-Grüneisen parameters. Besides, as a useful offshoot a new relation connecting the isobaric temperature variation of Grüneisen parameter with the corresponding enthalpy change is also obtained. Further, the possibility that suitably parametrised enthalpy or specific heat data can be made use of in obtaining reliable estimates of thermal expansion or vice versa is also demonstrated in this work. In all, a simple and consistent thermodynamic framework connecting elastic and thermal properties is presented. The applicability of the theoretical framework discussed in the present study towards the cause of estimation, extrapolation, and an integrated assessment of thermal cum elastic property data is exemplified, by taking thoria as the illustrative case study. In particular, a self-consistent estimate of its bulk modulus at high temperatures has been obtained using experimental enthalpy and molar volume data.
International Journal of Engineering Science, 2017
A set of six, independent, stress/strain, conjugate pairs are derived: One for dilation, two for squeeze, and three for shear. They follow from a Gram-Schmidt factorization of the deformation gradient. Theories for elastic solids are derived in terms of these conjugate pairs. Anisotropy is introduced through bijective maps between tensor components and constituents of the basis. The boundary value problems of simple tension and uniform pressure are used to illustrate the effects of anisotropy, as predicted by a Hooke-like material model.
International Journal of Trend in Scientific Research and Development, 2020
Expression for the Bulk modulus and its Pressure derivatives have been derived and reduced to the limit of infinite pressure. The Pressure dependence of thermal expensively and the Grüneisen Parameter both are determined using the formulations which satisfy the thermodynamic constraints at infinite pressure. Values of Bulk modulus and its Pressure derivative are also obtained for the entire range of Temperatures and Pressures considered in the present study. We have also investigated the Thermo elastic Properties of MgO at high Temperature and High Pressures using the results based on the EOS. The method based on the calculus in determinates for demonstrating that on the physically acceptable EOS Satisfy the identities for the pressure derivatives of bulk modulus Materials.