Showing papers on "Plate theory published in 2023"

The application of the nonlocal theory and various shear strain theories for bending and free vibration analysis of organic nanoplates

Abstract: This is the first study of the static bending and free vibration response of organic nanoplates based on the combination of nonlocal theory with various shear strain theories, where the theory of shear deformation of the plate has the advantage of requiring no shear correction factor. The equilibrium equation of the plate is derived using Hamilton’s principle, the analytic solution is derived using the Navier solution form, and the finite element technique is implemented using a quadrilateral element with four nodes and six degrees of freedom for each node. Moreover, this is the first work to calculate for nanoplates with a nonlocal parameter whose value varies with plate thickness. This study’s credibility has been established by comparing it to previously published findings, which are utilized to validate the results of analytical and numerical calculations, respectively. In addition, the study investigates how a range of components impact the displacement and stress response of organic nanoplates. The findings of the study indicate that there are some circumstances in which it is not feasible to disregard the nonlocal parameter while doing calculations for organic nanostructures.

Magnetic Control of Vibrational Behavior of Smart FG Sandwich Plates with Honeycomb Core via a Quasi‐3D Plate Theory

Abstract: Herein, vibrational behavior of functionally graded (FG) smart piezoelectric sandwich plates with honeycomb core resting on viscoelastic foundation under the effect of 2D magnetic field is presented. By varying the magnetic‐field magnitude and its direction, the vibrations can be controlled. The plate is composed of two FG piezoelectric layers bonded with honeycomb structure as a mid‐layer. Magnetic Lorentz force will be deduced via Maxwell's relations. A new quasi‐3D plate theory considering the shear and normal deformations is incorporated to evaluate the displacements. The governing equations of motion are introduced via Hamilton's principle. Galerkin technique is considered to solve the motion equations for different boundary conditions. Influences of the magnetic field, boundary conditions, electric voltage, and core thickness on the eigenfrequency of the FG sandwich piezoelectric plate are illustrated. It is found that considering the effect of the magnetic field on the smart devices increases their vibrations, which may lead to an increment in the energy harvested from them. Further, when the magnetic field is applied along the length of the rectangular plate, the vibrations are reduced and vice versa.

Piezoelectric Patches for Deflection Control of Functionally Graded Carbon Nanotube-Reinforced Composite Plates

Abstract: In the present work, a smart structure is being investigated, where a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plate is equipped with piezoelectric actuators to provide vibration control. Due to their high mechanical properties coupled with lightweight, FG-CNTRCs are mainly used in the aerospace industry and in advanced engineering applications. The CNTs have a linear and non-linear distribution along the thickness of the plate and are distributed according to five configurations, namely: UD, FG-X, FG-O, FG-A and FG-V. The first order shear deformation (FOSD) theory is considered in the formulation of a 9-node quadratic finite element with 5 degrees-of-freedom per node, and an additional degree of freedom is provided for the piezoelectric layer. The model developed in this study assesses the free vibration behavior and controls the nanocomposite plate deflection through the electromechanical coupling factor piezoelectric. In addition, it investigates: (i) the effect of the plate configuration, (ii) the CNT volume fraction, (iii) the CNT destruction patterns, (iv) the linear and nonlinear distribution of CNTs, (v) the number of CNTRC ply, (vi) the boundary conditions and (vii) the dimensions with different locations of actuators. The results obtained show the first natural frequencies for all configurations, which are considered to be in good agreement with those available in the literature and illustrate that the effective stiffness of the nanocomposite plates can be improved further when the reinforcement is dispersed according to the FG-X pattern. In addition, for the case of the deflection control analysis, results indicate that the distributed piezoelectric layers (actuators) attenuate the deflection of the CNTRC to the desired tolerance. It is noted that patches with partial coverage compared to the case of total coverage of piezoelectric layers require more electrical power to reach the same level of attenuation. The developed numerical model is intended to be used in a variety of potential advanced engineering applications.

Integrated effects of auxeticity and pyro-coupling on the nonlinear static behaviour of magneto-electro-elastic sandwich plates subjected to multi-field interactive loads

Abstract: This work presents a detailed investigation of the effect of auxeticity synergised with the pyro-coupling behaviour of multiphase magneto-electro-elastic (M-MEE) composites using a finite element (FE) framework. The nonlinear deflection and bending problems of sandwich plates with auxetic core and M-MEE skins subjected to multi-physics load (electric, magnetic and thermal) are probed. The plate kinematics is governed by Reddy’s higher-order shear deformation theory (HSDT). The nonlinear relation between the strains and displacements is established through von-Karman’s nonlinear relations. The temperature profiles are considered linearly and nonlinearly varying across the thickness of the plate. Various parametric studies presented in this article highlight the influence of the auxetic cell inclination angle, rib-length ratio, rib thickness, the plate-to-core ratio etc., associated with the pyro-coupling on the nonlinear deflection and bending of sandwich plates. The results reveal that the plate deflection, bending behaviour and degree of pyro-coupling significantly depend on the auxetic unit cell dimensions and magnitude of electro-magnetic loads. The significant influence of temperature profiles on the pyro-coupling are witnessed at lower auxetic cell angle. The nonlinear deflections of the sandwich plate and hence the potentials developed tend to improve with the lower values of plate-to-core ratio and rib thickness. The prominent outcomes of this work related to integrated effects of auxeticity and pyro-coupling are not yet reported in the open literature and are deemed to be utilised as a future reference.

Buckling of multilayered CNT/GPL/fibre/polymer hybrid composite plates resting on elastic support using modified nonlocal first-order plate theory

Abstract: In the following research, the buckling of multilayered CNT/GPL/Fibre/Polymer hybrid composite Levy-type nanoplates resting on Winkler–Pasternak support is investigated using modified nonlocal first-order plate theory. The different layers of the plate are assumed to be reinforced with functionally graded carbon nanotube composite or functionally graded graphene platelets composite. Modified nonlocal first-order plate theory is used to capture the molecular effects at nanoscale and extract the buckling equations. The along-thickness variation of reinforcing nanocomposites may be uniform or functionally graded, and functionality can be linear or nonlinear based on specific functions presented by scientists. Using a closed-form analytical solution, these partial differential governing equations are changed to a set of coupled ordinary differential equations that may be solved for the Levy-type boundary conditions (i.e., two opposite edges with simply supported and two other with edges arbitrary). The obtained results may be used as a useful benchmark for validation of other works developed in the future.

Peeling of Finite-Length Plates From an Elastomeric Foundation: A 1-D Cylindrical Bending Solution

Abstract: Quasi-static peeling of a finite-length, flexible, horizontal, one-dimensional (1-D) plate (strip, thin film) from a horizontal, thin, elastomeric layer (foundation) is considered. The displaced end of the plate is subjected to an upward deflection or to a rotation. The top of the interlayer is perfectly bonded to the plate, and its lower surface is bonded to a rigid, flat substrate. A transversality (debonding) condition is derived for peeling, based on the common fracture mechanics approach. Whereas debonding from a Winkler foundation can be expressed in terms of the displacement (or equivalently the foundation stress ) at the bond termination, the sixth-order formulation involves a more complex debonding criterion. Transversality relationships are used to describe this limit state (here the onset of debonding) in terms of co-state variables, herein the deflection and slope at the peel front. In the analysis, bending is assumed to be paramount, linear Kirchhoff-Love (classical) plate theory is used to model the deformation, and therefore displacements are assumed to be small. The foundation is linearly elastic and incompressible. The effects of the work of adhesion, the length of the plate, and the initial nonbonded length of the plate are investigated. The results are compared to those for a Winkler foundation. By replacing the shear modulus of the interlayer by viscosity, and displacements by their time derivatives, the results are expected to apply to viscous liquid interlayers as well.

Classical and homogenized expressions for the bending solutions of FGM plates based on the four variable plate theories

Abstract: Exact correspondence relations between the bending solutions of a simply supported rectangular functionally graded material plate based on the four variable refined higher-order shear deformation theory and those of the corresponding reference homogenous Kirchhoff plate based on the classical plate theory are derived analytically for the material properties varying continuously in the thickness direction. The deflection, stress components, the resultant forces and bending moments of a thick functionally graded material plate are expressed analytically in terms of the deflection of the reference homogenous Kirchhoff plate with the same geometry, loadings and boundary constraints. Consequently, the bending solution of a functionally graded material plate based on the higher-order shear deformation theory is simplified as calculations of three scaling factors which can be easily determined analytically for the specified material gradient profile, the shear stress shape function and the aspect ratio of the functionally graded material plate, because the solution of the reference homogenous Kirchhoff plate can be easily found even in the text book. As examples, particular solutions for a functionally graded material plate subjected to both uniformly and sinusoidally distributed loads are presented, which illustrate the validity of this new approach. Accuracy of the present solutions are demonstrated by comparing them with those obtained by different palate theories with different shear stress shape functions available in the literature. The analytical solutions can be used as benchmarks to check numerical solutions of static bending of functionally graded material plates based on different higher-order shear deformation theories.

Nonlinear dynamics analysis of a graphene laminated composite plate based on an extended Rayleigh–Ritz method

Abstract: Linear and nonlinear vibration characteristics of the graphene platelet-reinforced composite (GPLRC) laminated plates are investigated. Based on the first-order shear theory and von Karman geometric nonlinearity, the energy expressions of the GPLRC laminates are established. The boundary elastic potential energy is established by penalty function method to simulate different boundary conditions. The linear and nonlinear frequencies of the GPLRC laminated plate are calculated by introducing boundary potential energy into Rayleigh–Ritz method. The convergence and accuracy of the method are verified by numerical examples, and the effects of different parameters on frequency are analyzed. Considering the cantilever boundary conditions, the nonlinear motion governing equations of the GPLRC laminated plate are obtained by Hamilton principle. The two-degree-freedom ordinary differential motion equations of the laminates are derived by Galerkin method. Considering the fundamental parameter resonance and 1:1 internal resonance, the amplitude–frequency response curves of the structure under transverse excitation are obtained. The effects of transverse excitation and damping coefficient on nonlinear vibration characteristics of the GPLRC laminated plates are investigated by numerical simulation.

Novel finite element modeling for free vibration and buckling analysis of non-uniform thickness 2D-FG sandwich porous plates using refined Quasi 3D theory

Abstract: A novel four-node quadrilateral element with eleven degrees of freedom per node is approximated using the C1-order non-conforming Hermite and Lagrange functions and the novel refined Quasi-3D plate hypothesis for buckling and free vibration analysis of non-uniform thickness bi-directional functionally graded sandwich porous (BFGSP) plates resting on variable elastic foundations (VEF) in a hygro-thermal environment. Material and mechanical properties that change in both the length and thickness directions with three different laws of porosity, which are made up of a fully ceramic core layer and two bi-directional functionally graded (2D-FG) material ones, may be used a lot in the aerospace engineering and military industries. In order to analyze the buckling and free oscillation behaviors shown by the plate with arbitrary boundary conditions, a series of mathematical methods created in Matlab’s software are used. The computation program’s correctness is verified by comparing numerical findings to dependable assertions. In addition, a comprehensive analysis of the impact of factors on the buckling and free oscillation responses is conducted. The findings reveal that the novel porosity patterns, the hygro-thermal environment, the elastic medium characteristics, and the boundary conditions have a substantial impact on the mechanical behaviors of the non-uniform thickness 2D-FG sandwich porous plates. The results of this article can be used as a useful reference for engineers when calculating and designing structures of this type in engineering practice.

Nonlinear dynamic analysis of the functionally graded graphene platelets reinforced porous plate under moving mass

Abstract: Recent studies demonstrated that porous materials could gain satisfying improvements in some mechanical properties by adding graphene platelet (GPL) reinforcements. Following this result, the present work exhibits a semi-analytical method to investigate the nonlinear dynamic characteristics of the functionally graded GPLs reinforced porous (FG-GPLRP) plate under a moving mass. Two types of boundary conditions, i.e., simply supported (SSSS) and clamped (CCCC) edges, are incorporated in the study. Based on the refined sinusoidal shear deformation theory (RSSDT) and von Kármán nonlinearity, the governing equations are transformed into a group of ordinary differential equations for the deflection of the plate. Then, the dynamic behaviours of the plate can be investigated by operating the fourth-order Runge–Kutta approach. After verification, several numerical examples are displayed to illustrate the effects of porosity coefficient, GPLs content, Winkler–Pasternak foundation, damping, initial imperfection, and compression stress on the moving-load-bearing capability of the plate. The obtained results demonstrate that, without harming its moving load capacity, it is possible to decrease the mass of the FG-GPLRP plate to a satisfying extent by altering the porosity and GPLs content.

An Analytical Solution for the Dynamics of a Functionally Graded Plate resting on Viscoelastic Foundation

Abstract: This paper deals with the dynamic response of Functionally Graded Material (FGM) plates resting on a viscoelastic foundation under dynamic loads. The governing equations are derived by using Hamilton’s principle using the classical plate theory and the higher-order shear deformation plate theory. Using state-space methods to find the closed-form solution of the dynamic response of functionally graded rectangular plates resting on a viscoelastic foundation. Numerical examples are given for displacement and stresses in the plates with various structural parameters and the effects of these parameters are discussed. The result of the numerical example shows a marked decrease in displacement and stresses as the coefficient of viscous damping is increased.

Static and free vibration analysis of functionally graded annular plates using stress-driven nonlocal theory

Abstract: This study presents analytical and numerical solutions of the bending and the free vibration analysis of a functionally graded annular nanoplate based on the stress-driven nonlocal theory. The nonlocal equation is obtained using the classical plate theory; the power law distribution is assumed to model changes in material properties throughout the thickness. The governing differential equation is analytically solved for different types of edge supports. The explicit analytical solution is obtained in terms of special functions, namely, hypergeometric and Meijer functions. Furthermore, based on the Galerkin technique, a finite element method is introduced to figure out the nanoplate’s flexural deformation and natural frequencies. The analytical and numerical solutions are compared with those available in the literature. The size effects in the nonlocal theory and FGM’s gradient properties have been discussed, considering clamped and simply supported boundary conditions. According to the stress-driven nonlocal model, size-dependent flexural responses of the plate demonstrate stiffening behavior with increasing nonlocal parameters. Which, in general, leads to a decrease in the value of transversal deformation and an increase in natural frequencies when compared to the plate’s local solutions.

Time-domain motion of a floating or obliquely submerged non-uniform elastic plate

Abstract: We consider the motion of a thin elastic plate with non-uniform thickness. The plate is either submerged and has some inclination with the vertical or is floating on the upper surface of the water. Green's function arising from the fourth-order boundary condition for the non-uniform plate (which we refer to as plate Green's function) is determined using two different methods in terms of the vibrating modes of the plate. These, in turn, are derived from the modes of a plate with constant thickness. The problem is finally reduced to a boundary integral equation involving the plate Green's function and the fundamental Green's function. This equation is hypersingular in the case of a submerged plate. A numerical solution to the integral equation is used to find results for elastic plates with variable thicknesses. The results are validated by comparing them with those of an elastic plate with uniform thickness. We also present simulations of the time-domain motion when the plate–fluid system is subject to an incident wave pulse using Fourier transform.

Asymmetric/Axisymmetric buckling of circular/annular plates under radial load using first-order shear deformation theory

Abstract: This paper proposes a mathematical method for the asymmetric buckling analysis of homogeneous and isotropic circular/annular plates under radial load based on the first-order shear deformation theory and nonlinear von Kármán relations. The buckling load is presented for different combinations of the free, clamped, and simply supported boundary conditions at the plate outer edges and different aspect ratios. The equilibrium equations which are five coupled nonlinear partial differential equations are extracted using the principle of virtual work and they are solved analytically using the perturbation technique. The stability equations which are a system of coupled linear partial differential equations with variable coefficients are obtained by employing the adjacent equilibrium criterion. The differential quadrature method is utilized to find the buckling load which is the eigenvalue of the stability equations. Also, the buckling load is examined using the classical plate theory as well. The sensitivity analysis investigates the effect of geometrical parameters on the buckling load. The results are compared with the obtained results from the classical plate theory, finite elements, and the results were reported in the other references. • By employing the virtual work and adjacent criteria, the equations of circular/annular plates are extracted. • The equations are based on the first-order shear deformation theory and von Kármán relations. • The equilibrium equations are solved using the perturbation technique. • The stability equations are solved using the averaging method and the differential quadrature method. • Also, the buckling load has been obtained based on the classical plate theory.

Nonlinear bending of FG metal/graphene sandwich microplates with metal foam core resting on nonlinear elastic foundations via a new plate theory

Abstract: Nonlinear bending of functionally graded metal/graphene (FGMG) sandwich rectangular plate with metal foam core resting on nonlinear elastic foundations is elucidated in this article. A new shear deformation plate theory is presented to formulate the displacement field. The nonlinear partial differential equations considering the small size effect are established via the principle of virtual work. The nonlinearity is considered by using Von Karman’s strain-displacement relations. While, the size effect is captured by employing the modified couple stress theory. The upper and lower layers are made of aluminum as a matrix that reinforced with graphene platelets (GPLs). The GPLs are functionally graded through the thickness of the face layers according to a new cosine rule. Moreover, the metal foam core is also made of aluminum containing porosities that uniformly distributed or functionally graded through the core thickness. The governing equations are solved based on the Galerkin and Newton’s methods. The obtained results are examined by introducing some comparison examples. In addition, several parametric examples are discussed including the effects of the porosity type, GPLs distribution type, core-to-face thickness, elastic foundation stiffness, side-to-thickness ratio, plate aspect ratio and material length scale parameter on the nonlinear deflection and stresses of FGMG sandwich plate with metal foam core.

Sound transmission loss in functionally graded porous metastructural plate with absorber

Abstract: In the present work, sound transmission loss (STL) of two plates functionally graded sandwich structures with a porous core is studied. The plates are modeled by first-order shear deformation theory. To obtain an acoustic metamaterial plate with the greatest STL, a functionally graded porous (FGP) plate with a tuned mass damper absorber is used. The obtained results show that using a FGP plate or using only an absorber is sufficient for increasing sound transmission loss, but the simultaneous use of them is effective in higher frequencies.

Vibrational and Acoustical Wave Propagation Through Fibreglass Plate for Different Boundary Conditions

Abstract: Fibreglass composite plates are used for applications ranging from aerospace and automotive industries to construction industry. Vibroacoustic properties of the fibreglass plate has been investigated for different boundary conditions. The theory of the plate vibration and sound radiation has been used to determine the deflection of the composite plate at different locations on its surface. 3-Dimentional (3D) deformations and their corresponding contour velocities of plate have been computed for a range of frequencies. It has been shown that clamping the plate at four edges delayed first resonance of the plate by 15 Hz and second resonance of the plate by 30 Hz. Vibroacoustic indicators and the radiation impedance matrix of the plate have also been calculated. The results of this study show that fibreglass plate could be used in building industry to dampen the vibration of machine and structural vibration of buildings, and to sandwich building structures to form panels in order to attenuate airborne sound and to lower noise transmission of structural borne sound

Buckling of Elastic Plates

Abstract: The classical theory of plate buckling is shown here to emerge from our dimension reduction procedure applied to incremental elasticity theory, concerned with the linearized theory or small deformations superposed upon large. Plate buckling theory emerges as the leading-order-in-thickness model when the underlying pre-stress scales appropriately with respect to thickness.

On simplified deformation gradient theory of modified gradient elastic Kirchhoff–Love plate

Abstract: A concise modified gradient elastic Kirchhoff–Love plate model with two length-scale parameters is proposed based on simplified deformation gradient theory. A sixth-order basic differential equation and boundary conditions applicable to arbitrary shapes are obtained by applying the principle of minimum potential energy and combining with the generalized strain energy related to classical strain, strain gradient and rotation gradient. By introducing couple stress, the classical bending stiffness corresponding to the fourth-order terms and force-related boundary conditions in the typical gradient elastic Kirchhoff–Love plate model are modified, and the bending deformation of thin plates can be described more flexibly. Under certain conditions, the modified gradient elastic Kirchhoff–Love plate model can be reduced to typical gradient elastic thin plate model, couple stress thin plate model and classical Kirchhoff–Love plate model. The bending boundary value problems of gradient elastic thin plates are further studied, and the specific modified classical boundary conditions and non-classical higher-order boundary condition acceptable to the sixth-order basic equation in Cartesian coordinates are derived. The analytical and numerical bending solutions to gradient Navier-type and Levy-type thin plates with various boundary conditions, including simply supported, clamped and free boundaries are presented, and the size-dependent bending stiffness that is jointly determined by geometric dimensions and length-scale parameters is defined to describe the size effect of thin plates comprehensively.