Full Elasticity Research Articles

Elastic Shape Analysis of Tree-Like 3D Objects Using Extended SRVF Representation.

How can one analyze detailed 3D biological objects, such as neuronal and botanical trees, that exhibit complex geometrical and topological variation? In this paper, we develop a novel mathematical framework for representing, comparing, and computing geodesic deformations between the shapes of such tree-like 3D objects. A hierarchical organization of subtrees characterizes these objects - each subtree has a main branch with some side branches attached - and one needs to match these structures across objects for meaningful comparisons. We propose a novel representation that extends the Square-Root Velocity Function (SRVF), initially developed for Euclidean curves, to tree-shaped 3D objects. We then define a new metric that quantifies the bending, stretching, and branch sliding needed to deform one tree-shaped object into the other. Compared to the current metrics such as the Quotient Euclidean Distance (QED) and the Tree Edit Distance (TED), the proposed representation and metric capture the full elasticity of the branches (i.e., bending and stretching) as well as the topological variations (i.e., branch death/birth and sliding). It completely avoids the shrinkage that results from the edge collapse and node split operations of the QED and TED metrics. We demonstrate the utility of this framework in comparing, matching, and computing geodesics between biological objects such as neuronal and botanical trees. We also demonstrate its application to various shape analysis tasks such as (i) symmetry analysis and symmetrization of tree-shaped 3D objects, (ii) computing summary statistics (means and modes of variations) of populations of tree-shaped 3D objects, (iii) fitting parametric probability distributions to such populations, and (iv) finally synthesizing novel tree-shaped 3D objects through random sampling from estimated probability distributions.

Fully Eulerian models for the numerical simulation of capsules with an elastic bulk nucleus

In this paper, we present a computational framework based on fully Eulerian models for fluid–structure interaction for the numerical simulation of biological capsules. The flexibility of such models, given by the Eulerian treatment of the interface and deformations, allows us to easily deal with the large deformations experienced by the capsule. The modeling of the membrane is based on a full membrane elasticity Eulerian model that is capable of capturing both area and shear variations thanks to the so-called backward characteristics. In the validation section several test cases are presented with the goal of comparing our results to others present in the literature. In this part, the comparisons are done with different well-known configurations (capsule in shear flow and square-section channel), and by deepening the effect of the elastic constitutive law and capillary number on the membrane dynamics. Finally, to show the potential of this framework we introduce a new test case that describes the relaxation of a capsule in an opening channel. In order to increase the challenges of this test we study the influence of an internal nucleus, modeled as a hyperelastic solid, on the membrane evolution. Several numerical simulations of a 3D relaxation phenomenon are presented to provide characteristic shapes and curves related to the capsule deformations, while also modifying size and stiffness parameter of the nucleus.

BROM: An extension of beam theory through model order reduction

Beam analysis played a crucial role in the design of structures throughout modern history. Classical beam theories rely on analytical derivations under certain a priori kinematic assumptions (and are accurate with low computational cost), but they are not amenable to, for example, orthotropic materials and arbitrary cross sections. On the other hand, full 3D linear elasticity theory solved via the Finite Element (FE) method provides solutions for a wide range of engineering problems at the expense of a high computational cost. Reduced-order models try to provide a theory as general as 3D linear elasticity but as efficient as classical beam theories by assuming the solution to be represented in a low-dimensional basis. Among the several options available in the literature, the domain decomposition and hyperreduction via Empirical Cubature Method (ECM), which we name ddROM [1], exhibits this compelling feature. However, ddROM applied to beam structures, may provide more Degrees of Freedom (DoF) per node than most standard commercial FE codes which only consider rigid body motion of the cross section. This paper presents a methodology, called beam Reduced Order Model (bROM), to integrate ddROM in standard FE codes by means of a condensation process followed by a regression procedure. Condensation gives the stiffness matrix, required by FE commercial codes, of a super-element of a given length. This is achieved by assembling several elements and finding the relationship between applied displacements and reactions (Lagrange multipliers). The condensation process is then used to create a database of stiffness matrices as a function of the beam length over which a regression is performed. Some numerical examples are first presented to validate the proposed methodology against the classical beam theory for the case of isotropic beams, and finally, two orthotropic beam showing the utility of the method in comparison with FE.

High temperature decomposition and age hardening of single-phase wurtzite Ti1−xAlxN thin films grown by cathodic arc deposition

Wurtzite TmAlN (Tm=transition metal) themselves are of interest as semiconductors with tunable band gap, insulating motifs to superconductors, and piezoelectric crystals. Characterization of wurtzite TmAlN is challenging because of the difficulty to synthesize them as single-phase solid solution and such thermodynamic, elastic properties, and high temperature behavior of wurtzite Ti1−xAlxN is unknown. Here, we investigated the high temperature decomposition behavior of wurtzite Ti1−xAlxN films using experimental methods combined with first-principles calculations. We have developed a method to grow single-phase metastable wurtzite Ti1−xAlxN (x=0.65, 0.75, 085, and 0.95) solid-solution films by cathodic arc deposition using low duty-cycle pulsed substrate-bias voltage. We report the full elasticity tensor for wurtzite Ti1−xAlxN as a function of Al content and predict a phase diagram including a miscibility gap and spinodals for both cubic and wurtzite Ti1−xAlxN. Complementary high-resolution scanning transmission electron microscopy and chemical mapping demonstrate decomposition of the films after high temperature annealing (950∘C), which resulted in nanoscale chemical compositional modulations containing Ti-rich and Al-rich regions with coherent or semicoherent interfaces. This spinodal decomposition of the wurtzite film causes age hardening of 1–2 GPa. Published by the American Physical Society 2024

Synthesis of Single Crystals of ε-Iron and Direct Measurements of Its Elastic Constants.

Seismology finds that Earth's solid inner core behaves anisotropically. Interpretation of this requires a knowledge of crystalline elastic anisotropy of its constituents-the major phase being most likely ε-Fe, stable only under high pressure. Here, single crystals of this phase are synthesized, and its full elasticity tensor is measured between 15 and 33GPa at 300K. It is calculated under the same conditions, using the combination of density functional theory and dynamical mean field theory, which describes explicitly electronic correlation effects. The predictive power of this scheme is checked by comparison with measurements; it is then used to evaluate the crystalline anisotropy in ε-Fe under higher density. This anisotropy remains of the same amplitude up to densities typical of Earth's inner core.

Phytoremediation of Potentially Toxic Elements from Contaminated Saline Soils Using Salvadora persica L.: Seasonal Evaluation.

Plants in coastal ecosystems are primarily known as natural sinks of trace metals and their importance for phytoremediation is well established. Salvadora persica L., a medicinally important woody crop of marginal coasts, was evaluated for the accumulation of metal pollutants (viz. Fe, Mn, Cu, Pb, Zn, and Cr) from three coastal areas of Karachi on a seasonal basis. Korangi creek, being the most polluted site, had higher heavy metals (HM's) in soil (Fe up to 17,389, Mn: 268, Zn: 105, Cu: 23, Pb: 64.7 and Cr up to 35.9 mg kg-1) and S. persica accumulated most of the metals with >1 TF (translocation factor), yet none of them exceeded standard permissible ranges except for Pb (up to 3.1 in roots and 3.37 mg kg-1 in leaves with TF = 11.7). Seasonal data suggested that higher salinity in Clifton and Korangi creeks during pre- and post-monsoon summers resulted in lower leaf water (ΨWo) and osmotic potential at full turgor (ΨSo) and bulk elasticity (ε), higher leaf Na+ and Pb but lower extractable concentrations of other toxic metals (Cr, Cu, and Zn) in S. persica. Variation in metal accumulation may be linked to metal speciation via specific transporters and leaf water relation dynamics. Our results suggested that S. persica could be grown on Zn, Cr and Cu polluted soils but not on Pb affected soils as its leaves accumulated higher concentrations than the proposed limits.

Individual and interactive responses of woody plants' biomass and leaf traits to drought and shade

AbstractAimWhile drought and shade are critical factors that determine seedling survival and growth, whether the responses of woody seedlings to drought are facilitated, aggravated or unaffected by shade remains uncertain.LocationGlobal.Time period1993 to 2021.Major taxa studiedWoody plants.MethodsWe compiled 58 studies (including 75 woody species from 33 families) in which water and light availability were simultaneously manipulated, and conducted a meta‐analysis on the acclimation and adaptive responses of biomass production and allocation and leaf traits to drought and/or shade.ResultsShade alleviated the adverse impact of drought on photosynthesis (A) by decreasing stomatal limitation and photoinhibition, and increasing leaf relative water content, and thus facilitated biomass production, but shade aggravated the effect of drought on osmotic adjustment as evidenced by the lack of responses of osmotic potential at the full turgor and modulus elasticity at the maximum turgor to drought under shade. The effects of drought and shade on biomass allocation and specific leaf area were orthogonal. Biomass responses to drought and its interaction with shade were least responsive for drought‐tolerators, but the individual and interactive effects of shade and drought on most traits showed little dependence on shade tolerance. Drought–shade interaction increased root biomass and root mass fraction more for deciduous angiosperms than for evergreen angiosperms.Main conclusionsBoth shade and drought had negative impacts on biomass production and photosynthesis, but their interaction had positive effects that were less significant for drought‐tolerators and evergreen angiosperms. In contrast to shade, drought increased biomass allocation to roots, but shade and drought had orthogonal effects on biomass allocation and leaf morphology. These acclimation and adaptive responses of woody seedlings to drought and shade individually and interactively have important implications for predicting woody plant growth and distribution under global climate changes.

Performance review of locking alleviation methods for continuum ANCF beam elements

The absolute nodal coordinate formulation (ANCF) is a nonlinear finite element approach proposed for the large deformation dynamics analysis of beam- and plate/shell-type structures. In the ANCF approach, elastic forces can be defined using three-dimensional elasticity-based continuum mechanics. This approach is often straightforward, and it makes it possible to use advanced material models in the ANCF framework. However, it has been pointed out in several studies that continuum ANCF-based elements with a full three-dimensional elasticity description can suffer from locking phenomena. In this study, a comparison between various combinations of locking alleviation techniques and their applicability to different ANCF beam variants is studied using numerical examples. Furthermore, the enhanced deformation gradient (EDG) technique, which has been proposed recently in finite element literature, is demonstrated for high-order ANCF beam elements. Based on the numerical tests, none of the currently available techniques are suitable for all types of ANCF elements. The paper also shows that the efficiency and accuracy of the techniques are case-dependent. For the ANCF beam element involving higher-order terms with respect to trapezoidal mode, however, the EDG-based techniques are preferable to reduce locking phenomena.

World oil price impacts on country-specific fuel markets: Evidence of a muted global rebound effect

Given the interconnectedness of global oil markets, significant changes in oil consumption in one country could have a spillover impact on international oil consumption. This concept, often referred to as the global rebound effect, estimates how changes in fuel consumption in an individual country influences world fuel prices and subsequent changes on world fuel consumption. Using monthly retail fuel price and consumption data, we build structural vector autoregressive (SVAR) models for 63 countries, which together represent nearly 70% of global refined oil consumption to examine the issue of the global rebound effect. We break out price and consumption elasticities by fuel type (i.e., gasoline versus diesel) and by fuel price policy groupings (i.e., liberalized/ad-hoc), as well as provide individual country and full sample price and consumption elasticities. We find muted impacts of changes in world crude oil prices on global oil consumption. These results suggest that in the short-run country-specific policies that influence oil consumption may not have significant global rebound effects.

Hierarchic models for the free vibration analysis of functionally gradient plates

Hierarchic models for the modal analyses of ceramic–metal functional gradient (FG) plates are introduced. This study was motivated by the fact that the free vibration analyses of those structures are usually conducted by the first- or higher-order shear deformation theories (SDT). Meanwhile, a hierarchic model does not only include these classical theories, but their model level varies from the first order to the 3D full elasticity. In fact, in the hierarchic models, the part of displacement field through the thickness is assumed a priori using polynomials with the desired highest order. Hence, they can be numerically implemented using 2D finite element or mesh-free method, by discretizing only the mid-surface of 3D structures. The global mass and global stiffness matrices of FG plates are computed by adopting the Gaussian quadrature rules for the mid-surface integral and the trapezoidal rule for the thickness-wise integral. The proposed hierarchic models are numerically illustrated and their characteristics are investigated. In addition, the modal characteristics of FG plates are parametrically examined to the key factors of gradient layer. The hierarchic models approach the same limit and show a sequence of model accuracy. Meanwhile, the modal responses of metal-ceramic FG plate structures are dependent of the volume fraction pattern and the relative thickness ratio of gradient layer.

A biharmonic polynomial airy stress function for the square-end adhesive layer and sandwich core

Accurate analysis of the stress, strain, and deformation fields in the square ended adhesive layer or square ended sandwich structure core requires a full elasticity solution. To this end a polynomial Airy stress function that satisfies all displacement boundary conditions, all stress free boundary conditions, and point equilibrium is proposed. The remaining requirement of a full elasticity solution, strain compatibility, is satisfied through a Galerkin approximation of the biharmonic equation on the Polynomial Airy stress function. To explore the utility of this Biharmonic Polynomial solution, the stress fields are explored for the case of idealized simple shear. The peel, shear, and longitudinal normal stress fields are found to be finite, differentiable, and convergent throughout the domain including the sharp corners of the elastic medium. The resulting stress components are compared to the Goland-Reissner Model, the Closed Form Higher Order (CFHO) model, and the Decoupled Biharmonic (DB) model. It is found that previous models significantly underestimate the peak peel stress, and with it the peak von Mises stress and the peak Principal stress values that drive the failure envelope for ductile and brittle materials, respectively.

Seismic anisotropy in deforming halite: evidence from the Mahogany salt body

SUMMARY We present unambiguous evidence that the Mahogany salt body, located in the Northern part of the Gulf of Mexico, is seismically anisotropic. Evidence of anisotropy comes from shear wave splitting data obtained from a vertical seismic profile VSP. The data set consists of 48 vertically aligned receivers in a borehole drilled through the salt body. Splitting analysis is performed on shear wave phases that are converted from compressional waves at the top and bottom of the salt body. The phase converted at the top of the salt layer shows a clear signature of seismic anisotropy, while the phase at the base of the salt layer shows negligible splitting. We investigate the possibility of rock salt halite LPO as a cause of the observed anisotropy. A finite element geomechanical salt deformation model of the Mahogany salt body is developed, where deformation history is used as an input to the texture plasticity simulation program VPSC. Assuming a halite salt body, a full elasticity model is then calculated and used to create a synthetic VSP splitting data set. The comparison between the synthetic and real VSP data set shows that LPO of rock salt can explain the observed anisotropy remarkably well. This is the strongest evidence to date of seismic anisotropy in a deforming salt structure. Furthermore, for the first time, we are able to demonstrate clear evidence that deforming halite is the most likely cause of this anisotropy, combining data set analysis and synthetic full wave form modelling based on calculated rock salt elasticities. Neglecting anisotropy in seismic processing in salt settings could lead to potential imaging errors, for example the deformation models show an averaged delta parameter of δ = –0.06, which would lead in a zero offset reflection setting to a depth mismatch of 6.2 per cent. Our work also show how observations of salt anisotropy can be used to probe characteristics of salt deformation.

3D Myocardial Mechanical Wave Measurements: Toward In Vivo 3D Myocardial Elasticity Mapping

ObjectivesThis study aimed to investigate the potential of a novel 3-dimensional (3D) mechanical wave velocity mapping technique, based on the natural mechanical waves produced by the heart itself, to approach a noninvasive 3D stiffness mapping of the left ventricle. BackgroundMyocardial fibrosis is recognized as a pathophysiological substrate of major cardiovascular disorders such as cardiomyopathies and valvular heart disease. As fibrosis leads to increased myocardial stiffness, ultrasound elastography measurements could provide important clinical information. MethodsA 3D high frame rate imaging sequence was implemented on a high-end clinical ultrasound scanner to achieve 820 volumes/s when gating over 4 consecutive cardiac cycles. Five healthy volunteers and 10 patients with various degrees of aortic stenosis were included to evaluate feasibility and reproducibility. Mechanical waves were detected using the novel Clutter Filter Wave Imaging approach, shown to be highly sensitive to the weak tissue displacements caused by natural mechanical waves. Results3D spatiotemporal maps of mechanical wave velocities were produced for all subjects. Only the specific mechanical wave at atrial contraction provided a full 3D coverage of the left ventricle (LV). The average atrial kick propagation velocity was 1.6 ± 0.2 m/s in healthy volunteers and 2.8 ± 0.8 m/s in patients (p = 0.0016). A high correlation was found between mechanical wave velocity and age (R2 = 0.88, healthy group), septal wall thickness (R2 = 0.73, entire group), and peak jet velocity across the aortic valve (R2 = 0.70). For 3 of the patients, the higher mechanical wave velocity coexisted with the presence of late gadolinium enhancement on cardiac magnetic resonance. ConclusionsIn this study, 3D LV mechanical wave velocities were visualized and measured in healthy volunteers and patients with aortic stenosis. The proposed imaging sequence and measurement technique allowed, for the first time, the measurement of full spatiotemporal 3D elasticity maps of the LV using ultrasound. (Ultrasonic markers for myocardial fibrosis and prognosis in aortic stenosis; NCT03422770)

Full wave sensitivity of SK(K)S phases to arbitrary anisotropy in the upper and lower mantle

SUMMARY Core-refracted phases such as SKS and SKKS are commonly used to probe seismic anisotropy in the upper and lowermost portions of the Earth’s mantle. Measurements of SK(K)S splitting are often interpreted in the context of ray theory, and their frequency dependent sensitivity to anisotropy remains imperfectly understood, particularly for anisotropy in the lowermost mantle. The goal of this work is to obtain constraints on the frequency dependent sensitivity of SK(K)S phases to mantle anisotropy, particularly at the base of the mantle, through global wavefield simulations. We present results from a new numerical approach to modelling the effects of seismic anisotropy of arbitrary geometry on seismic wave propagation in global 3-D earth models using the spectral element solver AxiSEM3D. While previous versions of AxiSEM3D were capable of handling radially anisotropic input models, here we take advantage of the ability of the solver to handle the full fourth-order elasticity tensor, with 21 independent coefficients. We take advantage of the computational efficiency of the method to compute wavefields at the relatively short periods (5 s) that are needed to simulate SK(K)S phases. We benchmark the code for simple, single-layer anisotropic models by measuring the splitting (via both the splitting intensity and the traditional splitting parameters ϕ and δt) of synthetic waveforms and comparing them to well-understood analytical solutions. We then carry out a series of numerical experiments for laterally homogeneous upper mantle anisotropic models with different symmetry classes, and compare the splitting of synthetic waveforms to predictions from ray theory. We next investigate the full wave sensitivity of SK(K)S phases to lowermost mantle anisotropy, using elasticity models based on crystallographic preferred orientation of bridgmanite and post-perovskite. We find that SK(K)S phases have significant sensitivity to anisotropy at the base of the mantle, and while ray theoretical approximations capture the first-order aspects of the splitting behaviour, full wavefield simulations will allow for more accurate modelling of SK(K)S splitting data, particularly in the presence of lateral heterogeneity. Lastly, we present a cross-verification test of AxiSEM3D against the SPECFEM3D_GLOBE spectral element solver for global seismic waves in an anisotropic earth model that includes both radial and azimuthal anisotropy. A nearly perfect agreement is achieved, with a significantly lower computational cost for AxiSEM3D. Our results highlight the capability of AxiSEM3D to handle arbitrary anisotropy geometries and its potential for future studies aimed at unraveling the details of anisotropy at the base of the mantle.

Crystallographic preferred orientation, magnetic and seismic anisotropy in rocks from the Finero peridotite, Ivrea-Verbano Zone, Northern Italy – Interplay of anisotropy contributions from different minerals

Mineral alignment can provide valuable information on a rock's geological history. Many structural, tectonic, or geodynamic studies rely on determining the preferred orientation of rock-forming minerals, either by direct texture determination or using anisotropy measurements as proxies. Robust interpretations of anisotropy require detailed understanding of its relationship with texture. This study investigates the texture, magnetic and seismic anisotropies of ultramafic rocks from the Ivrea-Verbano Zone (IVZ), Northern Italy, with a special focus on the interplay between anisotropy contributions carried by different minerals. Texture was obtained from electron backscatter diffraction, magnetic anisotropy from low- and high-field anisotropy of susceptibility measurements, and seismic anisotropy from P- and S-wave velocities along the three principal directions of the macroscopic fabric, using 22 mm diameter cores. Texture-based models of magnetic anisotropy agree well with corresponding measurements. Larger variability is observed for seismic anisotropy. Possible explanations are (1) velocities were measured in three directions, insufficient to determine the full elasticity tensor, and (2) seismic anisotropy depends on grain boundaries, pores, and cracks in addition to crystallographic preferred orientation. The contributions to magnetic and seismic anisotropy of each constituent mineral can interfere positively, leading to a larger overall anisotropy, or negatively, resulting in a weaker anisotropy. Seismic anisotropy is mostly controlled by olivine, whereas magnetic anisotropy can be dominated by olivine, pyroxene, or hornblende. This study illustrates that, although the relationship between texture and anisotropy can be complex, it is possible to quantitatively predict magnetic, and to a lesser degree seismic, anisotropy from texture data. Even though both magnetic and seismic anisotropies are common proxies for texture, they are not coaxial, and no simple relationship between the degree of magnetic and seismic anisotropy was found. These results underline the importance of understanding different minerals' contributions to anisotropy in upper mantle rocks.

Physical Realization of Elastic Cloaking with a Polar Material.

An elastic cloak is a coating material that can be applied to an arbitrary inclusion to make it indistinguishable from the background medium. Cloaking against elastic disturbances, in particular, has been demonstrated using several designs and gauges. None, however, tolerate the coexistence of normal and shear stresses due to a shortage of physical realization of transformation-invariant elastic materials. Here, we overcome this limitation to design and fabricate a new class of polar materials with a distribution of body torque that exhibits asymmetric stresses. A static cloak for full two-dimensional elasticity is thus constructed based on the transformation method. The proposed cloak is made of a functionally graded multilayered lattice embedded in an isotropic continuum background. While one layer is tailored to produce a target elastic behavior, the other layers impose a set of kinematic constraints equivalent to a distribution of body torque that breaks the stress symmetry. Experimental testing under static compressive and shear loads demonstrates encouraging cloaking performance in good agreement with our theoretical prediction. The work sets a precedent in the field of transformation elasticity and should find applications in mechanical stress shielding and stealth technologies.

Determination of the Complete Elasticity of Nephila pilipes Spider Silk.

Spider silks are remarkable materials designed by nature to have extraordinary elasticity. Their elasticity, however, remains poorly understood, as typical stress–strain experiments only allow access to the axial Young’s modulus. In this work, micro-Brillouin light spectroscopy (micro-BLS), a noncontact, nondestructive technique, is utilized to probe the direction-dependent phonon propagation in the Nephila pilipes spider silk and hence solve its full elasticity. To the best of our knowledge, this is the first demonstration on the determination of the anisotropic Young’s moduli, shear moduli, and Poisson’s ratios of a single spider fiber. The axial and lateral Young’s moduli are found to be 20.9 ± 0.8 and 9.2 ± 0.3 GPa, respectively, and the anisotropy of the Young’s moduli further increases upon stretching. In contrast, the shear moduli and Poisson’s ratios exhibit very weak anisotropy and are robust to stretching.

Competitive Analysis among Multi-product Firms

We analyze and study competition in differentiated product market using public data source. Understanding competitive market structure is critical for firms to assess how their products compete against other firms in a given market. In this paper, we estimate consumer demand, extend clout and vulnerability framework, and study competition among multi-product manufacturers in differentiated product market. For our empirical analysis, we adopt choice-based aggregate demand model and estimate consumer demand while accounting for unobserved product characteristics. Once we estimate consumer demand, we compute full price elasticity matrix and investigate intra- and inter- manufacturer substitutions among consumers. This research offers a framework for marketers to analyze and understand market structures, leading them to informed decisions.

Competitive Analysis among Multi-product Firms

We analyze and study competition in differentiated product market using public data source. Understanding competitive market structure is critical for firms to assess how their products compete against other firms in a given market. In this paper, we estimate consumer demand, extend clout and vulnerability framework, and study competition among multi-product manufacturers in differentiated product market. For our empirical analysis, we adopt choice-based aggregate demand model and estimate consumer demand while accounting for unobserved product characteristics. Once we estimate consumer demand, we compute full price elasticity matrix and investigate intra- and inter- manufacturer substitutions among consumers. This research offers a framework for marketers to analyze and understand market structures, leading them to informed decisions.

Modeling dislocations with arbitrary character angle in face-centered cubic transition metals using the phase-field dislocation dynamics method with full anisotropic elasticity

In this study, we present a phase-field dislocation dynamics (PFDD) method that includes full anisotropic elasticity. We apply it to calculate the equilibrium core structures of dislocations with arbitrary character angle in eight face-centered cubic transition metals. The calculations investigate the effects of the gradient energy density in the total energy density and the choice of the averaging scheme to determine the isotropic equivalent elastic moduli (i.e., Voigt, Reuss, and Hill). We show that the addition of the gradient energy term increases the intrinsic stacking fault (ISF) widths for the edge and screw dislocations in most of the metals studied here, but decreases the ISF widths for the edge dislocations in four metals: Ir, Ni, Pd, and Rh. The analysis indicates that among the three isotropic averaging schemes, the Voigt isotropic equivalent modulus best predicts the ISF widths of the edge dislocations and the Reuss scheme for the ISF widths of the screw dislocations, compared to the full elastic anisotropy. Finally, a critical character angle ( ∼ 60∘) is revealed, at which the PFDD simulations with full elastic anisotropy and those with the isotropic Hill average predict the same ISF width. Our work advances the basic understanding of the elastic anisotropic effects on the equilibrium dislocation core structures and can help guide the choice of isotropic averaged moduli.