Nonlinear Strain-displacement Relations Research Articles

Buckling Analysis of Functionally Graded Shallow Spherical Shells Under External Hydrostatic Pressure

The aim of this paper is to determine the critical buckling load for simply supported thin shallow spherical shells made of functionally graded material (FGM) subjected to uniform external pressure. A metal-ceramic functionally graded (FG) shell with a power law distribution for volume fraction is considered, where its properties vary gradually through the shell thickness direction from pure metal on the inner surface to pure ceramic on the outer surface. First, the total potential energy functional is obtained using the first-order shell theory of Love and Kirchhoff, Donnell-Mushtari-Vlasov kinematic equations and Hooke''''''''''''''''s Law. Then, equilibrium equations are derived through the minimization of the total potential energy functional by employing the Euler equations. The stability equations are derived by application of the adjacent - equilibrium criterion. As the nonlinear strain-displacement relations are employed, so the presented analysis is nonlinear with high accuracy. The Galerkin method is used to determine the critical buckling load. The present problem is also analyzed numerically by simulating it in Abaqus software. For validation, the present analytical results are compared with the present numerical results and with the known data in the literature. Also, the effects of some important geometrical and mechanical parameters on the hydrostatic buckling pressure are investigated.

Higher-Order Stability Analysis of Imperfect Laminated Piezo-Composite Plates on Elastic Foundations Under Electro-Thermo-Mechanical Loads

This article provides a fully analytical approach for nonlinear equilibrium path of rectangular sandwich plates. The core of structure is made of symmetric cross-ply laminated composite and the outer surfaces are piezoelectric actuators which perfectly bonded to inner core. The structure is subjected to electro-thermo-mechanical loads simultaneously. One side of plate is rested on Pasternak type elastic foundation. The equilibrium equations of plate are derived based on the higher-order shear deformation theory of Reddy taking into account initial geometrical imperfection, nonlinear strain-displacement relations of von-Karman, temperature dependent properties, and different types of boundary conditions. Some numerical examples are presented to verify the accuracy of the proposed formulation. The effects of various parameters such as voltage on actuators, elastic foundation, imperfection, and pre-load condition on the buckling and postbuckling behaviors are studied. As an important finding of current research, there may be exists bifurcation point for imperfect plates by applying voltage on actuators.

Nonlinear transient dynamic analysis of laminated composite parabolic panels of revolution with variable thickness resting on elastic foundation

In this article, nonlinear transient behavior of laminated composite Parabolic Panels of Revolution Structure(s) (PPRS) with variable thickness resting on elastic foundation is investigated using Generalized Differential Quadrature (GDQ) method. Winkler-Pasternak model is used to represent elastic foundation. Linear, arch, sine and cosine thickness functions are used to express variable thickness. In transient analyses, First Order Shear Deformation Theory (FSDT) is used to consider the transverse shear effects. Nonlinearity is taken into account using Green-Lagrange nonlinear strain-displacement relations considering deepness effect. Virtual work principle is used to derive the equations of motion. Partial derivatives in the equation of motion are expressed with GDQ method and time integration is carried out using Newmark average acceleration method. Several problems are solved and compared with finite element results in order to validate the proposed method. After validation, effects of thickness functions, thickness variation parameter, geometric characteristic parameter of PPRS, boundary conditions, elastic foundation parameters as well as composite lamination scheme on nonlinear transient dynamic behaviour of PPRS are investigated.

Nonlinear Vibration Analysis of Prestressed Double Layered Nanoscale Viscoelastic Plates

In the present study, the nonlinear flexural vibration behavior of a double layered prestressed viscoelastic nanoplate under shear in-plane load is investigated based on nonlocal elasticity theory. Using nonlinear strain-displacement relations, the geometrical nonlinearity is modeled. Both nonlocal plate theory and Hamilton’s principle are utilized for deriving the governing equations. The differential quadrature method (DQM) is employed for the computation of nonlinear frequency of the nanoplate. The detailed parametric study is conducted, focusing on the influences of small scale, aspect ratio of the plate, Winkler and Pasternak effects, van der Walls (vdW) interaction, temperature, the effect of pre-stress under shear in-plane load, and the viscidity of the plate. The influence of the viscoelastic coefficient is also discussed. The plots for the ratio of nonlinear to linear frequencies versus maximum transverse amplitude for double layered viscoelastic nanoplate are presented.

Dynamic analysis of varying speed rotating pretwisted structures using refined beam theories

This paper investigates the dynamic modeling and analysis of rotating pretwisted structures with varying speed, considering complex geometries and setting angles. For thin structures with large curvature and pretwist, a modeling approach with high accuracy and efficiency is needed to capture their highly coupled vibratory behaviors. Therefore, there exists a clear demand for refining beam models with only few additional parameters to describe the shell-like performance. In this paper, we propose a novel three dimensional (3D) pretwisted beam element formulation based on the Carrera unified formulation (CUF) theory. The formulation is implemented to the coupled vibration studies of varying speed rotating pretwisted structures by describing the displacement field in rotating coordinate systems. Using nonlinear strain-displacement relations and Hamilton's principle, a 3D pretwisted beam model is derived on the basis of refined beam theories. Also the inertial, centrifugal and Coriolis effects are taken into account, so that the established model is able to reveal coupled dynamic responses, including in-plane, out-ofplane and torsional deformations. Numerical simulation is performed to study the dynamic characteristics and time responses of a fan blade and a space boom. Results show that this method accurately captures the periodic motions and unstable motions of rotating structures. Compared with traditional theories and finite element analysis (FEA) software, the proposed method significantly reduces the computational cost by using self-adaptive model order and element number. In summary, the proposed method can be used to study dynamics of varying speed rotating pretwisted structures in a precise, flexible and numerically economical fashion.

Nonlinear transverse vibrations of angle-ply laminated composite piezoelectric cantilever plate with four-modes subjected to in-plane and out-of-plane excitations

In this paper, the nonlinear vibrations of the laminated composite piezoelectric cantilever plate subjected to the transverse and in-plane excitations are investigated. Based on the Reddy’s third-order plate theory and von Karman nonlinear strain-displacement relation, the nonlinear partial differential governing equations of motion are established for the laminated composite piezoelectric cantilever plate under combined the transverse and in-plane excitations by applying Hamilton’s principle. Employing Galerkin’s approach, we discretize the continuous nonlinear dynamic equations of motion into the ordinary differential equations with four modes. Substituting the actual geometric and physical parameters of the laminated composite piezoelectric cantilever plate into the governing equation of motion, we use numerical method to investigate the effects of the lay-up parameters and the geometric parameters on the dimensionless natural frequencies of the cantilever plate. The amplitude-frequency response curves and the basin boundary diagram of two coexisting steady-state solutions are plotted for the laminated composite piezoelectric cantilever plate. The nonlinear dynamic responses of the laminated composite piezoelectric cantilever plate under combined the transverse and in-plane excitations are investigated by changing the lay-up parameters, the transverse excitation, the voltage excitation and the in-plane excitation. The periodic, the quasi-periodic, the chaotic motions and the nonlinear stiffness characteristics can be found. The nonlinear dynamic behaviors help us to optimize the structural and physical parameters of the laminated composite piezoelectric cantilever plate under combined the transverse and in-plane excitations.

A parametric study on geometrically nonlinear behavior of curved beams with single and double link rods, and supported on moving boundary

The paper presents a parametric study on effect of several geometric parameters on stress and deformation behavior of non-uniform curved beam with moving boundary. The parameters, considered in the present study, are height to span ratio of curved beam, ratio of radius of roller support to height of beam and types of support connection mechanisms. Effect of these parameters is observed for different curved profile shapes. Non-uniform initial curvature and combined effects of bending, stretching and warping of beam cross-section due to shear deformation are considered in the derivation of nonlinear strain-displacement relation in body fitted curvilinear frame. Based on the derived nonlinear kinematics and assumptions of large deformation but small strain and moderate initial curvature, nonlinear system governing equation is derived through variational principle based minimization of total potential energy method. An iterative numerical solution algorithm is developed and implemented in the computational environment of MATLAB® software to solve the nonlinear governing equation. The complicating effect coming from moving boundary imposes a kinematic constraint on the solution field. Hence, the solution methodology not only ensures energy balance of the system, but also satisfies shear force balance at each incremental load step. The geometric nonlinear incremental curved beam model is validated with deflection and stress results of commercial finite element software ABAQUS®. The geometry of master leaf spring of a previously reported study is considered as curved beam for the purpose of validation. After the validation study, results are generated for different curved profiles with two different combinations of support connection mechanisms and different height to span ratio for each type of them. In addition, to observe effect of roller radius, results are generated with different roller radius to height ratio for each type of the two curved beam systems. Investigations are made on the effects of the geometric parameters, support connection mechanisms and curved profile shape on large deflection characteristics of curved beam and the present study leads towards several meaningful observations.

Nonlinear bandgap properties in a nonlocal piezoelectric phononic crystal nanobeam

Applying nonlocal elasticity theory, von Kármán type nonlinear strain-displacement relation and plane wave expansion (PWE) method to Euler-Bernoulli beam, the calculation method of band structure of a nonlinear nonlocal piezoelectric phononic crystal (PC) nanobeam is proposed and formulized. In order to investigate the properties of wave propagating in the nanobeam in detail, band gaps of first four orders are picked, and the corresponding influence rules of electro-mechanical coupling fields, nonlocal effect and geometric parameters on band gaps are studied. During the researches, external electrical voltage and axial force are chosen as the influencing parameters related to electro-mechanical coupling fields. Scale coefficient is chosen as the influencing parameter corresponding to nonlocal effect. Length ratio between materials PZT-4 and epoxy and height-width ratio are chosen as the influencing parameters of geometric parameters. Moreover, all the influence rules are compared to those in linear nanobeam. The results are expected to be of help for the design of micro and nano devices based on piezoelectric periodic nanobeam.

Theory and experiment of nonlinear vibrations and dynamic snap-through phenomena for bi-stable asymmetric laminated composite square panels under foundation excitation

This paper theoretically and experimentally investigates the dynamic snap-through phenomena and the nonlinear vibrations of the bi-stable asymmetric laminated composite square panel under the foundation excitation. We propose a new method to describe the dynamic snap-through phenomena of the bi-stable asymmetric laminated composite square panel by using the time-varying principle curvatures. The boundary conditions of the bi-stable asymmetric laminated composite square panel are clamped at the center of the panel and free at four edges. The mode shapes are obtained by employing Rayleigh-Ritz method and Lagrangian multipliers method. According to the double curvature type of von-Karman nonlinear strain-displacement relations, the third-order shear deformation theory and Hamilton principle, the nonlinear partial differential governing equations of motion are derived for the bi-stable asymmetric laminated composite square panel. The nonlinear ordinary differential governing equation of motion is obtained by using Galerkin method. Numerical simulations are used to study the nonlinear vibrations and the dynamic snap-through phenomena of the bi-stable asymmetric laminated composite square panel including the effect of the structural damping by using the 4-order Runge-Kutta method. Analyzing the phase portrait and time history of the principle curvature, the dynamic snap-through phenomena of the bi-stable asymmetric laminated composite square panel are observed through changing the amplitude of the foundation excitation. The vibration experiments are also finished to find the dynamic snap-through phenomena and the nonlinear vibrations of the bi-stable asymmetric laminated composite square panel. It is known that the theoretical and the experimental results are the qualitative agreement and the experimental results provide a reference for the application of the bi-stable asymmetric laminated composite square panel in the aeronautic and mechanical engineering fields.

Pore pressure and porosity effects on bending and thermal postbuckling behavior of FG saturated porous circular plates

Postbuckling and nonlinear bending of a geometrically perfect circular plate arisen by the temperature of one side of a axisymmetric circular plate made from a poroelastic solid whose matrix pores have been saturated by fluid have been numerically analyzed. The plate porosity varies continuously through the plate thickness according to some given specific functions. The postbuckling and nonlinear bending configurations of respectively clamped and simply-supported plates, as well as the critical postbuckling temperature, have been obtained under the influence of the plate one side surface temperature, thickness, type of pore distribution, porosity and compressibility of fluid confined by the pores. Equilibrium equations of the plate have been derived on the basis of the classical plate theory, Love–Kirchhoff hypothesis and the Sander’s nonlinear strain–displacement relationship. They have been discretized via differential quadrature method. The numerical results appropriately coincide with the relevant literature, namely the results of saturated porous and mathematically equivalent plates made from functionally grated materials.

Buckling and postbuckling of dielectric composite beam reinforced with Graphene Platelets (GPLs)

Buckling and postbuckling behaviors of graphene platelet (GPL) reinforced dielectric composite beams are investigated through theoretical formulation. The effective material properties of the GPL reinforced composite (GPLRC) as required for structural analysis, i.e. tensile modulus, dielectric constant and Poisson's ratio are obtained by using effective medium theory and rule of mixture. Governing differential equations for the composite beam are established through Timoshenko beam theory, von Kármán nonlinear strain-displacement relationship and principle of virtual work. Governing equations are numerically discretized and solved by employing differential quadrature method (DQM), through which several parameters affecting the buckling performances are quantitatively identified. The results demonstrate there exists a critical GPL concentration, above which the electrical field significantly affects the beam's buckling and postbuckling behaviors. The dielectric beam's buckling performances are very sensitive to AC (alternating current) frequency within a certain range. Moreover, it is found the dielectric beam's buckling and postbuckling performances comprehensively depend on the concentration and aspect ratio of GPLs. The present work is envisaged to provide guidelines to develop GPL-based smart composites and structures.

Static and dynamic flame model effects on thermal buckling: Fixed-roof tanks adjacent to an ethanol pool-fire

Storage tank fires (such as pool-fires) often occur in the petrochemical tank farms. Flame pulsation is an important characteristic of the turbulent flames observed in pool-fires. Current cylindrical solid flame models inadequately predict the thermal radiation of pool-fires because of the ignorance of the flame pulsation effect. In this study, the thermal buckling behavior and fire resistance of a fixed-roof Q345 steel tank with a stepped thickness exposed to a neighboring ethanol pool-fire based on the flame pulsation model is numerically investigated. The influence of smoke generated by the combustion process which can reduce thermal radiation fluxes is taken into account. Geometric and material nonlinearity which considers nonlinear strain-displacement and stress-strain relation respectively is used on finite element analysis. Results show that the thermal buckling mode of the cylindrical tank wall is elastic buckling, and the thermal buckling behavior is non-linear. The fire resistance of the fixed-roof steel tank increases significantly as the vertical fire location increases, and the fire resistance decreases with the burning tank diameter increasing. Moreover, the fire resistance of the fixed-roof steel tank for a cylinder-cone combined flame based on the flame pulsation model is larger than that for a cylindrical flame. The study can be used to optimize the steel structure design of oil tanks for resisting pool-fires, and thus reduce the loss caused by fire accidents in tank farms.

Nonlinear strain gradient analysis of nanoplates embedded in an elastic medium incorporating surface stress effects

Nonlinear vibration of nano graphene plates with considering surface effects is studied in this paper based on the nonlocal strain gradient theory and von Karman geometric nonlinearity. The isotropic nanoplate is assumed to lie on an elastic foundation with the simply supported boundary conditions. Both Winkler-type and Pasternak-type models are utilized to simulate the interaction of the nano graphene with a surrounding elastic medium. Due to the increase in the surface-to-volume ratios at smaller scales, the surface elasticity theory of Gurtin and Murdoch is developed to study the effects of surface properties which are the basis for size-dependent behaviors. The governing equation of motion can be obtained by von Karman nonlinear strain-displacement relationship and the nonlinear frequency is obtained analytically using the perturbation approach. Moreover, two moveable and immoveable in-plane conditions are analyzed. The presented method is verified by comparing the results with their counterparts reported in the open literature and a good agreement is observed for two different boundary conditions. Finally, the effects of various parameters such as nonlocal parameter, material characteristic parameter, residual surface tension, mode number, temperature change and elastic medium coefficients for two kinds of in-plane conditions are discussed.

Nonlinear forced vibration of pre-stressed graphene sheets subjected to a mechanical shock: An analytical study

In this paper, the nonlinear forced vibration of single layered graphene sheet including the pre-Stressed effect is studied based on nonlocal elasticity theory. The graphene sheet is located on a viscoelastic foundation based on Kelvin-Voigt model and exposed to thermo-magnetic-mechanical loads. A particle with constant velocity and concentrated load moves on the graphene sheet and applies mechanical shock to it. At first, using nonlinear strain-displacement relations, the geometrical nonlinearity is modeled. Besides, nonlocal plate theory and Hamilton's principle are utilized for deriving the governing equation. In the second step, using Galerkin method, the partial differential equation is transformed to the ordinary differential equation. Then, governing equation is solved based on multiple time scales method. Finally, frequency-response equations under sub-harmonic and super-harmonic stimulation are studied. Emphasizing the effect of nonlinearity, the results for the nondimensional nonlinear frequency versus nondimensional amplitude, the nondimensional phase angle and nondimensional velocity for single layered graphene sheet are plotted. At the end, numerical results are compared with results in the other researches. The results emphasize that the velocity of the nano particle and force amplitude are responsible to make resonance in the system based on a SLGS. Besides, force amplitude can be effective to intensify the effect of the mechanical shock in behavior of SLGS.

Large displacement transient analysis of FGM super-elliptic shells using GDQ method

Current work concentrates on large displacement dynamic analysis of super-elliptic (SE) shells made of functionally graded materials (FGMs) employing generalized differential quadrature (GDQ) technique. SE shells can be in quasi-rectangular, elliptical or cylindrical shell forms according to the parameters in super-ellipse formulation. In this paper, large displacements are considered through Green-Lagrange nonlinear strain-displacement relationships derived for SE shells with full nonlinearity in transverse shear. Present solution is based on first-order shear deformation theory (FSDT). Virtual work principle and GDQ method are utilized to derive equation of motion and to express spatial derivatives existing in equation of motion, respectively. Newmark average acceleration method is employed in the solution of equation of motion. By solving various FGM super-elliptic (FGM-SE) shell problems, effects of FGM material properties (using different ceramic/metal pairs like Alumina/Steel (Al2O3/Steel), Zirconia/Aluminum (ZrO2/Al), Alumina/Aluminum (Al2O3/Al), Zirconia/Monel (ZrO2/Ni-Cu) and Silicon Nitride/Steel (Si3N4/Steel)), SE geometric characteristics (ellipticity and ovality) and boundary conditions on dynamic response are investigated.

Nonlinear dynamics of flexible nanopositioning systems with geometrical imperfection

In this paper, the nonlinear dynamics of flexible beams with geometrical imperfection and concentrated end-mass is investigated. Based on the Euler–Bernoulli beam theory and nonlinear strain–displacement relations, governing equations are obtained. The Galerkin method is employed to discretize the nonlinear partial differential equations of motion. The effect of system parameters on both the linear and nonlinear dynamics of the system are studied based on the numerical solution of the set of nonlinear ODEs with coupled terms. The results show that dynamic behavior of the system is affected significantly by the geometrical imperfection. It is revealed that the geometrical imperfection results in jump instability and sub-harmonics. Since, geometrical imperfections are inevitable in manufacturing process of these equipments, the effects of imperfections should be considered in design prior to the manufacturing of nanopositioning systems.

Effect of carbon nanotube waviness on smart damping of geometrically nonlinear vibrations of fuzzy-fiber reinforced composite plates

In this article, we present a finite element model for the three-dimensional analysis of smart constrained layer damping of geometrically nonlinear vibrations of laminated fuzzy-fiber reinforced composite plates. The three-dimensional fractional derivative constitutive relation is implemented for the viscoelastic layer. The constraining layer of the smart constrained layer damping treatment is composed of the vertically/obliquely reinforced 1–3 piezoelectric composites. The von Kármán–type nonlinear strain–displacement relations are used to incorporate the geometric nonlinearity in the model. The main aim of this article is to numerically investigate the effect of carbon nanotube waviness on the nonlinear smart damping. Several thin laminated substrate fuzzy-fiber reinforced composite plates with straight carbon nanotubes and wavy carbon nanotubes with different waviness in different planes are considered with various boundary conditions and stacking sequences to numerically compute their effect on smart damping. The performance of the obliquely reinforced 1–3 piezoelectric composites is discussed and the efficacy of the present smart finite element model in terms of active control authority is also presented.

Active Control of Geometrically Nonlinear Vibrations of Laminated Composite Beams Using Piezoelectric Composites by Element-Free Galerkin Method

In this article, active control of geometrically nonlinear transient vibrations in laminated composite beams has been demonstrated using an advanced piezoelectric composite (PZC) layer as the active layer with a proper control strategy. A nonlinear mesh free (MF) model is derived for analyzing the dynamic characteristics of the laminated composite beams integrated with the PZC patch based on the element-free Galerkin (EFG) method within the framework of a layer wise beam theory considering transverse shear deformations. The Von Kármán type nonlinear strain–displacement relationships are considered for formulating the coupled electromechanical nonlinear MF model. Both in-plane and transverse actuation by the PZC layer have been taken into consideration for deriving the model. Various geometrical shapes of the PZC layers have been considered for the analysis to understand the geometrical shape dependency of the PZC layer in attenuating vibration in the structure. The numerical results indicate that the PZC layer significantly improves the dynamic characteristics of the laminated beams by suppressing the geometrically nonlinear transient vibrations induced in them. It has been also noticed that the irregular shaped PZC patches like triangular PZC patches perform better as compared to the regular rectangular PZC patches in alleviating vibration the structure.

A coupled nonlinear continuum model for bifurcation behaviour of fluid-conveying nanotubes incorporating internal energy loss

A coupled continuum model incorporating size influences and geometric nonlinearity is presented for the coupled motions of viscoelastic nonlinear nanotubes conveying nanofluid. A modified model of nanobeams incorporating nonlocal strain gradient effects is utilised for describing size influences on the bifurcation behaviour of the fluid-conveying nanotube. Furthermore, size influences on the nanofluid are taken into account via Beskok–Karniadakis theory. To model the geometric nonlinearity, nonlinear strain–displacement relations are employed. Utilising Hamilton’s principle and the Kelvin–Voigt model, the coupled equations of nonlinear motions capturing the internal energy loss are derived. A Galerkin procedure with a high number of shape functions and a direct time-integration scheme are then employed to extract the bifurcation characteristics of the nanofluid-conveying nanotube with viscoelastic properties. A specific attention is paid to the chaotic response of the viscoelastic nanosystem. It is found that the coupled viscoelastic bifurcation behaviour is very sensitive to the flow velocity.

On the numerical investigation of size and surface effects on nonlinear dynamics of a nanoresonator under electrostatic actuation

This study aims to investigate the influences of nanostructure parameter and surface elasticity parameters on the nonlinear vibration of a nanoelectromechanical system under double-sided electrostatic actuation. For this, the effects of size dependency and surface energy are modeled through applying the consistent couple-stress theory together with the Gurtin–Murdoch elasticity theory. Taking into account the midplane stretching effect for doubly clamped boundary conditions, the nonlinear strain–displacement relationship is considered based on the Euler–Bernoulli beam assumption. Hamilton’s principle is utilized in order to establish the governing differential motion’s equation, and reduced-order model is obtained through implementing Galerkin’s procedure. Bifurcation diagrams are plotted to capture the steady-state response of the system with varying the nondimensional parameter, the ratio of AC to DC voltage amplitude. The influences of the length-scale parameter, surface elasticity modulus and density, and residual surface stress on the system dynamic response have been explored. The results reveal that the pull-in excitation frequency is highly influenced by these parameters, and also the interval length of the bifurcation parameter corresponding to the periodic and chaotic motions is extremely shifted by the amount of couple-stress and residual surface stress parameters.