Symmetric Boundary Research Articles

Buckling and postbuckling behaviors of symmetric/asymmetric double-beam systems

Double-beam systems composed of two interconnected parallel beams have become basic mechanical components of modern engineering structures. Focusing on the asymmetric features in boundary conditions and other structural parameters, this paper studies the buckling and postbuckling behaviors of a double-beam system supported on an elastic foundation. The nonlinear governing differential equations of the system are derived by using the Hamilton's principle, and discretized via the differential quadrature method. Then the pseudo-arc length algorithm is adopted to overcome difficulties of numerical continuation around bifurcation points. Compared with single beam systems, double-beam systems exhibit some peculiarities like multivalued critical buckling loads, in-phase and out-of-phase postbuckling paths. Based on the proposed procedures, these buckling and postbuckling behaviors are investigated in detail. For symmetric boundary cases of the two sub-beams, the multivalued critical buckling loads of the system compose a cluster of critical curves that intersect at a point determined only by the critical loads of single beams. Hence a simple criterion based on single beam critical loads is proposed to quickly estimate whether or not the axial loads would trigger the buckling of the system. However for asymmetric boundary cases, the cluster of critical curves locate in a region that depends on the intrinsic double-beam characteristics, and the criterion is revised accordingly. Also for asymmetric systems, some unique features of the out-of-phase postbuckling paths are detected. It is found that their stabilities are quite different from those of symmetric systems, and that strengthening the system stiffness may even trigger the out-of-phase buckling more easily.

Exact simulation for the first hitting time of Brownian motion and Brownian bridge

We propose an exact simulation scheme for the first hitting time of Brownian motion. Our method is applicable to expanding boundaries, which are linear or piecewise linear. We first develop an algorithm for symmetric linear boundaries, and then extend it to other boundaries using a localization technique. The extended algorithm iteratively invokes a symmetric-case algorithm, and we prove that the expected number of iterations is uniformly bounded. We apply our scheme to simulate the first hitting time of a Brownian bridge.

Symmetry properties of the ground state of the system of interacting spinless bosons

We perform the symmetry analysis of the properties of the ground state of a finite system of interacting spinless bosons for the three most symmetric boundary conditions (BCs): zero BCs with spherical and circular symmetries, as well as periodic BCs. The symmetry of the system can lead to interesting properties. For instance, the density of a periodic Bose system is an exact constant: ρ(r) = const. Moreover, in the case of perfect spherical symmetry of BCs, the crystalline state cannot produce the Bragg peaks. The main result of the article is that symmetry properties and general quantum-mechanical theorems admit equally both crystalline and liquid ground state for a Bose system of any density.

Evolution of Symmetrical Grain Boundaries under External Strain in Iron Investigated by Molecular Dynamics Method

In the present work, the evolution of atomic structures and related changes in energy state, atomic displacement and free volume of symmetrical grain boundaries (GB) under the effects of external strain in body-centered cubic (bcc) iron are investigated by the molecular dynamics (MD) method. The results indicate that without external strain, full MD relaxations at high temperatures are necessary to obtain the lower energy states of GBs, especially for GBs that have lost the symmetrical feature near GB planes following MD relaxations. Under external strain, two mechanisms are explored for the failure of these GBs, including slip system activation, dislocation nucleation and dislocation network formation induced directly by either the external strain field or by phase transformation from the initial bcc to fcc structure under the effects of external strain. Detailed analysis shows that the change in free volume is related to local structure changes in these two mechanisms, and can also lead to increases in local stress concentration. These findings provide a new explanation for the failure of GBs in BCC iron systems.

Influence of Chromium Atoms on the Shear-Coupled Motion of [110] Symmetric Tilt Grain Boundary in α-Iron: Atomic Simulation

Shear-coupled grain boundary motion (SCGBM) is an important mechanism of plastic deformation, especially in the cases of ultrafine-grained or nanocrystalline materials at low temperatures. Much research work has been focused on the geometric rules of coupling, the grain boundary migration mechanisms, or the temperature effect of SCGBM, but the effect of the alloy atoms is seldom involved. In this work, molecular dynamics (MD) simulations were carried out to examine the SCGBM of the Σ17[110](223) and Σ9[110](221) grain boundaries (GBs) in iron-chromium alloys containing from 1 at.% to 9 at.% Cr. A constant shear velocity corresponding to 10 m/s parallel to the boundary plane was applied to the models. Our simulation results indicate that the critical stress of GB migration reduces due to the addition of Cr atoms for the Σ17(223) GB. As for the Σ9(221) GB, sliding occurs simultaneously with coupling in the shear process when the atomic amount of Cr reaches 3%. This phenomenon was also observed in the Σ9(221) GB in pure Fe when the temperature was elevated to 300 K, which was studied in our previous simulation work. The existence of new structural units was demonstrated to be responsible for the sliding of the grain boundary.

Why do compact grain boundary complexions prevail in rock-salt materials?

Over 40 years of studies of grain boundaries in ionic rock-salt materials have left the community divided. On one hand, numerical simulations systematically predict open, hollow structural units to be the ground state. On the other, most recent experiments report compact structures to be the norm. To reconcile modelling with experimental evidence, we investigate the stability of three high-angle symmetric tilt grain boundaries in magnesium oxide MgO with respect to the presence of charge-neutral vacancy pairs. We demonstrate that although open structural units are energetically the most favourable, they are easily destabilized by vacancies. It follows that open complexions can never be at equilibrium with the surrounding grains at finite temperature. On the contrary, compact structural units can accommodate a wide range of defects concentrations, and are much more resilient with respect to the absorption of vacancies. These results highlight the limitations of studies that consider only the ground state, and stress the importance of accounting for the presence of other defects when modelling grain boundaries in ionic materials.

Erratum: Central mean temperature scaling in compressible turbulent channel flows with symmetric isothermal boundaries [Phys. Rev. Fluids 7 , 044606 (2022)

Erratum: Central mean temperature scaling in compressible turbulent channel flows with symmetric isothermal boundaries [Phys. Rev. Fluids <b>7</b> , 044606 (2022)

Numerical simulation for the comparison of thermal and plastic material flow behavior between symmetrical and asymmetrical boundary conditions during friction stir welding

Numerical simulation for the comparison of thermal and plastic material flow behavior between symmetrical and asymmetrical boundary conditions during friction stir welding

Tunable asymmetric transmission of Lamb waves in piezoelectric bimorph plates by electric boundary design

The multi-mode and dispersion natures of elastic guided waves bring big challenges for wave manipulation, and also provide more possibilities for smart device design. In this study, a piezoelectric bimorph plate connected to a periodic arrangement of electric circuits is proposed, and both propagating and evanescent guided Lamb waves are investigated through the complex band structure resulting form using different shunting capacitor or inductor distributions. Results show that keeping and breaking the symmetry of the electric boundary condition can, respectively, lead to decoupling or hybridizing of multi-mode Lamb waves. Accordingly, the asymmetric transmission at a desired frequency can be realized by combining the symmetric and asymmetric electric boundary conditions. Harmonic analysis shows that numerical results are in good agreement with the prediction of complex band structure. Actively changing the electric boundary conditions can be used for guided wave propagation in piezoelectric bimorph plates to easily control the system without changing the structure’s geometry. The framework presented herein offers enhanced capabilities for controlling guided wave propagation for engineering applications and to develop nondestructive testing techniques.

On the propensity of lath martensite intervariant boundaries to hydrogen absorption using molecular statics simulation

In the current study, the influence of boundary character (i.e., tilt, twist and mixed) on hydrogen absorption was, for the first time, investigated for lath martensite using molecular statics simulations. The energies of the martensitic intervariant boundaries varied widely from 306 to 1109 mJ/m2 depending on their character. The symmetric tilt and quasi-tilt boundaries revealed the lowest and highest boundary energies, respectively. This was, to some extent, reflected by distinct boundary unit structure/s and their complexity within a given intervariant boundary type. Interestingly, the intervariant boundary energy was not the main criteria to regulate the propensity of the boundary to hydrogen absorption. Rather, absorption depended on a distinct arrangement of unit structure/s within a given boundary plane, providing different interstitial sites for hydrogen. Indeed, the propensity for hydrogen atom/s was largely determined by the number of inequivalent interstitial sites, which resulted from the interaction of wide varieties of boundary unit structures. The intervariant boundary that consisted of regular arrangements of atoms with fewer boundary unit structure variety exhibited less tendency to H/H2 absorption (e.g., 60°/[111¯] symmetrical tilt boundary). On the contrary, the presence of a substantial number of diverse unit structures within the boundary plane significantly enhanced the tendency for hydrogen segregation (e.g., 51.73°/[116¯11] quasi-tilt and 10.53°/[01¯1¯] twist intervariant boundaries). The current findings provide a fundamental basis for the engineering of the intervariant boundary network within the martensitic microstructure to significantly enhance the hydrogen embrittlement resistance. This is critical for steels used in severe environments, such as hydrogen storage and pipelines for hydrogen transportation.

On Σ9 grain boundary plane orientation

Past measurements of the orientation of the ∑9 grain boundary plane have consistently shown that it adopts a tilt orientation. However, in different circumstances, a wide range of symmetric and asymmetric tilt boundaries are observed. Here, a stereological interpretation of electron-backscatter diffraction data was used to measure the orientation of the ∑9 grain boundary plane in Fe-30Ni. The grain boundary plane orientation was found to be influenced by the types of boundaries meeting the ∑9 at triple points. ∑9 boundaries attached to random high angle boundaries showed a nearly continuous range of symmetric and asymmetric tilt boundaries. ∑9 boundaries attached to ∑3 boundaries selectively adopted a range of lower energy grain boundaries with grain boundary normals between (001) and (11¯1) orientations. As the coherency of the attached ∑3 boundaries increased, the grain boundary plane orientation maximum was found at the (11¯4)//(11¯4) symmetric tilt orientation, thought to be minimum energy for this disorientation. The distribution of ∑9 grain boundary plane orientations depends on the fraction of ∑3 boundaries and how close they are to the coherent twin orientation.

Small-scale laterally-unrestrained corrugated web girders: (II) Parametric studies and LTB design

As shown in the companion paper, innovative tests of corrugated web girders (CWGs) were performed followed by validation of the finite element (FE) model. Accordingly, this paper continues investigating the lateral–torsional buckling of small-scale corrugated web girders through parametric studies. This is to expand the available data required to unveil their behaviour and strength. The layout of the tested girders is considered herein, where the webs of small corrugation dimensions are under pure bending moment with symmetrical boundary conditions. This study deals with the linear and nonlinear buckling analyses of the lateral–torsional buckling (LTB) of CWGs using ABAQUS software. Accordingly, 108 models are conducted, considering the effects of the girder length, web overall dimensions, corrugation dimensions, flange dimensions and different steel grades. The results show that increasing the girder length decreases consequently the ultimate flexural strength and stiffness of CWGs. With regard to the web overall dimensions, increasing the web depth is found to have a significant effect on the strength of CWGs in contrast to the effect of the web thickness value, which has insignificant effect on strength. Furthermore, the results revealed that increasing the flange dimensions increases subsequently the strengths of CWGs. In addition, it is noted that the lower the steel grade, the more effectively the material is used especially for girders failing elastically. Additionally, the critical and design LTB strengths of small-scale CWGs with small corrugation dimensions are evaluated. The comparisons show that the critical LTB stress requires modifications to accord well with FE results. Furthermore, comparisons of the ultimate FE strengths with those of Lindner’s equation (1990), EC3 (2004) and AISC (2010) are provided. The comparisons exhibit that the predictions of Lindner’s equation (1990) are suitable, while the predictions of EC3 (2004) and AISC (2010) are highly conservative. Therefore, modifications are made for EC3 (2004) and AISC (2010) design models to provide more appropriate design strengths for CWGs with small corrugation dimensions for use in buildings.

Arrangement and Decomposition of Grain Boundary Dislocations: Two-Mode Phase-Field Crystal Simulation

The grain-boundary dislocation arrangement and decomposition during constant-volume deformation of a nanoscale bi-crystal system in fcc-structured materials were studied by using the two-mode phase-field crystal (2PFC) method. The effects of different grain boundary misorientations (GBMs) and tensile deformation directions on the dislocation arrangement and decomposition are analyzed. In three different symmetrical tilt grain boundaries evaluated by PFC, the atomic density profile of grain boundaries changed periodically at equilibrium. The initial grain boundary dislocation arrangement of the three samples is almost the same when tensile deformation is applied to the samples in the x- or y- direction, and all are symmetrically arranged in a “bowknot ”structure. The stress at the grain boundary is concentrated with the increase of strain, and dislocation decomposition can effectively reduce the stress concentration. The time steps of dislocation decomposition at grain boundaries decreases with increasing strain rate. This work facilitates the application of PFC in the analysis of grain-boundary mechanics in an extended range of materials.

Atomistic investigation on grain boundary effect on helium segregation and clustering in iron

Molecular dynamics simulation was applied to study the behavior of helium segregation and clustering near different grain boundaries (GBs) in α-Fe. The 〈110〉 and 〈100〉 family symmetric tilt grain boundaries (STGBs) covering a wide range of GB energy and misorientation angle were investigated. The present results indicate that the major factor of GB characteristic effects on helium behavior at the grain boundary are the GB energy and structure. Generally, the proportion of helium that is bound to the grain boundary is higher for high-energy grain boundaries than for low-energy grain boundaries. However, the three grain boundaries with the highest energy lose their original structure units and become disordered under the combined effects of high temperature and helium bubble-induced stress field, forming a diffusion “barrier” for He interstitials to enter the GB plane, thus lead to the slight decrease of the proportion of segregated helium. Four typical He migration patterns at different grain boundaries are found, which depends on the GB structure and energy. In general, with the increase of GB energy, the average diameter of He-V clusters at the GB becomes smaller, and the cluster density becomes larger.

A consistent finite-strain plate model for wrinkling of stretched anisotropic hyperelastic films

Stretch-induced wrinkles usually occur in a thin, clamped–clamped, hyperelastic film and eventually disappear upon excess stretching, with wrinkling direction being perpendicular to the stretching direction within isotropic elasticity framework. Here, we consider in-plane anisotropy induced by infilling fibers in thin films, which significantly affects the orientation and amplitude of wrinkles. To precisely predict the isola-center bifurcation (“birth” and “death” of wrinkles) and to capture wrinkling morphology evolution under large in-plane anisotropic deformations, we develop a consistent finite-strain plate model by accounting for anisotropic hyperelastic constitutive laws. Advantages of this plate model lie in its asymptotic consistency with the 3D field equations and its applicability to large deformations. A new type of rotational symmetric boundary conditions is proposed to reduce the computational cost and to achieve robust convergence. We reveal that the fiber orientation plays a critical role in the wrinkling and restabilization behavior of stretched films. The fibers parallel or perpendicular to the stretching direction can resist the appearance of wrinkles, while the oblique fiber orientation (15°∼75°) can advance wrinkle formation. Besides, oblique wrinkling patterns can be regulated by fiber/matrix shear modulus ratio, and we draw phase diagrams on stability boundary to demonstrate the interactive parametric effects. Our results could provide quantitative guidance to design wrinkle-tunable surfaces for fiber-composite or biomimetic films.

Studying crack propagation along symmetric tilt grain boundary with H segregation in Ni by MD simulation

Hydrogen ingression in metals generally causes catastrophic failure. The H-induced fracture surface mostly exhibits intergranular feature, suggesting that grain boundary (GB) is the dominant crack propagation path. In this paper, the grand canonical Monte Carlo method was first performed to obtain equilibrated H distribution around various symmetric tilt grain boundaries (STGBs) in nickel. Then, atomistic simulations were conducted to study the influence of H segregation on the Mode I crack propagation behavior. In the absence of hydrogen, significant crack tip blunting and transgranular fracture are observed, implyinga tendency toward ductile transgranular fracture. However, in the case of H segregation, it is found that dislocation emission from the crack tip is enhanced but the crack tip still remains highly sharp. Moreover, crack propagation is dominated by alternate GB dislocation emission and GB cleavage, implying a tendency of ductile–brittle alternating intergranular fracture behavior. The H-inhibited GB migration and H-enhanced GB dislocation emission are believed to be the key mechanisms governing the H-assisted crack propagation along the GB.

First-principles prediction of electron grain boundary scattering in fcc metals

The electron reflection probability r at symmetric twin boundaries Σ3, Σ5, Σ9, and Σ11 is predicted from first principles for the eight most conductive face-centered cubic (fcc) metals. r increases with decreasing interplanar distance of atomic planes parallel to the boundary. This provides the basis for an extrapolation scheme to estimate the reflection probability rr at random grain boundaries, which is relatively small, rr = 0.28–0.39, for Cu, Ag, and Au due to their nearly spherical Fermi surfaces, but approximately two times higher for Al, Ca, Ni, Rh, and Ir with a predicted rr = 0.61–0.72. The metal resistivity in the limit of small randomly oriented grains with fixed average size is expected to be proportional to the materials benchmark quantity ρoλ × rr/(1 − rr), where ρo and λ are the bulk resistivity and bulk electron mean free path, respectively. Cu has the lowest value for this quantity, indicating that all other fcc metals have a higher resistivity in the limit of small randomly oriented grains. Thus, the conductivity benefit of replacement metals for narrow Cu interconnect lines can only be realized if the grains are larger than the linewidth or exhibit symmetric orientation relationships where r &amp;lt; rr.

Atomistic migration mechanisms of [formula omitted] symmetric tilt grain boundaries in magnesium

Grain boundary (GB) is an important microstructure and plays a vital role in the mechanical properties of polycrystalline materials by GB migration and sliding. In this work, molecular dynamic (MD) simulations were performed to investigate the migration mechanisms of [12¯10] symmetric tilt grain boundaries (STGBs) in magnesium. A total of 15 STGBs with the rotation angle θ from 0° to 90° were studied under a pure shear loading. The results show that the GB migration mechanisms are significantly influenced by the GB structure. For small angle STGBs (θ<28°), the GB migration is mediated by twin nucleation from GB and subsequent twin growth. For large angle STGBs (θ>83°), the GB migration is achieved by the glide of GB dislocations. The medium angle STGBs (28°<θ<83°), which are the majority of studied STGBs, were observed to be transformed into twin boundary (TB) by emitting lattice dislocations/stacking faults (SFs) during migration. The migration mechanisms for medium angle STGBs can be explained by two rules: GB decomposition and emission of lattice dislocations/SFs. This work provides atomic mechanisms on the GB migration, which are important for understanding the GB behaviors and mechanical properties in magnesium.

Aeroelastic Topology Optimization of Wing Structure Based on Moving Boundary Meshfree Method

The increasing structural flexibility of large aircraft leads to significant aeroelastic effects. More efficient topology optimization techniques are required for the design to further take advantage of aeroelasticity and obtain lightweight structures. This paper proposes a moving boundary meshfree topology optimization that combines the Galerkin method of weighted residuals and non-uniform rational B-splines (NURBS). The solution domain is described by the control points of NURBS and its property is calculated adaptively with an integration subtraction technique. The minimal compliance is searched for using the globally convergent method of moving asymptotes (GCMMA) by designing the locations of control points as subject to volume and flux constraints. The method is first applied to a typical two-dimensional design example with symmetric boundary conditions. The results show that the shape constraints can be conveniently applied, and smoother boundaries are obtained with fewer parameters. Then, a three-dimensional wing structure with asymmetric boundary conditions is optimized. A three-dimensional flight load that combines the high-order-panel and meshfree methods is employed to calculate the elastic loads and update asymmetric external loads during the optimization process. The designed wing satisfies engineering requirements and the presented method can solve the practical topology optimization problems of three-dimensional structures.

Exploring silicon [0 0 1] small angle symmetric tilt grain boundaries: Structures, energies and stress fields

A high efficient hybrid Monte Carlo and molecular dynamics (MC/MD) approach was carried out to reproduce silicon [001] small angle symmetric tilt grain boundaries at the atomic level. In the present paper, we concentrated on the variations of grain boundary properties under various misorientation angles, from which three critical misorientation angles were proposed and four grain boundary stages were divided to illustrate the structural transition of grain boundaries from small to large angle and the corresponding characteristics. The roles of dislocation stress fields in shaping some special grain boundary properties, such as dislocation structures and defect sink, were discussed. We also calculated three critical parameters (dislocation core radius, dislocation core energy and grain boundary width) about structures and energies of these studied grain boundaries, which would benefit the subsequent grain boundary engineering. Further investigations indicated that the three critical misorientation angles could be converted to the average distances of adjacent dislocations, which equalled to 6-, 2- and 1-times of dislocation core radius. Compared with the published literatures and classical theories, the simulations showed well accordance in structural units, stress fields and elastic strain energies.