Correlations of the temperature and pressure dependencies of the elastic constants of zircon

The temperature dependencies of the single crystal elastic constants and the isotropic elastic moduli of zircon were reevaluated using the ultrasonic in-phase frequencies vs. temperature data and the thermal expansion coefficients. The bulk moduli of zircon at different temperatures were also calculated using the recently derived analytical expression by Garai and Laugier (J. Appl. Phys. 101, 023514 (2007)) utilizing the Anderson–Grüneisen parameter (δ) obtained from the pressure dependencies of the single crystal elastic constants. The temperature derivatives of the bulk modulus of zircon evaluated from the temperature derivatives of the single crystal elastic constants agree well with the corresponding values calculated from the analytical expressions utilizing the pressure derivative of the bulk modulus (δ). The results reveal good correlations between the ultrasonic measurements of the pressure and temperature derivatives of the single crystal elastic constants of zircon.     

Orthotropic elastic constants for polyimide film

The orthotropic constants of polyimide film have been characterized using the theory of elasticity of an anisotropic material. Experimental techniques coupled with the mechanics of orthotropic materials are used to determine all 9 independent orthotropic elastic constants (3 tensile moduli, 3 shear moduli, and 3 Poisson's ratios) and 3 coefficients of thermal expansion. Vibrational holographic interferom‐etry is used to determine the orthotropic axes of symmetry. For this polyimide film, the two principal axes coincided with the machine and transverse directions. It is also used to evaluate the 2 in‐plane Poisson's ratios by measuring residual stresses in 2‐D and 1‐D square membranes. Using other instruments such as a high pressure gas dilatometry apparatus, a tensile tester, a pressure‐volume‐temperature apparatus, a thermornechanical analyzer, and a torsion pendulum, the 7 other orthotropic constants and the 3 coefficients of thermal expansion are determined.     

Systematics of elastic and thermodynamic properties of super-hardness cubic boron nitride under high pressure

The elastic constants of cubic boron nitride (c–BN) are calculated by the first-principles method at ambient condition. The dependence of the elastic constants, the bulk modulus B, and the shear modulus G on the applied pressure are presented. The calculated results are in good agreement with the comparable experimental and theoretical values. The deviation from the Cauchy relation and the elastic anisotropy are investigated. The normalized elastic constants c_ij ′ are introduced to investigate elasticity of c–BN in more detail. Moreover, the thermodynamic properties (Debye temperature, melting temperature) and sound velocity have been investigated under high pressure for the first time.     

Inverse determination of elastic constants of composite materials

Vibration testing, combined with a numerical method, is a potential alternative approach for determining elastic constants of materials because of its nondestructive characteristic, single test, and producing average properties. To simplify the modeling processes and to find better results, the numerical method of combining finite element analysis and a hybrid genetic algorithm is proposed to inversely determine the elastic constants from the vibration testing data in this work. As verified from the available data of literatures, the repeatability and accuracy of the proposed inverse determination method are confirmed. Four composite specimens with different stacking sequence and thickness are used to determine their elastic constants. As applied to obtain the four or five elastic constants of composite materials, the performance of the proposed method is excellent. To obtain the four elastic constant for thin composite plates, it is better to use unidirectional and thick enough laminates. To determine the five elastic constants for thick composite plates, it is suggested to use unidirectional laminates with the length‐to‐thickness ratio less than or equal to 50. To conquer the effect of measurement error or missing frequencies, it is necessary to carefully exclude these frequencies in the objective function, and reasonable results could be obtained by the proposed method. POLYM. COMPOS., 2009. © 2008 Society of Plastics Engineers     

Evaluation of the elastic constants of nanoparticles from atomistic simulations

We present an approach to estimate the elastic constants of molecules and nanoparticles, based on the analysis of thermal fluctuations from Monte Carlo (MC) or molecular-dynamics (MD) atomistic simulations. The method and the force-field used for these calculations have been tested by the calculation of Young's modulus of a graphite sample along the basal plane; the calculated value was found to be 1.07 TPa, in very good agreement with the experimentally determined one of 1.02 TPa.The results on a carbon-based nanotube indicate that for the longitudinal direction of the particle, the value of the elastic constant is on the order of 400 GPa. The elastic constant of the considered nanotube in the radial direction is significantly lower, the predicted values being in the range 4-7 GPa.The method was also applied to the elastic constants of a type of siloxane-based nanostructure, whose longitudinal elastic constant (30 GPa) is an order of magnitude lower than the corresponding value for the carbon-based nanotube.     

Single-crystal elastic constants of natural ettringite

The single-crystal elastic constants of natural ettringite were determined by Brillouin spectroscopy at ambient conditions. The six non-zero elastic constants of this trigonal mineral are: C₁₁ = 35.1 ± 0.1 GPa, C₁₂ = 21.9 ±0.1 GPa, C₁₃ = 20.0 ± 0.5 GPa, C₁₄ = 0.6 ± 0.2 GPa, C₃₃ = 55 ± 1 GPa, C₄₄ = 11.0 ± 0.2 GPa. The Hill average of the aggregate bulk, shear modulus and the polycrystal Young's modulus and Poisson's ratio are 27.3 ± 0.9 GPa, 9.5 ± 0.8 GPa, 25 ± 2 GPa and 0.34 ± 0.02 respectively. The longitudinal and shear elastic anisotropy are C₃₃/C₁₁ = 0.64 ± 0.01 and C₆₆/C₄₄ =0.60 ± 0.01. The elastic anisotropy in ettringite is connected to its crystallographic structure. Stiff chains of [Al(OH)₆]³⁻ octahedra alternating with triplets of Ca²⁺ in eight-fold coordination run parallel to the c-axis leading to higher stiffness along this direction. The determination of the elastic stiffness tensor can help in the prediction of the early age properties of cement paste when ettringite crystals precipitate and in the modeling of both internal and external sulfate attack when secondary ettringite formation leads to expansion of concrete.     

Evaluation of elastic constants and high temperature tensile behaviour with damage assessment of the SiC seal-coated 2.5D Cf/SiC composites having multilayered interphase and Si-B-C added matrix

Elastic constants and tensile behaviour of chemical vapour infiltration processed 2.5D C_f/SiC composites possessing multilayered (PyC/SiC)_n=4 interphase, Si-B-C containing matrix and SiC seal-coating have been evaluated with microstructural examination and damage assessment. The strength obtained as ～187 ± 2 MPa in tensile tests at 27 °C is increased by ～18% and ～22% at 1000 °C and 1250 °C, respectively due to reduced thermal stress and increased strength of load-sharing C-fibres, which are protected from oxidation till failure by a self-healing borosilicate layer. The damage evolving during tension tests has been quantified by relating it to decrease of stress-strain slope with strain. Higher (6–8 times) elastic constants measured along fibre-axes than that obtained transversely, indicate significant anisotropy. Owing to matrix cracking with fibre-debonding and pull-out, the fibre-oriented elastic constants of tensile-fractured samples are significantly lower than those of as-received composites, and the difference scales with temperature, whereas negligible change is observed perpendicular to the fibre axes.     

The elastic constants of moulded polymeric composites

The paper makes an approach to the determination of the five independent elastic constants of uniaxial symmetric material with the ultrasonic immersion technique. Only five ultrasonic wave velocities are required during measurement. The elastic constants of the moulded composites comprised of poly(phenylene sulfide), poly(aryl ether sulfone) with cardo side groups, short carbon fiber and carbon black are measured by using the suggested treatment. The results exhibit a close relationship between the elastic properties and the structure of the materials.     

First‐Principles Study of Elastic,Structural, Electronic,Thermodynamical, and Optical Properties of Yttria (Y2O3) Ceramic in Cubic Phase

Structural, elastic, optical, thermodynamical, and electronic properties of yttrium oxide compound in cubic phase have been studied using the full‐potential augmented plane waves (FP‐LAPW) within density functional theory (DFT) framework. Four different approximations were used for exchange‐correlation potentials terms, comprised Perdew–Burke–Ernzerhof generalized parameterization of gradient approximation (GGA‐PBE), Wu–Cohen (WC‐GGA), local‐density approximation (LDA), and new approximation modified Becke and Johnson (mBJ‐GGA). The structural properties such as equilibrium lattice parameter, bulk modulus and its pressure derivative have been obtained using optimization method. Moreover, Elastic constants, Young's modulus, shear modulus, Poisson's ratio, sound velocities for longitudinal and shear waves, Debye average velocity, Debye temperature, and Grüneisen parameters have been calculated. Obtained structural, elastic and other parameters are consistent with experimental data. Moreover pressure dependence of the elastic moduli was studied. From electronic calculations, it has been found that the band gap was 5.7 eV at Г point in the Brillouin zone using mBJ‐GGA approximation. Optical properties, such as the dielectric function, refractive index, extinction index, and optical band gap, were calculated for radiation up to 14 eV. In addition, the unique type of bonding in Y₂O₃ was discussed by three method including effective charge, B/G ratio, and charge density distribution.     

Bending modes, elastic constants and mechanical stability of graphitic systems

The thermodynamic and mechanical properties of graphitic systems are strongly dependent on the shear elastic constant C₄₄. Using state-of-the-art density functional calculations, we provide the first complete determination of their elastic constants and exfoliation energies. We show that stacking misorientations lead to a severe lowering of C₄₄ of at least one order of magnitude. The lower exfoliation energy and the lower C₄₄ (more bending modes) suggest that flakes with random stacking should be easier to exfoliate than the ones with perfect or rhombohedral stacking. We also predict ultralow friction behaviour in turbostratic graphitic systems.     

Self-consistent calculations of elastic constants of polycrystalline graphite

This paper reports the application of a self-consistent integral equation technique to the calculation of the elastic constants of polycrystalline graphite. The self-consistent expressions used in this paper are restricted to spherical grains and pores and neglect correlation among grains and pores. A model of graphite based upon the assumption that the grains are cracked crystals was used with the self-consistent expressions to account for the elastic constants of three isotropic or mildly anisotropic graphites to within approximately 6% and for the elastic constants of a more anisotropic graphite to within approximately 20%.     

Evaluation of the complex material constants of piezoelectric ceramics in the thickness vibration mode

A new noniterative method, for determining the dielectric, piezoelectric and elastic constants, in complex form, for piezoceramic materials, in the thickness extensional mode, was proposed.This method is very flexible, as it is based on the standard approach to determine the elastic stiffness, followed by the measurement of the impedance at two frequencies to calculate the dielectric and piezoelectric constants, by solving a system of two equations.The new method was tested on materials with various thickness coupling factors (k_t = 4.5–60%) and mechanical quality factors (Q_mt = 20–1600), proving very good accuracy for all of them.The accuracy of standard method was also evaluated for the same materials.     

Experimentally- and theoretically-evaluated ultimate 3-dimensional elastic constants of trans-1,4-polyisoprene α and β crystalline forms on the basis of the newly-refined crystal structure information

The ultimate 3-dimensional elastic constants and compliance tensors have been evaluated for trans-1,4-polyisoprene (Gutta percha) α and β crystalline forms experimentally and theoretically. The theoretical estimation was performed by the molecular mechanics method on the basis of the newly-refined atomic coordinates obtained by quantitative analysis of the crystal structures using many numbers of X-ray reflections from the highly-oriented and highly-pure α and β crystalline samples. The calculated Young's modulus along the chain axis was 85.8 GPa for the α form and 72.3 GPa for the β form, in good agreement with the experimentally-obtained X-ray crystallite moduli, 72.4 GPa and 61.4 GPa, respectively.     

Relation of Single-Crystal Elastic Constants to Polycrystalline Isotropic Elastic Moduli of MgO: II, Temperature Dependence

The technical adiabatic elastic moduli E_[hkl] and G_hkl of single crystals of magnesium oxide were measured over the temperature range 298° to about 1600°K by a Förster-type resonance method. These data were compared with the low-temperature values (80° to 560°K) of the principal elastic constants c_ij and coefficients Sy reported by Durand. Combining Durand's data and the present data, the elastic moduli for single-crystal magnesium oxide were evaluated for the temperature range 80° to 1600°K. Young's modulus and the shear modulus of densely formed isotropic polycrystalline magnesium oxide were measured over the temperature range 298° to 1600°K. The data on the elastic constants of the single crystals were compared with the measured elastic moduli of the isotropic polycrystalline magnesium oxide on the basis of the Voigt-Reuss-Hill approximation. The temperature dependence of the elastic moduli was fitted into the expression M = M_c— BT exp (—T_c/T) suggested by Wachtman et al. ; mean deviations were less than 0.4% for the temperature range considered. The significance of the present data is discussed with particular emphasis on the following points: (1) the temperature variation of the elastic modulus is a function of thermal expansion, (2) the temperature dependence of the elastic modulus can be well described by the foregoing expression for a wide range of temperature, (3) the expression gives a value of the elastic modulus at 0°K, and (4) it may be possible to make use of measurements on the elastic properties of a densely sintered polycrystalline material to obtain information heretofore obtainable only from the corresponding single-crystal data.     

A method of determining the elastic properties of diamond-like carbon films

The elastic properties of diamond-like carbon (DLC) films were measured by a simple method using DLC bridges which are free from the mechanical constraints of the substrate. The DLC films were deposited on a Si wafer by radio frequency (RF) glow discharge at a deposition pressure of 1.33 Pa. Because of the high residual compressive stress of the film, the bridge exhibited a sinusoidal displacement on removing the substrate constraint. By measuring the amplitude with a known bridge length, we could determine the strain of the film which occurred by stress relaxation. Combined with independent stress measurement using the laser reflection method, this method allows the calculation of the biaxial elastic modulus, E/(1−ν), where E is the elastic modulus and ν is Poisson's ratio of the DLC film. The biaxial elastic modulus increased from 10 to 150 GPa with increasing negative bias voltage from 100 to 550 V. By comparing the biaxial elastic modulus with the plane–strain modulus, E/(1−ν²), measured by nano-indentation, we could further determine the elastic modulus and Poisson's ratio, independently. The elastic modulus, E, ranged from 16 to 133 GPa in this range of the negative bias voltage. However, large errors were incorporated in the calculation of Poisson's ratio due to the pile up of errors in the measurements of the elastic properties and the residual compressive stress.     

Linear dielectric thermodynamics: A new universal law for optical,dielectric constants

The thermodynamics of linear dielectric are formally developed to explore the isothermal and adiabatic temperature-pressure dependence of dielectric constants. The refractive index of optical materials is widely measured in the literature: it is both temperature and pressure dependent. The argument to establish the dielectric constant's isentropic temperature dependence is a thermodynamic one and is thus applicable to all physical models that describe electron clouds and electronic resonances within materials. The isentropic slope of the displacement field vs the electric field at all temperatures is described by an adiabatic dielectric constant in an energy-per-unit mass system. This slope is shown through the electronic part of the entropy to be unstable at high temperatures due to the change in the curvature of the temperature dependence of the dielectric constant. The electronic entropy contribution for optical, thermo-electro materials has negative heat capacities which are unacceptable. The dielectric constant's temperature and pressure dependence is predicted to be only dependent on the specific volume so isentropes are always positive. A new universal form for the dielectric constant follows from this hypothesis: the dielectric constant is proportional to the square root of the specific volume for fully dense solids.     

Sound velocities and elastic properties of PbTiO3 and PbZrO3 under pressure: First principles study

The elastic constants and sound velocities as a function of pressure for perovskite materials PbTiO₃ (PTO) and PbZrO₃ (PZO) were investigated by first principles calculations. Under ambient pressure, the calculated structural parameters were calculated and found to be in good agreement with known values. To study properties under pressure, PTO and PZO were calculated at several reduced volumes, each of which corresponds to the system under pressure. The C₁₁, C₁₂ and C₄₄ elastic constants are all found to increase with pressure for the pressure range studied. Because the sound velocities are directly derived from the elastic constants, the relationships between the sound velocities and pressure also follow similar trends. The longitudinal modes are all larger than those of the transverse modes.     

Ultrasonic measurements of the elastic stiffness constants of oriented polyethylene

High density polyethylene (Rigidex 9), a high density copolymer of polyethylene and poly(butene-1) (Rigidex 2000), and low density polyethylene (Alkathene WJG 11) were oriented by hot drawing. The crystalline texture, as determined by wide- and low-angle X-ray diffraction, was a highly oriented chain axis in the draw direction with random orientation transversely and with lamellae surfaces perpendicular to the draw direction. Elastic stiffness constants were measured by a contact-probe ultrasonic pulse technique at 2.5 MHz both before and after annealing in a temperature range which did not significantly alter the crystalline texture. Assuming orthorhombic symmetry the nine stiffness constants of Rigidex 2000 and the three longitudinal and three shear stiffness constants of Rigidex 9 were measured after drawing and after subsequent annealing. Only the longitudinal constants of Alkathene were measured, as shear waves could not be transmitted. The longitudinal stiffness in the draw direction was markedly affected by drawing and by annealing, while the crystalline texture remained substantially unchanged; by comparison the other stiffness constants showed little change. Drawn Rigidex 9 reached a tensile modulus in the draw direction of 69 GPa. The results are compared with ‘static’ and low frequency measurements reported in the literature.     

Experimental and DFT insights into elastic,magnetic, electrical,and thermodynamic properties of MAB-phase Fe2AlB2

The elastic, magnetic, electrical, and thermodynamic properties of bulk polycrystalline Fe₂AlB₂ are reported, with further theoretical insights provided through first-principles calculations. DFT simulations, along with an appropriate empirical method to treat magnetic contributions, reproduce well the experimental heat capacity, phonon thermal conductivity, and thermal expansion. The shear, Young's and bulk moduli all decrease linearly from 107.7, 264.5, and 162.4 GPa at 298 K to 91.4, 227.7, and 149.6 GPa at 1273 K, respectively, while the Poisson's ratio shows a weak dependence on temperature, hovering around 0.23-0.26. The mechanical damping, Q⁻¹, is found to be a weak function of temperature up to 973 K, but above this temperature increases sharply, peaking at a specific temperature that becomes higher with increasing resonant frequency. The phonon contribution plays a dominant role in the measured heat capacity from 4.2 to 1000 K, while the magnetic contribution becomes significant around the Curie temperature (301 K). The electronic contribution increases with temperature above 100 K. Furthermore, the standard enthalpy, entropy, and Gibbs free energy are calculated as 16.60 kJ/mol, 92.92 J/(mol·K) and −11.09 kJ/mol, respectively. The thermal conductivity increases linearly with increasing temperature from RT to 1173 K, up to around 13.0 W/(m·K) at 1323 K after hovering at around 7.8 W/(m·K) in the low-temperature range. The electrons play a dominant role in heat conduction, while the phonons contribute only a little to the thermal conductivity. The dilatometric coefficient of thermal expansion of Fe₂AlB₂ is measured as 13.5 × 10⁻⁶ K⁻¹ in the temperature range RT-1348 K.