Elastic Constant Tensor Research Articles

Trench parallel anisotropy and large delay times: Elasticity and anisotropy of antigorite at high pressures

Using theoretical methods, we calculate the full elastic constant tensor and equation of state of antigorite up to 10 GPa, a pressure range that encompasses its experimentally observed stability field. At ambient conditions, the elastic constant tensor reveals significantly large acoustic anisotropy (38% in VP; 35% in VS) compared to the dominant mantle phase olivine. The shear anisotropy is enhanced upon compression. Upon compression, the full elastic constant tensor reveals anomaly at ρ ∼ 2.69 gm cm−3. At ρ < 2.69 gm cm−3, we find that the pressure‐volume results for antigorite are well represented by a third order Birch‐Murnaghan formulation, with K0 = 52 GPa, K0′ = 8.5 and V0 = 3037 Å3 (ρ0 ∼ 2.48 gm cm−3). Our results on elasticity and anisotropy at conditions relevant to the mantle wedge indicates that a 10–20 km layer of antigorite might account for the observed shear polarization anisotropy with fast polarization parallel to the trench and associated large delay times of 1–2 sec observed in some subduction zone settings.

High-pressure behavior of iron carbide (Fe7C3) at inner core conditions

[1] Carbon is a plausible light element candidate in the Earth's core owing to its cosmic abundance and its chemical affinity for iron. Recent experimental studies on Fe-C phase relations at high pressures have demonstrated that Fe7C3 iron carbide is a likely candidate for the Earth's inner core. Using electronic structure calculations, we determine the equation of state, the full elastic constant tensor, and the sound wave velocities for Fe7C3, up to inner core pressures. We find that Fe7C3 is ferromagnetic (fm) at low pressure, and that its compression behavior is well represented by a third-order Birch Murnaghan finite strain expression with V0fm = 9.1 A3/atom, K0fm = 231 GPa, and K′0fm = 4.4. Under compression the magnetic moments of the Fe atoms gradually decrease, and at ∼67 GPa the magnetic moment is lost. The high-pressure nonmagnetic phase (nm) has distinct finite strain parameters with V0nm = 8.8 A3/atom, K0nm = 291 GPa, and K′0nm = 4.5. Calculated elastic constants show softening associated with the loss of magnetization. In addition, we have conducted nuclear resonant inelastic X-ray scattering experiments on 57Fe enriched Fe7C3 at 1 bar and 300 K. On the basis of our nuclear resonant inelastic X-ray scattering spectra we have derived a Debye sound velocity of 3.18 km/s. The experimentally determined value is in good agreement with the computational predictions, based on athermal single elastic constants. The static P wave velocity at inner core pressures agrees well with seismological constraints, whereas the S wave velocity is greater by 30%. On the basis of the density of Fe7C3 at inner core conditions, we predict that the maximum possible carbon content of the inner core is around 1.5 wt %.

Calculation of X-Ray Stress Factors Using Vector Parameterization and Irreducible Representations for SO(3) Group

In the presence of texture, the concept of X-ray elastic constants as well as Sin2ψ law is inapplicable and the X-ray stress factors (XSF) connecting average strain and stress have to be used [1-2]. The SO(3) vector parameterization with smart composition law [3-4] proved to be a powerful tool for handling transformations between reference systems used in XSF calculation. Decomposition of the 4-th rank elastic constant tensor on the SO(3) irreducible representations (IR) allows one to highlight the symmetry properties and to separate isotropic and anisotropic parts. Joint use of the vector parameterization and IR decomposition enables to obtain transparent analytical expressions for XSF in case of textures described by preferred spherical/fiber components.

Decomposıtıon of elastıc constant tensor ınto orthogonal parts

In this paper, we have elaborated on the decomposition methods such as irreducible decomposition, orthonormal tensor basis, harmonic and spectral decomposition for elastic constant tensor. Irreducible decomposition and orthonormal tensor basis methods are developed by using the results of existing theories in the literature. As examples to each decomposition method, we give results for the decomposition of elastic constant tensor in triclinic symmetry as well as materials with isotropic and transversely isotropic symmetry. Numerical examples serve to illustrate and verify each of the four decomposition methods. These examples are used to compare the decomposition methods explicitly. As a result of comparison process, it is stated that the spectral method is a non-linear invariant decomposition method that yields non-linear orthogonal parts contrary to the other three methods which are linear invariant decomposition methods. It is also shown that total scalar (isotropic) part is decomposed into two physically meaningful orthogonal parts by irreducible decomposition, orthonormal tensor basis and spectral methods. While in harmonic decomposition method, decomposition of total scalar part is not orthogonal. We propose that it is possible to make these parts orthogonal to each other. International Journal of Engineering, Science and Technology, Vol. 2, No. 6, 2010, pp. 22-46

Elasticity of AlFeO3and FeAlO3perovskite and post-perovskite from first-principles calculations

[1] We use state-of-the-art ab initio calculations based on the generalized gradient approximation of the density functional theory in the planar augmented wavefunction formalism to determine the elastic constants tensor of perovskite and post-perovskite with formulas AlFeO3 and FeAlO3 in which Fe or Al respectively occupy only octahedral sites, for the stable magnetic configurations. The phase transition between perovskite and post-perovskite is associated with a site exchange, during which Fe from the inter-octahedral site in perovskite moves into the octahedral site in post-perovskite. Following this transition path the elastic moduli show positive jumps, considerably larger than for MgSiO3. The phase transition is marked by a positive jump of 0.04 km/s (0.33%) in the velocity of the compressional waves and by a negative jump of −0.15 km/s (−1.87%) in shear wave velocity. We find that the effects of the Mg + Si Al + Fe substitution on the seismic properties of MgSiO3 perovskite and post-perovskite depend on the crystallography of the substitution, namely the position the exchanged cations take in the structure.

Elasticity of (K,Na)AlSi 3O 8 hollandite from lattice dynamics calculations

We compute the elastic constants tensor and the seismic properties of KAlSi 3O 8 and K 0.8Na 0.2AlSi 3O 8 up to the ferroelastic transition using density-functional theory and density-functional perturbation theory in the ABINIT implementation. We observe a softening of the tetragonal shear with pressure that precedes the ferroelastic transition. The Reuss shear moduli become negative at respectively 23 GPa and 13 GPa for the two compositions considered in here. The ferroelastic transition is associated with a strong decrease of the horizontal shear wave velocities and a corresponding increase of the seismic anisotropy. The presence of Na enhances these features.

First-principles elastic and thermal properties ofTiO2: a phonon approach

Elastic and thermal properties of theTiO2 lattice in anatase and rutile phases were studied in the framework of density functionalperturbation theory within the local density approximation (LDA) and generalized-gradientapproximation (GGA). The full elastic constant tensors of the polymorphs were calculatedby linear fits to the acoustic branches of the phonon band structure near the center of thefirst Brillouin zone in symmetry directions of the crystals. It was observed that the elasticconstants within the GGA are in better agreement with experiment. In addition, the Borneffective charges, dielectric tensor, heat capacity, mean sound velocity and Debyetemperature were calculated. The heat capacity at room temperature and theDebye temperature within the LDA are in better agreement with the experimentalresults. Therefore, using the phonon band structure and the density of states,one can obtain the important mechanical and thermal properties of materials.

Ab initio calculation of the elastic properties and the lattice dynamics of the AgBr1−xClx alloy

The lattice dynamics and elastic properties of the ternary AgBr 1− x Cl x alloy have been studied using first-principles calculations. These are done using the density-functional perturbation theory (DFPT) and employing the virtual-crystal approximation (VCA). We study the variation of the optical phonon frequencies ( ω TO and ω LO ) and of the acoustical phonon frequencies ( ω TA and ω LA ), the high-frequency dielectric coefficient ( ε ∞ ), the dynamic effective charge ( Z ∗ ) and the elastic constants ( C 11 , C 12 , C 44 ) as a function of the composition ( x ). We find that the phonon frequencies as well as the elastic constants follow a quadratic law in x and agree well with the available experimental results. The elastic constant tensors for simple cubic 8-atom supercells with x = 0.25, 0.5 and 0.75 have been computed and are in good agreement with those obtained from VCA and experiments.

Elasticity of phase‐X at high pressure

[1] Using a plane-wave pseudopotential method, we calculate the equation of state and elasticity of phase-X [(K,Na)2Mg2Si2O7] up to 30 GPa, which encompasses its experimentally observed stability field. We find that the pressure-volume results for anhydrous phase-X are well represented by a third order Birch-Murnaghan formulation, with K0 = 132 GPa, K′0 = 4.4 and V0 = 284.6 A3 for the potassium (K) end member and K0 = 128 GPa, K′0 = 4.0 and V0 = 265.9 A3 for the sodium (Na) end member. The full elastic constant tensor reveals significantly larger stiffness along the [001] direction than in the (100) plane. We attribute such stiffness to the 1800 angle of Si-O-Si units and their orientation parallel to the [001] direction. Assuming ideal behavior between Na and K phase-X, the shear modulus softens with sodium content as G = [84 − 0.14 × Na(wt%)] GPa.

Ab initio calculation of the elastic properties and the lattice dynamics of the ZnxCd1−xSe alloy

The lattice dynamics and elastic properties of the ternary ZnxCd1−xSe alloy have been studied using first-principle calculations. These are done using the density-functional perturbation theory (DFPT) within local density approximation (LDA) and employing virtual-crystal approximation (VCA). We study the variation of the optical phonon frequencies (ωTO and ωLO), the high-frequency dielectric coefficient (ε∞), the dynamic effective charge (Z*) and the elastic constants (C11, C12, C44) as a function of the composition (x). We found that the ωTO and ωLO follow a quadratic law in x and agree well with the experiment. The obtained ε∞, Z*, C11, C12 and C44 have a quadratic form with x. The elastic constant tensors for simple cubic eight-atom supercells with x = 0.25, 0.5 and 0.75 have been computed and are in good agreement with those obtained from VCA.

Elliptical cracks arbitrarily oriented in 3D-anisotropic elastic media

An elliptical crack in an infinite anisotropic elastic medium is considered. For a polynomial external stress field, an efficient numerical algorithm of the solution of the problem is developed. The discontinuity of the displacement field on the crack surface (the crack opening vector) is presented in the form of 2D-regular integrals that can be evaluated numerically for any crack orientation with respect to the principal axes of the anisotropy of the medium. The integrand functions in these integrals are expressed via the Fourier transform of the Green function of the medium. The cases of constant and linear external stress fields are considered in detail. Simplifications for particular crack orientations in the cases of orthotropic and transversely isotropic media are indicated. The stress intensity factors are obtained in the form of regular integrals that depend only on the Fourier transform of the Green function of the medium. An equation for the tensor of effective elastic constants of an anisotropic media containing a random set of elliptical cracks is presented.

Structure and elasticity of serpentine at high-pressure

Serpentines occur in the subduction zone settings, both along the slab and within the mantle wedge, they are candidates for transporting water in to the deep earth. Their presence is manifested by serpentine mud volcanoes, high electrical conductivities, magnetic and seismic anomalies. Using theoretical methods, we predict a pressure induced structural transformations in serpentine. The transformations are related to the behavior of the silicate framework and misfit between octahedral and tetrahedral layers. As the structure is compressed, the octahedral layer and tetrahedral layers are compressed at different rates. At 7 GPa, the misfit between the layers vanishes. This causes non-linear pressure dependence of tetrahedral rotational angle. This is also manifested by the onset of anomalous pressure dependence of the elastic constants c 11, c 33, c 12, c 13. Beyond 7 GPa, the misfit between the layers grows again reaching extremum at 22 GPa. This is also manifested by discontinuity in average Si–O bond length, volume of tetrahedron and re-orientation of hydroxyl vector. The symmetry of the crystal-structure however, remains unaffected. Evidence of pressure-induced hydrogen bonding is absent in serpentine, as evident from reduction of O–H bond length upon compression. Results of compression for the low-pressure regime ( P < 7 GPa) is well represented by a fourth order Birch–Murnaghan finite strain expression with K 0 = 79 GPa, K 0′ = 12 and K 0″ = − 2, where K is the bulk modulus, prime indicates pressure derivatives, and O refers to zero pressure. Our best estimates of K 0, K 0′ and the Grüneisen parameter, γ at 300 K and zero pressure based on our results are: 61 GPa, 17, and 0.77, respectively. At low pressures, serpentine structure is anisotropic with c 11 ~ 2.4 × c 33. The pressure derivative of elastic constants ( ∂c ij / ∂P) are such, that around 22 GPa c 11~ c 33. An elastic instability ( c 66 < 0) at somewhat higher pressures (> 50 GPa) is also noted. The elastic constant tensor reveals large acoustic anisotropy (41% in V P ) and seismic wave velocities that are significantly higher than those inferred from experiments on serpentinites.

Elastic Constants and Related Mechanical Properties of the Monoclinic Polymorph of the Carbamazepine Molecular Crystal

Brillouin scattering has been used to probe the acoustic phonons of the monoclinic (P21/c) polymorph of the drug carbamazepine (CBZ III). By sampling a variety of acoustic phonons, the complete elastic constant tensor has been determined for this CBZ polymorph. The diagonal elastic constants c11, c22, c33, c44, c55 and c66 are 10.89, 11.47, 11.32, 3.68, 0.85 and 2.89 GPa, respectively. The elastic constants of CBZ III are effectively governed by nondirected dispersive type interactions similar to aromatic systems with delocalized pi bonds. The bulk modulus and linear compressibility have been calculated from the compliance matrix. The relative strength of intermolecular interactions in different directions is estimated from the linear compressibility plotted in three crystallographic planes. Cauchy's ratio, calculated from the elastic constants, suggests that many-body forces of dispersive interactions contribute significantly to the lattice anharmonicity of CBZ III.

On approximate symmetries of the elastic properties and elliptic orthotropy

The theory of anisotropic elasticity was originally motivated by applications to crystals, where geometric symmetries hold with high precision. In contrast, symmetries of the effective elastic responses of heterogeneous materials are usually approximate due to various imperfections of microgeometry. A related issue is that available data on the elastic constants may be incomplete or imprecise; it may be appropriate to select the highest possible elastic symmetry that fits the data reasonably well. Some of these problems have been discussed in literature in the context of specific applications, primarily in geomechanics. The present work provides a systematic discussion of the related issues, illustrated by examples on the effective elastic properties of heterogeneous materials. We also discuss a special type of orthotropy typical for a variety of heterogeneous materials – elliptic orthotropy – when the fourth-rank tensor of elastic constants can be represented in terms of a certain symmetric second-rank tensor. This representation leads, in particular, to reduced number of independent elastic constants.

First-principles calculations of the mechanical and structural properties ofGaNxAs1−x: Lattice and elastic constants

Using first-principles total energy calculations, we have investigated the lattice and elastic constants in $\mathrm{Ga}{\mathrm{N}}_{x}{\mathrm{As}}_{1\ensuremath{-}x}$ alloys as functions of the nitrogen concentration $x$. Ab initio density functional theory is used to find the relaxed atomic configurations and elastic constants for substitutional nitrogen in supercells containing up to 128 atoms. The effect of ${\mathrm{N}}_{2}$ and NAs split interstitial substitutions (ISs) on the mechanical properties has also been calculated in cells up to 64 atoms. Atomic positions and geometries in all supercells have been relaxed, and the elastic constant tensors for simple cubic 8-atom supercells have been computed. Deviation of the lattice constants from Vegard's law due to IS is shown to be substantial. We provide analytical expressions for the concentration dependence of the lattice and elastic constants in $\mathrm{Ga}{\mathrm{N}}_{x}{\mathrm{As}}_{1\ensuremath{-}x}$ alloys, including normal and split-interstitial substitutions, and find the ratio ${r}_{b}$ of the bowing coefficients of the elastic constants ${C}_{11}$ and ${C}_{12}$.

Ab-initio simulation of elastic constants for some ceramic materials

Athermal elasticity for some ceramic materials (α-Al2O3, SiC (α and β phases), TiO2 (rutile and anatase), hexagonal AlN and TiB2, cubic BN and CaF2, and monoclinic ZrO2) have been investigated via density functional theory. Energy-volume equation-of-state computations to obtain the zero pressure equilibrium volume and bulk modulus as well as computations of the full elastic constant tensor of these ceramics at the experimental zero pressure volume have been performed. The present results for the single crystal elasticity are in good agreement with experiments both for the aggregate properties (bulk and shear modulus) and the elastic anisotropy. In contrast, a considerable discrepancy for the zero pressure bulk modulus of some ceramics evaluated from the energy-volume fit to the computational zero pressure volume has been observed.

A Review of an Optimal Design Problem for a Plate of Variable Thickness

We revisit a classic design problem for a plate of variable thickness under the model of Kirchhoff. Our main contribution has two goals. One is to provide a rather general existence result under a main assumption on the structure of the tensor of material constants. The other focuses on providing a minimal number of additional design variables for a relaxation of the problem when that assumption on the tensor of elastic constants does not hold. In both situations, the cost functional can be pretty general.

Anisotropic Carbon Phases Prepared from Fullerite C70 under High Pressure

We present the structural and ultrasonic study of carbon phases prepared by quenching after heating of fullerite C70 in the temperature range 300–1100°C at pressures 4 and 7.5 GPa. The main aspect of the work concerns the structural and elastic anisotropy of samples resulted as consequence of quasi hydrostatic pressure with an additional pressure component along the axis of pressure load. Structural anisotropy correlates with anisotropy of the tensor of effective elastic constants taken for the medium with single axis of anisotropy. Comparison of the data obtained with anisotropy effects, earlier observed in the phases synthesized from C60 in the similar quasi hydrostatic conditions, clarifies the role of non‐spherical symmetry of C70 molecule.

Low temperature elastic and anelastic properties of an i-AlPdMn single quasicrystal

The complete elastic constant tensor, i.e. the real and imaginary parts C ijkl and Q i j k l − 1 , of icosahedral AlPdMn quasicrystal were measured at megahertz frequencies, from 4 K to room temperature, using electromagnetic acoustic resonance. The temperature dependence of C ijkl was analyzed using an Einstein oscillator model and a quasiharmonic approximation. The overall mode Grüneisen parameter γ is 2.0, in agreement with a previous thermal expansion measurement. Internal friction Q i j k l − 1 behaved normally throughout the measured temperature range. These results suggest that the lattice anharmonicity of i-AlPdMn is similar to usual crystalline materials despite its significant difference in structure.

Nonaffine correlations in random elastic media

Materials characterized by spatially homogeneous elastic moduli undergo affine distortions when subjected to external stress at their boundaries, i.e., their displacements from a uniform reference state grow linearly with position , and their strains are spatially constant. Many materials, including all macroscopically isotropic amorphous ones, have elastic moduli that vary randomly with position, and they necessarily undergo nonaffine distortions in response to external stress. We study general aspects of nonaffine response and correlation using analytic calculations and numerical simulations. We define nonaffine displacements as the difference between and affine displacements, and we investigate the nonaffinity correlation function and related functions. We introduce four model random systems with random elastic moduli induced by locally random spring constants (none of which are infinite), by random coordination number, by random stress, or by any combination of these. We show analytically and numerically that scales as where the amplitude is proportional to the variance of local elastic moduli regardless of the origin of their randomness. We show that the driving force for nonaffine displacements is a spatial derivative of the random elastic constant tensor times the constant affine strain. Random stress by itself does not drive nonaffine response, though the randomness in elastic moduli it may generate does. We study models with both short- and long-range correlations in random elastic moduli.