Elastic constants of lead-bismuth alloys

A study of ultrasonic velocities and internal friction has been carried out in Pb-Bi alloys in the concentration range of 0 to 49.5 atomic % Bi using the composite oscillator technique. From the velocity and density data a set of elastic constants namely, Young’s modulus, rigidity modulus, bulk modulus and Poisson’s ratio are estimated. The results are interpreted in terms of the phase changes occurring in the alloy system. Internal friction is found to be more sensitive than the elastic constants to the phase changes.     

Ultrasonic studies on bismuth-tin alloys

A study of ultrasonic velocities and internal friction in Bi-Sn alloys was carried out in the concentration range of 0–8·3 atomic percent of tin using the composite oscillator technique, and a set of elastic constants was calculated from the velocity and density data. The elastic constants as well as the internal friction were found to decrease abruptly in the region of solid solution. This peculiar variation is attributed to the semiconducting properties of this alloy system in this concentration range. The temperature variation of Young's modulus was studied for some of the concentrations and Young's modulus was found to decrease linearly with increasing temperature.     

Multiscale modeling of graphene- and nanotube-based reinforced polymer nanocomposites

A combination of molecular dynamics, molecular structural mechanics, and finite element method is employed to compute the elastic constants of a polymeric nanocomposite embedded with graphene sheets, and carbon nanotubes. The model is first applied to study the effect of inclusion of graphene sheets on the Young modulus of the composite. To explore the significance of the nanofiller geometry, the elastic constants of nanotube-based and graphene-based polymer composites are computed under identical conditions. The reinforcement role of these nanofillers is also investigated in transverse directions. Moreover, the dependence of the nanocomposite?s axial Young modulus on the presence of ripples on the surface of the embedded graphene sheets, due to thermal fluctuations, is examined via MD simulations. Finally, we have also studied the effect of sliding motion of graphene layers on the elastic constants of the nanocomposite.     

Analysis of Single-Walled Carbon Nanotubes Using a Chemical Bond Element Model

提出了一种纳米尺度的有限元方法,碳纳米管中的碳-碳化学键被模拟为键单元．按照平衡关系,根据有限元理论,作用于每个碳原子上的作用力可以写成键单元的刚度矩阵与每个碳原子位移的乘积．在分子力学的基本假设下,键单元刚度矩阵的每个元素可以写为分子力学中力场常数的函数,这样建立起了宏观力学方法(有限元)与纳米尺度力学方法(分子力学)之间的联系．应用该方法模拟了扶椅型与锯齿型单壁碳纳米管的力学行为从而验证了该方法的有效性．分析结果说明单壁碳纳米管的弹性模量与管厚度的选取直接相关．此外,弹性模量对所选取的分子力学中的力场常数非常敏感,管的弹性模量显示出对半径的尺度依赖性,但是管长度对弹性模量的影响小到可以被忽略．     

Monocrystal elastic constants in the ultrasonic study of welds

For studying welds ultrasonically, the importance of knowing the material's single-crystal elastic constants, the C_ijs, is explained. Where these constants are not known, some guidelines are given for estimating them from polycrystalline elastic constants such as Young's modulus and the shear modulus.The important case of [001] fibre texture is considered. Being transversely isotropic, this case exhibits five macroscopic elastic constants, which are related to the three cubic elastic constants: C₁₁, C₁₂, C₄₄. From these five constants the angular variations of Young's modulus, the torsional modulus, and the sound velocities can be computed. For the same [001] fibre texture, results are given for a standard well-characterized material — copper, where the C_ijs are well known.     

Effective elastic properties of piezoelectric composites with radially polarized cylinders

Effective elastic properties of piezoelectric composites containing an infinitely long, radially polarized cylinder embedded in an isotropic non-piezoelectric matrix are theoretically investigated under an external strain field. Analytical solutions of elastic displacement and electric potentials are exactly derived, and the effective elastic responses are formulated in the dilute limit. Meanwhile, a vanishing piezoelectric response mechanism is revealed in the piezoelectric composite containing radially polarized cylinders. Furthermore, it is shown that the effective elastic properties can be enhanced (or reduced) due to the increase of the piezoelectric (or dielectric) constants of the cylinders.     

Determination of elastic constants of Na0.5Bi0.5TiO3 from ab initio calculations

Elastic constants and bulk modulus for the tetragonal, rhombohedral, and cubic phase of Na_0.5Bi_0.5TiO₃ crystal were calculated from the first principles. From the calculated elastic constants, other structural properties such as bulk modulus, shear modulus, Young's modulus, and Poisson's ratio can be derived using respective relationships from Voight–Reuss–Hill approximation; bulk modulus was calculated as an example in this article. It was shown that elastic constants show different behavior for compression and elongation. The different values of elastic constants have been calculated for the direction parallel to the bismuth layer (crystallographic a(b)-axis) and the perpendicular direction (crystallographic c-axis). It seems to be caused by bismuth layer structure oxides of Na_0.5Bi_0.5TiO₃ crystal.     

First-principles study of the structural, elastic, electronic, optical, and vibrational properties of intermetallic Pd2Ga

The structural, elastic, electronic, optical, and vibrational properties of the orthorhombic Pd₂Ga compound are investigated using the norm-conserving pseudopotentials within the local density approximation in the frame of density functional theory. The calculated lattice parameters have been compared with the experimental values and found to be in good agreement with these results. The second-order elastic constants and the other relevant quantities, such as the Young's modulus, shear modulus, Poisson's ratio, anisotropy factor, sound velocity, and Debye temperature, have been calculated. It is shown that this compound is mechanically stable after analysing the calculated elastic constants. Furthermore, the real and imaginary parts of the dielectric function and the optical constants, such as the optical dielectric constant and the effective number of electrons per unit cell, are calculated and presented. The phonon dispersion curves are derived using the direct method. The present results demonstrate that this compound is dynamically stable.     

First-principles study of the structural, elastic, electronic, optical, and vibrational properties of intermetallic Pd2Ga

The structural,elastic,electronic,optical,and vibrational properties of the orthorhombic Pd2Ga compound are investigated using the norm-conserving pseudopotentials within the local density approximation in the frame of density functional theory.The calculated lattice parameters have been compared with the experimental values and found to be in good agreement with these results.The second-order elastic constants and the other relevant quantities,such as the Young’s modulus,shear modulus,Poisson’s ratio,anisotropy factor,sound velocity,and Debye temperature,have been calculated.It is shown that this compound is mechanically stable after analysing the calculated elastic constants.Furthermore,the real and imaginary parts of the dielectric function and the optical constants,such as the optical dielectric constant and the effective number of electrons per unit cell,are calculated and presented.The phonon dispersion curves are derived using the direct method.The present results demonstrate that this compound is dynamically stable.     

Elastic constants of nonionic anharmonic crystals

Taking into account the entire anharmonicity of the crystal an exact relation between the long-wavelength limit of the static self-energy and the isothermal elastic constants is established. The result can be reformulated as a sum rule for the displacement correlation function. The diagram technique is also employed to discuss the equation for the elastic constants in the pseudoharmonic approximation as well as to derive the integral equations for the quasiparticle parameters.     

First Principles Investigation of Fluorine Based Strontium Series of Perovskites

Density functional theory is used to explore structural, elastic, and mechanical properties of SrLiF₃, SrNaF₃, SrKF₃ and SrRbF₃ fluoroperovskite compounds by means of an ab-initio Full Potential-Linearized Augmented Plane Wave (FP-LAPW) method. Several lattice parameters are employed to obtain accurate equilibrium volume (V_o). The resultant quantities include ground state energy, elastic constants, shear modulus, bulk modulus, young's modulus, cauchy's pressure, poisson's ratio, shear constant, ratio of elastic anisotropy factor, kleinman's parameter, melting temperature, and lame's coefficient. The calculated structural parameters via DFT as well as analytical methods are found to be consistent with experimental findings. Chemical bonding is used to investigate corresponding chemical trends which authenticate combination of covalent-ionic behavior. Furthermore electron density plots as well as elastic and mechanical properties are reported for the first time which reveals that fluorine based strontium series of perovskites are mechanically stable and posses weak resistance towards shear deformation as compared to resistance towards unidirectional compression while brittleness and ionic behavior is dominated in them which decreases from SrLiF₃ to SrRbF₃. Calculated cauchy's pressure, poisson's ratio and B/G ratio also proves ionic nature in these compounds. The present methodology represents an effective and influential approach to calculate the whole set of elastic and mechanical parameters which would support to understand various physical phenomena and empower device engineers for implementing these materials in numerous applications.     

Ab initio calculations of the elastic, electronic, optical, and vibrational properties of PdGa compound under pressure

The structural, elastic, electronic, optical, and vibrational properties of cubic PdGa compound are investigated using the norm-conserving pseudopotentials within the local density approximation (LDA) in the framework of the density functional theory. The calculated lattice constant has been compared with the experimental value and has been found to be in good agreement with experimental data. The obtained electronic band structures show that PdGa compound has no band gap. The second-order elastic constants have been calculated, and the other related quantities such as the Young’s modulus, shear modulus, Poisson’s ratio, anisotropy factor, sound velocities, and Debye temperature have also been estimated. Our calculated results of elastic constants show that this compound is mechanically stable. Furthermore, the real and imaginary parts of the dielectric function and the optical constants such as the electron energy-loss function, the optical dielectric constant and the effective number of electrons per unit cell are calculated and presented in the study. The phonon dispersion curves are also derived using the direct method.     

Theoretical relationship between sound velocity and the physical properties of submarine sediment

Density and elastic modulus change ratios are introduced to describe the sound velocity of submarine sediment.The density change ratio is a composite parameter describing the sound velocity.It is expressed by three physical parameters:porosity,solid phase density and seawater density.The elastic modulus change ratio is also a composite parameter of sound velocity.It is expressed by three physical parameters,including porosity,solid phase modulus and seawater bulk modulus.The sound velocity formula can be developed into a Taylor polynomial formula of these two composite parameters.The change in the two composite parameters constitutes the sound velocity surface,which contains the complete information regarding velocity properties and sediment characteristics.The one-parameter velocity formula is a curve on the velocity surface.Each porosity-velocity empirical formula,which represents various sea locations and conditions,is transformed to a standard form.This result is the product of a reference velocity and a modulation function.Comparisons of the numerical calculation and measurements show that the derived modulation functions yield similar results.The difference between the velocity formula derived in this paper and the Wood velocity formula is due to the elastic modulus models.     

Prediction study of elastic properties under pressure effect for filled tetrahedral semiconductors LiZnN, LiZnP and LiZnAs

First principle calculations of elastic properties under pressure of the filled tetrahedral semiconductors LiZnN, LiZnP and LiZnAs are presented, using the pseudo-potential plane-waves approach based on density functional theory, within the local density approximation. Elastic constants, bulk modulus, Young’s modulus and Poisson’s ratio are calculated at zero pressure. A linear dependence of the bulk modulus and elastic constants with applied pressure is found. As the experimental elastic constants are not available for LiZnX, we have also calculated the elastic constants of GaN, GaP and GaAs, the binary analogues of LiZnN, LiZnP and LiZnAs, respectively, for checking the reliability and accuracy of our predicted results for LiZnX. The obtained results agree well with the available experimental data.     

Stiffness matrix invariants to validate the characterization of composite materials with ultrasonic methods

This paper presents a method of testing the ultrasonic measurements of the stiffness matrix, and the identification of the anisotropic behaviour, of composite materials. Some linear combinations of elastic constants are invariants for a rotation around an axis of symmetry. If the stacking sequence is the only parameter which changes in a set of long-fibre composites made of the superimposition of plies, the composites must own these invariants. So, PEEK-carbon fibre composite samples were constructed in this way to measure their elastic properties by immersion and contact ultrasonic methods, and to compare the results with predicted invariants. By changing the stacking sequence of plies three anisotropic models are tested: orthotropic, hexagonal and quadratic. Measurements of ultrasonic velocities in various planes of propagation permit the identification of the elastic constant and invariants. From the invariance of these linear combinations, the precision of the three-dimensional effective moduli can be estimated.     

Measurement of the dynamic elastic constants of short isotropic cylinders

A method based on a single test is proposed to characterize the elasticity of an isotropic homogeneous material in the shape of a cylinder of any slenderness (length-diameter) ratio. Firstly, the Rayleigh-Ritz method is used to determine the natural frequencies of the cylinders vibrating axisymmetrically. The study is focused on cylindrical samples with diameter and length of similar magnitude so that the shear modulus and the Poisson ratio can be calculated simultaneously. Subsequently, the theoretical results for cylinders of slenderness ratio between 0.1 and 3 are analyzed in order to obtain the data required to determine the elastic constants from one of the two lowest measured natural frequencies and their quotient. The analysis of the results demonstrates that any slenderness ratio is useful in the calculation of the elastic constants, although in some cases the third natural frequency should be used. Furthermore, the influence of the length-diameter quotient on the sensitivity of the method is analyzed by evaluating the systematic uncertainties for both dynamic elastic constants. Finally, the method is experimentally tested by characterizing two steel cylinders with slenderness ratios 0.1 and 1, respectively. The results demonstrate that uncertainties for both Poisson ratio and the shear modulus are smaller when the slenderness ratio is 1.     

Mechanical properties,Born effective charge tensors and high frequency dielectric constants of the eight phases of BaTiO<Subscript>3</Subscript>

We have preformed the first-principles calculations for the mechanical properties, Born effective charge tensors and high frequency dielectric constants of the eight phases of BaTiO₃. The independent elastic constants, bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio were obtained, which were consistent with the available theoretical and experimental values. The mechanical stability and brittle/ductile behaviors of the eight phases of BaTiO₃ have been discussed. The calculated results indicated that the eight phases were all mechanically stable and behaved in a brittle manner. The calculated Born effective charge tensors shown the covalent Ti–O bond and ionic Ba–O bond. Moreover, the high frequency dielectric constants have been given.     

Elastic Constants of Superconducting MgB2 from Molecular Dynamics Simulations with Shell Model

The elastic constants of superconducting MgB₂ are calculated using a molecular dynamics method (MD) with shell model. The lattice parameters, five independent elastic constants, equations of state (EOS), Debye temperature, and bulk modulus of MgB₂ are obtained. Meanwhile, the dependence of the bulk modulus B, the lattice parameters a and c, and the unit cell volume V on the applied pressure are presented. It is demonstrated that the method introduced here can well reproduce the experimental results with a reasonable accuracy.     

Complete mode identification for resonance ultrasound spectroscopy

This study is devoted to deducing exact elastic constants of an anisotropic solid material without using any advance information on the elastic constants by incorporating a displacement-distribution measurement into resonant ultrasound spectroscopy (RUS). The usual RUS method measures free-vibration resonance frequencies of a solid and compares them with calculations to find the most suitable set of elastic constants by an inverse calculation. This comparison requires mode identification for the measured resonance frequencies, which has been difficult and never been free from ambiguity. This study then adopts a laser-Doppler interferometer to measure the displacement-distribution patterns on a surface of the vibrating specimen mounted on pinducers; comparison of the measured displacement distributions with those computed permits us to correctly identify the measured resonance frequencies, leading to unmistakable determination of elastic constants. Because the displacement patterns are hardly affected by the elastic constants, an exact answer is surely obtained even when unreasonable elastic constants are used as initial guesses at the beginning of the inverse calculation. The usefulness of the present technique is demonstrated with an aluminum alloy and a langasite crystal.     

First-principles calculations for the elastic properties of Ni-base model superalloys: Ni/Ni3Al multilayers

A model system consisting of Ni[001](100)/Ni3Al[001](100) multi-layers are studied using the density functional theory in order to explore the elastic properties of single crystal Ni-based superalloys. Simulation results are consistent with the experimental observation that rafted Ni-base superalloys virtually possess a cubic symmetry. The convergence of the elastic properties with respect to the thickness of the multilayers are tested by a series of multilayers from 2γ′+2γ to 10γ′+10γ atomic layers. The elastic properties are found to vary little with the increase of the multilayer’s thickness. A Ni/Ni3Al multilayer with 10γ′+10γ atomic layers (3.54 nm) can be used to simulate the mechanical properties of Ni-base model superalloys. Our calculated elastic constants, bulk modulus, orientation-dependent shear modulus and Young’s modulus, as well as the Zener anisotropy factor are all compatible with the measured results of Ni-base model superalloys R1 and the advanced commercial superalloys TMS-26, CMSX-4 at a low temperature. The mechanical properties as a function of the γ′ phase volume fraction are calculated by varying the proportion of the γ and γ′ phase in the multilayers. Besides, the mechanical properties of two-phase Ni/Ni3Al multilayer can be well predicted by the Voigt-Reuss-Hill rule of mixtures.