Determination Of Third-order Elastic Constants Research Articles

Determination of third-order elastic constants using change of cross-sectional resonance frequencies by acoustoelastic effect

This investigation presents a new measurement technique for third-order elastic constants using the acoustoelastic effect. The cross-sectional resonance frequencies of a cylindrical rod are measured while the rod is stressed under tensional load. The Murnaghan's third-order elastic constants of the rod are determined by matching experimentally obtained resonance frequencies with those numerically obtained by a finite element model. Electromagnetic acoustic transducers (EMATs) are employed to measure cross-sectional resonance frequencies, enhancing the repeatability to minimize the variation of the contact condition between the transducer and the specimen. The present method was applied to specimens made of carbon steel (JIS-S20C) and aluminum (Al7075-T651), whose third-order elastic constants were calculated. It should be emphasized that the estimated Murnaghan's constants obtained by the proposed method showed very small coefficients of variation (CVs), less than 6.54%, for repeated tests conducted on several specimens, showing excellent repeatability and reproducibility that is almost unattainable by conventional methods using wave speeds or higher harmonics of an input wave.

Experimental Determination of Third-Order Elastic Constants of Diamond

To determine the nonlinear elastic response of diamond, single crystals were shock compressed along the [100], [110], and [111] orientations to 120 GPa peak elastic stresses. Particle velocity histories and elastic wave velocities were measured by using laser interferometry. The measured elastic wave profiles were used, in combination with published acoustic measurements, to determine the complete set of third-order elastic constants. These constants represent the first experimental determination, and several differ significantly from those calculated by using theoretical models.

Second harmonic generation of shear waves in crystals

Nonlinear self-interaction of shear waves in electro-elastic crystals is investigated based on the rotationally invariant state function. Theoretical analyses are conducted for cubic, hexagonal, and trigonal crystals. The calculations show that nonlinear self-interaction of shear waves has some characteristics distinctly different from that of longitudinal waves. First, the process of self-interaction to generate its own second harmonic wave is permitted only in some special wave propagation directions for a shear wave. Second, the geometrical nonlinearity originated from finite strain does not contribute to the second harmonic generation (SHG) of shear waves. Therefore, unlike the case of longitudinal wave, the second-order elastic constants do not involve in the nonlinear parameter of the second harmonic generation of shear waves. Third, unlike the nonlinearity parameter of the longitudinal waves, the nonlinear parameter of the shear wave exhibits strong anisotropy, which is directly related to the symmetry of the crystal. In the calculations, the electromechanical coupling nonlinearity is considered for the 6 mm and 3 m symmetry crystals. Complement to the SHG of longitudinal waves already in use, the SHG of shear waves provides more measurements for the determination of third-order elastic constants of solids. The method is applied to a Z-cut lithium niobate (LiNbO3) crystal, and its third-order elastic constant c444 is determined.

Third-order elastic constants determination in soda–lime–silica glass by Brillouin scattering

Brillouin light scattering allows the measurement of sound velocity and elastic moduli in transparent materials. The ability to select a small scattering volume and to use specific scattering configurations gives important information about the gradient and anisotropy of mechanical properties. Brillouin experiments are often used to measure the second-order elastic constants. When a high accuracy in frequency measurements is achieved, Brillouin scattering may allow the determination of third-order elastic constants in pre-stressed media. Hence, Brillouin scattering provides in principle, a method for the analysis of stress fields in tempered glasses. In order to validate the technique, samples of float soda-lime–silica glass submitted to controlled stresses by four-point flexion were investigated. The results show the expected profile for the velocity of sound waves propagating in directions parallel and perpendicular to the surface of the samples, respectively. They allow the determination of several third-order elastic constants of the investigated glass. This technique was applied to several samples of tempered glass corresponding to different values of the surface stress. The main result is the observation of the expected general trend, namely, through the thickness of the sample, a parabolic variation of the sound velocity whose amplitude increases with the magnitude of the surface stress.

Material nonlinearities in quartz determined by the transit-time method using direct current field interactions

The experimental data [J. Appl. Phys. 60, 1465–1471 (1986)] on dc field interactions with quartz determined by the transit-time method can provide 17 nonlinear parameters (including 5 fundamental constants) as an independent contribution toward the determination of the third-order electromechanical nonlinearities of quartz. The least-squares estimates of these parameters have been obtained. Their standard errors range from several percent to over 100%. With the assistance of the third-order nonlinear constants determined earlier, 17 estimates of the fundamental nonlinear constants have been obtained from the same data set. They include all electroelastic, electrostrictive, and the third-order permittivity constants of quartz. Their dependence of the third-order elastic constants is fully disclosed. Rigorous interpretation of the meaning of earlier results removes seeming conflicts between the current and the earlier estimates. Reasons are given why the type of experimental data analyzed in this paper is not suited to the determination of third-order elastic constants.

Determination of third-order elastic constants from stress-shifted resonance frequencies, observed by diffraction of light. Examples: aluminium alums of K, NH4, Cs and CH3NH3

Resonance frequencies of thick plane-parallel plates are easily detected by diffraction of light with an accuracy better than 1 p.p.m. The method, which has been earlier employed for the determination of thermoelastic properties, proved to be sufficiently sensitive also for the measurement of stress-induced shifts of resonance frequencies, and is therefore suited for the determination of third-order elastic constants. The technique is easier to handle and much less expensive than other methods in use today. The complete sets of third-order elastic constants for KAl(SO4)2.12H2O, NH4Al(SO4)2.12H2O, CsAl(SO4)2.12H2O and CH3NH3Al(SO4)2.12H2O have been measured. All constants possess negative values. These alums exhibit a behaviour similar to that of the structurally related caesium halides, with the exception of ammonium alum which shows anomalously small constants.

Theory of Ultrasonic Pulse Measurements of Third-Order Elastic Constants for Cubic Crystals

The equations of motion for an elastic nonisotropic solid are reduced to a form useful for determination of third-order elastic constants by means of ultrasonic pulse distortion measurements. Values of the coefficients in the reduced equations of motion are tabulated for two cubic materials. The tables are used to show that for cubic materials one should be able to measure C111, C112, and C166 with reasonable accuracy. An argument is given which shows that the equations of motion for a single plane wave in a cubic crystal depend on the five parameters C111, C112, C166, (2C144+C123), and (½C144+C456) instead of all six third-order elastic constants.

Interactions Between Elastic Waves in an Isotropic Solid

Interactions between elastic waves in an isotropic solid are studied in the elastic-continuum approximation. The analysis is carried out completely in a wave-packet formalism, i.e., scattering of wave packets by wave packets. The maximum amplitude (or intensity) and width of the scattered wave packet are expressed in terms of the maximum amplitudes, frequencies, widths, polarizations, and relative propagation directions of the primary-wave packets. The polarization relations and frequency ranges for the allowed interaction processes are obtained; these are essentially identical to the ones given by Jones and Kobett. The results are shown to be in good order-of-magnitude agreement with the experiments of Rollins. Possible application of elastic-wave scattering to the determination of third-order elastic constants is discussed.