Nonlinear Strain-displacement Relations Research Articles

Modeling Method of Meshing Stiffness of High Speed Spur Gears

As one of the most important dynamic excitation sources, the gear time-varying mesh stiffness is the key parameter of gear system dynamic model. So how to calculate the gear mesh stiffness accurately is of great importance. Based on Euler beam theory, this paper proposes a primitive calculation algorithm (PCA) to calculate the time-varying mesh stiffness of spur gears. A simplified gear model is developed based on nonlinear strain-displacement relationships, and a finite element program is used to simulate the gear's meshing stiffness. The rationality of this method was verified through finite element analysis. The results indicate that mesh stiffness plays a crucial role in controlling mesh deformation. In view of this, this study lays a theoretical foundation for further analysis of the vibration and noise of gears.

Nonlinear buckling and free vibration analysis of auxetic graphene origami composite beams under nonuniform thermal environment

This study examines the thermo-mechanical behavior of auxetic metamaterial beams enhanced by graphene origami (GOri) under spatially varying nonuniform temperature distributions (SVTD). Utilizing Timoshenko beam theory considering von-Kármánn type nonlinear strain–displacement relationship, GOri beams are modeled as layered structures. The Ritz method is employed to solve equilibrium equations, analyzing the impact of GOri distribution patterns, content, and folding degree on post-buckling and vibration paths. The effects of five SVTDs, three end conditions, and three GOri distribution patterns on buckling, post-buckling behavior, and nonlinear free vibration characteristics are explored. Findings reveal that the parabolic temperature distribution with peak temperatures at beam ends (P-MAE) results in higher critical temperatures and nonlinear free vibration frequencies. This research provides crucial insights into the design and optimization of GOri-enabled metamaterial structures in complex thermal environments, highlighting the significant influence of nonuniform temperature distributions along the beam’s length.

A semi-analytical methodology for predicting the vibroacoustic response of functionally graded nanoplates under thermal loads

In this investigation, the analysis of the nonlinear vibroacoustic and sound transmission loss behaviors of nanoplates made of functionally graded materials considering the nonlocal strain gradient theory is presented. It is presumed that the properties of the functionally graded nanoplate are in the form of the simple power law scheme and continuous along the thickness, under nonlinear temperature rise and incident oblique acoustic wave as well as the first-order shear deformation theory. For this purpose, applying Hamilton’s principle, the nonlinear partial differential equations of motion are derived by the displacement field function approach and by considering the von Kármán nonlinear strain-displacement relations. To solve the equations, using the Galerkin method, the nonlinear partial differential equations of motion lead to the Duffing equation. Then, applying the homotopy analysis method, the equation of the transverse movement of the nanoplate is solved semi-analytically to obtain the nonlinear frequencies. Lastly, the nonlinear vibration and acoustic response of functionally graded nanoplate are investigated by considering the variation of the significant factors such as material gradient indices, nonlocal parameters, length scale parameters, aspect ratio, dimensionless amplitude, nonlinear temperature changes, and sound characteristics of the functionally graded nanoplate.

Nonlinear effect on stable state and snap-through bistability of square composite laminate

Bistable structures with the elastic instability caused by buckling are confirmed to perform significantly in micro-electromechanical systems, metamaterials, energy harvester, vibration isolation and morphing structures. The target of this paper is to explore the dynamic responses of cross-ply bistable composite laminate, focusing on the nonlinear effect of the single- and double-well vibration. Both theoretical and finite element (FE) methods are employed to simulate the nonlinear vibration and dynamic snap-through of bistable composite laminate under the foundation excitation at the center. In the theoretical model, the governing equations are established via Lagrange's equation based on the first-order shear deformation theory, von Karman nonlinear strain-displacement relation and Rayleigh-Ritz method. The fourth-order Runge-Kutta method is adopted to solve the governing equations, and the numerical results are validated by FE model. Subsequently, the details of the dynamic responses are analyzed to identify the nonlinear effects in the form of bifurcation diagram, phase portrait, time history, Poincare maps and amplitude spectrum. The dynamic responses are examined for a series of excitation parameters in both time and frequency domain. Through fixed frequency and frequency sweep tests, the nonlinear phenomenon of the single- and double-well vibrations are analyzed including superharmonic resonance, stiffness softening, hysteresis phenomenon, various periodic and chaotic vibration. The diverse responses to external inputs contribute significantly to efficiently predicting mechanical behaviors in real-world conditions, thereby offering indispensable theoretical support for structural design applications.

Chaos and resonance of nonlinear FG shallow shells reinforced by oblique stiffeners

This paper presents an analysis of the nonlinear dynamics of functionally graded (FG) shallow shells that are simply supported, reinforced with oblique stiffeners, and subjected to external excitation. The properties of these oblique stiffened functionally graded (OSFG) shallow shells vary across their thickness, following a power law distribution based on the volume fractions. Employing the first-order shear deformation theory (FSDT) coupled with the von Kármán nonlinear strain-displacement relations, the derivation of the nonlinear equations of motion is accomplished through the application of Hamilton's principle. The model of FG shallow shells reinforced with oblique two-series stiffeners is developed, allowing for the angles of the two stiffeners to be similar or different from each other. In this context, the employment of the first-order shear deformation theory (FSDT) results in the enhancement of the complexity in modeling the oblique stiffeners. Galerkin's method is applied to discretize the given equations, converting them into a nonlinear dynamical system with two degrees of freedom. This system incorporates both quadratic and cubic nonlinearities and considers the effects of external excitation. The analysis of this research particularly focuses on the resonant scenario involving principal resonance and 1:1 internal resonance. Through the asymptotic perturbation method, a four-dimensional nonlinear averaged equation is obtained. For the first time, a numerical investigation of the research reveals critical insights into the behavior of the system, such as waveforms, phase diagrams, and Poincaré maps, highlighting the effects of different angles of the stiffeners and variations of material parameters on the nonlinear dynamics of these shells.

Nonlinear thermo-electro-mechanical responses and active control of functionally graded piezoelectric plates subjected to strong electric fields

Functionally graded piezoelectric plates (FGPPs) are often exposed to extreme thermal environments and subjected to large amplitude excitation and strong electric fields. Accurately predicting the nonlinear behaviors of FGPPs under large thermo-electro-mechanical loads remains a considerable challenge. This paper proposes a comprehensive nonlinear coupled analysis model that considers geometric nonlinearity, piezoelectric nonlinearity, and the temperature dependence of piezoelectric parameters. The total Lagrangian incremental finite element equations for FGPPs under thermo-electro-mechanical loads are derived using Hamilton's principle, by employing the first-order shear deformation theory and the von Kármán nonlinear strain-displacement relationship. The accuracy of the model is validated through comparative studies. Based on this multi-nonlinear model, we further investigate the effects of geometric nonlinearity, piezoelectric nonlinearity, and temperature dependence on the nonlinear static bending, dynamic responses, and active control of FGPPs. This study demonstrates that considering these factors enables accurate prediction of the static, dynamic, and active control behaviors of FGPPs.

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.

Non-linear primary resonance analysis of stiffened FGM conical panels resting on viscoelastic foundation

This article investigates the non-linear forced vibrations and primary resonance analysis of stiffened FGM conical panels (OCPs) resting on a viscoelastic foundation utilizing a semi-analytical method. The classical shells theory and von Kármán nonlinear strain–displacement relations are used to model of OCPs. The equations of motion are derived by applying Hamilton’s principle. Then, the dimensionless equations of motion are established and converted into dimensionless nonlinear ordinary differential equations by applying Galerkin’s method. The solution of the ordinary differential equations is carried out based on the multiple-scale method for analyzing the vibration behavior. In this regard, the necessary relations for the nonlinear primary resonance and internal resonance of stiffened FGM-OCPs are obtained. To validate the suggested methodology, first, the results are verified with various previous research. Second, a numerical method based on the fourth-order Runge-Kutta’s approach is used for verifying the nonlinear vibration analysis. Finally, changes in various parameters such as viscoelastic foundation coefficients, stiffeners, temperature, semi-vertex angle and subtended angle are examined by helping the frequency response plots.

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.