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Syntactic foams made of hollow particles embedded in a matrix material are widely used as buoyancy materials for underwater marine structures, where both low density and high mechanical properties are desired. In most engineering materials, however, the reduction in density is often accompanied with the decrease in stiffness and strength. Such a dilemma forces us to strike a balance between them for different application scenarios, which necessitates a profound understanding of the structure-property correlations in syntactic foams. In this paper, we investigated the mechanical properties of syntactic foams subjected to triaxial compressive loading. Taking particle crushing as the primary failure mode, we develop an analytical analysis on the compressive strength of syntactic foams. For syntactic foams with high strength under hydrostatic pressure, they may fail readily in the presence of deviatoric stress. In addition, we identify the critical conditions associated with other failure modes, including particle-matrix interface delamination and particle buckling. This study not only sheds light on the structure-property relationship in syntactic foams for deepsea usage, but also provides guideline for the design and optimization of syntactic foams, which would be of great value for the development of high-performance underwater structures. • Combining engineering practice and application, the boundaries of mechanical properties for syntactic foams were explored. • Ashby diagrams showing the mechanical property vs. density relationships for syntactic foams were provided. • The critical conditions associated with interface delamination and particle buckling in syntactic foams were identified. [ABSTRACT FROM AUTHOR]

The possible emergence of a localized buckled configuration for a simple supported linear elastic beam on an elastic foundation subject to axial compression is analyzed numerically. Beams on a softening foundation and, for comparison purposes, beams on a linear elastic foundation are considered. The critical bifurcation mode involves periodic lateral displacements for both foundations. Calculations are carried out for imperfections that can have a component in the shape of the critical periodic bifurcation mode as well as a component with a localized shape. For beams on a softening foundation, the buckled configuration obtained in the calculations depends on the magnitudes of the periodic and localized imperfection components: (i) for a wide range of imperfections, a rather abrupt growth of localized lateral displacements occurs after a load maximum, even if the imperfection magnitudes are such that the critical periodic mode is suppressed; or (ii) if the magnitude of the periodic imperfection component is sufficiently large, the buckled configuration with a localized imperfection involves lateral displacements that are oscillatory but not periodic. For beams on a linear elastic foundation, a calculation with a sufficiently large magnitude localized imperfection gives a localized buckled configuration under increasing load. For beams on both types of foundation, the main features of the lateral displacements in the localized buckled configuration are similar, for example, involving both positive and negative lateral displacements even if the localized imperfection only involves non-negative deflections. • A beam on a softening foundation with periodic and localized imperfections. • A localized buckled shape can occur by a bifurcation after a load maximum. • A localized buckled shape can occur without any periodic imperfection. • A large periodic imperfection acts to suppress emergence of the localized shape. • A localized buckled shape can develop with no softening under increasing load. [ABSTRACT FROM AUTHOR]

This study is the first of its kind to apply an analytical technique to offer an accurate solution for the static buckling and static bending of functionally graded (FG) sandwich nanobeams, in which the core layer is an auxetic honeycomb. One notable aspect of this work is the use of third-order shear deformation theory for the computations. It is worth noting that the displacement field of the nanobeam is contingent upon a single parameter. This approach enables the resolution of stability and static bending issues encountered in nanobeams including auxetic honeycomb cores, while also providing a realistic depiction of the beam's mechanical behavior. The two surface layers are composed of different materials, each of which has mechanical characteristics that change in accordance with the exponential function law. The precise solution described in this work is very adaptable and may be used to investigate the mechanical behavior of beams with different boundary constraints, taking into consideration the effect of the nonlocal parameter. When the angle α of the cell varies, the honeycomb core's mechanical characteristics, such as Poisson's coefficient and elastic modulus E , vary continually. This results in static bending and buckling responses that are distinct from those of other auxetics. This study also found it intriguing that the wall angle α has a value of around 90° and that sandwich nanobeams have the highest load capacity. In addition, numerical calculations have been performed to demonstrate the influence of certain geometrical parameters, materials, and boundary constraints on the critical buckling load and maximum deflection of the nanobeam. • This study is the first of its kind to apply an analytical technique for the static buckling and static bending of FG sandwich nanobeams, in which the core layer is an auxetic honeycomb. • This study offers calculation findings demonstrating the effect of geometrical factors, materials, and boundary conditions on the ultimate load response and maximum displacement of FG sandwich nanobeams. • It is discovered that a certain value for the wall angles of the core layer cell yields the highest possible performance from this beam. [ABSTRACT FROM AUTHOR]

In this paper we have discussed the frictional buckling behavior of a slender rod constrained in a horizontal cylinder. Buckling and post-buckling behaviors including the onset of helical buckling and subsequent axial force transfer are interpreted in a new way. Helical buckling and concomitant low force transfer efficiency which would prevent further axial extended reach is important to engineering structures, such as coiled tubing operations in petroleum industry. By means of the beam-column method, new governing differential equation of frictional buckling are derived to predict critical buckling force. We also investigate the transfer efficiency of axial force quantifying the effect of axial friction on rod lockup phenomenon in post-buckling analyses by perturbation solution. The coupling effect between axial force and friction is solved for the first time in this paper. The bending moment at the lock-up limit which would cause the rod to rupture is also analyzed. Finally, we have compared the proposed model with experimental results to verify its validity. Through these analyses, we have provided another interpretation to the existing experiment and numerical simulation results of frictional buckling. The derived results are simple and convenient for a problem in industrial practice. • New buckling differential equations of frictionally, cylindrically restrained rod are derived. • Critical buckling loads of frictional buckling are solved from the new buckling equations. • Analytical solutions about axial force transfer efficiency are derived by perturbation solution. • Good agreement is found between the theory and experimental results. [ABSTRACT FROM AUTHOR]

Abstract A finite strip method was developed for buckling and free vibration analysis of piezoelectric laminated composite plates based on various plate theories such as Zigzag, Refined plate and other higher order shear deformation theory by variation of transverse shear strains through plate thickness in the form of parabolic, sine and exponential. The plate edge conditions are considered to be simply supported and the polynomial shape functions are used to evaluate the in–plane and out–of–plane deflection and rotation of the normal cross section of plates in transverse direction. Numerical results were obtained based on various shear deformation plate theories to verify the proposed method. The effects of length to thickness ratio and fiber orientation of cross–ply and angle–ply laminate were presented. Also, the effect of different electrical conditions on the critical buckling load under in–plane forces and strains is investigated and electrical buckling load and natural frequency of piezoelectric laminate composite plates are calculated through numerical examples. In addition, the new results on the effect of piezoelectric layers thickness, placement of piezoelectric layers and also the effect of length to thickness ratio in the interaction of biaxial in–plane loading on critical buckling load were studied. Highlights • The buckling load of CC electrical conditions, unlike of OC, is close to elastic solutions. • The rate of decreasing in buckling load of OC is higher than CC with locating piezoelectric layers close to mid-surface. • In anti-sym angle-ply with locating the piezoelectric layers closer to the mid-surface the buckling load would increase. • The natural frequency would be increased or decreased with locating pzt in mid-surface for different thickness of plate. • The linear behavior is obtained for all electric boundary conditions and length to thickness ratio for biaxial loading. [ABSTRACT FROM AUTHOR]

Abstract The buckling (eigenvalue) problem of biaxially compressed laminated plates and shallow cylindrical panels having two symmetric piezoelectrical (PZT) patches on the top and the bottom of laminates is considered. The analysis is carried out with the use of the classical laminate theory and of the first order shear deformation theory. The variable thickness of structures (the local positions of PZT patches) is described by piecewise constant step functions in one direction or in both directions due. Three different methods of the solution of the linear eigenvalue problem are proposed: the exact analytical solution, the approximate solution based on the definition of the Rayleigh quotient and the numerical 3D finite element analysis. For the approximate Donnell's theory of shallow panels two variational formulations of the eigenvalue problem are derived in the form of the Hu-Washizu functional (the Airy stress functions and transverse normal displacements) and in the form of the Legendre functional (displacements). The influence of geometric parameters of composite panels and PZT patches, piezoelectric effect, external electric voltage and laminate configurations (symmetric angle-ply and cross-ply laminates) on buckling characteristics are discussed in detail. The analysis demonstrates evidently that the use of the local piezopatches should be considered as the buckling problem for structures with the non-uniform thickness distribution. The appropriate use of the local PZT patches should be always combined with the appropriate choice of the best (optimal) laminate configuration. The formulation system developed is suitable to other shell theories and to account for the analysis of thermal effects or the imperfection sensitivity. [ABSTRACT FROM AUTHOR]

Time depended deformation and critical buckling load of viscoelastic thick plates were studied using finite strip method with the trigonometric functions in longitudinal direction and the polynomial functions in transverse direction. The plates were considered to be thick and the third order shear deformation theory was used to consider the effect of shear stresses in thickness. The mechanical properties of the material were considered to be linear viscoelastic by expressing the relaxation modulus in terms of Prony series. Time history of maximum deflection of viscoelastic plates subjected to transverse loading and unloading on plates was calculated using a fully discretized formulation. In addition, the critical in–plane load of plates was calculated by a nonlinear procedure in different times of loading. Moreover, the effect of thickness and the interaction of biaxial in–plane loading on critical load of plate were studied. [ABSTRACT FROM AUTHOR]

In terms of the interaction between buckling and snap-through for shallow structures affected by geometrical imperfections results have shown that, after the nonlinear bifurcation, load reductions occurs accompanied with a predictable tendency. In order to do it, the tracing of interaction domains and interaction curves presented in recent papers by the same Authors, has been proposed. In this study, every simple structural arrangement has been analyzed with regards to different shallowness ratios, slenderness, restraining conditions and imperfection patterns. It has been observed how the driving parameters could be led back to two original interaction ( Δ ) and form ( Ω ) factors. This procedure could shed new light on the application of the more burdensome imperfection pattern for the instability of geometrically nonlinear structures. Possible implications of these results have been presented, with a special attention to sensors and structural identification. The study accomplished numerous numerical simulations and showed good agreement with the theoretical expected results. [ABSTRACT FROM AUTHOR]

This paper presents a theoretical study and finite element simulation for the buckling of a thin film on the compliant substrate. First, we develop a continuum mechanics approach for large deformation buckling analysis based on minimizing the total energy of the film/substrate structures, and considering the precise curvature of the buckled film and the Poisson's ratio of the substrate. The predicting results using this proposed theory agree quite well with previous experimental results. Then, we make a modification for the model to simplify the expressions for the wavelength and amplitude of the buckled geometry. Furthermore, considering a thin Si film on shape memory polymer (SMP) substrate, we investigate the buckling behavior of the thin film through theoretical analysis and finite element method. Through the investigation, it is found that the evolution rate of the buckling geometry of thin film depends on the temperature of the SMP substrate, and the buckling geometry changes faster at a higher temperature. Finally, a programmed method to control the buckling of thin Si film on the SMP substrate is proposed and is realized with finite element simulation in ABAQUS. [ABSTRACT FROM AUTHOR]

This article explores the local and global stability of CNT reinforced composite (CNTRC) sandwich shell panels consist of softcore material. The shell panel's face sheets are reinforced with functionally graded nanotube particles. The effective properties of CNTRC face sheets are determined considering the extended rule of mixture technique, and a multi-layer approach is used to examine the local and global stability phenomena. The material properties of the face sheets are assumed to be temperature-dependent. The formulation account for the core compressibility effect by considering higher-order polynomial terms for transverse core displacement. The governing equations for each layer of the sandwich panel are derived by minimizing the total potential energy of the system. The Galerkin's technique is adopted to obtain a set of non-linear algebraic equations from the derived partial differential equations. Global buckling and wrinkling loads of sandwich shell panels under thermomechanical loadings are obtained by solving standard eigenvalue problem. The nonlinear responses for sandwich shell panels subjected to mechanical and thermal stresses are traced through the arc-length approach and iterative approach, respectively. The obtained responses elucidate the influence of CNT gradation, volume fractions and core thickness on the local and global stability behaviour of CNTRC sandwich shell panels. • Semi-analytical higher order formulation of sandwich panels with FG-CNTRC face sheets is presented. • Buckling and wrinkling characteristics are elucidated for soft core sandwich panels. • Influences of temperature-dependent materials properties are studied under thermomechanical loadings. [ABSTRACT FROM AUTHOR]

The present study employs the non-polynomial shear deformation theory within the framework of isogeometric analysis to study the vibration and buckling behavior of carbon nanotubes reinforced composite plates. Owing to the higher aspect ratio of carbon nanotubes, they are prone to agglomeration in matrix. Therefore, to analyze the effect of carbon nanotubes agglomeration in composite plates, a two-parameter micromechanics model is used to determine the effective mechanical properties of the composite plate. Current study opts isogeometric analysis as numerical method, as it can represents the complex shape geometries, and provides higher-order element continuity. Higher-order element continuity is desirable for implementation of non-polynomial shear deformation theory. By the virtue of non-uniform rational B-splines, isogeometric analysis turns out to be better analysis numerical method for complex and shell structures. Governing isogeometric element equations are derived using Hamilton's principle. A detailed parametric study is also performed to investigate the effect of degree of agglomeration, thickness to length/radius ratio, different boundary conditions, and complex shape cutouts in a composite plate. The presented formulation is validated across analytical and finite element method solutions with available literature. • Implemented isogeometric framework to achieve higher-order element continuity, desirable for implementing IHSDT. • Employed the inverse hyperbolic shear deformation theory for the real displacement and stress distribution. • Considered the effect of CNT agglomeration in composite's effective properties using a two-parameter micromechanics model. • Carried out vibration and buckling analysis of carbon nanotube reinforced composite plates with complex shape cutouts. [ABSTRACT FROM AUTHOR]

We study the nonlinear forced dynamics of a bistable buckled beam. Depending on the forcing frequency and amplitude, we observe three different regimes: (i) small intra-well oscillations in the neighborhood of one of the equilibria, (ii) transient snap-through ending into intra-well oscillations, (iii) persistent dynamic snap-through. We build experimentally and numerically phase-diagrams determining the forcing amplitude and frequency leading to each of the three regimes. The experiments leverage an original setup based on the use of the electromagnetic Laplace forces. The controlled flow of an electric current through a metallic beam immersed in an electromagnetic field is at the origin of the electromechanical coupling. This non-invasive excitation system allows us to easily tune the forcing frequency and amplitude. The results of our numerical model, based on a weakly nonlinear geometrical approximation and a three-mode Galërkin expansion for the space discretization, are in excellent agreement with the experimental findings. We show that higher-order modes, often neglected in the modal models of the literature, have a major influence on the nonlinear dynamics. • Original setup for the dynamic non-invasive actuation using the Laplace force • Experimental and numerical characterisation of the snapping regimes • The simplified modal model quantitatively predicts the snapping regimes • The intensity of the electric current controls the force amplitude and frequency [ABSTRACT FROM AUTHOR]

In this study, the buckling and free vibration characteristics of three-phase randomly distributed carbon nanotube (CNT) reinforced fiber composite (RD-CNTRFC) beams subjected to in-plane compressive loadings and thermal environment are discussed in-depth through a semi-analytical approach. Displacement-based governing equations of motion are derived using Lagrange equation considering higher-order shear deformation theory (HSDT). The effective material properties of RD-CNTRFC are determined in two stages; firstly, effective properties of hybrid matrix (CNTs + Polymer) are evaluated using the Eshelbhy-Mori-Tanaka approach. Finally, overall effective properties of CNTRFC are estimated by implementing different homogenization techniques. The influences of temperature-dependent material properties and CNT-agglomeration are included in the derived formulation. The buckling loads and natural frequencies of RD-CNTRFC beams are computed using a typical eigenvalue solution. The influence of various boundary conditions, CNT mass fraction, CNT-agglomeration, length-to-thickness ratio, and various ply sequences are also addressed. • Semi-analytical higher order formulation of three-phase RD-CNTRFC beams is presented. • Buckling and vibration characteristics are elucidated for different boundary conditions. • Influences of CNT-agglomeration and temperature-dependent properties of materials are included. [ABSTRACT FROM AUTHOR]

A novel analytical framework for the inclusion of active damaging processes in a structural stability analysis is presented. The framework considers quasi-static deformation processes and is formulated in terms of a discrete coordinate approach, employing an extended total potential energy functional, where the current damage state is expressed in terms of geometric configuration and applied loading. Within the current framework, an efficient analysis of structures prone to buckling instability and material damage is enabled particularly where damage initiates once structural instability has been triggered. Composite panels with multiple damage mechanisms (e.g. delaminations, matrix cracks) are studied to showcase its application. [Display omitted] • Structural stability analysis framework including material damage is presented • Nonlinear buckling behaviour of damaged structures can be efficiently modelled • Analyses of composite panels with delaminations and matrix cracks are conducted • Comparisons with more exacting finite element models yield excellent results [ABSTRACT FROM AUTHOR]

This research considers different methods in sensitivity analysis and uncertainty quantification applied to cylindrical tubes subject to three types of thermal load representation, namely, uniform, linear, and nonlinear in the radial direction. Sensitivity analysis techniques are used to quantify and rank the parameters that are most influential in altering the critical buckling temperatures, the post-buckled configuration amplitudes, and the natural frequencies in the pre- and post-buckling regimes. The uncertainty quantification and sensitivity analysis strategies show a great potential and usefulness in terms of determining the most influential input parameters for cylindrical tubes subject to thermal loads. Based on the set of nominal parameters in this study, it is shown that the length is the most sensitive parameter in altering the critical thermal buckling load. Additionally, it is demonstrated how uncertainty in the thermal load representation can lead to over or underestimation in both sensitivity analysis and uncertainty quantification findings. The outcome of this analysis can be utilized by other researchers in the design and optimization of cylindrical tubes in thermal environment. • Cylindrical structures subjected to three types of thermal loads are investigated. • Critical buckling temperatures and pre-/post-buckled responses are studied. • Sensitivity analysis and uncertainty quantification methods are utilized. • Uncertainty is present in material and geometric properties for all thermal loads. • The importance of the input parameters is determined through sensitivity analysis. [ABSTRACT FROM AUTHOR]

Nanocomposite reinforced with functionalized graphene is a novel class of high-performance materials with great potential in developing advanced lightweight structures in a wide range of engineering applications. However, accurate estimation of its material properties at different temperature conditions remains a great challenge as existing micromechanics models fail to capture the effects of chemical functionalization and temperature. This paper develops machine learning (ML)-assisted micromechanics models by employing genetic programming (GP) algorithm and molecular dynamics (MD) simulation to address this key scientific problem. The well-trained ML-assisted Halpin-Tsai model and rule of mixture can accurately and efficiently predict the temperature-dependent material properties including Young's modulus, Poisson's ratio, coefficient of thermal expansion (CTE), and density of hydrogen-functionalized graphene (HFGr) reinforced copper nanocomposites with high coefficients of determination (R2). Then, the buckling behavior of functionally graded (FG) HFGr nanocomposite beams is studied with the aid of the ML-assisted micromechanics models. A detailed parametric study is performed with a particular focus on the effects of hydrogenation percentage, graphene content, and temperature on the buckling performance of the FG-HFGr beam. Results show that bonding more hydrogen functional groups on the HFGr can effectively improve the buckling resistance of the beam. • Temperature-dependent mechanical properties of hydrogen-functionalized graphene (HFGr)/Cu nanocomposites are investigated. • The molecular dynamics datasets of temperature-dependent material properties are trained utilizing genetic programming algorithm. • Halpin-Tsai model and rule of mixture are modified to estimate the material properties of HFGr/Cu nanocomposites. • Buckling of FG-HFGr nanocomposite beams in thermal environments are studied using the ML-assisted micromechanics models. [ABSTRACT FROM AUTHOR]

In this paper, a semi-analytical higher-order finite strip method is developed based on the nonlocal strain gradient theory (NSGT) for buckling analysis of orthotropic nanoplates. NSGT contains two material length scale parameters related to the nonlocal and strain gradient effects. To consider the effect of the strain gradient in the governing equation of the plate in the transverse direction, the higher-order polynomial shape functions (higher-order Hermitian shape functions) are utilized to assess the second derivatives, in addition to the displacement and first derivatives. Also, some numerical study is presented on the effects of different factors such as boundary conditions, nonlocal and strain gradient parameters, aspect ratio, and different types of in-plane loading for the isotropic and orthotropic rectangular nanoplate to verify the proposed formulation. In the following, a relation for the nanoplates based on the nonlocal strain gradient theory using the Navier method is extracted and presented for preliminary comparisons. According to the proposed relation, it is shown that in simply supported nanoplates, if the non-local parameter is equal to the second power of the strain gradient parameter, the responses obtained for the nanoplates are equal to the locally available responses. • A semi-analytical higher-order finite strip method is developed for buckling analysis of thin orthotropic nanoplates. • The study is based on the nonlocal strain gradient theory (NSGT). • The effects of nonlocal and strain gradient parameters, aspect ratio, and various types of in-plane loading are considered. • Numerical results contrary to Navier's theory are presented for different boundary conditions. • In this study, a relation for the nanoplates based on the NSGT using the Navier method is extracted and presented. [ABSTRACT FROM AUTHOR]

In this research effort, the general nonlocal elasticity theory is applied to an axially-loaded Euler-Bernoulli nanobeam with varying boundary conditions. In applying the general nonlocal elasticity theory, a modified version of Eringen's nonlocal elasticity theory, a sixth-order partial differential equation is obtained for the transverse governing equation of motion. Obtaining the sixth-order equation created the need for additional non-classical beam boundary conditions. These boundary conditions are determined using the weighted residual approach and are physically interpreted as higher-order force and moment terms. This study investigates the buckling and dynamic responses of nanobeams using a Galerkin-based approximation with different trial mode shapes, as it relates to the exact solution. As such, comparisons are made between using three distinct types guess functions in the approximation, namely, the exact unloaded classical mode shapes, exact unloaded Eringen nonlocal mode shapes, and exact unloaded general nonlocal mode shapes. Unloaded mode shapes refer to the mode shapes in which the applied axial load is neglected or set to zero. Additional studies are conducted focusing on direct comparison between Eringen nonlocal and general nonlocal beams for both equal and unequal nonlocal parameters. It is shown that selection of an appropriate guess function that satisfies all boundary conditions is necessary for the approximate solution. Additionally, the response of Eringen and general nonlocal nanobeams with equal nonlocal parameters and hinged-hinged boundary conditions is identical using both the exact and approximate solutions. Finally, it is demonstrated that the general nonlocal dynamic response follows a different trajectory than the Eringen nonlocal dynamic response as a function of the applied axial load. The findings of this investigation may be used by other researchers to simplify analysis procedures and to better understand the fundamental relationships existing in the size-dependent governing equations and boundary conditions for nanobeam-based systems. • A Galerkin-based approximation is used to characterize higher-order nanobeams. • Eringen's and the general nonlocal theory are used for nanoscale phenomena. • Buckling and dynamic analyses are performed for end-constrained beams. • Proper guess functions or mode shapes are needed to solve the approximation. • The exact and approximate solutions are compared through a convergence analysis. [ABSTRACT FROM AUTHOR]

Rapid development of technology puts forward higher performance requirements for advanced materials with multi-field coupling stability. The porous functionally graded materials, which is well developed with mature design, tools, can be customized according to specific application through nanostructures. In this work, the influences of hygro-thermo-mechanical coupling loadings on buckling behaviors of porous bi-directional functionally graded (2D FG) Timoshenko nanobeam is investigated through the numerical study. The governing equation and boundary conditions are derived on the basis of two phase local/nonlocal strain gradient theory employing the principle of virtual work, and a processing method for Clamped-Free (C–F) boundary condition is proposed. By employing generalized differential quadrature method (GDQM), the theoretical solution of buckling behavior of the Timoshenko nanobeam is obtained, with the implementation of comparison and convergence studies to demonstrate the accuracy and effectiveness. On the best of the authors' knowledge, it is novel to investigate the buckling response of 2D FG Timoshenko nanobeam under hygrothermal environment using the two local/nonlocal strain gradient theory. Herein, the influence of nonlocal parameter, strain gradient theory, aspect ratio, axial gradient index, thickness gradient index, hygrothermal loadings and local volume fraction on the critical buckling load under various natural boundary conditions are acquired and discussed. The numerical results indicate that the present model can capture the buckling response of porous 2D FG Timoshenko nanobeams under hygrothermal environment, exhibiting consistent softening/hardening effects under all boundary conditions and modes. [Display omitted] • An comprehensive analytical model of porous 2D FG Timoshenko micro/nanobeamconsidering the small size effect is proposed. • The hygrothermal effect on buckling stability of 2D FG micro/nanobeam is studied. • An efficient and accurate GDQM for hygrothermal buckling stability is proposed. [ABSTRACT FROM AUTHOR]

As a widely used non-destructive buckling experimental technique, the vibration correlation technique (VCT) is able to determine the buckling load without reaching the collapse point. Following the experimental procedure, the numerical implementation of VCT can be easily realized. That is to say, the numerical VCT (NVCT) can be used for predicting buckling load. In this paper, the derivation of NVCT formulas is firstly presented for cylindrical shells under axial compression load. Aiming at accelerating the repeated eigenvalue analysis in NVCT, the combined approximation (CA) method is combined with NVCT. Then, the computational procedure of CA-NVCT is provided. Aiming at the verification of the effectiveness of the CA-NVCT, three analysis examples are carried out including variable-stiffness composite cylindrical composite shell, isotropic cylindrical shell and composite cylindrical shell with various types of imperfections. Finally, an optimization design is carried out to improve the buckling load of a composite cylindrical shell against imperfections based on the CA-NVCT. In comparison to results obtained by buckling experiments and typical buckling numerical methods, the high prediction efficiency and accuracy of the CA-NVCT in buckling analysis and optimization are verified. • Numerical VCT (NVCT) formulas are derived for axially loaded cylindrical shells. • Combined approximation method is combined with NVCT to develop CA-NVCT. • Five illustrative examples are carried out for comparison and verification. • The high prediction accuracy and efficiency of the CA-NVCT is verified. [ABSTRACT FROM AUTHOR]

This paper presents an analytical model for the nonlinear analysis of fragmented beams — beams which are assembled from separate fragments (blocks) without any binding material or connector. The structural integrity in these structures is provided by the shapes of the fragments and the axial compression applied at the boundaries. However, this compression also introduces a destabilizing mechanism in the form of P-Delta effect, which may lead to a loss of the global stability. In addition, the absence of binding allows the fragments to move and rotate independently, which causes detachments between them. In this paper, this coupled nonlinear behavior is thoroughly studied by applying geometrically nonlinear kinematic relations at the fragments and nonlinear traction laws at the interfaces between them. In order to capture the localization of stresses in the partially separated fragments, the high-order beam theory is implemented. The obtained nonlinear governing equations are solved numerically and the results are compared with ones obtained by finite element simulations. It is demonstrated that the progressive detachment between the fragments yields a localization of stresses at the contact areas. These detachments and stress localizations impair the bending stiffness of the beam, and together with the axial compression, present a destructive coupled mechanism. This mechanism controls the strength and stability of the fragmented beam. Finally, a parametric study on the effects of the axial compression and boundary conditions on the strength and energy absorption capacity of the beams is presented. • Analytical model for the nonlinear analysis of fragmented beams. • Numerical study on the coupling of the detachments, geometrical nonlinearity, and stress localization effects. • High-order beam theory. [ABSTRACT FROM AUTHOR]

Analytical formulations and solutions for the stability analysis of simply supported Functionally Graded Material (FGM) sandwich plates hitherto not reported in the literature based on two higher-order refined computational models available in the literature are presented. These computational models are based on Taylor's series expansion of the displacements in the thickness coordinate and incorporate the realistic parabolic distribution of transverse strains through the plate thickness. One of them with twelve degrees-of-freedom considers the effects of both transverse shear and normal strain/stress while the other with nine degrees-of-freedom includes only the effect of transverse shear deformation. In addition another higher-order model and the first-order model developed by other investigators and available in the literature are also considered for the evaluation purpose. For mathematical modeling purposes, the Poisson's ratio of the material is considered as constant whereas Young's modulus is assumed to vary through the thickness according to the power law function. The governing equations of equilibrium for buckling analysis are obtained using the Principle of Minimum Potential Energy (PMPE). Solutions are obtained in closed form using Navier's technique by solving the eigenvalue problem. The comparison of the present results with the available elasticity solutions and the results computed independently using the first-order and another higher-order theory available in the literature shows that the higher-order refined theory with 12 degrees-of-freedom predicts the critical buckling load more accurately than all other theories considered in this paper. After establishing the accuracy of prediction, extensive numerical results for FGM sandwich plates using all the models are presented which will serve as a benchmark for future investigations. [ABSTRACT FROM AUTHOR]

This paper provides a review of models and solutions for the buckling and postbuckling of beams available from the 50s of the last century to date. Beams with axially unrestrained (movable) ends and restrained (immovable) ends are covered. In each class, the formulation of the nonlinear buckling problem for the buckling loads and the postbuckling states is discussed and the underlying assumptions are highlighted. For relatively large-amplitude buckling of beams with movable ends, approximate analytical solutions up to the third order are presented and compared with the exact solutions expressed in terms of elliptic integrals. For beams with immovable ends, buckling involves midplane stretching that makes the nonlinear buckling problem takes the same form of the standard eigenvalue problem and, hence, exact solutions are affordable. This review combines the research outcomes on buckled beams from two scientific viewpoints: the structural dynamics and the nonlinear vibration viewpoints, respectively. Moreover, it presents in one place the formulation and the exact solutions for the buckling of beams to serve as an on-demand resource for researchers concerned with the buckling and postbuckling of beams. • Presents a comprehensive review on the modeling of buckling and postbuckling of thin elastic beams. • Beams with sliding ends and immovable ends are considered. • Exact solution for the buckling loads and postbuckling amplitude are presented. • The review covers the period since the 50s of the last century to date. [ABSTRACT FROM AUTHOR]

A completely new analytical method to solve the fourth-order governing partial differential equations with variable coefficients is proposed for the buckling of cylindrical shells subjected to distributed external pressure. Unlike the present energy method in literatures, pressure distribution needs not be assumed beforehand and be simple, which is arbitrary in this manuscript. The present novel method makes the buckling analysis for the complicated external pressure distributions (including discontinuous loads) possible. Furthermore, general buckling load formulas related to geometrical sizes of the shell and pressure distribution functions are obtained by the combined perturbation method and Fourier series expansion for the first time. The first and second order asymptotic terms of buckling loads are directly given for the purpose of engineering application. Based on the presented method, three types of external pressure loads (namely, linear liquid pressure, wind type pressure and discontinuous stepwise external pressure)are considered. Validity and accuracy are demonstrated by comparing with the results in published literatures for linear liquid pressure and wind type pressure, and it is also the first time to derive the buckling load for a shell under discontinuous stepwise external pressure. Parametric discussion is also performed to investigate the effects of load distribution types and parameters on buckling resistance capacity. • A completely new analytical method is proposed for the buckling analysis of cylindrical shells under arbitrarily distributed external pressure. • Buckling load formulae considering varying pressure are presented for the first time. • The validity and accuracy of the study are confirmed. • The presented results are sufficiently precise and show significant advantages. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new first-order shear deformation theory is presented for functionally graded sandwich plates composed of functionally graded face sheets and an isotropic homogeneous core. By making a further assumption to the existing first-order shear deformation theory, the number of unknowns and governing equations of the present theory is reduced, thereby making it simple to use. In addition, the use of shear correction factor is no longer necessary in the present theory since the transverse shear stresses are directly computed from the transverse shear forces by using equilibrium equations. Equations of motion are derived from Hamilton's principle. Analytical solutions for bending, buckling and free vibration analysis of rectangular plates under various boundary conditions are presented. Verification studies show that the present first-order shear deformation theory is not only more accurate than the conventional one, but also comparable with higher-order shear deformation theories which have a greater number of unknowns. [Copyright &y& Elsevier]

Abstract: In this paper, the effect of local defects, viz., cracks and cutouts on the buckling behaviour of functionally graded material plates subjected to mechanical and thermal load is numerically studied. The internal discontinuities, viz., cracks and cutouts are represented independent of the mesh within the framework of the extended finite element method and an enriched shear flexible 4-noded quadrilateral element is used for the spatial discretization. The properties are assumed to vary only in the thickness direction and the effective properties are estimated using the Mori-Tanaka homogenization scheme. The plate kinematics is based on the first order shear deformation theory. The influence of various parameters, viz., the crack length and its location, the cutout radius and its position, the plate aspect ratio and the plate thickness on the critical buckling load is studied. The effect of various boundary conditions is also studied. The numerical results obtained reveal that the critical buckling load decreases with increase in the crack length, the cutout radius and the material gradient index. This is attributed to the degradation in the stiffness either due to the presence of local defects or due to the change in the material composition. [Copyright &y& Elsevier]

Abstract: In this paper, the buckling problem of thin rectangular functionally graded plates subjected to proportional biaxial compressive loadings with arbitrary edge supports is investigated. Classical plate theory (CPT) based on the physical neutral plane is applied to derive the stability equations. Mechanical properties of the FGM plate are assumed to vary continuously along its thickness according to a power law function. The displacement function is considered to be in the form of a double Fourier series whose derivatives are determined using Stokes' transformation. The advantage of this method is capability of considering any possible combination of boundary conditions with no necessity to be satisfied in the Fourier series. To give generality to the problem, the plate is assumed to be elastically restrained by means of rotational and translational springs at the four edges. Numerical examples are presented, and the effects of the plate aspect ratio, the FGM power index, and the loading proportionality factor on the buckling load of an FGM plate with different usual boundary conditions are studied. The present results are compared with those have been previously reported by other analytical and numerical methods, and very good agreement is seen between the findings indicating validity and accuracy of the proposed approach in the buckling analysis of FGM plates. [Copyright &y& Elsevier]

This paper aims to propose an equivalent cylinder radius for the buckling behavior of relatively thick steel conical shells subjected to axial compression. This is both experimental and numerical/theoretical work. Test data on this subject is scarce and the existing design recommendations have not been validated experimentally, especially in the plastic buckling region. In order to authenticate the appropriateness of these existing rules and suggest others, collapse tests were conducted at the University of Liverpool and Universiti Teknikal Malaysia Melaka on cones and their equivalent cylinder counterparts. The first set of experiments consisted of two nominally identical cones and one equivalent cylinder. These mild steel models were CNC-machined, with the cones having the following geometric parameters: r 2 = 99 mm, r 1 = 49 mm, h = 100 mm, β = 26.56°, wall thickness of 3 mm. The equivalent cylinder was machined using the DNV CV 30.1 equivalent cylinder radius. The results show that the equivalent cylinder overestimates the conical models by 35% and 44% respectively. The second set of four steel models consisted of: two cones having, r 2 = 70 mm, r 1 = 40 mm, h = 112 mm, β = 15°, wall thickness of 2 mm, and two equivalent cylinders designed according to 'ECCS 2008 equivalent cylinder radius'. All four shells were manufactured in pairs using the traditional welding process. It has become evident here that the experimental collapse load predicted by the equivalent cylinders underestimated that of the cones by about 9%. Furthermore, numerical calculation indicated that the equivalent cylinder radius, suggested by the ECCS 2008, mimics its conical counterpart with good accuracy for cones with cone angle, 5° ≤ β ≤ 20°. However, for cones with large cone angle, β > 20°, one obtains large discrepancies between the cone and the equivalent cylinder. This highlights the weakness of the ECCS 2008 recommended equivalent cylinder radius. Finally, a new equivalent cylinder radius for axially compressed cones is suggested with better accuracy as compared to the ECCS current recommendation, and this in turn could be useful for design purposes. • Propose a new equivalent cylinder radius for buckling behavior of axially compressed relatively thick steel conical shells. • The result highlights the weakness of the ECCS 2008 recommended equivalent cylinder radius for cone angle, β > 20°. • The result confirms that the proposed equivalent cylinder radius is safe for designing the cones with different geometries. [ABSTRACT FROM AUTHOR]

A buckling solution capable of accounting for prebuckling stress field heterogeneities is proposed. Oftentimes the same structural elements that restrict the out-of-plane displacement of the edges of a plate may also constrain their in-plane movements and cause heterogeneous prebuckling stress fields. The effect of these constraints and the consequent prebuckling stress field heterogeneity is vastly overlooked in the context of approximate analytical buckling solutions. The novelty of this work lies in basing the eigenvalue buckling analysis on a calculated prebuckling in-plane stress field. The in-plane stress field is determined by conducting a displacement-based linear elasticity analysis. In both elasticity and eigenvalue buckling analyses, the deformations of the anisotropic laminate were approximated using a first-order shear deformation theory. The equilibrium and instability equations were derived using the principle of stationary total potential energy and by employing the Ritz method. The derived formulation along with arbitrarily selected sets of boundary conditions was incorporated into a Matlab program and used for demonstrating the effect of in-plane constraints on prebuckling and buckling behavior of plates. Through a parametric study, the effect of the geometric configuration of plates on their buckling strength for selected symmetrical laminates was investigated. • An approximate analytical eigenvalue buckling solution is developed based on a calculated prebuckling stress field. • The in-plane elasticity problem is solved for obtaining the prebuckling distribution of stress resultants. • The effect of stress field heterogeneity caused by in-plane constraints is incorporated in the buckling analysis. • The applicability and accuracy of the Ritz method in solving the in-plane elasticity problem is vastly studied. [ABSTRACT FROM AUTHOR]

Abstract: This article deals with the mathematical modelling of the buckling and post-buckling response of structural systems composed of quasi-brittle materials such as concrete (as found for instance in reinforced concrete columns). More specifically, columns with softening characteristics are of concern in this study. The moment-curvature constitutive law is based on continuum damage mechanics arguments (CDM theory). Then, the instability problem can be referred to as the Elastica problem for the case with a continuum damage mechanics constitutive law, or, for short, it may be referred to as the Continuum Damage Mechanica problem. It is numerically shown for the parameters of interest, that results from geometrically exact and second-order analyses for small rotations are almost equivalent. The instability of the imperfect softening system is associated with a limit load which decreases with the imperfection considered. Such columns are shown to be typically imperfection sensitive. Furthermore, a load-imperfection relationship is derived (similar to Koiter''s power law), that can be useful in structural design contexts. An asymptotic expansion is performed to obtain a closed-form analytical solution of the limit load imperfection rule. A comparison with exact numerical values for the continuous column shows good agreement with the asymptotic expression results. The need for including non-locality formulations in a damage localization zone is finally discussed for post-buckling analysis in the presence of curvature softening. The adopted local bending-curvature constitutive law leads to the unloading Wood''s paradox. To cover the propagation phenomenon of localization, some non-locality is included in a generalized formulation of the principle of virtual work. However, non-locality is not necessary for limit load calculations since limit loads occur for curvatures in the local hardening regime. [Copyright &y& Elsevier]

Abstract: This study focuses on the buckling of cylindrical shells under external pressure with general axisymmetric thickness imperfections. Firstly, the result of uniform thickness imperfection and recent study result of axisymmetric modal thickness imperfection are reviewed briefly. Secondly, asymptotic formulas of the buckling load, which are in terms of the thickness imperfection parameter up to arbitrary order, are determined through asymptotic analyses. The analyses are based on a system of linearized governing partial differential equations of perfect shells with variable thickness. Thirdly, the effects led by three patterns of thickness imperfections on the buckling of the laterally pressured cylindrical shells, which are uniform, axisymmetric modal and parabolic, are respectively investigated. Buckling loads of uniform and axisymmetric modal thickness imperfections coincide with the known results. In addition to the analytic investigation, numerical analyses and comparisons are also carried out. The analytical study results presented in the paper can be utilized to evaluate the stabilities of cylindrical shells under external pressure with general axisymmetric imperfections in thickness, once the thickness imperfection is known. [Copyright &y& Elsevier]

Abstract: In this paper, we analytically calculate the in-plane elastic properties (linear-elasticity and elastic buckling) of a new class of bio-inspired nano-honeycomb materials possessing a hierarchical architecture. Incorporating the surface effect, modifications to the classical results for macroscopic and nonhierarchical honeycombs are proposed, and the results are compared with those in the literature. A parametrical analysis reveals the influences of two key geometrical parameters on the overall elastic properties. We discuss the relevant mechanical properties, e.g. stiffness efficiency (stiffness-to-density ratio) and strength efficiency (strength-to-density ratio), which are indices reflecting the mechanical efficiency of materials, and discover that the structural strength can be optimized. The developed theory allows us to design a new class of nano materials with tailored mechanical properties at each hierarchical level and could be useful for many applications. [Copyright &y& Elsevier]

Abstract: In this paper, investigation on buckling and post-buckling behaviors of a laminated cylindrical shell of functionally graded material (FGM) with the piezoelectric fiber reinforced composite (PFRC) actuators subjected to thermal and axial compressed loads is presented. Based on the Donnell assumptions, the material properties of the FGM layer vary smoothly through the laminated cylindrical shell thickness according to a power law distribution of the volume fraction of constituent materials. In the present study, a numerical procedure for the laminated cylindrical shell is used based on the Ritz energy method and the nonlinear strain–displacement relations. Some useful discussion and numerical examples are presented to show various effects of temperature field, volume fraction and geometric parameters on the buckling and post-buckling behaviors of the laminated cylindrical shell with PFRC. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate both pre- and post-buckling behaviors of multi-walled carbon nanotubes and multi-walled carbon nanopeapods by incorporating into the applied forces of a prescribed beam equation both van der Waals interactions between the adjacent walls of the nanotubes and the interactions between the fullerenes and the inner wall of the nanotube. Two beam theories are employed. First, we utilize Donnell’s equilibrium equation to derive an axial stability condition for the multi-walled carbon nanotubes and multi-walled carbon nanopeapods. We then determine analytically the critical forces for single-walled and double-walled nanotubes and nanopeapods. Given the outer nanotube of a fixed radius, we observe that the critical force and strain derived from the axial buckling stability criterion decrease as a result of the molecular interactions between the adjacent layers of the nanotubes and the molecular interactions between the embedded fullerenes and the inner carbon nanotube, which is in agreement with existing literature. Next, we utilize an Euler–Bernoulli beam equation incorporating the curvature effect to obtain the post-buckled axial bending displacement for the multi-walled nanotubes and nanopeapods. We find that the interactions between molecules generate an inward force, which tends to resist any applied forces. While the inward force induced by the fullerenes to the inner wall of the nanotube vanishes as we increase the applied force, the inward force induced by the layers increases as the applied force increases. The main contribution of this paper is the incorporation of both van der Waals interactions and the curvature effect into prescribed beam theories to accurately measure the critical forces and the buckled displacements of multi-walled nanotubes and nanopeapods subject to a small external force. Our analysis is relevant to future nano devices, such as biological sensors and measuring devices for small forces arising from electrical charges or Casimir forces. [Copyright &y& Elsevier]

Abstract: The governing equilibrium equations for strain gradient elastic thin shallow shells are derived, considering nonlinear strains and linear constitutive strain gradient elastic relations. Adopting Kirchhoff’s theory of thin shallow structures, the equilibrium equations, along with the boundary conditions, are formulated through a variational procedure. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient shallow thin shell theories. Those terms highly increase the stiffness of the structures. When the curvature of the shallow shell becomes zero, the governing equilibrium for the plates is derived. [Copyright &y& Elsevier]

Abstract: Line pipes have anisotropic mechanical properties, such as tensile strength, ductility and toughness. These properties depend on both prestrain during the cold forming process and on the anisotropy of the mother plates. In this study, a phenomenological model combining isotropic and kinematic hardening is developed to represent anisotropic hardening behavior of high strength steel line pipes. The model is adjusted on experiments carried out on smooth and notched axisymmetric bars and plane strain specimens. The model is used to simulate bending tests carried out on large pipes containing a geometric imperfection. Numerical results suggest that prestraining in pipe forming process significantly affects the bending capacity of pipes. [ABSTRACT FROM AUTHOR]

Abstract: Bending of strain gradient elastic thin beams is studied adopting Bernoulli-Euler principle. Simple linear strain gradient elastic theory with surface energy is employed. The governing beam equations with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin beams. Those terms are missing from the existing strain gradient beam theories. Those terms increase highly the stiffness of the thin beam. The buckling problem of the thin beams is also discussed. [Copyright &y& Elsevier]

Abstract: In this paper, the influence of various vacancy defects on the critical buckling loads and strains in carbon nanotubes under axial compression is investigated via a new structural model in ABAQUS software. The necessity of desirable conditions and expensive tests for experimental methods, in addition to the time expenditure required for atomic simulations, are the motivation for this work, which, in addition to yielding accurate results, avoids the obstacles of the previous methods. In fact, this model is a combination of other structural models designed to eliminate the deficiencies inherent in individual approaches. Because the present model is constructed in the CAE space of ABAQUS, there is no need to program for different loading and boundary conditions. A nonlinear connector is considered for modeling of stretching and torsional interactions, and a nonlinear spring is used for modeling of the angle variation interactions. A Morse potential is employed for stretching and bending potentials, and a periodic type of bond torsion is used for torsion interactions. The effect of different types of vacancy defects at various locations on the critical buckling loads and strains is studied for zigzag and armchair nanotubes with various aspect ratios (Length/Diameter). Comparison of our results with those of buckling of shells with cutouts indicates that vacancy defects in the carbon nanotubes can most likely be modeled as cutouts of the shells. Finally, results of the present structural model are compared with those from molecular dynamics (MD) simulation and show good agreement between our model and the MD model. [Copyright &y& Elsevier]

Abstract: In this paper, the free vibration and buckling of laminated homogeneous and non-homogeneous orthotropic truncated conical shells under lateral and hydrostatic pressures are studied. At first, the basic relations, the modified Donnell type dynamic stability and compatibility equations have been obtained for laminated orthotropic truncated conical shells, the Young''s moduli and density of which vary piecewise continuously in the thickness direction. Applying superposition and Galerkin methods to the foregoing equations, the buckling pressures and dimensionless frequency parameter of laminated homogeneous and non-homogeneous orthotropic conical shells are obtained. The appropriate formulas for single-layer and laminated cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Finally, the effects of the number and ordering of layers, the variations of conical shell characteristics, together and separately variations of the Young''s moduli and densities of the materials of layers on the critical lateral and hydrostatic pressures, and frequency parameter are found for different mode numbers. The results are compared with other works. [Copyright &y& Elsevier]

Abstract: Buckling behaviors of axially compressed functionally graded cylindrical shells with geometrical imperfections are investigated in this paper using Donnell shell theory and the nonlinear strain-displacement relations of large deformation. The analysis is based on the nonlinear prebuckling consistent theory. Both the prebuckling effects and the temperature-dependent material properties are taken into account. The buckling condition for imperfect functionally graded cylindrical shells is obtained by using the Galerkin method. Numerical results show various effects of imperfection, structural type, power law exponent, temperature and dimensional parameters on buckling. The present theoretical results are verified by those in literature. [Copyright &y& Elsevier]

Abstract: Nonlinear buckling and postbuckling analysis of single-walled carbon nanotubes subjected to torsional load are examined by continuum mechanics elastic shell model. The boundary layer theory of shell buckling and the singular perturbation technique is applied to solve the governing equations based on the Karman–Donnell-type nonlinear differential equations. The buckling loads and postbuckling equilibrium paths are obtained theoretically. For more detailed descriptions on the postbuckling behaviors of single-walled carbon nanotubes, postbuckling relations between twist, end-shortening, transverse deflection and the torque are investigated for nanotubes with different geometric parameters. Numerical results obtained show that single-walled carbon nanotubes under the torsional load have unstable postbuckling behavior, and the postbuckling equilibrium path is higher for the thinner nanotube than that of the thicker nanotubes. Furthermore, a new conclusion drawn from the postbuckling analysis on single-walled carbon nanotubes with different Young''s modulus and the wall thickness is significant to definite the mechanical parameters of single-walled carbon nanotubes. [Copyright &y& Elsevier]

This paper presents a comprehensive study on bending, vibration and buckling behaviours of the zigzag and armchair nanobeams. Based on a third order shear deformation beam theory and the doublet mechanics, a finite element model is proposed and employed to solve the problems of zigzag and armchair nanobeams with various boundary conditions for the first time. The verification is performed by comparing the numerical results with those from the previous studies with respect to various boundary conditions. A number of numerical examples on zigzag and armchair nanobeams with four boundary conditions have been carried out. The effects of material length scale parameters, aspect ratio, nanobeam model and boundary conditions on the displacements, natural frequencies and buckling loads of nanobeams are investigated in details. Some new results, which are not available in open literature, are provided as references for the future studies. [Display omitted] • The structural behaviors of the zigzag and armchair nanobeams are presented. • Doublet mechanics and finite element formulation are used. • S–S, C–S, C–C and C–F boundary conditions are analyzed for the first time. [ABSTRACT FROM AUTHOR]

Abstract: The thermal effect on axially compressed buckling of a double-walled carbon nanotube is studied in this paper. The effects of temperature change, surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model with thermal effect is presented for axially compressed buckling of a double-walled carbon nanotube embedded in an elastic matrix under thermal environment. Based on the model, an explicit formula for the critical axial stress is derived in terms of the buckling modes of the shell and the parameters that indicate the effects of temperature change, surrounding elastic medium and the van der Waals forces. Based on that, some simplified analysis is carried out to estimate the critical axial stress for axially compressed buckling of the double-walled carbon nanotube. Numerical results for the general case are obtained for the thermal effect on axially compressed buckling of a double-walled carbon nanotube. It is shown that the axial buckling load of double-walled carbon nanotube under thermal loads is dependent on the wave number of axially buckling modes. And a conclusion is drawn that at low and room temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube increase as the value of temperature change increases, while at high temperature the critical axial stress for infinitesimal buckling of a double-walled carbon nanotube decrease as the value of temperature change increases. [Copyright &y& Elsevier]

Abstract: An elastic double-shell model is presented for the buckling and postbuckling of a double-walled carbon nanotube subjected to axial compression. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual elastic shell and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on the Karman–Donnell-type nonlinear differential equations. The van der Waals interaction between the inner and outer nanotubes and the nonlinear prebuckling deformations of the shell are both taken into account. A boundary layer theory of shell buckling is extended to the case of double-walled carbon nanotubes under axial compression. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. Numerical results reveal that the single-walled carbon nanotube and the double-walled carbon nanotube both have an unstable postbuckling behavior. [Copyright &y& Elsevier]

Abstract: The vibration and buckling characteristics of sandwich plates having laminated stiff layers are studied for different degrees of imperfections at the layer interfaces using a refined plate theory. With this plate theory, the through thickness variation of transverse shear stresses is represented by piece-wise parabolic functions where the continuity of these stresses is satisfied at the layer interfaces by taking jumps in the transverse shear strains at the interfaces. The transverse shear stresses free condition at the plate top and bottom surfaces is also satisfied. The inter-laminar imperfections are represented by in-plane displacement jumps at the layer interfaces and characterized by a linear spring layer model. It is quite interesting to note that this plate model having all these refined features requires unknowns only at the reference plane. To have generality in the analysis, finite element technique is adopted and it is carried out with a new triangular element developed for this purpose, as any existing element cannot model this plate model. As there is no published result on imperfect sandwich plates, the problems of perfect sandwich plates and imperfect ordinary laminates are used for validation. [Copyright &y& Elsevier]

Abstract: Stitching has been used as through-thickness reinforcement to reduce the effects of delamination. In stitching, the delamination will be held by stitches in the form of crack/interface bridging. In the present work, the reinforcement of stitching threads is assumed to provide continuous linear restoring tractions opposing the delamination opening. A generalized mathematical model is developed to study the buckling analysis of two layer delaminated beams with bridging by using Rayleigh–Ritz energy method. The delaminated beam is analyzed as four interconnected beams using the delamination as their boundary. Lagrange multipliers are used to enforce the boundary and continuity conditions between the junctions of the interconnected beams. The developed mathematical model is solved as an eigenvalue problem in which the lowest eigenvalue gives the buckling load. Effective-bridging modulus, a new nondimensionalized parameter, is introduced to study the influence of bridging on the delamination buckling. It is shown that bridging strongly influences the buckling load of the delaminated beams and a monotonic relation is observed between the buckling load and the effective-bridging modulus. Parametric studies in terms of delamination sizes and locations along spanwise and thicknesswise positions on the buckling load have been carried out. The bridging is found to be effective for shallow delaminations of moderate length, and for deep and long delaminations. Spanwise positions of delamination strongly influence the buckling loads. In addition, an analytical model for obtaining upper bounds of the buckling load is developed by using Euler–Bernoulli beam theory. Effective-slenderness ratio, a new nondimensionalized parameter is defined and it is found to be controlling the buckling mode configurations, i.e., local, global and mixed modes. [Copyright &y& Elsevier]

The torsional buckling of a double-walled carbon nanotube embedded in an elastic medium is studied in this paper. The effects of surrounding elastic medium and van der Waals forces between the inner and outer nanotubes are taken into account. Using continuum mechanics, an elastic double-shell model is presented for the torsional buckling of a double-walled carbon nanotube. Based on the model, a condition is derived in terms of the buckling modes of the shell and the parameters describing the effect of van der Waals interaction and surrounding elastic medium. A simplified analysis is also carried out estimate the critical torque for torsional buckling of the double-walled carbon nanotube. [Copyright &y& Elsevier]

The buckling and collapse of empty and liquid-filled thin-wall cylindrical tubes resting on a horizontal or inclined plane is considered. The deflection is due to the action of gravity causing the tube to deform under the influence of its own weight, or due to a negative transmural pressure pushing the tube inward on the outside. Classical thin-membrane theory is used to formulate a boundary-value problem describing the shell deformation, and the results illustrate families of deformed shapes of inextensible shells with point or segment contact occurring between the shell and the supporting surface or between two collapsed sections of the shell. The computed two-dimensional deformed shapes are used to reconstruct the three-dimensional shape of a slowly collapsing fluid-conveying vessel in the absence of significant hydrostatic pressure variations over the cross section. [Copyright &y& Elsevier]

Buckling problem of the elastic and viscoelastic rotationally symmetric thick circular plate with a penny-shaped crack is investigated. It is supposed that the crack edges have a small initial rotationally symmetric imperfection. The lateral boundary of the plate is clamped and the clamp compresses this plate circumferentially and inwards by a fixed radial displacement. The investigations are carried out in the framework of the exact geometrically non-linear equations of the theory of viscoelasticity and as a buckling criterion the case for which the initial imperfection of the crack edges start to increase indefinitely is taken. Numerical results are obtained using the Laplace transform and finite element method and are compared with the known ones for elastic composites. [Copyright &y& Elsevier]

In this paper, we present a deep autoencoder based energy method (DAEM) for the bending, vibration and buckling analysis of Kirchhoff plates. The DAEM exploits the higher order continuity of the DAEM and integrates a deep autoencoder and the minimum total potential principle in one framework yielding an unsupervised feature learning method. The DAEM is a specific type of feedforward deep neural network (DNN) and can also serve as function approximator. With robust feature extraction capacity, the DAEM can more efficiently identify patterns behind the whole energy system, such as the field variables, natural frequency and critical buckling load factor studied in this paper. The objective function is to minimize the total potential energy. The DAEM performs unsupervised learning based on generated collocation points inside the physical domain so that the total potential energy is minimized at all points. For the vibration and buckling analysis, the loss function is constructed based on Rayleigh's principle and the fundamental frequency and the critical buckling load is extracted. A scaled hyperbolic tangent activation function for the underlying mechanical model is presented which meets the continuity requirement and alleviates the gradient vanishing/explosive problems under bending. The DAEM is implemented using Pytorch and the LBFGS optimizer. To further improve the computational efficiency and enhance the generality of this machine learning method, we employ transfer learning. A comprehensive study of the DAEM configuration is performed for several numerical examples with various geometries, load conditions, and boundary conditions. • Deep autoencoder based energy method (DAEM) with tailored activation function. • Stable and accurate results without gradient vanishing/exploding problems. • Unsupervised DAEM applied to Kirchhoff plates. [ABSTRACT FROM AUTHOR]