Von Karman Nonlinear Theory Research Articles

Nonlinear vibrations of doubly curved micropanels reinforced by graphene nanoplates in length direction under non-uniform thermal loading

Structures made of composites are quite common in different industries entailing automotive, marine, and aircraft industries, thanks to their favorable lightweight and strength. This study investigated the nonlinear vibrations of the various shapes of size-dependent graphene nanoplatelets reinforced micropanels under non-uniform thermal loading. In diverse ways in the axial direction, the material characteristics of graphene nanoplates (GPLs) alter. Taylor’s series expansion terms in curvature coordinates are used to illustrate the displacement field. Nonlinear von Karman theory and Hamiltonian principle work together to derive the nonlinear form of the pertinent equations and boundary conditions. Strain gradient theory with consideration for inherent length scale characteristics is investigated to account for size effects. By discretizing the spatial domain equations, deriving Duffing-type equations, and using Kronecker and Hadamard products, the governing equations and numerous nonlinear boundary edges are resolved. Using the techniques and answers outlined above, the results section is produced. There are four sections for this research. The findings are cross-checked with those from other studies that have been published in the first section. The effects of geometric and physical factors are shown in Parts 2, 3, and 4 on the nonlinear to linear frequency curve, the nonlinear frequency curve, and the sensitivity of the nonlinear frequency-to-size effect (SNFSE).

Buckling analysis of super-light composite cylinders with auxetic core and isotropic facing sheets with variable thickness: An analytical approach

An analytical method is presented to determine the axial buckling load of composite cylindrical shells with honeycomb core layer and nonuniform thickness using the first-order shear deformation theory, the nonlinear von Karman theory, and the Hooke law relations. The composite shell consists of two isotropic inner and outer layers with non-uniform thicknesses and one honeycomb core with constant thicknesses and adjustable Poisson's ratio. The equilibrium equations are a system of nonlinear differential equations with variable coefficients, and they are derived by employing the virtual work principle. The equilibrium equations have been solved analytically using the matched asymptotic expansion method of the perturbation technique. The stability equations extracted using the adjacent criterion include a system of linear homogenous equations with variable coefficients. By solving these equations, the buckling load is obtained analytically. A parametric study investigates the effects of different geometrical and mechanical parameters such as the honeycomb structure on the buckling load. The metamaterial honeycomb core layer has an adjustable Poisson's ratio, and the effect of negative/positive Poisson's ratio has been studied on the buckling behavior. The accuracy of the presented method is studied by comparing the results with the finite element method and some other references in special cases.

Nonlinear impulsive and vibration analysis of nonlocal FG-CNT reinforced sandwich plate by considering agglomerations

This paper is focused on impulsive and subsequent vibration response of a nano-scale composite sandwich plate considering both nonlocal effect and nanoparticle agglomerations. In particular, the core is homogeneous polymethyl methacrylate (PMMA) and the face sheet is PMMA matrix with functionally graded carbon nanotubes (short for FG-CNT). The material properties are given by Eshelby-Mori-Tanaka's approach taking into account of CNT agglomerations. The equations of motion for the FG-CNT reinforced nanoplate are derived based on nonlocal strain gradient theory, Reddy's higher order shear deformation theory and von-Karman nonlinear deformation theory. The impulsive and vibration problems for simply supported FG-CNT plates are solved using Galerkin method and Newton iteration method where the solutions have been verified with the literature. By considering impulsion of step load, exponential load and sinusoidal load and, it is found that weaker size effect, fewer CNT agglomerations, higher CNT content, smaller functional gradient, thinner core, bigger aspect ratio and larger length thickness ratio can enhance the stiffness of the nanoplate and thus will induce smaller center displacement during the load process and higher free vibration frequency after unloading, regardless of load type. However, the subsequent vibration amplitude is significantly influenced by the impulsive load type. Among the material parameters, minimizing size effect and agglomerations could greatly improve the structural stiffness whereas the improvement by varying functional gradient is very limited. It is the first time that a FG-CNT reinforced sandwich plate, especially considering nonlocal effect and nanoparticle agglomerations, is treated in the context of impulsive and subsequent vibration problems. The solutions presented in this paper could provide a benchmark for the design and manufacture of such a composite nanoplate.

Nonlinear vibration suppression of S-S and C-C laminated composite cylindrical shells with magnetostrictive strips

The focus of this study is on the control of nonlinear vibration of laminated composite cylindrical shells under S–S and C–C boundary conditions through magnetostrictive strips. The shells are modeled considering the effects of shear deformation and rotary inertia while geometrical nonlinearity is modeled using von Karman nonlinear theory. In order to obtain ordinary differential equation of the system, Ritz method and beam model are hired. The combination of multiple scales method and modal analysis is used for obtain of the vibration characteristics of the shells. Some results of literature are employed to investigate the validity of the results of this study. Lyapunov's indirect method is used to study asymptotical stability of the results of systems around zero equilibrium point. Finally, the effects of several parameters on the linear and nonlinear vibration characteristics are reported.

Computer simulation of the nonlinear static behavior of axially functionally graded microtube with porosity

Static analysis of microstructures, including bending and buckling, is widely practiced in the fabrication and creation of applications such as actuation, sensing, and energy recovery. This article aims to inquire about the static behavior of non-uniform and imperfect microtubes through a numerical solution. Based on the modified couple stress theory, the first-order shear deformation theory and Von-Karman nonlinear theory, and employing the energy conservation method, the linear and nonlinear governing equations are derived. The porosity-dependent material in both ceramic and metal phases makes the functionally graded materials which are varied along tube length, moreover, cross-sections are also considered uniform and nonuniform via three valuable functions. Finally, the linear and nonlinear equations are solved utilizing the generalized differential quadrature method (GDQM) coupled with the numerical iteration method.

Tuned deep neural network for computer simulation of chaotic performance of the electromechanical annular system coupled with a piezoelectric patch

In this paper, an attempt is made to extend a nonlinear two-dimensional model for large oscillation and chaotic characteristics of the annular plate coupled with a piezoelectric patch subject to external excitation. For improving the mechanical properties, the core of the electrically structure is reinforced with graphene nanoplatelets. Von-Karman nonlinear theory and Hamilton’s principle are taken into consideration for the exact derivation of the general nonlinear governing equations and boundary conditions of the axisymmetric annular plate coupled with a piezoelectric patch. For developing a precise solution approach, the generalized differential quadrature element method, and Multiple scales technique, are eventually used. The results demonstrate that radius ratio, applied voltage, thickness to radius ratio, and harmonic load have an important role in the nonlinear large oscillation, and chaotic motion of the graphene platelets reinforced composite (GPLRC) annular plate coupled with a piezoelectric patch. The golden and fundamental result of the current research is that, as the thickness of the piezoelectric patch increases, the system's nonlinear frequency decreases, and the area of instability in responses and maximum amplitude or the peak point of the backbone curve of the smart annular plate increases.

The study of the nonlinear vibration of S-S and C-C laminated composite conical shells on elastic foundations through an approximate method

This paper investigates the nonlinear vibration responses of laminated composite conical shells surrounded by elastic foundations under S-S and C-C boundary conditions via an approximate approach. The laminated composite conical shells are modeled based on classical shell theory of Love employing von Karman nonlinear theory. Nonlinear vibration equation of the conical shells is extracted by handling Lagrange method. The linear and nonlinear vibration responses are obtained via an approximate method which combines Lindstedt-Poincare method with modal analysis. The validation of this study is carried out through the comparison of the results of this study with results of published literature. The effects of several parameters including the constants of elastic foundations, boundary conditions, total thickness, length, large edge radius and semi-vertex angle on the values of fundamental linear frequency and curves of amplitude parameter versus nonlinear frequency ratio for laminated composite conical shells with both S-S and C-C boundary conditions are investigated.

Nonlinear frequency and chaotic motion of the honeycomb higher-order disk with graphene nanoplatelets face sheets subjected to harmonic excitation via two-dimensional analysis

Honeycomb structures are one type of structure that has the geometry of a honeycomb to allow the minimization of the amount of used material to reach minimal material cost and minimal weight. For this issue, in the current analysis, an attempt is made to develop the nonlinear mathematical model for the chaotic and large-amplitude motions of a sandwich disk with graphene nanoplatelet face sheets honeycomb core and subjected to external excitation. Using Hamilton’s principle, higher-order shear deformation theory (HSDT), and the Von-Karman nonlinear theory, the nonlinear governing equation is derived. The generalized differential quadrature method (GDQM) and perturbation approach (PA) are eventually used to develop a precise solution approach. This article’s fundamental and golden results are that for clamped–clamped boundary conditions, the softening of the system in zone 2 is less than zone 1, whereas the instable responses in zone 2 are more than zone 1. Also, for the sandwich disk with clamped–clamped boundary conditions, the system’s dynamic behavior tends to be more harmonious and less chaotic as the fiber’s angle increases. Therefore, increasing the fiber angle reduces the system’s chaos with clamped edges, which is very important for future works.

On the chaotic behavior of graphene-reinforced annular systems under harmonic excitation

In this study, a mathematical derivation is made to develop a nonlinear dynamic model for the nonlinear frequency and chaotic responses of the graphene nanoplatelets (GPLs)-reinforced composite (GPLRC) annular plate subject to an external harmonic load. Using Hamilton’s principle and the von Karman nonlinear theory, the nonlinear governing equation is derived. For developing an accurate solution approach, generalized differential quadrature method (GDQM) and perturbation approach (PA) are finally employed. Various geometrically parameters are taken into account to investigate the chaotic motion of the annular plate subject to a harmonic excitation. The fundamental and golden results of this paper could be that the chaotic motion and nonlinear frequency of the annular plate are hardly dependent on the value of the length to thickness ratio (lGPL/wGPL) of the GPLs. Moreover, utilizing GPLs in the shapes close to square (lGPL/wGPL = 1) presents higher frequency of the annular plate. Also, increase in lGPL/tGPL indicates that using GPLs with lower thickness relative to its length provides better frequency response

Nonlinear dynamics of fluid conveying double-walled nanotubes incorporating surface effect: A bifurcation analysis

The aim of this work is to analyze the effect of the residual surface stress on the dynamic instability of the fluid conveying double-walled nanotubes (DWNTs). To achieve this goal, the theory of surface effect, nonlocal elasticity and nonlinear von Karman theory are combined which lead to new insight in bifurcations of fluid conveying DWNTs. The conveying fluid is also modeled and analyzed based on modified Navier–Stokes relation considering Knudsen number. In addition, Galerkin technique and multiple time scales method is applied to solve the equation. By means of the bifurcation analysis, it is shown that a discontinuous bifurcation is always accompanied by a jump. Finally, stable and unstable regions in dynamic instability of DWNTs are addressed. We close this paper by stating that the residual surface stress enhances the stiffness of DWNTs. Furthermore, the results indicate and emphasize that the nanofluid velocity as an external excitation can significantly affect the instability behaviors of the system.

The effect of geometry on the performance of amplified flexure-guided actuators

This study presents an analytical model of a flextensional actuator comprised of a piezoceramic rod or stack transducer integrated into a flexible mechanical structure. It converts and amplifies the longitudinal in-plane strains in the transducer into the out-of-plane bending of metal beams. Two curvilinear beams mounted by means of two hinged links with an offset to a centrally located piezoceramic transducer create a monolithic hinge lever mechanism. The idea behind that solution is to reduce the bending stiffness at the hinges, while maintaining enough axial rigidity of the active beams and obtain greater magnification of the out-of-plane displacement. As the main purpose of this work is to estimate the influence of piezoelectric force on the performance of flextensional actuators of different geometric shapes and mechanical parameters, two transducers have been considered. An analytical model of the actuator has been developed on the basis of the stationary value of the total potential energy principle with the use of the von Karman non-linear strains theory. Moreover, the application of structure prestressing is considered to avoid undesired stretching of the piezoceramic transducer. A vast number of results exhibit the mechanical responses of the actuator of different geometry, initial midspan rise and beam material properties to piezoelectric stimulation. The analytical method described here may be adopted for other types of flextensional actuators. Additionally, a numerical model of flexure-guided transducer is presented to show that the analytical solution can provide a set of valuable design parameters in less time than FEM simulations.

Nonlinear forced vibrations of nanocomposite-reinforced viscoelastic thick annular system under hygrothermal environment

Mechanical systems especially annular disks have many applications in different fields such as engineering, agriculture, and medicine. In this regard, the large amplitude vibrations of multi-scale hybrid nanocomposite annular system (MHCAS) under hygrothermal environment and subjected to mechanical loading. Also, the structure is covered with elastic foundation. The matrix material is reinforced with carbon nanotubes or carbon fibers at the nano- or macro-scale, respectively. The displacement strain of nonlinear vibration of the multi-scale laminated disk via third-order shear deformation theory and using von Karman nonlinear shell theory is obtained. Hamilton’s principle is employed to establish the governing equations of motion, which is solved by the generalized differential quadrature method and perturbation method. Finally, the results show that increasing the value of the and parameters and the rigidity of the boundary conditions lead to increase in the frequency of the structure. Besides, when the value of the nonlinearity parameter () is positive and negative, the dynamic behavior of the system tends to have hardening and softening behaviors, respectively, and could not be seen any effects from parameter on the maximum amplitudes of resonant vibration of the MHCAS. The last but not the list by increasing the rigidity of the structure, the system can be susceptible to have unstable responses. The presented outputs can be used in the structural health monitoring.

RETRACTED ARTICLE: Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM

In this research, a mathematical derivation is made to develop a nonlinear dynamic model for the nonlinear frequency and chaotic responses of the multi-scale hybrid nano-composite reinforced disk in the thermal environment and subject to a harmonic external load. Using Hamilton’s principle and the von Karman nonlinear theory, the nonlinear governing equation is derived. For developing an accurate solution approach, generalized differential quadrature method (GDQM) and perturbation approach (PA) are finally employed. Various geometrically parameters are taken into account to investigate the chaotic motion of the viscoelastic disk subject to harmonic excitation. The fundamental and golden results of this paper could be that in the lower value of the external harmonic force, different FG patterns do not have any effects on the motion response of the structure. But, for the higher value of external harmonic force and all FG patterns, the chaos motion could be seen and for the FG-X pattern, the chaosity is more significant than other patterns of the FG. As a practical designing tip, it is recommended to choose plates with lower thickness relative to the outer radius to achieve better vibration performance.

Chaotic responses and nonlinear dynamics of the graphene nanoplatelets reinforced doubly-curved panel

In this article, a mathematical derivation is made to develop a nonlinear dynamic model for the nonlinear frequency, and chaotic responses of the graphene nanoplatelet (GPL) reinforced composite (GPLRC) doubly-curved panel subject to an external harmonic load. Using Hamilton's principle and the Von-Karman nonlinear theory, the nonlinear governing equations are derived. For developing an accurate solution approach, generalized differential quadrature method (GDQM) and perturbation approach (PA) are finally employed. The results show that GPL's pattern, radius to length ratio, harmonic load, and thickness to length ratio have important role in the chaotic motion of the doubly-curved panel. The fundamental and golden results of this paper is that the chaotic motion and nonlinear frequency of the panel is hardly dependent on the value of the smaller radius to length ratio (R1/a parameter) and viscoelastic foundation. It means that by increasing the value of R1/a parameter, and taking into account the viscoelastic foundation, the motion of the system tends to show the chaotic motion. Moreover, for GPL-A, GPL-V, and GPL-UD patterns, when the value of the R1/a parameter or the curvature shape of the doubly-curved panel increases, the chaoticity in motion response improves while for the GPL-O pattern, this matter reverses.

Nonlinear vibration of a deploying laminated Rayleigh beam with a spinning motion in hygrothermal environment

The free vibration of a deploying laminated beam with spinning motion in hygrothermal environment is studied. The model of this system is given within the framework of the Rayleigh beam theory and the von Karman nonlinear strain theory. The nonlinear dynamic equilibrium equation of the cantilever boundary condition is established based on Hamilton's principle with considering the combined effects of the axial motion, transverse vibration, spinning motion, and hygrothermal environment. The frequencies of the system are numerically determined by using the Newmark method. The detailed parameter analyses are provided to investigate the effects of the deploying speed, the spinning speed, the temperature variation, the moisture concentration, and the fiber direction angle on frequency and the corresponding sensitivity.

Effects of surface tension of graphene sheet on impact and rebound behavior of colliding nanoparticle

Abstract In the present work, the classical theory of plate, the von Karman nonlinear theory and the nonlocal theory of elasticity are adopted to analyze the impact between a nanoparticle and an orthotropic rectangular nanoplate with surface tensions. Employing an analytical contact law to explain the von der Waals forces, the equations of motion are solved for the indenter-plate system. As an application, this approach is taken for a nanoparticle colliding with a simply supported single-layered graphene sheet and the rebound behavior of the particle is discussed. Then, the influences of surface tension and the nonlocal parameter on the dynamics response of the particle are investigated. When enough surface tension is applied, the membrane theory gives an acceptable estimate for the restitution behavior of the indenter otherwise, the estimates are associated with a maximum error of 15%. The effects of disregarding the flexural stiffness of the graphene sheet and ignoring the attraction part of the von der Waals forces are also examined. Studying the dimensional effects on the system behavior revealed that optimum dimensions of the plate maximize its capacity for energy absorption when it is employed as an energy absorber against colliding particles. Finally, a contact law for large nanoparticles is proposed and then is applied to an example including an SLGS colliding with a multi-atom indenter.

Nonlinear dynamics and vibration of reinforced piezoelectric scale-dependent plates as a class of nonlinear Mathieu–Hill systems: parametric excitation analysis

This work is motivated by little research in the nonlinear dynamic instability of the reinforced piezoelectric nanoplates. This paper, using an analytical approach, presents bifurcations in the nonlinear dynamic instability of the reinforced piezoelectric nanoplates caused by the parametric excitation. An axial parametric load is applied to excite the system, while the reinforced piezoelectric nanoplate is under an applied electric voltage, simultaneously. The governing equations of motion for the reinforced piezoelectric nanoplate embedded on a visco-Pasternak foundation are derived using the nonlocal elasticity theory, Hamilton’s principle, and nonlinear von Karman theory. A class of nonlinear the Mathieu–Hill equation is established to determine the bifurcations and the regions of the nonlinear dynamic instability. The numerical results are performed, while the emphasis is placed on investigating the effect of the applied electric voltage, visco-Pasternak foundation coefficients, and the parametric excitation. It is found that the damping coefficient is responsible of the bifurcation point variation, while the amplitude response depends on the term of the natural frequency.

Investigation for electro-thermo-mechanical vibration of nanocomposite cylindrical shells with an internal fluid flow

In the present, based on geometrical nonlinearity in von Karman sense and classical thin shell theory (CLT), the vibrational analyses of conveying-fluid functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical shells with piezoelectric layers in thermal environment are taken into account by an analytical approach. The uniform and functionally graded CNTs are used to reinforce through the thickness of the shell. The fluid flow in the shell is mentioned as non-viscous, incompressible, isentropic and irrotational. Furthermore, piezoelectric layers are bonded onto the outer surface of the cylindrical shell to act as an actuator. The equations of motion are derived by using the CLT, von Karman nonlinearity theory, the fluid velocity potential and then solved by Galerkin's technique. Moreover, the fourth-order Runge-Kutta method is used to resolve the differential equations. The fundamental frequencies, responses of nonlinear vibration as time histories and bifurcation diagram are studied in this paper. Furthermore, the effects of CNT volume fraction, piezoelectric actuator, thermal environment, geometrical parameters, elastic medium and the fluid flow velocity are carefully analyzed. The obtained results that are validated with those of other studies can be used as benchmark solutions for an analytical approach serving in further research.

Nonlinear free vibration analysis of embedded flexoelectric curved nanobeams conveying fluid and submerged in fluid via nonlocal strain gradient elasticity theory

In this study, harmonic balance method is employed to solve nonlinear equation of flexoelectric nanobeams by initial curvature and embedded in an elastic medium conveying viscous fluid and submerged in fluid. Fluid is assumed to be incompressible, laminar and Newtonian. Assuming Euler–Bernoulli’s beam theory with simply supported ends based on nonlocal strain gradient theory and Von-Karman nonlinear strain theory come to the nonlinear differential equation of motion. Using Galerkin method, the nonlinear partial differential equation reduces to ordinary differential equation. The effects of nonlocal and strain gradient parameters, maximum amplitude, nonlinear foundation, initial curvature, flexoelectric effect and … on the real and imaginary parts of the nonlinear natural frequencies are investigated. By increasing flexoelectric coefficient, natural frequency and critical velocity decreases and divergence and flutter instability occur earlier and the stability range becomes smaller. .

Buckling and post-buckling analyses of functionally graded graphene reinforced piezoelectric plate subjected to electric potential and axial forces

The buckling and post-buckling behaviors are analyzed for the multilayer functionally graded graphene platelets reinforced piezoelectric (FG-GRP) plates. The FG-GRP plates are subjected to the external electric potential and axial forces, including the uniaxial loading and biaxial loading. The graphene platelets (GPLs) disperse uniformly and parallelly in each graphene platelets reinforced piezoelectric composite (GRPC) layer, but they spread grading across the thickness of the FG-GRP plates. The effective Young’s modulus of each layer for the FG-GRP plates is calculated by the Halpin-Tsai parallel model. The rule of the mixture is employed to predict the Poisson’s ratio, effective mass density and piezoelectric properties of each layer of the FG-GRP plates. The governing equations of motion for the FG-GRP plates are obtained by the first-order shear deformation plate theory, von Karman nonlinear theory and principle of virtual displacements. To obtain the buckling and post-buckling behaviors of the FG-GRP plates with different boundary conditions, the differential quadrature (DQ) method and a direct iterative technique are combined to solve the governing equations of motion for the FG-GRP plates. The impacts of the external electric voltage, distribution pattern, volume fraction, piezoelectric properties, length-to-thickness of the GPLs and geometry of the plates on the critical buckling load and post-buckling equilibrium paths of the FG-GRP plates are discussed in detailed. It is clearly illustrated that the GPLs have a significantly enhancing influence on the buckling and post-buckling strength of the FG-GRP plates.