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The vibration and buckling characteristics of the sandwich plate with a honeycomb core and functionally graded material (FGM) face sheet have been evaluated in this paper. The honeycomb core, also known as conventional honeycomb and auxetic honeycomb, is modeled based on positive and negative Poisson’s ratio. Material properties of face sheets varied according to simple power-law functionally graded material (P-FGM). Hamilton’s principle has been employed to derive the equation of motion, and Navier’s method is used to solve the plate problem. Three different plate configurations are used to study the effect of the thickness layer. In addition, the outcome based on span-to-thickness ratio, aspect ratio, geometric parameters, and volume fraction exponent of honeycomb structure on frequency parameter and critical buckling load is examined and exhibited for three different plate configurations. The validation of the present formulation is ascertained by comparing it with other available results. Some novel results are presented for different angles of the unit cell that can be useful as a validation study for the forthcoming research on sandwich honeycomb rectangular plates. It is observed that the thickness of the honeycomb layer plays a significant role in affecting the behavior of the sandwich plate. Besides, the auxetic structure is highly sensitive to high excitation frequency applications compared to the conventional honeycomb structure.

This paper reports a detailed numerical investigation on the buckling aspects of laminated composite shell panels of three forms (cylindrical, spherical and hyperbolic paraboloid) with five support conditions exposed to linearly varying in-plane edge load employing eight nodded isoparametric finite element formulation. The impacts of different parameters including ply orientation, load factor, shell forms, aspect ratio, modulus ratio, curvature ratio, width-to-thickness ratio and angle of lamination on the buckling load of shell panels are examined. It is found that the various parameters addressed in this study have a remarkable impact on the buckling phenomena of laminated composite shell panels. Further, a comparison is made showing the effects of five types of compressive edge loads like uniform, triangular, parabolic, partial edge load and point load on the buckling phenomena of laminated shell panels with respect to five support conditions.

In this paper, the bending, buckling, and free vibration of functionally graded porous (FGP) beams are studied based on two beam theories (with or without considering thickness stretching, respectively). The effect of thickness stretching is obtained by comparing the results of the two theories. Two symmetrical distributions and one asymmetrical distribution of pores are considered. Both Young’s modulus and mass density of the FGP beams are in gradient variation in the thickness direction. The governing equations are constructed using Hamilton’s principle. The analytical solutions are obtained by Navier’s method. The effects of slenderness ratios, pore distribution, porosity and thickness stretching on FGP beams have been investigated. The results show that the inhomogeneity of FGP beams in the thickness direction is positively correlated with the effect of thickness stretching.

This paper is concerned with the electro-mechanical buckling analysis of different kinds of smart sandwich shells with functionally graded porous core and piezoelectric sensor–actuator face sheets. Different types of shallow shells with double curvature including convex shells, cylindrical shells and concave shells are analyzed. It is assumed that effective properties of the porous core are functionally graded to vary as a special function of the thickness parameter. The equilibrium equations are established for the doubly-curved smart sandwich shells using the energy method. The stability equations governing the equilibrium position of the smart sandwich shells are obtained as a set of coupled partial differential equations. Closed-form expressions are generated to obtain the critical buckling loads of the smart sandwich shells using the airy stress function. Sandwich shells under different types of electro-mechanical loadings including axial, lateral and hydrostatic pressures are analyzed. These numerical results show the effects of feedback gain, porosity coefficient and piezoelectric layer thickness on the buckling resistance of these smart sandwich structures.

This paper presents a novel analytical solution for the buckling load in 3D of columns that fills a gap in the existing literature by accounting for shear deformation and spatial effects. The latter means that a column can be supported and buckle in any direction. We build on the foundational work of Simo and use a 3D beam model to derive a set of equations that form the basis for our analytical solution. The paper outlines the axioms of the Simo model, presents the corresponding equations, and describes the linearization procedure in details. By linearizing the equations we obtain a set of algebraic equations with boundary conditions, which we manipulate into an analytical solution for the buckling load. An illustrative example using a column with an elliptical cross-section shows how different boundary conditions and the spatial orientation of the column influence the buckling load. This research work not only contributes to the verification of numerical programs, but also provides engineers with a valuable tool for optimizing structural configurations.

Thin-walled composite cylindrical shells are nowadays widely used as primary structures in the aerospace industry. The so-called variable stiffness (VS) composite cylindrical shells with flexible stiffness tailoring characteristics have been made possible by existing AFP techniques. Considering the inherent imperfection sensitivity property of axially compressed thin-walled cylindrical shells, the buckling behaviors of the axially compressed VS composite cylindrical shells with initial geometric imperfections and delamination imperfections are investigated respectively in this paper. The design buckling loads of the axially compressed VS composite cylindrical shells are first determined by the probing method, which is verified by comparing with the existing test data and numerical simulation results. Additionally, a parametric study is conducted to investigate the effects of delamination imperfections on the buckling loads. The constraint delamination sizes of the VS composite cylindrical shells based on the design buckling load in this paper are given. Furthermore, the effects of continuously variable fiber angles on the buckling loads are investigated considering the imperfection sensitivity. The specific VS composite cylindrical shells with both high- buckling loads and low-imperfection sensitivity are obtained. Results indicate that the effects of imperfection sensitivity need to be carefully considered in buckling design of the axially compressed VS composite cylindrical shells.

Although the finite integral transform (FIT) method has been developed for buckling analysis of rectangular thin plates, the existing formulation involves solving complex nonlinear determinantal equations, leading to potentially questionable numerical results. To address this challenge, a new FIT-based formulation is established that transforms the solution of a complex nonlinear equation into that of a straightforward generalized eigenvalue problem. New analytical solutions for buckling of orthotropic rectangular thin plates with rotational restraints are obtained. The solution procedure is implemented via the following four steps: imposing the two-dimensional FIT on the governing equation; inputting deflection conditions of four edges, by which the relations between the transformed deflections and specific unknowns are provided; imposing the one-dimensional FIT on the elastic conditions to replace the unknowns with the transformed deflections, a generalized eigenvalue problem is constructed; determining the final analytical solutions by solving the generalized eigenvalue problem. Comprehensive highly accurate buckling load/mode solutions for typical rotationally restrained orthotropic plates are presented as benchmarks. The effects of boundary conditions, rotational fixity factors, and loading ratios on the buckling behaviors of orthotropic plates are expediently investigated adopting the present solutions.

This paper aims to develop a size-dependent numerical solution for a comprehensive study on the buckling responses of laminated functionally graded (FG) carbon nanotubes (CNTs)-reinforced composite microplates under linear and nonlinear in-plane compressive loading. According to the new modified couple stress theory and isogeometric analysis (IGA), in conjunction with the higher order plate theory, the size-dependent governing motion equations including material length scale parameter for buckling analysis of CNTs-reinforced laminated composite microplates is established. The research considers five types of loading shapes: uniform, triangular, trapezoidal, sinusoidal, and parabolic. The distribution of CNTs varies across the layer’s thickness, either uniformly or functionally patterns. The material properties of the composite layer are derived by using the New rule of mixture. The accuracy and reliability of the present solution are verified by comparing them with existing models. Different numerical examples illustrate the influences of various parameters on buckling responses of laminated FG-CNTs-reinforced composite microplates. These parameters include size-dependent effects, aspect ratio, volume fraction and distribution of CNTs, fiber orientation, lamination scheme, and boundary conditions. Additionally, the study considers a complex geometry of laminated microplates with flower cutouts to provide a comprehensive analysis.

In this paper, the buckling of a simply supported stepped periodic column is studied using an analytical method. The column is composed of biperiodic cells of stepped Euler–Bernoulli continuous segments. The deflection solution in each cell can be expressed from the resolution of a fourth-order differential equation. After expressing the continuity conditions between each cell, it is possible to relate the solution of each cell with respect to its neighbors. The differential eigenvalue problem of the bi-periodic structure is converted into a linear difference eigenvalue problem associated to the coefficients of the expressed solution in each cell. A transcendental equation for the buckling load of the continuous biperiodic column is obtained from the resolution of the discrete linear difference eigenvalue problem. This transcendental equation is valid whatever the number $N$ of bi-periodic cells, with $N$ larger than 2. This general expression is corroborated with the buckling values obtained using a direct method for few cells ($N=2$ and $N=3$ for instance). The behavior of the stability limit for large $N$ values is also specifically studied. It is shown that the bi-periodic column asymptotically converges toward a homogenized Euler–Bernoulli column with equivalent stiffness calibrated from Reuss’s averaging method. More refined beam models are also derived using asymptotic arguments. The buckling load converges toward the one of a gradient beam model for sufficiently large number $N$ of cells, which can be equivalently derived from a second-order homogenized beam theory. The convergence of this second-order homogenized beam model toward the equivalent homogenized Euler–Bernoulli column (obtained from Reuss’s averaging method) is from below, as also reported for the exact solution of the biperiodic continuous column. A comparison is also carried out for large values of $N$ with a nonlocal Euler–Bernoulli model, which has the same order of accuracy as obtained from the gradient beam model (second-order homogenized beam model).

Si field emission arrays (FEAs) were produced using simple optical lithography and plasma dry etching. By optimizing plasma etching conditions, we achieved uniform ultra-sharp emitters with 50 nm radius, 3.6 μm height, 60^° cone angle, and 1.38×10⁶tips/cm² packing density. For the fabricated FEAs, it was found that turn-on voltage was 850 V and the field emission current was 24 μA at 1100 V. It was also found that field enhancement factor γ of the fabricated Si FEAs was approximately 58. Nanomechanical characterization of Si necked tips array was performed by nanoindentation technology. The critical buckling load and critical stress of the Si necked tips array were 570 μN and 1.192 GPa, respectively.

Cross-sectional deformation of multiwalled carbon nanotubes under isotropic radial pressure is investigated in a realm of continuum elastic approximation. The nanotube we assumed is subjected to the embedment into an elastic medium and stiffener insertion into the core cavity. Combination of the two reinforcement manipulations is found to cause kaleidoscopic mode changes in the radial corrugation, in which the cylindrical walls exhibit wavy patterns along the circumferential direction. Physical consequences of the diverse corrugation patters are also discussed.

In this article, we address the problem of Euler's buckling instability in a charged semi-flexible polymer that is under the action of a compressive force. We consider this instability as a phase transition and investigate the role of thermal fluctuations in the buckling critical force. By performing molecular dynamic simulations, we show that the critical force decreases when the temperature increases. Repulsive electrostatic interaction in the finite temperature is in competition with thermal fluctuations to increase the buckling threshold.

Generating and manipulating valley polarization in a controlled method is significant. The inherently broken centrosymmetry of the buckled honeycomb structures gives it both ferroelectricity and valley degree of freedom, which provides an opportunity to realize electrically controlled valley polarization. In the first step, we explored the origin of buckling. The hexagonal structure is polar due to buckling of the surface, but the degree of buckling and the energy barrier to switching electric polarization are determined not solely by the chemical composition. We combined the electronegativity difference, bond length and the distribution of charge density to describe quantificationally the polarity of chemical bonds. It shows the characteristics of relatively long bond-length but relatively small electronegativity-difference. For exploring the ferroelectricity of buckling structures and the behavior of ferroelectric (FE) control of the valley degree of freedom, the $\beta$-GaP is used as a model system to elucidate the strain effect on FE behavior and the magnetic proximity effect on the polarization and switching of valley. We found that the spontaneous polarization is positively correlated with the electronegativity difference within a certain range, and the compression strain can effectively manipulate spontaneous polarization and switch barrier. A combination of the magnetic proximity effect and the inversion of electric polarization can generate and switch valley polarization effectively.

Design of mechanical metamaterials is typically realized by repeating microstructured building blocks or unit cells. Microstructures of these unit cells can be identical, whereas individual design of each cell and various combinations of unit cells definitely offer more freedoms and possibilities for combinatorial design of metamaterials. Unfortunately, this combinatorial design problem is prohibitively challenging, if not impossible, due mainly to its huge number of combinatorial cases. This paper poses and addresses the combinatorial optimization of a metabeam, aiming at maximizing its critical buckling load. The problem was conceptualized and solved by combination of ML accelerated surrogate modeling and optimization algorithm, and buckling and post-buckling performance of the optimal design was validated by high-fidelity simulations and experiments. The efforts provide efficient tools for combinatorial design of mechanical metamaterials. We publicly share all the data and codes for implementation.

Mechanical metamaterials are valued for their diverse properties and potential applications. Due to the instability and large deformability of soft mechanical metamaterials (SMMs), geometric reorganization will occur and lead to some unusual properties. It is possible to change the properties of materials by varying the parameters. Conventional SMMs contain a periodic distribution of holes with the same size and shape, which can be changed to a lesser extent. Periodic dispersion of regular through-hole patterns of various sizes or shapes into elastomers, resulting in metamaterials with more mechanical functionality and deformation scenarios. In this paper, we investigated the influence of parameters on the buckling mechanical behavior of SMMs and the buckling mechanical behavior of structures with multiple sizes and geometric shapes. The parameters studied include geometric parameters (pore shape, porosity and area ratio) and physical parameters (Poisson’s ratio and compression mode). Simulation of the buckling behavior of SMMs uses the finite element method. The finite element software ABAQUS is used, taking into account the almost incompressible characteristics of materials, the triangular quadratic plane strain hybrid element is selected (CPE6H). Numerical calculation gives the following results: Area ratio, pore shape and compression mode have obvious effects on buckling behavior, but Poisson’s ratio has little effect; the influence of parameters on the buckling critical strains varied for SMMs with various pore shapes; very different buckling behaviors will result from swapping out the pattern of holes with the same size or shape for holes with two different sizes or shapes; the expression of buckling behavior is also varied when the mix of hole shapes is modified. These findings demonstrate that the design parameters may be used to achieve the desired buckling behaviors. This is a new method that can be used to control the deformation of structures; modify the properties of the SMMs without changing stiffness; simplify the structures without significantly changing the material properties. The design path of mechanical metamaterials is increased.

Spatially chaotic bifurcations of an elastic web of links are investigated. We numerically construct the global bifurcation diagrams uniquely describing the buckled states, and show that the exponential growth of the number of equilibrium branches with the size of the web indicates spatial chaos. The types of bifurcations from the trivial equilibrium branch are also determined, and we show that cusp catastrophes of any order can appear. This observation relates the buckling of the elastic web of links to the buckling of rods with finite shear and infinite bending and normal stiffness.

We analyze from both the physical and the analytical viewpoints the equation

The paper investigates the sound signals radiated by cicadas and study why their "songs" are so loud. The sound pulses emanating from a class of insects are believed to come from small oscillators such as cicada tymbals, a vibrating drum-like membrane with some initially stored energy and a resonating air sac — the abdomen. The system is very efficient in a way that the resulting sound is very loud. The paper determines a region close to the cicada where the sound signals have strong nonlinear characteristics. Just outside this region the propagation of the sound signals are modeled by the Mendousse–Burgers' equation. The sound production mechanism is investigated in terms of "buckling" phenomenon to determine the sound pulses in atmospheric air medium. The resulting numerical results are very encouraging when compared with the data from the microphone readings.

We demonstrate a simple and cost-effective approach to realize two combined surface features of different scales together, namely submillimeter-sized protrusion array and microwrinkles, atop a polystyrene shape-memory polymer. Two different types of protrusions, namely flat-top protrusion and crown-shaped protrusion, were studied. The array of protrusions was produced by the Indentation-Polishing-Heating (IPH) process. Compactly packed steel balls were used for making array of indents. A thin gold layer was sputter deposited atop the polymer surface right after polishing. After heating for shape recovery, array of protrusions with wrinkles on the top due to the buckling of gold layer was produced.

Tantalum (Ta) films deposited on glass substrates have been prepared by a direct current magnetron sputtering method, and buckling patterns induced by residual compressive stress are investigated in detail. When the film thickness increases, the buckling morphologies evolve from straight-sided buckle network to wavy or wormlike wrinkles gradually, and finally change into telephone cord buckles. The geometrical parameters of the buckling patterns are found to increase linearly with the film thickness. Based on the geometrical parameters of the buckling patterns, the mechanical properties of the Ta films are also discussed in the frame of continuum elastic theory.