Components Of Elastic Tensor Research Articles

Elastic and acoustic characteristics of a three-dimensional carbon supracrystal (C)CTO

The force constants of the central and noncentral interactions of carbon atoms in a three-dimensional supracrystal (C)CTO have been calculated using the ab initio and Harrison bond-orbital methods. The components of the elastic rigidity tensor and the propagation velocity of elastic waves in the supracrystal have been calculated. The results are close in magnitude to corresponding characteristics of diamond.

Young’s modulus and the Poisson’s ratio of planar and nanotubular supracrystalline structures

A method has been proposed for calculating the two-dimensional Young’s modulus and the Poisson’s ratio for planar and nanotubular structures through the components of the two-dimensional elastic rigidity tensor obtained by numerical methods. The method has been tested for graphene and two-dimensional supracrystalline sheets.

Surface acoustic waves and elastic constants of Cu14%Al4%Ni shape memory alloys studied by Brillouin light scattering

A single crystal Cu14%Al4%Ni shape memory alloy in austenitic phase was studied by Brillouin spectroscopy to analyse angular dispersion of Rayleigh surface waves in the planes (0 0 1) and (1 1 0). Measurements were made at room temperature using a tandem-type Fabry–Perot interferometer (JRS Scientific Instruments). Experimental results permitted determination of three independent elastic constants and anisotropic Young's modulus. In the (1 1 0) plane the pseudo-surface waves were found to occur, which—in the context of the past literature data—is unusual for cubic crystals of anisotropy coefficient higher than one. The elastic tensor components determined were used for simulation of surface phonons by the finite element method. The surface phonon velocities obtained were compared with the experimental results and the results of calculations based on the available analytical expressions for the phonon velocities for high-symmetry directions. Particular attention was paid to the relation between the elastic constants (anisotropy coefficient) and the depth of surface phonon penetration.

Thermodynamic and mechanical properties of lanthanum–magnesium phases from density functional theory

Thermodynamic and mechanical properties of the six known phases in the La–Mg phase diagram, viz. LaMg, LaMg2, LaMg3, La5Mg41, La2Mg17, and LaMg12, and their elemental antecedents, Mg and La, are computed with density functional theory (DFT) using the PBE and PBEsol exchange-correlation functionals. Phase stability analyses show that both LaMg2 and La5Mg41 are metastable at low temperatures which is consistent with experiments and vibrational spectra. We generalize an existing approach for computing the crystallographic dependence of Young's modulus and Poisson's ratio, which is presently limited to cubic systems, to address any space group symmetry using 0K elasticity tensor components (Cij) from DFT. Isothermal and isentropic Cij(T) are computed with the quasiharmonic approximation (QHA) as are the linear thermal expansion of the cubic compounds, the average linear thermal expansion for the non-cubic compounds, the bulk modulus, and the constant pressure heat capacity. A critical comparison of theoretical results from the PBE and PBEsol functionals is made with available experimental data.

Symmetry-adapted calculations of strain and polarization fields in (111)-oriented zinc-blende quantum dots

We present expressions for the elastic and first-order piezoelectric tensor in (111)-oriented III-V zinc-blende semiconductors. Moreover, an equation for the second-order piezoelectric polarization vector in these systems is derived. Together these expressions provide an efficient route to calculate built-in potentials and strain fields in (111)-oriented zinc-blende nanostructures. Our detailed analysis provides insight into the key parameters that modify strain and built-in fields in a (111)-oriented zinc-blende system compared to a conventional (001) structure. We show that the calculated strain field in a (111)-oriented quantum dot displays the correct ${C}_{3v}$ symmetry of the underlying crystal structure, even though we use a continuum-based approach and the quantum dot geometry is higher in symmetry than ${C}_{3v}$, e.g., ${C}_{\ensuremath{\infty}v}$. This behavior originates from an in-plane angle dependence of certain elastic tensor components in the (111)-zinc-blende system. In addition, we compare the elastic and the first-order piezoelectric tensor of the (111)-zinc-blende systems with the corresponding quantities in a wurtzite structure and point out similarities and differences. This comparison provides, for example, insight into the sign of the shear piezoelectric coefficient ${e}_{15}$ in the wurtzite system, which is still under debate in the literature. Our analysis indicates ${e}_{15}<0$, in accordance with recent experimental and theoretical results.

Homogenisation of fibrous composites under stochastic ageing

The main aim of this paper is to present an application of the generalised stochastic perturbation technique to model the stochastic ageing processes of fibre-reinforced periodic composite materials in terms of their effective properties. The response surface method is used to detect, for various time moments of the ageing process, the analytical interrelations between the effective elasticity tensor components and the composite parameters exposed to the ageing phenomena. A numerical procedure based on the finite-element code MCCEFF and the system MAPLE is used to model ageing of glass–epoxy composites, but it may find general application in the reliability analysis of other composite structures.

Thermodynamic and Elastic Properties of the Phases Appearing in the Lightweight La-Mg Alloys

Lattice parameters, electronic and vibrational energies, enthalpies of formation at 0 and 298 K, the elasticity tensor components,Cij, polycrystalline bulk, shear and Young’s moduli based on the Hill criteria were computed for LaMg, LaMg2, LaMg3, and La5Mg41, La2Mg17, and LaMg12. The quasiharmonic approximation was used to compute the linear thermal expansion tensor, aij(T), and the constant pressure heat capacity,Cp(T). Elastic anisotropy was analyzed in terms of the Young’s moduli dependence on crystallographic direction. Results were compared with available experimental data and discussed in terms of the applications of materials from the La-Mg system.

Nonlinear piezoelectricity in electroelastic energy harvesters: Modeling and experimental identification

We propose and experimentally validate a first-principles based model for the nonlinear piezoelectric response of an electroelastic energy harvester. The analysis herein highlights the importance of modeling inherent piezoelectric nonlinearities that are not limited to higher order elastic effects but also include nonlinear coupling to a power harvesting circuit. Furthermore, a nonlinear damping mechanism is shown to accurately restrict the amplitude and bandwidth of the frequency response. The linear piezoelectric modeling framework widely accepted for theoretical investigations is demonstrated to be a weak presumption for near-resonant excitation amplitudes as low as 0.5 g in a prefabricated bimorph whose oscillation amplitudes remain geometrically linear for the full range of experimental tests performed (never exceeding 0.25% of the cantilever overhang length). Nonlinear coefficients are identified via a nonlinear least-squares optimization algorithm that utilizes an approximate analytic solution obtained by the method of harmonic balance. For lead zirconate titanate (PZT-5H), we obtained a fourth order elastic tensor component of c1111p=−3.6673×1017 N/m2 and a fourth order electroelastic tensor value of e3111=1.7212×108 m/V.

Ab initiothermodynamic and elastic properties ofAGaH4hydrides (A=Li, Na, K, Rb, and Cs)

Systematic properties of the $A{\text{GaH}}_{4}$ alkali gallium hydrides ($A=\text{Li}$, Na, K, Rb, and Cs) are investigated within density functional theory. Seven ground-state crystal structures are identified, with two energetically indistinguishable structures found for both ${\text{LiGaH}}_{4}$, whose structure is as yet undetermined experimentally, and ${\text{CsGaH}}_{4}$. Born effective charge tensors, static and high-frequency dielectric tensors, and phonon dispersion relations incorporating longitudinal-optical/transverse-optical mode splittings are computed. Our results indicate that ${\text{LiGaH}}_{4}$ and ${\text{NaGaH}}_{4}$ have technologically interesting standard enthalpies of formation near $\ensuremath{-}30\text{ }\text{kJ}/\text{mole}$ ${\text{H}}_{2}$. We find, however, that ${\text{LiGaH}}_{4}$ is thermodynamically unstable with respect to both LiGa and LiH, providing a possible explanation for its challenging synthesis. The Born stability criteria are evaluated with the computed elasticity tensor components, ${C}_{ij}$. All seven structures are found to be both elastically and vibrationally stable.

Threefold Increase in the Young’s Modulus of Graphite Negative Electrode during Lithium Intercalation

Density functional theory (DFT) is used to reveal that the polycrystalline Young’s modulus of graphite triples as it is lithiated to . This behavior is captured in a linear relationship between and lithium concentration suitable for continuum-scale models aimed at predicting diffusion-induced deformation in battery electrode materials. Alternatively, Poisson’s ratio is concentration-independent. Charge-transfer analyses suggest simultaneous weakening of carbon–carbon bonds within graphite basal planes and strengthening of interlayer bonding during lithiation. The variation in bond strength is shown to be responsible for the differences between elasticity tensor components, , of lithium–graphite intercalation (Li-GIC) phases. Strain accumulation during Li intercalation and deintercalation is examined with a core–shell model of a Li-GIC particle assuming two coexisting phases. The requisite force equilibrium uses different Young’s moduli computed with DFT. Lithium-poor phases develop tensile strains, whereas Li-rich phases develop compressive strains. Results from the core–shell model suggest that elastic strain should be defined relative to the newest phase that forms during lithiation of graphite, and Li concentration-dependent mechanical properties should be considered in continuum level models.

Distinguishing Elastic Shear Deformation from Friction on the Surfaces of Molecular Crystals

Elastic deformation on the surfaces of molecular crystals can be imaged using a variant of lateral force microscopy in which the tip is scanned parallel to the cantilever axis. The shear force transverse to this direction has a distinctly different origin than the friction force as determined by the tip velocity and temperature dependence of the cantilever torque. An elastic deformation model for the tip-sample interaction predicts the crystallographic anisotropy of the transverse shear contrast, establishing its connection with the relative magnitude of the in-plane elastic tensor components.

First-principles study of elastic and phonon properties of the heavy fermion compoundCeMg

Elasticity tensor components, Cij, the crystallographic dependence of Poisson’s ratio, the phase stability, and vibrational spectraare computed for nonmagnetic and magnetic CeMg (1:1 Ce:Mg) structures using densityfunctional theory. Results from both the generalized gradient approximation (GGA), and theGGA+U, based upon an effective on-site Coulomb potential,Ueff, are investigated. The GGA low energy structure, with wavevector along [110], disagrees withexperiment, while the [100] structure from experiment is predicted as the 0 K structure in theGGA+U. Accurate estimation of the 20 K Néel temperature can only be achieved with smallUeff,which suggests that CeMg is not a strongly correlated system. For all CeMg structures investigated, wefind C11≈C44; this is consistent with the near equivalency of transverse and longitudinal soundspeeds. The origin of this behavior is the negative stretching force constants forthe interaction between the second- and third-nearest-neighbor Mg and Ce ions,respectively. Results are compared with neutron scattering experiments at 30 and 110 K.

Reconstruction of the layer anisotropic elastic parameters and high‐resolution fracture characterization from P‐wave data: a case study using seismic inversion and Bayesian rock physics parameter estimation

ABSTRACTWide‐azimuth seismic data can be used to derive anisotropic parameters on the subsurface by observing variation in subsurface seismic response along different azimuths. Layer‐based high‐resolution estimates of components of the subsurface anisotropic elastic tensor can be reconstructed by using wide‐azimuth P‐wave data by combining the kinematic information derived from anisotropic velocity analysis with dynamic information obtained from amplitude versus angle and azimuth analysis of wide‐azimuth seismic data. Interval P‐impedance, S‐impedance and anisotropic parameters associated with anisotropic fracture media are being reconstructed using linearized analysis assuming horizontal transverse anisotropy symmetry. In this paper it is shown how additional assumptions, such as the rock model, can be used to reduce the degrees of freedom in the estimation problem and recover all five anisotropic parameters. Because the use of a rock model is needed, the derived elastic parameters are consistent with the rock model and are used to infer fractured rock properties using stochastic rock physics inversion. The inversion is based on stochastic rock physics modelling and maximum a posteriori estimate of both porosity and crack density parameters associated with the observed elastic parameters derived from both velocity and amplitude versus angle and azimuth analysis. While the focus of this study is on the use of P‐wave reflection data, we also show how additional information such as shear wave splitting and/or anisotropic well log data can reduce the assumptions needed to derive elastic parameter and rock properties.

Low-temperature acoustic properties of single-crystal hafnium

Five independent adiabatic elastic tensor components c ij of single-crystal Hf (0.3% Zr) have been precisely measured in the range 78–300 (c 11 and c 33) or 78–160 K (c 12, c 13, and c 44) at 50 MHz using an ultrasonic pulse technique. The corresponding linear absorption coefficients α ij (T) have been measured in the same temperature ranges. The c ij data have been used to determine adiabatic compressibilities β∥ and β⊥, and Young’s moduli E ∥ and E ⊥ along and across the c axis, respectively. Near 140 K, we have revealed an anomaly in c 33(T) due to changes in the phonon spectrum of single-crystal α-Hf. The observed anomalies in temperature-dependent α11 and α33 are tentatively attributed to grown-in dislocations.

Subject-Specific p-FE Analysis of the Proximal Femur Utilizing Micromechanics-Based Material Properties

Novel subject-specific high-order finite element models of the human femur based on computer tomographic (CT) data are discussed with material properties determined by two different methods, empirically based and micromechanics based, both being determined from CT scans. The finite element (FE) results are validated through strain measurements on a femur harvested from a 54-year-old female. To the best of our knowledge, this work is the first to consider an inhomogeneous Poisson ratio and the first to compare results obtained by micromechanics-based material properties to experimental observations on a whole organ. We demonstrate that the FE models with the micromechanics-based material properties yield results which closely match the experimental observations and are in accordance with the empirically based FE models. Because the p-FE micromechanics-based results match independent experimental observations and may provide access to patient-specific distribution of the full elasticity tensor components, it is recommended to use a micromechanics-based method for subject-specific structural mechanics analyses of a human femur.

Crystal structure and elastic constants of Dharwar cotton fibre using WAXS data

Wide-angle X-ray scattering (WAXS) recordings were carried out on raw Dharwar cotton fibres available in Karnataka. Using this data and employing linked atom least squares (LALS) method, we report here the molecular and crystal structure of these cotton fibres. Employing structural data, we have computed elastic moduli tensor components of these fibres. From these investigations, it turns out that the intrinsic strains present in the fibre arise due to hydrogen bonds and not covalent bonds, which is a significant result.

Higher Order Elastic Constants, Gruneisen Parameters and Lattice Thermal Expansion of Lithium Niobate

The second and third‐order elastic constants and pressure derivatives of second‐ order elastic constants of trigonal LiNbO3 (lithium niobate) have been obtained using the deformation theory. The strain energy density estimated using finite strain elasticity is compared with the strain dependent lattice energy density obtained from the elastic continuum model approximation. The second‐order elastic constants and the non‐vanishing third‐order elastic constants along with the pressure derivatives of trigonal LiNbO3 are obtained in the present work. The second and third‐order elastic constants are compared with available experimental values. The second‐order elastic constant C11 which corresponds to the elastic stiffness along the basal plane of the crystal is less than C33 which corresponds to the elastic stiffness tensor component along the c‐axis of the crystal. The pressure derivatives, dC′ij/dp obtained in the present work, indicate that trigonal LiNbO3 is compressible. The higher order elastic constants are used to find the generalized Gruneisen parameters of the elastic waves propagating in different directions in LiNbO3. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with available reported values.

Thermal Vacancy Effects on the Thermodynamic Properties and Stability of Fullerites

We study the influence of thermal vacancies on equilibrium thermodynamic properties of the high-temperature phase of fullerites taking into account the strong anharmonicity of the lattice vibrations. Treating a crystal with point defects as a quasi-multicomponent system and using the correlative method of the unsymmetrized self-consistent field and the Girifalco interaction potential for the molecular subsystem, we have obtained the vacancy formation parameters for the C(60) fullerite. We also take into account the divacancies. The influence of the lattice defects on the specific heats of fullerites is negligible, since a dominant contribution to them is given by intramolecular degrees of freedom. We have calculated the contributions of vacancies to various thermodynamic properties, the volume thermal expansion coefficient, the isothermal bulk modulus, and the components of the isothermal elastic tensor, depending on the temperature and pressure. Near the estimated triple point, these contributions run to more than 10% and increase still further at a metastable region. We also discuss the influence of defects on the thermodynamic stability of fullerites.

Elastic Properties of Structural Phases in Shape Memory Alloys Investigated by Resonant Ultrasound Spectroscopy

Elastic constants of solids were, until recently, evaluated predominantly by pulse-echo ultrasonic methods which are based on measuring the speed of ultrasonic waves propagation in solids. Resonant ultrasound spectroscopy (RUS) is a relatively novel method in which all components of elastic tensor are determined from measured resonance frequencies of a freely vibrating specimen. The RUS technique has been employed in this work to investigate temperature dependence of the elastic properties of the parent austenite phase in CuAlNi shape memory alloy single crystals. This phase exhibits very high elastic anisotropy (anisotropy factor A 12) and softening the shear coefficient C0 upon cooling when approaching the Ms transformation temperature. The complications (need for large number of resonant frequencies) emerging when one tries to determine all elastic constants of highly elastically anisotropic materials by the RUS technique are discussed. It is concluded that the shear coefficients C0 and C44, which are the most important for shape memory alloys, are, nevertheless, determined reliably.

Higher Order Elastic Constants, Gruneisen Parameters and Lattice Thermal Expansion of Trigonal Calcite

The second‐ and third‐order elastic constants of trigonal calcite have been obtained using the deformation theory. The strain energy density derived using the deformation theory is compared with the strain dependent lattice energy obtained from the elastic continuum model approximation to get the expressions for the second‐ and third‐order elastic constants. Higher order elastic constants are a measure of the anharmonicity of a crystal lattice. The seven second‐order elastic constants and the fourteen non‐vanishing third‐order elastic constants of trigonal calcite are obtained. The second‐order elastic constants C11, which corresponds to the elastic stiffness along the basal plane of the crystal is greater than C33, which corresponds to the elastic stiffness tensor component along the c‐axis of the crystal. First order pressure derivatives of the second‐order elastic constants of calcite are evaluated. The higher order elastic constants are used to find the generalized Gruneisen parameters of the elastic waves propagating in different directions in calcite. The Brugger gammas are evaluated and the low temperature limit of the Gruneisen gamma is obtained. The results are compared with available reported values.