Third-order Elastic Constants Research Articles

Seismic scattering inversion for multiple parameters of overburden-stressed isotropic media

Subsurface reservoirs are compressed by overburden stress resulting from the gravity of overlying masses, and the resulting stress changes significantly affect the seismic reflection responses generated at the reservoir interfaces. Several exact reflection coefficient equations have been well established to delineate the role that in situ stress plays in altering the energy transition and amplitude of seismic reflection responses. These exact equations, however, cannot be effectively used in practice due to their intricate formulations and the difficult geophysical estimation for the third-order elastic constants (3oECs) embedded in reflection coefficients. Based on the theories of nonlinear elasticity and elastic wave inverse scattering, we derive an approximate seismic reflection coefficient equation for overburden-stressed isotropic media in terms of the P-wave modulus, shear modulus, density, and a defined stress-related parameter (SRP). The SRP is a combined quantity of elastic moduli, 3oECs, and overburden stress, which can be naturally treated as a dimensionless stress-induced anisotropy parameter. Its inclusion effectively eliminates the need for 3oECs information when using our equation to estimate the desired reservoir properties from seismic observations. By comparing our equation to the exact one, we confirm its validity within the moderate-stress range. Then, we introduce a Bayesian inversion approach incorporating the new reflection coefficient equation to estimate four model parameters. In our approach, the Cauchy and Gaussian distribution functions are used for the a priori probability and the likelihood distributions, respectively. The synthetic tests from two well-log data sets and a field example demonstrate that four parameters can be reasonably inverted using our approach with rather smooth initial models, which illustrates the feasibility of our inversion approach.

Nonlinear, elastic, piezoelectric, electrostrictive, and dielectric constants of lithium tantalate

Three third-order dielectric constants, 8 electrostrictive constants, 13 third-order piezoelectric constants, and 14 third-order elastic constants for lithium tantalate (LiTaO3) single crystals, which are mainly used in surface acoustic wave (SAW) devices, were determined as part of basic research to realize SAW devices with a low degree of nonlinearity. The third-order dielectric constants were determined by measuring the change in capacitance along selected directions when an alternating electric field was applied to the crystal. The electrostrictive constants were determined by measuring the change in capacitance when static stress was applied. Some of the piezoelectric constants were determined directly from the change in sound velocity due to the application of an alternating electric field, whereas others were obtained from the measured dielectric constants, electrostrictive constants, and the change in sound velocity due to an alternating electric field. In addition, the elastic constants (compliance) were determined using the three aforementioned determined nonlinear constants and the measured small-amplitude ultrasonic velocity change as a function of applied static stress. The measurements performed in the present study are more advanced than those reported for LiNbO3 single crystals in 1987 due to the adoption of the d-form nonlinear piezoelectric equation (instead of the e-form) and a higher degree of precision using dynamic measurement methods. The use of the d-form of the nonlinear piezoelectric equation allows many nonlinear constants to be determined independently from other nonlinear constants, eliminating measurement errors associated with these other constants. The d-form nonlinear constants obtained in the present study were converted to e-form constants to make them applicable to the analysis of nonlinear phenomena in piezoelectric devices.

Padé acoustoporoelasticity finite-difference simulation of elastic wave propagation in prestressed porous media

Abstract Acoustoelasticity has significant implications in seismic exploration, enabling the inference of geological structures and the properties of oil and gas reservoirs through rock physical acoustoelastic parameters, as well as elucidating the propagation of elastic waves in subsurface rocks. In traditional acoustoporoelastic numerical simulations, third-order elastic constants are derived from the Taylor expansion of the strain energy function. However, under conditions of high effective stress, the Taylor approximation can lead to divergent elastic wave velocities. To address this limitation, we propose employing the Padé approximation for the expansion of the strain energy function, as it provides superior accuracy compared to the Taylor approximation. By calibrating the Padécoefficients using experimental data for various rock types, the resultant acoustoelastic constants exhibit a reasonable theoretical limit on elastic wave velocities as effective stress increases. This method allows for a more precise characterization of velocity variations with stress, especially under high-pressure conditions. The acoustoporoelasticity numerical simulation is mainly limited to the 2D cases, so we consider confining prestressed mode in the 3D cases.

Elastic, thermophysical and ultrasonic investigation of tin monochalcogenides

This work explores the investigation of tin monochalcogenides i.e. SnX (X: S, Se, Te) at room temperature (300[Formula: see text]K). The mechanical, thermophysical and ultrasonic properties of SnX (X: S, Se, Te) have been evaluated using the computed values of second- and third-order elastic constants (SOECs and TOECs). The SOECs and TOECs have been obtained using Born–Mayer potential model at 0[Formula: see text]K and 300[Formula: see text]K. The brittleness behavior of tin monochalcogenides is detected by Pugh’s ratio. It is observed that phonon–phonon interaction mechanism is dominant in tin monochalcogenides leading to high ultrasonic attenuation. The obtained temperature dependence behavior of tin monochalcogenides is validated via comparison with available works of literature data in previous works done experimentally as well as theoretically by others.

Acoustoelastic characterization of aluminum plates using zero group velocity Lamb modes

Acoustoelasticity, a characteristic of material anharmonicity, gives rise to a link between wave propagation velocity and the stress state in materials. Ultrasonic techniques to monitor this coupling, particularly with high sensitivity and in a noncontact manner, can have widespread application both in the quantification of applied and residual stress and in the characterization of nonlinear material behavior through measurement of higher order elastic constants. Here, we use a laser ultrasonic technique to excite and detect zero group velocity (ZGV) Lamb wave resonances in aluminum plates under uniaxial loading. A laser line source is used to excite these resonances at different orientations with respect to the applied load and the signals are detected using an interferometer. The effects of stress and source orientation on ZGV resonance frequencies are validated using the theory of acoustoelastic Lamb wave propagation. In addition, a model-based inversion technique is used to extract Murnaghan’s third-order elastic constants from measurements of the stress dependence of the first two ZGV modes generated parallel and perpendicular to the applied load. Laser generation and detection of ZGV resonances is shown to be an effective and powerful approach for the noncontact and nondestructive acoustoelastic characterization of elastic waveguides.

Acoustoelastic characterization of plates using zero group velocity Lamb modes

Acoustoelasticity, a characteristic of material anharmonicity, gives rise to a link between wave propagation velocity and the stress state in materials. Ultrasonic techniques to monitor this coupling, particularly with high sensitivity and in a noncontact manner, can have widespread application both in the quantification of applied and residual stress and in the characterization of nonlinear material behavior through measurement of higher order elastic constants. Here, we use a laser ultrasonic technique to excite and detect zero group velocity (ZGV) Lamb wave resonances in aluminum plates under uniaxial loading. A laser line source is used to excite these resonances at different orientations with respect to the applied load, and the signals are detected using an interferometer. The effects of stress and source orientation on ZGV resonance frequencies are validated using the theory of acoustoelastic Lamb wave propagation. In addition, a model-based inversion technique is used to extract Murnaghan's third-order elastic constants from measurements of the stress dependence of the first two ZGV modes generated parallel and perpendicular to the applied load. Laser generation and detection of ZGV resonances is shown to be an effective and powerful approach for the noncontact and nondestructive acoustoelastic characterization of elastic waveguides.

Stress and frequency dependence of wave velocities in saturated rocks based on acoustoelasticity with squirt-flow dissipation

SUMMARY We perform seismic and ultrasonic measurements in carbonate and shaley sandstone samples as a function of differential pressure. The velocities show a strong frequency and pressure dependence. The dispersion disappears with increasing pressure and the squirt flow in turn inhibits the pressure dependence. To model these effects, we combine the Gurevich's squirt-flow model with the Mori–Tanaka scheme and the David Zimmerman model, and extend it with third-order elastic constants, to obtain a frequency-dependent acoustoelasticity model. Comparisons between measurements from this study and literature and modelling results show that the P-wave velocity increases non-linearly first and then nearly linearly, dominated by crack closure and acoustoelasticity, respectively. The pressure dependence of wave velocities is reduced by liquid substitution and further by the squirt-flow mechanism. The effects of fluid properties and crack closure on P-wave velocity decrease with differential pressure. The results will feed a new model and help better understanding the wave propagation in pre-stressed rocks at different scales.

Large difference of lattice thermal conductivity between thermoelectric compounds Cu3VSe4 and Tl3VSe4: Elasticity, anharmonicity and chemical bonding from first-principles calculation

The elasticity has a great impact on thermoelectric efficiency and working service life as thermoelectric device. In this work, the large difference of lattice thermal conductivity (κL) between thermoelectric materials Cu3VSe4 and Tl3VSe4 was investigated by calculating their second-order (Cij) and third-order elastic constants (Cijk). The mechanical properties and thermal expansion coefficients were also evaluated. These results show that Cij of Cu3VSe4 and Tl3VSe4 are relatively overall weak. Particularly, in contrast to Cu3VSe4, Tl3VSe4 displays larger negative Cijk, revealing its stronger intrinsic anharmonicity. The relatively smaller elasticity of Tl3VSe4 derives from its weaker Tl-Se and V-Se chemical bonds as compared with that of Cu3VSe4. Finally, the increased anharmonicity results in a non-linearly mechanical behavior and reducing the κL with the rise of temperature. The research provides insight into the relationship of elasticity, anharmonicity and thermal expansion and κL in view of elasticity and will be useful for designing and optimizing of thermoelectric materials with high performance.

Elastic and thermodynamic properties of B2-MgX intermetallics: A high throughput first-principles investigation

Precipitation strengthening is an effective approach in Mg alloys, and hence, it is worthwhile investigating the properties of the Mg-containing intermetallics. In the present work, the thermodynamic and mechanical properties of B2-MgXs (69 different elements) are calculated by high-throughput first-principles calculations. The static properties, including equilibrium volume, equilibrium energy, and bulk modulus, and thermodynamic properties, including linear thermal expansion coefficient (LTEC), are predicted. Meanwhile, the second-order elastic constants (SOECs) and third-order elastic constants (TOECs) are calculated using an efficient strain-stress method. The consistency between our results and the limited experimental or previous theoretic work confirms the high reliability of the present work. Using SOECs and equilibrium energy, the stability of MgXs is assessed from mechanics, energy and lattice dynamics, and 33 (out of 69) stable compounds are screened out. According to Young's and shear modulus, the MgXs are divided into four classes, and the first two classes present high modulus, high ductility, and reasonable anisotropy, equipping those compounds as promising candidates for precipitation strengthening in Mg alloys. In addition, the pressure and temperature dependence of SOECs is estimated using TOECs and LTEC. Furthermore, the present work provides abundant fundamental data for the design of new Mg alloys.

Temperature correction in acoustoelastic coefficient measurements

Acoustoelastic coefficients (AECs) of a material are used for determining third-order elastic constants (TOECs), measuring residual stress, and evaluating damage and deterioration by analyzing the change of wave velocity with stress. However, wave velocity measurements are highly sensitive to ambient or load-induced temperature changes. In this study, we proposed a temperature correction method that uses the measured temperature history and the temperature–velocity relationship of the test specimen. Acoustoelastic tests were performed in various ambient temperature conditions on three types of metals: 6063-T5 aluminum alloy (Al6063), 304 stainless steel (SS304), and 1018 low-carbon steel (S1018). Before temperature correction, the AEC measurements varied significantly, due to changes in the temperature of the specimen caused by both the ambient environment and mechanical loading. The results after the correction demonstrate that the proposed self-temperature correction method reduces temperature induced error, enabling highly accurate AEC measurements.

Anisotropic seismic velocity variations in response to different orientations of tidal deformations

SUMMARY Microcracks or micropores in rocks cause the elastic moduli to change with the applied strain owing to the nonlinear elasticity of the geomaterial, which causes temporal changes in the seismic wave velocity. Thus, variations in seismic wave velocity can be used as a proxy for understanding the strain or stress variations in the crust, which are crucial for figuring out the dynamics of the fault zones and volcanic domains. According to the theory of nonlinear elasticity, the second- and third-order elastic constants and strain tensors contribute to the strain derivative of seismic wave velocity changes. Although laboratory experiments have estimated third-order elastic constants for rock samples, the in situ values of those constants for the crust are difficult to obtain. In this study, seismic velocity changes were investigated in different directions of tidal deformations to provide constraints on the third-order elastic constants in the shallow crust by applying a seismic interferometry method to ambient noise records. We observed that negative velocity changes were of larger magnitude in the station-pair direction parallel to the tidal strain’s direction. Nonlinear elasticity in shallow crust may cause anisotropic velocity variations in response to tidal deformations. Our results highlight the use of velocity change measurements in different directions of tidal strain to constrain nonlinear elastic parameters on a field scale.

Theoretical investigation on the elastic and mechanical properties of high-entropy alloys with partial replacement of Sc in Hf0.25Ti0.25Zr0.25Sc0.25−x Al x (x ≤ 15 %)

Abstract The theoretical assessment of mechanical and elastic properties is used to analyze the distinctive properties of high entropy alloys (HEAs) at room temperature. Using Lennard–Jones potential model, the second order elastic constants (SOECs) and third order elastic constants (TOECs) have been determined for the HEAs Hf0.25Ti0.25Zr0.25Sc0.25−x Al x (x ≤ 15 %) in their hexagonal close-packed (hcp) phases. SOECs have been used to calculate mechanical constants, Poisson’s ratio, Pugh’s ratio, Kleinman’s parameter. In order to determine the anisotropic behaviour of the selected HEAs, the elastic anisotropy has also been computed at room temperature. All the HEAs under consideration have anisotropy parameters that are not equal to one, indicating anisotropic behaviour. Later, the Grüneisen parameters were estimated for the chosen HEAs Hf0.25Ti0.25Zr0.25Sc0.25−x Al x (x ≤ 15 %) along longitudinal and shear modes of wave propagation. Analysis of the research results reveals the inherent properties of HEAs.

First-principles calculation of the pressure derivative of the bulk modulus from second- and third-order elastic constants

First-principles calculation of the pressure derivative of the bulk modulus from second- and third-order elastic constants

Acoustoelastic Mori-Tanaka model for third-order elastic constants of fractured rocks

Insights into the stress-dependent elastic moduli of fractured rocks are important in monitoring geopressure and tectonic stress. This can be addressed by the theory of acoustoelasticity that describes elastic nonlinearity due to the cubic strain-energy function through third-order elastic constants (TOECs). The acoustoelastic TOECs are strictly valid for an isotropic homogeneous medium. The extension to fractured rocks remains largely unaddressed. A group of fractures (ellipses) is embedded into a homogeneous background medium subjected to a uniform confining pressure. We formulate an acoustoelastic Mori-Tanaka (MT) model for the effective TOECs of fractured rocks which can be defined as the weighted average of individual TOECs between the background and fractures in terms of respective volume fractions and stiffnesses. For homogeneously distributed fractures, the stress-induced strains are isotropic in the background and in the fractures. For aligned fractures, the resulting anisotropic strains over the background and fractures make the rock a vertical transverse isotropic medium. The effective elastic moduli and TOECs of fractured rocks depend on background elastic moduli, fracture contents, aspect ratios, and fracture orientations. The prestressed aligned fractures induce nonlinear effects on the effective TOECs that increase with increasing fracture contents and decreasing aspect ratios. We validate the acoustoelastic MT model by experimental data and investigate its limitation by comparing it with finite-element simulations.

Evaluation of Early-Stage Fatigue Damage in Metal Plates Using Quasi-Static Components of Low-Frequency Lamb Waves

Abstract Nonlinear Lamb waves including second harmonic and acoustic-radiation-induced quasi-static components (QSC) have a potential for accurately evaluating early-stage fatigue damage. Most previous studies focus on second-harmonic-based techniques that require phase velocity matching and are hard to isolate interferences from ultrasonic testing systems. The aforementioned requirement and deficiency limit applications of the second-harmonic-based techniques. In this study, a QSC-based technique of low-frequency Lamb waves is proposed for early-stage fatigue damage evaluation of metal plates, which does not need to require phase velocity matching and can remove interferences from ultrasonic testing systems. Both in simulations and in experiments, the primary Lamb wave mode at a low frequency that meets approximate group velocity matching with the generated QSC is selected. In finite element simulations, different levels of material nonlinearities by changing the third-order elastic constants are used to characterize levels of fatigue damage. Numerical results show that the magnitude of the generated QSC pulse increases with the levels of fatigue damage. Early-stage fatigue damage in aluminum plates with different fatigue cycles is further experimentally evaluated. The generated QSC pulse is extracted from received time-domain signals using the phase-inversion technique and low-pass digital filtering processing. The curve of the normalized relative acoustic nonlinearity parameter versus the cyclic loading number is obtained. Numerical simulations and experimental results show that the early-stage fatigue damage in aluminum plates can effectively be evaluated using the QSC generated by low-frequency Lamb waves.

Peridynamic modeling of nonlinear surface acoustic waves propagating in orthotropic materials

Due to the elastic nonlinearity of the material, high-amplitude surface acoustic waves undergo nonlinear evolution during propagation and may lead to material failure. To enable the acoustical quantification of material nonlinearity and strength, a comprehensive understanding of this nonlinear evolution is necessary. This paper presents a novel ordinary state-based nonlinear peridynamic model for the analysis of the nonlinear propagation of surface acoustic waves and brittle fracture in anisotropic elastic media. The relationship between seven peridynamic constants and second- and third-order elastic constants is established. The capability of the developed peridynamic model has been demonstrated by predicting surface strain profiles of surface acoustic waves after propagating in the silicon (111) plane and the 〈112¯〉 direction. On this basis, the nonlinear wave-induced spatially localized dynamic fracture is also studied. The numerical results reproduce the main features of nonlinear surface acoustic waves and fracture observed in experiments.

Evaluation of material surface properties using one-way collinear mixing Rayleigh waves

Nonlinear Rayleigh waves are useful for material surface damage detection. Traditional non-destructive testing (NDT) techniques based on higher harmonics of Rayleigh waves are inevitably interfered with by the nonlinearity of measurement systems. To overcome this disadvantage, an approach based on one-way collinear mixing Rayleigh waves (MRWs) is proposed to evaluate the material surface properties. A wedge with two different incidence angles for the excitations of transverse and longitudinal waves is designed to generate one-way collinear MRWs. The one-way collinear MRWs can be generated on the whole surface of the material, and propagate along the propagation way under one excitation. Numerical simulations of one-way collinear MRWs under different surface properties, which are characterized using different third-order elastic constants, are studied on a semi-infinite aluminum. Experiments are also conducted on the aluminum specimen, whose surface is corroded by hydrochloric acid so as to change the material surface properties. Both numerical and experimental results show that the one-way collinear MRWs are generated effectively, and the defined relative acoustic nonlinearity parameter increases with the propagation distance as well as the severity of the material surface damages. The results verify the effectiveness of the proposed MRW-based method and prove that it would be a promising NDT technique for the accurate evaluation of material surface properties.

First-principles study of electronic and optical properties of novel 2D TiOS monolayer and bilayer—Dimensionality reduction opens up a band gap in TiOS

The finding of advanced functional materials with superior properties to existing ones to develop cutting-edge technologies for societal advancement is indispensable. We identify a new two-dimensional (2D) TiOS, which opens up a band gap due to the lowering of the dimensionality employing first-principles computations within the framework of density functional theory (DFT) and beyond. Electronic structure estimations reveal that bulk TiOS is metal whereas the 2D-TiOS possesses a band gap of ∼4.5 eV within GW approximation, which is higher than the band gap of 2D-InOF and 2D transition metal dichalcogenides. The computed phonon dispersion curves show that the 2D-TiOS is dynamically stable. The 2D-TiOS has a very high absorption coefficient in the 0–50 eV. Third-order elastic constants (TOECs) of this 2D-TiOS are achieved using DFT and within the adiabatic-connection fluctuation–dissipation theorem in the random phase approximation. Surprisingly, we find a band overlap, indicating that TiOS/TiOS bilayer is a metal. If the incident light frequency surpasses the plasma frequency (60.00 eV), and then the bilayer turns out to be transparent. Our findings suggest that this 2D sheet is a promising alternative for nanotechnology and optoelectronic devices.

3D acoustoelastic FD modeling of elastic wave propagation in prestressed solid media

AbstractSeismic exploration of deep oil/gas reservoirs involves the propagation of seismic waves in high-pressure media. Traditional elastic wave equations are not suitable for describing such media. The theory of acoustoelasticity establishes the dynamic equation of wave propagating in prestressed media through constitutive relation using third-order elastic constants. Many studies have been carried out on numerical simulations for acoustoelastic waves, but are mainly limited to 2D cases. A standard staggered-grid (SSG) finite-difference (FD) approach and the perfectly matched layer (PML) absorbing boundary are combined to solve 3D first-order velocity-stress equations of acoustoelasticity to simulate wave propagating in 3D prestressed solid medium. Our numerical results are partially validated by plane-wave analytical solution through the comparison of calculated and theoretical P-/S-wave velocities as a function of confining prestress. We perform numerical simulations of acoustoelastic waves under confining, uniaxial and pure-shear prestressed conditions. The results show the stress-induced velocity anisotropy in acoustoelastic media, which is closely related to the direction of prestresses. Comparisons to seismic simulations based on the theory of elasticity illustrate the limitation of conventional elastic simulations for prestressed media. Numerical simulations prove the significant effect of prestressed conditions on seismic responses.

Thermo-acoustoelastic effect of Rayleigh wave: Theory and experimental verification

Previous studies showed that the thermally-induced ultrasonic bulk wave velocity change could be used to measure acoustoelastic coefficients and third-order elastic constants of elastic materials. This method is naturally immune from the ambient temperature effect and has improved sensitivity and a simpler test setup than the conventional acoustoelastic test. However, Rayleigh wave is preferred for thick components or structures with only one accessible surface. In this work, the thermo-hyperelastic constitutive equation, along with acoustoelastic theory, is used to derive the expression of the thermo-acoustoelastic coefficient (TAEC) of Rayleigh wave. The numerical relationship between the TAEC of Rayleigh wave and Murnaghan constants (l, m and n) are given for common metals. The TAEC expressions for Rayleigh wave and shear wave are similar, and both are dominated by the constant m. The TAEC of Rayleigh wave was measured on an aluminum 6061 specimen using the thermal modulation experiment in a temperature range of 22 ∼35 °C. The measured TAEC shows good agreement with the theoretical calculation. Then the third-order elastic constants were calculated based on TAECs of bulk waves and Rayleigh wave.