Determination Of Elastic Constants Research Articles

Determination of Diffraction Elastic Constants of Gamma-Double-Prime in Polycrystalline Ni-Base Superalloys

In this paper we introduce a new method to measure diffraction elastic constants of gamma double prime phase in Ni-base superalloys using in-situ neutron diffraction. During aging heat treatment of the alloys, stress is applied so that gamma double prime variants can be selected, which facilitates accurate fitting of the diffraction peaks. Then the Miller Indices reflections from different gamma double prime variants can be used to separate the overlapping gamma and gamma double prime diffraction peaks. The measured values agree with the elastic constants calculated using Kroner and Reuss models.

Elastic constants determination of anisotropic materials by depth-sensing indentation

Depth-sensing indentation is a useful and powerful tool for the mechanical characterization of materials at the micro and nano scale. This technique allows the determination of the Young modulus from the analysis of the load-penetration depth curve according to specific theoretical models. One of the most used models is that one proposed by Oliver and Pharr. However, when a material with anisotropic mechanical properties is tested, Oliver and Pharr’s theory is no longer suitable to describe the contact mechanics between the indenter tip and the tested material. This paper provides an overview of the theoretical models developed for the evaluation of the elastic constants of anisotropic materials through depth-sensing indentation. Specifically, the cases of generally anisotropic and orthotropic materials are described in order to cover the entire range of anisotropy. Examples on how these models can be applied for the mechanical characterization of generally anisotropic topological insulators and transversely isotropic pyrolytic carbon are also reported. This topical overview represents a useful tutorial for the evaluation of the elastic constants of anisotropic materials by depth-sensing indentation by shading light on the contact mechanics at the micro and nano scale.

Diffraction-based determination of single-crystal elastic constants of polycrystalline titanium alloys.

Single-crystal elastic constants have been derived by lattice strain measurements using neutron diffraction on polycrystalline Ti-6Al-4V, Ti-6Al-2Sn-4Zr-6Mo and Ti-3Al-8V-6Cr-4Zr-4Mo alloy samples. A variety of model approximations for the grain-to-grain interactions, namely approaches by Voigt, Reuss, Hill, Kroener, de Wit and Matthies, including texture weightings, have been applied and compared. A load-transfer approach for multiphase alloys was also implemented and the results are compared with single-phase data. For the materials under investigation, the results for multiphase alloys agree well with the results for single-phase materials in the corresponding phases. In this respect, all eight elastic constants in the dual-phase Ti-6Al-2Sn-4Zr-6Mo alloy have been derived for the first time.

Understanding the origin of half‐metallicity and thermophysical properties of ductile La 2 CuMnO 6 double perovskite

For first time, the magneto-electronic structure with thermoelectric and mechanical properties of lanthanum-based double perovskite La2CuMnO6 are investigated, using first-principle methods. Generalized gradient approximation and modified Becke-Jhonson potentials are integrated to figure out exchange-correlation potential. The alloy stabilizes in cubic structure with ferromagnetic nature and determined structural parameters are consistent with experimental results. The band profile reveals the half-metallic character, which is further confirmed by calculated electronic conductivities of up and down spin channels. The effect of pressure on the structural and electronic profile is demonstrated here. The analysis of the transport properties portrays that the highest value of 0.39 is achieved for figure of merit at higher temperatures. The mechanical stability of La2CuMnO6 is established, by determination of elastic constants. The calculated elastic parameters specify the ductile behavior of alloy with high melting temperature. The efficient thermoelectric parameters with half-metallic and ductile character suggest the likelihood of applications of alloy to design hard spintronic devices or potential thermoelectric materials.

On the determination of the third-order elastic constants of homogeneous isotropic materials utilising Rayleigh waves

This paper presents a new method for determining the third-order elastic constants (TOECs) of a homogeneous isotropic material utilising the acoustoelastic effect associated with Rayleigh waves. It is demonstrated that the accuracy of the evaluation of TOECs can be substantially improved by supplementing the classical equations of acoustoelasticity, which describe the effect of applied stress on bulk wave speeds, with the nonlinear characteristic equation for the propagation of Rayleigh waves in pre-stressed media. The developed method can be readily implemented for Structural Health Monitoring applications; for example, the measurement of applied stresses based on the acoustoelastic effect, or the monitoring of near-surface microstructural damage based on the change in magnitude of the TOECs.

Microstructural and Mechanical Properties of Binary Ti-Rich Fe–Ti, Al-Rich Fe–Al, and Ti–Al Alloys

Three series of binary, FeTi (Ti-rich), FeAl and TiAl (Al-rich) alloy samples were produced in an argon arc furnace. An annealing treatment of 72 h at 1000 °C was applied to the samples, giving rise to different equilibrium microstructures depending on chemical composition. Their mechanical properties were studied through the determination of elastic constants that measure the stiffness of the elaborated materials. Young’s modulus of the binary alloys was determined using Resonance Ultrasonic Vibration (RUV). The accuracy of this technique was demonstrated. A scanning electron microscope (SEM) with an energy dispersive spectrometer (EDS) and X-ray diffraction (XRD) made it possible to identify intermetallic compounds FeTi and Ti, FeAl and Fe, and TiAl and Ti in respective systems Fe–Ti, Fe–Al, and Ti–Al. The link between their composition, microstructure, and elastic properties was established.

Elastic and optoelectronic properties of CaTa2O6 compounds: Cubic and orthorhombic phases

Using first principles density functional theory (DFT) simulations, the structural, electronic, optical and elastic properties of CaTa2O6 oxide for cubic and orthorhombic phases are studied by highly accurate (FP-LAPW) method within the GGA + U approximation. The calculated lattice parameters are consistent with available experimental data. The electronic band structure calculations have shown that the band gaps in CaTa2O6 are equal to 3.08 eV and 4.40 eV for the cubic and orthorhombic structures, respectively. For both the phases the main optical properties, e.g., absorption coefficient, dielectric constant, energy loss function, reflectivity are calculated and discussed in detail in the spectral range 0–14 eV. Cubic and orthorhombic phases exhibit significantly different optical characteristics. The electronic bonding characters of CaTa2O6 with different symmetries are explored via charge density distribution mapping. Strong covalent bonding character dominates in both the phases of CaTa2O6. Elastic properties of CaTa2O6 for cubic and orthorhombic phases are also investigated. The stress strain method is used for the determination of elastic constants in both the phases. The bulk modulus, shear modulus, Young's modulus, along with the important elastic anisotropy factors and Poisson's ratio are studied in detail.

Elastic Constants Determination of Composite Plates Using Measured Strains

A strain-based elastic constant identification method is proposed to determine the elastic constants of fiber-reinforced composite rectangular laminates using three measured strains of the plates subjected to uniaxial load testing. In the proposed method, the measured normal strains in 0°, and 45°, and 90° directions, respectively, of the plate made of one composite material subjected to uniaxial tensile testing are used to identify four elastic constants of the constituent composite material via a two-level optimization approach. The objective function used for constructing the two-level optimization problem consists of the sum of the differences between the experimental and theoretical predictions of the three strain components and a strain restraining function, which is used to help even up the effects of the measured strains on the identified elastic constants. The accuracy of the proposed method has been verified via an experimental approach.

Determination of elastic constants of Inconel-625 superalloy, using laser-based ultrasonic

This paper deals with the determination of the elastic constants of Inconel-625 from the analysis of laser-generated ultrasonic bulk waves. A pulsed Nd/YAG laser (1064 nm) is used for ultrasonic generation in a thick stepped Inconel-625 sample, and a He–Ne laser is used for heterodyne detection of the laser-generated signals. Ultrasonic signals obtained at epicenter and at off-epicenter position of the detection points are analyzed using wavelet transforms. Here, the identification of pressure as well as shear waves and the estimation of their velocities are done successfully. From the estimated velocities, the elastic constants (Young’s modulus, shear modulus, bulk modulus, Lames constant and Poisson’s ratio) are calculated. The agreement of these constants with the standard values confirms the identification of shear waves at off-epicenter position. The work presented in this paper brings out the applicability of the study of laser-generated bulk waves for the determination of elastic constants of any bulk material.

Determination of rigidities, stiffness coefficients and elastic constants of multi-layer graphene sheets by an asymptotic homogenization method

This work is concerned with the analysis of bending and torsional rigidities of multi-layer graphene sheets (MLGSs) based on the Kalamkarov’s general asymptotic homogenization composite shell model. Also, the effective stiffness coefficients and elastic constants of MLGSs are estimated with this analytical method. The unit cell with both in-plane and out-of-plane interactions is assumed in this model. A MLGS as a homogeneous honeycomb network sheet with the periodic hexagonal unit cell is considered here in which the layers are held together by different densities of van der Waals interactions. The stiffness coefficients, elastic constants and rigidities of MLGS are found by considering different densities of the van der Waals forces and different number of layers. The results show good agreements in comparison with other experiments and numerical solutions. It is found that the homogenization method gives the ability to create promise analytical approach that can be used for other nanostructures.

An approximate solution of the spherical indentation on a generally anisotropic elastic half-space

The indentation of a generally anisotropic elastic half-space by a rigid spherical indentor is considered. We derive an approximate equation for the eccentricity of the ellipsoidal contact region and an approximate expression for the indentation modulus. Our approximate formulae are first validated by ABAQUS-based numerical simulations, and then specialized to the particular case of a fibre-reinforced composite half-space. Numerical results are presented to show the dependence of the eccentricity and indentation modulus on the compressibility and fibre strength of the composite material. Possible application of our approximate formulae to the determination of elastic constants of fibre-reinforced materials is discussed.

Determination of surface properties and elastic constants of FCC metals: a comparison among different EAM potentials in thin film and bulk scale

Three independent elastic constants C11, C12, and C44 were calculated and compared using available potentials of eight different metals with FCC crystal structure; Gold, Silver, Copper, Nickel, Platinum, Palladium, Aluminum and Lead. In order to calculate the elastic constants, the second derivative of the energy density of each system was calculated with respect to different directions of strains. Each set of the elastic constants of the metals in bulk scale was compared with experimental results, and the average relative error was for each was calculated and compared with other available potentials. Then, using the Voigt-Reuss-Hill method, approximated values for Young and shear moduli and Poisson’s ratio of the FCC metals in the bulk scale were found for each potential. Furthermore, to observe the surface effects on the metals in nanoscale, surface elastic constants of the thin films of the metals have been calculated. In the study of the thin films of materials in nanoscale, the number of surface atoms is considerable compared to all atoms of the object. This leads to an increase in the surface effects, which influence the elastic properties. By considering this fact and employing related definitions and equations, the properties of the thin films of the metals were calculated, and the surface effects for different crystallographic directions were compared. Subsequently, in some cases, comparisons among characteristics of the metals in the thin film and bulk material were made.

Inverse determination of elastic constants of a hyper-elastic member with inclusions using simple displacement/length measurements

There are many engineering parts and natural soft members, which consist of several phases with different mechanical properties. Identification of material parameters of multi-material members is an important issue that has attracted considerable attention. In this article, a finite-element-based inverse method for identification of material parameters of a body made of several hyper-elastic materials is presented. In the presented inverse method, the unknowns are computed using a few simple displacement/length measurements in a few elasto-static experiments. The inverse analysis can be carried out without any full-field measurements. Both plane strain and plane stress conditions are considered. In the plane strain case, displacements of several sampling points on the boundary of the problem domain are used as measured data. In the plane stress case, some characteristic lengths of the member in the deformed configuration are used as measured data. A cost function in terms of the differences between calculated and measured values is defined and the Gauss–Newton method is used to minimize it. The multi-material sensitivity analysis in the optimization process is made by analytical differentiation of the weak form of the problem. Both numerical and experimental studies are made to demonstrate the effectiveness and accuracy of the proposed methods.

Elastic and structural properties of low silica calcium aluminosilicate glasses from molecular dynamics simulations

The elastic constants and the structural of low silica calcium aluminosilicate glasses with a low content of silica (5–25 mol%) and [CaO]/[Al2O3] =2, are investigated using molecular dynamics simulations. In particular, their elastic constants are calculated using two methods: a standard approach at zero temperature and another method designed to work at finite temperature. We show that while they both reproduce qualitatively the experimental behavior, only the method taking temperature into account provides a quantitative agreement with the experimental values. Therefore, taking temperature into account in the determination of elastic constants seems necessary to give a satisfactory account of the mechanical properties of these materials. In accordance with the experimental results, our molecular dynamics calculations show values of elastic constants that decrease with increasing amount of silica in the vitreous network. Then, the structure of the glasses is analyzed in term of pair distribution functions, oxygen kinds, and the parameter of short-range order, and discussed in correlation with the available experimental data. An overall good agreement is obtained with the calculated structure and the experimental data.

Specific features of structural, electronic, optical and elastic properties of the cubic calcium pyroniobate Ca2Nb2O7 crystals

Using first principles density functional theory (DFT) calculation, the structural, electronic, optical and elastic properties of cubic Ca2Nb2O7 pyrochlore oxide is studied by highly accurate full potential augmented plane wave method with GGA+U approximation. The calculated lattice parameter is in good agreement with available experimental data. The electronic band structure calculations reveal that Ca2Nb2O7 has direct energy band. For the understanding of optical properties of Ca2Nb2O7, absorption coefficient, dielectric constant, energy loss function, reflectivity, refractive index, extension coefficient and optical conductivity are also calculated for 0–12 eV. A study of elastic properties for cubic pyrochlore oxide (Ca2Nb2O7) is also conducted within a frameowork of DFT calculations. The stress strain method is used for the determination of elastic constants (C11, C12 and C44) in cubic phase. Hence, bulk, shear and Young's modulus along with Debye temperature, elastic anisotropy factors and Poisson's ratio are calculated successfully. Furthermore, pressure dependence of transverse and longitudinal wave velocities for cubic pyrochlore oxide (Ca2Nb2O7) are also calculated for the first time.

Stress-stress fluctuation formula for elastic constants in the NPT ensemble.

Several fluctuation formulas are available for calculating elastic constants from equilibrium correlation functions in computer simulations, but the ones available for simulations at constant pressure exhibit slow convergence properties and cannot be used for the determination of local elastic constants. To overcome these drawbacks, we derive a stress-stress fluctuation formula in the NPT ensemble based on known expressions in the NVT ensemble. We validate the formula in the NPT ensemble by calculating elastic constants for the simple nearest-neighbor Lennard-Jones crystal and by comparing the results with those obtained in the NVT ensemble. For both local and bulk elastic constants we find an excellent agreement between the simulated data in the two ensembles. To demonstrate the usefulness of the formula, we apply it to determine the elastic constants of a simulated lipid bilayer.

Characterization of elastic constants of anisotropic composites in compression using digital image correlation

Experimental determination of elastic constants of anisotropic composite laminates in all orthogonal directions is generally a complex process. In this paper a simple direct technique to determine a broad set of elastic moduli is presented based on compression testing of a prism sample. Digital image correlation is used to measure the full-field deformations that allow the determination of Young’s moduli and all six Poisson’s ratios for the three orthogonal directions based on a single sample. Finite element model is used in evaluation of the effect of friction on the measured properties. In addition to quantitative characterization of the material properties, local strain mapping is used in qualitative evaluation of the sample structures.

Shear horizontal wave dispersion in nanolayers with surface effects and determination of surface elastic constants

In this work, the shear horizontal (SH) wave dispersion in two dissimilar nanolayers is investigated by using the surface elasticity theory in which the surface effects are featured by surface elastic constants. It is found that the SH wave dispersion shows distinct dependence on the nanolayer thickness as well as the surface elastic constants. The larger the surface elastic modulus and/or the smaller the thickness, the higher the phase velocity. In particular, as the wave frequency approaches zero, the analytical relation between the phase velocity in the first mode dispersion and the surface elastic constants is deduced. Thereby, a facile method is suggested to determine the surface elastic constants from the phase velocity of SH waves scattered in nanolayers.

A Gradient Taguchi Method for Engineering Optimization

To balance the robustness and the convergence speed of optimization, a novel hybrid algorithm consisting of Taguchi method and the steepest descent method is proposed in this work. Taguchi method using orthogonal arrays could quickly find the optimum combination of the levels of various factors, even when the number of level and/or factor is quite large. This algorithm is applied to the inverse determination of elastic constants of three composite plates by combining numerical method and vibration testing. For these problems, the proposed algorithm could find better elastic constants in less computation cost. Therefore, the proposed algorithm has nice robustness and fast convergence speed as compared to some hybrid genetic algorithms.

Determination of elastic constants of a single-crystal topaz and their temperature dependence via sphere resonance method

Water and halogens in ocean floor sediments transported by a descending slab might play important roles in geodynamic processes. Imaging subducted sediments through seismological observations requires a thorough understanding of elastic properties of sediment origin hydrous minerals. Topaz is a sediment origin hydrous mineral, which is formed at the depth of 250–350km on a cold subducting slab. We determined elastic constants and their temperature derivatives of a natural single-crystal of topaz (Al1.97SiO4(F1.56, OH0.42)) at the temperature from 271.5 to 312.7°K by using the sphere-resonance method. Elastic constants at an ambient temperature (T=291.9°K) are C11=281.21(1)GPa, C22=346.23(9)GPa, C33=294.99(9)GPa, C44=108.49(1)GPa, C55=132.47(1)GPa, C66=130.32(1)GPa, C12=121.48(3)GPa, C13=80.94(3)GPa and C23=81.77(2)GPa. Since our sample [Al2SiO4(F1.56,OH0.42)] was relatively rich in fluorine, only small differences in elastic constants can be seen between our sample and fluorine end member. Elastic constants of OH-rich topaz should be experimentally investigated to understand the influence of F-OH substitution on elasticity of topaz. All the elastic constants decrease linearly with increasing temperature. The temperature derivatives are dC11/dT=−0.014(3)GPa/°K, dC22/dT=−0.010(7)GPa/°K, dC33/dT=−0.021(5)GPa/°K, dC44/dT=−0.011(1)GPa/°K, dC55/dT=−0.016(2)GPa/°K, dC66/dT=−0.0101(2)GPa/°K, dC12/dT=−0.0041(6)GPa/°K, dC13/dT=−0.001(2)GPa/°K and dC23/dT=−0.002(1)GPa/°K. The isotropic seismic velocities in topaz are distinctly higher than those in olivine at 10GPa and 300–1400°K. There should be a strong velocity contrast between the overlying mantle and the thin sediment-origin layer at the depth around 300km. A seismological technique like the receiver function technique should be applied to detect a thin layer of topaz in a cold subduction zone.