Order Shear Deformation Plate Theory Research Articles

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.

Buckling and Post-buckling of Sandwich Plate with Functionally Graded Graphene Origami Reinforced Core Layer in Hygrothermal Environment

The buckling and post-buckling of magneto-electric-elastic (MEE) sandwich plate are presented. The core layer consists of epoxy matrix reinforced with graphene origami auxetic metal-material in which the volume fraction of graphene origami is assumed to vary the thickness direction in three different distributions. For MEE face sheets, the proportions of ferroelectric and ferromagnetic components are chosen equally. The Reddy's higher order shear deformation plate theory forms the basis for establishing the governing equations. The axial compressive loading- deflection amplitude relation and the expression of critical buckling loading are obtained through the application of Galerkin method. The method's accuracy is validated by comparing its results with those published by other authors. The influences of various parameters on the buckling and post-buckling of the sandwich plate are investigated in details. 

Vibration analysis of magneto-electro-elastic sandwich plate with auxetic graphene reinforced metal matrix composite core

This research addresses analytical solutions for vibration performance of simply supported sandwich plate subjected to coupled mechanical-electric-magnetic-thermal loads. The plate is made of graphene reinforced metal matrix composite (GRMMC) core and magneto-electro-elastic (MEE) face sheets. Three graphene distribution patterns, FG-X, FG-O, and UD are considered for evaluating the reinforcement efficiency of graphene. The Reddy’s higher order shear deformation plate theory and Hamilton’s principle are employed to formulate the equations of motion of the sandwich plate. Then, using double trigonometric functions for variables and applying Bubnov–Galerkin procedure, the system of nonlinear second-order differential equations is constructed to reveal the natural frequency, frequency ratio, and dynamic response of the sandwich plate. The influences of graphene distribution patterns and volume fraction, magnetic and electric potentials, temperature change, and volume fraction of piezoelectric phase are discussed in details. This research provides basis for the design of sandwich composite structures, especially smart structures and devices.

Natural frequency analysis of sandwich plate with auxetic honeycomb core and CNTRC face sheets using analytical approach and artificial neural network

In this paper, an analytical investigation is presented on the natural frequency analysis of sandwich plate resting on Winkler-Pasternak type elastic foundations in thermal environment. The sandwich plate consists of an auxetic honeycomb core with negative Poisson's ratio, which is assumed to perfectly bond to two carbon nanotube reinforced composite (CNTRC) face sheets to ignore potential interface effects or imperfections in bonding. The material properties of CNTRC are assumed to be temperature dependent. The volume fraction of carbon nanotube (CNT) varies through the thickness dimension according to linear functions with five different distribution patterns of CNT including UD, FG-O, FG-V, FG-Λ and FG-X. The governing equations of motion are derived based on the Reddy's first order shear deformation plate theory and Hamilton's principle, and the expression of the natural frequency is performed via Galerkin method. Artificial neural network (ANN) model is established to accurately predict the natural frequency of the sandwich plate on the basis of 1450 data points collected from analytical results. The effects of various material and geometrical parameters on the natural frequency of the sandwich plate are investigated in details.

Nonlinear dynamic stability analysis of CNTs reinforced piezoelectric viscoelastic composite nano/micro plate under multiple physical fields resting on smart foundation

This study analyzes the nonlinear dynamic stability of carbon nanotube reinforced composite (CNTRC) piezoelectric viscoelastic nano/micro plate under time dependent harmonic compressive biaxial mechanical loading. The nano/micro plate is single-layer with a uniform and rectangular cross section. Polyvinylidene fluoride (PVDF) is the material used for the nano/micro plate. Also, it is located on a nonlinear viscoelastic piezoelectric foundation made of Zinc Oxide (ZnO). Single-walled carbon nanotubes (SWCNTs) have been used to strengthen nano/micro plate. Using Kelvin-Voigt model, the material properties of the system are assumed to be viscoelastic. Nano/micro plate is subjected to 2D magnetic field and electric potential. Magnetic field effects are incorporated using Maxwell’s relations. An alternating and direct voltage is applied to the nano/micro plate and also smart foundation in thickness direction. Eshelby-Mori-Tanaka approach is used to estimate the effective elastic properties. Additionally, follower force and uniform thermal gradient is applied to nano/micro plate in this study. The nonlinear strain–displacement relations are based on the Von-Kármán theory. According to classical, first order, third order, sinusoidal, parabolic, hyperbolic and exponential shear deformation plate theories, a new formulation is proposed incorporating surface stress effects through the Gurtin-Murdoch elasticity theory. In order to consider small scale parameters, this research is based on modified strain gradient theory (MSGT). Using the energy method and Hamilton’s principle, the non classical governing equations are derived. For nano/micro plate, simply supported boundary conditions are considered. For the purpose of solving differential equations, an analysis based on Galerkin procedure and finally the Bolotin method will be used to obtain the dynamic instability region (DIR). It is the objective of this study to investigate the impact of such parameters as small-scale parameters, alternating and direct applied voltage, intensity of magnetic field, surface effects, thermal environment and static load factor on the DIR. This study revealed that taking into account the smart foundation reduces the area of dynamic instability region by more than 60% for a constant intensity of magnetic field. The stability responses are also more convergent with the smart foundation. Additionally, the range is moved toward higher excitation frequencies and greater system stability. Validation of answers based on reliable scientific articles is one of the final stages of this research. These results can be used to design nano/micro electromechanical systems.

Buckling analysis of stiffened functionally graded multilayer graphene platelet reinforced composite plate with circular cutout embedded on elastic support subjected to in-plane normal and shear loads

Buckling analyses of orthogonally stiffened functionally graded graphene platelet reinforced composite plate with circular cutout supporting on Winkler elastic foundation is conducted according to the third order shear deformation plate theory and the finite element method to achieve the desired solutions. Material properties of the composite plate are evaluated using Rule of mixture and Halpin-Tsai approach. Graphene platelet reinforcements and Poly methyl methacrylate matrix are considered as the two components of the composite media. The material of the stiffeners is selected the same as that of the polymer matrix. In one efficient case, adding a set of stiffeners increased the critical buckling load by 41.8 % with only a 10.3 % increase in weight of the structure. Influence of broad range of factors such as GPLs nanofillers distribution pattern and weight fraction, Winkler-type elastic foundation, uniaxial and biaxial normal and shear loads, plate thickness and aspect ratio, width, height and number of longitudinal and transverse stiffeners are reported in several tabular and graphical data in detail.

Green’s function for thick laminated composite plates with coupled stretching-bending and transverse shear deformation

In this paper, the Green’s function of a thick laminated composite plate is derived for the first time in the literature. The plate can be unsymmetric with respect to the mid-plane such that the coupling of in-plane stretching and out-of-plane bending deformations occurs. Also, the plate is thick enough that the transverse shear deformation cannot be ignored. Through the plane wave decomposition method, the partial differential governing equations based upon the first order shear deformation plate theory can be reorganized into the ordinary differential equations in terms of the transformed variable. With the governing equations reorganized into matrix form, the Green’s functions are expressed in terms of matrix exponential. Through eigen-decomposition, the matrix exponential including its related integrals can be evaluated explicitly. The explicit solutions of Green’s functions in transformed domain are obtained accordingly. In order to improve the accuracy and efficiency of numerical integration for the required transform integrals, a special quadrature rule which can eliminate the integral singularity is suggested. To have a fair verification and to extend the applicability of newly derived Green’s functions, the associated boundary element method is also introduced in this study.

Natural Frequencies of Soft-Core Composite Sandwich Plates Reinforced by FG Graphene Platelets

Superior mechanical properties such as ultrahigh stiffness and strength along with amazing physical properties have led to the use of nano graphene platelets (GPLs) as a reinforcement in polymer composite structures such as sandwich plates. In this research, free vibration analysis of rectangular sandwich plates with functionally graded (FG) GPL-reinforced face sheets and a soft core is investigated. A high-order three-layer accurate sandwich plate theory is employed to model the soft-core sandwich structure. Third order shear deformable plate theory is applied for functionally graded graphene nano-composite face sheets while second and third order polynomials are used for displacements of the soft core. In this theory, the transverse flexibility of the core as well as the continuity conditions of transverse shear stresses are assumed. Hamilton’s principle is applied to extract the motion equations and the corresponding analytical solutions are presented. The effects of materials properties and geometrical parameters on the natural frequencies of the sandwich plates are studied. A comparison of the present results in particular cases with those of published studies confirms the accuracy of the present analysis. Results show that the natural frequencies of the sandwich plates are enhanced by reinforcing the face sheets with GPLs.

Theoretical and experimental investigations on steady-state responses of rotor-blade systems with varying rotating speeds based on a new nonlinear dynamic model

Rotor-blade systems are major units in turbine structures, whose dynamic responses will affect structural safety, energy conversion efficiency, overall durability and service performance. During high-speed rotation, periodic disturbances near resonant regions may lead to a large deflection of flexible blades and an unstable rotation of rotor-blade systems. Due to the coupling effect of rotation and vibration, the distributions of mass/inertia, damping and stiffness are related with varying rotation speed and local deformation in rotor-blade systems, which has not been concerned in conventional harmonic balance methods. In this paper, a modified harmonic balance method is proposed to obtain dynamic responses of rotor-blade systems with a comprehensive model in incremental form, which considers geometric nonlinearity, centrifugal and gyroscopic effects of flexible blades and nonlinear constraints between rotor and multiple blades. Based on Reddy’s high order shear deformation plate theory, dynamic equations for a rotating composite blade with a pre-setting angle is established with consideration of von Kármán type geometrical nonlinearity, varying rotating speeds and equivalent aerodynamic loads. A coupling motion equation for rotor-blade systems is derived out by assembling corresponding elemental equilibrium and joint nonlinear constraints. Steady-state responses are obtained by using a modified harmonic balance method which accounts nonlinearities of mass, damping and stiffness. Numerical and experimental studies for rotor-blade systems are carried out to validate the effectiveness of the proposed model. Results show that a higher rotating speed leads to larger natural frequencies of rotor-blade systems, and a more evidently coupling effect between motion and vibration. Meanwhile, joint constraints, damping effect and load distributions have evident influences on vibration amplitudes and resonant points of system’s steady-state responses.

Buckling analysis of thick plates using a single variable simple plate theory

Buckling analysis of thick plates has been carried out herein by using a single variable simple plate theory. Theory used herein is a third order shear deformation plate theory which uses a single displacement function for the complete formulation of plates. Plate formulation is governed by only one governing differential equation. Governing equation of the theory has close resemblance to that of Classical Plate Theory. Thus, plate problems can be solved in the similar lines as in case of classical plate theory. Plate theory used herein does not require a shear correction coefficient. To check the efficacy of the theory buckling analysis of simply supported thick rectangular plates is carried out. Critical buckling loads for simply supported plates are evaluated and the results obtained are compared to other shear deformation plate theories. Buckling load results are found to be in good agreement with other plate theory results.

Bending and Buckling of FG-GRNC Laminated Plates via Quasi-3D Nonlocal Strain Gradient Theory

To improve the structural stiffness, strength and reduce the weight of nanoplate structure, functionally graded (FG) graphene-reinforced nanocomposite (GRNC) laminated plates are exploited in this paper. The bending and buckling behaviors of FG-GRNC laminated nanoplates are investigated by using novel quasi-3D hyperbolic higher order shear deformation plate theory in conjunction with modified continuum nonlocal strain gradient theory, which considered both length and material scale parameters. The modified model of Halpin–Tsai is employed to calculate the effective Young’s modulus of the GRNC plate along the thickness direction, and Poisson’s ratio and mass density are computed by using the rule of mixture. An analytical approach of the Galerkin method is developed to solve governing equilibrium equations of the GRNC nanoplate and obtain closed-form solutions for bending deflection, stress distributions and critical buckling loads. A detailed parametric analysis is carried out to highlight influences of length scale parameter (nonlocal), material scale parameter (gradient), distribution pattern, the GPL weight fraction, thickness stretching, geometry and size of GPLs, geometry of the plate and the total number of layers on the stresses, deformation and critical buckling loads. Some details are studied exclusively for the first time, such as stresses and nonlocality effect.

Dispersion and band gaps of elastic guided waves in the multi-scale periodic composite plates

This paper presents dispersive behaviors and band gap characteristics of the multi-scale periodic composite plate reinforced with grapheme platelets (GPLs) and fibers. The plate consists of a laminated composite structure and an isotropic part. The material properties are measured by the layer-wise distribution of graphene platelets (GPLs) in nanocomposite matrix and fiber content in the laminated composite. Based on the first order shear deformation plate theory, Bloch theorem and the higher-order spectral elements, a semi-analytical approach is proposed for wave analysis of the periodic laminated composites. Considering the periodic boundary and analytic integration through the thickness, elastodynamic wave equation is established on the middle surface of the periodic unit cell, and then transformed into a one-dimensional problem by the Fourier transformation along the wave propagation direction, which improves the calculation efficiency. A second-order polynomial eigenvalue problem with regard to wave dispersion is obtained. Besides, the accuracy of the proposed approach is analyzed and the layer number of the nanocomposite laminate is determined. Various significant parameters including weight content of GPLs, fiber volume fraction and lattice size of the plate are taken into account to investigate guided wave propagation in the multi-phase periodic composite plates. The results indicate that full band gaps can be realized in the periodic multi-scale composite phononic crystal plates and partial band gaps appear in both in-plane and out-of-plane wave modes. The addition of reinforcements can efficiently regulate the center frequency and width of full and partial band gaps by changing GPLs content and fiber volume. Moreover, the variation of lattice parameter of the periodic composite plate has significant impact on the first full band gap. This work can provide guidelines to the developments of novel composite structures, acoustic devices and the applications in structural health monitoring.

Static buckling analysis and geometrical optimization of magneto-electro-elastic sandwich plate with auxetic honeycomb core

The nonlinear static buckling analysis of magneto-electro-elastic sandwich plate on Pasternak-type elastic foundations subjected to the mechanical, thermal, electric and magnetic loadings is presented in this paper. The sandwich plate is composed of an auxetic honeycomb core with negative Poisson’s ratio and two face sheets made of magneto-electro-elastic​ material. The system basic equations are derived based on the Reddy’s higher order shear deformation plate theory taking into account the effect of von Kármán the kinematic nonlinearity and initial imperfection. The form of possible solutions and electric, magnetic potentials are chosen as trigonometric functions based on two cases of boundary conditions. The relationship between axial compressive loading and dimensionless deflection amplitude is determined by using the Galerkin method. For optimization problem, the Bees algorithm is applied to obtain the maximum value of critical buckling load of the sandwich plate which depends on five geometrical and material parameters. The effects of elastic foundations, temperature increment, geometrical parameters and electric and magnetic potentials on the stability characteristics are investigated in numerical results. The accuracy and reliability of present approach is confirmed by comparisons with the existing results in the literature.

Dynamic characterization of hybrid composite based on flax/E-glass epoxy composite plates

The fundamental investigation on the dynamic characterization of Flax/E-glass hybrid composite plates was done and the mathematical model was developed for Flax/E-glass hybrid composite using Finite element method. In this experimental set up vaccum bagging process was implemented in order to fabricate prototypes of Flax/Epoxy, E-glass/Epoxy, Flax/E-glass epoxy hybrid composites and to conduct experimental investigation on dynamic properties. The first order shear deformation plate theory was used in this study for developing the mathematical model. Modal analysis was done on each composite plate for three times and average was taken between the values obtained to minimize the error. By this natural frequency of Flax/Epoxy, E-glass/Epoxy and Flax/E-glass epoxy hybrid composite and natural frequencies were found out experimentally by modal testing. The effect of natural frequencies on composite material, with respect to boundary condition, ply angle and aspect ratio were studied and the experimental results were validated with respect to mathematical modeling Mode Shapes with effect of natural frequencies on composite material, ply angle, aspect ratio, boundary conditions were studied, for all laminates and it was found that for all laminates CCCC boundary condition has high natural frequencies compared to other models. Hence it was observed that experimental results obtained through modal testing were very close to the Matlab results.

Vibration analysis of auxetic laminated plate with magneto-electro-elastic face sheets subjected to blast loading

Mathematic modeling and analytical approach are presented for nonlinear vibration analysis of laminated plate with auxetic honeycomb core and magneto-electro-elastic face sheets supported by Pasternak-type elastic foundations. The laminated plate is subjected to the simultaneous action of blast, thermal, electric and magnetic loadings. The mechanical properties of auxetic core including negative Poisson’s ratio are assumed to depend on the geometrical parameters of the unit cells and elastic modulus of original material. The volume fraction of two components of each magneto-electro-elastic face sheet is chosen equally. The nonlinear motion equations and the geometrical compatibility equation are established by using Reddy’s higher order shear deformation plate theory taking into account the coupling between elastic, electric and magnetic fields. The analytical vibration solutions for the auxetic laminated plate can be obtained by using Galerkin and fourth-order Runge – Kutta methods. The numerical results are conducted to investigate the effect of geometrical and material parameters, elastic foundations, temperature increment, magnetic and electric potentials on the vibration characteristics of the auxetic laminated plate.

Thermo-mechanical postbuckling analysis of sandwich plates with functionally graded auxetic GRMMC core on elastic foundations

This paper investigates the thermo-mechanical postbuckling behaviors of sandwich plates with auxetic graphene-reinforced metal matrix composite (GRMMC) core resting on elastic foundations. Two cases of the compressive postbuckling of sandwich plates subjected to uni-axial compression in thermal environments and the thermal postbuckling of sandwich plates subjected to a uniform temperature rise are considered. The auxetic GRMMC core having negative Poisson's ratio (NPR) is assumed to be piece-wise functionally graded (FG) by changing the volume fraction of graphene through the thickness domain of the core. The material properties of both metal face sheets and the GRMMC core are assumed to be temperature dependent. The Reddy’s third order shear deformation plate theory and the von Kármán-type kinematic nonlinearity are used to formulated the governing equations of the sandwich plate which account for the plate-foundation interaction and the thermal effects. The postbuckling solutions are obtained by employing a two-step perturbation approach. The results reveal that the compressive postbuckling and thermal postbuckling behaviors of sandwich plates with auxetic GRMMC core are significantly influenced by the FG pattern, the face sheet-to-core-to-face sheet thickness ratio, and foundation stiffness.

Panel flutter analysis of cracked functionally graded plates in yawed supersonic flow with thermal effects

In the present paper, an isogeometric analysis formulation along with the Nitsche technique is implemented based on the first order shear deformation plate theory to investigate the aero-thermo-elastic panel flutter behavior of a functionally graded (FG) plate containing cracks in the supersonic flow field. A FG material is considered with temperature-dependent mechanical properties distributed in the plate thickness according to the power law. The temperature distribution in the plate thickness is calculated through solution of the one-dimensional steady-state conductive heat transfer equation while the surface temperatures are kept fixed. The physical contact and the crack growth phenomena are overlooked in the present research. The panel flutter behavior is investigated under combination of crack orientation, crack length, thermal conditions, flow direction, FG volume fraction index and boundary conditions. The accuracy and quality of the present isogeometric formulation is shown in comparison with those available in literature.

Nonlinear thermal dynamic buckling and global optimization of smart sandwich plate with porous homogeneous core and carbon nanotube reinforced nanocomposite layers

The analytical investigation for the nonlinear thermal dynamic buckling of smart sandwich plate subjected to mechanical, thermal and electric loadings is presented. The sandwich plate is composed of a porous homogeneous core, two carbon nanotube reinforced composite (CNTRC) layers and two piezoelectric face sheets. Basic equations are derived based on the Reddy's higher order shear deformation plate theory and Hamilton's principle in which the initial imperfection and Pasternak-type elastic foundations are included. The external pressure is assumed to be uniformly distributed on the surface of the sandwich plate and depend on time according to the linear functions. The nonlinear dynamic response, the frequency – amplitude relation are obtained by using the Galerkin and Runge – Kutta methods and the critical dynamic buckling load is determined by using Budiansky – Roth criterion. Bees Algorithm is used to determine the maximum value of natural frequency of smart sandwich plate and the corresponding optimum values of geometrical and material parameters. The effects of geometrical parameters, CNT volume fraction, elastic foundations, temperature increment, initial imperfection and porosity coefficient on the nonlinear vibration and dynamic buckling of the smart sandwich plate are considered specifically. The numerical results are also compared with existing results using different theories.

Examination of thermal postbuckling behavior of temperature dependent FG-GRMMC laminated plates with in-plane negative Poisson’s ratio

Auxetic composite laminates are a new type of engineering materials that have unique features for important potential applications. This paper examines the effect of in-plane negative Poisson’s ratio (NPR) on the thermal postbuckling behaviors of graphene-reinforced metal matrix composite (GRMMC) plates. The plates rest on an elastic foundation and are subjected to a uniform temperature rise. The GRMMC layers with different volume fractions of graphene reinforcement can be arranged to achieve piece-wise functionally graded (FG) patterns across the plate thickness and the material properties of the GRMMC layers are temperature-dependent. The Reddy’s third order shear deformation plate theory and the geometric nonlinearity of von Kármán-type are applied to formulate the thermal postbuckling equations for GRMMC laminated plates. The nonlinear problem can be solved by a two-step perturbation approach. Parametric study is performed for (±10)5T and (±10)3T GRMMC laminated plates possessing in-plane NPR. The results reveal that the buckling temperatures for (±10)5T and (±10)3T plates are significantly enhanced with an FG-X pattern for the plates. We found that due to the combined effect of FG and in-plane NPR, the thermal postbuckling strength of FG-X (±10)3T plate is higher than that of FG-X (±10)5T plate.

Buckling analysis of nanoplates based on a generic third-order plate theory with shear-dependent non-isotropic surface stresses

In this paper, a general third-order shear deformation plate theory (GTSDPT) is employed to investigate the buckling behaviors of rectangular nanoplates. A new model with non-isotropic surface modulus that depends on in-plane and transverse shear strains, such as that for most crystalline faces, is established to analyze the significance of non-isotropic surface effects on buckling behaviors. The governing equations and the corresponding boundary conditions are established by using the principle of minimum potential energy. The analytical critical buckling load solutions are obtained for nanoplates with different boundary conditions. The proposed model is verified by comparing buckling results against the ones from existing plate theories. Using both Reddy’s third order shear deformation plate theory (TSDPT) and GTSDPT, the influence of plate thickness, length-to-thickness ratio, surface modulus, residual surface stress, bulk modulus and boundary conditions on the buckling behavior of nanoplates with surface stress effects is analyzed in detail. It is observed that the buckling results of the nanoplates from TSDPT and GTSDPT exhibit some differences due to the presence of shear stresses on the interface between surface layers and the bulk of the plates. The TSDPT theory may not be able to capture shear stress effect and the GTSDPT theory may need to be employed in this case. Finally, the influence of surface stresses on the buckling behaviors of nanoplates made of gold, nickel, silver and copper and subjected to uni- and bi-axial compressive loads is investigated using the GTSDPT theory.