Annular Sector Plates Research Articles

Annular sector plates made of FG graphene origami-enabled auxetic metamaterial under bending and buckling loading: A VDQ-based approach

The behaviors of annular sector plates made of functionally graded graphene origami-enabled auxetic metamaterials (FG-GOEAMs) under the action of buckling and bending loads are investigated herein using a numerical approach. Multiple GOEAM layers with graphene origami (GOri) content are considered for the plate which changes in layer-wise patterns. Moreover, the material properties of the plate are calculated using novel genetic programming-assisted micromechanical models. In order to derive the governing equations, the Mindlin plate theory in conjunction with Hamilton’s principle is utilized within the framework of the variational differential quadrature (VDQ) method. The vector-matrix representation of the presented formulation can be beneficial from the viewpoint of implementing numerical approaches. The governing equations are then discretized and solved based on VDQ matrix differential and integral operators so as to obtain the critical buckling load and maximum lateral deflection of plates with different edge conditions. The effects of GOri content, folding degree and distribution pattern on the results are studied. It is revealed that increasing the GOri content leads to the negative Poisson’s ratio (NPR) effect which respectively reduces and increases the lateral deflection and the critical buckling load.

Thermal buckling of the concrete structures reinforced by nanocomposites: Introducing advanced deep neural networks for thermal buckling problem

The thermal buckling behavior of concrete annular sector plates reinforced with nanocomposites is investigated in this work. The thermal buckling issue is stated using mathematical modeling, taking into account the geometric linearity and material heterogeneity caused by the nanocomposite reinforcement. The governing differential equations that are obtained from the theoretical model are solved using the discrete singular convolution (DSC) technique. To generate accurate buckling load predictions, the DSC method’s great accuracy in handling complicated boundary conditions and discontinuities is used. Furthermore, using mathematical modeling to create extensive datasets, the research aims to train deep neural networks (DNNs) to anticipate thermal buckling reactions. These datasets provide a strong basis for DNN training as they include a wide range of factors, such as geometric configurations, loading situations, and material properties. By combining DNN predictions with DSC solutions, this unique technique for solving this difficult issue seeks to improve the accuracy and efficiency of thermal buckling analysis. The findings show how well sophisticated mathematical methods and machine learning models work together, opening the door to creative solutions for the design and analysis of concrete structures reinforced with nanocomposite under complicated stress.

Dynamic stability analysis of CFRP sandwich structure reinforced by advanced nanocomposites via both machine learning method and mathematical simulation

This work introduces a quasi-3D refined theory for assessing the transient dynamic reactions of a sandwich annular sector plate subjected to mechanical shock loading. The sandwich structure has two facesheets composed of graphene origami (GOri)-enabled auxetic metal metamaterials (GOEAMs) and a core composed of carbon fiber reinforced polymer. The auxetic quality of the annular plates is mainly determined by the quantity of graphene and the level of folding in the GOri material. The elements are assessed layer by layer across the thickness of the plates. Micromechanical models supported by genetic programming may be used to predict the position-dependent Poisson’s ratio and other material parameters. The concept of Hamiltonian is used to deduce the governing equations of the structure. The equations of motion, which vary with time, are solved using the numerical solution process and the Laplace transform. A comprehensive parametric study is carried out to examine the influence of various geometric and physical parameters on the time-dependent behavior of annular sector plates. The current mathematical modeling results are being compared to the findings of previous works, as well as a machine learning technique. By using this machine learning methodology, it is feasible to computationally solve differential equations at a reduced expense, while simultaneously surmounting the challenge of formulation. To use machine learning techniques, a dependable dataset acquired from either experimental or numerical analysis is necessary. The dataset was created using the quantitative findings of the investigation. Furthermore, the machine learning technique demonstrates its capacity to provide very precise outcomes when predicting the transient behavior of the existing structure under novel loading and boundary circumstances.

Free vibration analysis of sandwich structure via point interpolation meshfree method based on polynomial basic function

The stability of circular and annular sector plate structures under aerodynamic pressure in supersonic conditions is crucial for aircraft components. Conventional uniform materials lead to heavy structures, which are undesirable for aircraft. This study explores the use of functionally graded materials to enhance flutter stability. It employs a refined logarithmic shear deformation theory to describe plate displacement, classical elasticity theory for constitutive equations, and Hamilton’s principle for equations of motion. Two methods, finite element, and Meshfree radial point interpolation, are used to solve these equations and verify the results. Parameter studies are conducted to assess the effects of grading index, Biot’s coefficient, span angle, and geometrical aspect ratio, revealing unexpected adverse effects of grading material on stability.

Dynamic responses of functionally graded origami-enabled auxetic metamaterial sector plate induced by mechanical shock: Application of innovative machine learning algorithm

For the first time, in the current work, the quasi-3D new refined theory (Q3D-NRT) is used to analyze the dynamic reactions of functionally graded (FG) annular sector plates made of graphene origami (GOri)-enabled auxetic metal metamaterials to the mechanical shock loading. The auxetic property of the annular plates is effectively controlled by graphene content and GOri folding degree that is graded across the thickness direction of the annular plates in a layer-wise manner, as Poisson’s ratio and other material properties are position-dependent and can be estimated by genetic programming-assisted micromechanical models (MM). Hamilton’s concept is used to find the structure’s governing equations. The differential quadrature technique and the Laplace transform are used at design sites to solve the time-varying equations of motion. The system response is translated from the Laplace domain to the time domain using a modified version of the Dubner and Abate approach. To investigate the impact of different geometrical and physical factors on the dynamic reactions of the annular sector plates, thorough parametric research is conducted. The findings of the present mathematical modeling are compared with those of the earlier publications and also with those of the machine learning approach. Utilizing this learning strategy, differential equations may be solved with very cheap computer costs while also overcoming formulation complexity. It needs a valid dataset from experimental or numerical analysis to use machine learning techniques. This dataset was compiled using the study’s numerical findings. Additionally, the machine learning approach demonstrates the capacity to deliver findings with a high degree of accuracy when predicting the mechanical characteristics of the existing structure under novel loading and boundary circumstances.

Review of solution methodologies for structural analysis of composites

This article reviews state-of-art solution methodologies for structural analysis of composites. The use of composites and advance composite in the design of structures has contributed to significant development in modeling and analysis of composite structures. Various modeling approaches as well as solution methodologies had been developed in the past for accurate and efficient response prediction of these structures. Our study focuses on the review of solution methodologies adopted for examined the bending, buckling and free vibrations of composites. These solution methodologies are classified as analytical, semi-analytical and numerical methods. While analytical solutions are restricted to specific set of boundaries and loading conditions, the numerical solutions find applications in complex geometries and general boundary conditions. With the development of computational technologies, the numerical solutions have gained popularity over a period of last three decades. In this article, the analytical approaches are reviewed in the framework of Navier technique, Levy Method, and symplectic superposition method while the numerical approaches are reviewed in the framework of Rayleigh–Ritz method, finite element method (FEM), meshfree methods, wavelets-based methods and differential quadrature method. The laminated composite plates/shells, CNT reinforced composite, FGM plates, circular and annular sector or sector plates have been studied in this review. An overview of mathematical formulation for the numerical methods is also presented.

Nonlinear large-amplitude vibration analysis of annular sector plates made of FGMs subjected to cooling shock

A numerical approach is developed in the present article to study the geometrically nonlinear vibrations of annular sector plates made of functionally graded materials (FGMs) due to being exposed to cooling shock. The first-order shear deformation plate theory (FSDT) is used in the modeling of plate, and the governing equations of motion are derived based on Hamilton's principle considering von Kármán nonlinear kinematic relations. It is also considered that the properties of FGMs are dependent on temperature and distribution of materials. According to the uncoupled thermoelasticity theory, the temperature distribution is obtained via 1D Fourier-type transient heat conduction equation. To numerically solve the problem, the generalized differential quadrature (GDQ) method and Newmark-beta integration scheme are utilized. Considering two different types of thermal loading as cooling shock, the effects of various parameters including thermal load rapidity time, geometry, magnitude of thermal load and material properties on the large-amplitude vibrations of annular sector plates are investigated.

Free vibration analysis of the sandwich structured composite annular plates: Numerical and experimental investigation

Sandwich structures are interesting structural elements due to their high stiffness-to-weight ratio, high fatigue resistance, and excellent vibration characteristics. The aerospace industry relies heavily on these types of structures considering the different optimized materials and shapes available for sandwich construction. In the present study, vibration aspects of annular sandwich structures with composite laminates and a 3D printed honeycomb core are investigated in both the numerical and experimental methods. Amongst the various shapes of sandwich structures, annular sector plates are predominantly used in the construction of modern bridges, aircrafts, and automobile parts. Furthermore, the investigation of annular sandwich plates featuring GFRP facesheet and 3D printed PLA cores are studied in a very limited extent. In order to obtain the sector plates experimentally, Glass Fiber Reinforced Polymer (GFRP) laminates are fabricated by hand layup techniques, and the 3D printing Poly-lactic Acid (PLA) honeycomb core is manufactured using Fused Deposition Modelling (FDM). The material properties of honeycomb core and composite skins are estimated with the help of Gibson formulation and rule of mixture, respectively. Based on available literature, the analysis approach is validated through numerical analysis using ANSYS Finite Element (FE) software. Also, the frequency responses of annular sandwich plate were measured in both experimental and numerical analysis and the results showed good agreement. However, in order to describe the effectiveness of annular composite sandwich plates, parametric studies are carried out in terms of natural frequencies and mode shapes with different configurations of core geometry, ply orientation, boundary constraints, and inner to outer radius ratio. The research outcome highlights that the optimum angle ply orientation, core height of 12.5 mm with clamped–clamped condition and inner to outer radius of 0.75 mm yields better structural stiffness.

Influence of Fiber Angle on the Steady State Periodic Response of Annular Sectorial Plates

Structural elements in the form of annular sectorial plates are widely used in aeronautical, biomedical, and marine engineering. When these components are exposed to dynamic load, large-amplitude vibrations occur. The vibration response analysis based on the linear strain-displacement relation typically yields a conservative estimate and can be used as a first approximation to the actual solution for thin structural elements. As a result, geometric nonlinearity must be incorporated for the efficient and fail-proof design of such elements. This paper shows non-linear and linear steady-state forced vibration responses of the annular sectorial plates. The governing equations of motion have been solved in the time domain by using a modified shooting method and an arc-length/pseudo-arc length continuation technique at the bifurcation point to obtain the complete response curve comprised of stable and unstable branches. This work investigates the effect of fibre angle on the non-linear steady-state forced vibration response of annular sectorial plates. The strain/stress fluctuation throughout the thickness of the laminated annular sectorial plate is determined to explain why hardening nonlinearity has increased. The cyclic fluctuation of the non-linear steady-state normal stress during a time period at the centre of the top and bottom surface is also provided in relation to the forcing frequency ratio of peak amplitude in the non-linear response. Due to the change in restoring forces, the frequency spectra reveal much increased harmonic involvement along with the fundamental harmonic for all fibre angles.

On the asymmetric dynamic response of viscoelastic sector plate made of FG polymer foam

This work provides an analytical investigation of the damped forced vibration behavior of viscoelastic annular sector plates made of porous polymer foam. For this aim, the motion equations are obtained through first-order shear deformation theory (FSDT) in conjunction with the energy method and the calculus of variations. Three distinct types of pore distribution are explored in the plate thickness such as porosity symmetric distribution type 1 (PSDT1), porosity symmetric distribution type 2 (PSDT2), and porosity non-symmetric distribution (PND). Then, the obtained relations are extended to constitutive equations by considering the standard linear solid (SLS) viscoelastic model. The perturbation technique and Fourier series are employed to solve the system of equations with variable coefficients. The equations with variable coefficients are changed to a system with constant coefficients by inserting a new variable, and the asymmetrically dynamic response is computed analytically in a closed-form solution. Then, the transient dynamic behavior of viscoelastic functionally graded porous (VFGP) annular sector plates is detailed for various types of radial and asymmetric circumferential loadings. Furthermore, a user-defined field (USDFLD) code is developed for evaluating the reliability of the analytical findings using finite element (FE) software. Finally, we have benchmarked the effectiveness of our framework by investigating several geometrical parameters, material properties, and loadings types.

Effects of Lamination Schemes on the Periodic Response of the Annular Sectorial Plates

A broad variety of aeronautical, biomedical, and military applications make use of structures in the form of annular sectorial plates. When in use, they are subjected to stress fluctuations and vibrations with considerable amplitudes. Geometric non-linearity is necessary for efficient design of such components, as the vibration responses obtained using the linear strain–displacement relation are often conservative and can only be utilized to approximate the genuine response. This study examined the influence of different lamination schemes on a non-linear/linear steady-state forced vibration response of annular sectorial plates. Both the modified shooting approach and the arc-lengthalgorithm technique were used for solving the governing equations of motion in the time domain. Frequency-response curves were drawn in their entirety, including both the stable and unstable portions. Sectorial plates of varying layer numbers are studied by analyzing their non-linear dynamic behavior by means of frequency response curves, periodic stress variation, phase plane plots, and frequency spectra. The number of layers shown to have an impact on steady-state responsiveness. In a hardening non-linear manner, the annular sector plates show a peak linear displacement amplitude that is significantly larger than the corresponding values in the non-linear analysis.

Inverse differential quadrature solutions for free vibration of arbitrary shaped laminated plate structures

An essential aspect of design of laminated plate structures in many engineering applications is the analysis of free vibration behaviour in order to model the structure for random excitations. In this regard, numerical solutions to the systems of high-order partial differential equations governing free vibration response of the structure become important. Direct approximation of such high-order systems are prone to error arising from the sensitivity of high-order numerical differentiation to noise necessitating the demand for improved solution techniques. In this work, a novel generalised inverse differential quadrature method is developed to study the dynamic behaviour of first-order shear deformable arbitrary-shaped laminated plates. The ensuing underdetermined system is operated upon by Moore-Penrose pseudo-inverse preconditioning to form a squared eigenvalue system. Free vibration solutions of square, skew, circular, and annular sector plates for different boundary conditions are obtained and validated against exact and numerical solutions in the literature and ABAQUS. It is demonstrated with numerous examples that iDQM solutions are in excellent agreement with exact solutions for square plates and the results for arbitrary shaped plates are comparable with solutions in the literature while saving up to 96% degrees of freedom required for ABAQUS solution. Finally, refined parametric studies conducted reveal that, subject to varying geometric configurations, iDQM solutions are numerically stable and potentially converge faster than DQM.

Numerical assessment of the nanocomposite sector annular plates with the aim of promoting their mixed-form thermal shock response

As practical research, this study assesses the resistance of the radially-graded graphene-platelets reinforced (RG-GPLR) nanocomposite annular sector plates against thermal shock. Three-dimensional declaration of the elasticity theory is the base for defining the governing equations of the system. Discrete singular convolution technique (DSCT) is applied as an efficient solution to determine the stress and defection variation of the clamped plate against the instantaneous thermal shock. For the first time, this study considers the effect of thermal shock induced by the radical temperature gradient, heat flux, and the mixed form. Laplace transform combined with the extended Dubner and Abate's technique to find the time history of the system's thermomechanical behavior. The accuracy of the solution is inspected through a comparative analysis between the present and published studies’ results. As a valuable outcome, it is revealed that the fluctuations of the system's response around their ultimate constant values would be damped faster when the applied heat flux is higher.

A series of multi-domain matched interface and boundary algorithms for dynamic and static responses of annular sectorial plates

A series of multi-domain matched interface and boundary (MDMIB) algorithms are proposed to analyze the free vibration and bending of annular sectorial thin plates. Through an in-depth understanding of the matched interface and boundary (MIB) method, six types of MDMIB algorithms are developed herein by using various domain decomposition techniques for handling bi-directional crossing interfaces due to non-homogeneous elastic foundations and discontinuous loads, which cannot be treated by the conventional MIB method. The rationale of the MDMIB method is to divide the entire computational domain into multiple subdomains that can be connected by using the interconnecting conditions between crossing interfaces and mesh lines. Annular sectorial plates, having non-uniform boundary conditions, partially supported elastic foundations, and patch loads, are all considered. Numerous examples are chosen to examine the computational performance of the MDMIB algorithms. For verification, the present results are checked against existing and finite element solutions. It is shown that the proposed MDMIB algorithms are highly effective to deal with the bi-directional interface problems.

Effect of in-plane loads on the natural frequencies of transversely isotropic thick annular sector plates resting on elastic foundation

This paper has studied the effects of in-plane load on the vibration of transversely isotropic annular sector plates resting on the Winkler and Pasternak elastic foundations. The governing coupled partial differential equations are obtained based on the third-order shear deformation plate theory. Based on Levy’s solution, the radial edges are assumed to be simply supported, and various boundary conditions are considered for circular edges. Therefore, nine different boundary conditions are considered here. At first, stability equations are analytically solved, and the critical buckling loads are obtained. Then, the natural frequency is extracted from the governing equations of motion subjected to in-plane loads. The effects of different in-plane loads on first natural frequencies of laminated thick annular sector plates with different aspect ratios, various thickness-length ratios, and some Winkler and Pasternak elastic foundation parameters are investigated. In addition, the effects of in-plane load on mode shape contour plots are presented for various boundary conditions. The obtained results show that the in-plane loads have a significant effect on the natural frequency of transversely isotropic annular sector plates. Also, increasing the in-plane force can affect the natural frequencies of the sector plate. For unique industrial designs, the natural frequency of the plate from the work area can be changed or prevent vibrations. Therefore, the resonance phenomenon can be easily changed and modified.

A series of elasticity solutions for flexural responses of functionally graded annular sector plates

This paper aims at investigating the flexural responses of transversely isotropic functionally graded (FG) annular sector plates subjected to biharmonic loads, where the material parameters are assumed to vary arbitrarily along the thickness direction and the cylindrical boundaries is simply supported at the two radial edges and arbitrary at the two circumferential edges. Based on the extended England-Spencer plate theory, a series of analytical solutions to FG annular sector plates are derived by virtue of the mid-plane displacements of the plate, which can be expressed by four analytic functions of the complex variable, and the corresponding unknown constants can be determined by the boundary conditions at two circumferential edges. In order to validate the proposed analytical solutions, a finite element (FE) model is built to obtain numerical solutions for comparison, which shows that the obtained deflection and stress components are coherent with those from the FE results. Furthermore, two typical biharmonic loads are taken in the numerical examples to discuss the effects of material gradient index, boundary conditions and thickness-to-radius ratio on the flexural responses of the plate in detail. This work gives exact 3D analytical solutions for the FG annular sector plates, which not only provide an in-depth understanding of their mechanical behaviors and a new theoretical route for the analysis of engineering structures, but also serve as benchmark to check analytical solutions based on any simplified plate theories or numerical solutions to the same problem.

Free vibration analysis of annular sector sandwich plates using first-order shear deformation theory

The present paper is concerned with the analysis of free transverse vibration of moderately thick annular sandwich sector plates of uniform thickness consisting of isotropic facings and core based on first-order shear deformation theory. The radial edges of the plate are taken as simply-supported (S) while the other two edges are considered to be under three combinations of clamped (C), and free (F) boundary conditions. The effects of transverse shear and rotatory inertia are included in the governing equations which are derived from Hamilton's principle. The differential quadrature method is used to discretise these equations and compute the lowest three eigenvalues of frequency parameter reported as the first three modes of vibration. The influence of boundary conditions, core thickness, facing thickness, inner-to-outer radii ratio, and sector angle on the frequency parameter of the plate is presented. Convergence of the results is also presented. Results are compared in special cases. Three-dimensional mode shapes for the first three modes are plotted.

Parametric vibration analysis of sector plates using perturbation technique

This paper aims to investigate the in-plane and out-of-plane free vibrations analysis of both solid and annular sector plates with time-dependent properties. The Fourier expansion and the perturbation technique are employed to solve the partial differential equations based on the first-order shear deformation theory. By defining a new parameter, the motion equations which have variable coefficients convert to equations with constant coefficients that lead to a closed-form solution for each order of the perturbation expansion. Due to the use of this displacement field, rapid convergence and high accuracy results can be easily obtained for both in-plane and out-of-plane natural frequencies. The influences of some key parameters including the inner to outer radius ratios, sector angle, thickness to radius ratios, and damping parameters on free vibration of the viscoelastic sector plates are illustrated and analyzed, which may serve as benchmark solutions in the future. Besides, the results are compared with the actual results from the literature, classic theory of plates, and finite element analysis. The present formulation yields many new and accurate results at a low computational cost for the study of mechanical and geometrical parameters.

Thermoelastic Analysis of Annular Sector Plate Under Restricted Boundaries Amidst Elastic Reaction

An analytical framework is developed for the thermoelastic analysis of annular sector plate whose boundaries are subjected to elastic reactions. The exact expression for transient heat conduction with internal heat sources is obtained using a classical method. The fourth-order differential equation for the thermally induced deflection is obtained by developing a new integral transformation in accordance with the simply supported elastic supports that are subjected to elastic reactions. Here it is supposed that the movement of the boundaries is limited by an elastic reaction, that is, (a) shearing stress is proportional to the displacement, and (b) the reaction moment is proportional to the rate of change of displacement with respect to the radius. Finally, the maximum thermal stresses distributed linearly over the thickness of the plate are obtained in terms of resultant bending momentum per unit width. The calculation is obtained for the steel, aluminium and copper material plates using Bessel's function can be expressed in infinite series form, and the results are depicted using a few graphs.

Analytical buckling response of sectorial porous plates integrated with piezoelectric layers

This paper is concerned with influence of porosity and mechanical/electrical boundary conditions on buckling of moderately thick, porous sector and annular sector plates integrated with piezoelectric layers. Presence of both porous and piezoelectric elements in a configuration provides new opportunities for manipulating the buckling load broadly. To demonstrate these possibilities, across the thickness of the core plate, varying porosity profiles are considered, yet the network of pores is assumed to be free of fluid. The governing equations of the sandwich plate under uniform in-plane mechanical loading are obtained by taking the Mindlin plate theory and von Kármán assumptions into account; and later the nonlinear nature of the problem is simplified through the use of the adjacent equilibrium criterion. Next, decoupled formulae for all the unknown kinematic variables are reached by introducing two transformation functions. The critical buckling loads are computed analytically by assuming simply supported radial edges and classical boundary conditions for the circumferential edges. Eventually, the effect of the mechanical/electrical boundary conditions, geometrical parameters and porosity is realized by comprehensive parametric studies.