Symmetric Directions Research Articles

Nonlinear structural strength of steel chain contacted with knuckle-shaped plate under lifting conditions

ABSTRACT The study focuses on evaluating the nonlinear structural strength of steel chains in lifting operations, particularly in contact with knuckle-shaped plates, under various loading conditions. Using finite element analysis (FEA), the research examines how different contact angles (0°, 45°, 90°) and materials (SS400, HT32, HT36, ASTM A356) influence the chain’s performance under tensile loading. The analysis categorises four main loading conditions, referred to as 0 degree through 45 degrees to symmetric direction, each representing different orientations of the chain relative to the lifted object. Among these, lifting at 90 degrees consistently showed the lowest ultimate strength across all materials and chain sizes. The primary cause of this weakness was identified as significant stress concentration at the central link, where the loading angle of 90 degrees led to early yielding. This critical loading condition not only reduced the chain’s load-bearing capacity but also increased the risk of premature failure due to localised stress buildup. The study further highlights that as the yield strength of the material increases, the difference in load capacity between non-symmetric 45 and 90 degrees becomes more pronounced, with 90 degrees lifting exhibiting lower strength across the board. For instance, ASTM A356, although generally stronger than SS400, still demonstrated a notable reduction in load capacity under 90 degrees lifting. Mesh convergence tests were performed to ensure the accuracy of the FEA results, confirming that 12 mesh elements were sufficient to accurately model stress distributions. The results also indicated that increasing the chain diameter could improve load-bearing capacity under most loading conditions, but even larger chains were still vulnerable under 90 degrees lifting. In conclusion, 90 degrees lifting presents a critical design challenge due to its inherent weakness, and mitigating stress concentration at the central link is key to improving chain durability and safety. This study’s findings offer valuable insights for engineers, particularly in the shipbuilding and offshore industries, on how to optimise chain design and improve safety in hoisting operations.

Relation Constrained Capsule Graph Neural Networks for Non-Rigid Shape Correspondence

Non-rigid 3D shape correspondence aims to establish dense correspondences between two non-rigidly deformed 3D shapes. However, the variability and symmetry of non-rigid shapes usually lead to mismatches due to shape deformation, topological changes, or data with severe noise. To finding an accurate correspondence between 3D dynamic shapes for the local deformation complexity, this article proposes a Relation Constrained Capsule Graph Network (RC-CGNet), which combines global and local features by encouraging the relation constraints between the embedding feature space and the input shape space based on the functional maps framework. Specifically, we design a Diffusion Graph Attention Network (DGANet) to segment the surface into parts with correct edge boundary between two regions. The Minimum Spanning Tree (MST) of geodesic curves among the singularities obtained from the segmented parts is added as relation constraints, which can compute isometric correspondences in both direct and symmetric directions. Besides that, the relation-and-attention constrained neural networks are designed to learn the shape correspondence via attention-aware CapsNet and functional maps under relation constraints. To improve the convergence speed and matching accuracy, we propose an optimized residual network structure based on the Nesterov Accelerated Gradient (NAG) to extract local features, and use graph convolution structure to extract global features. Moreover, a lightweight Gated Attention Module (GAM) is designed to fuse global and local features to obtain a richer feature representation. Since the capsule network has better spatial reasoning ability than the traditional convolutional neural network, our novel network architecture is a dual-route capsule network based on Routing Attention Fusion Block (RAFB), filtering low-discriminative capsules from a holistic view by exploiting geometric hierarchical relationships of semantic parts. Experiments on open datasets show that our method has excellent accuracy and wide adaptability.

Prediction of threshold displacement energies in TiC by ab initio molecular dynamics simulation method

The ab-initio molecular dynamics (AIMD) simulation method is used to compute the threshold displacement energies (Ed) of the Ti atom and the C atom in TiC. The Ed is calculated along the three high symmetric crystallographic directions, namely, [100], [110], and [111] and the weighted average of Ed is found to be 33 eV and 32 eV for Ti and C sublattice, respectively. The effect of small deviations from the high symmetry directions is also studied and the weighted average value of Ed along the 2° off-symmetric directions is found to be 35 eV for both the Ti atom and the C atom. Furthermore, the values of Ed are calculated in the presence of vacancies in C sites, which are often encountered in TiCx. In this case, small increases in the magnitude of weighted average values of Ed are observed, and these are found to be 39 eV and 34 eV for Ti and C, respectively. Anti-site defects are formed in all cases of Ti primary knock-on atom (PKA), however, no anti-site defect is formed along the high symmetry direction for C PKA. The calculated Ed values are explained by the structural and chemical properties of TiC. The projected values of Ed would be valuable parameters for modeling irradiation damage in TiC.

Persistence of Spin Coherence in a Crystalline Environment.

We analyze quantum interference in the triplet-exciton pair generated by singlet exciton fission in a molecular crystal and introduce transport-induced dephasing (TID) as a key effect that can suppress the expected fluorescence quantum beats when the triplet-exciton wave function can localize on inequivalent sites. TID depends on the triplet-exciton hopping rate between inequivalent sites and on the energy shifts among the stationary states of the entangled triplet pair in different spatial configurations. The theoretical model is confirmed by experiments in rubrene single crystals, where triplet pairs remain entangled for more than 50ns but quantum beats are suppressed by TID within a few nanoseconds when the magnetic field is misaligned by just a few degrees from specific symmetric directions. Our experiments deliver the zero-field parameters for the rubrene molecule in its orthorhombic lattice and information on triplet-exciton transport, in particular, the triplet-exciton hopping rate between inequivalent sites, which we evaluate to be of the order of 150ps in rubrene.

PH-NODE: A DFPT and finite displacement supercell based python code for searching nodes in topological phononic materials

Exploring the topological physics of phonons is fundamentally important for understanding various practical applications. Here, we present a density-functional perturbation theory and finite displacement supercell based Python 3 software package called PH-NODE for efficiently computing phonon nodes present in real material through a first-principles approach. The present version of the code is interfaced with the WIEN2k, Elk, and ABINIT packages. In order to benchmark the code, six different types of materials are considered, which include (i) FeSi, a well-known double-Weyl point; (ii) LiCaAs, a half-Heusler single-type-I Weyl topological phonon (TP); and (iii) ScZn, coexisting nodal-line and nodal-ring TPs; (iv) TiS, six pairs of bulk Weyl nodes; (v) CdTe, type-II Weyl phonons; (vi) CsTe, coexisting TP and quadratic contact TP. In FeSi, the node points are found at Γ(0,0,0) and R(0.5,0.5,0.5) high symmetric points. Also, there are 21 energy values at which the node points are situated, corresponding to the full Brillouin zone. For LiCaAs, the previously reported literature claims that there is a node point along the W-X high symmetry direction between the highest longitudinal acoustic and the lowest transverse optical branch, while in our DFT calculations, a gap of 0.17 meV is found. Furthermore, ScZn hosts six nodal-ring TPs phonons at the boundary planes of the Brillouin zone in the vicinity of the M high-symmetric point. In addition to this, straight-line TPs are also found along the Γ-X and Γ-R high symmetric directions. Moreover, for TiS, six Weyl node points (WP1, WP2, WP3, WP4, WP5 and WP6) are found along H-K high-symmetric direction. In CdTe, it is found that Weyl points are located along the X-W high-symmetry direction. In the case of CsTe, a TP and a quadratic contact TP are found along the Γ-X direction and at the R high-symmetry point, respectively. The results obtained from the PH-NODE code are in good agreement with the experimentally and theoretically reported data for each material. Program summaryProgram title: PH-NODECPC Library link to program files:https://doi.org/10.17632/sjydzn49nw.1Licensing provisions: GNU General Public License 3.0Programming language: Python 3External routines/libraries: Math, Time, NumPy, SciPyNature of problem: Searching for the phonon-node points corresponding to the given number of phonon-branch using Nelder-Mead's simplex approach.Solution method: We present a density-functional perturbation theory and finite displacement supercell based Python 3 software package called PH-NODE for efficiently computing phonon nodes present in real material through a first-principles approach.

Learning solid dynamics with graph neural network

Deep learning has shown great promise in solid physic dynamic simulation. By incorporating physical laws, recent works have further improved performance. However, existing methods rarely conform to macrophysics and incur computational costs. Furthermore, the velocity direction features of the current model have not been decomposed, forcing the model to learn vector operations. To address the aforementioned problems, we propose a Solid-Graph Network (Solid-GN), designed for more accurate and efficient learning of solid dynamics. Firstly, we design a novel and cost-effective symmetric direction encoding system that achieves macroscopic equivariance in 2D space by representing same relative directional relationships among particle pairs and their flipping version. Secondly, we propose a message-passing mechanism incorporated with the contact process, facilitating an accurate description of interactions through the decomposition of normal and tangential effects. Lastly, four novel datasets for solid dynamics with non-uniform radius particles are released, thereby enabling more complex and realistic physical simulations. Experimental evaluations conducted on five datasets demonstrate our model's performance concerning predictive accuracy, macroscopic equivariance, and computational cost over the state-of-the-art ones.

Molecular Dynamics Simulation of Bauschinger Effect in Cu Nanowire with Different Crystallographic Orientation

In this study, the Bauschringer Effect (BE) resulting from tension-compression deformation applied to nanowires obtained by placing Cu atoms in , and highly symmetric crystallographic directions was investigated using the Molecular Dynamics (MD) simulation method. The forces between atoms were determined from the gradient of the Embedded Atom Method (EAM) potential function, which includes many-body interactions. It was determined that there is an asymmetry between the stress-strain curves obtained as a result of the tension and compression deformation process applied to the model system. From this asymmetry, it was determined that the yield stress obtained in the drawing process for nanowire with crystallographic orientation was greater than the yield strain obtained as a result of the compression process. In contrast, the opposite was found for nanowires with crystallographic orientation and . In addition, after the yield strain value is exceeded as a result of the drawing process applied to the model nanowire system, compression deformation process was applied at different pre-strain values. The existence of the Bauschinger Effect (BE), which is expressed as the yield strength value as a result of forward loading corresponding to the tension operation, is smaller than the yield value obtained as a result of the compression process in which the loading is removed, was determined. To clarify the effect of BE on Cu nanowires with different crystallographic orientations, Bauschinger Stress parameter (BSP) and Bauschinger Parameter (BP) values were calculated.

Fractional structure and texture aware model for image Retinex and low-light enhancement

This paper proposes a fractional structure and texture aware Retinex (FSTAR) model for image decomposition with application to low-light enhancement. First, a novel structure aware measure called Maximum Fractional Difference (MFD) is introduced, which is the maximum of fractional differences of the input image in eight symmetric directions. Then fractional structure and texture aware maps are built based on MFD and Huber function from robust statistics, which are used as weighted matrices in the regularization terms of reflectance and illumination components. To ensure the intrinsic physical constraints of Retinex on two components, two extra penalty terms are incorporated into the objective function of FSTAR to penalize deviations of the estimated components from the corresponding constraints. The FSTAR is solved via an alternating iterative algorithm; i.e., the objective function is minimized with respect to one variable with another variable fixed, and this process is done alternately until termination condition is met. Qualitative and quantitative evaluations of FSTAR on three low-light image datasets, compared to ten state-of-the-art methods, show its superior performance in Retinex decomposition and low-light image enhancement.

Study of the growth mechanism of a self-assembled and ordered multi-dimensional heterojunction at atomic resolution

Multi-dimensional heterojunction materials have attracted much attention due to their intriguing properties, such as high efficiency, wide band gap regulation, low dimensional limitation, versatility and scalability. To further improve the performance of materials, researchers have combined materials with various dimensions using a wide variety of techniques. However, research on growth mechanism of such composite materials is still lacking. In this paper, the growth mechanism of multi-dimensional heterojunction composite material is studied using quasi-two-dimensional (quasi-2D) antimonene and quasi-one-dimensional (quasi-1D) antimony sulfide as examples. These are synthesized by a simple thermal injection method. It is observed that the consequent nanorods are oriented along six-fold symmetric directions on the nanoplate, forming ordered quasi-1D/quasi-2D heterostructures. Comprehensive transmission electron microscopy (TEM) characterizations confirm the chemical information and reveal orientational relationship between Sb2S3 nanorods and the Sb nanoplate as substrate. Further density functional theory calculations indicate that interfacial binding energy is the primary deciding factor for the self-assembly of ordered structures. These details may fill the gaps in the research on multi-dimensional composite materials with ordered structures, and promote their future versatile applications.Graphical

Ultrafast charge transfer in mixed-dimensional WO3-x nanowire/WSe2 heterostructures for attomolar-level molecular sensing

Developing efficient noble-metal-free surface-enhanced Raman scattering (SERS) substrates and unveiling the underlying mechanism is crucial for ultrasensitive molecular sensing. Herein, we report a facile synthesis of mixed-dimensional heterostructures via oxygen plasma treatments of two-dimensional (2D) materials. As a proof-of-concept, 1D/2D WO3-x/WSe2 heterostructures with good controllability and reproducibility are synthesized, in which 1D WO3-x nanowire patterns are laterally arranged along the three-fold symmetric directions of 2D WSe2. The WO3-x/WSe2 heterostructures exhibited high molecular sensitivity, with a limit of detection of 5 × 10−18 M and an enhancement factor of 5.0 × 1011 for methylene blue molecules, even in mixed solutions. We associate the ultrasensitive performance to the efficient charge transfer induced by the unique structures of 1D WO3-x nanowires and the effective interlayer coupling of the heterostructures. We observed a charge transfer timescale of around 1.0 picosecond via ultrafast transient spectroscopy. Our work provides an alternative strategy for the synthesis of 1D nanostructures from 2D materials and offers insights on the role of ultrafast charge transfer mechanisms in plasmon-free SERS-based molecular sensing.

A chaotic differential evolution and symmetric direction sampling for large-scale multiobjective optimization

As large-scale multiobjective optimization problems (LSMOPs) contain many decision variables, most existing large-scale multiobjective optimization algorithms (LSMOEAs) require millions or even tens of millions of function evaluations to obtain high-quality solutions, which is unaffordable for many real-world problems. Moreover, few studies are dedicated to solving LSMOPs under limited function evaluations. This article proposes a Chaotic Differential Evolution and Symmetric Direction Sampling for large-scale multiobjective optimization (CDE-SDS). First, based on the decision variable information of nondominated solutions, a chaotic differential evolution strategy is adopted to accelerate the convergence of the population towards the Pareto front. Second, a symmetric direction sampling strategy is employed, which generates sampling solutions by constructing symmetric search directions in a large-scale decision space. Thus, the high-dimensional decision space can be explored more fully. The two strategies optimize the population alternately to balance exploitation and exploration. The performance of CDE-SDS is compared with seven state-of-the-art large-scale multiobjective evolution algorithms on an artificial test suit LSMOP with 500 to 5000 decision variables and a real-world time-varying ratio error estimation problem with 3000 decision variables. Experimental results show that the performance of CDE-SDS significantly outperforms the seven compared algorithms in terms of both diversity and convergence of solutions under limited function evaluations.

Anisotropic Interlayer Dzyaloshinskii–Moriya Interaction in Synthetic Ferromagnetic/Antiferromagnetic Sandwiches

AbstractThe interfacial Dzyaloshinskii–Moriya interaction (DMI) in ferromagnetic/non‐magnetic‐metal bilayers is essential to stabilize chiral spin textures for potential applications. Recent works reveal that the interlayer DMI is beneficial to designing 3D chiral spin textures that possess fundamental importance and the associated technological promises. Here, the interlayer DM constants are determined quantitatively in synthetic ferromagnetic/antiferromagnetic Pt/Co/Pt/Ru/Pt/Co/Ta structures. The results demonstrate that the interlayer DMI shows uniaxial anisotropic characteristics. The first‐principles calculations elucidate that the anisotropic interlayer DMI is induced by the in‐plane symmetry breaking along two high symmetric directions, which favors the magnetization of adjacent ferromagnetic layers canting in different directions. The anisotropic interlayer DMI is also confirmed by spin‐orbit torque driven asymmetric magnetization switching. Moreover, the interlayer DMI can be tuned by the Ru‐layer‐thickness and beneficial to designing 3D spin textures for future spintronic devices.

CA-CGNet: Component-Aware Capsule Graph Neural Network for Non-Rigid Shape Correspondence

3D non-rigid shape correspondence is significant but challenging in computer graphics, computer vision, and related fields. Although some deep neural networks have achieved encouraging results in shape correspondence, due to the complexity of the local deformation of non-rigid shapes, the ability of these networks to identify the spatial changes of objects is still insufficient. In this paper, we design a Component-aware Capsule Graph Network (CA-CGNet) to further address the features of embedding space based on the component constraints. Specifically, the dynamic clustering strategy is used to classify the features of patches produced by over-segmentation in order to further reduce noise interference. Moreover, aiming at the problem that existing routing ignores the embedding relationship between capsules, we propose a component-aware capsule graph routing to fully describe the relationship between capsules, which regards capsules as nodes in the graph network and constrains nodes through component information. Then, a knowledge distillation strategy is introduced to improve the convergence speed of the network by decreasing the parameters while maintaining accuracy. Finally, a component pair constraint is added to the functional map, and the component-based semantic loss function is proposed, which can compute isomeric in both direct and symmetric directions. The experimental results show that CA-CGNet improves by 10.21% compared with other methods, indicating the accuracy, generalization, and efficiency of our method on the FAUST, SCAPE, TOSCA, and KIDS datasets.

How Do Colloidal Nanoparticles Move in a Solution under an Electric Field?: In Situ Light Scattering Analysis.

Understanding the dynamics of colloidal nanoparticles (NPs) in a solution is the key to assembling them into solids through a solution process such as electrophoretic deposition. In this study, newly developed in situ analysis with light scattering is used to examine NP dynamics induced by a non-uniform electric field. We reveal that the symmetric directions of moving NP aggregates are due to dielectrophoresis between the cylindrical electrodes, while the actual NP deposition is based on the charge of NPs (electrophoresis). Over time, the symmetry of the dynamics becomes less evident, inducing feeble deposition as the less-ordered dynamics become stronger. Eventually, two separate deposition mechanisms emerge as the deposition rate decreases with the change in the NP dynamics. Furthermore, we identify the vortex-like NP motion between the electrodes. These in situ analyses provide insights into the electrophoretic deposition mechanism and NP behavior in a solution under an electric field for fine film construction.

Rectifying Nonreciprocal Perfect Absorber Based on Generalized Effective-Medium Theory for Composite Magnetic Metamaterials

In this work, we demonstrate the implementation of a nonreciprocal perfect absorber (NPA) made of composite magnetic metamaterials (MMs) consisting of an array of dielectric core loaded (DCL) ferrite rods with either hollow or dielectric cores. The NPA can be functionalized as a PA for the incident beam at a specified direction, while at the symmetric direction the absorption is very weak so that a strong reflection is observed due to the excitation of nonreciprocal magnetic surface plasmon. Interestingly, it is shown that the material loss might be beneficial to the absorption, but it will result in the degradation of nonreciprocal performance. For the delicately designed MMs, only a very small material loss is necessary and simultaneously ensures the high nonreciprocal performance of NPA. To interpret the high quality of NPA, we developed a generalized effective-medium theory for the composite MMs, which shows the direct consequence of the DCL ferrite rods with optimized core size and core permittivity. The partial wave analysis indicates that the nonreciprocal dipole resonance in DCL ferrite rod plays a crucial role in improving the nonreciprocity. The narrow band feature and the angular sensitivity make the NPA promising for the diode-like functionalities. In addition, by controlling the magnitude and orientation of bias magnetic field both the operating frequency and the nonreciprocity can be flexibly controlled, adding an additional degree of freedom. The concept proposed in this research is promising for microwave photonics and integrated photonics.

Crustal anisotropy beneath southeastern Tibet inferred from directional dependence of receiver functions

The present study reveals depth-dependent crustal anisotropic signatures beneath southeastern Tibet. It is used to characterize the rheology of major faults and shear zones, which is important for understanding crustal deformation and geodynamic processes beneath the study area. The depth-dependent anisotropic orientations have been investigated based on the directional dependence of radial and tangential receiver functions (RFs). To achieve our objective, we first computed 3683 good-quality P-RFs from 174 teleseismic earthquakes (M ≥ 5.5) recorded within epicentral distance range of 30∘ to 90∘ at 70 seismic stations of the Eastern Syntaxis experiment (XE Network). After that, we employed the harmonic decomposition technique at each seismic station to retrieve the first (k = 0), second (k = 1) and third (k = 2) degree harmonics from the RF dataset. Our study characterizes the type (horizontal or plunging) of the symmetry axis. The anisotropic axes of the upper crust (0–20 km) appear to vary from approximately N-S to NE-SW. They are usually orthogonal to orientations of major faults and suture zones in the region, implying the effect of structure-induced anisotropy. It can be explained by regularly oriented cracks or macroscopic structure alignment along the major faults. The anisotropic orientations of the middle crust (20–40 km) are NE-SW to E-W direction, reflecting a different pattern than those estimated in the upper crust. The lower crustal (40–70 km) anisotropic pattern (E-W or ESE-WNW direction) exhibits distinct orientations than the upper and middle crust. The crystal preferred orientations (CPO) of the mica and amphibole minerals are likely to cause anisotropy observed at mid-to-lower crustal depth ranges, emphasizing the role of ductile deformation due to material movement towards the east underneath southeastern Tibet. Our results, along with S(K)KS and direct S-waves splitting signatures, suggest mid-to-lower crust and lithospheric mantle material movement towards the east, while the discrepancies in anisotropic symmetric axes directions may be indicative for the partial coupling between the crust and upper mantle beneath the region.

Application of Computer-Aided Precious Metal Materials in Electrochemistry of Ceramic Jewelry Design

In order to solve the electrochemical application of precious metal materials in ceramic jewelry design, a computer-aided design method of precious metal materials in ceramic jewelry was proposed. First, the potential energy surface analyzed in this paper is composed of the sum of disomy and trisomy, with different proportions of disomy and trisomy. Therefore, in the fitting process, the above two relations can be used to measure the contribution of the two-body term and the three-body term, respectively; secondly, from the viewpoint of lattice dynamics, precious metals are of special significance, because they are simple fcc lattice and large and pure single crystals can be obtained. Therefore, their accurate phonon dispersion curves can be obtained experimentally; finally, the phonon dispersion curves along the highly symmetric vector direction are given, and the calculated results are in good agreement with the experimental data. This shows that the new analytical potential energy surface accurately reproduces the interactions inside these crystals. The new analytical potential energy surfaces of the three noble metals correctly reproduce the macroscopic properties of the system, including elastic modulus, cohesive energy, and phonon dispersion curve, as well as important surface features. For all three systems, the order of the surface energies of the unreconstituted surfaces is (110) > (100) > (111), which is also correct. At the same time, the new potential energy surface gives the correct relaxation behavior of the unreconstituted surface.

Wind Forces and Flow Patterns of Three Tandem Prisms with a Small Height–Width Ratio

Wind tunnel tests and large eddy simulations were conducted to investigate the dependency of wind forces and flow patterns on the spacing (S) for three tandem prisms with a small height–width ratio H/W = 0.4. At the spacing ratio S/W = 0.7, mean and root-mean-square drag of downstream prisms have large local peaks, and their magnitudes are larger than those at adjacent spacing ratios; these should be noted to ensure the safety and economy of the wind-resistant design of prism-like low-rise buildings. These phenomena are different from that of a small group of tandem prisms with a large H/W and a large group of tandem prisms with a small H/W. At S/W = 0.7, tap pressure time histories of downstream prisms are non-stationary with abrupt changes, but wind force time histories of downstream prisms are stationary, unlike a small group of tandem prisms with a large H/W, where both tap pressure and win d force time histories are non-stationary. Above phenomena at S/W = 0.7 are attributed to a special asymmetric time-averaged wake regime, which has two modes with symmetric wake flow directions and they irregularly switch. The duration of each mode is ruleless. This special wake regime was not observed in previous studies on tandem prisms.

Structural, electronic, elastic, phonon and thermoelectric properties of Heusler-structured intermetallic HfCu2In: Using density functional theory

Structural, electronic, elastic, vibrational and thermoelectric properties of HfCu2In Heusler-structured intermetallic compound in MnCu2Al-type structure with space group Fm-3m has been studied by using ab-initio density functional theory for the first time. The calculated band structure shows the metallic nature of the compound. HfCu2In is ductile which was affirmed by the computed values of the Poisson's ratio, Cauchy's pressure and Pugh's ratio. Thermoelectric properties such as electrical conductivity, Seebeck-coefficient, electronic thermal conductivity and power factor are measured with the BoltzTrap software. The calculated phonon dispersion curve contains positive frequencies in all symmetric directions in the first Brillouin zone, indicating the dynamical stability of the compound in a cubic MnCu2Al-type structure. Furthermore, near the zone center, Raman and infrared phonon modes for alloy have been explored, indicating that the estimated phonon spectra are accurate. The present calculated results are compared with those of the other Heusler alloys of a similar type.

Diversity of the bifurcations and deformations on films bonded to soft substrates: Robustness of the herringbone pattern and its cognate patterns

In this study, we investigate the diversity of the bifurcations and deformations during evolution of periodic patterns on compressed films bonded to compliant substrates. Three-dimensional finite element analysis is performed assuming that the first bifurcation has either the hexagonal or square (i.e., checkerboard) dimple mode. Step-by-step eigenvalue buckling analysis is performed to explore sequential bifurcations on the bifurcated paths. It is found that at the second bifurcation, a rectangular checkerboard or stripe mode occurs depending not only on the Young's modulus ratio of the film and substrate, but also on the magnitude of the imperfection prescribed by the first bifurcation mode. Different bifurcation modes give a family of herringbone deformation patterns with different dimensions, i.e., the evolutional process is multiple and robust. Further, superposition of the identical modes in symmetric directions elucidates the existence of distinctive patterns cognate with the herringbone pattern, including a variety of experimentally observed patterns.