First-order Shear Deformation Plate Theory Research Articles

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.

Wave propagation analysis in pre-stressed incompressible hyperelastic multi-layered plates using a plate theory

Wave propagation analysis of pre-stressed incompressible hyperelastic thin multi-layered plates is developed based on the first-order shear deformation plate theory (FSDT) with second order thickness extensibility. Variables describing the thickness-direction deformation are obtained from the incompressibility condition. The incremental theory of nonlinear elasticity is employed in conjunction with a plate theory. The multi-layered plate is assumed to be symmetric in the thickness direction. Since the plate theory has not been used for the wave propagation analysis of hyperelastic plates, the results of flexural wave modes are compared with results of the straight-crested flexural waves based on the three-dimensional (3D) elasticity theory available in the literature and the results for in-plane wave modes are compared with those developed in this study using the plane-stress elasticity theory. The dispersion curves of both extensional and flexural waves for single-layer and three-layer plates are presented and the effects of different parameters including the homogenous pre-stretch, the wave propagation direction and the plate initial thickness on the dispersion curves are discussed in detail. The results indicate that the phase speed of the extensional waves depends on the wave propagation angle due to the induced anisotropy. As the overall shear modulus of the three-layer plate increases, the phase velocity increases. Also, the effect of pre-stretch and thickness on phase speed of flexural waves is not pronounced for high frequencies.

Free vibration and nonlinear dynamic response of sandwich plates with auxetic honeycomb core and piezoelectric face sheets

This article analyses the nonlinear dynamic behavior and the free vibration of a piezoelectric auxetic honeycomb sandwich plate resting on a Pasternak elastic substrate. The plate includes two face sheets of piezoelectric material and an auxetic honeycomb core with a negative Poisson ratio. The first-order shear deformation plate theory, together with the von Kármán type nonlinear strain-displacement relationship, is used to establish the nonlinear equations of motion. Linear electric potential change with piezoelectric layer thickness is assumed. Two types of boundary conditions, namely immovable and movable simply supported, are considered. The displacement-time and acceleration-displacement curves under dynamic loads are determined by using the Galerkin and Runge-Kutta fourth-order methods. Verification examples have been performed and confirmed the accuracy of the model. Finally, several novel investigations have been conducted to evaluate the effect of material, geometrical parameters, and elastic substrate coefficients on the frequency and nonlinear dynamic characteristics of piezoelectric auxetic honeycomb sandwich plates for both open-circuit and closed-circuit operation.

Propagation characteristics of guided waves in CNTRCs plates resting on elastic foundations in a thermal environment

This paper investigates the wave propagation behavior of carbon nanotube-reinforced composites (CNTRCs) plates resting on elastic foundations in a thermal environment. The first-order shear deformation plate theory is applied to establish the mathematical model for the CNTRC plates. The effective material properties are calculated using the rule of mixture. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the CNTRC plates are derived using Hamilton’s principle. Then, the equations of motion are discretized by the Galerkin method and solved analytically by the eigenvalue method. Finally, the effects of various parameters including the temperature variations, the volume fraction of carbon nanotubes (CNTs), elastic foundations and the CNTs’ distribution patterns on the dispersion relations are investigated, it can be found that these parameters have significant effects on the wave propagation of CNTRCs plates.

Isogeometric stiffness matrix formulations for buckling analysis of multi-layered shells based on lamination parameters for solid-shell elements

Multi-layered composite shells are extensively used in many industrial fields since they have superior performance in applications. As load-carrying components, buckling failure is one of the most common failure modes of multi-layered shells. Hence, the layup design of multi-layered shells based on optimal buckling factors is widely concerned, where lamination parameters are commonly used in the representation of stiffness. However, the common formulations of laminate stiffness are based on the Classical Laminate Plate Theory (CLT) or the First-order Shear Deformation Plate Theory (FSDT). This paper aims at developing stiffness formulations based on 3D continuum solid theory. The accuracy and efficiency of the stiffness formulations derived in this study are analysed and compared to those obtained by layer-by-layer integration and summation of stiffness calculations.

Flexural wave manipulation in perforated metamaterial plates with acoustic black holes interconnected by piezoelectric studs

Aiming at achieving lower and broader bandgaps (BGs) and stronger collimation effect, this research proposes a novel perforated metamaterial plate with acoustic black holes (ABHs) interconnected by piezoelectric studs for flexural wave manipulation. Based on the differential quadrature element method together with the first-order shear deformation plate theory, the governing equations of the metamaterial plate are derived. The proposed model is validated by comparing with finite element simulation results. Finally, the wave propagation characteristics including the BGs, equi-frequency contours (EFCs), and group velocities are obtained. The results reveal that the introduction of the stud embedded with piezoelectric patches brings wider complete BGs and stronger collimation effect at lower frequencies. The boundary frequencies and bandwidth of BGs and the preferential direction of wave energy flow are tunable by adjusting geometry parameters of the unit cell. The balance between ABH and piezoelectric effects should be kept to enhance the collimation by generating the caustic lobes. Utilizing the BG and self-collimation mechanism, flexural wave propagation can be manipulated by creating point or line defects in the metamaterial plates. This research provides guidance on the design of metamaterial structure with ABHs and piezoelectric material to achieve broadband vibration absorption and directional wave propagation.

Extension–bending coupling phenomena and residual hygrothermal stresses effects on the Energy Release Rate and mode mixity of generally layered laminates

This article presents a novel analytical solution to the delamination problem of interface deformable generally layered (asymmetric and unbalanced) composite laminates. The governing equation describing the interface forces and moments is obtained for the reduced form of the first-order shear deformation theory of plates (FSDT) under plane strain assumptions and accounts for the extension–bending coupling phenomena and residual hygrothermal stresses, both of which have been neglected in the past. The interface forces and moments are included in a novel generalization of the J-integral which is used for the estimation of the Energy Release Rate (ERR). The analytical results of the ERR and the mode mixity exhibit a great agreement with the results of Finite Element (FE) models developed using the Cohesive Zone Method (CZM) and the Virtual Crack Closure Technique (VCCT) for a Double Cantilever Beam (DCB) test of a generally layered fibre metal laminate.