Components Of Elastic Tensor Research Articles

Machine learning-assisted investigation of anisotropic elasticity in metallic alloys

This study aims to develop a machine learning (ML)-based method for predicting the elasticity tensor of anisotropic metallic alloys with 21 independent constants. In this respect, the elastic moduli of a batch of additive-manufactured specimens with various orientations, determined by Tait–Bryan angles, are utilized as input data, while the anisotropic elastic tensor are targeted for prediction. Our primary findings indicate the efficacy of the radial basis function neural network (RBFNN) as a robust ML model for elasticity tensor prediction. Notably, the model demonstrates slightly higher efficiency in predicting elasticity tensor components aligned with principal axes compared to off-diagonal components. Moreover, with increasing anisotropy, the model assigns greater contribution to shear and orthotropic components, optimizing prediction performance. Additionally, a case study using Inconel 718 further illustrates the model's effectiveness in capturing the elasticity tensor. Importantly, it is suggested that the proposed ML model offers advantages over experimental methods, such as impulse excitation-based procedures, and iterative numerical simulations, by reducing sample preparation requirements and saving time.

First principles study on the elastic properties of two-dimensional Janus ZrXY (X/Y = Cl, Br, and I, X ≠ Y)

In recent years, a novel two-dimensional semiconductor ZrX2 (X = Cl, Br, and I) has been found to have excellent optoelectronic properties and has attracted much attention. In this paper, the elastic properties of two-dimensional Janus ZrXY (X/Y = Cl, Br, and I, X ≠ Y) are studied by first principles, including elastic parameters, such as elastic tensor components, Young’s modulus, Poisson’s ratio, and stability. Research has found that the lattice parameters of two-dimensional Janus ZrXY are still influenced by the atomic radius. The Janus structure shows excellent dynamic stability both before and after its construction. Based on the elastic theory, the mechanical stability of the two-dimensional Janus ZrXY was proved indirectly. After constructing the two-dimensional Janus ZrXY structure with different planes, the elastic tensor component increases to a certain extent, and Young’s modulus and Poisson’s ratio increase, but the anisotropy of Young’s modulus and Poisson’s ratio decreases significantly.

Mechanical properties of 2D metal–organic and covalent–organic frameworks with non trivial topological band dispersion

Using density functional theory (DFT), we investigate mechanical properties of a few 2D metal–organic frameworks (MOFs) and covalent–organic frameworks (COFs) having Dirac and flat bands. These porous materials have become a subject of great captivation because of their physical stability, distinctive structural characteristics and large surface to volume ratio. The inherent porosity of these frameworks gives rise to many fascinating and occasionally surprising phenomena, which makes them potential candidates for technological applications. For reliable usage of MOFs/COFs in functional nanodevice and practical application, it is quite imperative to investigate their mechanical properties. Thus, herein a particular attention is paid to study elastic deformation of few 2D MOFs and COFs having non trivial topological band dispersion in the regime with linear dependency of stress upon strain. Specially, we consider different types of deformation and find all the components of elastic tensor from the stress–strain and energy–strain curves. These findings may provide useful information to fabricate the MOFs/COFs based devices by lowering the number of experiments.

Stresses at grain boundaries: The maximum incompatibility stress in an infinitely extended elastic bicrystal under uniaxial loading

In a material under stress, grain boundaries may give rise to stress discontinuities. Stress localization is crucial to materials' behavior such as segregation, precipitation, and void nucleation. Here, the stress state at a grain boundary perpendicular to a uniaxial external stress is studied systematically. The grain boundary with the most extreme stress discontinuity is determined for cubic materials within the elastic limit for a bicrystal model. Additionally, grain boundaries with negligible stress discontinuity are identified. The influence of the elastic tensor components, C11, C12, and C44, and grain orientation is studied quantitatively.

Shear properties of MgO inferred using neural networks

Abstract. Shear properties of mantle minerals are vital for interpreting seismic shear wave speeds and therefore inferring the composition and dynamics of a planetary interior. Shear wave speed and elastic tensor components, from which the shear modulus can be computed, are usually measured in the laboratory mimicking the Earth's (or a planet's) internal pressure and temperature conditions. A functional form that relates the shear modulus to pressure (and temperature) is fitted to the measurements and used to interpolate within and extrapolate beyond the range covered by the data. Assuming a functional form provides prior information, and the constraints on the predicted shear modulus and its uncertainties might depend largely on the assumed prior rather than the data. In the present study, we propose a data-driven approach in which we train a neural network to learn the relationship between the pressure, temperature and shear modulus from the experimental data without prescribing a functional form a priori. We present an application to MgO, but the same approach works for any other mineral if there are sufficient data to train a neural network. At low pressures, the shear modulus of MgO is well-constrained by the data. However, our results show that different experimental results are inconsistent even at room temperature, seen as multiple peaks and diverging trends in probability density functions predicted by the network. Furthermore, although an explicit finite-strain equation mostly agrees with the likelihood predicted by the neural network, there are regions where it diverges from the range given by the networks. In those regions, it is the prior assumption of the form of the equation that provides constraints on the shear modulus regardless of how the Earth behaves (or data behave). In situations where realistic uncertainties are not reported, one can become overconfident when interpreting seismic models based on those defined equations of state. In contrast, the trained neural network provides a reasonable approximation to experimental data and quantifies the uncertainty from experimental errors, interpolation uncertainty, data sparsity and inconsistencies from different experiments.

Identification of Linear Elastic Fracture Mechanics Parameters Through the Atomistic Simulation Based Analysis

The contribution presents atomistic modeling of the proximate crack tip and notch stress fields through molecular dynamics method. The study is aspired to comparing atomistic stress distributions for monocrystalline fcc copper and aluminum with continuous stress fields obtained on the basis of classical continuum fracture mechanics. Atomistic simulations were implemented for pure tensile loadings and mixed (I+II mixed loadings) mode loadings in Large-scale Atomistic/ Molecular Massively Parallel Simulator (LAMMPS). In atomistic computations specimens with central cracks and single edge notches made of crystalline fcc copper and aluminum under mixed mode loadings are considered. The elastic tensor components of the selected crystalline materials are obtained via molecular dynamics simulations and the circumferential distributions of stress fields in the straight away proximity of the crack and notch tips in aluminum and copper are constructed. The main regularities arising from the comparative analysis of the atomistic and continuum approaches are revealed. The major conclusion that can be reached after atomistic modeling is that the stress apportionments given by classical linear fracture mechanics qualitatively and quantitatively describe the molecular dynamics stress distributions.

Determination of the temperature dependences of the coefficients of the elastic compliance tensor and the elasticity tensor in the tetragonal phase BaTiO3

A regression analysis of the temperature dependences of the elastic compliance coefficients in the tetragonal phase BaTiO3 has been carried out. Two functions are proposed that have critical temperatures, one of which corresponds to the second to the rest of the coefficients From the obtained analytical dependences, the temperature dependences of the elasticity tensor components in the tetragonal phase are calculated. Formulas for determining the effective coefficient of elasticity and the effective coefficient of elastic compliance are proposed. It is shown that has minims in the near of structural phase transitions in BaTiO3.

Micromechanics Approach for the Overall Elastic Properties of Ceramic Matrix Composites Incorporating Defect Structures

Abstract The initial mechanical response of ceramic matrix composites (CMCs) is linear until the proportional limit. This initial response is characterized by linear elastic properties, which are anisotropic due to the orientation and arrangement of fibers in the matrix. The linear elastic properties are needed during various phases of analysis and design of CMC components. CMCs are typically made with ceramic unidirectional (UD) or woven fiber preforms embedded in a ceramic matrix formed via various processing routes. The matrix processing of interest in this work is the polymer impregnation and pyrolysis (PIP) process. As this process involves pyrolysis to convert a preceramic polymer into ceramic, considerable volume shrinkage occurs in the material. This volume shrinkage can lead to significant defects in the final material in the form of porosity of various size, shape, and volume fraction. These defect structures can have a significant impact on the elastic and damage response of the material. In this paper, a multiscale micromechanics modeling framework is developed to study the effects of processing-induced defects on the linear elastic response of a PIP-derived CMC. A combination of analytical and computational micromechanics approaches is used to derive the overall elastic tensor of the CMC as a function of the underlying constituents and/or defect structures. It is shown that the volume fraction and aspect ratio of porosity at various length scales play an important role in accurate prediction of the elastic tensor. Specifically, it is shown that the through-thickness elastic tensor components cannot be predicted accurately using the micromechanics models unless the effects of defects are considered.

Anisotropic elastic parameter estimation from multicomponent ground-motion observations: a theoretical study

SUMMARY We investigate the potential of multicomponent, single-point ground-motion observations (displacement, rotation and strain) to allow the estimation of near-receiver anisotropic elastic parameters. Based on full-space, plane-wave propagation analysis, we demonstrate that in (locally homogeneous) anisotropic media, the wave propagation direction and the velocities of quasi-P and quasi-S waves can—in principle—be determined from three components of displacements and three components of rotations. Mimicking the situation of a borehole setting, we formulate an inverse problem, estimating the full elastic tensor from multidirectional observations. We show that in the presence of noise it is beneficial to observe additionally a longitudinal strain component (e.g. along the borehole), further constraining the predominantly quasi-P related elastic tensor components.

Machine learning elastic constants of multi-component alloys

The present manuscript explores application of machine learning methods for determining elastic constants and other derived mechanical properties of multi-component alloys. A number of machine learning models, including linear regression, neural network and random forest based models, are trained and tested on a dataset of binary alloys generated using density functional theory (DFT) calculations and spanning over a large number of elemental species in the periodic table. Starting with a wide range of simple and easily accessible compositionally-averaged elemental features, a correlation-based feature selection strategy was used to systematically down-select a set of most relevant features towards the prediction of the elasticity tensor components. The true predictive performance and the associated uncertainties of the models were established by testing on unseen data and bootstrapping, respectively. A single and pair-wise feature partial dependence analysis was performed to visualize the average property trends in the multi-dimensional feature space in order to further understand the achieved predictive performance. The utility of the trained model is further demonstrated by obtaining sufficiently accurate yet highly efficient approximations for bulk modulus, Young’s modulus, shear modulus and Poisson’s ratio for alloys beyond the binary space (i.e., two-component alloys) on which the model was originally trained. More importantly, we test and validate the predictive performance of the developed model directly against the experimentally measured elastic constants of technologically relevant multi-component alloys (such as, Ni- and Ti-based alloys). Finally, utility of such a data-enabled route is demonstrated by predicting the possible range of various elastic properties for vast composition space available within the five component Ni-Cr-Fe-Mo-W alloy system in a high-throughput manner.

Deep learning of the aftershock hysteresis effect based on the elastic dislocation theory

Abstract. This paper selects fault source models of typical earthquakes across the globe and uses a volume extending 100 km horizontally from each mainshock rupture plane and 50 km vertically as the primary area of earthquake influence for calculation and analysis. A deep neural network is constructed to model the relationship between elastic stress tensor components and aftershock state at multiple timescales, and the model is evaluated. Finally, based on the aftershock hysteresis model, the aftershock hysteresis effect of the Wenchuan earthquake in 2008 and Tohoku earthquake in 2011 is analyzed, and the aftershock hysteresis effect at different depths is compared and analyzed. The correlation between the aftershock hysteresis effect and the Omori formula is also discussed and analyzed. The constructed aftershock hysteresis model has a good fit to the data and can predict the aftershock pattern at multiple timescales after a large earthquake. Compared with the traditional aftershock spatial analysis method, the model is more effective and fully considers the distribution of actual faults, instead of treating the earthquake as a point source. The expansion rate of the aftershock pattern is negatively correlated with time, and the aftershock patterns at all timescales are roughly similar and anisotropic.

Comparative analysis of temperature dependencies of the elastic compliance and dielectric permittivity tensor components in BaTiO3 single crystal

The regression analysis of the temperature dependences of the components of the elastic compliance tensor in the tetragonal phase and the inverse permittivity coefficients in the cubic and tetragonal phases is performed in single crystal BaTiO3. Two modified logarithmic functions having a critical temperature are proposed that approximate well the experimental results of in two phases and in a tetragonal phase. It is found that the dielectric constant has a maximum in the region of Tc, and the maximum of is shifted to the region of higher temperatures. There is a large correlation (0.99) between the changes the elastic compliance coefficient and the dielectric constant It is concluded that the anomalies in the elastic and dielectric properties are due to softening of the lattice when approaching a first-order phase transition.

In-situ high energy X-ray diffraction study of the elastic response of a metastable β-titanium alloy

In the present work, the elastic behavior of a metastable β-Ti alloy, Timetal-18, is studied in the β-quenched condition using in-situ far-field (ff) and near-field (nf) high energy synchrotron X-ray diffraction microscopy (HEDM) techniques and full-field crystal elasticity simulation. HEDM enables assessment of the 3D microstructure as well as the complete (6 components) elastic strain tensor and crystallographic orientation of each grain in the gauge volume of a polycrystalline aggregate. Using this elastic strain and orientation data from ~100 grains, a straightforward strategy to obtain the single crystal elastic constants (SEC) is demonstrated. The elastic moduli thus obtained are (in GPa): C11 = 118 ± 5, C12 = 79 ± 3, C44 = 39 ± 4, with a Zener anisotropy ratio (A) of 2.0 ± 0.4. The nf-HEDM 3D microstructure was used to instantiate a virtual microstructure, as input into the parallelized code, Micromechanical Analysis of Stress-strain Inhomogeneities with fast Fourier transform (MASSIF), in order to simulate the grain level constitutive response. The predicted evolutions of the elastic strain tensor components are shown to be in good agreement with the measured values. However, grain-by-grain comparisons emphasize the strong effect of elastic anisotropy, suggesting it may be even higher than A = 2. The accuracy of the approach and implications for modeling the grain-level constitutive response in the plastic regime are discussed.

Determination of temperature dependence of elastic coefficients in ferroelastics under 4/m F 2/m second-order phase transition

The effect of macrosymmetry conservation in the polydomain ferroelastic crystal BiVO4 on the temperature dependence of the elasticity tensor components is analyzed. It is found that the average elasticity tensor for two orientation states in the ferroelastic phase corresponds to the paraelastic phase symmetry. It is shown that redistribution of the values (one increase due to the other) between the components of the elasticity tensor C11 and C22, C13 and C23, C16 and C26, C44 and C55 ​​takes place as the spontaneous deformation changes. The functions describing the temperature dependences of the elasticity tensor components for the second-order phase transition 4/m F 2/m in the ferroelastic and paraelastic phases suggested.

К РАСЧЕТУ ВНУТРЕННИХ НАПРЯЖЕНИЙ ОТ МЕЗОДЕФЕКТОВ, НАКАПЛИВАЮЩИХСЯ НА ГРАНИЦАХ РАЗДЕЛА ПРИ ПЛАСТИЧЕСКОЙ ДЕФОРМАЦИИ ТВЕРДЫХ ТЕЛ

A new method of calculation of elastic stress fields from internal interfaces (intergranular and interphase boundaries) of plastically deformed polycrystals is proposed. As elementary sources of stress fields, rectangular boundary segments containing uniformly distributed segments of dislocation families accumulating on these segments during plastic deformation are considered. It is shown that the elastic stresses field from a segment with arbitrary geometry of plastic flow and segment orientation can be represented as a superposition of fields from four families of continually distributed dislocation segments with tangential and normal components of the Burgers vector. Analytical expressions for the elastic stress tensor components from each of the four families are obtained. In the limiting case, when the length of dislocation segments tends to infinity, the resulting expressions for the components of the elastic stress field transform known expressions for rotational and shear mesodefects. If dislocation segments have a normal burgers vector, then the stress field is equivalent to the stress field from the biaxial dipole of wedge disclinations. In the case of tangential components of the burgers vector, the field is equivalent to the stress field of the planar mesodefect. As an example of the method application, the calculation of internal stress fields in the plastically deformed crystal containing uniformly distributed cubic particles of the second phase is given. It is shown that the intensity of internal stresses at a given value of strain increases with increasing volume fraction of particles. It is found that for a given volume fraction of the second phase, the internal stresses does not depend on their size.

Optical, mechanical, and photoelastic anisotropy of biaxially stretched polyethylene terephthalate films studied using transmission ellipsometer equipped with strain camera and stress gauge

ABSTRACTThe anisotropic properties of polyethylene terephthalate film resulting from its manufacturing process are quantitatively investigated in terms of its optical, mechanical, and photoelastic aspects. Transmission ellipsometers and a Jones‐matrix‐based analysis software together with a 4 × 4 Berreman‐matrix‐based analysis software are adopted to determine the wavelength‐dependent in‐plane birefringence, the principal refractive indices, and the orientation of the optical axis. Mechanical anisotropy is characterized in terms of the elastic compliance tensor components using the measured azimuthal angle‐dependent Young's modulus and Poisson's ratio. From the measured variation of the wavelength‐dependent in‐plane birefringence as a function of tensile stress, the dispersive photoelastic coefficients are obtained for a few sample azimuthal angles, and the components of the photoelastic tensor are determined. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019, 57, 152–160

Intramedullary pinning of a non-homogeneous adaptive elastic bone hollow cylinder: A theoretical implant–bone interface investigation

A metal rod forced into the medullary cavity of a bone, known as an intramedullary pinning, is considered as the quickest and cheapest form of oblique fracture repair available for use in the femur, tibia, metatarsal bones, humerus, ulna and metacarpal bones. Implants endurance as well as a good clinical tolerance depends on the recovery of a physiological stress/strain distribution within bone after implantation. What is well known is that the human femur, along the entire femoral diaphysis, exhibited statistically significant anatomic variation in elastic magnitude and anisotropy and the main question treated here is: does bone's inhomogeneity, expressed here by the proportionality to some power of the hollow bone cylinder radius r with respect to the elastic tensor components, affects substantially the implant–bone interfacial pressure as well as the implant–bone interfacial strains and stresses both immediately and a longtime after insertion?

REDUCED ISOTROPIC CRYSTAL MODEL WITH RESPECT TO THE FOURTH-ORDER ELASTIC MODULI

Using a reduced isotropic crystal model the relationship between the fourth-order elastic moduli of an isotropic medium and the independent components of the fourth-order elastic moduli tensor of real crystals of various crystal systems is found. To calculate the coefficients of these relations, computer algebra systems Redberry and Mathematica for working with high order tensors in the symbolic and explicit form were used, in light of the overly complex computation. In an isotropic medium, there are four independent fourth order elastic moduli. This is due to the presence of four invariants for an eighth-rank tensor in the three-dimensional space, that has symmetries over the pairs of indices. As an example, the moduli of elasticity of an isotropic medium corresponding to certain crystals of cubic system are given (LiF, NaCl, MgO, CaF2). From the obtained results it can be seen that the reduced isotropic crystal model can be most effectively applied to high-symmetry crystal systems.

Orthotropic elastic moduli of biological tissues from ultrasound-assisted diffusing-wave spectroscopy.

We obtain vibro-acoustic (VA) spectral signatures of a remotely palpated region in tissue or tissue-like objects through diffusing-wave spectroscopy (DWS) measurements. Remote application of force is through focused ultrasound, and the spectral signatures correspond to vibrational modes of the focal volume (also called the ROI) excited through ultrasound forcing. In DWS, one recovers the time evolution of mean-square displacement (MSD) of Brownian particles from the measured decay of intensity autocorrelation of light, adapted also to local particles pertaining only to the ROI. We observe that the plateau of the MSD-versus-time curve has noisy fluctuations when ultrasound is applied, which disappear when forcing is removed. It is shown that the spectrum of fluctuations contains peaks corresponding to some of the modes of vibration of the ROI. This enables us to measure the vibrational modes carried by VA waves. We also show recovery of components of the orthotropic elastic tensor pertaining to the material of the ROI from the measured vibrational modes. We first recover the elastic constants for agar slabs, which are verified to be isotropic. Thereafter, we repeat the exercise on fat recovered from pork back tissue, which, from these measurements, is seen to be orthotropic. We validate some of our present measurements through independent runs in a rheometer. The present work is the first step taken, to the best of our knowledge, to characterize biological tissue on the basis of the anisotropic elasticity property, which may potentially aid in the diagnosis and tracking of the progress of cancer in soft-tissue organs.

Approximate Artery Elasticity Using Linear Springs

The mechanical properties of arteries play an essential role in the study of the circulatory system dynamics, which has been becoming increasingly important in the treatment of cardiovascular diseases. Similarly, when building virtual reality simulators, it is crucial to have a tissue model able to respond in real time. The aim of this work is to linearize an artery model to calculate the stiffness of springs. Arteries with three tissue layers (Intima, Media, and Adventitia) are considered and, starting from the stretch-energy density, some of the elasticity tensor components are calculated. The artery is discretized by a two dimensional mesh where the nodes are connected by three kinds of linear springs (one normal and two angular ones). The model linearizes and homogenizes the material response, but it still contemplates the geometric nonlinearity. Comparisons showed a good match with a nonlinear model and with a standard two-dimensional finite element model, when the artery undergoes a stretch in the circumferential and axial directions. The agreement is also good if the arterial tissue undergoes bending. Finally, the Intima layer shows the biggest deviation from linearity when there is a large deformation in the axial direction. If the arterial stretch varies by 1% or less, then the agreement between the linear and nonlinear models is trustworthy.