Temperature dependent ultrasonic properties of aluminium nitride

Hexagonal wurtzite structured aluminium nitride has been characterized by the theoretical calculation of ultrasonic attenuation, ultrasonic velocity, higher order elastic constants, thermal relaxation time, acoustic coupling constants and other related parameters in temperature range 200-800 K for wave propagation along the unique axis of the crystal. Higher order elastic constants of AlN at different temperatures are calculated using Lennard-Jones potential for the determination of ultrasonic attenuation. A decrease in ultrasonic velocity with temperature has been predicted, which is caused by reduction in higher order elastic constants with temperature. The temperature dependent ultrasonic properties have been discussed in correlation with higher order elastic constants, thermal relaxation time, thermal conductivity, acoustic coupling constants and thermal energy density. Anomalous behaviour of the attenuation is found at 400 K. On the basis of attenuation, the ductility and performance of AlN have been studied.     

Higher order elastic constants of alkali halides

The Coulomb, van der Waals and repulsive lattice sums occurring in the higher order elastic constants up to sixth order have been calculated for the rocksalt and cesium chloride structures. Numerical values of the static elastic constants up to sixth order based on a rigid ion model with van der Waals and Born-Mayer type central force interaction between first and second nearest neighbors are calculated for several alkali halides representing both structure types. Fair agreement with the available experimental third and fourth order elastic constant data is found.     

关于岩石三阶弹性常数的超声测量方法

油气储层岩石的三阶弹性常数反映了该岩石的速度应力敏感性,是利用声波测井或地震资料进行原位地应力反演的基本参数。本文阐述了利用声弹性理论在实验室测量岩石三阶弹性常数的原理与方法,并给出了部分实验结果。这些参数将为研究声波在预应力声场中传播规律的提供基础数据,同时也为定量分析利用交叉偶极声波测井评价地应力的精度提供了依据。不同岩石的三阶弹性常数较大差异表明,通过速度各向异性进行应力反演时必须考虑岩石本身非线性的差异。     

High pressure equation of state and sound velocities to fourth order elastic constants. Application to sodium chloride to 270 kbars

Using the theory of small deformation superposed on a finite strain, we expand the strain energy to fourth order in terms of the strain invariants, and derive an athernal expression for the equation of state and the longitudinal and shear velocities of sound of a homogeneous Isotropic material to fourth order elastic constants. We evaluate reported velocity data in sodium chloride by Voronov and Grigorev to 80 kbar in terms of our formulation, to predict the equation of state to 270 kbar. Experiments performed in our laboratory (Frankel, Rich, and Homan) to 270 kbar which use ultrasonic interferometry to obtain resonance frequency intervals in the specimen were also evaluated. Both results are in good agreement with Decker's equation of state.     

Elastic constants of solids at high pressures

The isothermal and adiabatic nth-order (n ?? 2) elastic constants of a loaded crystal are defined. These constants fully determine the behavior of solids at an arbitrary load and are controlled by both an interatomic interaction and an applied load. Expressions that relate these constants (of the second, third, and fourth order) to Brugger elastic constants of the corresponding order, which are only determined by an inter-atomic interaction, are found for cubic symmetry crystals under hydrostatic pressure. These expressions are used to calculate the equation of state and the second- and third-order elastic constants of bcc tantalum at T = 0 K over a wide pressure range (0?C600 GPa) using an electron density functional method. The results of calculating the equation of state and the second-order elastic constants agree with available experimental data and the calculation results obtained in other works.     

Ultrasonic wave propagation in IIIrd group nitrides

The ultrasonic attenuation in hexagonal structured (wurtzite) third group nitrides (GaN, AlN and InN) has been evaluated at 300 K for an ultrasonic wave propagating along the unique axis of the crystal. Higher order elastic constants of these materials are calculated using the Lennard-Jones potential for the determination of ultrasonic attenuation. The ultrasonic velocity, Debye average velocity, thermal relaxation time and acoustic coupling constant are evaluated along the z-axis of the crystal using the second order elastic constants and other related parameters. The contributions of the elastic constants, thermal conductivity, thermal energy density, ultrasonic velocity and acoustic coupling constant to the total attenuation are studied. On the basis of the ultrasonic attenuation, it can be concluded that the AlN is more ductile than either GaN or InN at 300 K. Orientation dependent characterization has been achieved by calculation of the orientation dependent ultrasonic velocity, Debye average velocity and thermal relaxation time for the materials.     

On the third- and fourth-order constants of incompressible isotropic elasticity

Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress-strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incompressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incompressible elasticity μ, ā, and D are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.     

Energy and elastic constants of lithium and aluminium

In the proposed model of the fourth order in the pseudopotential including the distortion effects of Fermi surface and the energies of the ground state, elastic constants of the second and third orders of lithium and aluminium are calculated. The agreement with the experiment is rather better than in the second order models.     

An improved approximation for the Rayleigh wave equation

We derive a simple expression, which gives an approximate value of the Rayleigh wave velocity in an isotropic solid. This approximation is five times better than that given by Viktorov. The velocity equation can be easily inverted in order to obtain an accurate determination of the elastic constants. This procedure can be worthwhile for elastic microanalysis of bulk materials by scanning acoustic microscopy.     

Analysis of anharmonic and thermoelastic properties of non-centrosymmetric crystals

In the present paper an analytical potential form is used for overlap repulsive energy, derived by Harrison from quantum mechanical considerations, along with the composite effect of three-body forces and intersublattice displacement. The short-range overlap parameters in Harrison's potential form have direct correlation with the valence state energies for outermost electrons. The potential model is applied to calculate the third and fourth order elastic constants, first and second pressure derivatives of second order elastic constants, Grüneisen parameter and its volume dependence, Anderson parameter, and thermal expansion coefficient for three non-centrosymmetric crystals, viz. CaF₂, SrF₂ and BaF₂. The calculated values of various physical quantities are found to be in good agreement with experimental data.The authors are grateful to Dr. Mansour Khalef, the Head of Physics Department, TNRC, Tajura (Tripoli) for the facilities and encouragements.     

Molecular Dynamics Simulations of the Elastic Anisotropy of Pd at Extreme Conditions

It is very interesting to discover the elastic properties of engineering material palladium, especially its elastic anisotropy along Hugoniot states. We here investigate the evolution of its high pressure and temperature(PT) elastic ansotropy along Hugoniot using molecular dynamics simulations based on accurate classical interatomic potential. In order to testify the validity of the interatomic potential of Pd in describing the high PT elastic properties, we calculate its isothermal and adiabatic elastic moduli using molecular dynamics method. The obtained data are in good agreement with experimental data. From the isothermal elastic constants, we deduce the Hugoniot acoustic velocities and find that the resulting data are in good agreement with experimental acoustic velocity data. Based on the reliable elastic constants, we further investigate the spacial elastic ansotropy along Hugoniot PT states. It is found that the spacial elastic anisotropy of Pd increases along Hugoniot states.     

Third order elastic constants of V3Si and V3Ge

General expressions have been derived for the second order elastic constants and third order elastic constants of the A-15 structure compounds with the nearest neighbour central interaction. The second order elastic constants, the third order elastic constants and the pressure derivatives of the second order elastic constants of V₃Si and V₃Ge are reported and compared with the available experimental measurements.     

Acoustic symmetry and the third order elastic constants of tetragonal TII (4/m) crystals

The mode equations which relate the measured hydrostatic and uniaxial pressure derivatives of the “natural” ultrasonic velocity to the second and third order elastic constants for the tetragonal TII (4/m) Laue group have been transformed from the crystallographic (XYZ) axial frame to a reference frame (k, k + π/2, Z) comprised of two acoustic symmetry axes and the fourfold axis. It is found that, if the acoustic symmetry axes do not shift appreciably with pressure, the mode equations for TII crystals in this reference frame are much simpler than those in the crystallographie axial frame (XYZ), having the same form as those for the higher symmetry TI Laue group. Hence 12 transformed third order elastic constants can be obtained experimentally for TII crystals by making measurements of the hydrostatic and uniaxial pressure derivatives of ultrasonic wave velocity of modes propagated in the (k, k + π/2, Z) axial set with the uniaxial stresses also applied in this reference frame. The approach has been used to determine the 12 transformed third order elastic constants of scheelite (CaWO₄).     

Higher order elastic constants and non-linear stress-strain relation for V3Si

Relations for the third and fourth order elastic constants are derived from a Landau-Devonshire type free energy function for crystals with the A15 structure which undergo the structural transformation at low temperature. Application to transforming V₃Si at 21° K gives in connection with experimental data for the pressure derivatives of the elastic constants unusually large numerical values for c₁₁₁, c₁₁₂, c₁₂₃ and c₁₁₁₁. The stress-' strain relation calculated from these data is in semiquantitative agreement with the directly measured, strongly non-linear behavior.     

The second and third order elastic constants of alkali metals

The second and third order elastic constants of the alkali metals have been calculated on the long wave method using the Heine-Abarenkov lacal model potential with different exchange-correlation corrections. It is found that the use of an exchange correlation correction which satisfies the compressibility sum rule leads to a good agreement between the calculated and measured second order elastic constants of the alkali metals Na, K, Rb and Cs. The shear elastic constants however come out correct even if the compressibility sum rule is violated by the exchange-correlation correction. The third order elastic constants and the pressure derivatives of the second order elastic constants and the pressure derivatives of the second order elastic constants calculated on the HA local potential are lower than the experimental values at room temperature. The discrepancy is pronounced for the heavier alkali metals. Similar calculations using the Wallace potential for Li, Na and K and the Schneider-Stoll potential for Rb give the pressure derivative in good agreement with experiment. In view of the important role by the exchange correlation correction, Suzuki's results calculated without taking this correction into account can only be accepted with some reservation.     

单向复合纤维材料弹性系数的超声测量

本文将单向复合纤维材料视作六角对称晶体，认为其弹性特征由五个独立的弹性系数来描述。利用Christoffel方程，我们得到这一各向异性介质中不同方向的声波速度与弹性系数的关系．我们还建立了一套计算机控制的实验测量系统，通过它对纤维板样品中的声速进行测量，反演得到了五个独立的弹性系数C11、C33、C13、C14和C12。理论计算得到的声速和实验测量值吻合较好，表明我们的方法是有实用意义的。     

Elastic Energy Separation at Impact Loading of Cylindrical Bodies Made of Isotropic and Anisotropic Materials

Russian Physics Journal - The determination of the elastic wave propagation velocity using ultrasonic methods for a calculation of the elastic constants imposes constraints on the geometry of...     

Higher order elastic constants and generalized Gruneisen parameters of elastic waves and low temperature thermal expansion of gadolinium

Expressions for the higher order elastic constants are derived using the sublattice displacements to the second degree in strains. These expressions are used to obtain the higher order elastic constants and their pressure derivatives in gadolinium. The higher order elastic constants are used to find out the generalized Gruneisen parameters of the elastic waves propagating in different directions in gadolinium. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with the available reported values.     

Ultrasonic characterization of Cu-Al-Ni single crystals lattice stability in the vicinity of the phase transition

Measurements of elastic constants of the austenite phase when approaching the phase transformation either upon cooling or stressing is of the crucial interest for the shape memory alloy field. Acoustic properties (wave velocity and also attenuation changes) of the Cu-Al-Ni single crystal were investigated in situ during stress-induced martensitic transformation at constant (room) temperature. The parent austenite cubic lattice of the Cu-Al-Ni exhibits very high elastic anisotropy (anisotropy factor A approximately 12). The measurements were made using nine combinations of (i) applied uniaxial compression in a given crystal direction, (ii) the wave propagation and (iii) polarization vectors. The chosen configurations are sufficient for evaluation of all independent third order elastic constants (TOEC). The longitudinal modes were also measured by the immersion technique, using the transducer pair in a water tank installed on the testing machine. The device works as "a ultrasonic extensometer" measuring a transverse strain of the specimen. The dependencies of both natural and initial wave velocities on the applied stress may be evaluated. Three elastic constants of the stress-induced martensite were determined. The elastic properties were found to vary with the increasing stress above the Ms transformation temperature, which is interpreted as a precursor for the martensitic transformation. The onset of the transformation was additionally identified from the acoustic emission measurement.     

Third-order elastic constants and pressure derivatives of the second-order elastic constants of β-tin

The second- and third-order elastic constants and pressure derivatives of second-order elastic constants of tetragonal β-tin have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second- and third-order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The 12 non-vanishing third-order elastic constants and the six pressure derivatives of the second-order elastic constants in tetragonal β-tin are obtained in the present work and are compared with the available experimental values. The second-order elastic constant C₃₃ obtained in the present study is in reasonable agreement with the experimental values. The third-order elastic constants are generally one order of magnitude greater than the second-order elastic constants as expected of a crystalline solid. The third-order elastic constant C₃₃₃ is higher in magnitude than all other values. This shows a greater anharmonicity of β-tin along the c-axis direction of the crystal.