Pressure Dependence Of Elastic Moduli Research Articles

A semi‐empirical approach to model pressure dependence of elastic moduli in granular media accounting for variations of coordination‐number and Poisson‐ratio

ABSTRACTThe effective medium theory based on the Hertz–Mindlin contact law is the most popular theory to relate dynamic elastic moduli (or elastic velocities) and confining pressure in dry granular media. However, many experimental results proved that the effective medium theory predicts pressure trends lower than experimental ones and over‐predicts the shear modulus. To mitigate these mispredictions, several evolutions of the effective medium theory have been presented in the literature. Among these, the model named modified grain contact theory is an empirical approach in which three parametric curves are included in the effective medium theory model. Fitting the parameters of these curves permits to adjust the pressure trends of the Poisson ratio and the bulk modulus. In this paper, we present two variations of the modified grain contact theory model. First, we propose a minor modification in the fitting function for the porosity dependence of the calibration parameters that accounts for non‐linearity in the vicinity of the critical porosity. Second, we propose a major modification that reduces the three‐step modified grain contact theory model to a two‐step model, by skipping the calibration parameter–porosity fit in the model and directly modelling the calibration parameter–pressure relation. In addition to an increased simplicity (the fitting parameters are reduced from 10 to 6), avoiding the porosity fit permits us to apply the model to laboratory data that are not provided with accurate porosity measurements. For this second model, we also estimate the uncertainty of the fitting parameters and the elastic velocities. We tested this model on dry core measurements from literature and we verified that it returns elastic velocity trends as good as the original modified grain contact theory model with a reduced number of fitting parameters. Possible developments of the new model to add predictive power are also discussed.

Elastic potential and pressure dependence of elastic moduli in fluid-saturated rock with double porosity

Models of seismic velocity dispersion and attenuation in porous rock are often based on quantitative relations between empirical moduli by Gassmann, Mavko-Jizba, Sayers-Kachanov, and others. All of these relations have a common origin in the concept of elastic potential; nevertheless, this concept itself has been insufficiently used in the context of multiple porosities and squirt flows. Regardless of the microstructure of a rock, knowledge of its macroscopic elastic potential reveals the complete set of parameters that are necessary and sufficient for characterizing its elasticity and obtaining rigorous equations of deformation. For isotropic rock with double (such as stiff and/or compliant) porosity, this parameter set contains six elements of the bulk-modulus matrix and six moduli for shear. The elastic matrices predict all observable elastic properties, such as bulk, shear and Young’s moduli, pore compressibilities, Skempton coefficients, consolidation parameters, Poisson’s ratios, static (Gassmann’s) and ultrasonic (Mavko-Jizba’s) undrained moduli, as well as the hypothetical high-pressure modulus [Formula: see text] and unrelaxed-frame moduli. By using laboratory observations with Westerly granite, all six elements of the elastic matrix for bulk deformation are inverted for exactly; three moduli for shear are determined with uncertainties due to insufficient data. The inverted matrices of the elastic moduli predict all observations (drained and low- and high-frequency undrained moduli) within measurement accuracy, with minor uncertainties for shear deformation. The secondary pores are approximately 50 times more compressible than are the primary ones, which explains their interpretation as “soft.” Most of the inverted elastic moduli vary with the confining pressure. Despite its usually assumed constancy, the high-pressure modulus [Formula: see text] exhibits an approximately 10% decrease with confining pressure from 0 to 100 MPa, which is opposite of the trend of the solid-grain modulus [Formula: see text].

High-temperature elasticity of polycrystalline orthoenstatite (MgSiO3)

Compressional and shear wave velocities of a polycrystalline specimen of MgSiO 3 orthoenstatite have been measured by ultrasonic interferometry to 1373 K at 300 MPa in an internally heated gas-medium apparatus. The elastic wave velocities and bulk and shear moduli vary linearly with temperature to 1073 K. Below 1073 K, the temperature derivatives of the elastic moduli [(∂KS/∂T) P = -28.3(7) MPa/K and (∂G/∂T) P = -14.5(1) MPa/K, respectively] determined in this study are consistent with averages of single-crystal elastic constants measured using Brillouin spectroscopy by Jackson et al. (2007). The measured temperature dependence of elastic moduli, along with pressure dependence of elastic moduli, thermal expansion and calorimetric data have been assimilated into a finite-strain equation of state of the type proposed by Stixrude and Lithgow-Bertelloni (2005). This analysis suggests significant revisions to the optimal values of the zero-pressure Grüneisen parameter γ 0 and its zero-pressure logarithmic volume derivative q 0 . The unusually high absolute values of (∂K/∂P) T and (∂K/∂T) P are related through the extrinsic part of the temperature derivative. Above 1073 K, a pronounced softening of the elastic wave velocities is observed, which is plausibly associated with a phase transformation for which there is microstructural evidence: The recovered specimen was found to have transformed to the low-pressure clinoenstatite polymorph.

Goddard rattler-jamming mechanism for quantifying pressure dependence of elastic moduli of grain packs

An analysis is presented to show how it is possible for unconsolidated granular packings to obey overall non-Hertzian pressure dependence due to the imperfect and random spatial arrangements of the grains in these packs. With imperfect arrangement, some gaps that remain between grains can be closed by strains applied to the grain packing. As these gaps are closed, former rattler grains become jammed and new stress-bearing contacts are created that increase the elastic stiffness of the packing. By allowing for such a mechanism, detailed analytical expressions are obtained for increases in bulk modulus of a random packing of grains with increasing stress and strain. Only isotropic stress and strain are considered in this analysis. The model is shown to give a favorable fit to laboratory data on variations in bulk modulus due to variations in applied pressure for bead packs.

The first principles study on PtC compound

We have studied structural, thermodynamic, elastic, and electronic properties of platinum carbide (PtC) in zinc-blende and rock-salt structures by performing ab initio calculations within the LDA approximations. Particularly, we have focused on the structural and the pressure dependence of elastic moduli and related quantities. The other basic key properties, such as the lattice constant, cohesive energy, the phase transition pressure, bulk modulus and its pressure derivative are also repeated and compared with the other available experimental and theoretical works.

Thermoelastic properties of minerals

The simple theory of equation of state recently developed by Kumar is used to investigate the temperature and pressure dependence of elastic moduli of MgO. The results are found to present good agreement with the experimental data. It is concluded that the Kumar formulation is far better than the Suzuki theory of thermal expansivity, and the Shanker formulation is not a new relation.

Ultrasound studies ofMgB2superconductor under hydrostatic pressure

The pressure dependence of sound velocities of longitudinal and shear wave were measured up to hydrostatic pressure of 0.5 GPa using a pulse-echo-overlap method on a polycrystalline single-phase ${\mathrm{MgB}}_{2}$ superconductor which was high-pressure synthesized and with a ${T}_{c}$ of 39 K. The large hysteresis of sound velocities was observed in the experiment. Because of weaker metal-boron bonding in ${\mathrm{MgB}}_{2},$ it is speculated that the large hysteresis is due to relaxation between Mg and B layers under hydrostatic pressure. The pressure dependence of elastic moduli, Debye temperature, and specific heat were calculated based on sound velocity data. The results indicate that all elastic moduli and thermodynamic parameter increase with the applied pressures except specific heat. The isothermal equation of state of ${\mathrm{MgB}}_{2}$ is established in terms of the Murnaghan form. From the sound velocities, the pressure coefficient of the average phonon frequency was obtained.

X-ray diffraction under non-hydrostatic conditions in experiments with diamond anvil cell: wüstite (FeO) as an example

The stress state in a solid specimen compressed between two flat and parallel anvil faces (in a diamond anvil cell, for example) is non-hydrostatic. The resulting lattice strains, measured by X-ray diffraction, exhibit certain features, which are absent if truly hydrostatic conditions prevail. We describe the method of analysis of lattice strain in non-hydrostatic stress states. This is based on the transformation of stress axis into crystal axis and further on the calculations of the strain in a given direction by taking into account the pressure dependence of elastic moduli. Application of the theory is demonstrated with an example of a study of wüstite (FeO) at pressures up to 18 GPa. We found that the pressure of phase transition is proportional to the uniaxial stress component, which rises from 10.1 GPa for compression without pressure medium to 15.4 GPa in Ar pressure medium. Extrapolation to zero stress conditions gives the value 16.8 GPa, which is in good agreement with the data of Fei (Y. Fei, in: M.D. Dyar, C. MacComman, M.W. Schaefer (Eds.), Mineral Spectroscopy, Geochemical Society, 1996 pp. 243–254) (powder diffraction data, compression in He pressure medium, P tr=17(1) GPa) and Shu (J. Shu, Eos Trans. Am. Geophys. Union 75 (1998) 203) (single crystal diffraction data, compression in He pressure medium, P tr=18(1) GPa).

Structure, elastic moduli, and thermodynamics of sodium and potassium at ultrahigh pressures

The equations of state at room temperature as well as the energies of crystal structures up to pressures exceeding 100 GPa are calculated for Na and K. It is shown that allowance for generalized gradient corrections in the density functional method provides a precise description of the equation of state for Na, which can be used for the calibration of a pressure scale. It is established that the close-packed structures and bcc structure are not energetically advantageous at high enough compressions. Sharply nonmonotonic pressure dependences of elastic moduli for Na and K are predicted, and melting temperatures at high pressures are estimated from various melting criteria. The phase diagram of K is calculated and found to be in good agreement with experiment.

Thermal expansion at low temperatures of the alkaline earth fluorides and PbF2

Thermal expansions have been measured from 2 to 90K on four compounds of fluorite structure. Data at the lowest temperatures lead to the following limiting values of the Gruneisen parameter: gamma 0thermal=1.0 (CaF2), 0.64 (SrF2), 0.23 (BaF2) and 0.5-0.8 (PbF2). These agree within limits of error with those calculated from the pressure dependence of elastic moduli. For the three alkaline earth fluorides gamma (T) increases monotonically with temperature from gamma 0 to gamma 293 approximately 1.5-1.8, but for PbF2 there is a marked maximum near 25K where gamma approximately=3.5, consistent with the effect of an optic mode of characteristic temperature theta E approximately=68K.

Structure in the upper lunar crust

Estimates are made of the degree of lithification and of structure densities which are compatible with lunar in situ seismic profiles in the top 30 km of the Moon. Estimates are based on comparison of results of passive and active lunar seismic experiments with the pressure dependence of elastic moduli for various classes of lunar samples. Competent rock, such as igneous rock or recrystallized breccias with crack porosity of not more than about 0.5 % are required to satisfy velocity profiles in the depth range 1-30 km. Velocity profiles in the upper 1 km are best satisfied by comminuted material to highly fractured lithic units. These estimates constrain those thermal and shock histories which are compatible with lunar seismic results. After crystallization, or recrystallization, rock below 1 km cannot have been exposed to more than moderate shock levels. In the uppermost 1 km, an unannealed and broken rock layer would imply low thermal conductivity resulting in possible temperatures at 1 km depth of several hundred kelvins.

The pressure dependence of elastic moduli of NiF 2 to 10 kbar

At room temperature, the pressure dependence of the elastic stiffness moduli of NiF 2 have been measured by pulse superposition method to 10 kbar. The observed strong decreasing behavior in the modulus C s = 1 2 (C 11 − C 12) is discussed.