Elastic Constant Tensor Research Articles

Mech2d: An Efficient Tool for High-Throughput Calculation of Mechanical Properties for Two-Dimensional Materials.

Two-dimensional (2D) materials have been a research hot topic in the passed decades due to their unique and fascinating properties. Among them, mechanical properties play an important role in their application. However, there lacks an effective tool for high-throughput calculating, analyzing and visualizing the mechanical properties of 2D materials. In this work, we present the mech2d package, a highly automated toolkit for calculating and analyzing the second-order elastic constants (SOECs) tensor and relevant properties of 2D materials by considering their symmetry. In the mech2d, the SOECs can be fitted by both the strain-energy and stress-strain approaches, where the energy or strain can be calculated by a first-principles engine, such as VASP. As a key feature, the mech2d package can automatically submit and collect the tasks from a local or remote machine with robust fault-tolerant ability, making it suitable for high-throughput calculation. The present code has been validated by several common 2D materials, including graphene, black phosphorene, GeSe2 and so on.

Computation of the thermal elastic constants for arbitrary manybody potentials in LAMMPS using the stress-fluctuation formalism

This paper describes the implementation of the stress-fluctuation technique into the LAMMPS code to compute the anisotropic thermal elastic constants tensor of materials. The implementation provides both methods for computing the analytical fluctuation expressions and also a generic numerical derivative method. The former makes the extension to new potentials straightforward, as it requires writing code only for the second derivatives of each energy term w.r.t. distance, angle, etc. The latter provides a generic interface to compute an accurate approximation of the elastic constants for any potential already implemented in LAMMPS. We show how both methods compare with the direct deformation computation in several test cases and discuss the implementation advantages and limitations.

The impact of the parameters of the constitutive model on the distribution of strain in the femoral head

The rapid spread of the finite element method has caused that it has become, among other methods, the standard tool for pre-clinical estimates of bone properties. This paper presents an application of this method for the calculation and prediction of strain and stress fields in the femoral head. The aim of the work is to study the influence of the considered anisotropy and heterogeneity of the modeled bone on the mechanical fields during a typical gait cycle. Three material models were tested with different properties of porous bone carried out in literature: a homogeneous isotropic model, a heterogeneous isotropic model, and a heterogeneous anisotropic model. In three cases studied, the elastic properties of the bone were determined basing on the Zysset-Curnier approach. The tensor of elastic constants defining the local properties of porous bone is correlated with a local porosity and a second order fabric tensor describing the bone microstructure. In the calculations, a model of the femoral head generated from high-resolution tomographic scans was used. Experimental data were drawn from publicly available database “Osteoporotic Virtual Physiological Human Project.” To realistically reflect the load on the femoral head, main muscles were considered, and their contraction forces were determined based on inverse kinematics. For this purpose, the results from OpenSim packet were used. The simulations demonstrated that differences between the results predicted by these material models are significant. Only the anisotropic model allowed for the plausible distribution of stresses along the main trabecular groups. The outcomes also showed that the precise evaluation of the mechanical fields is critical in the context of bone tissue remodeling under mechanical stimulations.

Generalized quasiharmonic approximation via space group irreducible derivatives

The quasiharmonic approximation (QHA) is the simplest nontrivial approximation for interacting phonons under constant pressure, bringing the effects of anharmonicity into temperature-dependent observables. Nonetheless, the QHA is often implemented with additional approximations due to the complexity of computing phonons under arbitrary strains, and the generalized QHA, which employs constant stress boundary conditions, has not been completely developed. Here we formulate the generalized QHA, providing a practical algorithm for computing the strain state and other observables as a function of temperature and true stress. We circumvent the complexity of computing phonons under arbitrary strains by employing irreducible second-order displacement derivatives of the Born-Oppenheimer potential and their strain dependence, which are efficiently and precisely computed using the lone irreducible derivative approach. We formulate two complementary strain parametrizations: a discretized strain grid interpolation and a Taylor series expansion in symmetrized strain. We illustrate our approach by evaluating the temperature and pressure dependence of select elastic constants and the thermal expansion in thoria (${\mathrm{ThO}}_{2}$) using density functional theory with three exchange-correlation functionals. The QHA results are compared to our measurements of the elastic constant tensor using time-domain Brillouin scattering and inelastic neutron scattering. Our irreducible derivative approach simplifies the implementation of the generalized QHA, which will facilitate reproducible, data-driven applications.

Anomalous elasticity of talc at high pressures: Implications for subduction systems

Talc is a layered hydrous silicate mineral that plays a vital role in transporting water into Earth’s interior and is crucial for explaining geophysical observations in subduction zone settings. In this study, we explored the structure, equation of state, and elasticity of both triclinic and monoclinic talc under high pressures up to 18 GPa using first principles simulations based on density functional theory corrected for dispersive forces. Our results indicate that principal components of the full elastic constant tensor C11 and C22, shear components C66, and several off-diagonal components show anomalous pressure dependence. This non-monotonic pressure dependence of elastic constant components is likely related to the structural changes and is often manifested in a polytypic transition from a low-pressure polytype talc-I to a high-pressure polytype talc-II. The polytypic transition of talc occurs at pressures within its thermodynamic stability. However, the bulk and shear elastic moduli show no anomalous softening. Our study also shows that talc has low velocity, extremely high anisotropy, and anomalously high VP/VS ratio, thus making it a potential candidate mineral phase that could readily explain unusually high VP/VS ratio and large shear wave splitting delays as observed from seismological studies in many subduction systems.

Shear softening and hardening of a two-dimensional Yukawa solid.

Langevin dynamical simulations are performed to study the elastic behaviors of two-dimensional (2D) solid dusty plasmas under the periodic shear deformation. The frequency- and strain-dependent shear moduli G(ω,γ) of our simulated 2D Yukawa solid are calculated from the ratio of the shear stress to strain in different orientations. The shear-softening and -hardening properties in different lattice orientations are discovered from the obtained G(ω,γ). The component of the elastic constant tensor corresponding to the shear deformation is also calculated, whose variation trend exactly agrees with the discovered shear-softening and -hardening features in different shear directions. It is also found that the shear modulus of the 2D Yukawa solid always increases monotonically with the frequency.

Measurement of the Anisotropic Dynamic Elastic Constants of Additive Manufactured and Wrought Ti6Al4V Alloys.

Additively manufactured (AM) materials and hot rolled materials are typically orthotropic, and exhibit anisotropic elastic properties. This paper elucidates the anisotropic elastic properties (Young’s modulus, shear modulus, and Poisson’s ratio) of Ti6Al4V alloy in four different conditions: three AM (by selective laser melting, SLM, electron beam melting, EBM, and directed energy deposition, DED, processes) and one wrought alloy (for comparison). A specially designed polygon sample allowed measurement of 12 sound wave velocities (SWVs), employing the dynamic pulse-echo ultrasonic technique. In conjunction with the measured density values, these SWVs enabled deriving of the tensor of elastic constants (Cij) and the three-dimensional (3D) Young’s moduli maps. Electron backscatter diffraction (EBSD) and micro-computed tomography (μCT) were employed to characterize the grain size and orientation as well as porosity and other defects which could explain the difference in the measured elastic constants of the four materials. All three types of AM materials showed only minor anisotropy. The wrought (hot rolled) alloy exhibited the highest density, virtually pore-free μCT images, and the highest ultrasonic anisotropy and polarity behavior. EBSD analysis revealed that a thin β-phase layer that formed along the elongated grain boundaries caused the ultrasonic polarity behavior. The finding that the elastic properties depend on the manufacturing process and on the angle relative to either the rolling direction or the AM build direction should be taken into account in the design of products. The data reported herein is valuable for materials selection and finite element analyses in mechanical design. The pulse-echo measurement procedure employed in this study may be further adapted and used for quality control of AM materials and parts.

On the effective elastic properties of a material with mutually perpendicular systems of parallel cracks

The effective properties of cracked solids are often expressed in terms of the crack density parameter or its tensor generalization, using the approximation of noninteracting cracks. This approximation remains accurate at sufficiently high crack densities, provided the location of cracks are random. The presented analysis confirms that the effective elastic moduli of a material with ordered fracture structures strongly depend on the linear dimensions of cracks and their mutual arrangement even at a constant crack density. A change in these parameters can cause a noticeable anisotropy of the effective properties of the material even when the eigenvalues of the crack density tensor are equal to each other. The effective elastic characteristics of a material with one doubly periodic system of parallel cracks are compared with those for a material with two mutually perpendicular systems of such cracks in a two-dimensional formulation. The calculations are carried out using the approximate method of M. Kachanov for determining the mean stresses at the cracks edges, applicable for large systems of interacting cracks. Analysis of the obtained results showed that the effective compliance of the material in a certain direction is largely determined by the effects of interaction (shielding and amplification) within a system of parallel cracks perpendicular to this direction. The interaction of this system of cracks with the perpendicular system has a weak effect on the indicated properties in the case of rectangular symmetry of the system. In this case, the interaction of mutually perpendicular systems of cracks leads to a violation of the symmetry of the tensor of effective elastic constants.

A FUNDAMENTAL SOLUTION FOR AN ANISOTROPIC PLATE ON AN INERTIAL FOUNDATION

The work is devoted to the construction of a new fundamental solution for a thin elastic anisotropic unbounded Kirchhoff plate on an inertial Winkler foundation. The fundamental solution is a function of the normal movement of two spatial coordinates and time in response to the impact of a single concentrated load, mathematically modeled by the Dirac delta function. The anisotropy model considered in this article has one plane of symmetry that geometrically coincides with the median plane of the plate and is characterized for the Kirchhoff plate model by six independent components of the tensor of elastic constants of the material. The problem statement includes the equations of motion of an anisotropic plate in displacements and initial conditions. The solution of the corresponding initial problem for the fundamental solution is constructed using integral Laplace transformations in time and two-dimensional integral Fourier transform in spatial coordinates. The inverse integral Laplace transform is found analytically with a preliminary analysis of the invertible function. The original fundamental Fourier solution is constructed using the numerical method of integrating rapidly oscillating functions. The parameters of numerical integration are calculated with a given accuracy when analyzing the quality of the desired functions by a continuous norm. For the constructed new fundamental solution, a numerical analysis of the nature of the propagation of nonstationary disturbances is carried out, and the influence of the parameters of the inertial Winkler foundation on the behavior of the normal deflection of an anisotropic plate in time at the point of action of the delta function is investigated. The asymmetric nature of unsteady plate oscillations consistent with the model of the elastic medium under consideration is demonstrated. The constructed fundamental solution makes it possible to investigate the nonstationary normal deflection of an anisotropic plate on an inertial basis under the action of arbitrary loads using an integral operator of the convolution type in spatial coordinates and time. In addition, the fundamental solution has versatility with respect to the properties of the plate materials and the parameters of the foundation, namely: isotropic, transversally isotropic and orthotropic materials can act as the plate material. In this case, the foundation may be inertialess or absent altogether.

Quantifying Mechanical Properties of Molecular Crystals: A Critical Overview of Experimental Elastic Tensors

AbstractThis review presents a critical and comprehensive overview of current experimental measurements of complete elastic constant tensors for molecular crystals. For a large fraction of these molecular crystals, detailed comparisons are made with elastic tensors obtained using the corrected small basis set Hartree–Fock method S‐HF‐3c, and these are shown to be competitive with many of those obtained from more sophisticated density functional theory plus dispersion (DFT‐D) approaches. These detailed comparisons between S‐HF‐3c, experimental and DFT‐D computed tensors make use of a novel rotation‐invariant spherical harmonic description of the Young's modulus, and identify outliers among sets of independent experimental results. The result is a curated database of experimental elastic tensors for molecular crystals, which we hope will stimulate more extensive use of elastic tensor information—experimental and computational—in studies aimed at correlating mechanical properties of molecular crystals with their underlying crystal structure.

PropSym: a tool to establish relationships between property constants for material property tensors of any order

The properties of crystalline materials can be described mathematically by tensors whose components are generally known as property constants. Tabulations of these constants in terms of the independent components are well known for common material properties (e.g. elasticity, piezoelectricity etc.) aptly described by tensors of lower rank (e.g. ranks 2–4). General relationships between constants of higher rank are often unknown and sometimes reported incorrectly. A computer program is developed here to calculate the property constant relationships of a property of any order, represented by a tensor of any rank and point group. Tensors up to rank 12, e.g. the tensor of sixth-order elastic constants c_{_{6}i\,j\,k\,l\,m\,n\,p\,q\,r\,s}, can be calculated on a standard computer, while ranks higher than 12 are best handled on a supercomputer. Output is provided in either full index form or a reduced index form, e.g. the Voigt index notation common to elasticity. As higher-order tensors are often associated with nonlinear material responses, the program provides an accessible means to investigate the important constants involved in nonlinear material modeling. The routine has been used to discover several incorrect relationships reported in the literature.

Quantifying Mechanical Properties of Molecular Crystals: A Critical Overview of Experimental Elastic Tensors.

This review presents a critical and comprehensive overview of current experimental measurements of complete elastic constant tensors for molecular crystals. For a large fraction of these molecular crystals, detailed comparisons are made with elastic tensors obtained using the corrected small basis set Hartree-Fock method S-HF-3c, and these are shown to be competitive with many of those obtained from more sophisticated density functional theory plus dispersion (DFT-D) approaches. These detailed comparisons between S-HF-3c, experimental and DFT-D computed tensors make use of a novel rotation-invariant spherical harmonic description of the Young's modulus, and identify outliers among sets of independent experimental results. The result is a curated database of experimental elastic tensors for molecular crystals, which we hope will stimulate more extensive use of elastic tensor information-experimental and computational-in studies aimed at correlating mechanical properties of molecular crystals with their underlying crystal structure.

First-principles calculation of electroacoustic properties of wurtzite (Al,Sc)N

We study the electroacoustic properties of aluminum scandium nitride crystals ${\mathrm{Al}}_{1\ensuremath{-}x}{\mathrm{Sc}}_{x}\mathrm{N}$ with the metastable wurtzite structure by means of first-principles calculations based on density functional theory. We extract the material property data relevant for electroacoustic device design, namely, the full tensors of elastic and piezoelectric constants. Atomistic models were constructed and analyzed for a variety of Sc concentrations $0\ensuremath{\le}x\ensuremath{\le}50$%. The functional dependence of the material properties on the scandium concentration was extracted by fitting the data obtained from an averaging procedure for different disordered atomic configurations. We give an explanation of the observed elastic softening and the extraordinary increase in piezoelectric response as a function of Sc content in terms of an element-specific analysis of bond lengths and bond angles.

First principles prediction of exceptional mechanical and electronic behaviour of Titanite (CaTiSiO5)

Titanite (CaTiSiO5) is a naturally occurring silicate phase, recently recognised as a potential material for immobilization of nuclear wastes, high-end ceramic lining and optical device development. The silicate undergoes a pressure-induced structural transition (P21/c to C2/c) at ~3.5 GPa, which we confirm from density function perturbation theory. Using first principles calculations pressure-dependent structural parameters of both the phases are presented. This article features the mechanical property of C2/c titanite characterized by a negative component (C36 = -16.41 GPa) of the elastic constant tensor (Cij), an enigmatic phenomenon seen not only in low symmetry (monoclinic), but also in the high symmetry system (cubic). We propose rotational kinematics of bonds, controlled by the valence charge distributions as a crucial atomic scale mechanism for structural collapse of the lattice under strain resulting in negative C36. This kinematic model allows us to predict the necessary and sufficient condition for the pressure dependent softening of their shear elastic constants (C44 and C55). The present study also sheds a new light upon the electronic properties of titanite, accounting for the intraband and interband transitions that influence the optical activity. Its anisotropic optical properties of titanite evaluated in the range 0–60 eV are marked by an attractive optical behaviour of this phase. It is transparent in the visible spectra, but showing excellent absorption and reflectivity in the UV region. We thus project titanite as an industrially potential UV shield material. Our theoretical estimate yields the highest value of anisotropic refractive index, 2.21 in [001].

Anısotropy Property of Dıfferent Types of Wood

The norm of elastic constant tensor and the norms of the irreducible parts of the elastic constants of different types of wood. The relation of the scalar parts norms and the other parts norms and the anisotropy of these woods is presented. The norm ratios are used as a criterion to present the anisotropy degree of the properties of these woods

Determination of macroscopic stress from diffraction experiments: A critical discussion

The paper is motivated by some inconsistencies and contradictions present in the literature on the calculation of the so-called diffraction elastic constants. In an attempt at unifying the views that the two communities of Materials Science and Mechanics of Materials have on the subject, we revisit and define the terminology used in the field. We also clarify the limitations of the commonly used approaches and show that a unified methodology is also applicable to textured materials with a nearly arbitrary grain shape. We finally compare the predictions based on this methodology with experimental data obtained by in situ synchrotron radiation diffraction on additively manufactured Ti-6Al-4V alloy. We show that (a) the transverse isotropy of the material yields good agreement between the best-fit isotropy approximation (equivalent to the classic Kröner's model) and the experimental data and (b) the use of a general framework allows the calculation of all components of the tensor of diffraction elastic constants, which are not easily measurable by diffraction methods. This allows us to extend the current state-of-the-art with a predictive tool.

Physical properties of prop-2-en-1-one based single crystals using molecular mechanics

A study of mechanical properties of prop-2-en-1-one based compounds using molecular dimension simulations was investigated. Several mechanical properties have been calculated such as elastic constant tensors, shear modulus, bulk modulus, Young’s modulus and Poisson’s ratio. The s-wave and p-wave velocities as well as static dielectric tensor were also calculated. A comparison of these properties with the number of carbon atoms have been carried out to obtain a correlation between molecular structure and mechanical properties and all calculations were performed using GULP program. Correlation between one of the physical parameter and molecular weight has been shown using Functional analysis method. The present study provides a concrete basis to discuss the dominant role played by elastic constants and the critical values associated with these lead to an extent of elastic displacement in the form of 3-dimensional shape which is novel in the context of single crystal analysis. Validation of these results is done in relation to number of carbon atoms and their experimental elastic constants measurements.

Constant electric bias dependence of wave propagation in a rotating piezoelectric crystal

Main theoretical results concerning bulk piezoelectric wave propagation subjected to the action of electric bias and rotation are presented, in which the higher-order tensors of piezoelectric, elastic, dielectric and electrostrictive constants are necessary. Some calculations for lithium niobate are performed revealing that the application of electric bias and rotation has considerable influence on the wave dispersion parameters.

In-situ characterization of growth of isothermal ω phase in metastable β-Ti alloy TIMETAL LCB

Metastable β-Ti alloys exhibit various solid-solid phase transitions. Our study is focused on the characterization of the diffusion controlled β→ωiso phase transition. The particles of ω phase play an important part in thermomechanical treatment since they serve as heterogeneous nucleation sites for precipitation of finely dispersed particles of hexagonal α phase. The in-situ observation of the growth of particles of ω phase could be difficult by conventional techniques. However, it was shown recently that the ω phase significantly influences the elastic constants of the material, and the different forms of ω phase have different effects on the elastic anisotropy, as well as on the internal friction coefficients. Therefore, the β→ω phase transformation could be in-situ observed by the precise measurement of the tensor of elastic constants. In this contribution, we present the study of the kinetics of the β→ωiso phase transformation by resonant ultrasound spectroscopy. The polycrystalline samples of TIMETAL LCB alloy were in-situ examined by this technique during isothermal and non-isothermal ageing at temperatures up to 300 °C.

Perfect acoustic bandgap metabeam based on a quadruple-mode resonator array

Solid structures guide a multitude of elastic modes of different polarizations including both compression and shear, and the nature of the elastic constant tensor implies a much richer behavior than in optics. Here, we introduce a metamaterial in the form of a rectangular cross section beam of a single isotropic material that can simultaneously suppress all elastic-wave polarizations in the beam over a range of frequencies in the kHz range. This is experimentally achieved by machining replicas of a subwavelength unit cell in an aluminum metabeam based on a planar resonator with interconnected ribs, showing complex vibrational degrees of freedom that allow it to couple to compressional, in-plane shear, flexural and torsional vibrations, that is, all four existing mode types. The result is a lightweight structure that can forbid all possible acoustic modes over the metamaterial bandgap frequency range, an exotic behavior that opens up diverse applications in easily manufacturable vibration isolation structures and acoustic wave control.