Nonlinear Dynamic Buckling Research Articles

Nonlinear thermo-mechanical dynamic buckling and vibration of FG-GPLRC circular plates and shallow spherical shells resting on the nonlinear viscoelastic foundation

Nonlinear thermo-mechanical dynamic buckling and vibration of FG-GPLRC circular plates and shallow spherical shells resting on the nonlinear viscoelastic foundation

Advanced virtual modelling aided stochastic nonlinear dynamic stability analysis of the GPLR-FGP plate in thermal environments

This paper explores the crucial yet challenging task of stochastic nonlinear dynamic buckling investigations of the GPLR-FGP plate under biaxial impacts in thermal environments with multi-dimensional uncertainties for the first time. An advanced virtual modelling technique, namely the Extended Support Vector Regression (XSVR) with a fresh S-spline polynomial kernel, is newly developed to alternatively depict the underpinned and sophisticated relationship between the inevitable system uncertainties and concerned nonlinear buckling response in an efficient fashion. Adequate statistical information of stability performance is furnished for the safety assessment and reliability-based design of the GPLR-FGP composite. The superior competence, advisable exactness, and appealing efficiency of the proposed uncertainty quantification framework are vividly demonstrated through benchmark tests and comprehensive numerical experiments against established machine learning algorithms. Moreover, the user-friendly modularity feature, and unique capabilities of expeditious information update and responsiveness highlight the versatile applicability of the proposed technique in realistic structural implementation under swiftly changing environments.

Nonlinear dynamic buckling analysis of the FG-GPLRC cylindrical and sinusoid panels with porous core under time-dependent axial compression

The nonlinear dynamic buckling responses of functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and sinusoid panels with porous core are presented in this paper. The governing formulations are established by applying the nonlinear higher-order shear deformation theory (HSDT). an approximation technique is used to determine the stress function for complexly curved panels. Euler-Lagrange equations can be used to achieve the nonlinear motion equation. Numerical investigations are considered using the Runge-Kutta method for dynamic responses, and using the Budiansky-Roth criterion for critical dynamic buckling loads. Some discussions on the dynamic buckling responses of panels with porous core can be achieved from the numerical examples.

Nonlinear Vibration and Dynamic Buckling of Complex Curved Functionally Graded Graphene Panels Reinforced with Inclined Stiffeners

This paper presents a new semi-analytical approach for the nonlinear vibration and dynamic responses of functionally graded graphene platelet-reinforced composite (FG-GPLRC) panels with complex curvature reinforced by orthogonal and/or inclined stiffeners in the thermal environment resting on the nonlinear viscoelastic foundation with the nonlinearities of von Kármán in the framework of the higher-order shear deformation theory (HSDT). The three curvature types of complexly curved panels are considered to be cylindrical, sinusoid, and parabola panels. Orthogonal and/or inclined stiffeners are modeled utilizing an improved smeared stiffener technique. The stress function form is estimated by applying the like-Galerkin method for complexly curved panels. By applying the Lagrange function and Euler–Lagrange equation, the nonlinear equations of motion of the panels are obtained. The viscous damping effects of the viscoelastic foundation are considered by utilizing the Rayleigh dissipation function. Numerical results are examined utilizing the Runge–Kutta method to acquire the time-deflection curves, and by utilizing the Budiansky–Roth criterion, the critical dynamic buckling loads are determined. From the investigated results, it is possible to evaluate the nonlinear vibration and buckling dynamic responses of three forms of panels with stiffeners.

Nonlinear vibration and dynamic buckling responses of stiffened functionally graded graphene‐reinforced cylindrical, parabolic, and sinusoid panels using the higher‐order shear deformation theory

AbstractThe nonlinear vibration and dynamic responses of functionally graded graphene‐reinforced composite (FG‐GRC) laminated cylindrical, parabolic, and sinusoid panels stiffened by FG‐GRC stiffeners in the uniformly distributed temperature variation are presented in this paper. An improved smeared stiffener technique is used to model the added stiffnesses of stiffeners to the total stiffnesses of panels. The higher‐order shear deformation shell theory (HSDT) with the geometrical nonlinearities of von Kármán is applied to establish the governing formulations. The stress function form is estimated using the approximated technique for complex curvature panels. Lagrange function and Euler‐Lagrange equations are applied, and the Rayleigh dissipation function is taken into account to obtain the nonlinear equation of motion. Numerical examples are investigated using the Runge‐Kutta method to obtain the dynamic responses of panels, and the critical dynamic buckling loads of panels are considered using the Budiansky‐Roth criterion. Some significant remarks on the nonlinear vibration and dynamic buckling responses of three types of stiffened panels can be recognized from the numerical examples.

Asymmetric Dynamic Buckling of Isotropic/ Anisotropic Spherical Caps

This paper deals with nonlinear asymmetric dynamic buckling of clamped isotropic/anisotropic spherical shells under suddenly applied pressure loads. The formulation is based on first-order shear deformation theory and Lagrange's equation of motion. The nonlinearity due to finite deformation of the shell considering von Karman's assumptions is included in the formulation. The buckling loads are obtained through dynamic response history using Newmark's numerical integration scheme coupled with a Newton-Raphson iteration technique. An axisymmetric curved shell element is used to investigate the dynamic characteristics of the spherical caps. The pressure value beyond which the maximum average displacement response shows significant growth rate in the time history of the shell structure is considered as critical dynamic load. Detailed numerical results are presented to highlight the influences of shell geometric parameter, orthotropicity, ply-angle, number of layers and asymmetric mode on the critical load of spherical caps.

Nonlinear dynamic buckling and multi-objective design optimisation of FG-GPLRP plates

Functionally graded graphene platelets reinforced porous (FG-GPLRP) composites have the advantages of tuneable properties, lightness, and enhanced mechanical capacity, which inspire the customised and intelligent multi-objective optimal structural designs. This study explores the nonlinear buckling response and resolves the multi-objective design optimisation of FG-GPLRP plates subjected to various biaxial impacts for the first time. Manifold practical influences on nonlinear dynamic stability and impulse-carrying capacity have been incorporated and investigated meticulously. By improving the state-of-the-art technique, the Competitive and Adaptive Learning Multi-objective Poplar Optimisation Algorithm (CALMOPOA) has been innovatively proposed for structural design optimisation. Through the developed framework, the multi-objective optimal designs of the FG-GPLRP structure possessing maximum nonlinear dynamic stability, impulse resistance ability, and contradictory minimum composite mass, have been resolved in the form of the Pareto front. In numerical studies, the CALMOPOA is compared against competitive and prevalent existing design optimisation algorithms, reflecting vast merits in superior optimal composite designs, more advantageous convergence rates and diversity characteristics. Moreover, the modularised feature of the presented structural design optimisation facilitates user-friendliness and practicality in realistic engineering applications. The proposed study can enable the demand-orientated customised, intellectualised, and modularised composite designs for the trending strategic blueprints.

Nonlinear dynamic buckling behavior of axially loaded functionally graded graphene-enhanced composite laminated cylindrical shells in thermal conditions

Nonlinear dynamic buckling behavior of axially loaded functionally graded graphene-enhanced composite laminated cylindrical shells in thermal conditions

Nonlinear dynamic instability and multi-objective design optimization of the GPLR-FGP plate under biaxial impacts

This paper investigates the nonlinear dynamic buckling and addresses the multi-objective design optimization of graphene platelets reinforced functionally graded porous (GPLR-FGP) composites under biaxial impacts, for the first time. Multifaceted influential effects on nonlinear instability and dynamic impulse resistance have been incorporated and examined meticulously. Two new multi-objective optimization algorithms, viz., Multi-objective Poplar Optimization Algorithm (MOPOA) and Adaptive Learning Multi-objective Poplar Optimization Algorithm (ALMOPOA), have been developed and innovatively proposed respectively, for composite design optimization. The MOPOA extends the state-of-the-art Poplar Optimization Algorithm to multi-objective problems. Through further ameliorating the MOPOA, the ALMOPOA is proposed incorporating manifold advisable strategies and novel improvements. Through the proposed framework, the multi-objective optimal designs of the GPLR-FGP composite have been established in forms of the Pareto front, with maximized nonlinear dynamic performance, impact carrying capacity, and contradictorily minimized structural weight. In numerical studies, the ALMOPOA is compared against the competitive existing optimization algorithms, revealing superior advantages with more surpassing optimal composite designs, more desirable convergence rates and diversity attributes.

Nonlinear dynamic stability analysis of axial impact loaded structures via the nonlocal strain gradient theory

In engineering applications, there has been an overwhelming tendency towards portability, miniaturization, and integration in recent years. To link the intrinsic size dependency feature of the small-scale structure with its structural stability, the nonlocal strain gradient theory, which captures the size effect in a more general size-dependent continuum-based model, is introduced to explore the nonlinear dynamic stability behaviour of nanoplates. Four types of axial impact loading configurations, namely, sinusoidal, exponential, rectangular, and damping, are considered. Some practical factors, such as Winkler-Pasternak elastic foundation and damping, are taken into account in the analysis. The equations of motion for the size-dependent initially imperfect plate are derived in the framework of the first-order shear deformation plate theory in conjunction with the Von Kármán nonlinear terms. Then the Airy stress function corresponding to simply supported nanoplate is introduced; then, by applying the Galerkin method, the obtained differential equations are addressed by the fourth-order Runge-Kutta algorithm. Subsequently, the specific value of the critical dynamic buckling load is determined by the Volmir criterion. Organic solar cells (OSCs), a type of emerging solar-to-electrical energy conversion nanodevice, are used as an illustrative example within the existing framework. The effects of size dependency in conjunction with the pulse load configuration, the initial imperfection, the elastic foundation, as well as the damping ratio on the nonlinear dynamic buckling behaviour of the OSC are thoroughly investigated.

Nonlinear forced vibration and dynamic buckling of FG graphene-reinforced porous arches under impulsive loading

Nonlinear forced vibration and dynamic buckling of fixed functionally graded graphene platelet-reinforced (FG-GPLRC) porous arches under impulsive loading are investigated. The porosity coefficient varies along the thickness of arch based on a power law distribution with uniform dispersion of graphene platelets (GPLs) in the whole arch. The FG-GPLRC porous arch is made of closed-cell metal foams whose effective materials properties are determined by the volume fraction distribution of materials together with Halpin–Tsai model. The nonlinear motion equations of the FG-GPLRC porous under impulsive loading are derived by employed Hamilton’s principle, and numerically solved by Runge–Kutta (RK) method combined with the differential quadrature method (DQM). Based on the Budiansky–Roth (B-R) criterion of dynamic buckling, the critical dynamic buckling load of the arch is also determined. Two case studies are conducted to show the determination of the dynamic buckling load of the arch, and good accuracy of the developed method in predicting the critical dynamic buckling load of both shallow and deep arches is also verified. Numerical results show that the nonlinear forced vibration and dynamic buckling of arches are quite sensitive to impulsive load duration. In addition, FG-GPLRC porous arches have significantly higher critical dynamic buckling load when undergoing instantaneous impact loading.

Nonlinear dynamic buckling and vibration of thermally post-buckled temperature-dependent FG porous nanobeams based on the nonlocal theory

In this paper, nonlinear dynamic snap-through buckling and vibration behavior of the thermally post-buckled functionally graded (FG) porous nanobeams subjected to static and sudden mechanical loads are investigated utilizing the nonlocal elasticity theory. The physical properties of the nanobeam are considered to be functions of temperature based on the Touloukian model. In addition, to describe the FG porous materials, two different patterns of porosity distribution are adopted using trigonometric functions through the thickness of the nanobeam. The equations of motion in conjunction with the von-Kármán nonlinear assumption are established in the framework of Hamilton’s principle. By employing the Chebyshev-Ritz procedure, the nonlinear equations are discretized for three types of edge supports. Following that, the cylindrical arc-length technique is employed to assess the vibrational responses of the post-buckled nanobeam during static snap-through buckling. To evaluate the nonlinear dynamic buckling of the graded nanobeam under a sudden dynamic load, the Newmark time integration scheme together with the Newton-Raphson iterative method are utilized. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. After validating the developed mathematical model, a comprehensive investigation is carried out to determine the role of various physical and geometrical parameters on the dynamic snap-through buckling and vibration characteristics of the post-buckled FG nanobeams.

Nonlocal nonlinear static/dynamic snap-through buckling and vibration of thermally post-buckled imperfect functionally graded circular nanoplates

The bi-stability characteristics of the post-buckled plates have demonstrated extensive potential applications for energy harvesting and vibration isolation. In this article, at first, thermal post-buckling, nonlinear bending, and vibration behavior of the functionally graded (FG) circular nanoplates in bifurcation buckling are investigated according to nonlocal elasticity theory. Then, the nonlinear static/dynamic snapping phenomena and free vibration responses of the thermally post-buckled nanoplate subjected to static and sudden types of mechanical load are presented. For this aim, the equations of motion in conjunction with the von-Kármán nonlinearity and geometrical imperfection are established in the framework of Hamilton's principle. In addition, two distinct cases of temperature distribution as well as two types of edge conditions are taken into consideration. The equations of motion are discretized using the Chebyshev-Ritz procedure along with three different numerical algorithms. To evaluate the nonlinear dynamic snap-through buckling, the Newmark time integration scheme is adopted. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. A set of parametric studies is presented to provide an insight into influences of the nonlocal parameter, geometrical imperfection, gradient index, and edge supports on the static and dynamic characteristics of the nanosystem.

Effect of Cutout on the Stability and Failure of Laminated Composite Cylindrical Panels Subjected to In-Plane Pulse Loads

In this investigation, the nonlinear dynamic buckling analysis and the failure analysis of laminated composite cylindrical (LCC) panel with different shapes of cutouts under the action of rectangular in-plane pulse loads are performed in the finite element framework. Cross-ply laminates which are balanced symmetric are considered in the investigation. The first ply failure load (FPFL) of the panel is evaluated and checked whether it occurs before the nonlinear dynamic buckling phenomenon considering Tsai–Wu failure criterion. Convergence and validation studies are undertaken, and the results are compared with those from the existing literature. The effects of loading duration, cutout area and cutout geometry on the panel are investigated in detail and results are reported. The results indicate that for the panel with cutout, its dynamic buckling load (DBL), in certain cases, compared to the static buckling load (SBL), can be lower even if the loading duration is half of its first natural period. Additionally, the vibration and the static buckling analyses of the panels are carried out as and when required.

Dynamic thermal buckling of spherical porous shells

The aim of present work is to address nonlinear dynamic thermal buckling of shallow spherical functionally graded porous shells subjected to transient thermal loading using the first order shear deformation theory (FSDT). A power-law distribution as well as cosine-type porosity distribution are used to model the variation of constituents through the shell thickness. Thermomechanical properties are assumed to be temperature dependent. Using Crank–Nicolson time marching scheme, an iterative procedure is employed to solve nonlinear transient heat conduction equation. For thermal boundary conditions, the outer surface of shells is kept at a reference temperature, while the inner surface experiences a sudden temperature rise. Geometrical type of nonlinearity in the sense of von-Karman is taken into account. The highly coupled nonlinear governing equations of motion are extracted by constructing the appropriate weak form and also using multi-term Ritz–Chebyshev method. The resulting ODEs are then reduced to a system of nonlinear algebraic equations by employing the well-known Newmark family of time integration schemes. The latter equations are solved by means of Newton–Raphson iteration procedure. Budiansky criterion is used to recognize critical parameters of dynamic instability of shells due to applied thermal shocks. Some comparison studies are conducted in order to verify the accuracy of results of the present work. Moreover, various parametric studies are performed to assess the influence of involved parameters.

Nonlinear dynamic stability analysis of clamped and simply supported organic solar cells via the third-order shear deformation plate theory

This study presents a semi-analytical framework to explore the nonlinear dynamic buckling characteristics of the imperfect organic solar cell (OSC) with clamped and simply supported restrains grounded on the third-order shear deformation plate theory (TSDT). The exerting axial displacement loading is divided into infinite and finite durations. Moreover, the spatially dependent Winker-Pasternak elastic foundation and damping effect are incorporated in the current analysis. Combining the von Kármán nonlinearity, the governing equations are derived with the aid of the Airy stress function. By applying the Galerkin method and fourth-order Runge–Kutta procedure, the dynamic buckling critical condition of the OSC subjected to these two kinds of loadings is determined by Budiansky–Hutchinson (B-H) criterion. Based on the numerical results, the influence of some critical parameters, including the geometrical dimension, boundary condition, loading configuration, initial imperfection, damping ratio, and elastic foundation coefficient are investigated. Not limited to the solar cell, this method is also suitable to the dynamic buckling behaviours of other laminated plates.

Non-linear static and dynamic buckling analysis of laminated composite cylindrical shell embedded in non-linear elastic foundation using the swarm-based metaheuristic algorithms

In this paper, by determining the optimal fiber's angles for maximum critical buckling load, the non-linear static and dynamic buckling analyses of laminated composite cylindrical shells resting on non-linear elastic foundation subjected to external loading are investigated. The laminated composite cylindrical with three, four, and five layers is considered. In the present study, first, the optimal fiber's angles are found using the swarm-based metaheuristic algorithms, then, the non-linear static and dynamic buckling analyses are carried out at optimum fiber directions. According to classical shell theory, von-Kármán equation, and Hooke's law, the stress-strain relations are derived for the laminated composite cylindrical shell. Then, based on the Galerkin's method, the discretized motion equation is obtained. For the optimization problem, the design variables are ply angles, and the objective function is maximizing the critical buckling load. Four recently developed swarm-based metaheuristic algorithms including, grey wolf optimizer (GWO), whale optimization algorithm (WOA), grasshopper optimization algorithm (GOA), and salp swarm algorithm (SSA), are utilized to solve these optimization problems. Based on the results, the optimization algorithms have improved the value of the critical buckling load significantly. Also, the WOA algorithm had better results than the others, while the GOA was the weakest.

The nonlinear dynamic buckling behaviour of imperfect solar cells subjected to impact load

Solar cells are becoming a strong competitor in the new energy market due to their superior ability to generate electricity in an environmentally friendly and sustainable manner. The present study is devoted to presenting a theoretical framework for nonlinear dynamic buckling behaviours of imperfect multilayer solar cells subjected to impact loading resting on the elastic foundation. Two types of solar cell models, namely, organic solar cell (OSC) and perovskite solar cell (PSC), with simply supported and clamped boundary conditions are investigated. Sinusoidal, exponential, rectangular, and damping pulse functions are considered Based on the first-order shear deformation plate theory, the nonlinearity are introduced with the aid of von Kármán theory. The equations of the dynamic system of the plate with the consideration of large-deflection are derived by the Galerkin method and then solved by the fourth-order Runge–Kutta methods. After validation, some parametric experiments are performed to explore the influences of the pulse duration, pulse function pulse amplitude, initial imperfection, boundary conditions, Winkler–Pasternak​ elastic foundation coefficients, and damping ratios on the dynamic stability of the structures.

Nonlinear Static and Dynamic Buckling Analysis of Imperfect Stiffened Double-Layer FGP Shallow Spherical Shells Embedded Within Elastic Foundations

In this work, an analytical method is presented to analyze the nonlinear static and dynamic buckling of imperfect stiffened double-layer functionally graded porous (FGP) shallow shells subjected to external pressure. The shallow spherical shell consists of two layers, where upper layer of shell is nonporous functionally graded (FG) material and the lower layer of shell is FG porous. This spherical shell is resting on elastic foundations, which consist of two-term Winkler–Pasternak’s model. Two FG porous distributions types consisting of symmetric porosity distribution (SPD) and nonsymmetric porosity distribution (NSPD) are investigated. The porous materials are assumed to vary through the thickness direction based on the assumption of a common mechanical feature of the open-cell foam. Utilizing the classical theory of shells considering geometrical nonlinearity based on von Kármán–Donnell relations and Hooke’s law, the governing equations of stiffened double-layer FG porous spherical shells are extracted. To discretize the partial differential equation of shell and obtaining the system equations of motion, the Galerkin method is applied. Regarding the discretized equation of motion, the explicit expressions for dynamic and static critical buckling load are determined. To investigate the dynamic behavior, the equation of motion is solved via a numerical method named Runge–Kutta approach, and then a presented approach by Budiansky–Roth is employed to obtain a critical load for the nonlinear dynamic buckling. The influence of various porosity distributions and coefficients of the shallow shell, elastic foundations, and geometrical characteristics on the nonlinear static and dynamic buckling analysis of imperfect stiffened double-layer FG porous shallow shells is examined.

Nonlinear thermal dynamic buckling and global optimization of smart sandwich plate with porous homogeneous core and carbon nanotube reinforced nanocomposite layers

The analytical investigation for the nonlinear thermal dynamic buckling of smart sandwich plate subjected to mechanical, thermal and electric loadings is presented. The sandwich plate is composed of a porous homogeneous core, two carbon nanotube reinforced composite (CNTRC) layers and two piezoelectric face sheets. Basic equations are derived based on the Reddy's higher order shear deformation plate theory and Hamilton's principle in which the initial imperfection and Pasternak-type elastic foundations are included. The external pressure is assumed to be uniformly distributed on the surface of the sandwich plate and depend on time according to the linear functions. The nonlinear dynamic response, the frequency – amplitude relation are obtained by using the Galerkin and Runge – Kutta methods and the critical dynamic buckling load is determined by using Budiansky – Roth criterion. Bees Algorithm is used to determine the maximum value of natural frequency of smart sandwich plate and the corresponding optimum values of geometrical and material parameters. The effects of geometrical parameters, CNT volume fraction, elastic foundations, temperature increment, initial imperfection and porosity coefficient on the nonlinear vibration and dynamic buckling of the smart sandwich plate are considered specifically. The numerical results are also compared with existing results using different theories.