Compliance Constants Research Articles

Improving equations of state calibrations in the toroidal DAC—The case study of molybdenum

We report an updated isothermal equation of state (EoS) of molybdenum (Mo) obtained by compression in beveled and toroidal diamond-anvil cells (DACs). For an improved compression environment, we developed a copper (Cu) pressure-transmitting medium (PTM) for the toroidal diamond-anvil cell samples, as it is a soft metal compared to Mo with a well calibrated EoS. A Ne PTM was used for the conventional beveled DAC samples. The unit-cell volumes of Mo were measured to 336(1) GPa in the Cu PTM and 231.2(6) GPa in the Ne PTM at room temperature. We additionally calculated elastic stiffness and compliance constants and evaluated the uniaxial stress of Mo and Cu with pressure. A new EoS for Mo is presented from data collected in all sample environments and compared to our theoretical predictions as well as previous compression studies of Mo. The (200) lattice plane of Mo produced the lowest volumes across the pressure range of this study for all compression environments, suggesting that it is less affected by nonhydrostatic stresses in the DAC compared to the other observed diffraction planes. The presented Mo EoS is compatible with extrapolations of EoS fits of Mo in helium (He) within ∼1% at 330 GPa. Results from this work demonstrate that compressing a sample in a softer metal in the toroidal DAC can improve the compression environment and result in measured sample volumes comparable to those collected in noble-gas media at multi-megabar conditions.

Elastic constants of graphane, graphyne, and graphdiyne

Graphyne (two-dimensional one-atom-thick carbon allotrope) a promising semiconductor distinct by the number of acetylenic bonds. The study of mechanical properties is of key importance for electronics, composite fabrication, and fundamental understanding of the mechanical behavior of 2D structures. Molecular dynamics and analytical calculations were used to obtain the elastic constants (stiffness and compliance constants, Young’s and shear modulus, Poisson’s ratio) for five graphynes. For the first time, the direct link between stiffness and compliance constants for 2D materials were presented. Among hexagonal graphynes, γ1 has the largest Young’s modulus (154 N/m). The deformation is attributed to the competition between the rotation and elongation of bonds in graphyne under loading. An analysis of stationary and extreme values of Young’s and shear modulus, Poisson’s ratio was carried out for the first time based on the search for the extremum of a function of one variable. A strong mechanical anisotropy was observed for orthorhombic graphynes (β3 and γ2). Orthorhombic graphynes possess extremely large in-plane Poisson’s ratio. Explanation of how elastic constants depend on structure density and atomic arrangement was given. The obtained results open new opportunities for the development of new nanomechanical devices based on such 2D materials.

Precise evaluation of the material constants of [011]-poled mangan-doped 0.32Pb(In1/2Nb1/2)O3-0.39Pb(Mg1/3Nb2/3)O3-0.29PbTiO3 single crystals

Recently, there has been a surge of interest in [011]-poled Mn: PIN-PMN-PT piezoelectric single crystals for high-power transducers. These crystals boast impressive electromechanical coupling coefficients, piezoelectric constants in the transverse mode, and excellent heat resistance due to their high mechanical quality factor. To effectively design transducers using these crystals, understanding their material constants is crucial. Our study focused on determining these constants by employing the resonance method. We created eleven different resonators, analyzed their impedance spectra, and derived seventeen elastic compliance, piezoelectric, and dielectric constants. The precision of the obtained material constants was enhanced by minimizing the difference between the impedance spectra found through finite element analyses and actual measurements of each resonator. The accuracy of these derived constants was validated by comparing the impedance spectra obtained from analysis with those measured experimentally. This novel approach demonstrated a capability to derive more precise material constants than prior studies. These refined material constants hold the potential to enable more precise and effective design of acoustic transducers constructed from the single crystals.

Текстура рекристалізації та анізотропії пружних властивостей листів сталі DC04

We studied steel sheets DC04 (0.06% C, up to 0.35% Mn, up to 0.40% Si, ~ 0.025% S and P) with a thickness of 0.95 mm as delivered. Sheets of A4 size were annealed in a laboratory oven (6000C in an argon atmosphere, hold for 1 hour). The structure of DC04 steel sheets after recrystallization annealing was studied. The microstructure of the steel sheets under study is presented from the side of the rolling plane and in the section of the sheet perpendicular to the direction. In the plane of the sheets, the grains are elongated; in the cross section, the grains are approximately equiaxed. Pole figures (PF) were constructed based on the results of electron backscatter diffraction (EBSD) on a LEO 1455 VP electron microscope at an accelerating voltage of 20 kV from the plane of the sheets and from the section of the sheet perpendicular to the rolling direction. To improve statistics, PF were constructed by averaging reflex stereographic projections from 20 different representative volumes of material relative to the rolling direction and transverse direction. The texture and anisotropy of Young's modulus in the plane and cross section of steel sheets DC04 after recrystallization annealing was studied using EBSD method. A connection has been obtained between ideal orientations that describe the texture in two mutually perpendicular planes and the corresponding integral characteristics of texture (ICT). Rectangular samples with a length of 100 and a width of 10 mm at different angles to the rolling direction every 150 to measure Young's modulus. Samples were processed in a bag to ensure uniform dimensions. Young's modulus was determined by the dynamic method from the frequency of natural transverse vibrations. Three batches of samples were used to construct Young's modulus anisotropy curves. The anisotropy of the Young's modulus in the plane of steel sheets, calculated from the ICT based on the results of EBSD data, is in good agreement with the results of direct measurements. The value of Young's modulus in the direction normal to the plane of the sheet and in the section plane in the direction normal to the plane of the sheet, calculated from the ICT and the values of the compliance constants of iron, coincide. Keywords: texture, pole figure, anisotropy, integral characteristics of texture, Young's modulus.

An Overview of Mechanical Properties of Diamond-like Phases under Tension.

Diamond-like phases are materials with crystal lattices very similar to diamond. Recent results suggest that diamond-like phases are superhard and superstrong materials that can be used for tribological applications or as protective coatings. In this work, 14 stable diamond-like phases based on fullerenes, carbon nanotubes, and graphene layers are studied via molecular dynamics simulation. The compliance constants, Young's modulus, and Poisson's ratio were calculated. Deformation behavior under tension is analyzed based on two deformation modes-bond rotation and bond elongation. The results show that some of the considered phases possess very high Young's modulus (E≥1) TPa, even higher than that of diamond. Both Young's modulus and Poisson's ratio exhibit mechanical anisotropy. Half of the studied phases are partial auxetics possessing negative Poisson's ratio with a minimum value of -0.8. The obtained critical values of applied tensile strain confirmed that diamond-like phases are high-strength structures with a promising application prospect. Interestingly, the critical limit is not a fracture but a phase transformation to the short-ordered crystal lattice. Overall, our results suggest that diamond-like phases have extraordinary mechanical properties, making them good materials for protective coatings.

Temperature dependence of complete sets of material constants of Pb(In1/2Nb1/2)O3–Pb(Mg1/3Nb2/3)O3–PbTiO3 crystals using different poling methods

The complete sets of material constants of <001>-oriented two wafers (HP: high-performance and LP: low-performance) using direct current poling (DCP) and alternating current poling (ACP) were investigated. ACP can improve the longitudinal and transverse piezoelectric properties (about 20% increase in both d31 and d33) and decrease the shear properties (about 40% decrease in d15). The compliance constants SEij of both ACP HP and LP samples have the similar change trend (the absolute of SE11, SE12, SE13, SE33, SE66 increasing, the SE44 decreasing), while most stiffness constants CEij of both ACP HP and LP samples show the different change trend. The values of CE11, CE12, CE13 of ACP LP samples increase by at least 50% while ACP HP samples decrease by about 5%. As temperature elevated, the dielectric properties (εT11/ε0, εT33/ε0), piezoelectric properties (d15, d31, d33) and electromechanical coupling coefficients (k15, k33) increased steadily, while the k31 keep stable.

Stress formulation of acoustoelasticity

Acoustoelasticity describes the relationship between elastic wave velocities and the initial stress present in a material. Traditional theories consider successive deformations involving small amplitude wave motion superimposed on an initially deformed material. Then, a constitutive relationship must be applied to relate the initial static deformation to the desired relationships involving initial stress, resulting in expressions of wave velocities involving a mix of elastic stiffness and compliance constants. In this article, a pure stress formulation is developed for acoustoelasticity. In this setting, the problem involves the superposition of a dynamic stress wave on an initially stressed material configuration, rather than the superposition of kinematic variables. The phase velocity of the stress wave is naturally related to the initial stress through only the compliance constants. Thus, compliances are the fundamental constants of acoustoelasticity.

Relaxed and local mode force constants: Linear ABC as a model system

Following on from Wilson’s FG approach to molecular vibrations, there have been two major developments: the exploitation of compliance constants and the introduction of local modes. Both have been heavily augmented by the application of increasingly sophisticated DFT and high-level calculations. Using the linear ABC molecule as a model, we demonstrate the congruence of the two approaches.

On the Incompressible Behavior in Weakly Nonlinear Elasticity

This article considers the influence of incompressibility on the compliance and stiffness constants that appear in the weakly nonlinear theory of elasticity. The formulation first considers the incompressibility constraint applied to compliances, which gives explicit finite limits for the second-, third-, and fourth-order compliance constants. The stiffness/compliance relationships for each order are derived and used to determine the incompressible behavior of the second-, third-, and fourth-order stiffness constants. Unlike the compressible case, the fourth-order compliances are not found to be dependent on the fourth-order stiffnesses.

How Strong is a Reverse Dative Bond? Compliance Constants as Unique Bond Strength Descriptors.

The concepts of bond strength and bond order are at the very heart of chemistry. Nevertheless, the study of local mechanical bond strengths in large polyatomic molecules is a nontrivial task. In a recent publication, Peters and Lehnert et al. [Inorg. Chem. 2020, 59 (20), 14967-14982] found a quite strong (reverse) dative Fe···B bond for a synthesized low-spin {FeNO}10 complex supported by an ambiphilic trisphosphineborane ligand. Their conclusion was, among other criteria, based on a surprisingly high Fe-B force constant of 1.56 mdyn/Å. Nevertheless, this high value might be based on numerical ambiguities of (rigid) real-space force constants, a flaw, that may affect many other combined spectroscopic and electronic structure studies.

Anharmonicity in compliance formalism: Potential constants and interaction coordinates

AbstractAt the harmonic level, compliance constants are the inverse force constant matrix elements. In the present work, conversion of anharmonic force constants to compliance constants has been revisited and several new variants of the existing equations have been suggested. The reverse conversion of anharmonic compliance constants to force constants is addressed. The change in the internal coordinate (Ri) for constrained optimization is written as a monotonic polynomial function of the change in another chosen constrained internal coordinate (Rk) and the expansion coefficients extracted “directly” are the interaction coordinates (ICs). New expressions for computing the cubic and quartic anharmonic ICs using compliance constants are presented and the obtained numerical values are compared with the ICs extracted by the “direct” method. The work also highlights an effective way of extracting the diagonal harmonic and anharmonic compliance and force/relaxed force constants characterizing the bond in diatomic molecules and pseudo diatomic fragments in a molecule. The ability to compute the mentioned numerical values is likely to open up new possibilities for useful chemical applications.

Anisotropy in the elasticity of α-U from first-principles calculations and nanoindentation

The anisotropic behavior of elastic properties of the low-temperature phase of uranium was studied by theoretical and experimental methods in this work. Firstly, based on the density functional theory , the elastic constants and compliance constants of α-U were obtained. Secondly, Young's modulus , bulk modulus , shear modulus, and Poisson's ratio of polycrystalline α-U were calculated using the Voigt-Reuss-Hill model. Then, according to the crystal symmetry of α-U, the spatial distribution characteristics of its Young's modulus were calculated and displayed. The Young's modulus of α-U in some specific crystal directions (including the typical deformation-induced twinning {1 3 0}, {1 7 2}) were also given. Nanoindentation method combined with electron backscatter diffraction (EBSD) technology were used to measure and characterize the grains with different crystal orientations , to obtain experimental values of Young's modulus of α-U in different crystal orientations. Also, the inverse pole figure of Young's modulus of α-U was given.

Acoustoelastic wave propagation in initially stressed solids using a pure stress formulation

Wave propagation in homogeneous elastic solids is governed by Cauchy's first law of motion, which relates the acceleration of material points to the divergence of stress. For Hookean solids, the stress is typically written in terms of displacements to form an equation of motion with displacement as the dependent variable, i.e., Navier's equation of elastodynamics. Previously, the authors considered wave propagation in anisotropic elastic solids by formulating the equation of motion using only stress as the dependent variable. Eigenvalue solutions provided the (stress) wave phase velocity in terms of elastic compliance constants. In this presentation, the stress formalism is extended and applied to the nonlinear problem of stress wave propagation in an initially stressed material (also known as acoustoelasticity). The stress formulation is viewed as a more natural fit over historical derivations, which were based on wave displacements superimposed on an initially strained material state. The stress formulation of elastodynamics is firstly reviewed. Then, the extension to stress wave propagation in an initially stressed material is given. The phase velocity of the stress wave is related to the initial stress through second- and third-order compliance constants (instead of elastic stiffnesses in traditional acoustoelasticity).

Intrinsic Dynamic and Static Nature of Halogen Bonding in Neutral Polybromine Clusters, with the Structural Feature Elucidated by QTAIM Dual-Functional Analysis and MO Calculations.

The intrinsic dynamic and static nature of noncovalent Br-∗-Br interactions in neutral polybromine clusters is elucidated for Br4–Br12, applying QTAIM dual-functional analysis (QTAIM-DFA). The asterisk (∗) emphasizes the existence of the bond critical point (BCP) on the interaction in question. Data from the fully optimized structures correspond to the static nature of the interactions. The intrinsic dynamic nature originates from those of the perturbed structures generated using the coordinates derived from the compliance constants for the interactions and the fully optimized structures. The noncovalent Br-∗-Br interactions in the L-shaped clusters of the Cs symmetry are predicted to have the typical hydrogen bond nature without covalency, although the first ones in the sequences have the vdW nature. The L-shaped clusters are stabilized by the n(Br)→σ*(Br–Br) interactions. The compliance constants for the corresponding noncovalent interactions are strongly correlated to the E(2) values based on NBO. Indeed, the MO energies seem not to contribute to stabilizing Br4 (C2h) and Br4 (D2d), but the core potentials stabilize them, relative to the case of 2Br2; this is possibly due to the reduced nuclear–electron distances, on average, for the dimers.

Linear Multiselenium Interactions in Dicationic Oligomers of 1,5-(Diselena)canes: Behavior of Semc σ(mc c-ne e) (6≤mc ≤16) Elucidated with QTAIM Dual Functional Analysis.

The intrinsic dynamic and static nature m c center‐n e electron interactions of the σ‐type σ(m cc‐n ee) were elucidated for the Se‐Se interactions in dicationic oligomers of Se(CH2CH2CH2)2Se (1 (Se, Se)) [n 2+ (Se, Se): n=1–8], especially for m c≥6, where n 2+ (Se, Se: n=1–8) are abbreviated by n 2+ (n=1–8), respectively. QTAIM dual functional analysis (QTAIM‐DFA) was applied to the interactions. Perturbed structures generated using coordinates derived from the compliance constants (Cii) were employed for QTAIM‐DFA. Each Se‐*‐Se in 1 2+ and 2 2+ has the nature of CT‐TBP (trigonal bipyramidal adduct formation through CT) and Cov‐w (weak covalent), respectively, which supply the starting points of the investigations. The asterisk emphasizes the existence of a bond critical point on the interaction. All Se‐*‐Se in 3 2+ are classified by the regular closed shell (r‐CS) interactions and characterized as CT‐MC (molecular complex formation through CT), which are denoted as r‐CS/CT‐MC, except for the central interaction, of which nature is r‐CS/CT‐TBP. Most interactions in 4 2+–8 2+ are r‐CS/t‐HBwc (typical‐HB with covalency) but some are pure‐CS/t‐HBnc (t‐HB with no covalency). The linear Se2n 2+ interactions in 2 2+–8 2+ seem close to those without any limitations, since the nature of Se‐*‐Se inside and outside of (CH2CH2CH2)2 are very similar with each other. The linear Se2n 2+ interactions in 3 2+–8 2+ are shown to be analyzed as σ(mcc‐nee: 6≤mc≤16), not by the accumulated σ(3c–4e).

Effects of Temperature and Pressure on Elastic Properties of Single Crystal Aluminum in Different Crystal Orientations

Molecular dynamics simulations are carried out to study the mechanical and elastic properties of single crystal aluminum with various crystal orientations at different temperatures and under different pressure. Transformations of lattice structures under high pressure, tensile and shearing loads are investigated, respectively. When the pressure increases to 1011 Pa, the face centered cubic (FCC) structure is completely transformed into body centered cubic (BCC) and other structures. The transformation from FCC to hexagonal close packed (HCP) and other structures takes place during tensile and shearing processes. Elastic compliance constants are calculated at different temperatures. Elastic modulus and shear modulus of different crystal orientations at different temperatures are calculated by the formula method, and compared with the results of molecular dynamics method. The decrease extent of elastic modulus in &lt;111&gt; crystal orientation is the smallest with the increase of temperature. The effects of high pressure on elastic modulus and shear modulus in different crystal orientations are also investigated. When the pressure exceeds 109 Pa, the elastic modulus and shear modulus in all crystal orientations begin to increase. Poisson's ratio in different crystal orientations at different temperatures and under high pressure are also calculated.

Stress formalism for elastic waves in anisotropic solids

Traditionally, the propagation of bulk elastic waves in anisotropic crystals or solids is governed by the displacement form of the elastodynamic wave equation. Phase velocities follow after transforming the wave equation into an eigenvalue problem, which gives the Christoffel equations, and solving for the roots of the characteristic polynomial. Alternatively, only a few researchers have considered a pure stress wave equation of motion to describe the propagation of waves in elastic solids. However, to our knowledge, the dual characteristic polynomial and its solutions based on a pure stress formalism has not been established previously. This presentation will outline the needed steps to reach and solve the dual characteristic polynomial. Phase velocity solutions follow from the formation of new principal invariants, which are functions of the elastic compliance constants and propagation directions. The dual formalism is proven consistent to the traditional displacement formalism through the use of sixth-rank Levi-Civitia identities. Lastly, possible extensions of this work will be highlighted.

First principles investigations of structural, elastic, mechanical, electronic and optical properties of triple perovskite Ba2K2Te2O9

We report the structural, elastic, mechanical, electronic and optical properties of Ba2K2Te2O9 (dibarium dipotassium nonaoxidoditellurate) triple perovskite compound by employing generalized gradient approximation (GGA) of first principles density functional theory (DFT) within Perdew-Burke-Ernzerhof (PBE) exchange correlation energy for the first time. This compound shows structural stability and found to be as a direct band gap semiconductor with 2.339 eV band gap energy. Further, elastic compliance constants, stiffness constants and other mechanical properties of Ba2K2Te2O9 were also computed with some optical properties of i.e. the dielectric function, optical conductivity, refractive index, extinction and absorption coefficients. According to Pugh ratio analysis Ba2K2Te2O9 is ductile and displays a more compressible tendency along a - and b -axis and less compressible behavior along the c -axis.

Comparative study on structural, electronic, optical and mechanical properties of normal and high pressure phases titanium dioxide using DFT

In this paper, a Self-consistent Orthogonalized linear combination of atomic orbitals (OLCAO) technique with a generalized gradient approximation such as Perdew–Burke–Ernzerhof Solid (GGA-PBE SOL) has been used to scrutinize the structural, optical, electronic and mechanical properties of normal pressure phase (Anatase and Rutile) and high pressure phase i.e., cubic (Fluorite and Pyrite) TiO2. Electronic and optical properties of normal pressure phases of TiO2 are also investigated using (Meta) MGGA-Tran and Blaha (TB09) and obtained results are a close approximation of experimental data. It is seen that the virtually synthesized structural parameter for cubic and tetragonal phases of TiO2 are consistent with experimental and theoretical data. From the effective mass of charge carriers (m*), it can be observed that pyrite TiO2 is having lower effective mass than the fluorite and hence shows higher photocatalytic activity than fluorite. Furthermore, it is seen that fluorite is more dense than anatase, rutile and pyrite TiO2. From the theoretical calculations on the optical properties, it can be concluded that optical absorption occursin the near UV region for high and normal pressue phases of TiO2. Again from the reflectivity characteristics R(ω), it can be concluded that TiO2 can be used as a coating material. Elastic constants, elastic compliance constants, mechanical properties are obtained for anatase, rutile, fluorite and pyrite TiO2. A comparison of the results with previously reported theoretical and experimental data shows that the calculated properties are in better agreement with the previously reported experimental and theoretical results.

Local vibrational force constants – From the assessment of empirical force constants to the description of bonding in large systems

The local vibrational mode analysis, originally introduced by Konkoli and Cremer, provides a physically sound platform for a comprehensive analysis of calculated or measured vibrational spectra and for providing detailed insights into chemical bonding and other structural features. In this work, we summarize the essentials of the local vibrational mode theory with a focus on local vibrational force constants and their relationship with compliance and relaxed force constants. Furthermore, we discuss how local vibrational force constants can be used (i) to assess the quality of empirically derived force constants, (ii) to disclose bonding features in large molecules, and (iii) to provide metal ligand force field parameters. Future applications will be suggested.