Nonlinear Strain-displacement Relations Research Articles

Interplay of photonic, electrical, and inertial loads on the stability of rotating sector perovskite sandwich plates with a GPL-based nanocomposite core

The bifurcation stability of sandwich sector plates, primarily constructed from lead halide perovskite skins known for their significant photostrictive and electrostrictive properties, is explored. These properties render them highly relevant for multiphysics applications. The influence of a photo-induced thermal environment on the behavior of these plates is also examined. A notable challenge, the inherent stiffness of these structures, is addressed by integrating a nanocomposite laminated core composed of a polymer matrix and graphene platelet (GPL) reinforcers. The GPLs are distributed throughout the core layers according to functionally graded models, significantly enhancing structural integrity. To effectively model the core environment, the Halpin-Tsai micromechanical rule is employed. The structural displacement field is modeled using the first-order shear deformation theory. Moreover, the von-Kármán geometrically nonlinear strain-displacement relations are applied. The constitutive relationships are governed by the theory of linear photo-thermo-electro-elasticity, providing a framework for the analysis of perovskite-based structures. The reorganization of bifurcation points from the pre-buckling route and the linearization of stability equations are performed using the adjacent-equilibrium criterion. The generalized differential quadrature (GDQ) method is utilized to solve the equilibrium equations of pre-buckling and the stability equations of buckling. This comprehensive investigation reveals the critical influence of photonic, electrical, and rotational stimuli on the stability characteristics of advanced perovskite-based sandwich sector plates, demonstrating potential advancements in multiphysics applications.

Nonlinear motion modeling and analysis of underwater cables under complex loads based on global nodal position finite element method in Hamiltonian formulation

The nonlinear dynamic response of underwater cable is essential for underwater equipment system. This paper presents an effective global nodal position finite element method in Hamiltonian formulation to investigate the nonlinear motion of underwater cable under complex loads. The method fully considers the axial, bending, torsional and shear deformation of underwater cable. The non-linear Green-Lagrange strain-displacement relationship is applied to obtain the strain energy. Furthermore, the fourth-order circular cross-section quadrature is introduced to integrate the volume of underwater cable element after deformation. The dynamic governing equations in Hamiltonian formulation are derived and converted into the corresponding canonical equations, which are calculated by the symplectic difference algorithm. Then an initial curved pendulum system, a 180° turn towed underwater cable system and the spiral turning motion of ROV are designed to verify the presented method. The results show that compared with the existing Hamiltonian global nodal position finite element method which ignores torsional and shear deformation, the proposed method is more accurate in describing the nonlinear motion of the underwater cable with axial, bending, torsional and shear deformation.

Nonlinear wave propagation in graphene incorporating second strain gradient theory

The exploration of graphene has attracted extensive interest owing to its significant structural and mechanical properties. In this research, we numerically investigate wave propagation in defect-free single-layer graphene, considering its geometrically nonlinear behavior through second-strain gradient elasticity. To capture the geometric nonlinearity, firstly, the nonlinear strain–displacement relations are introduced. The governing equation and associated boundary conditions are derived using Hamilton’s principle. Then, the weak form, including the element matrices, is established. The eigenvalue problem is solved for 2D wave propagation through periodic structures theory. Finally, the dynamical properties such as band structures, mode shapes, energy flow, and wave beaming effects are analyzed. The numerical results reveal that the geometric nonlinearity through the second strain gradient influences the wave propagation characteristics in graphene. The findings are significant and contribute to the understanding of graphene’s dynamic response, with implications for the engineering applications of graphene-based nanostructures.

Tension buckling and postbuckling of nanocomposite laminated plates with in-plane negative Poisson’s ratio

Abstract Mechanical metamaterials with negative Poisson’s ratio (NPR) have emerged as a novel class of engineering material, and have attracted increasing attention in various engineering sectors. Most studies available on the buckling problem of laminated plates with positive or NPR are those under uniaxial compression. Here, we report that the buckling phenomenon may occur for auxetic nanocomposite laminated plates under uniaxial tension when the unloaded edges of the plates are immovable. Two types of nanocomposites are considered, including graphene/Cu and carbon nanotube/Cu composites. Governing equations of the auxetic nanocomposite laminated plates are formulated based on the framework of Reddy’s higher-order shear deformation theory. In modeling, the von Kármán nonlinear strain–displacement relationship, temperature-dependent material properties, thermal effects, and the plate–substrate interaction are considered. The explicit analytical solutions for postbuckling of auxetic nanocomposite laminated plates subjected to uniaxial tension are obtained for the first time by employing a two-step perturbation approach. Numerical investigations are performed for tension buckling and postbuckling behaviors of auxetic nanocomposite laminated rectangular plates with in-plane NPR rested on an elastic substrate under temperature environments.

A multi-physical coupling isogeometric formulation for nonlinear analysis and smart control of laminated CNT-MEE plates

This paper presents a novel and comprehensive numerical approach for nonlinear analysis and smart damping control in laminated functionally graded carbon nanotube reinforced magneto-electro-elastic (FG-CNTMEE) plate structures, taking into account multiple physical fields. The approach employs a multi-physical coupling isogeometric formulation based on Non-Uniform Rational Basis Spline (NURBS) to analyze the behavior of laminated FG-CNTMEE plate structures. It incorporates a refined zig-zag theory to accurately capture the Von Kármán nonlinear strain–displacement relationship, as well as the magneto-electro-elastic coupling properties of laminate layers. To achieve nonlinear damped responses, the smart constrained layer damping (SCLD) treatment is applied. Furthermore, to enable smart control of structures using the time-dependent passive/semi-active damping mechanism of viscoelastic materials, the formulation is transformed into the Laplace domain using the Golla–Hughes–McTavish model. The results are then converted back to the time domain through inverse techniques. The proposed approach has been rigorously validated through verification work and demonstrated its effectiveness in simulating and controlling FG-CNTMEE plates in two numerical examples.

Nonlinear dynamic stability analysis of CNTs reinforced piezoelectric viscoelastic composite nano/micro plate under multiple physical fields resting on smart foundation

This study analyzes the nonlinear dynamic stability of carbon nanotube reinforced composite (CNTRC) piezoelectric viscoelastic nano/micro plate under time dependent harmonic compressive biaxial mechanical loading. The nano/micro plate is single-layer with a uniform and rectangular cross section. Polyvinylidene fluoride (PVDF) is the material used for the nano/micro plate. Also, it is located on a nonlinear viscoelastic piezoelectric foundation made of Zinc Oxide (ZnO). Single-walled carbon nanotubes (SWCNTs) have been used to strengthen nano/micro plate. Using Kelvin-Voigt model, the material properties of the system are assumed to be viscoelastic. Nano/micro plate is subjected to 2D magnetic field and electric potential. Magnetic field effects are incorporated using Maxwell’s relations. An alternating and direct voltage is applied to the nano/micro plate and also smart foundation in thickness direction. Eshelby-Mori-Tanaka approach is used to estimate the effective elastic properties. Additionally, follower force and uniform thermal gradient is applied to nano/micro plate in this study. The nonlinear strain–displacement relations are based on the Von-Kármán theory. According to classical, first order, third order, sinusoidal, parabolic, hyperbolic and exponential shear deformation plate theories, a new formulation is proposed incorporating surface stress effects through the Gurtin-Murdoch elasticity theory. In order to consider small scale parameters, this research is based on modified strain gradient theory (MSGT). Using the energy method and Hamilton’s principle, the non classical governing equations are derived. For nano/micro plate, simply supported boundary conditions are considered. For the purpose of solving differential equations, an analysis based on Galerkin procedure and finally the Bolotin method will be used to obtain the dynamic instability region (DIR). It is the objective of this study to investigate the impact of such parameters as small-scale parameters, alternating and direct applied voltage, intensity of magnetic field, surface effects, thermal environment and static load factor on the DIR. This study revealed that taking into account the smart foundation reduces the area of dynamic instability region by more than 60% for a constant intensity of magnetic field. The stability responses are also more convergent with the smart foundation. Additionally, the range is moved toward higher excitation frequencies and greater system stability. Validation of answers based on reliable scientific articles is one of the final stages of this research. These results can be used to design nano/micro electromechanical systems.

Mechanical and thermal postbuckling of functionally graded graphene origami-enabled auxetic metamaterials plates

Postbuckling under high-temperature biaxial compression and thermal postbuckling under uniform thermal loading of functionally graded (FG) graphene origami (GOri)-enabled auxetic metamaterials (GOEAMs) plates are investigated. The plates consist of multiple layers of GOEAMs with varying levels of GOri content in the layers across the direction of the thickness. The nonlinear governing equations of the plate are based on the first-order shear deformation theory (FSDT) and the von Kármán type nonlinear strain–displacement relationship. The mechanical and thermal postbuckling equilibrium paths of FG-GOEMA plates on the elastic foundation are determined using the two-step perturbation method. The effects of the elastic foundation, temperature, GOri content, GOri distribution pattern, and degree of GOri folding on the mechanical and thermal postbuckling of FG-GOEMA plates are discussed.

An analytical study on vortex-induced vibrations of cylinders with variable cross-section in the range of low Reynolds numbers

In the current paper, vortex-induced oscillations of circular cylinders with variable have been investigated. Considering the von-Kármán nonlinear strain-displacement relations and using the displacement coupling model to simulate the system and fluid-structure interaction, the nonlinear differential equations governing the motion have been derived using Hamilton’s principle. To apply the effect of vortices on the system vibrations, the nonlinear Van der Pol oscillator equation has been used. Then, using the Galerkin method, differential equations with partial derivatives are discretized, and nonlinear differential relations are solved by the Runge-Kutta approach. Finally, the response of the system, the phase curves, and the changes in the maximum amplitude according to the fluid velocity are obtained for various parameters, and the results have been discussed. Regarding the obtained results, changes in the cross-section of the beam have a significant effect on the lock-in region and also the maximum deflection of the beam. The maximum dynamic deflection for θ = 2° occurs at the velocity u = 2 , which is equal to 0.927. Also, the lock-in phenomenon is created for the beam with θ = 2.5° and θ = 3° at the velocities u = 2.78 and u = 4.23 , respectively.

An Accurate Solution for Buckling and Nonlinear Analysis of a Sandwich Beam with Functionally Graded Material by Considering Zigzag Displacements

In this paper, critical buckling analysis and nonlinear load-deflection curve for sandwich beam with functionally graded material are presented based on refined zigzag theory (RZT). By using the variational principle, the equilibrium equations in buckling analysis are given based on the RZT formulations as well as the nonlinear strain-displacement relations. The solutions are also derived for eigenvalue problems in critical buckling load calculations and for load-deflection relations with initial geometric imperfection. The solutions are presented analytically, and the mathematical properties during the derivation process have been proven in order to keep the mathematical rigor. The present analytical RZT critical buckling loads are validated by the RZT FEM, which is the finite element solutions of the sandwich beam meshed by the beam elements based on RZT. These solutions are also compared by commercial software ANSYS, resulting that this approach can obtain an accurate critical buckling load. Various parameters such as aspect ratio, thickness ratio and modulus ratio are considered to investigate their effects on the critical buckling loads. The present results are compared to the beams with higher-order shear deformation theory (HSDT). From the comparisons of the RZT and the HSDT results, it is seen that both theories approach to CBT for slender beam. The results show that the HSDT overestimates the stiffness in the load-deflection curve. It is shown that the RZT exhibits the zigzag displacements at high accuracy, resulting in accurate calculation in critical buckling loads, mode shapes and nonlinear load-deflection curves than HSDT. The superiority of the RZT solutions is presented especially for the case of FGM sandwich beam with soft middle layer.

Energy harvesting using variable stiffness piezolaminated shallow shell panels exposed to hygrothermal environments

The effect of hygrothermal environment on the energy harvesting capabilities of piezolaminated shallow shell panels made of variable stiffness composite layers (VSCL) is investigated. The use of curvilinear fibers to tailor the characteristics of the energy harvester is also explored. To model the deformation of the harvester, a finite element model based on a first-order normal shear deformation theory (FNSDT) is developed. Nonlinear strain–displacement relations based on Green–Lagrange strains are considered for nonuniform stress distribution due to temperature and moisture. A nine noded isoparametric element is used to discretize the domain. The effects of temperature and moisture concentration change on Power frequency response functions (FRFs), Voltage FRFs and Motion FRFs are presented. Power FRFs, Voltage FRFs and Motion FRFs are plotted for optimum load resistance when the shell structure is excited at short and open circuit frequencies. The current study is expected to help design and optimize the vibrational energy harvester to power sensors used in different automotive, aerospace, nuclear applications, subjected to elevated temperature and moisture conditions.

Free vibration and nonlinear dynamic response of sandwich plates with auxetic honeycomb core and piezoelectric face sheets

This article analyses the nonlinear dynamic behavior and the free vibration of a piezoelectric auxetic honeycomb sandwich plate resting on a Pasternak elastic substrate. The plate includes two face sheets of piezoelectric material and an auxetic honeycomb core with a negative Poisson ratio. The first-order shear deformation plate theory, together with the von Kármán type nonlinear strain-displacement relationship, is used to establish the nonlinear equations of motion. Linear electric potential change with piezoelectric layer thickness is assumed. Two types of boundary conditions, namely immovable and movable simply supported, are considered. The displacement-time and acceleration-displacement curves under dynamic loads are determined by using the Galerkin and Runge-Kutta fourth-order methods. Verification examples have been performed and confirmed the accuracy of the model. Finally, several novel investigations have been conducted to evaluate the effect of material, geometrical parameters, and elastic substrate coefficients on the frequency and nonlinear dynamic characteristics of piezoelectric auxetic honeycomb sandwich plates for both open-circuit and closed-circuit operation.

On the Static Instability of FG-GNP-Reinforced Composite Cylindrical Shells Under Thermo-Mechanical Loading

The nonlinear buckling response of laminated composite cylindrical shells reinforced with graphene nanoplatelets (GNPs) is studied in this paper. The functionally graded (FG) shell reinforced by GNPs is analytically studied under external pressure and uniform temperature rise loadings. It is also assumed that the GNP-reinforced laminated composite shell is in contact with an elastic foundation. Various types of profiles are employed for the GNP distribution patterns in the shell thickness including 10 nanocomposite layers. The nonlinear strain-displacement relations of the shallow cylindrical panel are established utilizing the third-order shear deformation shell theory. Governing equilibrium equations of the laminated GNP-reinforced composite shell are formulated employing the principle of virtual displacement. The coupled system of nonlinear differential equations is solved analytically for the hinged–hinged and fixed–fixed boundaries of the shell using a perturbation-based technique. Correctness of presented formulations and obtained solutions is proved by comparisons with results from previous studies for an isotropic cylindrical shell. Novel numerical results reveal that the material properties, geometrical characteristics and load parameters significantly affect on the buckling behavior of laminated composite cylindrical shells.

Nonlinear parametric excitation and dynamic stability of auxetic honeycombs core with CNTRC face sheets sandwich plate

This study aims to analyze the Nonlinear vibration and instability of a sandwich plate with an Auxetic Honeycomb core and a Carbon Nanotube Reinforced Composite (CNTRC) face layer on a viscous elastic foundation under parametric excitation. In the analytical model, the Hamilton principle and nonlinear strain-displacement relations determined by Von Karman theory and the first shear deformation plate theory (FSDT) are used. The governing equations are solved using the Galerkin method, which involves expanding the displacement field based on orthogonal functions. Once the displacement field has been expanded, a multiple-scale method is used to solve the equation of motion. The results show that the bifurcation diagrams and stability of sandwich plates depend on several parameters, including the geometric dimension of the honeycomb core, the material properties of the CNTRC face layer, and the temperature increment. The results of this study will likely be helpful in optimizing the design of sandwich plates for engineering applications, including the aerospace and automotive industries, where sandwich structures are widely utilized because of their high strength-to-weight ratios and excellent energy absorption capabilities.

Auto-parametric resonance of flexible viscoelastic beams under interaction between longitudinal and transverse modes

For a system encompassing subsystems, dynamic modal interaction may experience unanticipated vibrations among subsystems that are excited indirectly. This paper focuses attention to a nonlinear auto-parametric resonance phenomenon of a flexible beam introduced via modal coupling of longitudinal forced resonance and transverse parametric resonance under longitudinal pulsatile excitation. The coupled resonance equations are first modeled based on the Euler-Bernoulli beam theory by incorporating nonlinear curvature, von Kármán's nonlinear strain-displacement relationship and the flexible beam model characterizing nonlinear inertia and displacement interaction. The Galerkin truncation-incremental harmonic balance method (IHBM) is then introduced to obtain both the stability boundary and amplitude-frequency bifurcation maps. Through the corresponding relationship between stability boundary and bifurcation diagram, the energy exchange between two modes is identified as well as the instability region and bifurcation topology can be tuned. It is found that the modal interaction greatly widens the instability region with at most two resonance bands of lateral resonance. By tuning structural and foundation damping, resonance can be controlled to occur only in one frequency band of the pair as desired. This research provides guidance to design broadband energy harvesting or vibration suppression equipment with tunable working frequency band for using the strategy of auto-parametric resonance.

Free vibration analysis of a skew sandwich plate with bamboo biocomposite and polylactic acid core under temperature and moisture conditions

This article investigates the impact of moisture and temperature on vibration characteristics of bio-composite skew-laminated composite sandwich (SLCS) plates. The bio-composite SLCS plates with bamboo face sheets and polylactic acid (PLA) cores are biodegradable, radiolucent, lightweight, high strength, and withstand vibrations. The coupled thermo-elastic and hygro-elastic finite element (FE) model of the SLCS plate is derived using the higher-order shear deformation theory (HSDT). An initial stress stiffness matrix is developed using the nonlinear strain-displacement relations to incorporate the effect of temperature and moisture in FE modeling. Temperature and moisture-dependent material properties of bamboo fiber-reinforced biocomposite (BFB) and PLA core are employed for the analysis. A comprehensive investigation has been carried out to study the impact of moisture and temperature situations with different geometrical parameters and various skew angles on the frequency of the SLCS plate. The results indicate that the biocomposite sandwich material has excellent potential for structural applications under different environmental conditions in various applications like food processing, and biomedical applications, including MRI and CT scan beds.

Free Vibration Analysis of Trapezoidal Bi-Stable Laminates Supported at the Elastic Midpoint of the Median Line

This study investigates the natural vibration of trapezoidal bi-stable laminates (TBL) with elastic supports at the midpoints of the median lines. Configuration of the midplane of the TBL is expressed by a polynomial with 17 parameters. Then, the first order shear deformation theory, curing temperature, and nonlinear strain displacement relations combining energy principles are applied to obtain the bi-stable shapes numerically. Three translational springs and two rotational springs are added at the midpoint of the median line in the trapezoidal bi-stable laminate to acquire elastic point supports. And, by varying the stiffness of the springs, arbitrary elastic point support boundary conditions can be achieved. Chebyshev polynomials are applied to characterize the mode shape function of the TBL. The vibration mode functions of the TBL are mapped to a square area under the new coordinate system by using the coordinate mapping method. Furthermore, the effects of geometry, layup sequence, and spring stiffness on the natural vibrations of the TBL are analyzed, which provides a reference for research in this field. The innovation and highlights lie in the following: (1) the natural frequencies and modes of trapezoidal bi-stable plates are solved; (2) arbitrary elastic support is achieved by a set of artificial springs; (3) the influences of spring stiffness, layer sequence, and trapezoidal base angle on the natural vibration of a trapezoidal bi-stable plate are studied.

Nonlinear dynamic study of non-uniform microscale CNTR composite beams based on a modified couple stress theory

This study aims at investigating the nonlinear dynamic behavior of microscale carbon nanotube reinforced (CNTR) composite Euler–Bernoulli beams with a non-uniform cross-section, based on a modified couple stress theory (MCST). The nonlinear partial differential equations (PDEs) of motion are established based on the Von-Karman nonlinear strain–displacement relationship and Hamiltonian principle. The coupled PDEs are reduced to a single PDE, by neglecting the effects of the axial inertia and considering two different types of boundary conditions (i.e. clamped–clamped and clamped–free). At the same time, the single PDE is reverted to a nonlinear ordinary differential equation (ODE) by means of the Galerkin approach, and it is solved by using a semi-inverse method and the method of multiple time scales (MTS) for a free and forced vibration analysis, respectively. A large systematic numerical analysis is here performed to check for the sensitivity of the nonlinear response of CNTR composite beams to different boundary conditions and reinforcement parameters, with useful scientific insights for further computational investigations on the topic.

Dynamic Analysis of FGM Plates with Variable Delamination Parameters by the Chebyshev–Ritz Method

Delamination is one of the failure modes of functionally graded materials (FGM), resulting in the critical change of vibration characteristics. The Chebyshev polynomials are commonly used as admissible functions to improve the computational efficiency of numerical algorithms and avoid the occurrence of ill-conditioned problems. This paper extends the Chebyshev–Ritz method to the free vibration analysis of delaminated FGM plates, in which the material variation through the plate thickness follows the exponential-law distribution. A plane crack that is considered to be perpendicular to the thickness direction penetrates through the width direction. Based on the region approach, the analysis of FGM plates with a mid-plane delamination is divided into four sub-regions. The kinetic energy and potential energy of each sub-region are derived by the thin plate theory and von Kármán nonlinear strain–displacement relation. The modal functions of the displacement fields of FGM plates can be constructed in accordance with the displacement continuity conditions of the delamination interface and the boundary conditions of such plates. The effects of asymmetric material distribution, delamination length ratio, Young’s modulus ratio, and boundary support on the vibration behavior of FGM plates are investigated. This semi-analytical study provides a reasonable theoretical basis for the behavior prediction and delamination identification of composite structures.

Reexamination for linear and nonlinear free vibration of porous sandwich cylindrical shells reinforced by graphene platelets

This paper reexamines the linear and nonlinear free vibration responses of porous sandwich cylindrical shells with graphene platelet (GPL) reinforcements surrounded by an elastic medium under temperature conditions. The graphene platelets reinforced composite (GPLRC) core is assumed to be multilayers, and each layer may have different values of porosity coefficient to achieve a piece-wise functionally graded pattern. By introducing an inhomogeneous model instead of the equivalent isotropic model (EIM), the Young’s moduli along with the shear modulus of porous GPLRC core are predicted through a generic Halpin–Tsai model in which the porosity is included. Thermo-mechanical properties of porous GPLRC core and metal face sheets are assumed to be temperature dependent. Motion equations of porous sandwich cylindrical shells reinforced by GPLs are formulated based on the Reddy’s third order shear deformation theory. In the modeling, von Kármán nonlinear strain-displacement relationships, shell-foundation interaction and thermal effect are also taken into account. The analytical solution for the nonlinear vibration problem is obtained by applying a two-step perturbation approach. Numerical studies are performed to compare the results obtained from the present model and the EIM. The results reveal that, in most cases, the difference of the fundamental frequency between the two models is over 10% and in that case the EIM is not suitable for the linear free vibration analysis of porous sandwich cylindrical shells reinforced by GPLs, whereas for the nonlinear-to-linear frequency ratio curves, the difference between the two models is less than 2% when the non-dimensional vibration amplitude reaches 1.0 and in that case the EIM may be valid in the analysis for the same shell.

Re-examination of nonlinear vibration and nonlinear bending of porous sandwich cylindrical panels reinforced by graphene platelets

Abstract This article re-examines the nonlinear vibration and nonlinear bending responses of porous sandwich cylindrical panels reinforced by graphene platelets resting on elastic foundations in thermal environments. The graphene platelet-reinforced composite (GPLRC) core is assumed to be of multilayers, and each layer may have different porosity coefficient values to achieve a piece-wise functionally graded pattern. By introducing an inhomogeneous model instead of the equivalent isotropic model (EIM), the Young’s moduli along with the shear modulus of the porous GPLRC core are predicted through a generic Halpin–Tsai model in which the porosity is included. The thermomechanical properties of metal face sheets and the porous GPLRC core are assumed to be temperature-dependent. Governing equations of motion for sandwich cylindrical panels with porous GPLRC core are formulated based on Reddy’s third-order shear deformation theory coupled with von Kármán nonlinear strain–displacement relationships. In the modeling, the panel–foundation interaction and the thermal effects are also considered. The analytical solutions for the nonlinear vibration and nonlinear bending problems are obtained by applying a two-step perturbation approach. Numerical studies are performed to compare the results obtained from the present model and the EIM. The results confirm that the EIM is not suitable for linear free vibration analysis of sandwich cylindrical panels with the porous GPLRC core, but the EIM may be valid for the cases of nonlinear vibration and nonlinear bending analyses of the same panel resting on Pasternak elastic foundations.