Elastic properties of graphene: The Keating model

The elastic properties of graphene have been described in terms of the Keating model. It has been shown that the two-dimensional structure of graphene is described by two independent elastic constants, like an isotropic solid. The Young’s modulus and the Poisson’s ratio have been determined. The results are compared with the experimental data obtained for graphite.     

Third-order elastic moduli of single-layer graphene

In the framework of the Keating model with allowance made for the anharmonic constant of the central interaction between the nearest neighbors μ, analytical expressions have been obtained for three third-order independent elastic constants c _ijk (μ, ζ) of single-layer graphene, where ζ = (2α − β)/(4α + β) is the Kleinman internal displacement parameter and α and β are the harmonic constants of the central interaction between the nearest neighbors and the noncentral interaction between the next-nearest neighbors, respectively. The dependences of the second-order elastic constants on the pressure p have been determined. It has been shown that the moduli c ₁₁ and c ₂₂ differently respond to the pressure. Therefore, graphene is isotropic in the harmonic approximation, whereas the inclusion of anharmonicity leads to the appearance of the anisotropy.     

DSC and elastic moduli studies on tellurite-vanadate glasses containing antimony oxide

xSb₂O₃-40TeO₂-(60 − x) V₂O₅ glasses with 0 ≤ x ≤ 10 (in mol%) have been prepared by rapid- melt quenching method. DSC curves of these ternary glasses have been investigated. The glass transition properties that have been measured and reported in this paper, include the glass transition temperature (T _g ), glass transition width (ΔT _g ), heat capacity change at glass transition (ΔC _P ) and fragility (F). Thermal stability, Poisson’s ratio, fragility and glass forming tendency of these glasses have been estimated, to determine relationship between chemical composition and the thermal stability or to interpret the structure of glass. In addition, Makishima and Makenzie’s theory was applied for determination of Young’s modulus, bulk modulus and shear modulus, indicating a strong relation between elastic properties and structure of glass. Generally, results of this work show that glass with x = 0 has the highest shear, bulk and Young’s moduli which make it as suitable candidate for the manufacture of strong glass fibers in technological applications; but it should be mentioned that glass with x = 8 has higher handling temperature and super resistance against thermal attack.     

Young’s modulus and internal friction in porous biocarbon white pine wood precursors

This paper reports on a study performed in the temperature range 100–293 K, in air and in vacuum, for the amplitude and time dependences of the Young’s modulus and the internal friction (ultrasound damping) of biocarbon precursors prepared from white pine wood at two pyrolysis (carbonization) temperatures of 1000 and 2400°C. The measurements have been conducted by the resonance technique with a composite vibrator on samples cut along and across the tree growth direction. The desorption of molecules of the external medium at low amplitudes of ultrasonic vibrations has been found to produce the pronounced influence on the effective elastic modulus and elastic vibration decrement. The data obtained from acoustic measurements of the amplitude dependences of the elastic modulus have been used to estimate the microplastic properties of the samples. It has been shown that increasing the carbonization temperature gives rise to noticeable changes in the Young’s modulus and internal friction, as well as to reduction of the microplastic stress σ _y of the biomaterial studied. The stress σ _y of the samples cut across the growth direction has been found to be substantially smaller than that of the “longitudinal” samples. The elastic and microplastic properties of precursors prepared from white pine wood have been compared with those of the white eucalyptus wood.     

Ab initio calculation of the electronic structure,Fermi surface,and elastic properties of the new 7.5-K superconductor Nb<Subscript>2</Subscript>InC

Ab initio calculations were performed to investigate electronic and elastic properties of the newly discovered 7.5 K superconductor: layered Nb₂InC. As a result, electronic bands, total and site-projected l—decomposed density of states at the Fermi level, shape of the Fermi surface for Nb₂InC were obtained for the first time. Besides, independent elastic constants, bulk modulus, compressibility, shear modulus, Young’s modulus, Poisson’s ratio together with the elastic anisotropy parameters and indicator of brittle/ductile behavior of Nb₂InC were evaluated and analyzed in comparison with the available data.     

The structural, elastic and thermodynamic properties of intermetallic compound CeGa2

The structural, elastic and thermodynamic characteristics of CeGa₂ compound in the AlB₂ (space group: P6/mmm) and the omega trigonal (space group: P-3m1) type structures are investigated using the methods of density functional theory within the generalized gradient approximation (GGA). The thermodynamic properties of the considered structures are obtained through the quasi-harmonic Debye model. The results on the basic physical parameters, such as the lattice constant, the bulk modulus, the pressure derivative of bulk modulus, the phase-transition pressure (P _t ) from P6/mmm to P-3m1 structure, the second-order elastic constants, Zener anisotropy factor, Poisson’s ratio, Young’s modulus, and the isotropic shear modulus are presented. In order to gain further information, the pressure and temperature-dependent behavior of the volume, the bulk modulus, the thermal expansion coefficient, the heat capacity, the entropy, Debye temperature and Grüneisen parameter are also evaluated over a pressure range of 0–6 GPa and a wide temperature range of 0–1800 K. The obtained results are in agreement with the available experimental and the other theoretical values.     

Ultrasonic measurement of the elastic properties of benzoyl glycine single crystals

Certain organic crystals are found to possess high non-linear optical coefficients, often one to two orders of magnitude higher than those of the well-known inorganic non-linear optical materials. Benzoyl glycine is one such crystal whose optical second-harmonic generation efficiency is much higher than that of potassium dihydrogen phosphate. Single crystals of benzoyl glycine are grown by solvent evaporation technique usingN, N-dimethyl formamide as the solvent. All the nine second-order elastic stiffness constants of this orthorhombic crystal are determined from ultrasonic wave velocity measurements employing the pulse echo overlap technique. The anisotropy of elastic wave propagation in this crystal is demonstrated by plotting the phase velocity, slowness, Young’s modulus and linear compressibility surfaces along symmetry planes. The volume compressibility, bulk modulus and relevant Poisson’s ratios are also determined. Variation of the diagonal elastic stiffness constants with temperature over a limited range are measured and reported.     

Elastic,electronic and thermodynamic properties of fluoro-perovskite KZnF3 via first-principles calculations

The elastic, electronic and thermodynamic properties of fluoro-perovskite KZnF₃ have been calculated using the full-potential linearized augmented plane wave (FP-LAPW) method. The exchange-correlation potential is treated with the generalized gradient approximation of Perdew-Burke-Ernzerhof (GGA-PBE). Also, we have used the Engel and Vosko GGA formalism (GGA-EV) to improve the electronic band structure calculations. The calculated structural properties are in good agreement with available experimental and theoretical data. The elastic constants C _ij are calculated using the total energy variation with strain technique. The shear modulus, Young’s modulus, Poisson’s ratio and the Lamé coefficients for polycrystalline KZnF₃ aggregates are estimated in the framework of the Voigt-Reuss-Hill approximations. The ductility behavior of this compound is interpreted via the calculated elastic constants C _ij . Electronic and bonding properties are discussed from the calculations of band structure, density of states and electron charge density. The thermodynamic properties are predicted through the quasi-harmonic Debye model, in which the lattice vibrations are taken into account. The variation of bulk modulus, lattice constant, heat capacities and the Debye temperature with pressure and temperature are successfully obtained.     

First principles study on the structural, electronic, and elastic properties of Na-As systems

We have performed the first principles calculation by using the plane-wave pseudopotential approach with the generalized gradient approximation for investigating the structural, electronic, and elastic properties Na-As systems (NaAs in NaP, LiAs and AuCu-type structures, NaAs₂ in MgCu₂-type structure, Na₃As in Na₃As, Cu₃P and Li₃Bi-type structures, and Na₅As₄ in A₅B₄-type structure). The lattice parameters, cohesive energy, formation energy, bulk modulus, and the first derivative of bulk modulus (to fit to Murnaghan’s equation of state) of the related structures are calculated. The second-order elastic constants and the other related quantities such as Young’s modulus, shear modulus, Poisson’s ratio, sound velocities, and Debye temperature are also estimated.     

Lattice mechanical properties of noble and transition metals

A model pseudopotential depending on an effective core radius but otherwise parameter free is used to study the interatomic interactions, phonon dispersion curves (inq and r-space analysis), phonon density of states, mode Grüneisen parameters, dynamical elastic constants (C ₁₁,C ₁₂ andC ₄₄), bulk modulus (B), shear modulus (C′), deviation from Cauchy relation (C ₁₂–C ₄₄), Poisson’s ratio (σ), Young’s modulus (Y), behavior of phonon frequencies in the elastic limit independent of the direction (Y ₁), limiting value in the [110] direction (Y ₂), degree of elastic anisotropy (A), maximum frequencyω _max, mean frequency 〈ω〉, 〈ω ²〉^1/2=(〈ω〉/〈ω ⁻¹〉)^1/2, fundamental frequency 〈ω ²〉, and propagation velocities of the elastic constants in Cu, Ag, Au, Ni, Pd, and Pt. The contribution of s-like electrons is calculated in the second-order perturbation theory for the model potential while that of d-like electrons is taken into account by introducing repulsive short-range Born-Mayer like term. Very recently proposed screening function due to Sarkar et al. has been used to obtain the screened form factor. The theoretical results are compared with experimental findings wherever possible. A good agreement between theoretical investigations and experimental findings has proved the ability of our model potential for predicting a large number of physical properties of transition metals.     

Effect of pressure on the elastic properties of silicon carbide

The pressure dependences of the second-order elastic constants C _ij and the velocity of sound in 3C-SiC and 2H-SiC crystals are calculated in the framework of the Keating model. The third-order elastic constants C _ijk for 3C-SiC are determined from the dependences of the second-order elastic constants C _ij on the pressure p.     

Amplitude dependence of the internal friction and Young’s modulus defect of polycrystalline indium

The dependences of the internal friction and the Young’s modulus defect of polycrystalline indium on the oscillatory strain amplitude have been studied over a wide range of temperatures (7–320 K) and oscillatory strain amplitudes (10⁻⁷−3.5 × 10⁻⁴) at oscillatory loading frequencies of about 100 kHz. It has been revealed that the amplitude dependences of the internal friction and the Young’s modulus defect include stages associated with the interaction of dislocations with point defects and the interdislocation interaction. The temperature range characterized by the formation of point-defect atmospheres (the Cottrell atmospheres) near dislocations in indium has been determined.     

Study of elastic properties of CeO2 by statistical moment method

The purpose of the present paper is to investigate the temperature and pressure dependences of the elastic properties of cerium dioxide using the statistical moment method (SMM). The equation of states of bulk CeO₂ is derived from the Helmholtz free energy, and the pressure dependences of the elastic moduli like the bulk modulus, B_T, shear modulus, G, Young’s modulus, E, and elastic constants (C11, C12, and C44) are presented taking into account the anharmonicity effects of the thermal lattice vibrations. In the present study, the influence of temperature and pressure on the elastic moduli and elastic constants of CeO₂ has also been studied, using three different interatomic potentials. We compare the results of the present calculations with those of the previous theoretical calculations as well as with the available experiments.     

Elastic constants of lead-bismuth alloys

A study of ultrasonic velocities and internal friction has been carried out in Pb-Bi alloys in the concentration range of 0 to 49.5 atomic % Bi using the composite oscillator technique. From the velocity and density data a set of elastic constants namely, Young’s modulus, rigidity modulus, bulk modulus and Poisson’s ratio are estimated. The results are interpreted in terms of the phase changes occurring in the alloy system. Internal friction is found to be more sensitive than the elastic constants to the phase changes.     

Structural, electronic and elastic properties of Ti2TlC, Zr2TlC and Hf2TlC

Using First-principle calculations, we have studied the structural, electronic and elastic properties of M₂TlC, with M = Ti, Zr and Hf. Geometrical optimization of the unit cell is in good agreement with the available experimental data. The effect of high pressures, up to 20 GPa, on the lattice constants shows that the contractions are higher along the c-axis than along the a axis. We have observed a quadratic dependence of the lattice parameters versus the applied pressure. The band structures show that all three materials are electrical conductors. The analysis of the site and momentum projected densities shows that bonding is due to M d-C p and M d-Tl p hybridizations. The M d-C p bonds are lower in energy and stiffer than M d-Tl p bonds. The elastic constants are calculated using the static finite strain technique. We derived the bulk and shear moduli, Young’s modulus and Poisson’s ratio for ideal polycrystalline M₂TlC aggregates. We estimated the Debye temperature of M₂TlC from the average sound velocity. This is the first quantitative theoretical prediction of the elastic properties of Ti₂TlC, Zr₂TlC, and Hf₂TlC compounds that requires experimental confirmation.      

Elasticity and inelasticity of biomorphic carbon,silicon carbide,and SiC/Si composite produced on the basis of medium density fiberboard

The amplitude and temperature dependences of the Young’s modulus and the internal friction (ultrasonic absorption) of biomorphic carbon, silicon carbide, and SiC/Si composite produced from medium density fiberboard (MDF) by pyrolysis (carbonization), followed by infiltration of molten silicon into the prepared carbon preform have been studied in the temperature range 100–293 K in air and under vacuum. The measurements have been performed by the acoustic resonance method with the use of a composite vibrator for longitudinal vibrations at frequencies of approximately 100 kHz. The data obtained by acoustic measurements of the amplitude dependences of the elastic modulus have been used for evaluating the microplastic properties of samples under study. It has been shown that the Young’s modulus, the decrement of elastic vibrations, and the conventional microyield strength of the MDF samples differ from the corresponding data for previously studied similar materials produced from natural eucalyptus, beech, sapele, and pine woods. In particular, the desorption of environmental molecules at small amplitudes of vibrations, which is typical of biomorphic materials based on natural wood, is almost absent for the MDF samples. The results obtained have been explained by different structures and the influence of pores and other defects, which, to a large extent, determine the mechanical characteristics of the biomaterials under investigation.     

Computation of collective modes and acoustic investigations at different temperatures of vitrous silica

This paper is mainly concerned with elastic and acoustic properties of vitrous silica besides the computation of phonon frequencies. Thus the phonon frequencies of vitrous silica have been calculated assuming the electronic bulk modulus,K _e, as equal to zero. New equations have been derived to relate the pressure derivatives of second order elastic constants to the acoustic Gruneisen’s parameters using both Bhatia-Singh’s parameters and Schofield’s equations. The calculated longitudinal and transverse Gruneisen’s parameters and the predicted absorption band spectra from Nagendranath’s equation and Bhatia Singh’s parameters are in good agreement with experiment. The calculated mean acoustic mode Gruneisen’s parameter evaluated from the pressure derivative of Nagendranath’s equation is also in good agreement with experiment. An erratum to this article is available at .     

Monocrystal elastic constants in the ultrasonic study of welds

For studying welds ultrasonically, the importance of knowing the material's single-crystal elastic constants, the C_ijs, is explained. Where these constants are not known, some guidelines are given for estimating them from polycrystalline elastic constants such as Young's modulus and the shear modulus.The important case of [001] fibre texture is considered. Being transversely isotropic, this case exhibits five macroscopic elastic constants, which are related to the three cubic elastic constants: C₁₁, C₁₂, C₄₄. From these five constants the angular variations of Young's modulus, the torsional modulus, and the sound velocities can be computed. For the same [001] fibre texture, results are given for a standard well-characterized material — copper, where the C_ijs are well known.     

Equation of state and temperature variation of the bulk modulus of the alkaline earth fluorides

The isotherms for the alkaline earth fluorides (CaF₂, SrF₂ and BaF₂) have been computed using the expression for the total free energy of a crystal in the quasiharmonic approximation. The theoretical points for SrF₂ and BaF₂ have been compared with the points derived from Bridgman’s experimental relation. The temperature variation of the isothermal bulk modulus of the alkaline earth fluorides has been worked out on the basis of Axe’s shell model. The theoretica points are compared with the points obtained from the temperature variation of the elastic constants data. It is found that the vibrational contribution to the temperature variation of the elastic constants in these crystals is significant.