Non-hydrostatic Conditions Research Articles

Bulk Modulus and Young's Modulus of Nanocrystalline γ‐Alumina

Compression measurements were performed for the first time on nanocrystalline γ‐alumina utilizing a diamond anvil cell (DAC) and the energy dispersive X‐ray diffraction method. The cubic unit cell (a= 0.7924 nm) for γ‐alumina was found to have a volume compression of about 2.4% over the pressure range from ambient to 3.8 GPa at room temperature under both hydrostatic and nonhydrostatic conditions. Using the first‐order Bridgman equation and the Birch equation of state, the isothermal bulk modulus (B0) was determined to be 162 ± 14 GPa and Young's modulus (E) was estimated to be 253 ± 22 GPa assuming a Poisson's ratio for γ‐alumina of 0.24 ± 0.2.

Pressure-induced structural modifications in Mg2GeO4-olivine: A Raman spectroscopic study

Raman spectra of Mg2GeO4-olivine were obtained from ambient pressure up to 34 GPa at ambient temperature. Under quasi-hydrostatic pressure conditions, the following modifications in the Raman spectra occur as pressure increases: 1) near 11 GPa, two sharp extra bands appear in the 600–700 cm−1 frequency range, and increase in intensity with respect to the olivine bands; 2) above 22 GPa, these two bands become very intense, and the number, position and relative intensity of the other vibrational bands drastically change; 3) the intensity of sharp bands progressively decreases above 25 GPa. The transformation occurs at lower pressures under non-hydrostatic conditions. During decompression to atmospheric pressure, the high-pressure phase partially reverts to olivine. These observations can be interpreted as the progressive metastable transformation from the olivine structure to a crystalline phase with four-fold coordinated Ge, in which the GeO4 tetrahedra are polymerized. We propose that the metastable high-pressure phase is a structurally disordered spinelloid close to the hypothethical ω- or ɛ*-phase, and forms by a shear mechanism assisted by the development of a dynamical instability in the olivine structure. Implications for the transformations undergone by olivines under far-from-equilibrium conditions (e.g. in subducting lithospheric slabs and in shocks) are discussed.

High-pressure deformation of calcite marble and its transformation to aragonite under non-hydrostatic conditions

We conducted deformation experiments on Carrara marble in the aragonite and calcite stability fields to observe the synkinematic transformation of calcite to aragonite, and to identify any relationships between transformation and deformation or sample strength. Deformation-induced microstructures in calcite crystals varied most significantly with temperature, ranging from limited slip and twinning at 400°C, limited recrystallization at 500°C, widespread recrystallization at 600 and 700°C, to grain growth at 800–900°C. Variations in confining pressure from 0.3 to 2.0 GPa have no apparent effect on calcite deformation microstructures. Aragonite grew in 10 −6-10 −7 s −1strain rate tests conducted for 18–524 h at confining pressures of 1.7–2.0 GPa and temperatures of 500–600°C. As in our previously reported hydrostatic experiments on this same transformation, the aragonite nucleated on calcite grain boundaries. The extent of transformation varied from a few percent conversion near pistons at 400°C, 2.0 GPa and 10 −4 s −1 strain rate in a 0.8 h long experiment, to 98% transformation in a 21-day test at a strain rate of 10 −7 s −7, a temperature of 600°C and a pressure of 2.0 GPa. At 500°C, porphyroblastic 100–200 μm aragonite crystals grew at a rate faster than 8 × 10 −1m s −1. At 600°C, the growth of aragonite neoblasts was slower, ≈6 × 10 −1 m s −1, and formed ‘glove-and-finger’ cellularprecipitation-like textures identical to those observed in hydrostatic experiments. The transformation to aragonite is not accompanied by a shear instability or anisotropic aragonite growth, consistent with its relatively small volume change and latent heat in comparison with compounds that do display those features.

Structural transformations of quartz and berlinite AlPO4 at high pressure and room temperature: a transmission electron microscopy study

Single crystals of α-quartz and α-berlinite AlPO4 have been compressed at high pressure and room temperature in a diamond anvil cell (DAC). The pressure-induced microstructures have been studied on recovered specimens using transmission electron microscopy. As previously reported, quartz is shown to exhibit an amorphous transition at high pressure (≈30 GPa). Under the markedly non-hydrostatic conditions of the present study, a wide mixed-phase regime in which amorphous lamellae form within the crystalline matrix is encountered at lower values of the mean stress. The amorphous lamellae are interpreted as shear lamellae. The formation of these shear lamellae as well as their habit planes are described by the evolution with pressure of shear moduli μ as computed in anisotropic elasticity. Our calculations also show instabilities at higher pressure of the elastic moduli (i.e. of the α-quartz structure) which are related to the amorphous transition. Berlinite exhibits a more ductile behavior with simultaneous dislocation activity and shear on amorphous lamellae which become pervasive at high pressure (≈10 GPa). These amorphous lamellae of berlinite do not revert to crystal when pressure is released.

Analysis of C60fullerite compression under non-hydrostatic pressure

Abstract An equation has been derived for the lattice strain that corresponds to the strain measured by X-ray diffraction in a specimen compressed non-hydrostatically in a diamond anvil cell. This equation has been used to discuss the observed difference between the volume compressions of C60 fullerite measured under non-hydrostatic and hydrostatic pressure conditions.

The Elastic Strain Energy of Coherent Ellipsoidal Precipitates in Anisotropic Crystalline Solids: Applications to the Aragonite-Calcite Transformation

ABSTRACTEshelby's theory has been applied to the aragonite-calcite transformation in order to calculate the elastic strain energy of coherent ellipsoidal precipitates in anisotropie solids. Using a quasiNewton's Method and a finite-difference gradient, the elastic strain energy was incorporated in order to calculate the activation energy ΔG* for homogeneous nucleation under hydrostatic and non-hydrostatic conditions.

Characterization of the thermal and thermomechanical behaviour of polyphase salt rocks by means of electrical conductivity and gas permeability measurements

Measurements of electrical conductivity and gas permeability in carnallite-halite-kieserite rocks were carried out under hydrostatic (2.5–24.0 MPa) and non-hydrostatic pressure conditions (strain rate: 10 −7–10 −8s −1) at temperatures up to 125°C in order to obtain information on the thermal and thermomechanical stability of hydrated carnallite (KMgCl 3) · 6H 2O and the corresponding state of rock microstructure. At hydrostatic pressure an anomalous increase in conductivity with increasing temperature was observed at 100°C which is attributed to a release of chemically bound water. Isothermal deformation of carnallitic rocks (non-hydrostatic pressure) is accommodated by the interaction of different deformation mechanisms (cataclastic flow, intracrystalline slip, mechanical twinning and dynamic recrystallization). Increasing temperature or increasing humidity on grain boundaries shifts the deformation behaviour from “brittle” to “ductile”, due to the onset of dynamic recrystallization and leads to a significant weakening of the rock. The brittle/ductile transition is indicated by characteristic signatures of the electrical conductivity and gas permeability/strain relations. The combination of both parameters allows one to separate the effects on conductivity caused by deformation-related changes in porosity from those caused by dehydration reactions.

Potential use of the ruby R2 line shift for static high-pressure calibration

Results of shock-wave experiments to examine ruby R-line shifts for uniaxial strain compression along the crystal a axis are summarized and compared with previous c-axis results [P. D. Horn and Y. M. Gupta, Phys. Rev. 39, 973 (1989)]. The present results show that R1-line shift with compression depends on both nonhydrostatic stresses and crystal orientation. Hence, pressure determination using R1-line shifts in a nonhydrostatic environment can lead to errors. In contrast, the R2-line shift depends only on density compression and is independent of nonhydrostatic stresses and crystal orientation. These findings suggest the use of R2-line shifts for pressure calibration under nonhydrostatic loading conditions.

Pressure dependence of the 4T2 and 4T1 absorption bands of ruby to 35 GPa.

The pressure dependence of the $^{4}$${\mathit{A}}_{2}$${\ensuremath{\rightarrow}}^{4}$${\mathit{T}}_{2}$ and $^{4}$${\mathit{A}}_{2}$${\ensuremath{\rightarrow}}^{4}$${\mathit{T}}_{1}$ absorption bands of ruby have been studied to 35 GPa using a luminescence excitation technique developed for use at high pressure in the diamond anvil cell. The quasihydrostatic (Ar medium) pressure dependence of the zero-phonon lines, the crystal-field splitting parameter (\ensuremath{\Delta}), and the Racah parameters B and C are determined. A significantly smaller variation of \ensuremath{\Delta} is found for nonhydrostatic stress conditions for a ruby powder compact, and the implications of this on the pressure calibrations, lifetime measurements, and excitation of the R lines at high pressure are discussed.

An emergency computation model for the wind field and diffusion during accidental nuclear pollutants releases

Since 1986, a simple computation model for a nuclear accident has been operating in the emergency information center of Japan Agency for Science and Technology. It was developed by introducing the variation method for wind and a random walk particle model for diffusion in 50–100 km scale. Furthermore, we developed a new model with dynamic equations and a diffusion equation to predict more accurately the wind and diffusion, including local thermal convection, as an improvement on the present system.In the new model, the momentum equation and the continuity equation are solved numerically in non-hydrostatic and incompressible conditions, using a finite difference technique. Then, the equation of thermal energy preservation is solved for potential temperature in the predicted wind field of every time step. Pressure is given as a deviation from the average value of each level by a numerical solution of a parabolic equation introduced from the momentum and continuity equations. The diffusion of nuclear pollutants is computed numerically by the finite difference method in the predicted wind field, using diffusion coefficients obtained from the predictive dynamic equations.These computations were verified with meteorological surveys and gas tracer diffusion experiments over flat land, along a sea shore and over a mountainous area. As the result of new model development, horizontal circulations and vertical convections can be computed in any mesh size from several tens of meters to several kilometers, while small vertical convections less than 1 km or so cannot be represented with the former hydrostatic circulation models.

THE PRESSURE CALIBRATION UP TO MBARS AND THE ACHIEVEMENT OF 5.5 MBARS UNDER HYDROSTATIC AND NONHYDROSTATIC CONDITION

Pressure calibrated up to 0.8 Mbar under hydrostatic condition and to 2 Mbars under non-hydrostatic condition. We find the red shift of ruby R fluorescence to 830nm at the highest pressure point in our diamond high pressure cell. The shift correspond to 5.5 Mbars by calibration.

Static compression of iron to 78 GPa with rare gas solids as pressure‐transmitting media

New data are presented for the room temperature, static compression of iron to 78 GPa with solid neon and argon as pressure‐transmitting media. X ray diffraction studies have been performed on a geophysically relevant material, for the first time to such pressures under quasihydrostatic conditions, in a diamond anvil cell. The hydrostatic technique leads to increased precision in the measurement of high pressures and has placed closer constraints on the equation of state of ε iron. From a linear least squares fit of a finite strain equation of state to the present data combined with earlier, low‐pressure data for metastable ε iron, the preferred values for the zero‐pressure isothermal bulk modulus, K0, and first pressure derivative, K0′, are 192.7 (±9.0) GPa and 4.29 (±0.36), respectively. The zero‐pressure volume for the ε phase is 6.687 (±0.018) cm3/mol. On the basis of the pressure‐volume curve calculated from fits of the finite strain equation of state, ε iron appears to be less compressible under nonhydrostatic conditions, but the differences are within the error of the nonhydrostatic experiment. The results also confirm that the absence of a soft medium in static compression experiments with the diamond anvil cell results in an overestimate of the unit cell volume (measured with the incident X ray beam parallel to the load axis) for pressures calculated with the nonhydrostatic ruby calibration scale. It is found that for ε iron, substantial compensation for this nonhydrostatic effect is implicit in the nonhydrostatic ruby pressure scale up to intermediate strains. The hydrostatic data and the ε iron isotherm derived from shock wave experiments on iron samples are in very close agreement.

Model line-shape analysis for the ruby R lines used for pressure measurement

An empirical model spectral line shape has been considered for the R1 and R2 fluorescence lines of ruby to investigate the possibility of improvements in the precision and the reproducibility of pressure measurements at high temperature using the ruby fluorescence technique. Other advantages, such as using the ruby method to obtain a simultaneous measurement of pressure and temperature or to obtain quantitative estimates of pressure distributions under nonhydrostatic conditions, have also been considered.

ELECTRICAL RESISTANCE MEASUREMENTS OF METALS TO 40 GPa IN THE DIAMOND CELL

A four-probe technique has been developed for measuring electrical resistances of metals to 40.0 GPa in a diamond-anvil cell. Several gasketing techniques are presented which provide the necessary support for the diamonds and the electrical leads at the diamond edges. Electrical resistance data are presented for iron and bismuth. A comparison is made between electrical resistance measurements made on bismuth under hydrostatic and non-hydrostatic conditions.

Raman study of rutile (TiO 2) at high pressures

The Raman spectra and polymorphism of rutile have been investigated under hydrostatic pressures up to 90 kbar at room temperature. A transition previously observed in rutile at 30 kbar in a Drickamer-type cell under nonhydrostatic conditions was observed to begin at approximately 70 kbar in a 4:1 mixture of methanol and ethanol. The small amount (10–20%) of the high-pressure phase synthesized from rutile, however, did not increase even though the sample was left under pressure for a period of 1 month at ambient temperature. On the basis of factor group analysis, in situ powder x-ray diffraction data, and comparison of the Raman spectrum of the high-pressure modification with that of TiO 2-II (α-PbO 2-type structure synthesized from anatase powder at 40 kbar and 400°C), it is evident that a high pressures rutile transforms irreversibly to TiO 2-II.

Mössbauer study of phase transitions under high hydrostatic pressures. I. The phase transition of Mg2Sn

Experimental results of the hydrostatic pressure influence on Mossbauer spectrum parameters are obtained over the pressure range including the area of structural phase transition. A linear increase of the Mossbauer effect probability (recoilless fraction) is accompanied by a linear decrease of the electron density at tin nuclei within the pressure range foregoing the phase transition. The electric resistance and the recoilless fraction of the new phase of Mg2Sn are lower, but the electron density at tin nuclei is greater than the initial phase ones. Hydrostatic conditions allow to fix clearly the diphasic transition area and to determine the influence of the pressure on the Mossbauer line position and on the recoilless fraction of the high pressure phase. The phase transition heat Q = 415 cal mol−1 is calculated using recoilless fractions of the high and low pressure phases at 25 kbar. The present results are qualitatively and quantitatively different from the results, obtained at nonhydrostatic conditions. [Russian Text Ignored].

Laboratory Investigation of Fracture Initiation Pressure and Orientation

Abstract The mechanics of hydraulic fracture initiation have been investigated by comparing laboratory experiments with theoretical predictions based on poro-elastic behavior. Experiments were conducted poro-elastic behavior. Experiments were conducted with 4-in. (10-cm) diameter cores containing spherical and cylindrical cavities and loaded in a triaxial cell under variable confining pressure, end load, and pore pressure. Experimental results agreed with theory for nonpenetrating fracturing fluid for limited ranges of hydrostatic confining stresses for four kinds of limestone rock. With penetrating fracturing fluids, the theory was penetrating fracturing fluids, the theory was confirmed only partially. Under nonhydrostatic stress conditions, reproducibility of measurements was too poor to evaluate the theory. Fracture orientation was controlled predominantly by stress conditions and cavity geometry. Notching of cylindrical cavities failed through notch extension only if the notch depth exceeded the value predicted approximately by a simple Griffith theory predicted approximately by a simple Griffith theory equation. Field applications of all results are discussed. Introduction This paper describes a combined theoretical/ experimental investigation of the mechanics of hydraulic fracture initiation. We considered fracture initiation pressure, fracture orientation, and mode of failure for various stress conditions and wellbore geometries. Our intention has been to consider theory applicable for both field and laboratory conditions, to test this theory with laboratory experiments, and to apply the results to interpretation of field data. The laboratory experiments were designed not to duplicate field conditions so much as to provide a critical test of the theory. Some field data are examined, but it is impractical to learn much about fracture initiation from field experiments because of the limited number of quantities that can be measured. The theory presented here is more a generalization of earlier work than a development of new theory. It provides a completely general treatment of fracture initiation in spherical and cylindrical cavities for poro-elastic materials. An extension of this theory poro-elastic materials. An extension of this theory to porous materials with nonelastic behavior already has been developed by Biot and will be referred to later. We begin by presenting theory for fracture initiation in spherical and cylindrical cavities. The theoretical results are followed by descriptions of laboratory experiments that test the equations for failure pressure in these geometries under various stress conditions, using penetrating and nonpentrating fracturing fluids. The effects of notching in cylindrical cavities then are considered, and a simple model based on Griffith crack theory is developed to explain experimental results. Field applications of all results then are discussed in detail. THEORY OF FRACTURE INITIATION The theory of hydraulic fracture initiation in rock materials has been treated in successive degrees of refinement. Cases of interest are hollow sphere and long hollow cylinder geometry with penetrating and nonpenetrating fracturing fluids. penetrating and nonpenetrating fracturing fluids. Refs. 1 through 7 cover various parts of the overall picture; Rice and Cleary give the most complete picture; Rice and Cleary give the most complete analysis. We present here an independent analysis based on Biot's theory for fluid saturated porous solids. Our analysis adds little that is new to the basic literature of fracture initiation theory. It is presented mainly to provide a way to analyze presented mainly to provide a way to analyze scaling effects between field results and our laboratory experiments. We start with Biot's stress-strain relations for a fluid saturated porous solid: (1) SPEJ P. 129

Shear stress enhancement of void compaction

Motivated by the success of the spherical model in predicting the volumetric compaction behavior of both porous rocks and metals to a hydrostatic pressure, we consider the applicability of the spherical model to nonhydrostatic loading conditions. Specifically, the spherical model is used to examine the influence of the presence of a shear stress on the volumetric compression of a porous solid. We first obtain the linear, elastic solution for a hollow sphere subject to homogeneous tractions on the outer boundary. Then, assuming that the matrix material is governed by the Drucker‐Prager yield criterion, we use the elastic solution to derive an analytic expression for the onset of yield in the hollow sphere. The expression for the initial yield surface shows that the presence of a shear stress hastens the onset of yield in the sphere in comparison to a hydrostatic loading condition. This result agrees well with experimental data which shows that, for porous solids, permanent crush‐up begins at a lower mean stress under a nonhydrostatic loading than when the applied loading is a hydrostatic pressure. At this point, due to difficulty in obtaining an analytic solution, we turn to a numerical scheme (finite element method) to extend the analysis of the hollow sphere problem into the elasto‐plastic range. The spherical model results clearly exhibit the experimental finding that the presence of a shear stress tends to enhance the volumetric compaction of porous solids in comparison to a hydrostatic loading condition. For both a porous rock and metal sample, agreement between the spherical model and experimental results is excellent.

Quasistatic deformation of porous beryllium and aluminum

Loading and unloading of two types of porous beryllium and a porous aluminum under conditions of uniaxial strain, proportional loading, and hydrostatic pressure indicate that yielding is dominated by porosity. Analysis of the data prior to yielding indicates that aspherical pores cause increased compressibility on initial loading. All materials exhibit enhanced compaction when loaded under nonhydrostatic stress conditions. Models which treat the collapse of spherical pores do not agree with the beryllium data, probably because of the influence of aspherical pores and pore-size distribution.

An Experimental Study of Nepheline-Kalsilite Exsolution

The nepheline-kalsilite exsolution reaction was studied isothermally between 400 and 700°C. Under nonaqueous conditions the mechanism involves nucleation of kalsilite and growth by diffusion of the alkalis. As predicted by simple nucleation theory, the nucleation rate and hence the over-all exsolution rate are strongly dependent on the supersaturation of the nepheline. A decrease in temperature at constant composition increases the supersaturation and thereby the nucleation rate. This increased nucleation rate is opposed by the decrease in the growth rate due to slower volume diffusion. At a supersaturation of more than 8–10 mole per cent the number of nuclei is large and the over-all exsolution rate is determined primarily by the growth rate. The activation energy for growth is 28 kcal/mole. An increase of two kilobars in the hydrostatic pressure has little effect on the kinetics of the reaction. Under nonhydrostatic conditions the exsolution rate increases significantly because the nucleation rate is faster. Under hydrothermal conditions the ‘exsolution’ rate is approximately two orders of magnitude faster due to a modification in the mechanism. Partial dissolution of the original solid solution in distilled water creates a condition of nonequilibrium in which the fluid is sodium-rich. Rapid alkali exchange eliminates this condition but produces the equilibrium compositions of the solids because kalsilite nucleates and grows in contact with the fluid. The experimental evidence for this mechanism includes X-ray diffraction data showing a gradual change in the composition of the initial supersaturated solid, essentially identical activation energies for growth under aqueous and nonaqueous conditions, and a lower percentage of oxygen isotope exchange than ‘exsolution’ in the same experiment.