Classical Plate Theory Research Articles

An asymptotic consistent couple stress theory for the three‐dimensional free vibration analysis of functionally graded microplates resting on an elastic medium

Based on the three‐dimensional (3D) consistent couple stress theory (CCST), we develop an asymptotic theory for the free vibration analysis of functionally graded (FG) microplates resting on an elastic medium. The material properties of the FG microplate are assumed to obey a power‐law distribution of the volume fractions of the constituents in the thickness direction, for which the effective material properties are estimated using the rule of mixtures. The interactions between the FG microplate and its foundation medium are simulated using a Winkler–Pasternak foundation. Carrying out the asymptotic expansion method, we obtain recursive sets of motion equations for various order problems. The CCST‐based classical plate theory is derived as a first‐order approximation of the 3D CCST. The asymptotic theory is shown to converge rapidly and to be in excellent agreement with the exact solutions for FG macroplates reported in the literature when the value of the material length scale parameter is assigned to be zero.

A uniformly-valid asymptotic plate theory of growth with numerical implementation

In this paper, we aim to develop a simplified and uniformly-valid finite-strain plate theory of growth. First, starting from a consistent finite-strain plate theory of growth proposed in our previous work, we specify the magnitudes of growth functions in five different cases and conduct systematic asymptotic analyses, from which the original plate theory can be reduced to some classical plate and membrane theories. Based on the asymptotic analyses, it is found that some terms in the original plate equations (which contain the high-order stress tensor S(2)) always have higher asymptotic orders. By dropping these terms, the plate equations can be simplified significantly. Then, a reduced finite-strain plate theory of growth is established, which is valid in a wide range of growth and mechanical loading conditions. The associated weak form of this uniformly-valid plate theory has also been derived for numerical implementation. To demonstrate the efficiency of this plate theory, it is implemented into a finite element software and applied to study four typical examples. To verify the accuracy of the 2D plate models, the corresponding 3D volume models have also been constructed for these examples. Through some comparisons, it is found that the numerical results obtained from the 2D and 3D models can fit each other quite well. Furthermore, the numerical calculations based on the plate models show obvious advantages in the aspects of computational efficiency, convergence rate and stability. In our opinion, the uniformly-valid asymptotic plate theory of growth proposed in the current work is adequate for studying the complex growth behaviors of thin hyperelastic plates.

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

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.

Effect of nonlocal parameters and Kerr foundation on nonlinear static and dynamic stability of micro/nano plate with graphene platelet reinforcement

It is a fact that size effects appear when studying the mechanical behavior of nano/microscopic materials. This paper, for the first time, applies an analytical method to determine the size effect dependence and Kerr foundation on nonlinear static and dynamic stability of micro/nano plate with graphene platelet reinforcement. For this purpose, the nonlocal elasticity theory along with classical plate theory is used to derive the basic equations. Both motion and geometric compatibility equations of the plate are deduced by introducing the Von Kármán strain–displacement relationship. Applying the analytical method (Bubnov–Galerkin and the stress function methods), the nonlinearity of buckling and postbuckling, the vibration and the dynamic response of micro/nano plate with graphene platelet reinforcement are analyzed. The fourth-order Runge–Kutta methods are implemented to solve the governing equation of the dynamic system. To prove the reliability of the formulas, the results of the present study in specific cases are compared with those of relevant published papers in the reference and good agreements are observed. In the final part, the effects of nonlocal parameters, Kerr foundation, initial imperfection, GPL weight fraction and geometrical parameters on the overall nonlinear static and dynamic stability of the micro/nano plate with graphene platelet reinforcement are evidently demonstrated.

Some closed form series solutions to peridynamic plate equations

Peridynamics is a generalized continuum theory which is developed to account for long range internal force/moment interactions. Peridynamic equations of motion are of integro-differential type and only several closed-form analytical solutions are available for elementary structures. The aim of this paper is to derive analytical solutions to peridynamic Kirchhoff type (shear rigid) plates equations for both static and dynamic cases. Applying trigonometric series with respect to in-plane coordinates, solutions for the deflection function of a rectangular simply supported plate are derived. The coefficients in the series are presented in a closed analytical form such that the equations of motion are exactly satisfied. For the dynamic case the solution is derived applying the variable separation with respect to the time and the in-plane coordinates. Several numerical cases are presented to illustrate the derived solutions. Furthermore, results of peridynamic plate equations are compared against results obtained from the classical plate theory. A very good agreement between these two theories is obtained for the case of the small horizon sizes which shows the capability of the presented approach.

Bending waves localized along the edge of a semi-infinite piezoelectric plate with orthogonal symmetry

We study the propagation of bending waves along the free edge of a semi-infinite piezoelectric plate within the framework of two-variable refined plate theory (TVPT, a high-order plate theory), Reissner-Mindlin refined plate theory (RMPT, a first-order plate theory), and the classical plate theory (CPT). The piezoelectric plate has macroscopic symmetry of orthogonal mm2 The governing equations are derived using Hamilton principle. The dispersion relations for electrically open and shorted boundary conditions at the free edge are obtained analytically. The difference in dispersion property between the three plate theories is analyzed. The numerical results show that the dispersion curves predicted by TVPT and RMPT are similar and have small difference over the complete frequency range, which means both the two theories are valid for the analysis of edge waves in a piezoelectric plate. But the wave velocity calculated by CPT is much larger than the two theories above and is no longer valid for high frequency and thick plate. The electrical boundary condition at the free edge has an insignificant effect on phase velocity and group velocity which can be ignored for the analysis of edge waves in a piezoelectric plate governed by bending deformation. The velocity of bending edge waves in a semi-infinite piezoelectric plate is positively related to that of Rayleigh surface wave in a traction-free piezoelectric half-space. The edge wave velocity can be enhanced when the piezoelectric plate is considered as one with weaker anisotropy.

Displacement and Stress Analysis of Thick Rectangular Clamped Plates Using Third-Order Energy Functional

This study analysed the stress and displacement responses of an all-clamped rectangular thick plate subjected to a uniformly distributed load using polynomial displacement functions. A polynomial shear deformation function was incorporated in the determination of the general governing equations; hence, there is no need for shear correction factors. Approximate Polynomial Displacement Functions w, u, and v for all the clamped thick rectangular plates subject to transverse loading were obtained. Numerical results were obtained using displacement equations obtained from the literature for the non-dimensional form of displacements and stresses of an all clamped (CCCC) plate at different aspect and span-depth ratios to determine the efficiency of this theory. The results obtained were validated as they showed good agreement with the results obtained by other researchers available in literature. Hence, this theory can be used as a reliable, concise, easy and dependable means for the stress and displacement analysis of thick plates. The results obtained also indicate that at a span-depth ratio of 100 and above, the classical plate theory (CPT) can be used to analyse plates, as the numerical values obtained are approximately equal to that of the CPT.

Vibration and stability analysis of a tensioned moving printed electronic laminated membrane under multiple working conditions

This paper investigated the vibration of a printed electronic laminated membrane subjected to pretension and air resistance in thermal environments. First, a mathematical model for the moving printed electronic laminated membrane is established, which considers the inhomogeneity of the tension at both ends of the membrane. The model equilibrium governing differential equation is obtained according to the classical laminate plate theory and Hamilton’s principle. The differential quadrature method is used for numerical calculation, and the effects of the aspect ratio, x-axis tension, thermoelastic coupling coefficient, dimensionless air resistance, and other parameters on the vibration of simply supported opposite sides and free opposite sides of a printed electronic laminated membrane are analyzed. The model’s accuracy is verified by comparison with the results reported in the literature. This study can provide theoretical guidance for setting printing equipment parameters and improving the transport stability of roll-to-roll printed flexible electronic laminated membranes.

Elastic buckling of simply supported bimetallic steel plates

Bimetallic steel plate is a laminated composite one consisting of metallurgically bonded cladding and substrate layers, such as the stainless-clad (SC) or titanium-clad (TC) ones. Due to significant corrosion resistance and cost efficiency, it has gained growing recognition as an alternative to stainless steel plate and is particularly suitable for offshore and marine structures. In this paper, the nominal elastic buckling stress of a bimetallic steel plate is derived by adapting the first-order shear deformation plate theory (FSDT) and compared with the solutions of classical plate theory (CPT). The plates are simply supported and subjected to uniaxial compression; two different types of cladding configurations are considered. The transverse shear stress distributions and shear correction factors are obtained by theoretical analyses. Finite element models of the bimetallic steel plates are developed by MATLAB and ABAQUS to validate the derivation. Parametric studies are then conducted to elucidate the effects of the cladding configuration, clad ratio, elastic modulus ratio and width-to-thickness ratio on the buckling stress. Based on the theoretical analyses, a series of simplified design formulae for the critical buckling stress and design requirements for interface shear strength requirements are proposed for further engineering applications. The research works may provide important bases for understanding and investigating the buckling behaviour of the bimetallic steel structures.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Nonlinear buckling analysis of FGM plates stiffened by different FGM stiffener types

In this paper, the nonlinear buckling behavior of functionally graded material (FGM) plates subjected to an axial compression is analytically investigated. Assuming that the plates are stiffened by FGM rectangular, I- and T-stiffeners. The formulations are established by using the classical plate theory considering the geometrical nonlinearities of von Karman. The Lekhnitskii’s smeared stiffener technique is developed for different types of FGM stiffeners. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves of plates. Numerical examples validate the effects of different types of FGM stiffeners, material and geometrical parameters on nonlinear behavior of plates.

Asymptotic analysis of 3D dynamic equations in linear elasticity for a thin layer resting on a Winkler foundation

Abstract A 3D dynamic problem for a thin elastic layer resting on a Winkler foundation is considered. The stiffness of the layer is assumed to be much greater than that of the foundation in order to allow low-frequency bending motion. The leading long-wave approximation valid outside the vicinity of the cut-off frequency is derived. It is identical to the classical Kirchhoff plate theory. A novel near cut-off 2D approximation is also established. It involves both bending and extension motions which are not separated from each other due to the effect of the foundation. The associated dispersion relation appears to be non-uniform over the small wavenumber domain.

Development of an analytical model for the flexural vibration of fish bone active camber structures with truncated, variable thickness partitions

The need for smooth, gapless mechanisms that offer superior aerodynamic performance has been the prime mover for a wide range of studies on the morphing concepts from the early years of the twenty-first century. In light of the importance of novel analytic means for inspecting such structures, this article describes a comprehensive dynamic model for the flexural vibration of a special type of morphing structure known as fish bone active camber (FishBAC). In view of the shortcomings of previous models, the developed procedure considers the variable thickness of the central spine of FishBAC as well as the possibility of arbitrary truncations in its numerous partitions to offer more accurate results. A general idea is first proposed based on the assumptions of classical plate theory for such partitioned structures. The equations are derived using the energy method and the assumption of artificial springs to satisfy the continuity of shear force and bending moment along the connecting lines between partitions. A robust Rayleigh-Ritz method along with a set of fast-converging admissible functions is presented to solve the governing equations of motion. Combined with the power of Mathematica and validated by other analytic models and finite element codes, this technique can be used to deal with a plethora of similar systems while paving the path for the dynamic analysis of FishBAC built around various airfoils.

Free vibration characteristics of viscoelastic nano-disks based on modified couple stress theory

This paper investigates the vibrational behavior of a viscoelastic and size-dependent nano-disk based on the modified couple stress theory (MCST). The material characteristics in nano-scale are modeled according to Zener viscoelastic constitutive relation. In addition, displacement components are defined based on classical plate theory. Leaderman integral is also used to determine the viscous parts of the stress tensor. Hamilton’s principle is utilized to derive the governing equations of motion for specifying the strain, kinetic energy, and viscous work. The obtained equations are discretized with the help of the Galerkin method and decoupled through the diagonalization procedure. Laplace transformation is employed to solve the resulting equations in differential–integral form. The damping ratio, the imaginary part and real part of the Eigen frequency of the considered nano-disk are calculated to investigate the effects of influential parameters on the nano-disk vibrational behavior. These parameters include nonlocal parameter boundary conditions, geometric constant, power constant, and element relaxation coefficient. Results obtained on different mode shapes indicate that increasing the dimensionless element relaxation coefficient is followed by a decrease in the imaginary part of the Eigen frequency regarding the energy dissipation as well as a decrease in the real part of the Eigen frequency. Furthermore, increasing the h/l ratio is accompanied by variations in the imaginary part, real part, and damping ratio. According to the results, the effect of damping on vibrational behavior of the nano disk is more distinguished for smaller values of h/l.

Nonlinear interaction of the damped large dynamic deformations of the Mooney–Rivlin hyperelastic plates with the viscoelastic and shear reactions of the supporting substrate

In this research, the interaction between the damped large deformation dynamic responses of the Mooney–Rivlin hyperelastic plates and the viscoelastic and shear characteristics of the supporting substrate (that constitute a visco-hyperelastic system) is investigated for various distributions and various time variations of the transverse loads and different boundary conditions. A viscoelastic Winkler–Pasternak model is chosen for the supporting substrate to account for the dissipative, shearing, and supporting features of the foundation. The governing equations of motion are obtained by using Hamilton’s principle, von Karman assumptions, left Cauchy–Green deformation tensor, and a modified classical plate theory whose accuracy is enhanced through the incorporation of an appropriate high-order incompressibility condition. The combination of the nonlinear and coupled motion equations and boundary conditions is solved by an iterative 2D differential quadrature (DQ) spatial-discretization and Newmark’s time-marching methods. The effects of the constitutive parameters of the hyperelastic material, magnitude and type of the distributed loads, the thickness and aspect ratios of the plate, parameters of the Winkler–Pasternak viscoelastic foundation, and boundary conditions on the dynamic lateral deflections of the plate are studied examined during the parametric studies. Results emphasize the role of the visco-hyperelasticity features interaction of the combined plate-substrate system on the vibration suppression and dynamic performance of the structure in different spatial and time-variation patterns of the distributed loads and various boundary conditions.

Effect of cracks on dynamical responses of double-variable-edge plates made of graphene nanoplatelets-reinforced porous matrix and sur-bonded by piezoelectric layers subjected to thermo-mechanical loads

This paper presents free vibration analyses and nonlinear dynamic responses of new complex edge contour non-rectangular plate structures with a part-through surface crack which the authors name as temporarily double-variable-edge (DVE) plates. The analytical results and numerical simulation are investigated in the Part I and it is expected that the experimental approach will be shown in Part II. The cracked plate made of a functionally graded porous (FGP) matrix structure and reinforced by graphene nano-platelets (GNPs) is covered by two piezoelectric layers and rested on a Winker-Pasternak elastic foundation. The material properties are determined by using the Halpin-Tsai micro-mechanical model, Gaussian Random Field (GRF) proposal, and the rules of mixtures. The governing equations are established based on the classical plate theory (CPT) and Galerkin's method which are convenient in analyzing the behaviors of complex edge contour plates. The obtained results are compared with those obtained from the existing Finite Element Method (FEM) to ensure the reliability and accuracy. In addition, the effects of geometrical shapes, crack parameters, applied voltages of piezoelectric layers, elastic foundations as well as thermal impacts are scrutinized. The obtained outcomes in the paper are promising for applications in the fields of aerospace, civil, and electronic engineering.

Analytical Investigation of Sound Radiation from Functionally Graded Thin Plates Based on Elemental Radiator Approach and Physical Neutral Surface

This paper analyzes the sound radiation behavior of a clamped thin, functionally graded material plate using the classical plate theory and Rayleigh Integral with the elemental radiator approach. The material properties of the plate are assumed to vary according to the power-law distribution of the constituent materials in the transverse direction. The functionally graded material is modeled using a physical neutral surface instead of a geometric middle surface. The effects of the power-law index, elastic modulus ratio, different constituent materials, and damping loss factor on the sound radiation of functionally graded plate are analyzed. It was found that, for the considered plate, the power-law index significantly influences sound power level and radiation efficiency. There exists a critical value of the power-law index for which the corresponding peak of sound power level is minimum. In a wide operating frequency range, approximately 500–1500 Hz, this research suggests that the radiation efficiency is lower for the power-law index equal to 0 and 1. However, for very low frequencies (less than 250 Hz), the power-law index does not affect radiation efficiency significantly. Further, as the modulus ratio increases, the sound power peak decreases for a given power-law index. For the given material constituents of the functionally graded plate, the different values of damping loss factors do not significantly influence radiation efficiency. However, the selection of material constituents affects the radiation efficiency peak.

Broadband Sound Insulation and Dual Equivalent Negative Properties of Acoustic Metamaterial with Distributed Piezoelectric Resonators.

Aiming at the unsatisfactory sound transmission loss (STL) of thin-plate structures in the low-mid frequency range, this paper proposes an acoustic insulation metamaterial with distributed piezoelectric resonators. A complete acoustic prediction model is established based on the effective medium method and classical plate theory, and the correctness is verified by the STL simulation results of the corresponding acoustic-structure fully coupled finite-element model. Moreover, the intrinsic relationship between the dual equivalent negative properties and STLs is investigated to reveal the insulation mechanisms of this metamaterial. Then, the influence of the geometric and material parameters on the double equivalent negative characteristics is studied to explore the broadband STL for distributed multi-modal resonant energy-dissipation modes in the frequency band of interest. The results show that the two acoustic insulation crests correspond to the dual equivalent negative performances, and the sound insulation in the low-mid frequency range is improved by more than 5 dB compared with that of the substrate, even up to 44.49 dB.

General closed-form expressions for the three-dimensional vibrations of elastic bodies using the Ritz method

A novel Ritz-β approach is proposed for accurately determining the vibrational frequencies of a body based on its three-dimensional elasticity. In the conventional Ritz method, when admissible 3D displacement fields are expressed as either simple algebraic, Chebyshev, or Legendre polynomials, the volumetric integrals are transformed into beta integrals through a change of variables and then furthered simplified to beta functions and gamma functions; the resulting stiffness and mass matrices are expressed in a closed-form and the computational time is significantly decreased. The proposed approach does not involve the intensive computational procedures employed in the classical finite element, finite difference, and spectral approximation theories. Furthermore, it does not involve the kinematic constraints employed in classical beam, plate, and shell theories. The correctness and accuracy of the proposed approach were confirmed through comprehensive convergence studies and comparisons with existing results.