Search

• Novel mechanical characteristics of metamaterials are studied by theoretical analysis. • Dissipated energy per buckling event is studied. • Static and dynamic(re)programming characteristics are studied. • Potential energy landscape of programmable structure is studied. Metamaterials derive promising properties mainly from an internal micromechanical mechanism, rather than their chemical composition. Many interesting mechanical behaviors can be obtained by orderly assembling buckling elements. Elastic programmable properties, corresponding potential energy landscape and dynamic dissipation mechanism of elastic beam structures are studied in detail. To accomplish this work, a combination of theoretical models and extensive numerical calculations are applied to investigate a single element as well as homogeneous and gradient structures. This comprehensive investigation not only provides insights into the complex internal buckling behaviors and mechanisms of a specific unstable structure but also establishes conclusions that are applicable to other unstable structures having similar buckling behaviors. The potential energy of a serial combination of unstable elements is extremely wiggly, enabling multiple equilibrium configurations and resulting in multistable transformations and energy dissipation with elastic deformation and loading timescale independence. To the best of our knowledge, the most of programming properties investigated in this paper, the complex energy landscape, the deformation of internal elements, and programmable dynamic responses of gradient unstable structures have not been reported in previous papers. [Display omitted] [ABSTRACT FROM AUTHOR]

• A novel multiscale analysis is developed to assess the buckling behavior of MWCNT-reinforced laminated composites. • Integration of microstructural factors like MWCNT clustering, alignment, and curvature in evaluating composite material performance. • Comprehensive assessment of the influence of moisture and temperature on the buckling performance of MWCNT composites. • Results successfully compared and validated with existing experimental and theoretical data. • Offers significant insights for the design and optimization of MWCNT-reinforced composites. This study introduces a detailed method for analyzing the buckling behavior of laminated composite structures strengthened with multi-walled carbon nanotubes (MWCNTs). We propose a multi-scale analysis that combines analytical and computational techniques to assess the mechanical performance of MWCNT-reinforced composites under combined moisture, temperature, and mechanical stress conditions. The Halpin-Tsai equations are used to calculate the overall stiffness properties of the nano-enhanced matrix, considering factors like MWCNT clustering, alignment, and curvature. Additionally, we incorporate the nanoscopic, size-dependent features of MWCNTs into our model. The Chamis micromechanical formulas are applied to determine the individual elastic properties of the nanocomposite layers, considering the impacts of temperature and moisture. We then explore how variables such as MWCNT content and size, along with temperature and moisture levels, influence the critical buckling load of MWCNT-based laminated composite beams and plates using our multi-scale model. Our results are successfully compared with existing experimental and theoretical data to validate our approach. The developed method offers significant insights for the design and optimization of MWCNT-reinforced composites, potentially benefiting various engineering fields, including aerospace and automotive industries. [ABSTRACT FROM AUTHOR]

• Semi-analytical method to study agglomerated CNT composite cylindrical panel considering non-local elasticity is presented. • Buckling and free vibration behaviours of agglomerated CNT nano composite panel under non-uniform edge load are presented. • Effects of non-local parameter, nature of agglomeration and non-uniform edge loadings are analysed. • The results obtained using the present approach are in good agreement with the results available in literature. A semi analytical nonlocal elasticity model to analyze the effect of non-uniform edge loads on static stability and free vibration characteristics of agglomerated carbon nanotubes (CNTs) reinforced nano cylindrical panels are presented. Effective material properties of the agglomerated CNT reinforced composite are obtained using a two-parameter micro-mechanics model while Eringen's non-local theory is used to account the size effect. Sinusoidal shear deformation theory is adopted to analyze the buckling and vibration parameters using Galerkin's approach. The accuracy of the proposed model is presented first by comparing the results in the literature. Then a comprehensive study is carried out to analyze the influence of various degrees of agglomeration (complete, partial), nature of edge load, and non-local effects on the buckling and free vibration response of CNT reinforced nano cylindrical panel. The results revealed that non-local size effect leads to a reduction in stiffness and thus reduces buckling and dynamic characteristics. Moreover, it is observed that critical buckling load varies with type of in plane load and reduction in natural frequency is different for different in plane loading conditions. [ABSTRACT FROM AUTHOR]

Nano-indentation instrumented tests are carried out at shallow depths on PAN-based and MPP-based carbon fibres. Indentation moduli are obtained by performing the tests at ten different measured orientations with respect to the fibre axis. They are used to identify the elastic constants of the fibres, assuming a transversely isotropic behaviour, by minimising a cost function between measured and estimated values. Inconstancies between the identified in-plane shear and transverse moduli and reported literature values are pointed out, and some drawbacks of the nano-indentation method are highlighted. An improved method taking into account the buckling mechanisms of crystallites at stake during the indentation process, and visible in the hysteretic behaviour of force-penetration nanoindentation curves, is proposed. It allows to identify values of elastic constants that are in accordance with literature values. These elastic properties of carbon fibres are in turn used to estimate the elastic properties of epoxy matrix composites containing these fibres. Very good agreement is found with experimentally available values of unidirectional ply properties. An excellent correlation between experiments and Finite Element Analyses of the indentation response of carbon fibres is eventually found. Image 1 [ABSTRACT FROM AUTHOR]

Abstract The stability of a spinning thin elastic disk has been widely studied due to its central importance in engineering. While the plastic deformation and failure of an annular disk mounted on a rigid and accelerating circular shaft are well understood, shear-induced elastic buckling of the disk due to this 'spin-up' is yet to be reported. Here, we calculate this buckling behavior within the framework of the Föppl–von Kármán equations and give numerical results as a function of the disk's aspect ratio (inner-to-outer radius) and Poisson's ratio. This shows that shear-induced elastic buckling can dominate plastic failure in many cases of practical interest. When combined with existing theory for plastic failure, the results of the present study provide foundation results for a multitude of applications including the characterization of accelerating compact disks and deployment of space sails by centrifugal forces. [ABSTRACT FROM AUTHOR]

As we enter the age of designer matter — where objects can morph and change shape on command — what tools do we need to create shape-shifting structures? At the heart of an elastic deformation is the combination of dilation and distortion or stretching and bending. The competition between the latter can cause elastic instabilities, and over the last fifteen years, these instabilities have provided a multitude of ways to prescribe and control shape change. Buckling, wrinkling, folding, creasing, and snapping have become mechanisms that when harmoniously combined enable mechanical metamaterials, self-folding origami, ultralight and ultrathin kirigami, and structures that appear to grow from one shape to another. In this review, I aim to connect the fundamentals of elastic instabilities to the advanced functionality currently found within mechanical metamaterials. [ABSTRACT FROM AUTHOR]

Graphical abstract Abstract Uniform pattern transformation can be observed in some structures with periodic arrays of pores at a critical compressive load because of buckling of the constituents of the structures. This pattern transformation can be exploited to design structures for various potential applications. Previous studies have focused on the instability of periodic porous structures of which the base materials were elastomers, and applications of these structures may be narrow because of the elastomer limitations of low melting temperature and stiffness. In addition, material failures such as plasticity and fracture were rarely discussed in previous studies. Here, we introduce metals as the base materials for some periodic metallic porous nanostructures (PMPNs). Our molecular dynamics simulation results show that PMPNs can exhibit pattern transformation at a critical strain because of buckling. In addition, we develop a simple formulation by incorporating the effect of surface on the Euler–Bernoulli beam theory to predict the critical load for the buckling of nanostructures. The prediction of our model is in good agreement with the molecular dynamics simulation results. When the applied strain is sufficiently large, the nanoscale metals experience dislocation-medicated plasticity. We also show that the pore shape of the PMPNs strongly affects the characteristics of the periodic metallic structures including the effective Young's modulus, critical strain for micro-buckling, and critical strain for plasticity. [ABSTRACT FROM AUTHOR]

Highlights • This work analyzes the non-linear response of periodic elastic lattice structures using moderate-rotation theory. • The periodicity of the structure is accounted for through a unit cell and a RVE. • We find that the choice of RVE size is important to capture the non-linear response. • The method is illustrated through several geometries and boundary conditions. Abstract Periodic elastic lattice materials are a class of cellular materials with unique properties that cannot be achieved with fully uniform solids. In this work we employ moderate-rotation theory, which is an expansion of linear elasticity that takes into account moderate displacements and rotations, to derive a fundamental multi-scale model that captures the non-linear response of infinitely periodic elastic lattice materials and allows the imposition of local constraints on lattice junctions. We begin by modeling the response of a single strut. Next, an infinitely periodic lattice structure is considered. The periodicity of the structure is accounted for through a unit cell (UC) and periodic lattice directions. A representative volume element (RVE) comprising one or more UCs is then defined. It is assumed that the RVE experiences the macroscopic displacement gradient. To determine the stress, we employ the principle of virtual work and homogenization methods. To illustrate the predictions of the model, we analyze two infinitely periodic lattice materials - a diamond lattice and a triangular lattice with rigid junctions. To validate our predictions, we compute the response for RVEs of several sizes until convergence is reached. [ABSTRACT FROM AUTHOR]

Abstract We investigate the quasi-static growth of elastic fibers in the presence of isotropic dry or viscous friction. An unusual form of destabilization beyond a critical length is described. In order to characterize this phenomenon, a new definition of stability against infinitesimal perturbations over finite time intervals is proposed and a semi-analytical method for the determination of the critical length is developed. The post-critical behavior of the system is studied by using an appropriate numerical scheme based on variational methods. We find post-critical shapes for uniformly distributed as well as for concentrated growth and demonstrate convergence to a figure 8 shape for large lengths when self-crossing is allowed. Comparison with simple physical experiments yields reasonable accuracy of the theoretical predictions. [ABSTRACT FROM AUTHOR]

Abstract Steel beams and girders with corrugated webs have found many applications, especially in industrial structures and bridges. This choice is backed by the many features, this class of structural members can offer, such as the high load carrying capacity before failure due to the web geometry. The failure of a corrugated web may occur due to material yielding or due to geometrical buckling. Three modes of shear buckling are typical for corrugated webs: local, global, and interactive shear buckling. In this paper, the shear failure mechanisms of trapezoidal corrugated webs are investigated. Nine, shear-critical, corrugated web panels were tested until failure. The new test results confirmed a previous observation that all panels achieved a residual strength of about half of the buckling shear strength, regardless of their geometry and most of them reached yielding before buckling takes place. Since the shear strength of corrugated webs is a function of several material and geometric parameters, a comprehensive global, density-based, sensitivity analysis was performed to estimate the relative contribution of each parameter to the behavior of shear strength through a previously developed mathematical model. The analysis revealed that the length of the flat folds is the most influential parameter on the shear strength of corrugated webs, while the modulus of elasticity is the least influential. The study was extended to determine ranges on which each parameter is non-influential. Highlights • Nine corrugated web steel panels were tested until failure; • The residual strength is not influenced by the mode of shear buckling; • The flat fold length and web thickness are the most influential parameters on the shear strength followed by the web height; • The steel modulus of elasticity is a non-influential parameter; • Ranges of non-influence for each parameter were determined at 0.05 and 0.001 confidence levels. [ABSTRACT FROM AUTHOR]

Highlights • Low-cycle fatigue life of Grade 300E and 500E reinforcing bars. • Effect of buckling on low-cycle fatigue behaviour of reinforcing bars. • Low-cycle fatigue model incorporating the effect of buckling. • Effect of mean strain ratio on fatigue response of bars. Abstract Low-cycle fatigue life of Grade 300E and 500E reinforcing bars including the effect of buckling is evaluated. Low-cycle fatigue tests are carried out on un-machined specimens subjected to constant axial strain loading with strain amplitude ranging from 1% to 5%. The experimental test observations suggest that buckling of reinforcing bars has detrimental effect on its fatigue life and increase in slenderness ratio of the reinforcing bars results in substantial reduction of their low-cycle fatigue life. Equations relating the fatigue life (i.e. number of reversals to failure) with total strain amplitude and total energy dissipated as a function of buckling parameter is proposed. Also, experimental tests are carried out to investigate the influence of mean strain ratio on low-cycle fatigue behaviour of reinforcing bars. Comparative evaluation of the fatigue life models with the results of experimental investigation shows that the proposed model can predict the fatigue life of buckled and unbuckled reinforcing bars with reasonable accuracy. [ABSTRACT FROM AUTHOR]

Abstract In the present investigation, the buckling behaviours of porous double-layered functionally graded nanoplates in hygrothermal environment are presented for the first time. The nonlocal strain gradient theory with two material scale parameters is developed to examine buckling behaviour much accurately. Based on the new first order shear deformation theory the equations of equilibrium are obtained from the principle of minimum potential energy. To simplify the equations of equilibrium and removing the bending-extension coupling, the buckling behaviours of FG nanoplates are investigated based on physical neutral surface concept. The equations of equilibrium are solved for various boundary conditions using Galerkin's method. The obtained results are compared with the results available in the literature to valid the correctness of present solution method. The effects of nonlocal parameter, strain gradient parameter, porosity volume fraction, power-law index, temperature change, humidity change and boundary conditions on critical buckling load are presented. [ABSTRACT FROM AUTHOR]

Highlights • A practical, educational augmentation for teaching elastic buckling. • Innovative use of 3D printing. • Measurement of critical loads. • Justification for various theoretical approximations. • Parameter variations and how geometry influences buckling. Abstract This is the third part in a trilogy of papers examining ways in which additive manufacturing can be used to facilitate the introduction of basic principles in structural analysis. Each paper uses 3D-printing and simple, but non-trivial, slender geometric forms, to provide a hands-on aspect to structural behavior in which flexure plays a dominant role. The first part dealt with linear structural analysis (Virgin, 2017), extended to dynamics and vibration in the second part (Virgin, 2017). The current paper focuses on slender structures in which compressive axial loading is the new ingredient and hence buckling becomes a central issue. This has similarities and differences with the two previous papers, but in all instances the role played by relatively high-precision 3D-printing opens the door to versatile and effective illustration, and the development of a deeper appreciation of structural phenomena. [ABSTRACT FROM AUTHOR]

Abstract This study considers the buckling of double nanofibers embedded in an elastic matrix based on an Euler-Bernoulli beam model. A scale dependent doublet mechanics theory is used in modelling of the double nanobeam system. Critical buckling loads are obtained using doublet mechanics and results are compared with the classical elasticity theory. The variation of critical buckling loads with different beam length, doublet separation distance and stiffness of the springs are investigated. Some mode shapes of the double nanobeam system are presented. [ABSTRACT FROM AUTHOR]

The elastic buckling of shell structures such as spherical shells subject to external pressure and cylindrical shells loaded in axial compression is highly sensitive to imperfections and often catastrophic. Recent studies of spherical shells have provided accurate quantitative results for the relation between the buckling pressure and the amplitude and shape of geometric imperfections and, additionally, quantitative results for the energy barrier that must be overcome to buckle the shell by extraneous loadings or disturbances when it is loaded to pressures below the buckling pressure. Results for the simultaneous interaction of imperfections and energy barriers for spherical shells under external pressure will be presented. Numerical studies for probing forces illustrate their use in determining the buckling energy barrier, and new experimental results on energy barriers obtained by others by probing spherical shells will be discussed and compared with predictions. It will be argued that while imperfections determine the buckling load of a shell, the energy barrier at loads below the buckling load supplies important additional information about the relative safety or precariousness of the shell to additional disturbances. Results for the energy barrier for perfect and imperfect spherical shells under external pressure provide important insights into the shell's robustness, or lack thereof, at pressures below the buckling pressure. In particular, the energy barrier trends provide critical insights into the low knockdown factor usually employed in establishing the design load of unstiffened spherical and cylindrical shells. These design loads are shown to correlate with conventional predictions provided that imperfection amplitudes scale as the shell radius. [ABSTRACT FROM AUTHOR]

This study examined the buckling of unstiffened cylindrical shells under external pressure. Six stainless steel cylindrical shell specimens were tested, with a length-to-radius ratio, L/R, ranging from 1 to 7. The wall thickness, diameter, axial length, and geometry of each specimen and the material properties of the corresponding sheets were measured. All cylindrical specimens were subjected to external pressure in a pressure chamber; the buckling load and final collapsed mode were recorded. This paper presents a comparison among theoretical calculations, finite element (FE) results, and experimental data for externally pressurized cylindrical shells. In the numerical calculations, true geometry, average wall thicknesses, and elastic–perfectly plastic modeling were considered. Deviation between theoretical and FE results was 0%to − 22%, and it increased with the length-to-radius ratio. Experimental results are consistent with FE results, with deviation of 2–9%, and final collapsed modes of all shells are consistent. [ABSTRACT FROM AUTHOR]

Many existing shape memory alloy (SMA) devices consist of slender beams and frames. To better understand SMA beam behavior, we experimentally examined the isothermal, room temperature response of superelastic NiTi rods and tubes, of similar outer diameters, subjected to four different modes of loading. Pure tension, pure compression, and pure bending experiments were first performed to establish and compare the baseline uniaxial and bending behaviors of rods and tubes. Column buckling experiments were then performed on rod and tube columns of several slenderness ratios to investigate their mechanical responses, phase transformation kinetics under combined uniaxial and bending deformation, and the interaction between material and structural instabilities. In all experiments, stereo digital image correlation measured local displacement fields in order to capture phenomena such as strain localization and propagating phase boundaries. Superelastic mechanical behavior and the nature of stress-induced phase transformation were found to be strongly affected by specimen geometry and the deformation mode. Under uniaxial tension, both the rod and tube had well-defined loading and unloading plateaus in their superelastic responses, during which stress-induced phase transformation propagated along the length of the specimen in the form of a high/low strain front. Due to the dependence of strain localization on kinematic compatibility, the high/low strain front morphologies differed between the rod and tube: for the rod, the high/low strain front consisted of a diffuse “neck”, while the high/low strain front in the tube consisted of distinct, criss-crossing “fingers.” During uniaxial compression, both cross-sectional forms exhibited higher transformation stresses and smaller transformation strains than uniaxial tension, highlighting the now well-known tension-compression asymmetry of SMAs. Additionally, phase transformation localization and propagation were absent under compressive loading. During pure bending, the moment-curvature response of both forms exhibited plateaus and strain localization during forward and reverse transformations. Rod specimens developed localized, high-curvature regions that propagated along the specimen axis and caused shear strain near the high/low curvature interface; whereas, the tube specimens exhibited finger/wedge-like high strain regions over the tensile side of the tube which caused nonlinear strain profiles through the thickness of the specimen that did not propagate. It was therefore found that classical beam theory assumptions did not hold in the presence of phase transformation localization (although, the assumptions did hold on average for the tube). During column buckling, the structures were loaded into the post-buckling regime yet recovered nearly-straight forms upon unloading. Strain localization was observed only for high aspect ratio (slender) tubes, but the mechanical responses were similar to that of rods of the same slenderness ratio. Also, an interesting “unbuckling” phenomenon was discovered in certain low aspect ratio (stout) columns, where late post-buckling straightening was observed despite continuous monotonic loading. Thus, these behaviors are some of the challenging phenomena which must be captured when developing SMA constitutive models and executing structural simulations. [ABSTRACT FROM AUTHOR]

This paper deals with the representative challenging buckling problem of a fully free plate under biaxial compression by a distinctive symplectic superposition method, which yields the benchmark analytic solutions by converting the problem to be solved into the superposition of two elaborated subproblems that are solved by the symplectic elasticity approach. The solution is advanced in the symplectic space-based Hamiltonian system rather than in the classic Euclidean space-based Lagrangian system, which shapes the main advantage of the method that a direct rigorous derivation is qualified for obtaining the analytic solutions, without any assumptions or predetermination of the solution forms. Comprehensive new analytic results for both the buckling loads and mode shapes are presented and validated by the finite element method. The fast convergence and accuracy of the method make it applicable to analytic modeling of more plate problems. [ABSTRACT FROM AUTHOR]

In the present work, elastic stability analysis of curved nanobeams is investigated using the differential constitutive law consequent to Eringen's strain-driven integral model coupled with a higher-order shear deformation theory accounting for through thickness stretching effect. The formulation developed here is general in the sense that it can be deduced to realise the influence of different structural theories and analyses of nanobeams. The governing equations derived are solved employing finite element method using a 3-nodes curved beam element. The model developed here is validated considering problems for which analytical/numerical solutions are available in the literature. For comparison purpose, results are also presented for various structural theories obtained from the present formulation. The influence of structural and material parameters such as thickness ratio, beam length, rise of the curved beam, boundary conditions, and size-dependent or nonlocal parameter are brought out on the buckling behaviours of curved nanobeams. It is observed that the type of buckling mode pertaining to the lowest critical value can be different depending on geometrical and internal material length scale parameter, and boundary conditions. [ABSTRACT FROM AUTHOR]

Thin-walled composite beams of complex sectional contour are widely used along with the traditional elements of reinforced structures. Based on the generalized theory of elasticity of thin-walled beams, the mathematical model of flexural–torsional deformation of open-profile thin-walled composite beams of non-uniform cross section under thermo-mechanical loading has been developed. To confirm the reliability of developed approach to determination of the spatial displacements of thin-walled composite beams of non-uniform cross section, several experimental studies were conducted. Based on the analysis of the range of typical profiles, for the further analysis we selected several compound cross sections for which the analytical dependences were obtained regarding displacements of their characteristic points. Our studies can serve as a base for development of the practical recommendations for the optimal designing and manufacturing of thin-walled anisotropic profiles, as well as proposing the measures to reduce the value of deformations caused by the production process. [ABSTRACT FROM AUTHOR]

We present an efficient numerical scheme based on asymptotic numerical method for continuation of spatial equilibria of special Cosserat rods. Using quaternions to represent rotation, the equations of static equilibria of special Cosserat rods are posed as a system of thirteen first order ordinary differential equations having cubic nonlinearity. The derivatives in these equations are further discretized to yield a system of cubic polynomial equations. As asymptotic-numerical methods are typically applied to polynomial systems having quadratic nonlinearity, a modified version of this method is presented in order to apply it directly to our cubic nonlinear system. We then use our method for continuation of equilibria of the follower load problem and demonstrate our method to be highly efficient when compared to conventional solvers based on the finite element method. Finally, we demonstrate how our method can be used for computing the buckling load as well as for continuation of postbuckled equilibria of hemitropic rods. [ABSTRACT FROM AUTHOR]

This paper is concerned with the effects of distortion on the lateral buckling of steel beams which have rectangular or square hollow flanges and folded plate webs. The hollow flanges potentially make very large contributions to the torsional stiffness, but flange distortion (in a shear mode) causes a significant reduction in the effective torsional stiffness and in the lateral buckling resistance. In this paper, the shear distortion of the hollow flanges is analysed, and the strain energy stored in a distorted flange is compared with the strain energy stored during uniform torsion of the flange. This comparison allows the development of a suitable parameter for use in the evaluation of the effect of flange distortion on the torsional stiffness of a hollow flange beam. Uniform torsion test results are used to confirm the significance of this parameter. Finite strip analyses of simply supported hollow flange beams in uniform bending show that there is a significant interaction between distortion and uniform torsion during lateral buckling, which may be approximated by combining their flexibilities. More accurate finite element analyses show that the finite strip predictions are conservative. An approximate method of predicting the effects of flange distortion on elastic lateral buckling is presented which shows good agreement with the finite element predictions. [ABSTRACT FROM AUTHOR]

In this study, a thin-walled polyhedral polymer pipe liner is proposed for the internal rehabilitation of a deteriorated/cracked underground circular metal pipe. The pipe liner is externally confined and subjected to the hydrostatic pressure of water seeped through the cracked pipe. The critical buckling pressure of the pipe liner is derived analytically based on the principle of minimum potential energy and compared with that of a cylindrical pipe liner. A finite element model of the pipe liner is established and analyzed to understand pressure-deformation equilibrium paths and the stability of post-buckling behavior. The analytical buckling pressure is in excellent agreement with the numerical results. The buckling pressure of a polyhedral liner increases with the increase of thickness-to-radius ratio and the decrease of the number of sides in polygon base shape. In comparison with the cylindrical liner, the polyhedral liner can increase buckling pressure up to 10 times but result in a less stable post-buckling behavior. [ABSTRACT FROM AUTHOR]

The Hencky bar-net model (HBM) has been shown to be a physical structural representation of the finite difference plate model (FDM). This paper extends the HBM to tackle buckling problems of rectangular plates under in-plane non-uniform loads. This class of problems requires a pre-buckling stress analysis involving plane-stress elasticity equations in order to establish the non-uniform stress distributions in the plate domain for input loads to the HBM. The converged results given by HBM compare well with existing reference solutions; thereby verifying the model. The HBM is also used to generate some new buckling solutions for rectangular plates under various point loads and partially distributed loads. [ABSTRACT FROM AUTHOR]

The various assessment methods of ultimate strength for hull girder of ships or offshore structures might lead to different results and computation time. The nonlinear finite element (FE) analyses include the implicit static analysis and explicit dynamic analysis, which both can consider the large deflection and material nonlinearity during the process of progressive collapse. Comparing with the implicit static analysis, the explicit dynamic analysis can consider the transient influence of time and avoid the convergence issue in iterative procedure. The object of the present paper is to figure out a reliable and suitable FE modelling in the explicit dynamic method, which could keep the balance of the acceptable accurate results and computation resources. Several influential factors on the collapse behaviours of hull girder are discussed including boundary conditions, geometric ranges of finite element model, element types, loading methods and loading time. The results of a Suezmax oil tanker and Reckling models assessed by the explicit dynamic method are compared with that by the other analytical methods or in the experiment. [ABSTRACT FROM AUTHOR]

This study investigates the stability of the structural systems assembled by system scaffolds. Loading tests were conducted on the indoor single-bay system scaffolds and the outdoor full-scale structural system assembled by system scaffolds. Test results were compared with the second-order inelastic analysis with semi-rigid joints to determine the reliability of various analytical parameters. The study reveals that, while the failure model of the single-bay two-storey system scaffold without diagonal braces is overall buckling failure of the structure, the failure model of the same structure with diagonal braces is buckling failure of the vertical standards between the upper and lower storeys. The critical load of the configuration with diagonal braces is about double that of the same structure without diagonal braces. At the moment of buckling failure of the outdoor full-scale two-row, two-bay, six-storey structural system assembled by system scaffolds, the vertical standards at the four corners appeared to have large horizontal displacements owing to without diagonal braces. However, the vertical standard in the center did not exhibit obvious horizontal deformation because it was reinforced with diagonal braces on four sides. In addition, the use of diagonal braces can significantly increase the load capacity of vertical standards. This study recommends that all vertical standards be reinforced with diagonal braces to ensure the load capacity of the structural systems assembled by system scaffolds. A second-order inelastic analysis with semi-rigid joints and various notional lateral forces is recommended to determine the critical loads of full-scale structural systems assembled by system scaffolds. [ABSTRACT FROM AUTHOR]

In this paper, a generalized discrete elastica including both bending and shear elastic interactions is developed and its possible link with nonlocal beam continua is revealed. This lattice system can be viewed as the generalization of the Hencky bar-chain model, which can be retrieved in the case of infinite shear stiffness. The shear contribution in the discrete elastica is introduced by following the approach of Engesser (normal and shear forces are aligned with and perpendicular to the link axis, respectively) and that of Haringx (shear force is parallel to end section of links), both supported by physical arguments. The nonlinear analysis of the shearable-bendable discrete elastica under axial load is accomplished. Buckling and post-buckling of the lattice systems are analyzed in a geometrically exact framework. The buckling loads of both the discrete Engesser and Haringx elastica are analytically calculated, and the post-buckling behavior is numerically studied for large displacement. Nonlocal Timoshenko-type beam models, including both bending and shear stiffness, are then built from the continualization of the discrete systems. Analytical solutions for the fundamental buckling loads of the nonlocal Engesser and Haringx elastica models are given, and their first post-buckling paths are numerically computed and compared to those of the discrete Engesser and Haringx elastica. It is shown that the nonlocal Timoshenko-type beam models efficiently capture the scale effects associated with the shearable-bendable discrete elastica. [ABSTRACT FROM AUTHOR]

This research presents an exhaustive analysis on the buckling behavior of two-dimensional functionally graded (2D-FG) tapered Euler–Bernoulli beams made of porous materials in nano- and micro-scales. The imperfect nanobeams are studied based on the Eringen's nonlocal elasticity theory and the governing equations of micro-scaled imperfect beams are derived using the modified couple stress theory. The 2D-FG composition of the material is considered using the power law function. The generalized differential quadrature method (GDQM) is used for the solution process. The buckling behavior of 2D-FG porous micro- and nano-beams is studied in different porosity volume fractions ( α ), FG indexes ( nx and nz ) and rates of cross section change along thickness ( β h ) and width ( β b ). [ABSTRACT FROM AUTHOR]

A three-dimensional (3D) asymptotic formulation is developed for the buckling analysis of simply-supported, single-layered nanoplates/graphene sheets (SLNP and SLGS) embedded in an elastic medium and under biaxial compressive loads. In the formulation, the Eringen nonlocal elasticity theory is used to capture the small length scale effect, and the interaction between the SLNP/SLGS and its surrounding medium is simulated using a Pasternak-type foundation. After performing the mathematical processes of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recursive sets of governing equations for various order problems. The nonlocal classical plate theory (CPT) is derived as a first-order approximation of the 3D nonlocal elasticity theory, and the governing equations for higher-order problems retain the same differential operators as those of nonlocal CPT, although with different nonhomogeneous terms. Some accurate nonlocal elasticity solutions of the critical load parameters of simply-supported, biaxially-loaded SLNP/SLGS with and without being embedded in the elastic medium are given to demonstrate the performance of the 3D asymptotic nonlocal elasticity theory. [ABSTRACT FROM AUTHOR]

As the parallel to the advancement of technology in nano-sizes, the importance of nanotubes is rising every day. The mostly worked and used nanotubes are Carbon Nanotubes (CNT) due to their superior mechanical and electrical properties. On the other hand, the technology always needs better materials with superior mechanical, electrical conductivity and thermal properties. After a couple years of working with Carbon nanotubes, scientists have discovered different types of nanotube such boron nitride and Silicon carbide nanotubes (SiCNTs). In this work, the stability of the Silicon carbide nanotube is investigated in the static buckling case with surface effect. Nonlocal continuum theory is also used in order to see the difference between two higher-order elasticity theories. The stability of nanotubes has an important role since it is used in high-tech equipment. Buckling behavior of SiCNTs is discussed by using the continuum model based on the Euler-Bernoulli beam theory for different boundary conditions in conjunctions with the surface effect and nonlocal elasticity theory. The harmonic differential quadrature method (HDQ) is used for numerical simulations. Some parametric values for critical buckling loads have been obtained with different geometrical quantities of SiCNTs. The size effects on results have been also investigated by the surface elasticity and Eringen's nonlocal elasticity parameters. [ABSTRACT FROM AUTHOR]

A thin metallic glass foil of 100 mg mass forming a sinusoidal arc behaves as non-conventional flat micro-spring withstanding loads 10 5 times higher than its load. Upon a normal load applied on the top of the arc, the foil deforms elastically leading to sinusoidal wavy patterns of higher order. The lifespan of the novel spring is higher than conventional low cycle springs and can potentially be further improved by eliminating surface and edge preparation induced defects. This unique behavior of metallic glass foils has the potential to revolutionize the field of springs and can be exploited for numerous applications. [ABSTRACT FROM AUTHOR]

Buckling of centrosymmetric anisotropic beams is studied within strain gradient theory. First, the three dimensional anisotropic gradient elasticity theory is outlined. Then the dimension of the three dimensional theory is reduced, resulting in Timoshenko beam as well as Euler–Bernoulli beam theories. The governing differential equations together with the consistent (classical and non-classical) boundary conditions are derived for centrosymmetric anisotropic beams through a variational approach. By considering von Kármán nonlinear strains, the geometric nonlinearity is taken into account. The obtained nonlinear formulation can be used to study the postbuckling configuration. The analysis of size effect on anisotropic beam structures is missing in the literature so far, while the present model allows one to characterize the size effect on the buckling of the centrosymmetric anisotropic micro- and nano-scale beam structures such as micropillars. As a specific case, the governing buckling equation is obtained for the more practical case of orthotropic beams. Finally, the buckling loads for orthotropic simply supported Timoshenko and Euler–Bernoulli beams as well as a clamped Euler–Bernoulli beam are obtained analytically and the effect of the internal length scale parameters on the buckling load is depicted. [ABSTRACT FROM AUTHOR]

Various bifurcation behaviors, including asymmetric axial buckling induced by end thrust, bifurcation under external pressure, and eversion without any loading, of a soft circular hollow cylinder (or cylindrical shell) composed of functionally graded incompressible elastomeric material are considered in a unified way. Based on the nonlinear elasticity theory of a deformable continuous body undergoing large deformation and the linearized incremental theory for a superimposed infinitesimal deformation, an efficient approach for buckling analysis is developed and applied to the cylinder subjected to a combination of axial end thrust and internal/external pressure. It is based on the state-space formalism, which naturally avoids the derivatives of instantaneous material constants, enabling a convenient and accurate numerical implementation. Along with the layerwise method, an analytical characteristic equation (i.e. the bifurcation criterion) governing the general asymmetric buckling of the cylinder is derived. Detailed parametric studies are then carried out for a pressurized soft functionally graded hollow cylinder using an incompressible Mooney-Rivlin material model. The effects of material gradient on bifurcation induced either by end thrust or external pressure are discussed through numerical examples. Not only does this study provide an efficient tool for buckling analysis of soft cylinders, some findings that are unique to a functionally graded material are also highlighted. All in all, it is found highly possible to tune the bifurcation behavior of soft elastomeric cylindrical shells by tailoring material composition and/or adjusting the pressures acting on the surfaces of the cylinder. [ABSTRACT FROM AUTHOR]

This paper investigates buckling characteristics of a curved functionally graded (FG) nanobeam based on nonlocal strain gradient elasticity theory accounting the stress for not only the nonlocal stress field but also the strain gradients stress field. The modeling of nanobeam is carried out via a higher order refined beam theory which captures shear deformation influences needless of any shear correction factor. Power-law model is adopted to describe continuous variation of material properties of curved FG nanobeam. The governing equations of nonlocal strain gradient curved FG nanobeam in the framework of refined hyperbolic beam model are obtained using Hamilton’s principle and solved implementing an analytical solution for simply-supported and clamped boundary conditions. To validate the present model, the results are compared with those of straight FG nanobeams by extending the radius of nanobeam to infinity. The effects of nonlocal parameter, length scale parameter, power-law exponent, boundary conditions and slenderness ratio on the buckling response of curved FG nanobeams are investigated. [ABSTRACT FROM AUTHOR]

The elastic buckling load of an axially compressed piecewise member was derived under various ideal boundary conditions based on the differential element method; then, the result was simplified for a symmetrical member. The theoretical results were found to be consistent with existing formulas for the elastic buckling load of uniform members. Then, the elasto-plastic buckling of an axially compressed piecewise member composed of steel and elastic materials was investigated and found to be quite different from that of a pure steel member, and a theoretical analysis approach was developed. In elasto-plastic buckling analysis, the load-bearing capacity of the steel section in the ultimate state is determined considering initial imperfections and steel hardening. Furthermore, the symmetric method for obtaining the load-bearing capacity of this member was obtained based on the ultimate state of the critical section, which is simple and reparative for design. The theoretical results of the elasto-plastic buckling analysis were consistent with experimental results, therein obtaining an average error of 4%. [ABSTRACT FROM AUTHOR]

A torsion differential equation previously used for analysing the elastic lateral buckling of simply supported doubly symmetric beams with distributed loads acting away from the centroidal axis omits an expected term and includes an unexpected term. A different equation is derived by two different methods, either by using the calculus of variations with the second variation of the total potential, or by considering the equilibrium of the deflected and twisted beam. Four different methods are used to find solutions for the elastic buckling of beams with uniformly distributed loads. Two of these solve the differential equations numerically, either by using a computer program based on the method of finite integrals, or by making hand calculations with a single term approximation of the buckled shape. These methods produce different solutions for the two torsion differential equations. The two other methods used are based on the energy equation for lateral buckling. The first of these uses hand calculations and a limited series for the buckled shape, while the second uses a finite element computer program based on cubic deformation fields. Both of these produce solutions which agree closely with the finite integral and approximate solutions for the different differential equation derived in this paper, but are markedly different from the solutions for the previously used equation. It is concluded that the previously used torsion differential equation is in error. [ABSTRACT FROM AUTHOR]

A simple nonlocal beam model is proposed to study buckling response of protein microtubules. The size-effect for buckling model of microtubules is considered by using the nonlocal continuum theory. Finite element procedure is used for solution of nonlocal differential equation of microtubules for elastic stability. The influence of the small length scale on the buckling value is examined for different geometric parameters. The effect of elastic matrix surrounded of microtubules is also examined and some benchmark results are presented. [ABSTRACT FROM AUTHOR]

A novel semi-analytical finite strip method is presented for buckling analysis of composite plate structures with boundary edges elastically supported. A set of unique Fourier series functions is introduced to represent the longitudinal variation of deflection along a strip, and they are capable of handling elastic edges with translational and rotational spring supports. The proposed hybrid method overcomes limitation of classical finite strip method only capable of handling simple end boundary conditions of structures, and it avoids the ill-conditioning when a set of standard Fourier series functions is used for buckling analysis. Accuracy and validity of the proposed method are demonstrated by the convergence and comparative studies in comparison with the numerical finite element method. As an example, the present method is applied to buckling analysis of a composite Z-stiffened panel under pure shear, and its capability and efficiency of treating different edge conditions in the panel skin and stiffeners are illustrated. [ABSTRACT FROM AUTHOR]

In this comparative study to estimate the values of critical buckling temperatures of carbon nanotubes, modelled as Timoshenko beam, two new nonlocal elasticity models based on Trefftz and Kounadis approaches are addressed. The previous results obtained from literature are compared with those obtained from two new models under simple boundary conditions. The comparative results reveal that the differences between critical buckling temperatures obtained from three nonlocal elasticity models is more pronounced,especially at the high modes and for the short nanotubes. [ABSTRACT FROM AUTHOR]

In this article, buckling response of functionally graded nano-plate based on exponential shear deformation theory is presented. The theory presented herein is built upon the classical plate theory. In this displacement-based, refined shear deformation theory, an exponential functions are used in terms of thickness co-ordinate to include the effect of transverse shear deformation and rotary inertia. The number of unknown displacement variables in the proposed theory are same as that in first order shear deformation theory. Nonlocal elasticity theory is employed to investigate effects of small scale on buckling of the functionally graded rectangular nano-plate. The material properties of the plate are assumed to vary according to the Power Law form in the thickness direction. The governing equations and corresponding boundary conditions are derived by implementing Hamilton's principle. To show the accuracy of the formulations, our research results in specific cases are compared with available results in the literature and a good agreement will be observed. Finally, the effect of various parameters such as nonlocal parameter, Power Law indexes, aspect ratio, and the thickness ratio on the non-dimensional critical buckling load of rectangular FG nano-plates are presented and discussed in detail. [ABSTRACT FROM AUTHOR]

A novel buckling model is formulated for the Bernoulli-Euler nanobeam resting on the Pasternak elastic foundation. The formulation is based on the local-nonlocal stress-driven gradient elasticity theory. In order to incorporate the size-dependency, the strain at each point is defined as the integral convolutions in terms of the stresses and their first-order gradients in all the points, accounting also for the local contribution. The differential form of the nonlocal constitutive equation, together with a set of constitutive boundary conditions, are used to define the buckling equation in terms of transverse displacement, which is solved in closed form. Both variationally consistent and the constitutive boundary conditions are imposed to calculate the buckling loads and the corresponding mode shapes. The predictions of the present model are in agreement with the results available in the literature for the carbon nanotubes based on the molecular dynamics simulations. Insightful results are presented for the first three buckling modes of local-nonlocal nanobeams considering the gradient effects. The distinctive feature of the present model is its capability to capture both stiffening and softening behaviors at the small-scales, which result in, respectively, higher and lower buckling loads of the nanobeams with respect to those of the large-scale beams. [ABSTRACT FROM AUTHOR]

A simple mechanical model for buckling behavior of boron nitride nanotube (BNNT) surrounded by an elastic matrix is presented. A nonlocal-continuum model is proposed for BNNT using the Euler–Bernoulli beam theory on an elastic matrix. The elastic matrix surrounded of the BNNT is modeled via linear spring model using the Winkler and Pasternak elastic foundation models. The equation is obtained by variational approach for buckling and has been solved by two different approaches. Separation of variables and method of discrete singular convolution are used for computations. The influences of some geometric parameters of BNNT on buckling behavior are investigated in detail. The effect of mode numbers and nonlocal parameter on buckling behavior of BNNT has also been investigated. Finally, some parametric results are presented for BNNT buckling. It is noticed that the present DSC approach can predict accurately the buckling loads for nano-scaled structures. [ABSTRACT FROM AUTHOR]

In this present study, Chebyshev collocation method is utilized to solve buckling and vibration problems of functionally graded (FG) sandwich beams resting on two-parameter elastic foundation including Winkler and shear layer springs. The faces of FG sandwich beam are assumed to be made by functionally graded materials (FGMs) composing of ceramic and metal phases and the core of the beam is made from homogenous material. Timoshenko beam theory is employed to construct the governing equations of motion in order to cover the significant effects of shear deformation and rotary inertia. The beams with various boundary conditions are considered to find out their critical loadings and natural frequencies. An accuracy of the present solutions is confirmed by comparing with some available results in the literature. Moreover, many important parametric studies of layer and beam thickness ratios, material volume fraction index, spring constants, etc. are taken into investigation. According to numerical exercises, it is revealed that the spring constants of elastic foundation have significant impact on buckling and vibration results of such beams. [ABSTRACT FROM AUTHOR]

Discrete Element Method (DEM) simulations are used to model the elastic properties of a continuous material. The preparation route of the particle packing is shown to have a significant effect on the macroscopic properties. We propose simple relations, generalized from the mean field solution, that are able to fit DEM results. These relations introduce only basic microstructural features such as the coordination number and the packing density. When the tangential to normal stiffness ratio increases above unity, the material becomes potentially auxetic. Buckling is also explored with DEM, and results on cylindrical bars are compared to the classic Euler results for critical stress. [ABSTRACT FROM AUTHOR]

The enabling emerging technologies such as nanotechnology increased the demand for small-size devices. The proper understanding of the nonclassical behavior of nanostructures is key for the design of these devices. As a result, the static and dynamic behavior of nanoscale beam structures has received a great attention in the past few years. This review aims at directing the light to research work concerned with bending, buckling, vibrations, and wave propagation of nanobeams modeled according to the nonlocal elasticity theory of Eringen. Due to the large body of references found in the literature, the authors chose to briefly present the key findings and challenges and direct light to possible future work. This review does not intersect with recent relevant reviews, which reflects its significance to readers. [ABSTRACT FROM AUTHOR]

Numerical simulations based on finite-strain elasticity and a phase-field model of dislocations reveal that dislocations are emitted at the crack-front delamination of a circular blister. It is shown that the phenomenon induces a sliding in the film-substrate interface that modifies the shape of the buckling structure. This phenomenon is theoretically quantified introducing an axisymmetric sliding into the Föppl and von-Kármán equations that describe the elastic behavior of the film. By extending the analytical investigations to the crack opening, it is shown that dislocation-induced sliding may stabilize the buckling-driven delamination of three-dimensional structures. [ABSTRACT FROM AUTHOR]

In this paper, the buckling analysis of FG circular/annular nanoplates under uniform in-plane radial compressive load with a concentric internal ring support and elastically restrained edges is studied using an exact analytical approach within the framework of nonlocal Mindlin plate theory. The material properties vary according to a power-law distribution of the volume fraction of the constituents whereas Poison's ratio is set to be constant. In solving this problem, the circular/annular FG nanoplate is first divided into an annular segment and a core circular/annular segment at the location of the internal ring support; accordingly solutions for two segments brought together by using the interfacial conditions. It is observed that an internal ring support can increase the buckling capacity, accordingly this capacity is maximized when the internal ring support is located at an optimal position. Furthermore, the effects of small scales on the maximum buckling load corresponding to the optimal radius of the internal ring support are investigated for various parameters such as radius and thickness of the FG nanoplate, boundary conditions, mode numbers and material properties. [ABSTRACT FROM AUTHOR]

By physical modelling we investigated the buckling and post-buckling behaviour of an elastic layer hosted in a viscous medium, as analogue of the elastic response of crustal rocks. The experiments were performed by embedding thin elastic layers of finite length in a linear viscous medium, and in two successive stages: a first stage of layer-parallel shortening in pure shear, followed by a second stage of post-buckling unfolding with zero velocity at the bounding walls. The experimental results show that the fold wavelength varies inversely with applied piston velocity, following an exponential function for which the higher the piston velocity the higher the number of waves. With cessation of layer-parallel shortening, the buckled layers underwent post-buckling relaxation in response to the elastic strain accumulated during the buckling stage. Such relaxation reduces the number of waves by successive elimination of the lower amplitude folds. The relaxation process gives rise to larger folds of higher amplitude in the centre of the elastic layer, concomitantly with the production of a train of gentle folds in the outer domains by outward motion of the ends of the elastic layer. This means that relaxation is more efficient away from the central domain. Comparison of our experimental results with available analytical solutions reveals significant discrepancies. Finally, we suggest that a similar process of relaxation of the elastic strain, although of lower amplitude, may explain the late stage open folding and the formation of overlying extensional basins observed in past orogens like the Caledonides in western Norway. [ABSTRACT FROM AUTHOR]