First-order Shear Deformation Plate Theory Research Articles

Predicting low-velocity impact on the CNTs-reinforced plate using machine learning models

Purpose The study aims to predict a low-velocity impact on a plate reinforced with carbon nanotubes (CNTs) using machine learning models. Design/methodology/approach The first-order shear deformation plate theory (FSDT) is used to express the plate displacements filed. The Hertz nonlinear contact law is used to predict the contact between impactor and plate. Using the energy method and Hamilton’s principle, the motion equations are extracted. The six main parameters considered as inputs to machine learning models are CNTs percentage, impactor radius, plate thickness, plate length and width, CNTs distribution profile and impactor initial velocity. These input parameters are used to predict two impact targets including contact force and contact time. Findings As the values of the targets are continuous, the machine learning task is considered a regression problem. Therefore, this study uses different regression models to predict the targets. These regression models include linear regression, stochastic gradient descent regressor, Bayesian regression, partial least squares regression, Gaussian process regression, multilayer perceptron regressor, support vector regression and decision tree regression. To validate the effectiveness of the regression models, experiments are designed based on different evaluation metrics. The results of the experiments demonstrate that the machine learning models achieve promising performance in predicting the contact force and contact time based on the input parameters. Originality/value Due to the volume of high numerical calculations of impact mechanics to reach the response, the targets of the impact problem are predicted using a variety of machine learning methods.

A non-uniform compound strip method for effective buckling analysis of stiffened structures

In this paper, a versatile compound strip method (CSM) with non-uniform spline functions is developed for buckling analysis of stiffened structures. The proposed method further extends the existing CSM by permitting the flexible modification of local knots to place the stiffeners along the strip arbitrarily. In addition, a reasonable knot arrangement can achieve an accurate solving procedure with improved convergence. The displacements of stiffeners are expressed compatibly by the fundamental parameters of skin according to the beam-plate model. Consequently, the reinforcement of stiffeners is naturally incorporated by adding the beam stiffness matrix to the corresponding strips. Based on the first-order shear deformation plate theory and non-uniform spline functions, the semi-analytical formulations of stiffened structures are established. The present method provides the possibility to achieve the local mesh refinement in the finite strip methods. The convergence and validation study is demonstrated through the comparisons with the results of the existing literature solution and finite element method. In conclusion, a more efficient and applicable CSM is proposed to analyze the buckling behaviors of stiffened structures.

Tunable Bending and Buckling Behaviors of Circular Plates Made of FG Graphene Origami-Enabled Auxetic Metamaterials

Based on the variational differential quadrature (VDQ) method, the bending and buckling characteristics of circular plates made of functionally graded graphene origami-enabled auxetic metamaterials (FG-GOEAMs) are numerically studied in this paper. It is considered that the plate is composed of multiple GOEAM layers with graphene origami (GOri) content that changes in layer-wise patterns. The results from genetic programming-assisted micromechanical models are also employed in order to estimate the material properties. The plate is modeled according to the first-order shear deformation plate theory whose governing equations are obtained using an energy approach in the context of VDQ technique. The governing equations are given in a new vector-matrix form which can be easily utilized in coding process of numerical methods. By means of VDQ matrix differential and integral operators, the governing equations are discretized and solved to calculate the lateral deflection and critical buckling load of plates under various boundary conditions. Selected numerical results are presented to investigate the influences of boundary conditions, GOri content, folding degree and distribution pattern on the buckling and bending behaviors of FG-GOEAM plates.

Influence of the CNT addition on the mechanical performance of three-phase composite plates

This work has analyzed the bending, free vibration and buckling performance of a three-phase (CNT/Polymer/Carbon fiber) composite plate as a function of the carbon fiber volume fraction and the CNT volume fraction, under several values of the plate aspect ratio (a/b) and the plate slenderness parameter (h/b), and considering the presence of defects due to the poor dispersion on the CNTs. Based on a two-steps homogenization technique, the mechanical properties of the carbon fibers reinforced –CNT enriched– polymeric matrix are computed. Then, the mechanical response of the plate is computed according to the first-order shear deformation plate theory solutions presented in Reddy’s books. The numerical results have revealed the presence, in three-phase (CNT/Polymer/Carbon fiber) composite plates, of a CNT volume fraction threshold above which the bending, free vibration and buckling performance of the plate decay with the carbon fiber volume fraction.

Modelling and analysis of large periodic origami structures for local vibrations

In this paper, a novel framework is proposed for the structural analysis of large origami structures. The first order shear deformation theory of plates (FSDT) is employed to consider the stiffness of the origami facets and the folds stiffness is modeled via six linear and rotational springs at the intersections of the facets. The framework enables to predict the response of the entire elastic origami tessellation, by solving for the response of a unit cell under periodic boundary conditions. As a case study, a composition of four facets that its repetition forms the entire domain of Miura-ori tessellations is chosen as the unit cell. Then, the response is generated through solving a system of 20 linear-coupled PDEs subject to 80 boundary conditions. The results are generated numerically using generalized differential quadrature (GDQ) and finite element (FE) analysis. Via use of the framework, for the first time the effects of microstructure properties on the local vibration response of different Miura-ori tessellations are studied. Specifically, the effects of the base material, the facets geometry and the folds stiffness on the local vibration response of the structure are studied. Particularly, we show the existence of mode crossings (different mode shapes being excited with the same frequency) in the local vibration response of different Miura-ori unit cells that is highly affected by the microstructural characteristics. The results generated here can be used for designing and tailoring origami structures for high-end applications.

TSDT vs CPT and FSDT for Free Vibration Analysis of Functionally Graded Incompressible Plates

This article studies vibrational behavior of incompressible functionally graded plates through utilizing classical, first-order shear, and third-order shear deformation plate theories. Plate material properties are presumed to differ continuously in the thickness direction concurringto a power law function. The motion and continuity equations are produced using three different plate theories, and then they are analytically solved for rectangular plates with simple supports utilizingNavier's method. The results are verified by obtaining the plate natural frequency for the power law index value equals zero and comparing them to those reported in previous works. It is shown that transverse vibrational analysis of the classical and first-order shear deformationplate theories for incompressible plates are not as precise as thecompressible ones. It is shown that this issue is due to the fact that according to those theories, unlike the higher order theories, the hydrostatic pressure cannot participate in carrying the bending loads. Consequently, the equivalent flexural stiffness and as a result flexural frequencies decrease. So, to analyze the vibrational behavior of functionally graded incompressibleplates, whether the plate is either thin or thick, higher order theories should be adopted. Also, it is demonstrated that TSDT is the simplest shear deformation plate theory for which the hydrostatic pressure can contribute to withstandig the bending moments. So, it can be a practical theory for free vibration analysis of functionally graded incompressible plates. Finally, this theory has been taken into consideration to analyze the vibrational behavior of rectangular plates made of functionally graded incompressible materials Also detailed parametric studies have been carried out.

Natural frequencies and modal shapes of folded sandwich plates made of porous core and FG-CNTRC coating layers resting on two parameters elastic foundation

Folded structures hold significant importance in aerospace technology due to their distinct design, effectively reconciling strength with a lightweight framework. Within aerospace applications, these structures fulfill diverse roles, serving as pivotal elements in aircraft wings, fuselage sections, and essential components of spacecraft and satellites. This study delves into the free vibration behavior of folded sandwich plates composed of a porous core and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) coatings, resting on a two-parameter elastic foundation. Leveraging first-order shear deformation plate theory (FSDT), elasto-dynamic equations are established for each individual plate segment. Utilizing the two-dimensional generalized differential quadrature method (2D-GDQM), these equations are discretized. Subsequently, by applying boundary conditions to all six edges, including traditional clamped, simply supported, and free configurations, and incorporating continuity conditions for displacements, forces, and moments at the joint borders, natural frequencies and modal shapes are extracted for the folded plate structure. The validity of the proposed formulation and results is demonstrated by comparing them with numerical examples for both composite flat and folded plates. This comparison provides strong support for the accuracy and reliability of the present study. The study comprehensively examines the influence of various parameters on natural frequencies, including modal shape visualizations under different boundary configurations, plate thickness, crank angle, porosity parameter, core thickness, foundation stiffness, and CNTs distribution type.

Free-Vibration Analysis for Truncated Uflyand–Mindlin Plate Models: An Alternative Theoretical Formulation

Plates are flat structural elements whose thickness is small in relation to the size of the surface. Their use may include engine foundations, reinforced concrete bridge elements or parts of various floating structures. Consequently, knowledge of their mechanical behavior under static and dynamic loads is of primary importance in engineering applications and of interest from a structural point of view. As a result, numerous works existing in the literature have investigated the mechanical properties of plates using various plate models, such as Reissner’s theory, Levinson’s theory, Kirchhoff’s theory and Mindlin’s theory, and their static and dynamic behavior has been examined. In the present paper the truncated Uflyand–Mindlin plate equation is proposed. According to Uflyand–Mindlin theory, an alternative theoretical formulation is presented for the free-vibration analysis of plates, and the equations of motion and the general corresponding boundary conditions are derived. This paper develops the truncated Uflyand–Mindlin plate equation, i.e., without the fourth-order derivative, by means of the direct method and variational formulation. The first-order shear deformable plate theory developed by Elishakoff, which takes into account rotational inertia and shear deformation and does not include a fourth-order time derivative, is variationally derived here. This derivation complements that performed by Mindlin some 70 years ago. The innovative aspect of the suggested strategy is that variational and direct methods for studying plate dynamics are analogous. Finding the third equation of the reduced Uflyand–Mindlin equations, the accompanying boundary conditions and their mathematical resemblance are the goals of the presented formulations. In order to solve the dynamic equilibrium problem of a truncated Uflyand–Mindlin equation via a variational formulation, it is demonstrated that the differential equations and the corresponding boundary conditions have the same form as those found using the direct technique. This paper successfully completes this task. Finally, in order to validate the effectiveness and correctness of the proposed procedure, a numerical example of the case of a plate simply supported at all four ends is proposed.

Nonlinear aero-thermo-elastic flutter analysis of stiffened graphene platelets reinforced metal foams plates with initial geometric imperfection

Due to its lightweight design and high load-bearing capacity, the stiffened plate structure is extensively applied in the passive control of aircraft flutter. However, the evolution mechanism of the nonlinear response of stiffened plates within the unstable region remains unclear. In particular, there is a lack of research on the nonlinear aero-thermo-elastic properties of stiffened plates with initial geometric imperfection. Inspired by this, this article investigates the stability and post-instability behavior of stiffened geometrical imperfect graphene platelets reinforced metal foams (GPLRMF) plates under combined thermal load and aerodynamic pressure. Graphene platelets (GPLs) and foam are distributed evenly or unevenly in the plates, while the material properties are calculated by modified Halpin-Tsai model and rule of mixture. Considering initial geometric imperfection, the models of plates and stiffeners are established by first-order shear deformation (FOSD) plate and Timoshenko beam theories, respectively. Using the linear piston theory to simulate supersonic aerodynamic pressure. The stability boundary of stiffened plates is determined by eigenvalue analysis. The Newton-Raphson approach is used to simulate the aero-thermo-elastic static response of stiffened plates, and the dynamic post-flutter behavior of the stiffened plate is determined through the Runge-Kutta method. Finally, the effects of initial geometric imperfection, material parameters (foam distribution type, porosity coefficient, GPLs weight fraction and GPLs distribution type), and stiffeners (size, position, and stiffening scheme) on the thermal aeroelastic behavior of stiffened GPLRMF plates are revealed. It can be found that the presence of initial geometric imperfection will significantly change the nonlinear flutter behavior of the stiffened plates.

Vibration analysis of functionally graded carbon nanotubes reinforced composite nanoplates

This work presents the analytical analysis for free linear vibration behavior of functionally graded-carbon nanotubes reinforced composite (FG-CNTRC) nanoplates in the framework of nonlocal strain gradient theory (NSGT) and the first-order shear deformation plate theory (FSDPT). The nanoplate is considered made of a mixture of an isotropic polymer matrix and reinforced carbon nanotubes (CNTs). Four different distributions of CNTs are examined including uniformly distributed and FG reinforcements (FG-O, FG-X, and FG-V). The governing equations of motion are established based on the Hamilton’s principle. The closed-form analytical solution for the natural frequency of FG-CNTRC nanoplates with simply supported all edges is carried out by using the Navier-type solution. The impact of some key parameters on the natural frequencies of FG-CNTRC nanoplates is also studied and discussed. The result shows that FG-CNTRC nanoplates reveal the softening- or hardening-stiffness effects depending on the relationship between the nonlocal parameter and the material length scale parameter. The aspect ratios of FG-CNTRC nanoplates, the volume fraction, and the distribution pattern of CNTs have also an important impact on the vibration behavior of FG-CNTRC nanoplates.

Panel flutter analysis of nano-hybrid laminated composite quadrilateral plates presuming curvilinear fibers

In the present article, the panel flutter behavior of nanocomposite hybrid laminated quadrilateral panel is studied. A three-phase carbon-nanotube (CNT)/polymer/fiber laminated composite is considered where the probable agglomeration of the CNT phase in the nano-matrix mixture is also addressed. Diverse use of fiber phase material in different plies is assumed in the context of hybrid laminated composite. A combination of nanocomposite polymer matrix, tow-steered curvilinear reinforcing fibers within the plies, and hybrid multi-material laminate is considered. The panel flutter behavior of the arbitrary quadrilateral hybrid laminated nanocomposite plate is formulated by a non-uniform rational B-splines-type isogeometric analysis method based on the first-order shear deformation plate theory. The nonideal mixture of the nanocomposite is also concerned about through applying a micromechanics approach. Representative results are obtained and compared with those available in the literature to demonstrate the quality of the proposed formulation. The effects of different parameters including the layup, mixture quality, boundary condition, geometry, and flow direction on the flutter behavior are investigated by performing parametric studies.

A third-order shear deformation plate bending formulation for thick plates: first principles derivation and applications

A third-order shear deformation plate bending formulation is presented in this study from the first principles. The derivation assumed a displacement field constructed using third-order polynomial function of the transverse (z) coordinate; and made to apriori satisfy the linear three-dimensional (3D) kinematics relations as well as the transverse shear stress free boundary conditions at the top and bottom plate surfaces. The formulation thus has no need for shear stress correction factors of the first-order shear deformation plate theories. The domain equations of equilibrium are obtained as a set of three coupled differential equations in terms of three unknown displacements. The system of coupled equations is solved for simply supported rectangular and square plates subjected to four cases of loading distributions: sinusoidal loading, uniformly distributed loading, linearly distributed loading and point load at the plate center. Navier’s double trigonometric series method is used to construct trial solutions for the three displacement functions such that the boundary conditions are satisfied identically. The integration problem is thus reduced to an algebraic problem and is solved for each considered loading. It is found that the present formulation gives exact results for the normal stresses σxx for sinusoidal and uniformly distributed loads. The study further showed that the results for deflection and stresses agreed with Krishna Murty’s higher order shear deformation plate theory results. The present formulation gave accurate results because of the inclusion of transverse normal strain effects in the formulation. The formulation gives a quadratic variation of the transverse shear stresses across the thickness in consonance with the theory of elasticity method.

Free Vibration Response of Functionally Graded Porous Metallic Plates Embedded with Piezoelectric Layers

The objective of this study is to determine the natural frequencies of Functionally Graded (FG) metallic plates comprising piezoelectric layers on both the top and bottom surfaces. The material characteristics of the FG plates are expected to exhibit a gradual variation along the thickness direction in accordance with a power-law model along with porosity. The governing equations are derived using the principle of virtual displacements, taking into account the first-order shear deformation plate theory. A commercial finite element programme in ANSYS Parametric Design Language (APDL) is developed to compute the natural frequency and mode shapes of functionally graded porous plates embedded with piezoelectric layers. The obtained natural frequencies results are used for different boundary condition to show their variations with respect to the constituent volume fractions, boundary condition, and piezoelectric thickness for the parameteric study. The present paper highlights some important characteristics of Functionally Graded Materials (FGM) plate embedded with piezoelectric layers that can be advantageous in the design of smart structures.

Mass lumping schemes fitted to MLS-based numerical manifold method in vibration of plates with cutouts using CPT and FSDT

It is critical to accurately extract the natural frequencies in the transverse vibration of plates using the classical plate theory (CPT) and the first-order shear deformation plate theory (FSDT). The moving least squares (MLS) based numerical manifold method (MLS-based NMM) in conjunction with a suitable diagonally lumped mass matrix is a reasonable choice because it can naturally treat plates with cutouts and easily formulate an H2 regular approximation based on MLS interpolation, as well as mitigate shear locking issues for vibration analysis of thin plates based on FSDT. Moreover, the mass lumping techniques involving the row-sum method, the diagonal scaling method, and the mathematically rigorous manifold-based method are extended and derived in the unified framework of MLS-based NMM for transverse vibration analysis of plates. Furthermore, the positive-definiteness of the lumped mass matrix (LMM) generated by the manifold-based mass method is explained in MLS settings, and the row-sum method is proven to be a special case of the manifold-based method. A comprehensive comparative study of different LMMs is performed in accordance with the consistent mass matrix (CMM) on extensive numerical benchmarks. Numerical results demonstrate the convergence properties of LMMs compared with CMM in MLS-based NMM for plate vibration based on CPT and FSDT. It can be observed that LMMs yield comparable performance compared with CMM. The row-sum method obtains accurate results in most cases but can not guarantee the positivity. In contrast, the manifold method can guarantee the positive-definiteness of LMM and is more accurate than the diagonal scaling method in most cases.

A refined first-order shear deformation theory for large dynamic and static deflection investigations of abruptly/harmonically pressurized/blasted incompressible circular hyperelastic plates

In this paper, nonlinear large-deformation dynamic and static responses of the suddenly and harmonically pressurized or blasted compliant incompressible hyperelastic circular plates are investigated through the development of a novel enhanced circular plate theory that in contrast to the available plate theories, utilizes transverse normal strains whose order is much higher than those of the in-plane strains. For this reason, no shear correction factor is needed for the new plate theory. The deformation gradient and Green's strain tensors are first determined based on the first-order shear deformation plate theory as predictors and then, corrected by simultaneously including the transverse compliance and volume conservation concepts in the neo-Hookean framework of hyperelasticity. The equations of motion are obtained using Hamilton's principle. The resulting highly nonlinear equations that include both large deformation and constitutive nonlinearities are solved by a hybrid iterative/updating technique that utilizes a combination of the differential quadrature (DQ) spatial-discretization and Newmark's time-marching methods, after an exhaustive order analysis for the identification of the extremely small terms. The effects of the changes in geometric parameters, material properties, and loading rate and type are investigated on the dynamic lateral deflection of the plate and the approximate fundamental natural frequency. Results reveal that the contribution of the higher vibration modes in the responses is much more remarkable in the high-rate and high-frequency loadings. Furthermore, the deformed shape of the plate that indicates the superimposed effects of the higher vibration modes is plotted for successive time instants.

Free vibration analysis of cracked composite plates reinforced with CNTs using extended finite element method (XFEM)

In this article, the free vibration behavior of cracked nanocomposite plates was investigated using the extended finite element method (XFEM), with a focus on examining the impact of different distribution patterns (UD, FG-X, FG-A, and FG-O) in the functionally graded (FG) carbon nanotube composite (FG-CNTRC) plate. The FG material properties are assumed to vary through the thickness direction according to different distributions. The governing equations are based on the first-order shear deformation plate theory (FSDT), and the nanocomposite plates are composed of a polymer matrix and the FG-CNTRC. The novelty of this article lies in studying the free vibration behavior of the CNTRC plate, considering all distribution patterns, using XFEM. The effective elastic modulus is computed using the modified Halpin Tsai model, while mass density and Poisson’s ratio are determined by employing the rule of mixture. A parametric study is conducted to investigate the effect of the crack length, crack position, width/thickness ratios, volume fraction, and power law index (Pin) of nanofillers on the natural frequencies of the CNTRC plate, offering valuable insights for different distribution patterns under various boundary conditions. The results demonstrate that the XFEM's accuracy and efficiency as a tool for the free vibration analysis of cracked nanocomposite plates.

A new trigonometric shear deformation plate theory for free vibration analysis of FGM plates with two-directional variable thickness

The present research aims to propose a new trigonometric shear deformation plate theory (TSDPT) for free vibration analysis of functionally graded plates with two-directional variable thickness. Trigonometric functions are utilized to determine the distribution of transverse shear stress in the new TSDPT. The thickness variation follows three distinct patterns: linear, concave, and convex. The governing equations of the composite plates are derived via the new theory and the Hamilton's principle. Then, via the Galerkin's method, the current study establishes a solution for determining natural frequencies of variable-thickness FGM plates. The obtained outcomes of FGM plates are calculated by using the MATLAB program. To confirm the accuracy of the current computational model, we conducted comparisons between the obtained results and data previously reported in the open literature. The results obtained from the new theory are compared not only for the case of a plate made of isotropic material but also in comparison to a plate made of functionally graded material. In both scenarios, the proposed theory showcases superior performance when juxtaposed against classical plate theory (CPT), first-order shear deformation plate theory (FSDPT), and other shear deformation plate theories, displaying significantly reduced error rates. Furthermore, graphical representations of the influences of three-parameter stiffness of Kerr foundation, boundary conditions, geometrical parameters, two different FGMs, FGM index, and mode numbers on the vibrational behaviors of FGM plates are provided.

Vibration Characteristics of Mistuned Blisk with Chordwise Variable Thickness Blades Subjected to Aerodynamic Prestress

In this paper, the influence of aerodynamic prestress on vibration characteristics of mistuned blisk is investigated by comparative study considering one-way fluid–structure interaction. It is found that the aerodynamic prestress changes twisted angle of the blade, increases natural frequencies of the mistuned blisk and affects the sensitivity of natural frequency to rotational velocity. To facilitate the mechanism study of these phenomena, a simplified blade model with chordwise variable thickness is established. The effects of Young’s modulus, rotational velocity and pre-twisted angle on natural frequency are investigated by the first-order shear deformation plate theory and Chebyshev–Ritz method. The results show that the natural frequency of the blade is positively correlated with the rotational velocity and pre-twisted angle. For different twisted angles, the same rotational velocity change can cause different natural frequency increments. In addition, the aerodynamic prestress can suppress the effect of Young’s modulus mistuning on natural frequency.

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.

Hygrothermally-Induced Vibration Analysis of Porous FGM Rectangular Mindlin Plates Resting on Elastic Foundation

The nonlinear vibration response of rectangular plates made of functionally graded porous materials (FGPs) induced by hygrothermal loading is investigated in this article using a numerical approach. The effect of elastic foundation on the vibrations is taken into account according to the Winkler–Pasternak model. Hygroscopic stresses produced due to nonlinear rise in moisture concentration are also considered. The temperature-dependent material properties of plate are computed based on the modified Voigt’s rule of mixture and Touloukian experiments for even and uneven distribution patterns of porosity. Within the framework of the first-order shear deformation plate theory and von-Kármán nonlinearity, Hamilton’s principle is utilized in order to derive the equations of motions. To achieve the temporal evolution of maximum lateral deflection of hygrothermally-induced plates, the generalized differential quadrature (GDQ) and Newmark integration methods are employed. Selected numerical results are presented to study the influences of temperature distribution, porosity volume fraction, moisture concentration, geometrical parameters, elastic foundation parameters and FG index on the geometrically nonlinear vibrations of FG porous plates with various boundary conditions.