Aggregate Shear Modulus Research Articles

Sound Velocity Anisotropy and Single‐Crystal Elastic Moduli of MgO to 43 GPa

AbstractSeismic anisotropy in the Earth's lower mantle likely results from a combination of elastic anisotropy and lattice preferred orientations of its main constituent minerals. As the second most abundant component of the lower mantle, ferropericlase has been widely studied, and the experimental results demonstrated, in general, a growing with pressure elastic anisotropy up to 1 Mbar. However, the unique measurements on the endmember (MgO) at comparable pressure conditions contradict the above observations and theoretical results. Here, time‐domain Brillouin scattering was applied to measure longitudinal sound velocities in single crystals of MgO compressed in diamond anvil cell. Velocities along two specific crystallographic directions, [100] and [111], were independently collected to 43 GPa. Applying the known bulk modulus, a complete set of single‐crystal elastic moduli, elastic anisotropy and aggregate shear modulus were derived. Our results revealed a steadily increasing with pressure elastic anisotropy at P > 20 GPa, consistent with the previous theoretical predictions and measurements on ferropericlase with moderate amounts of iron.

Debye temperature of CaO under high pressure up to 65.2 GPa

In the present work we used the experimental relative volume unit cell and the elastic stiffness constants measured by Speziale et al. (Journal of Geophysical Research, Vol. 111, (2006), pp. B02203 (12 pages)) using the Radial X-Ray Diffraction at high pressure from 5.6 up to 65.2 GPa to investigate the effect of high pressure on the bulk modulus, aggregate shear modulus, elastic wave velocities and Debye temperature of calcium oxide (CaO) ceramic material in cubic rock-salt (B1) phase. Our obtained values show that the bulk modulus increases monotonously with increasing pressure, while the aggregate shear modulus, the mean elastic wave velocity and the Debye temperature of CaO material varied non-linearly and non-monotonously with increasing pressure from 5.6 up to 65.2 GPa.

Theoretical Prediction of Mechanical Properties of BxAl1−xSb Ternary Semiconducting Alloys

Abstract The present work aims to predict the elastic constants and other significant properties of ordered BxAl1-xSb (0 ≤ x ≤ 1) ternary semiconducting alloys. We report the initial results of the elastic stiffness constants, the bulk modulus, the aggregate shear modulus, the Cauchy ratio, the aggregate Young’s modulus, the Born ratio, the isotropy factor, the fracture toughness and the longitudinal, transverse and average sound velocities. The Debye temperature and the melting point were also predicted using two different empirical expressions. Except the Cauchy ratio, which decreases with enhancing boron content x, all other physical quantities of BxAl1-xSb alloys increase gradually and monotonically with increasing of boron concentration x in the range 0-1. Our obtained data for BSb and AlSb binary semiconducting compounds are discussed and analyzed in comparison with experimental and other theoretical values of the literature. Generally, our data for BSb and AlSb are in good agreement with other results reported previously in literature. Indeed, our obtained value (335.82 K) of the Debye temperature for AlSb compound overestimates the result (328.6 K) reported by Salehi et al. by around 2.03%, while that (1520 K) of the melting point for BSb overestimates the result (1500 K) reported recently by Bioud et al. by around 1.34%. Furthermore, to the best of our knowledge, no theoretical or experimental data were reported in the literature on the elastic constants and other properties for BxAl1-xSb alloys to compare with them.

Influence of elastic anisotropy on measured sound velocities and elastic moduli of polycrystalline cubic solids

Cubic solids such as NaCl, crystalline argon, or H2O-ice VII exhibit significant elastic anisotropy strongly increasing upon compression. As earlier recognized for solid argon and H2O-ice (both exhibiting Zener ratio A > 1), longitudinal sound velocities of their polycrystals, VLav, measured using Brillouin light scattering (BLS) or pulse-echo ultrasonics are much closer to VL⟨111⟩ than to VL⟨100⟩, the VL-extremes in any cubic single crystal. Here, we experimentally confirm, using the technique of time-domain Brillouin scattering, the same tendency for NaCl exhibiting the opposite anisotropy type, A < 1. To understand this tendency, we modelled orientational distribution and the frequency of occurrence of VL values in texture-free polycrystalline samples of NaCl and solid argon. We found a remarkable and predictable asymmetry of VL distributions with maxima at VL⟨110⟩ that is always much closer to VL⟨111⟩. This asymmetry persists in BLS peaks but can be obscured in experiments. In the case of solid argon at 49 GPa, the asymmetry can lead to a moderate deviation of experimental VLav from VLH (obtained from elastic-stiffness constants Cij applying the Hill approximation) by ∼7%. The latter can cause, however, a significant overestimation of the aggregate shear modulus by δG/G ∼ 50% or of the bulk modulus by δB/B ∼ 20% if just one BLS peak of longitudinal modes is detectable. A similar analysis, performed for transverse sound velocities, VT and VTav, has shown that by the use of a BLS spectrum showing peaks of both longitudinal and transverse modes, overestimation of B is similarly high but that of G is much less dramatic.

Confined compression and torsion experiments on a pHEMA gel in various bath concentrations

The constitutive behaviour of cartilaginous tissue is the result of complex interaction between electrical, chemical and mechanical forces. Electrostatic interactions between fixed charges and mobile ions are usually accounted for by means of Donnan osmotic pressure. Recent experimental data show, however, that the shear modulus of articular cartilage depends on ionic concentration even if the strain is kept constant. Poisson-Boltzmann simulations suggest that this dependence is intrinsic to the double-layer around the proteoglycan chains. In order to verify this premise, this study measures whether--at a given strain--this ionic concentration-dependent shear modulus is present in a polymerized hydroxy-ethyl-methacrylate gel or not. A combined 1D confined compression and torque experiment is performed on a thin cylindrical hydrogel sample, which is brought in equilibrium with, respectively, 1, 0.1 and 0.03 M NaCl. The sample was placed in a chamber that consists of a stainless steel ring placed on a sintered glass filter, and on top a sintered glass piston. Stepwise ionic loading was cascaded by stepwise 1D compression, measuring the total stress after equilibration of the sample. In addition, a torque experiment was interweaved by applying a harmonic angular displacement and measuring the torque, revealing the relation between aggregate shear modulus and salt concentration at a given strain.

Single-crystal elastic properties of (Mg0.987, Fe0.013)O to 9 GPa

The single-crystal elastic moduli of (Mg 0.987 ,Fe 0.013 )O were measured by Brillouin spectroscopy in a diamond-anvil cell at high pressures to 9 GPa at room temperature. The ambient-pressure single-crystal elastic moduli are (1) C 11 = 291.2(3.0) GPa; (2) C 12 = 96.1(2.0) GPa; and (3) C 44 = 151.9(2.0) GPa. From the single-crystal moduli, the aggregate elastic moduli are calculated to be adiabatic bulk modulus K S0 = 161.1(3.0) GPa, the Voigt bound of the shear modulus is G V = 130.0(2.0), and the Reuss bound G R = 124.2(2.0) GPa, giving a Voigt-Reuss-Hill average G = 127.1(2.0) GPa. We find that the addition of 1.3 mol% of Fe has a surprisingly large effect on the aggregate shear modulus, decreasing the room-pressure value by 2.4% as compared to Brillouin data for periclase (MgO) measured with the same technique. The adiabatic bulk modulus also decreases by 1.3%, although this decrease is within the mutual uncertainties of the measurements. Our results confirm significant non-linearity in single-crystal elastic moduli C 11 and C 44 and the aggregate shear modulus G of magnesiowustite in the Mg-rich end. The pressure derivative of the bulk modulus K S ′ = 4.2(2), as determined by a third-order finite-strain fit, is about 9% higher than the Brillouin results for the MgO end-member, whereas the pressure dependence of the shear modulus G ′ = 2.3(1) is found to be identical to that of periclase. The measurements demonstrate that even a small amount of Fe (1.3 mol%) has a measurable effect on the elastic properties of MgO-FeO solid solutions.

X‐ray diffraction study of the single‐crystal elastic moduli of ε‐Fe up to 30 GPa

Room temperature investigations of the single‐crystal elastic moduli and anisotropy of the ε phase of iron are performed up to 30.3 GPa using the radial X‐ray diffraction technique. The accuracy of the calculated elastic moduli has improved compared to previous measurements using similar techniques because of an increase in accuracy of the measurement, confinement of the sample to limit the effect of plasticity, and better calibration of the stress conditions. The aggregate shear modulus that we obtain is in good agreement with a variety of other experimental deductions but differs from first‐principles calculations. The effects of the calibration of stress and micromechanical model on the deduction of elastic moduli and elastic anisotropy are discussed in detail. The anisotropy we obtain has the same order of magnitude as first‐principles calculations but the direction is reversed, with a weaker amplitude that previous measurements.

Elastic properties of intermetallic AB2 and AB5 hydrogen-absorbing compounds

The elastic properties of AB2 and AB5 hydrogen-absorbing intermetallic compounds are reviewed. The relevance of these properties to hydrogen absorption is discussed briefly. In many cases the temperature dependence of the elastic constants of the AB2 compounds is anomalous in that the moduli do not decrease monotonically with increasing temperature. This unusual behavior appears to be associated with electronic structure effects in a number of cases. The absorption of hydrogen by polycrystalline TaV2 has a profound effect on the aggregate shear modulus, changing both the magnitude and temperature dependence significantly. This effect of hydrogen on the mechanical properties appears to occur via effects on the electronic structure. Few measurements of elastic properties have been reported for hydrogen-absorbing AB5 intermetallic compounds. The LaAlxNi5−x system has been investigated by means of measurements on polycrystals prepared by hot isostatic pressing. The moduli decrease about 8% as x increases from 0 to 1.

Aggregate shear moduli of iron up to 90 GPa and 1100 K

We demonstrate that the mean-square atomic vibrations or Debye-Waller temperature factors of cubic (bce and fcc) and hcp-metals can be related to aggregate shear moduli at different pressures and t ...

Estimation of Single-Crystal Elastic Moduli from Polycrystalline X-Ray Diffraction at High Pressure: Application to FeO and Iron

X-ray diffraction data obtained under nonhydrostatic compression of a polycrystalline sample yield an estimate of the single-crystal elasticity tensor of the material when analyzed using appropriate equations. The analysis requires as input the aggregate shear modulus from independent measurements. The high-pressure elastic moduli of face-centered-cubic FeO, body-centered-cubic iron ( $\ensuremath{\alpha}$-Fe), and the pressure-induced hexagonal close-packed iron ( $\ensuremath{\epsilon}$-Fe) are obtained. This analysis currently provides the only method of determining single-crystal elasticity tensors in the megabar pressure range and of studying elasticity of very high-pressure phases.

Equation of state, elastic properties, and stability of CaSiO3 perovskite: First principles (periodic Hartree‐Fock) results

The cubic CaSiO3 perovskite phase is possibly the third most abundant mineral in the lower mantle. In addition to its geochemical significance, CaSiO3 perovskite provides a useful system with which to test the application of quantum chemistry to mantle silicates. Here, the electronic structure, equation of state, and elastic properties of cubic CaSiO3 perovskite are calculated using the periodic Hartree‐Fock formalism. Cubic CaSiO3 is found to be stable relative to an orthorhombic structure to at least 106 GPa. Calculations using a moderately extended basis set give an equation of state for CaSiO3 in close agreement with experiment (V0 = 44.96 Å3, K0 = 300 GPa, and K0′ = 4). The aggregate shear modulus of CaSiO3 perovskite is found to be 209 GPa which is comparable to that of MgSiO3 perovskite. The high seismic velocities of CaSiO3 perovskite means it will act as a seismic complement to magnesiowustite in the lower mantle. The free energy, as a function of pressure, of stishovite and B2‐CaO have also been calculated at the same level of theory. From those results, it is found that cubic CaSiO3 perovskite is stable relative to the free oxides at pressures up to 130 GPa (the core‐mantle boundary).

Elastic properties of hedenbergite

The single‐crystal elastic moduli of hedenbergite (CaFeSi2O6) hare been measured at 20°C and 1 bar using Brillouin spectroscopy. The moduli are (in gigapascals): C11 = 222, C22 = 176, C23 = 249, C44 = 55, C55 = 63, C66 = 60, C12 = 69, C13 = 79, C33, = 86, C15 = 12, C25 = 13, C35 = 26, C46 = −10. The comparison of elastic properties among Mg‐Fe‐Ca bearing pyroxenes, known as quadrilateral pyroxenes, reveals only weak variations with changes in composition. Of the four quadrilateral pyroxenes, orthoferrosilite has elastic properties distinctive from the others. The principal differences among these pyroxenes are due to subtle structural differences. In particular, the mechanical linkage between the M2 polyhedral chains in clinopyroxenes enhances the importance of the cation in this site. In contrast to the orthopyroxenes, the aggregate shear modulus μ of the calcium‐bearing clinopyroxenes (diopside and hedenbergite) exhibits no dependence on the amount of iron (Fe2+) present in the structure, while the ratio K/μ does. As a result, the compressional and shear acoustic velocities of the calcium‐bearing clinopyroxenes show a smaller dependency on iron content than do the orthopyroxenes.