Isotropic Elastic Properties Research Articles

A microarchitecture design methodology to achieve extreme isotropic elastic properties of composites based on crystal symmetries

The present contribution describes an optimization-based design technique of elastic isotropic periodic microarchitectures with crystal symmetries aiming at the realization of composites with extreme properties. To achieve this goal, three consecutive procedures are followed: (i) a series of inverse homogenization problems with symmetry constraints, (ii) a correlation analysis between symmetries and effective elastic properties of the attained microarchitectures, and (iii) the pattern resemblance recognition of these topologies and their redesign, by adopting microstructures with two length-scales, through optimized parametric geometries. This paper is devoted to assessing the third procedure because the first two procedures have been evaluated in previous works of the authors, and here they are only summarized. By applying the methodology, two plane group symmetries are assessed to define two families of 2D periodic parameterized microarchitecture. Once the parameters have been optimized, the resulting composites achieve elastic isotropic properties close to the whole range of the theoretically estimated bounds. Particularly, an unprecedented microstructure attaining the theoretical maximum stiffness is reported. Starting from these parameterized topologies, simple, one-length scale, and easily manufacturable geometries are defined. One of the so-designed microarchitectures has been manufactured and tested, displaying an effective Poisson’s ratio of − 0.90 simultaneously with a high shear modulus.

Micromechanical modeling over two length-scales for elastic properties of graphene nanoplatelet/graphite fiber/polyimide composites

The transversely isotropic elastic properties of the multi-scale composites made of unidirectional T650-35 graphite fibers embedded in graphene nanoplatelets (GNPs)-enriched PMR-15 polyimide resin are evaluated. A multi-step micromechanical model consisted of generalized method of cells (GMC) and Halpin-Tsai approaches is hierarchically developed. The main advantage of this micromechanical methodology lies in its ability to consider the size and the agglomerated state of GNPs. The numerical results of the model are compared to the existing experimental data and a quite good agreement is found. The influences of amount, size, and agglomerated degree of GNPs, volume fraction of graphite fibers on the Young's moduli, Poisson's ratios, and shear moduli along the axial and transverse directions are investigated. The results clearly demonstrate an improvement in the elastic properties of the graphite fiber-reinforced multi-scale composites by the uniform dispersion of nano-graphene fillers in the polyimide resin. Furthermore, the change of GNP size can notably affect the multi-scale composite properties. Thinner GNPs would be preferable to improve the mechanical characteristics. However, the nano-graphene agglomeration drastically reduces the stiffening effect of the GNPs.

Molecular dynamics simulation of transversely isotropic elastic properties of carbon nanocones

In this paper, transversely isotropic elastic properties of carbon nanocones are studied using molecular dynamics simulation implemented in the large-scale atomic/molecular massively parallel simulator (LAMMPS). All atomic interactions are calculated based on the Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) potential energy functions. To determine the five independent elastic constants, four distinct loading conditions, i. e. uniaxial tensile, longitudinal torsion, in-plane biaxial stretching, and in-plane shear are imposed. The results reveal that Young’s and axial shear moduli are dependent on the apex angle of carbon nanocones, while the effect of the length on them is negligible. Furthermore, the in-plane bulk modulus and in-plane shear constant of these structures increase as their apex angle increases.

AVO inversion using pseudoisotropic elastic properties

Amplitude variation with offset (AVO) inversion of an anisotropic data set is a challenging task. Nonnegligible differences in the anisotropy parameters between the various lithologies make the seismic data AVO response completely different from the isotropic synthetic seismogram. In this case, it is difficult to invert for VP/VSand density consistent with well-log data. AVO inversion using pseudoisotropic elastic properties is a practical solution to this problem. Verification of this method was performed using data from an offshore Western Australia field. It was found that wavelet extraction and density inversion are improved significantly by replacing the isotropic elastic properties with the pseudoisotropic properties. Inverted density shows reasonable quality and therefore can be included in the reservoir characterization study. Postinversion analyses can be performed effectively on the pseudoisotropic elastic properties because crossplot analysis shows the increased separation of different lithofacies due to contrasts in anisotropy parameters. This result could have significant implications for other fields, as shale constitutes most of the overburden in conventional oil and gas fields and often shows strong elastic anisotropy.

Optimal grading of TPMS-based lattice structures with transversely isotropic elastic bulk properties

ABSTRACT In this work, a topology optimization (TO) based framework for functional grading of triply periodic minimal surfaces (TPMS) based lattice structures is developed, implemented and demonstrated. Material interpolation laws of the gyroid, G-prime and Schwarz-D surfaces are derived by numerical homogenization for transversely isotropic elasticity and are represented as convex combinations of solid isotropic material with penalization (SIMP) and rational approximation of material properties (RAMP) models. These convex combinations are implemented in the TO-based compliance problem with new upper and lower bounds on the density variables representing the volume fraction limits of the lattices. The lower bound on the density variables is treated by introducing a sigmoid filter in the optimization loop forcing densities below the lower boundary towards zero. The optimal density solution is represented by Shepard interpolations or radial basis function networks, which, in turn, are utilized for the thickness grading of the TPMS-based lattices. In addition, the global boundary of the lattice structure is identified by support vector machines. Finally, a standard triangle language (STL) file is generated from the implicit surfaces by using marching cubes, which is utilized for further studies by nonlinear finite element analysis and to set up 3D printing of the optimal component quickly. The framework is demonstrated for the established L-shaped benchmark and the well-known General Electric engine bracket.

First Principles Density Functional Theory Prediction of the Crystal Structure and the Elastic Properties of Mo2ZrB2 and Mo2HfB2

The Molybdenum rich ternary alloys Mo-M-B (M = Zr, Hf) contain, next to the Mo solid solution (bcc Mo with small amounts of Zr or Hf as substitutional atoms), the binary borides Mo2B, MB and MB2. Recently, it was found that there is also ternary Mo2MB2, but the crystal structure and further properties are currently unknown. Density functional theory (DFT) calculations were used not only to predict the crystal structure of the Mo2MB2 phases, but also to estimate the isotropic and anisotropic elastic properties like bulk, shear and Young’s modulus, as well as the Vickers hardness of these new borides. Several known crystal structures that fulfill the criterion of the chemical composition were investigated, and the AlMn2B2 type structure seems to be the most stable crystal structure for Mo2HfB2 and Mo2ZrB2 as there are no signs of electronic or dynamic instability. Regarding the elastic properties, it was found that Mo2HfB2 shows higher elastic moduli and is less elastically anisotropic than Mo2ZrB2.

Composite tree-like re-entrant structure with high stiffness and controllable elastic anisotropy

Critical drawback of poor stiffness in traditional auxetic materials limits them to be applied where negative Poisson’s ratio behavior and excellent load-bearing capacity are simultaneously desired. In this work, a general prescription for designing auxetic lattice structures with high stiffness and controllable isotropic/anisotropic elastic properties is proposed. Based on it, we rationally design a class of novel auxetic lattice structures whose deformation is mainly governed by the stretching mechanisms and conducted an investigation on a tree-like re-entrant structure. A combination of theoretical predictions, numerical simulations and tensile test experiments have been carried out to gain a comprehensive understanding of the structural in-plane elastic mechanical properties along both the principal axes as well as in the off-axis directions. Besides, theoretical model for designing tree-like re-entrant structure with isotropic elastic properties is established and experimentally verified with specimens fabricated with CFRP composite. The results of this research has proven the tree-like re-entrant structure exhibits excellent effective Young’s modulus, shear modulus and obvious negative Poisson’s ratio effect along omni-direction. Investigation on the directional dependence of structural effective elastic properties has further indicated the proposed structure presents a great potential for designing auxetic material with controllable anisotropic elastic properties. In particular, the proposed structure is capable in achieving isotropic design with any negative Poisson’s ratio between −1 and 0. The design philosophy presented in this paper provides a general prescription for solving the challenge of poor stiffness in traditional auxetic materials and paves a new way for isotropic auxetic design.

Topology optimization for transversely isotropic materials with high-cycle fatigue as a constraint

We propose a topology optimization method for design of transversely isotropic elastic continua subject to high-cycle fatigue. The method is applicable to design of additive manufactured components, where transverse isotropy is often manifested in the form of a lower Young’s modulus but a higher fatigue strength in the build direction. The fatigue constraint is based on a continuous-time model in the form of ordinary differential equations governing the time evolution of fatigue damage at each point in the design domain. Such evolution occurs when the stress state lies outside a so-called endurance surface that moves in stress space depending on the current stress and a back-stress tensor. Pointwise bounds on the fatigue damage are approximated using a smooth aggregation function, and the fatigue sensitivities are determined by the adjoint method. Several problems where the objective is to minimize mass are solved numerically. The problems involve non-periodic proportional and non-proportional load histories. Two alloy steels, AISI-SAE 4340 and 34CrMo6, are treated and the respective as well as the combined impact of transversely isotropic elastic and fatigue properties on the design are compared.

Microscale approximation of the elastic mechanical properties of randomly oriented rock cuttings

This study examined a method for approximating the transversely isotropic (TI) elastic mechanical properties and bedding plane orientations of randomly oriented rock cuttings. Microindentation testing was conducted on multiple rock cuttings with unknown bedding orientations to obtain their experimental indentation moduli. The measured indentation moduli were assumed to be functions of both the bedding orientations and the intrinsic TI mechanical properties of the rock cuttings. This assumption holds due to the anisotropic stress history and preferred horizontal alignment of the rock fabric along the bedding direction, as well as the presence of plate-like clay particles with intrinsic TI mechanical properties. An anisotropic contact mechanics solution was then utilized to predict both the TI elastic mechanical properties and bedding plane orientations of the cuttings using a constrained inverse algorithm that minimizes the error between the predicted indentation modulus (a function of both the predicted elastic constants and the orientation of the cuttings) and the experimental indentation modulus. Several constraints were imposed on the inverse algorithm to mathematically bound the TI stiffness matrix and optimization results. A Monte Carlo simulation was also incorporated into the inverse algorithm to consider the effects of uncertainties in the experimental results. The results obtained from the proposed indentation–inverse algorithm approach show good agreement with the results obtained using non-invasive ultrasonic pulse velocity measurements on a 2.5 cm cube sample.

Transversely isotropic elastic properties of multi-walled boron nitride nanotubes under a thermal environment

The temperature-dependent transversely isotropic elastic properties of multi-walled boron nitride nanotubes (MWBNNTs) were determined using molecular dynamics simulations with a three-body Tersoff potential force field. These elastic properties were calculated by applying the four different loading conditions on MWBNNTs: uniaxial tension, torsional moment, in-plane biaxial tension and in-plane shear. The effect of chirality, number of layers and aspect ratio (AR) were taken into consideration. The results reveal that the elastic constants of MWBNNTs decrease as their number of layers increase. The elastic moduli of MWBNNTs do not depend on the AR, but are a function of chirality. Furthermore, the effect of temperature on the transversely isotropic elastic constants of MWBNNTs was studied. Higher temperature considerably affects the mechanical properties of MWBNNTs. For instance, the reduction in the values of axial Young’s, longitudinal shear, plane-strain bulk and in-plane shear moduli of MWBNNTs was found to be by approximately 10% due to the increase in temperature. The results reveal that the mechanical properties and failure behavior of MWBNNTs significantly depend on the number of layers, chirality and temperature. The finding of this work can be utilized for engineering MWBNNT-based advanced nanocomposite structures for specific application under thermal environment.

Modeling and simulation of a gauge shaped beam coupled with macro fiber composite for energy harvesting application

This paper presents a novel unimorph piezoelectric cantilever with non-conventional geometry named as gauge shaped beam to investigate stress, Eigen frequencies and electric potential for improved vibration energy scavenging. The harvester is made of macro fiber composite (MFC) as the active piezoelectric layer bonded to a brass substrate. Complete transversely isotropic elastic and piezoelectric properties are assigned to the composite structure in a three-dimensional finite element model using COMSOL Multiphysics. Dynamic behaviour of the proposed structure was simulated and the results obtained were compared with traditional rectangular geometry. The results proved that the gauge shaped beam has a higher stress distribution around the gauge length, enhanced electric potential and lower resonance frequencies than the conventional rectangular structure which makes it better for the design of piezoelectric based vibration energy harvesters.

Theoretical investigation of the crystallographic structure, anisotropic elastic response, and electronic properties of the major borides in Ni-based superalloys

ABSTRACT In Ni-based superalloys, it is usually found that borides can strengthen the grain boundaries, thereby resulting in an increase in mechanical strength and high-temperature creep properties. Due to their importance and prevalence in Ni-based superalloys, this study employs first-principles methods to investigate the crystallographic structure, anisotropic elastic response, and electronic properties of the major borides, such as M2B, M5B3 and M3B2 (M: Cr, Mo, W), respectively, which is necessary for the assessment of complex mechanical responses of Ni-based superalloys. The results demonstrate that the studied borides are all thermodynamically and mechanically stable. Among the M x B y binary borides analysed, Cr x B y exhibits the largest shear modulus, Young’s modulus, and Vicker hardness values, and these properties increase with the increase of B contents. The studied borides display nearly isotropic elastic properties except for W5B3 and W3B2. The electronic structure analysis of M x B y shows that the strong hybridisation between M-d and B-p orbitals leads to these borides exhibiting higher theoretical hardness, and the overlapping peaks of M-d and B-p orbitals move to a lower energy area with the increase of B contents, which leads to the increase of shear and Young’s moduli of M x B y . Furthermore, for M3B2 borides, the Cr-B bonds and Cr–Cr bonds are much stronger than the W-B & Mo-B bonds, and W-W & Mo-Mo bonds, respectively, which leads to Cr x B y yielding the largest values of elastic moduli.

Anisotropic Elastic Properties of Battery Anodes

Negative battery electrodes based on metals exhibit theoretical energy densities that surpass those of the intercalation-based anodes used in present-day Li-ion batteries. Nevertheless, metal anodes have a propensity to form dendrites during charging; thus, use of interfacial protection schemes or solid electrolytes (SE) may be necessary for these systems to be practical. The efficacy of these schemes are influenced by the interplay of the mechanical properties of the anode with those of the SE. To aid in the design of robust anode/SE interfaces, the present study employs Density Functional Theory calculations to assess the elastic properties of eight candidate anode materials: Li, Na, K, Ca, Mg, Zn, Al, and Si. The elastic constants, Young’s modulus, and shear modulus are predicted as a function of temperature within the quasi-harmonic approximation. Anisotropy is assessed by resolving the moduli as a function of crystallographic direction. The alkali metals (Li, Na, and K) are predicted to have the smallest elastic moduli overall, and their moduli decrease with increasing atomic number. Regarding anisotropic behavior, Al and Mg are predicted to exhibit highly isotropic elastic properties, while the alkali metals are highly anisotropic. In the cubic systems, the crystallographic directions exhibiting extrema in the elastic properties are diametrically opposed: under axial loading the stiffest (most compliant) orientation is 〈111〉 (〈100〉), while in shear 〈100〉 (〈111〉) is the stiffest (most compliant). Importantly, the maximum anisotropic shear modulus of some metals is observed to be more than twice as large as their respective polycrystalline values. In these cases the polycrystalline properties may be a poor approximation to the elastic behavior. Accounting for this anisotropy, the resistance to dendrite initiation of several classes of solid electrolytes is discussed.

Особенности процессов деформирования монокристаллического никелида титана при действии импульса всестороннего сжатия

Using the example of single-crystal titanium nickelide, the importance of taking into account the volume compressibility anisotropy when calculating the processes of elastoplastic deformation in materials with cubic symmetry of elastic properties is shown. It is shown that the process of uniform volumetric deformation corresponds to the process of non-uniform stress state for materials with cubic symmetry of properties for some orientations of the calculated coordinate system relative to the directions of the main crystallographic axes. The index surface of volume compressibility (or the reciprocal of it - the compression modulus) has a non-spherical shape and is a function of Euler angles. This is shown for the first time by solving a model problem - determining the stress and strain states of a spherical body made of single-crystal titanium nickelide under the action of a comprehensive compression pulse. In the general case of orienting the calculated coordinate system relative to the directions of the main crystallographic axes, an initially spherical body of single-crystal titanium nickelide is deformed into a biaxial ellipsoid under the influence of a comprehensive compression pulse.

Two-level elastic-viscoplastic model: application to the analysis of the crystal anisotropy influence

Multilevel crystal plasticity models have been widely used to study the processes of inelastic deformation of polycrystalline materials over the last 15-20 years. Anisotropy of plastic strains in crystallites is usually taken into account at the mesoscale, while their elastic properties are often assumed to be isotropic. The objective of this work is to assess the differences in the stress-strain characteristics (especially, residual mesoscopic stresses) by taking into account the anisotropy of elastic properties of materials calculated for the isothermal deformation of polycrystals with various types of crystallitelattice symmetry within a representative macrovolume. For this purpose, the results obtained by the authors were compared with the data received for the material with isotropic elastic properties and obtained using the Voigt-Reuss-Hill averaging procedures. The results of the analysis for the stress-strain state of polycrystalline samples with FCC, BCC, and HCP lattices obtained in a simple sheartest(up to the accumulated strain of 50%) are presented. The statistical two-level model developed on the basis of the geometrically nonlinear elasto-viscoplasticity theoryis used to perform calculations. In such constitutive models, the main fundamental relation is the elastic law written in the rate relaxation form in terms of the measures of the stress rates and strain rates, being independent of the choice of a reference frame (or superimposed rigid motion). It is shown that the analysis of the anisotropy effect has a noticeable impacton the characteristic macrovolume stress-strain state only at the initial stage of deformation. Subsequently, for deformations exceeding 1-1.5%, the difference becomes insignificant. At the same time, the results of calculations of the residual mesoscopic stresses (i.e., the stresses after unloading a representative macrovolume), which have a significant effect on the strength characteristics of materials, taking into account the crystallite anisotropy, turned out to be significantly different from those obtained under the isotropy hypothesis.

Study of the Structural, Elastic, Electronic and Optical Properties of the Ternary Acetylides A2MC2 (A = Na, K) and (M = Pb, Pt)

We studied the ternary acetylides A2MC2 (A = Na, K) and (M = Pb, Pt) with trigonal structures by using the density functional theory (DFT) as implemented in the CASTEP code. The calculated lattice parameters and atomic fractional coordinates were in agreement with previous calculations and experimental data. A set of isotropic elastic parameters and related properties, namely the module and shear moduli, Young’s modulus, Poisson’s ratio, average sound velocity and Debye temperature were numerically estimated in the frame work of the Voigt-Reuss-Hill approximation. The absorption, energy-loss and dielectric function were calculated. We studied the main contribution to the optical spectra from the transition from the top four valence bands towards the lower three first one of conduction based on the electronic structures.

Rock physics of the Wolfcamp Formation, Delaware Basin

Production from wells in organic-rich shales often shows considerable lateral variation. Reliable predrill methods to characterize the lateral heterogeneity of such reservoirs are required to optimize the trajectory of lateral wells in these low-permeability reservoirs. Petrophysical interpretation of measured well logs provides information on mineral, porosity, and kerogen content. Combining the results of petrophysical analysis with P-wave, S-wave, and density logs allows generation of a probability density function (PDF) for each of the different significant lithofacies. The PDFs are applied to the P- and S-impedance from prestack seismic amplitude variation with offset inversion to predict the spatial variation in the distribution of lithofacies and associated probability for the Wolfcamp Formation in an area covered by a 3D seismic survey in the Delaware Basin, West Texas. An anisotropic rock-physics model for the Wolfcamp Formation allows the effect of complex mineralogy, organic carbon concentration, and porosity on the P- and S-impedance to be investigated. Kerogen inclusions and pores act to increase Thomsen’s anisotropy parameter [Formula: see text] relative to [Formula: see text], and there is a competition between clay matrix anisotropy and inclusion shape anisotropy in determining the anisotropy of the rock. Inclusions with isotropic elastic properties act to decrease the anisotropy due to the dilution effect, but this decrease is partially offset by the increase in anisotropy due to the anisotropic shape of the inclusions. Application of the model to the determination of minimum horizontal stress indicates that kerogen-rich siliceous shales have the lowest value of minimum horizontal stress, whereas silica-rich calcareous shales, mixed siliceous shales, and clay-rich siliceous shales have higher values and may therefore act as barriers for the vertical growth of hydraulic fractures.

Failure, crack initiation and the tensile strength of transversely isotropic rock using the Brazilian test

Brazilian tests are used to determine tensile strengths of rocks. Failure mechanisms, and the strengths determined, are influenced by the configuration of the load contacts. Most research on how to interpret Brazilian tests is only valid for isotropic rocks. This study addresses Brazilian tests on transversely isotropic rocks, experimentally and analytically, to identify failure conditions for a range of load contact types and anisotropy angles (representing orientations of the transversely isotropic planes with respect to direction normal to the loading). Stress fields are determined analytically and coupled with a failure criterion to infer locations where failure and crack initiations occur. The analytical results are compared with experimental observations using digital image correlation (DIC) for transversely isotropic Hawkesbury sandstone. It is found that anisotropy angles affect the locations of the crack initiations, with agreement between the experimental results and analytical solutions being observed. The study goes on to show that, for a wide range of transversely isotropic rocks and for planes parallel and perpendicular to the loading directions, cracks will always initiate at disc centres as long as the contact area is large (corresponding to a subtended angle at the disc centre of at least 25∘). Failure and cracking at a disc centre enables reliable determination of the indirect tensile strength. The equation for strength, including a stress concentration factor for a range of transversely isotropic elastic properties, is presented for use by practitioners and laboratory technicians.

Effect of saw blade geometry on crack initiation and propagation on the lateral cortical hinge for HTO: Finite element analysis

IntroductionThe hinge plays a primary role in the hold and healing of a high tibial osteotomy (HTO). Weakening of the hinge is a risk factor for failure. The aim of our study was to determine whether the geometry of the saw blade's cutting edge impacts crack initiation or propagation on the hinge. HypothesisA certain cutting edge geometry exists that will reduce this risk. Materials and MethodsA finite element model with transverse isotropic elastic bone properties was created. A 1.27-mm thick saw cut (full thickness in anteroposterior direction) was made leaving a 1cm lateral cortical hinge. Three different cutting edge geometries were compared: rectangular, U-shaped, V-shaped. Opening of the osteotomy was done over 1mm for 1 s by a load applied distally with the proximal portion fixed. In the first simulation, no crack was initiated at the hinge, while in the second simulation, the beginnings of a 2mm crack angled upward at 15° was added. These two simulations were used to identify whether a local stress riser was present at the hinge. This information was used to calculate the energy release rate to the hinge, which corresponds to the energy needed to initiate and propagate a crack on the hinge. ResultsIn the first simulation (no crack initiation), a rectangular saw blade geometry resulted in the lowest local stress concentration. In the second simulation (with crack initiation), the U-shaped geometry resulted in the lowest local stress concentration. The U-shaped geometry had the lowest energy release rate, meaning that it was the least likely to initiate and propagate a crack on the lateral cortical hinge. Discussion/ConclusionKeeping the inherent limitations related to computer modelling in mind, our findings show that a U-shaped cutting edge is least likely to initiate or propagate a crack since it has the lowest energy release rate. This confirms our hypothesis. Level of evidenceV, expert opinion.

Shear anisotropy-driven crystallographic orientation imaging in flexible hexagonal two-dimensional atomic crystals

Hexagonal two-dimensional (2D) atomic crystals commonly reveal intrinsically isotropic elastic properties, but stretching or bending deformation can lead to their mechanical symmetry breaking. So far, little work has been done on strain-induced in-plane anisotropic shear behaviors of such 2D atomic crystals. Here, in theory, we predict the appearance of in-plane shear stiffness anisotropy under uniaxial normal tension strain in monolayer molybdenum disulfide. We verify experimentally such a shear characteristic based on friction-driven stretch deformation during a contact scan by employing transverse shear microscopy, and we demonstrate the visualization of anisotropic shear deformation as a function of crystallographic orientation. The present work provides deep insights into flexibility governed interactions among friction, deformation, and in-plane elastic characteristics in 2D atomic crystals.