Third-order Elasticity Research Articles

Second- and Third-Order Elastic Constants of Aluminum and Lead

Second- and Third-Order Elastic Constants of Aluminum and Lead

Second- and Third-Order Elastic Constants of Aluminum and Lead

The second- and third-order elastic constants of Al and Pb are calculated as the second and third derivatives of the binding energy with respect to the finite deformation parameter. The binding energy is derived from a local pseudopotential by use of second-order perturbation theory. It is shown that the binding energy satisfied not only the diagonal equilibrium condition, but also the off-diagonal equilibrium condition, i.e., the first derivative of the binding energy with respect to a volume change as well as with respect to shear deformations is zero. Accordingly, the present method of calculation is based on a stable lattice model. The results of the present calculation of the third-order elastic constants of Al are found to be in qualitative agreement with the experimental data obtained by Thomas. The complete experimental set of the third-order elastic constants of Pb is not yet available; however, the calculated pressure derivatives of the second-order elastic constants are in agreement with the experimental data of Miller and Schuele. On the other hand, the method initiated by Leigh, which is based on a rigid-band model, cannot reproduce the experimentally observed third-order elastic constants of Al. This is an indication that the pseudopotential method is to be preferred to the rigid-band method as far as the calculation of second- and third-order elastic constants is concerned.

Second- and Third-Order Elastic Constants of Alkali Metals

The second- and third-order elastic constants of lithium, sodium, potassium, and rubidium in the bodycentered cubic structure are calculated. The relationship between Brugger elastic constants and Fuchs elastic constants is worked out. The Brugger elastic constants, which are defined as the derivatives of the energy with respect to the Lagrangian strain, are widely used to express experimental results. The Fuchs elastic constants, which are defined as the derivatives of the energy with respect to homogeneous expansion and volume-conserving homogeneous shear, are often more convenient for calculations in terms of atomistic considerations, and are particularly convenient for calculations with pseudopotentials. They are used here to calculate the contribution of the band-structure energy to the elastic constants using the local pseudopotential proposed by Ashcroft. This pseudopotential contains the core radius as the only adjustable parameter. The contribution of the band-structure energy to the elastic constants is represented as a summation of two kinds of derivatives of the wave-number characteristics over the reciprocal lattice points---those with respect to homogeneous expansion and those with respect to volume-conserving homogeneous shear. The core radius which gives the best fit to the experimental second-order elastic constants agrees with that determined by Ashcroft from data on the Fermi surface or on the resistivity of liquid metals. The band-structure energy term is found to make a small contribution to the second-order elastic constants but an indispensable contribution to the third-order elastic constants.

Third-Order Elastic Constant c. in Lithium Fluoride

Journal of the American Ceramic SocietyVolume 51, Issue 10 p. 603-603 Third-Order Elastic Constant c. in Lithium Fluoride J. R. ARCHER, J. R. ARCHERSearch for more papers by this authorW. E. MOODY, W. E. MOODYSearch for more papers by this authorA. L. STANFORD JR., A. L. STANFORD JR.Search for more papers by this author J. R. ARCHER, J. R. ARCHERSearch for more papers by this authorW. E. MOODY, W. E. MOODYSearch for more papers by this authorA. L. STANFORD JR., A. L. STANFORD JR.Search for more papers by this author First published: October 1968 https://doi.org/10.1111/j.1151-2916.1968.tb13333.xCitations: 4 At the time this work was done the writers were, respectively, research assistant, professor of ceramic engineering, and associate professor of physics, Georgia Institute of Technology, Atlanta, Georgia 30332. J. R. Archer is now research engineer, Glassrock Products, Inc., Atlanta, Georgia 30318. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article.Citing Literature Volume51, Issue10October 1968Pages 603-603 RelatedInformation

Morse-Potential Evaluation of Second- and Third-Order Elastic Constants of Some Cubic Metals

Morse-Potential Evaluation of Second- and Third-Order Elastic Constants of Some Cubic Metals

Morse-Potential Evaluation of Second- and Third-Order Elastic Constants of Some Cubic Metals

The second- and third-order elastic constants of Al, Cu, Ag, Au, Na, and K have been calculated in the framework of a central-force model. The interatomic potential was expressed in terms of the Morse potential. The parameters in the potential were determined by using the lattice parameter, bulk modulus, and cohesive energy. The range of the interatomic potential was cut off after 176 and 168 neighbors in the case of face-centered and body-centered cubic metals, respectively. The evaluated second- and third-order elastic constants were used to calculate the pressure derivatives of the second-order elastic constants. These pressure derivatives compare fairly well with the experimental values. The agreement between experimental values and theoretical values of third-order elastic constants is less satisfactory. Theory predicts ${\mathrm{C}}_{123}$ to be positive for all the fcc metals and negative for bcc metals. The available experimental data on the third-order elastic constants of Cu, Ag, and Au show that ${\mathrm{C}}_{123}$ is positive for Ag and negative for Cu and Au. The possible implications of this result are discussed.

Calculation of Lattice-Parameter Changes with Hydrostatic Pressure from Third-Order Elastic Constants

Part I describes the calculation of the first and second pressure derivatives of the lattice parameters from the second- and third-order elastic constants (or from the second-order constants and their pressure derivatives). These derivatives determine only the linear and quadratic terms in the expansions of the lattice parameters as power series in the pressure. The quadratic expressions obtained by truncating the power series are not suitable for extrapolation beyond a few tens of kilobars. Part II gives extrapolation formulas that turn out to be far better than the truncated power-series expansions. For quartz, magnesium, and cadmium, the resulting extrapolation from ultrasonic data taken at low pressures is found to be in reasonable agreement with x-ray determinations of the lattice parameters out to well over 100 kbar. Appendix A includes derivations of all the formulas needed to make the extrapolation from ultrasonic and thermal data. Complications arise not only from the distinction between adiabatic and isothermal coefficients, but also from the distinction between “effective” and “thermodynamic” coefficients.

Relation between Third-Order Elastic Moduli and the Thermal Attenuation of Ultrasonic Waves in Nonconducting and Metallic Crystals

The background loss of all materials, and sometimes the principal loss in single crystals, is connected with the conversion of acoustic waves into thermal phonons. One source of conversion is the thermoelastic loss resulting from the flow of heat from the compressed (hotter) part of the wave to the expanded (cooler) part. This source accounts for about half the thermal attenuation in a metal, but only about 4% of that in an insulator. Another source of thermal attenuation—first suggested by Akhieser—is connected with the separation of phonon-mode temperatures by a suddenly applied stress, followed by an equilibration of these temperatures with a relaxation time τ. This source has been called “phonon viscosity.” A method for evaluating this source of attenuation was proposed in previous papers [W. P. Mason and T. B. Bateman, J. Acoust. Soc. Am. 36, 644–652 (1964); W. P. Mason, Physical Acoustics (Academic Press Inc., New York, 1965), Vol. 3B, Chap. VI, pp. 235–286]. It involves the use of the third-order elastic moduli to determine the energy stored by the phonon-mode temperature separations, together with a relaxation time τ to equilibrate this energy. Recently, new determinations of these third-order moduli have been made for NaCl, KCl, MgO, and Y3Fe5O12 (YIG). The present paper shows that the formulas predict the measured attenuations (longitudinal and shear waves) within about 25%. Silicon, without oxygen, has a higher thermal conductivity than silicon with dissolved oxygen. This is reflected in the higher attenuation of oxygen-free material, in agreement with theory. A theoretical evaluation of the attenuation is made for a longitudinal wave propagated along a (111) axis. For metals, it is shown that the “phonon-viscosity” attenuation is in the same order as thermoelastic attenuation. A method for evaluating this loss in single-crystal lead is devised. The attenuation measured over a temperature range is consistent with the sum of the thermoelastic attenuation and the “phonon-viscosity” attenuation.

Measurement of the third-order elasticity constants for steel by the ultrasonic method

Measurement of the third-order elasticity constants for steel by the ultrasonic method

Measurement of Third-Order Elastic Moduli of Yttrium Iron Garnet

The third-order elastic moduli of yttrium iron garnet (YIG) have been determined from measurements of ultrasonic velocity. A pulsed ultrasonic system which uses a phase comparison principle to measure velocity versus applied stress is described. In units of 1012 dyn/cm2 and for 28°C, the third-order moduli are: C111=−23.3±0.8C112=−7.17±0.6C123=−0.33±1.3C144=−1.48±0.29C155=−3.06±0.14C456=−0.97±0.16.Thermodynamic definitions for the moduli [K. Brugger, Phys. Rev. 133, A1611 (1964)] have been used.

Measurements of the compressibility of solids

In high pressure research, static megabar pressures are typically produced by compression of a thin sample by two diamonds in various types of diamond anvil cells. This process is accompanied by large plastic deformation (sample thickness is reduced by a factor of 30), and finite elastic deformation of a sample and even the diamond. A thermodynamically consistent system of equations for large elastic and plastic deformation of an isotropic material obeying nonlinear elasticity and pressure dependent yield condition is formulated. The Murnaghan elasticity law and pressure-dependent J2 plasticity are utilized. The finite-strain third-order elasticity law for cubic crystals is utilized for diamond. A computational algorithm is presented with emphasis on the stress update procedure and derivation of the consistent tangent moduli. It is implemented as a user material subroutine in the finite element code ABAQUS. Material parameters for a rhenium sample, as an example, and a diamond are calibrated based on the experimental and atomistic simulation results in the literature. The evolution of the stress and strain tensor fields in the sample and diamond is studied up to a pressure of 300 GPa. Good correspondence between numerical and experimental pressure distributions at the diamond-sample contact surface is obtained. Because there is a significant scatter of the magnitude of reported third-order single-crystal elastic moduli for diamond, their effect on strains and stresses is studied in detail. With the smaller third-order elastic moduli, the phenomenon of cupping of the diamond-sample contact surface is reproduced, which plays an important role in increasing maximum pressures for a given anvil geometry. The results provide important insight into the mechanical response in diamond anvil cells, interpretation of materials properties under extreme conditions from heterogeneous fields, and optimum design of cells for reaching the maximum static pressure in a volume sufficient for the desired measurements.