Von Karman Nonlinearity Research Articles

Nonlinear bending analysis of matrix cracked functionally graded graphene origami-enabled auxetic metamaterial plates using the C0-HSDT meshless method

The auxetic metamaterials characterized by negative Poisson’s ratio (NPR) are attracting significant interest due to their extraordinary and captivating mechanical properties. This paper presents a meshless model based on the Improved Moving Kriging (IMK) interpolation to investigate the nonlinear bending of functionally graded (FG) graphene origami (GOri)-enabled auxetic metamaterials (GOEAM) plates with matrix cracks. The model is based on the C0-higher order shear deformation theory ( C0-HSDT) framework with seven variables and incorporates the von-Karman nonlinearity. The material properties of FG-GOEAM plate are determined by a genetic programming (GP)-assisted micromechanics model, while the degraded stiffness of cracked layers is incorporated via a self-consistent micromechanics model. The convergence and effectiveness of the proposed method is examined by comparing it with existing literature results. Firstly, the convergence and validity of the method presented in this paper are verified through comparison with existing literature. Then, A comprehensive study is conducted to examine key parametric effects on the tunability of nonlinear bending deflection and normal stress in matrix-cracked FG-GOEAM plates. The results show that higher GOri content enhances the stiffness of the FG-GOEAM plate, leading to lower deflection, whereas increased GOri folding degree results in greater deflection. Additionally, the deflection and in-plane stress of the matrix-cracked FG-GOEAM plate are higher than those of the intact plate, while shear stress is lower, with these effects becoming more pronounced as crack density increases.

A novel approach for buckling optimization of stiffened piezolaminated composite plates

This study presents a novel approach for optimizing the critical buckling load of the stiffened piezolaminated composite plates using a self-adaptive special relativity search (SASRS) algorithm. The fiber orientations in the layers are considered as design variables. The stability equations are first derived based on the classical laminated plate theory (CLPT) with von Karman nonlinearity. The optimization problem is implemented in MATLAB. The optimization results are obtained to indicate the effecs of various boundary conditions, applied voltages, grid shapes, angles of diagonal ribs, and plate aspect ratios on the optimal design. Numerical investigations are carried out for 10-ply symmetrical stiffened piezolaminated plates. The robustness and efficiency of the proposed SASRS algorithm in maximizing the critical buckling load of the stiffened piezolaminated plates are demonstrated by comparing the optimized results from the standard special relativity search (SRS) algorithm. It is found that the proposed optimization method has better performance than the SRS algorithm.

On the measurement of the nonlinear dynamics of sandwich sector plate surrounded by the auxetic concrete foundation: Introducing a machine learning algorithm for nonlinear problems

The measurement of the nonlinear dynamics of sandwich rotary sector plates is crucial for measurement engineers as it enables the precise analysis and understanding of the behavior of advanced composite structures under dynamic conditions. These measurements help in identifying and mitigating potential issues related to vibration, stability, and fatigue, which are critical in industries like aerospace, automotive, and civil engineering. Accurate data on nonlinear dynamics aids in the optimization of design, enhancing performance, durability, and safety. Furthermore, it supports the development of predictive maintenance strategies, reducing downtime and operational costs, and contributing to the advancement of engineering materials and techniques. So, in the current work, for the first time, nonlinear dynamics of sandwich rotary sector plate via a mathematical modeling and machine learning algorithm is presented. First, due to a lack of information on the nonlinear dynamics of sandwich rotary sector plates, a dataset for training, testing, and validating the machine learning algorithm is presented. For this purpose, using Hamilton’s principle, Von-Karman nonlinearity, and first-order shear deformation theory (FSDT), the nonlinear governing equations are obtained. Also, due to increasing stiffness and stability, an auxetic concrete foundation covers the sandwich structure. After that using the two-dimensional generalized differential quadrature method (GDQM) and Newmark’s time integration method (NTIM), the nonlinear equations are solved. This research provides valuable insights for engineers in designing advanced composite structures with improved dynamic properties. The results also contribute to the broader understanding of nonlinear dynamic interactions in complex material systems, paving the way for innovative applications in various engineering fields.

Nonlinear vibrations and critical angular velocity information of the high-speed rotating eccentric disk made of Gori-metamaterials: Introducing data-driven solution for solving the nonlinear problems

Eccentric discs in rotational motion are commonly utilized in various technical fields, including gas turbine engines, flywheels, gears, and brakes. So, improving its critical angular velocity and frequency characteristics is a challenging issue for engineers. So, in this work for the first time, nonlinear vibrations and buckling analysis of the high-speed rotating eccentric disks using mathematical simulation and data-driven solutions are presented. One of the suggestions for improving its mechanical properties is considering metamaterials in the construction of the eccentric disk. Metamaterial is a novel synthetic material that has distinctive physical and mechanical capabilities that are unattainable in natural materials due to its well-designed structure. The properties of the eccentric disks are controlled by the amount of graphene and the degree of folding of graphene origami (GOri) across the thickness of the eccentric disks. These properties, such as Poisson's ratio, vary depending on the position and can be estimated using micromechanical models assisted by genetic programming (GP). Using von-Karman nonlinearity, transformed differential quadrature method (TDQM), and Newton's method the nonlinear governing equations are obtained and solved, respectively. The results show that when the radius ratio of the rotating eccentric disk increases by 8%, the critical point is decreased from 420 HZ to 370 HZ, a reduction of around 12 %. Another suggestion for improving its mechanical properties is controlling the geometrics and physics of the presented structure according to the designer's purposes.

Electroelastic wave dispersion in the rotary piezoelectric NEMS sensors/actuators via nonlocal strain gradient theory

This article introduces a computational means for investigating the electroelastic nonlinear wave dispersion traits of the nano-dimension sandwich pipe, which is composed of a core formed of a bi-directional functionally graded (Bi-FG) material, together with a piezoelectric sensor/actuator. A combination of Hamilton’s principle, first-order shear deformation, along with Von-Karman nonlinearity, is used for modeling and obtaining the nonlinear governing equations of the nano-sized sandwich pipe connected to a piezoelectric part. The nonlinear governing equations to obtain the nonlinear phase velocity of the current system are determined using a combination of analytical and multiple scales approaches. Due to some computational cost for choosing the precise values of both the nonlocality factor and length scale of the nanopipe in the laboratory, for the first time in this research, with the aid of COMSOL multi-physics finite element simulation, the results are verified, and new outcomes for obtaining the exact functions for nonlocal and length scale factors is presented. In addition, an artificial neural network (ANN) is utilized in this study for the prediction of the results. The mathematical and finite element results were used to train the ANN. A newly presented optimization algorithm is exploited for the first time for optimizing the ANN parameters concluding higher accuracy of the ANN predictions. Consequently, to explore the influences of the location of the piezoelectric patch, nonlocality and length scale factors, and applied voltage parameter on the phase velocity characteristics of the nano-dimension sandwich pipe made of a Bi-FG core and electrically patch, an effort is performed. As an applicable result that can be useful in the related nano-industries, the current work presents exact nonlocal and length scale functions for different conditions.

The random thermo-elastic nonlinear vibration analysis of porous FGM nano-beams using the first order perturbation theory

The objective of present work is to study the nonlocal nonlinear thermo-elastic vibrations of porous nanobeams made up of functionally graded material (FGM) with random material properties. The mechanical properties of the nano-beam are calculated using a porosity based power law model by assuming the cosine type porosity distributions. The randomness present in the material properties cause the mechanical characteristics of the nano-beams to be random, hence first order perturbation theory has been employed to take care of this randomness. The Reddy’s beam theory in association with the Eringen’s non classical elasticity has been used to account for nonlocal effects, and Von-karman nonlinearity has been employed to take care of geometric nonlinearity. The governing equations are derived from the minimum potential energy principle and solved them using the finite element method. To validate the present model, a convergence and comparison study has been performed. Further, the effects of amplitude ratio, aspect ratio, uncertain material properties, size parameter, porosity, thermal environment, and gradation of material on the stochastic behavior of the porous FGM nano-beams are demonstrated. As a result of the present research, the significance of considering thermal environment, porosity, and random material properties has been revealed for the design of nanodevices and systems to be more reliable.

Dynamic instability analysis of timoshenko FG sandwich nanobeams subjected to parametric excitation

According to the nonlocal theory, the dynamic instability and vibration of functionally graded (FG) porous sandwich nanobeams resting on a visco-elastic foundation under axial harmonic load are studied. In this paper the Timoshenko and nonlocal continuum theory are employed to take shear deformation, rotational bending and small-scale effects into account. The governing equations of motion are extracted using the nonlinear Von-Karman, and Hamilton approaches. Then, to turn the partial differential equations (PDEs) into ordinary differential equations (ODEs) in the case of equations of motion, the method of Galerkin is employed, followed by the use of the multiple time scale method to solve the resulting equations. According to the results of this study, it can be claimed that the porosity ratio is more effective than porosity distribution and nonlocal parameters on the amplitude response and dynamic instability regions. Moreover, to examine thermal increments, foundation characteristics, slenderness, and thickness ratios, the bifurcation diagrams are drawn and discussed. It is found that foundation and geometric characteristics are of the highest significance in the nonlinear response of the Timoshenko nanobeams. The main intention of this study is to present a holistic idea about the porous materials used in sandwich structures which can be used in many composite structures in aircraft and automobiles.

Thermal Postbuckling of Heated Uniform Columns Considering Green Nonlinearity: A Novel Liner Finite Element Formulation

Thermal postbuckling behavior of uniform, heated columns is presented, by using a novel linear Finite Element (FE) formulation. In this investigation, the general Green axial nonlinear strain- displacement relation is used, instead of the popularly used simpler von-Karman nonlinearity, which is a subset of Green nonlinearity. In the earlier complex nonlinear FE formulations, time consuming iterative or step-by-step methods are used to obtain the solution for thermal postbuckling. In the novel FE formulation, normally used FES , for performing linear buckling analysis, is proposed to obtain thermal postbuckling loads. The nodal degrees of freedom are deflection and its first derivative with respect to the axial coordinate. The geometric nonlinearity is incorporated through the tensile loads induced, with axially restrained ends of the column, due to large deflections. The effectiveness of the novel FE formulation is demonstrated, from the numerical results obtained, in terms of the ratio of thermal postbuckling to buckling loads, for specified reference deflection and slenderness ratios, of the columns with different boundary conditions. The numerical results reveal some interesting phenomena of thermal postbuckling behavior of columns.

Bifurcation Point Analysis of a Magnetostrictive Sandwich Composite Plate Subjected to Magnetic Field and Axial Force

This article studies parametric vibration and dynamic instability of a rectangular and symmetric magnetostrictive sandwich composite plate (MSCP) on a visco-Pasternak medium. The MSCP consists of three layers; a magnetostrictive layer considers the core and composites as its upper and lower faces. The MSCP subjected to temperature change, parametrically exciting force, and magnetic load is studied with consideration to geometrical von Karman nonlinearity. Based on the energy method and first shear deformation theory (FSDT), Hamilton’s principle is used to achieve the system’s governing equations and boundary conditions. In the next step, the partial differential equation is transformed into ordinary differential equations by applying the Galerkin technique. Then the equation of motion is solved using the multiple-scale method. Numerical results illustrate the stability of the sandwich plate is significantly related to the magnetostrictive parameters. In addition, the effects of significant parameters, such as the effect of amplitude response and parametric excitation or detuning parameter, coupled with the effect of foundation, thickness ratio, aspect ratio, and temperature increment on vibration characteristics, bifurcation points behavior and stability of the systems are charted, plotted and discussed. The innovation of this article is the use of magnetostrictive material in sandwich plates and the development of its mathematical relationships. It is anticipated that the results of this research can contribute to the development of future smart structural applications subjected to in-plane axial forces.

Nonlinear thermo-mechanical buckling and postbuckling of sandwich FG-GPLRC spherical caps and circular plates with porous core by using higher-order shear deformation theory

The main aim of this research is to establish the algorithm for nonlinear thermo-mechanical buckling of sandwich functionally graded graphene platelet reinforced composite (FG-GPLRC) shallow spherical caps and circular plates with porous core under external pressure and/or uniformly distributed thermal load according to the higher-order shear deformation theory considering the von Karman nonlinearities. Sandwich spherical caps and circular plates are made by the porous core and two FG-GPLRC coatings and are assumed to be rested on an elastic foundation modeled by the Pasternak model. The equilibrium equation system in the form of nonlinear algebra can be approximately obtained using the Ritz energy method. The critical buckling loads and postbuckling curves can be explicitly determined. The effects of material parameters, geometrical parameters, porous core, and elastic foundation on thermo-mechanical buckling of sandwich spherical caps and circular plates with porous core and FG-GPLRC coatings are investigated and discussed in detail in the numerical investigation section.

Dynamical and chaotic analyses of single-variable-edge cylindrical panels made of sandwich auxetic honeycomb core layer in thermal environment

This paper proposes a new structure of the cylindrical panel called single-variable-edge (SVE) panel resting on a Winkler–Pasternak elastic foundation to form a mechanical system. The panel is made of sandwich structures with a negative Poisson’s ratio auxetic honeycomb core layer. The classical shell theory and Von-Karman nonlinearity assumption are used to establish the motion equation, then Galerkin’s method is applied to obtain the characteristics of the system. The comparisons to the previous article and the finite element analysis are shown in the validation section to confirm the reliability and accuracy of the present calculating procedure. In the results section, the dynamical characteristics of thermo-mechanical systems are investigated through the impact of materials and geometrics. In addition, the dynamics and chaos phenomena of the SVE panel under the critical external load are presented in three phases: stability, the beginning of stabilization and completely chaos state. The new structure has great potential in practical engineering application and is expected to contribute knowledge to the plate/shell research field.

Effect of Fibers Orientation on the Nonlinear Dynamic Performance of Laminated Composite Plate under Different Loading In-plane

Non-linear dynamic analysis of a cross-ply laminated composite with fibre spacing plates under in-plane loading was presented. The mathematical formulation is based on first-order shear deformation theory and von-Karman non-linearity. Eight-node isoperimetric quadrilateral elements with five degrees of freedom per node were used to maintain geometric nonlinearity. In this investigation, a variety of fiber spacings and fibre orientations were used for the purpose of studying the effect of improvements on the behaviour of samples with this type of load. The dynamic equilibrium equations were solved using the Newmark integration technique. The non-linear dynamic analysis addressed several functions of changing fibre spacing with various changes in volume percentage and diverse fibre orientations. Fibre orientation, volume fraction fluctuation, and fibre distribution significantly affected laminated composite plates' non-linear dynamic behaviour. This study showed that the combination of improvements does not give a clear vision of the ideal improvement, and the best case for fibre distribution and the best case for layer rotation and combination should be studied to know the effect of the ameliorations one on the other and distinguish the impact of the transverse loading pattern in one direction on rotating the sample layers with axial load in one direction, so that changing the fibre distribution is more effective in the behaviour and stability of the plate by taking advantage of the orientation change, as demonstrated in this paper. Doi: 10.28991/CEJ-2022-08-12-03 Full Text: PDF

On the stochastic vibration analysis of the geometrically nonlinear graded cellular curved panels with material stochasticity

In this paper, an effort was made to study the free flexural vibration analysis of the geometrically nonlinear graded cellular curved panels in a stochastic domain. The nonlinear finite element formulations were developed using von Karman nonlinearity hypotheses based on higher-order shear deformation theory, including Lame's parameter. The first-order perturbation technique (FOPT) was employed to assess the stochastic vibration behavior of the graded cellular panels considering randomness in the material properties. The graded cellular curved panels are considered to have three distinguished porosity distributions. The properties of the graded cellular curved panels are continuously varying in the thickness direction as a function of porosity coefficient and mass density. The influences of material stochasticity, porosity coefficient, porosity distribution type, and geometric parameters have been included in the formulations. The detailed convergence and comparison studies were conducted to demonstrate the accuracy and efficiency of the present formulation and highlight the importance of the present study in stochastic environments. The various numerical examples for the second-order statistics (i.e., mean and standard deviation) of the linear frequency parameters (λ) and nonlinear frequencies of the graded cellular structures have been presented which may be considered as the guidelines for future studies in this area of research.

Geometrically nonlinear analysis for flexure response of FGM plate under patch load

Present work explores quadratic extrapolation (QE) technique for large deformation flexure study of FGM plate with five variables higher-order shear deformation theory (HSDT). Rule of mixtures is utilized in achieving effective material properties across the thickness. Governing differential equations (GDEs) and boundary conditions have been acquired using the principle of virtual work. Quadratic extrapolation (QE) and Newton-Raphson (NR) linearization techniques are considered and compared with respect to computational time. Von-Karman type nonlinearity is used in the present work. Multiquadric radial basis function (RBF) based meshfree technique discretizes GDEs in displacement form. FGM plates with different boundary conditions are investigated under different patch loadings. Present results are validated with some previously published results after a convergence study. Some new results are produced for the flexural response of the FGM plate under patch loads. Effect of types of loads, grading index, patch load location, and boundary conditions are presented.

On size-dependent nonlinear forced dynamics of MRE-cored sandwich micro-pipes in presence of moving flow and harmonic excitation

This paper is dedicated to examine the size-dependent nonlinear forced dynamics of MRE-cored sandwich micro-pipes in existence of external distributed load and internal flow. For this aim, Euler-Bernoulli beam theory, considering von-Karman nonlinearity, together with the modified couple stress theory (MCST) are exploited to attain a mathematical formulation of the problem. Employing Galerkin’s approach with consideration of two axial and transverse modes, the obtained nonlinear partial differential equations are discretized, from which nonlinear ordinary equations are achieved. Thereafter, static condensation method is utilized in order to reduce the number of the obtained ordinary equations. The acquired nonlinear equations are then solved according to the method of multiple scales, which leads to nonlinear natural frequencies and frequency-response curves of the system. Throughout the numerical analysis, the effect of a smart MRE core existence on the system’s response, considering various boundary conditions, is explored. Additionally, a numerical investigation is implemented to highlight the influences of MRE-related intrinsic properties namely the intensity of the applied magnetic potential and MRE core thickness on the natural frequencies, frequency-response and load-response curves of the system. Totally, the substantial role of the MRE core and its intrinsic characteristics on the dynamic response of the system is explored, which implies that, this smart core could be a good candidate in such sandwich structures in order to control and boost the dynamic properties and response of the mechanical systems.

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Linear and nonlinear free vibration analysis of functionally graded porous (FGP) nanobeam with four different porous distribution patterns is performed on the basis of stress-driven two-phase local/nonlocal integral model. The strain is defined according to von Karman nonlinearity, and the differential governing equations as well as standard boundary conditions are derived through the Hamilton’s principle. The integral relation between nonlinear strain and nonlocal stress is transformed equivalently into differential form with constitutive boundary conditions. Neglecting the nonlinear terms, the bending deflection and moment are derived and expressed explicitly through the Laplace transformation with six unknown constants for linear free vibration, and the nonlinear characteristic equation for vibration frequency and linear vibration mode shape can be calculated through standard and constitutive boundary conditions. Two analytic methods including Ritz–Galerkin and perturbation methods based on both local and nonlocal linear vibration mode shape are applied to study the linear and nonlinear free vibration of nonlocal FGP nanobeams under different boundary conditions Meanwhile, direct numerical method based on general differential quadrature method and Newton’s iterative process are utilized to obtain the numerical solutions of nonlinear vibration frequencies. The influence of nonlocal parameters and vibration amplitude on the free vibration frequencies is investigated numerically. The numerical results show that influence of nonlocal parameters on linear vibration mode shape is not significant. Local linear vibration mode shape based Ritz–Galerkin​ and perturbation methods would overestimate and underestimate the linear free vibration of nonlocal simply-supported and clamped–clamped nanobeams. Nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods could provide more accurate prediction on nonlinear vibration frequencies than local mode shape based methods when W(0.5) is small. However, the relative error increases quickly with the increase of W(0.5). Based on the results obtained in this study, local and nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods cannot provide accurate prediction on nonlinear vibration frequencies when the vibration magnitude is big, since linear vibration mode shape assumption is not suitable for strong nonlinear vibration.

Effect of Porosity on the Thermal Buckling Analysis of Power and Sigmoid Law Functionally Graded Material Sandwich Plates Based on Sinusoidal Shear Deformation Theory

This work examines the effect of porosity distributions on thermal buckling analysis of functionally graded material (FGM) sandwich plates. To consider the porosity effect, five different types of distribution models, even, uneven, logarithmic uneven, linear uneven, and sinusoidal uneven are considered. It is assumed that the FGM faces of the sandwich plate are porous while the ceramic core is nonporous. To investigate the thermal buckling behavior of porous FGM sandwich plates, four different types of thermal loads, such as uniform, linear, nonlinear, and sinusoidal temperature rise along the thickness direction are considered. Effective material properties and thermal expansion coefficients of FGM sandwich plates are evaluated based on Voigt’s micromechanical model considering power law FGM (P-FGM) and sigmoid function FGM (S-FGM). The analytical solution is carried out using Hamilton’s variational principle considering the von Karman nonlinearity. The equilibrium and stability equations are derived based on sinusoidal shear deformation theory (SSDT). Numerical results are obtained to observe the influence of different porosity distributions, porosity coefficients, thermal loadings, and geometrical parameters over critical thermal buckling temperature.

Nonlinear flutter analysis of a bend-twist coupled composite wind turbine blade in time domain

Modern wind turbine blade having large slender structure are vulnerable to aeroelastic instability. Aeroelastic tailoring through bend-twist stiffness coupling is an effective way to improve the aeroelastic performance of such large blades. Effect of geometric nonlinearity is significant in these blades. This study presents a time domain aeroelastic analysis followed by a detailed parametric study of large composite wind turbine blade considering von-Karman nonlinearity in the strain–displacement relationship. Flutter limits of three MW sized namely, NREL 5 MW, SNL 61.5 and SNL 100–00 wind turbine blades considering and without considering geometric nonlinearity are compared. A parametric study of SNL 61.5 is conducted by changing the lamination sequence in the spar cap and all section of blade considering the symmetric and asymmetric skins. It has been observed that both nonlinear and linear structural models i.e., models considering and without considering geometric nonlinearity predict highest flutter speed at the same blade configuration and lamination sequence. However, there is a significant difference in the flutter speeds predicted by these models.

Nonlinear Free Vibration Analysis of In-plane Bi-directional Functionally Graded Plate with Porosities Resting on Elastic Foundations

This paper deals with the nonlinear free vibration analysis of in-plane bi-directional functionally graded (IBFG) rectangular plate with porosities which are resting on Winkler–Pasternak elastic foundations. The material properties of the IBFG plate are assumed to be graded along the length and width of the plate according to the power-law distribution, as well as, even and uneven types are taken into account for porosity distributions. Equations of motion are developed by means of Hamilton’s principle and von Karman nonlinearity strain–displacement relations based on classical plate theory (CPT). Afterward, the time-dependent nonlinear equations are derived by applying the Galerkin procedure. The nonlinear frequency is determined by using modified Poincare–Lindstedt method (MPLM). Numerical results are obtained in tabular and graphical form to examine the effects of some system key parameters such as porosity coefficients, distribution patterns, gradient indices, elastic foundation coefficients, aspect ratio and vibration amplitude on the nonlinear frequency of the porous IBFG plate. To validate the analysis, the results of this paper have been compared to the published data and good agreements have been found.

An analytical approach of nonlinear buckling behavior of longitudinally compressed carbon nanotube‐reinforced (CNTR) cylindrical shells with CNTR stiffeners in thermal environment

AbstractNonlinear buckling and postbuckling of longitudinally compressed carbon nanotube‐reinforced (CNTR) cylindrical shells stiffened by longitudinal or circumferential CNTR stiffeners in thermal environments surrounded by elastic medium are presented in the present study. Five linear distributions of CNT are considered for the shell‐stiffener structure system and they are modeled by innovation homogenization technique for CNTR stiffeners. Based on the classical Donnell thin shell theory with von Karman's nonlinearities and the Galerkin procedure, the governing equations can be built to analyze the critical buckling compression and postbuckling compression‐deflection and compression‐shortening behavior. The noticeable effects of volume fraction of CNTs, shell‐foundation interaction stiffnesses, uniformly distributed temperature, and geometric properties of stiffened cylindrical shells on the critical buckling compression and compression‐deflection and compression‐shortening postbuckling behaviors of stiffened CNTR cylindrical shells are obtained and remarked in numerical examinations.