Third-order Elastic Constants Research Articles

First-principles calculations to investigate electronic band structure, optical and mechanical properties of new CaFCl monolayer

Two-dimensional materials present new opportunities to venture into largely unexplored areas of material property space. They are appropriate for several electronic powers, photonics, and sunlight device applications. In this framework, leveraging density functional theory and even beyond, we investigate the structural, vibrational, electronic, optical, and elastic properties of novel CaFCl thin film. We have found that the CaFCl sheet is dynamically stable because its spectrum contains no imaginary frequencies. The direct band gap energy value of our GW findings is around 9.10 eV, which really is higher than the band gaps of ScOI and InOF oxyhalides thin films. We have found that the external electric fields immediately affect the structure of the layer. The band gap of the CaFCl structure is reduced rapidly by increasing the strength of the electric field. Additionally, our Bethe–Salpeter theory calculations demonstrate that when the excitation light rate is greater than the resonance frequency (8.00 eV), the substance becomes translucent. We have also used the method of homogeneous deformation, which is coupled with density functional theory and random phase approximation total-energy computations to produce a comprehensive scheme for studying the second-and third-order elastic constants for the CaFCl layer. A polynomial fit to the estimated energy-strain connection yielded these constants. Our results reveal that this 2D monolayer is a promising material for nanophotonics and optoelectronics.

Pressure Dependent Elastic and Thermo-Physical Properties of U Ternaries with ZrNiAl Structure Compounds

bstract: The Thermophysical and ultrasonic aspects of U- ternaries are examined theoretically under high pressure. In order to determine the second- and third-order elastic constants of U- ternaries (URhAl and ThRhAl) compounds under pressure conditions (0-45GPa), the potential model technique has been used. The assessed second-order elastic constants are used along with the pressure-dependent ultrasonic velocities, thermal relaxation time, and other significant and important thermophysical data to determine the orientation. It is observed that thermal relaxation time and mechanical & thermal properties exhibit an increasing trend with increasing pressure for U-ternaries URhAl and ThRhAl compounds. The effect of increasing pressure on the URhAl and ThRhAl compounds shows up as an increase in elastic constants, ultrasonic velocities, ultrasonic attenuation, and Debye temperature. The ratio B/G increases with pressure which indicates that the materials are ductile in nature. We have used the approximate values of SOECs, density, and lattice properties in calculating the melting point, ultrasonic velocity, and thermal conductivity.

Third-Order Elastic Constants of Germanium and Silicon using Adiabatic-Connection Fluctuation-Dissipation Theorem in Random Phase Approximation

Third-Order Elastic Constants of Germanium and Silicon using Adiabatic-Connection Fluctuation-Dissipation Theorem in Random Phase Approximation

Decoupled acoustoelasticity equations and stable FD modeling for wave propagation in prestressed media

Seismic exploration of deep oil/gas reservoirs involves wave propagation in high-pressure media. Acoustoelasticity with the third-order elastic constants can account for nonlinear strain responses due to stresses of finite magnitude. Since current seismic exploration depends primarily on P waves, further development in prospecting technology to handle deep reflection data will benefit from the ability to decouple the elastic waves from high-pressure deep formations. We decouple the first-order velocity-stress acoustoelasticity equations for wave propagation under isotropic (confining) and anisotropic (uniaxial and pure shear) prestress conditions. A rotated staggered-grid finite-difference (RSG-FD) method is used to solve the decoupled acoustoelasticity equations with an unsplit convolutional perfectly matched layer (CPML) absorbing boundary. Numerical examples demonstrate the significant impact of prestressed conditions on seismic responses in both phase and amplitude. The stress-induced seismic anisotropy with orthotropic characteristics is strongly related to the orientation of prestresses. We compare the coupled and the decoupled wavefields to validate the proposed decoupling method. Investigation of the influence of prestresses on the acoustoelastic decoupling illustrates that the isotropic prestress does not affect the decoupling, but the anisotropic prestresses significantly affect the decoupling accuracy.

EFFECT OF PRESSURE ON ELASTIC CONSTANTS AND RELATED PROPERTIES OF RARE-EARTH INTERMETALLIC COMPOUND TBNIAL

The Lennard-Jones potential approach is used to investigate the effect of pressure on the ultrasonic and elastic properties of the rare-earth ternary TbNiAl intermetallic compound. The second- and third-order elastic constants of TbNiAl are considered using the potential model. The pressure-dependent higher-order elastic constants are studied, and it is observed that the elastic constants of the TbNiAl compound increased monotonously with pressure. The hexagonal TbNiAl compound is mechanically stable up to the pressure 20 GPa according to the Born elastic stability criteria. The Voigt-Reuss-Hill approach is used to compute such elastic parameters as Young’s modulus, bulk modulus, Poisson’s ratio, and shear modulus in the pressure range 0-45 GPa. Hardness, melting temperature, and anisotropy are also determined for the intermetallic TbNiAl compound. The pressure-dependent velocities and attenuation of ultrasonic waves in this ternary compound are evaluated. The computation results are also satisfactory in estimating the Debye temperature and thermal conductivity Kmin under different pressure. It is observed that TbNiAl has a significant anisotropy at zero pressure, which becomes stronger as the pressure increased. This ternary compound behaves as its purest form at higher pressure and is more ductile, which is demonstrated by the minimum attenuation.

First-principles calculations to investigate third-order elastic constant, anharmonicity and temperature dependent second elastic constant of thermoelectric materials Cu3MSe4(M = V and Nb)

The research on material‘s mechanics is critical in thermoelectric. Herein, we study the second(Cij), third(Cijk) order elastic constants, anisotropic and pressure and temperature dependent elasticity of Cu3MSe4(M = V and Nb) utilizing density functional theory. The elasticity of Cu3MSe4(M = V and Nb) is anisotropic. In addition, the calculated Cij suggests that these materials show relatively weak mechanical properties as thermoelectric device, and the ratio of bulk modulus(B) versus shear modulus(G) suggest that they are brittle. Cijk and pressure induced Cij reveal that Cu3NbSe4 has stronger anharmonic strength than that of Cu3VSe4. Finally, the thermal expansions are evaluated via quasi-harmonic approximation, and the temperature dependent elastic constants Cij indicate that Cu3NbSe4 has weak temperature dependent than that of Cu3VSe4. This research is dedicated to the design of thermoelectric devices and discovering novel thermoelectric materials with high performance in view of mechanical properties.

MyElas: An automatized tool-kit for high-throughput calculation, post-processing and visualization of elasticity and related properties of solids

Elasticity is one of the most fundamental mechanical properties of solid. In high-throughput design of advanced materials, there is an imperative demand for the capability to quickly calculate and screen a massive pool of candidate structures. A fully automatized pipeline with minimal human intervention is the key to provide high efficiency to achieve the goal. Here, we introduce a tool-kit MyElas that aims to address this problem by forging all pre-processing, elastic constant and other related property calculations, and post-processing into an integrated framework that automatically performs the assigned tasks to drive data flowing through parallelized pipelines from input to output. The core of MyElas is to calculate the second and third order elastic constants of a solid with the energy-strain method from first-principles. MyElas can auto-analyze the elastic constants, to derive other related physical quantities. Furthermore, the tool-kit also integrates a visualization function, which can, for example, plot the spatial anisotropy of elastic modulus and sound velocity of monocrystalline. The validity and efficiency of the toolkit are tested and bench-marked on several typical systems. Program summaryProgram title: MyElasCPC Library link to program files:https://doi.org/10.17632/jbz2s9xsmv.1Code Ocean capsule:https://codeocean.com/capsule/4149434Licensing provisions: GNU General Public License 3Programming language: Python 3.XExternal routines/libraries: Numpy [1], Spglib [2], Matplotlib [3]Nature of problem: Through the first-principal calculation, the second-order and third-order elastic constants of solid materials are automatically calculated, and the post-processing and visualization of single crystal and polycrystalline physical properties are carried out.Solution method: Firstly, the required strain structure is automatically generated through the space group of materials and the corresponding strain matrix. Secondly, the energy of the structure is calculated by first-principles calculation software such as VASP [4]. Thirdly, the relationship between energy and strain is polynomial fitted to deduce the elastic constant of the material. Fourth, automatically derive the relevant physical properties. Finally, the distribution of mechanical modulus and sound velocity in spherical space is visualized.Additional comments including restrictions and unusual features: Many modules of the software can be embedded into other software through simple modification, such as visualization module. The software supports finite temperature calculation with electronic temperature. In addition, the software supports the derivation of corresponding elastic constants from phonon spectrum data generated by Phonopy [5], Alamode [6] and TDEP [7] software through long-wave limit approximation and Christoffel equation.

Poro-acoustoelasticity finite-difference simulation of elastic wave propagation in prestressed porous media

Insights into wave propagation in prestressed porous media are important in geophysical applications, such as monitoring changes in geo-pressure. This can be addressed by poro-acoustoelasticity theory, which extends the classical acoustoelasticity of solids to porous media. The relevant poro-acoustoelasticity equations can be derived from anisotropic poroelasticity equations by replacing the poroelastic stiffness matrix with an acoustoelastic stiffness matrix consisting of second-order and third-order elastic constants. The theory considers the poroelasticity equations to be nonlinear due to the cubic strain-energy function with linear strains under finite-magnitude prestresses. A rotated staggered-grid finite-difference method with an unsplit convolutional perfectly matched layer absorbing boundary is used to solve a first-order velocity-stress formulation of poro-acoustoelasticity equations for elastic wave propagation in prestressed porous media. Numerical solutions are partially verified by computing the velocities of fast P wave, slow P wave, and S wave as a function of hydrostatic prestress and are compared with the exact values. Numerical simulations of wave propagation are carried out for the model of poro-acoustoelastic homogeneous space under three states—prestress confining (hydrostatic), uniaxial, and pure shear—and for the model of two poro-acoustoelastic homogeneous half-spaces in the planar contact under confining (hydrostatic) prestress. The resulting wavefield snapshots show fast P-wave, slow P-wave, and S-wave propagations in poro-acoustoelastic media under loading prestresses, which illustrate that the stress-induced velocity anisotropy is of orthotropy strongly related to the orientation of prestresses. These examples demonstrate the significant impact of prestressing conditions on seismic responses in velocity and anisotropy.

Effects of fracture-surface geometries on the third-order acoustoelastic constants for aligned fluid-saturated fractures

SUMMARYThe pressure sensitivity of stiffness in fractured rocks is closely related to fracture-surface geometries. The resulting stress-dependence of stiffness can be represented by the third-order elastic constants (3oECs). Fracture surfaces are generally rough at various scales, and can significantly affect the 3oECs of pre-stressed fractures as well as the wave-induced fluid flow (WIFF) induced by the Biot slow P-wave between fractures and the background medium. The WIFF usually depends on the fracture width relative to the slow P-wavelength and the fracture-surface roughness. We generate various fracture-surface geometries at different scales of random roughnesses parametrized by the surface standard deviation (SSD) of fracture-surface heights. With theoretical analyses and numerical simulations, we investigate the effect of fracture-surface geometries on the stress- and frequency-dependent stiffness through the 3oECs for pre-stressed rocks with aligned fractures. For the elastic wave in the low-frequency regime of Biot theory with the fracture scale much less than the wavelength, the induced WIFF significantly enhances the effect of fracture-surface geometries on the 3oECs and P- and S-wave moduli. The stiffness of fractured rocks increases with increasing SSDs, yielding a high sensitivity to pre-stresses. Toward the high-frequency limit, however, the fluid diffusion between fractures and the porous background decreases, which reduces the influence of fracture-surface roughnesses with the 3oECs much less than that in the low-frequency regime. The resulting P-wave modulus of aligned fluid-saturated fractures approximates to the background value.

An iterative method to evaluate one-dimensional pulsed nonlinear elastic wavefields and mixing of elastic waves in solids

Over the last 15 years, literature on nondestructive testing has shown that the generation of higher harmonics and nonlinear mixing of waves could be used to obtain the nonlinearity parameters of an elastic medium and thereby gather information about its state, e.g., aging and fatigue. To design ultrasound measurement setups based on these phenomena, efficient numerical modeling tools are needed. In this paper, the iterative nonlinear contrast source method for numerical modeling of nonlinear acoustic waves is extended to the one-dimensional elastic case. In particular, nonlinear mixing of two collinear bulk waves (one compressional, one shear) in a homogeneous, isotropic medium is considered, taking into account its third-order elastic constants ( A, B, and C). The obtained results for nonlinear propagation are in good agreement with a benchmark solution based on the modified Burgers equation. The results for the resonant waves that are caused by the one-way and two-way mixing of primary waves are in quantitative agreement with the results in the literature [Chen, Tang, Zhao, Jacobs, and Qu, J. Acoust. Soc. Am. 136(5), 2389-2404 (2014)]. The contrast source approach allows the identification of the propagating and evanescent components of the scattered wavefield in the wavenumber-frequency domain, which provides physical insight into the mixing process and explains the propagation direction of the resonant wave.

Acoustoelastic FD Simulation of Elastic Wave Propagation in Prestressed Media

Insights into wave propagation in prestressed media are important in geophysical applications such as monitoring changes in geo-pressure and tectonic stress. This can be addressed by acoustoelasticity theory, which accounts for nonlinear strain responses due to stresses of finite magnitude. In this study, a rotated staggered grid finite-difference (RSG-FD) method with an unsplit convolutional perfectly matched layer absorbing boundary is used to solve the relevant acoustoelastic equations with third-order elastic constants for elastic wave propagation in prestressed media. We partially verify our numerical simulations by the plane-wave theoretical solution. Comparisons of theoretical and calculated wave velocities are conducted for both P-wave and S-wave as a function of hydrostatic prestresses. We discuss several aspects of the numerical implementation. Numerical acoustoelasticity simulations for wave propagation in single- and double-layer models are carried out under four states of prestresses, confining, uniaxial, pure-shear, and simple-shear. The results display the effective anisotropy of elastic wave propagation in acoustoelastic media, illustrating that the prestress-induced velocity anisotropy is of orthotropic features strongly related to the orientation of prestresses. These examples demonstrate the significant impact of prestressed conditions on seismic responses in both phase and amplitude.

Enhancing efficiency and scope of first-principles quasiharmonic approximation methods through the calculation of third-order elastic constants

A novel computational strategy is presented to calculate from first principles the coefficient of thermal expansion and the elastic constants of a material over meaningful intervals of temperature and pressure. This strategy combines a novel implementation of the quasiharmonic approximation to calculate the isothermal-isochoric linear and nonlinear elastic constants of a material, with elementary equations of nonlinear continuum mechanics. Our implementation of the quasiharmonic approximation relies on finite deformations, the use of nonprimitive supercells to describe a material, a recently proposed technique to calculate generalized mode Gr\"uneisen parameters, and the numerical differentiation of the stress tensor to calculate both second- and third-order elastic constants. The combination of this method with nonlinear continuum mechanics is shown to yield accurate predictions of lattice parameters and linear elastic constants of a material over finite intervals of temperature and pressure, at the cost of calculating isothermal second- and third-order elastic constants for a single reference state. Here, the validity and limits of our novel methods are assessed by carrying out calculations of MgO based on classical interatomic potentials. To demonstrate potential, our methods are then used in conjunction with a density functional theory approach to calculate thermal expansion and elastic properties of silicon, lithium hydrate, and graphite.

A new acoustoelastic model for velocity-hydrostatic-pressure in rocks based on general acoustoelastic theory for elastic solids

Acoustoelastic model based on classic acoustoelastic theory of solids fails to describe rock acoustoelasticity, where wave velocities in rocks has complex non-linear relations with the applied hydrostatic pressure. In order to extend acoustoelastic model into better prediction for rock acoustoelasticity, a new acoustoelastic model is presented based on general acoustoelasticity of elastic solids having complicated constitutive law of rocks, which combines third-order elastic constants of compact rocks with the exponential terms for the influence of pore/cracks in rocks. The data for P- and S-wave velocities of diabase and greywacke samples subjected to the pressure up to 400 MPa are taken as examples to validate the presented model. The results show that the presented model could not only predict rock acoustoelasticity but also be linked with frequently-used empirical model and meso-mechanical model. The intrinsic connection among mechanical parameters, acoustical parameters, and meso-mechanical parameters could be established clearly, which will benefit the determination of high-resolution constitutive law and the compliant porosity by the measured ultrasonic-wave velocities of rock samples.Article HighlightsGeneral acoustoelasticity of elastic solids having complicated constitutive law yields a new acoustoelastic model of rock acoustoelasticity.New model predicts better non-linear characteristics of rock acoustoelasticity.New model benefits for high-resolution evaluation of constitutive law and meso-mechanical parameters for rocks.

Guided acoustic waves at periodically structured edges: Linear modes and nonlinear generation of Lamb and surface waves

Acoustic waves are investigated which are guided at the edge (apex line) of a wedge-shaped elastic body or at the edge of an elastic plate. The edges contain a periodic sequence of modifications, consisting either of indentations or inclusions with a different elastic material which gives rise to high acoustic mismatch. Dispersion relations are computed with the help of the finite element method. They exhibit zero-group velocity points on the dispersion branches of edge-localized acoustic modes. These special points also occur at Bloch-Floquet wavenumbers away from the Brillouin zone boundary. Deep indentations lead to flat branches corresponding to largely non-interacting, Einstein-oscillator like vibrations of the tongues between the grooves of the periodic structure. Due to the nonlinearity of the elastic media, quantified by their third-order elastic constants, an acoustic mode localized at a periodically modified edge generates a second harmonic which partly consists of surface and plate modes propagating into the elastic medium in the direction vertical to the edge. This acoustic radiation at the second-harmonic frequency is investigated for an elastic plate and a truncated sharp-angle wedge with periodic inclusions at their edges. Unlike nonlinear bulk wave generation by surface acoustic waves in an interdigital structure, surface and plate mode radiation by edge-localized modes can be visualized directly in laser-ultrasound experiments.

Динамика решетки и нелинейный отклик ОЦК-титана на деформации при высокой температуре в методе ab initio молекулярной динамики

The lattice dynamics and the response to deformations of the bcc titanium lattice at high temperatures are studied using the method of ab initio molecular dynamics. The results of calculation of phonon spectra using the harmonic Hamiltonian and the Hamiltonian containing third-order force constants are compared. It is shown that the lattice dynamics of bcc titanium cannot be correctly described without taking into account the anharmonic terms. Stress-strain curves are calculated for a given deformation scheme η1= η11= η, η6=2η12= η. For this deformation scheme, analytical expressions are given for the dependence of the nonzero components of the stress tensor on the magnitude of the deformation, taking into account the elastic constants up to the fourth order inclusive. All second-order elastic constants, three third-order elastic constants, and four fourth-order elastic constants of bcc titanium are calculated.

Pressure-Dependent Elastic, Mechanical, and Ultrasonic Analysis of CaAuBi Compound

The Lennard-Jones potential model is used to investigate the effects of pressure on the elastic and ultrasonic properties of CaAuBi half-Heusler compound. Potential model technique approaches is used to evaluate second- and third-order elastic constants of CaAuBi compound at various pressures (0-15 GPa). The pressure dependence of elastic constants is studied and it has been observed that the elastic constants of the half-Heusler CaAuBi compound increase monotonically as pressure is increased. The hexagonal half-Heusler CaAuBi compound is mechanically stable at different pressures according to Born's elastic stability criteria. The Voigt-Reuss-Hill method was used to compute elastic parameters such as Young's modulus Y, shear modulus G, bulk modulus B, and Poisson's ratio ν under various pressures. For the provided pressure range, the second-order elastic constants were also utilized to determine ultrasonic velocities along with z-axis at various angles. The half-Heusler CaAuBi compound's hardness, ultrasonic attenuation, melting temperature, and anisotropy are also determined. The computation have been also satisfactory in estimating the thermal conductivity k(min) and Debye average velocity under varied pressure. Keywords: elastic properties, thermal conductivity, ultrasonic properties, thermo-physical properties.

Cubic nonlinearity parameter measurement and material degradation detection using nonlinear ultrasonic three-wave mixing

Early detection of accumulated damage or material degradation in structures is important to ensure their structural safety. Nonlinear ultrasonic techniques are widely used to measure the quadratic nonlinearity parameter that represents the third-order elastic constants of materials for material degradation detection. In addition, there are ongoing efforts to exploit both the third- and fourth-order elastic constants that describe the cubic nonlinearity parameter to detect material degradation. This study develops a nonlinear ultrasonic three-wave mixing technique to generate and measure third-order combined harmonics (TOCH) in plate-like structures and to measure cubic nonlinearity parameter. The proposed three-wave mixing technique generates three primary Lamb waves in the target structure and measures the TOCH produced by nonlinear cross-interaction of the primary Lamb waves. A theoretical model is developed to describe the generation of TOCH in a nonlinear elastic and homogeneous plate, and the effectiveness of the theoretical model is validated experimentally. Measurements of the TOCH are conducted for intact and degraded aluminum specimens with different degradation levels. Because inherent material nonlinearity and material degradation alter the third- and fourth-order elastic constants of a structure, the three-wave mixing technique for measuring the TOCH can be used to identify the inherent material nonlinearity and material degradation. In particular, the experimental results indicate that the proposed technique is more sensitive to early-stage material degradation than existing nonlinear ultrasonic techniques such as two-wave mixing, and third harmonic generation techniques.

The Third-Order Elastic Constants and Mechanical Properties of 30° Partial Dislocation in Germanium: A Study from the First-Principles Calculations and the Improved Peierls–Nabarro Model

The elastic constants, core width and Peierls stress of 30° partial dislocation in germanium has been investigated based on the first-principles calculations and the improved Peierls–Nabarro model. Our results suggest that the predictions of lattice constant and elastic constants given by LDA are in better agreement with experiment results. While the lattice constant is overestimated at about 2.4% and most elastic constants are underestimated at about 20% by the GGA method. Furthermore, when the applied deformation is larger than 2%, the nonlinear elastic effects should be considered. And with the Lagrangian strains up to 8%, taking into account the third-order terms in the energy expansion is sufficient. Except the original γ—surface generally used before (given by the first-principles calculations directly), the effective γ—surface proposed by Kamimura et al. derived from the original one is also used to study the Peierls stress. The research results show that when the intrinsic−stacking−fault energy (ISFE) is very low relative to the unstable−stacking−fault energy (USFE), the difference between the original γ—surface and the effective γ—surface is inapparent and there is nearly no difference between the results of Peierls stresses calculated from these two kinds of γ—surfaces. As a result, the original γ—surface can be directly used to study the core width and Peierls stress when the ratio of ISFE to the USFE is small. Since the negligence of the discrete effect and the contribution of strain energy to the dislocation energy, the Peierls stress given by the classical Peierls–Nabarro model is about one order of magnitude larger than that given by the improved Peierls–Nabarro model. The result of Peierls stress estimated by the improved Peierls–Nabarro model agrees well with the 2~3 GPa reported in the book of Solid State Physics edited by F. Seitz and D. Turnbull.

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.

Nonlinear Guided Wave Tomography for Detection and Evaluation of Early-Life Material Degradation in Plates.

In this paper, the possibility of using nonlinear ultrasonic guided waves for early-life material degradation in metal plates is investigated through both computational modeling and study. The analysis of the second harmonics of Lamb waves in a free boundary aluminum plate, and the internal resonance conditions between the Lamb wave primary modes and the second harmonics are investigated. Subsequently, Murnaghan’s hyperelastic model is implemented in a finite element (FE) analysis to study the response of aluminum plates subjected to a 60 kHz Hanning-windowed tone burst. Different stages of material degradation are reflected as the changes in the third order elastic constants (TOECs) of the Murnaghan’s model. The reconstructed degradations match the actual ones well across various degrees of degradation. The effects of several relevant factors on the accuracy of reconstructions are also discussed.