Elastic Stiffness Tensor Research Articles

On the Identification of Orthotropic Elastic Stiffness Using 3D Guided Wavefield Data.

Scanning laser Doppler vibrometry is a widely adopted method to measure the full-field out-of-plane vibrational response of materials in view of detecting defects or estimating stiffness parameters. Recent technological developments have led to performant 3D scanning laser Doppler vibrometers, which give access to both out-of-plane and in-plane vibrational velocity components. In the present study, the effect of using (i) the in-plane component; (ii) the out-of-plane component; and (iii) both the in-plane and out-of-plane components of the recorded vibration velocity on the inverse determination of the stiffness parameters is studied. Input data were gathered from a series of numerical simulations using a finite element model (COMSOL), as well as from broadband experimental measurements by means of a 3D infrared scanning laser Doppler vibrometer. Various materials were studied, including carbon epoxy composite and wood materials. The full-field vibrational velocity response is converted to the frequency-wavenumber domain by means of Fourier transform, from which complex wavenumbers are extracted using the matrix pencil decomposition method. To infer the orthotropic elastic stiffness tensor, an inversion procedure is developed by coupling the semi-analytical finite element (SAFE) as a forward method to the particle swarm optimizer. It is shown that accounting for the in-plane velocity component leads to a more accurate and robust determination of the orthotropic elastic stiffness parameters.

Modelling and simulation of fracture in anisotropic brittle materials by the phase-field method with novel strain decompositions

When brittle and quasi-brittle materials, are concerned, it is essential to account for the fundamental fact that they behave differently according as they undergo tensile or compressive loadings. Due to this fact, in modelling and simulating the fracture of brittle and quasi-brittle materials by the phase-field method (PFM), the strain tensor is usually decomposed into a tensile part and a compressive part, and it is argued that only the strain energy related to the tensile part controls the nucleation and propagation of cracks. The present work improves the phase-field method by incorporating into it a novel strain decomposition strategy giving rise to strain decompositions being orthogonal in the sense of the natural inner product with the elastic stiffness tensor acting as a metric. The improved phase-field method involving only a scalar phase-field variable is applied to model and simulate the fracture of brittle anisotropic materials. The results thus obtained are compared with the relevant analytical and numerical ones reported in the literature. It turns out from these comparisons that the improved phase-field method is accurate and efficient.

Stability of intermediate states in the surface lithiation process of silicene

We have studied using density functional theory the Li-silicene system, mainly in the saturated region (0.11>X(Li)<1.0) which is critical to understand the solid electrolyte interface layer (SEI) formation. The electronic and vibrational properties of the intermediate states of lithiated silicene were calculated. Two models were used for the Li atoms on the silicene surface: Li on top of Si (perpendicular) and Li on top of the hexagon center (centered). The centered configurations have lower Eb values up to X(Li) = 0.33. However, the vibrational properties showed instabilities for lithium in the hexagon center. An intermediate state at X(Li) = 0.38 was found to be dynamically stable. A metallic to semiconductor transition is observed at X(Li) = 0.33, and X(Li) = 0.46. Few ordered vacancies or vacancies in the dilute region exhibit vibrational stability and gap opening. Elastic stiffness tensor for different concentrations of Li in the silicene-Li system were calculated, a critical concentration of X(Li)=0.333 was determined for the change of the mechanical properties behavior. silicel has a slightly higher 2D Young‘s moduli than silicene. However, vacancies in silicel produce a substantial decrement in mechanical properties.

Molecular dynamics modelling of amorphisation induced change in the mechanical properties of β-Li2TiO3

ABSTRACT The objective of this study is to understand the effect of irradiation on the mechanical properties of Lithium metatitanate (β-) at the molecular level. Computational samples with different levels of damage are subjected to tensile and compressive loads to study changes in elastic stiffness tensor and the crushing strength. Three levels of displacement ratios are chosen; stoichiometric, where atom types are chosen at random, secondly only Lithium atoms are displaced, and finally, atoms to be displaced are chosen in the ratio 60% Li 20% O 20% Ti. Further, local structural changes due to the interaction of stress and radiation are also analysed. It is observed that for stoichiometric and 60% Li, 20% Ti and 20% O displacement cases, the material becomes isotropic at around ≃0.5 dpa. However, for explicit Li displacement, the material continued to show an anisotropic response. This observation indicates that the collapse of the Ti/O sublattice was essential for amorphisation and deteriorating stiffness. However, it was found that very small disturbances in the O/Ti sub-lattice were sufficient to cause a drastic reduction in the crushing strength. Therefore, while small damages below 0.5 dpa may not drastically change the stiffness of the material, it could alter the crushing strength significantly.

Anisotropic temperature-dependent elastic constants and thermal conductivities of TRISO particle coatings

Tristructural isotropic (TRISO) particles show great promise as a candidate fuel for use in several next-generation high-temperature nuclear reactor designs due to their structural integrity and fuel performance at high temperatures and burnups. Computational codes exist that can simulate TRISO fuel performance characteristics and failure probabilities under extreme conditions which require knowledge of the TRISO coatings’ thermophysical properties. The thermophysical descriptions of the TRISO particle's layers (i.e., buffer, pyrolytic carbon, and silicon carbide) currently used in fuel performance codes, however, assume that many of these properties are constant with respect to temperature or texture. In this paper, we obtain the full elastic stiffness tensors of the carbon and silicon carbide layers, which have transversely isotropic symmetry. The calculated elastic properties of each of the coating layers are in remarkable agreement with the current models at room temperature and correct orientations. Additionally, the textured 3C-SiC layer was found to exhibit novel auxetic behavior above 1500 °C. The anisotropic temperature-dependent thermal conductivities of the carbon layers were calculated using acoustical Grüneisen-Debye theory which are in excellent agreement with current models at room temperature and correct orientations. These texture- and temperature-dependent relationships can be incorporated into the thermophysical description of TRISO particles in order to more accurately model fuel performance and failure probabilities under extreme conditions in forthcoming high-fidelity computational simulations.

Molecular dynamics simulation for chlorinated butyl rubber composite reinforced by dendritic amino‐terminated aromatic polyamide

AbstractIn this study, we explore the application of dendrimers in damping modification of chlorinated butyl rubber (CIIR) through molecular dynamics simulation. The dendrimer is dendritic polyamide G2 with polar end groups. The results showed that the composite with 9‐equivalent of G2 exhibited significant damping characteristics. The small difference of solubility indicates that G2 had good compatibility with CIIR. The cohesive energy density and bonding energy are improved. Under the same external force, the hydrogen bond consumes more energy. Besides, the maximum free volume fraction also decreased. The molecular chains of G2/CIIR (9/91) were closer. Thus, the internal friction increased and damping performance was enhanced. In terms of the mechanical properties, the elastic stiffness tensors of the material were almost constant. The glass transition temperature moved to the low‐temperature region. This may indicate a better damping performance in the low‐temperature environment. This study provides a basis for improving CIIR damping performance by using dendrimer.

Dependence of the Elastic Stiffness Tensors of PETN, α‐RDX, γ‐RDX, ϵ‐RDX, ϵ‐CL‐20, DAAF, FOX‐7, and β‐HMX on Hydrostatic Compression

AbstractThe dependence of the components of the elastic stiffness tensors (or elastic constants) of the organic explosives PETN, RDX, CL‐20, DAAF, FOX‐7, and HMX on hydrostatic pressure up to 10 GPa have been computed using dispersion‐corrected density functional theory. We report the evolution of lattice parameters and the non‐zero stiffnesses for the tetragonal, orthorhombic, and monoclinic crystal symmetries. Linear and quadratic dependencies of the components of the elastic stiffness tensors on volumetric compression and hydrostatic pressure are tabulated for use in single crystal plasticity models.

Parameterization on formation free energy of dislocation loops up to 1100 K in bcc iron

This work provides a simple analytical formula with a set of best-fit coefficients for evaluating formation free energies of prismatic dislocation loops as a function of temperature in body-centered-cubic iron. The parameterization is made based on the anisotropic elastic self-energies and corresponding non-linear core-energies of edge dislocations. The variation of these two energy terms with temperature is estimated by including temperature-dependent elastic stiffness tensors in the anisotropic elasticity calculation, and based on the stability of C15 clusters and dislocation loops, respectively. The new analytical formula explicitly contains temperature and is valid for temperatures up to 1100 K.

Atomistic vs. continuum models of nanoporous elastic solid: Stress fields, size-dependent effective stiffness and surface constants

The method of classical molecular dynamics (MD) is applied to study the stress fields and effective elastic stiffness tensor of nanoporous materials. A continuum model for a nanoporous solid involving a surface layer of sub-nanometer thickness is proposed. Systematic comparisons of the unit cell atomistic MD results are performed with two continuum models, one based on the finite element analysis and another based on the multipole series expansion. This comparison validates the continuum models, shows their applicability limits, and provides reliable identification of the surface elastic constants.

Crystal mechanics-based thermo-elastic constitutive modeling of orthorhombic uranium using generalized spherical harmonics and first-order bounding theories

In earlier works, a mathematical procedure for invertible microstructure-property linkages was developed using computationally efficient spectral methods for polycrystalline cubic and hexagonal metals. This paper formulates such invertible microstructure–property linkages for orthorhombic polycrystalline metals relying on the generalized spherical harmonics (GSH) spectral basis. The procedure is used to compute property closures of orthorhombic polycrystals. The closures represent the complete set of theoretically possible combinations of effective properties for a selected material. The procedure relies on the first-order bounding theories and considers orientation distribution functions (ODFs) as the main microstructural descriptor influencing homogenized properties. Numerous examples of these closures involving second-rank thermal expansion and fourth-rank elastic stiffness tensorial properties over a broad range of temperatures are presented for α-uranium (α-U). In doing so, certain key properties of these closures are exploited to facilitate their computation with drastically reduced computational effort. Along with the recently developed GSH-based interpolation procedure for ODFs from coarsely spaced experimental measurement grids to finely spaced finite element mesh resolution grids presented in Barrett et al., the developed computationally efficient ODF-effective property linkages are used to establish a crystal mechanics-based simulation framework coupled with the finite element method (FEM). The ODF dependent thermal expansion and elastic stiffness tensors are efficiently calculated at every integration point and used by the FEM to predict the overall distortion of a hemispherical part made of α-U during heating. It is shown that the developed framework can be used to simulate microstructurally heterogeneous components under thermo-mechanical loadings in a computationally efficient manner.

A critical assessment of interatomic potentials for modelling lattice defects in forsterite Mg_2SiO_4 from 0 to 12\xa0GPa

Five different interatomic potentials designed for modelling forsterite Mg_2SiO_4 are compared to ab initio and experimental data. The set of tested properties include lattice constants, material density, elastic wave velocity, elastic stiffness tensor, free surface energies, generalized stacking faults, neutral Frenkel and Schottky defects, in the pressure range 0-12 GPa relevant to the Earth’s upper mantle. We conclude that all interatomic potentials are reliable and applicable to the study of point defects. Stacking faults are correctly described by the THB1 potential, and qualitatively by the Pedone2006 potential. Other rigid-ion potentials give a poor account of stacking fault energies, and should not be used to model planar defects or dislocations. These results constitute a database on the transferability of rigid-ion potentials, and provide strong physical ground for simulating diffusion, dislocations, or grain boundaries.

ElATools: A tool for analyzing anisotropic elastic properties of the 2D and 3D materials

We introduce a computational method and a user-friendly code with a terminal-based graphical user interface (GUI), named ElATools, developed to analyze mechanical and anisotropic elastic properties. ElATools enables facile analysis of the second-order elastic stiffness tensor of two-dimensional (2D) and three-dimensional (3D) crystal systems. It computes and displays the main mechanical properties including the bulk modulus, Young's modulus, shear modulus, hardness, p-wave modulus, universal anisotropy index, Chung-Buessem anisotropy index, log-Euclidean anisotropy parameter, Cauchy pressures, Poisson's ratio, and Pugh's ratio, using three averaging schemes of Voigt, Reuss, and Hill. It includes an online and offline database from the Materials Project with more than 13,000 elastic stiffness constants for 3D materials. The program supports output files of the well-known computational codes IRelast, IRelast2D, ElaStic, and AELAS. Four types of plotting and visualization tools are integrated to conveniently interface with GNUPLOT, XMGRACE, view3dscene and plotly libraries, offering immediate post-processing of the results. ElATools provides reliable means to investigate the mechanical stability based on the calculation of six (three) eigenvalues of the elastic tensor in 3D (2D) materials. It can efficiently identify anomalous mechanical properties, such as negative linear compressibility, negative Poisson's ratio, and highly-anisotropic elastic modulus in 2D and 3D materials, which are central properties to design and develop high-performance nanoscale electromechanical devices. Moreover, ElATools can predict the behavior of the sound velocities and their anisotropic properties, such as acoustic phase/group velocities and power flow angles in materials, by solving the Christoffel equation. Six case studies on selected material systems, namely, ZnAu2(CN)4, CrB2, δ-phosphorene, Pd2O6Se2 monolayer, and GaAs, and a hypothetical set of systems with cubic symmetry are presented to demonstrate the descriptive and predictive capabilities of ElATools. Program summaryProgram Title:ElAToolsCPC Library link to program files:https://doi.org/10.17632/njh92x66c6.1Developer's repository link:https://github.com/shahramyalameha/ElAToolsLicensing provisions: GNU General Public Licence 3.0Programming language: Fortran 90External routines/libraries:LaPack, Blas, and Plotly Javascript libraries, GNUPLOT, XMGRACE, view3dscene.Nature of problem: Identifying anisotropic elastic properties of 2D and 3D materials, and calculating acoustic phase and group velocities in homogeneous solids.Solution method: Second-order elastic stiffness tensor analysis using transformation law and calculations of the elastic surfaces properties. Solving the Christoffel equation eigenvalue problem using diagonalization and calculations of the sound velocities.Additional comments: Documentation available at https://yalameha.gitlab.io/elastictools/index.html

Strain states and relaxation for alpha-(Al_xGa_{1-x})_2O_3 thin films on prismatic planes of alpha-Al_2O_3 in the full composition range: Fundamental difference of a- and m-epitaxial planes in the manifestation of shear strain and lattice tilt

Pseudomorphic and relaxed alpha-(Al_xGa_{1-x})_2O_3 thin films are grown by combinatorial pulsed laser deposition in the entire composition range on prismatic a- and m-plane alpha-Al_2O_3 substrates. Pseudomorphic growth on m-plane sapphire has been achieved for x ge 0.45. A distinct difference between the a- and m-epitaxial plane is observed in reciprocal space map measurements being in agreement with continuum elasticity theory for rhombohedral heterostructures. While pseudomorphic layers on m-plane sapphire show a pronounced shear strain e'_5 along the c-axis direction, relaxed layers exhibit a global lattice tilt in the same direction. Both effects are not present on the a-epitaxial plane. Out-of-plane lattice constants as well as e'_5 are modeled as function of x employing elasticity theory, confirming theoretical values of the elastic stiffness tensor for alpha-Ga_2O_3, especially the non-zero value of the C_{14} component. Possible pyramidal slip systems for strain relaxation in c-axis direction are examined to explain and numerically model the difference in lattice tilt for the two substrate orientations.Graphic abstract

The ‘effective modulus’ in cubic single crystals subjected to full transverse constraint

The orientation dependence of the ‘effective modulus’ relating the applied stress to the applied strain for a single crystal of a triclinic material subjected to full transverse constraint is specified as a general problem for which special solutions apply when conditions of symmetry exist. When the crystal is cubic and the direction of applied stress is along <100>, <110> or <111> directions, the ‘effective modulus’ is simply the elastic stiffness tensor cij resolved along the direction under consideration. For all other orientations of cubic single crystals, this ‘effective modulus’ can be shown analytically to be less than the elastic stiffness tensor resolved along the direction under consideration.

A damage model for the frictional shear failure of brittle materials in compression

Damage models have been successfully employed for many decades in the modelling of tensile failure, where the crack surfaces separate as a crack grows. The advantage of this approach is that crack trajectories can be computed simply and efficiently on a fixed finite element mesh without explicit tracking. The development of damage models for shear failure in compression, where the crack faces slide over each other subject to friction, is a non-trivial extension of this approach. A major difference is that part of the material stiffness in the damaged region must be retained to avoid interpenetration of the crack faces. This problem is resolved here by employing an anisotropic modification to the elastic stiffness tensor in the damaged region. This has the benefit of driving frictional cracks into the correct orientation, according to the Mohr–Coulomb failure criteria, but three issues remain. The first is that the shear discontinuity introduces some spurious stress perturbations around crack regions that are narrow (less than 3-4 elements wide). This is ameliorated by Helmholtz smoothing to allow efficient simulation on a coarse mesh. The second is that the complementarity of shear stress means that the shear stiffness is removed normal to the crack interface as well as parallel to it, and the third is that there are two potentially active failure planes at each point. Both these latter two issues are resolved by the introduction of a novel failure plane selection variable, which regulates either single plane or dual plane failure, and prevents the growth of erroneous cracks normal to the crack face. Both local and non-local models are investigated for linear and exponential strain softening responses. Unlike the non-local model, the local model demonstrates some mesh-size dependence, but it still retains some properties of interest, in that it supports narrower cracks and more rapidly forms a preference for the growth of a single crack when there are a number of competing cracks. The model is implemented in commercial finite element package COMSOL Multiphysics v5.5 and validated against two benchmark simulations: biaxial compression and the failure of a 45° slope. The correct crack angles, stipulated by the Mohr–Coulomb friction angle, are correctly reproduced, as is the post-failure residual frictional force in biaxial compression. The effect of the shear fracture energy on the force–displacement response is investigated, demonstrating successful simulation of the range of material behaviour expected in geological samples, from broad ranged gradual collapse to sharp, almost instantaneous failure.

Effect of elastic anisotropy on thermally induced distortions of a laser beam in single cubic syngony crystals with radial cooling. Part III

We report a study of thermally induced depolarisation of a laser beam in single cubic syngony crystals of 23 and m3 symmetry groups with an anisotropic elastic stiffness tensor. For active elements in the form of a long rod and a thin disk with radial cooling under uniform volume pumping, we investigate the effect of anisotropy of elastic properties on the dependence of the degree of depolarisation in the crystal on the orientation of the crystallographic axes, as well as on the position of the critical orientation in the crystallographic coordinate system.

Effect of elastic anisotropy on thermally induced distortions of a laser beam in single cubic syngony crystals with radial cooling. Part II

We report a study of thermally induced depolarisation of a laser beam in single cubic syngony crystals of 432, , and m3m symmetry groups with an anisotropic elastic stiffness tensor. For active elements in the form of a long rod and a thin disk with radial cooling under uniform volume pumping, we investigate the effect of anisotropy of elastic properties on the dependence of the degree of depolarisation in the crystal on the orientation of the crystallographic axes, as well as on the position of the critical orientation in the crystallographic coordinate system.

Elastoplastic source model for microseismicity and acoustic emission

SUMMARYThe microseismic events can often be characterized by a complex non-double couple source mechanism. Recent laboratory studies recording the acoustic emission during rock deformation help connecting the components of the seismic moment tensor with the failure process. In this complementary contribution, we offer a mathematical model which can further clarify these connections. We derive the seismic moment tensor based on classical continuum mechanics and plasticity theory. The moment tensor density can be represented by the product of elastic stiffness tensor and the plastic strain tensor. This representation of seismic sources has several useful properties: (i) it accounts for incipient faulting as a microseismicity source mechanism, (ii) it does not require a pre-defined fracture geometry, (iii) it accounts for both shear and volumetric source mechanisms, (iv) it is valid for general heterogeneous and anisotropic rocks and (v) it is consistent with elasto-plastic geomechanical simulators. We illustrate the new approach using 2-D numerical examples of seismicity associated with cylindrical openings, analogous to wellbore, tunnel or fluid-rich conduit and provide a simple analytic expression of the moment density tensor. We compare our simulation results with previously published data from laboratory and field experiments. We consider four special cases corresponding to ‘dry’ elastically homogeneous and elastically heterogeneous isotropic rocks, ‘dry’ transversely isotropic rocks and ‘wet’ isotropic rocks. The model highlights theoretical links between stress state, geomechanical parameters and conventional representations of the moment tensor such as Hudson source type parameters.

Elastic stiffness coefficients of thiourea from thermal diffuse scattering.

The complete elastic stiffness tensor of thiourea has been determined from thermal diffuse scattering (TDS) using high-energy photons (100 keV). Comparison with earlier data confirms a very good agreement of the tensor coefficients. In contrast with established methods to obtain elastic stiffness coefficients (e.g. Brillouin spectroscopy, inelastic X-ray or neutron scattering, ultrasound spectroscopy), their determination from TDS is faster, does not require large samples or intricate sample preparation, and is applicable to opaque crystals. Using high-energy photons extends the applicability of the TDS-based approach to organic compounds which would suffer from radiation damage at lower photon energies.

Elastic Coefficients of β-HMX as Functions of Pressure and Temperature from Molecular Dynamics

The isothermal second-order elastic stiffness tensor and isotropic moduli of β-1,3,5,7- tetranitro-1,3,5,7-tetrazoctane (β-HMX) were calculated, using the P21/n space group convention, from molecular dynamics for hydrostatic pressures ranging from 10−4 to 30 GPa and temperatures ranging from 300 to 1100 K using a validated all-atom flexible-molecule force field. The elastic stiffness tensor components were calculated as derivatives of the Cauchy stress tensor components with respect to linear strain components. These derivatives were evaluated numerically by imposing small, prescribed finite strains on the equilibrated β-HMX crystal at a given pressure and temperature and using the equilibrium stress tensors of the strained cells to obtain the derivatives of stress with respect to strain. For a fixed temperature, the elastic coefficients increase substantially with increasing pressure, whereas, for a fixed pressure, the elastic coefficients decrease as temperature increases, in accordance with physical expectations. Comparisons to previous experimental and computational results are provided where possible.