Non-classical Plate Model Research Articles

A microstructure-dependent Kirchhoff plate model based on a reformulated strain gradient elasticity theory

A new microstructure-dependent non-classical model for Kirchhoff plates is developed by using a reformulated strain gradient elasticity theory that incorporates both the strain gradient and couple stress effects. The equation of motion and the boundary conditions are simultaneously obtained through a variational formulation based on Hamilton’s principle. The new plate model contains one material constant to account for the strain gradient effect and one material length scale parameter to capture the couple stress effect. The newly developed non-classical plate model includes the plate model incorporating the couple stress effect alone and the plate model based on the classical elasticity as two special cases. To illustrate the new model, the buckling, static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. The numerical results reveal that the presence of the strain gradient and couple stress effects leads to reduced plate deflections, enlarged critical buckling loads and increased natural frequencies. These microstructure effects are significant when the plate is very thin, but they are diminishing as the plate thickness increases. These predicted trends of the size effects at the micron scale agree with those observed experimentally.

A comparison for the non-classical plate model based on axial buckling of single-layered graphene sheets

In this study, the size effect on the axial buckling behavior of single-layered graphene sheets embedded in elastic foundation is studied. Eringen’s non-local elasticity equations are incorporated into first-order shear deformation, higher-order shear deformation, and classical plate theories. Values of Winkler and Pasternak moduli parameters, side lengths of square SLGSs, non-local parameter and mode numbers are obtained from different non-local plate theories. It is shown that the axial buckling behavior is strongly dependent on non-local parameters and moduli which are different for different numbers and side lengths. Furthermore, we find that non-locality is more influential in first-order shear deformation than other non-local plate theories, especially in certain ranges of non-local parameters. Sensitivity indices are determined based on the Sobol method in their corresponding physical ranges.

Non-classical plate model for FGMs

In this work, we developed a non-classical plate model for the analysis of the behavior of microplates prepared from functionally graded materials (FGMs). It was assumed that the material properties of FGM microplates vary along the direction of thickness and could be estimated using Mori–Tanaka homogenization technique. A variational approach based on Hamilton’s principle was employed to achieve motion equations and corresponding boundary conditions simultaneously. Unlike classical plate theory, the proposed plate model contained three material length scale parameters which allowed size effect to be taken into account effectively. Moreover, the proposed non-classical plate model could be divided into classical and modified couple stress plate models when all material length scale parameters were set at zero. To approve the applicability of the proposed size-dependent plate model, analytical solution for free vibration problem of a simply supported FGM microplate was obtained according to the Navier solution, where generalized displacements were stated as multiplication of undetermined functions with known trigonometric functions to identically satisfy simply supported boundary conditions for all edges. The results obtained by the proposed non-classical plate model were compared with those obtained from reduced ones relevant to different values of material property gradient index and dimensionless length scale parameter. It was found that the differences observed between the sets of predicted natural frequencies obtained from different types of plate models were more significant for lower values of dimensionless length scale parameter.

Static Analysis of a Nano Plate by Using Generalized Differential Quadrature Method

In this paper, static analysis of a nano rectangular plate subjected to uniform distributed load is studied based on modified couple stress theory (MSCT) by using Generalized Differential Quadrature (GDQ) Method. The inclusion of an additional material parameter enables the new plate model to capture the size effect. The new non-classical plate model reduces to the classical plate model when the length scale parameter is set to zero. In deriving of the governing equations, the minimum total potential energy principle is used. In the solution of the governing equations, the DQM method is used for simply-supported nano plate within the Kirchhoff-Love plate theory. In the numerical results, the influences the material length scale parameter and the dimension parameters of the plate on the static deflection of the nano plate is presented. Also, the difference between the classical theory (CT) and MSCT is investigated for static responses of nano plate.

A non-classical Mindlin plate model incorporating microstructure, surface energy and foundation effects.

A non-classical model for a Mindlin plate resting on an elastic foundation is developed in a general form using a modified couple stress theory, a surface elasticity theory and a two-parameter Winkler-Pasternak foundation model. It includes all five kinematic variables possible for a Mindlin plate. The equations of motion and the complete boundary conditions are obtained simultaneously through a variational formulation based on Hamilton's principle, and the microstructure, surface energy and foundation effects are treated in a unified manner. The newly developed model contains one material length-scale parameter to describe the microstructure effect, three surface elastic constants to account for the surface energy effect, and two foundation parameters to capture the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the new model includes the Mindlin plate models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases, recovers the Kirchhoff plate model incorporating the microstructure, surface energy and foundation effects, and degenerates to the Timoshenko beam model including the microstructure effect. To illustrate the new Mindlin plate model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulae derived.

Size-Dependent Geometrically Nonlinear Forced Vibration Analysis of Functionally Graded First-Order Shear Deformable Microplates

AbstractIn this paper, a non-classical plate model capturing the size effect is developed to study the forced vibration of functionally graded (FG) microplates subjected to a harmonic excitation transverse force. To this, the modified couple stress theory (MCST) is incorporated into the first-order shear deformation plate theory (FSDPT) to account for the size effect through one length scale parameter, only. Strong form of nonlinear governing equations and associated boundary conditions are obtained using Hamilton's principle. The solution process is implemented on two domains. The generalized differential quadrature (GDQ) method is first employed to discretize the governing equations on the space domain. A Galerkin-based scheme is then applied to extract a reduced set of the nonlinear equations of Duffing-type. On the second domain, through a time differentiation matrix operator, the set of ordinary differential equations are transformed into the discrete form on time domain. Eventually, a system of the parameterized nonlinear equations is acquired and solved via the pseudo-arc length continuation method. The frequency response curve of the microplate is sketched and the effects of various material and geometrical parameters on it are evaluated.

A non-classical model for circular Kirchhoff plates incorporating microstructure and surface energy effects

A new non-classical model for circular Kirchhoff plates subjected to axisymmetric loading is developed using a modified couple stress theory, a surface elasticity theory and Hamilton’s principle. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation. The new plate model contains a material length scale parameter to capture the microstructure effect and three surface elasticity constants to describe the surface energy effect. The current non-classical plate model includes the plate models considering the microstructure influence or the surface energy effect alone as special cases and recovers the classical elasticity-based Kirchhoff plate model when both the microstructure and surface energy effects are suppressed. To demonstrate the new model, the static bending problem of a clamped solid circular Kirchhoff plate subjected to a uniform normal load is analytically solved by directly applying the general formulas derived. The numerical results reveal that the deflection of the clamped circular plate predicted by the current Kirchhoff plate model is smaller than that predicted by the classical elasticity-based model, but the former approaches the latter with the increase of the plate thickness. It is also found that the difference between the plate deflection predicted by the model incorporating the surface energy effect and that predicted by the classical model is significant when the plate thickness is sufficiently small. However, the difference is diminishing as the plate thickness increases. These predicted size effects at the micron scale agree with the general trends observed in experiments.

A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects

A new non-classical Kirchhoff plate model is developed using a modified couple stress theory, a surface elasticity theory and a two-parameter elastic foundation model. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the microstructure, surface energy and foundation effects. The new plate model contains a material length scale parameter to account for the microstructure effect, three surface elastic constants to describe the surface energy effect, and two foundation moduli to represent the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the newly developed plate model includes the models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases and recovers the Bernoulli–Euler beam model incorporating the microstructure, surface energy and foundation effects. To illustrate the new model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection of the simply supported plate with or without the elastic foundation predicted by the current model is smaller than that predicted by the classical model. Also, it is observed that the difference in the deflection predicted by the new and classical plate models is very large when the plate thickness is sufficiently small, but it is diminishing with the increase of the plate thickness. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the elastic foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. These predicted trends of the size effect at the micron scale agree with those observed experimentally. In addition, it is shown both analytically and numerically that the presence of the elastic foundation reduces the plate deflection and increases the plate natural frequency, as expected.

Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory

The buckling and vibration responses of nanoplates made of functionally graded materials (FGMs) subjected to thermal loading are studied in prebuckling domain with considering the effect of surface stress. To accomplish this purpose, Gurtin–Murdoch elasticity theory is incorporated into the classical plate theory to develop a non-classical plate model including the surface effects. The material properties of FGM nanoplate are considered to be graded in the thickness direction on the basis of the power law function. Hamilton׳s principle is utilized to derive size-dependent governing differential equations of motion and associated boundary conditions. Selected numerical results are presented to indicate the importance of surface stress effect. It is revealed that in the presence of surface stress effect, the influence of material property gradient index on the critical thermal buckling load is more prominent for FGM nanoplates with lower length-to-thickness ratios. Also, by increasing the natural frequency of FGM nanoplate, the role of surface stress effect in the value of critical thermal buckling load is more prominent.

Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory

Abstract A Mindlin microplate model based on the modified strain gradient elasticity theory is developed to predict axisymmetric bending, buckling, and free vibration characteristics of circular/annular microplates made of functionally graded materials (FGMs). The material properties of functionally graded (FG) microplates are assumed to vary in the thickness direction. In the present non-classical plate model, the size effects are captured through using three higher-order material constants. By using Hamilton's principle, the higher-order equations of motion and related boundary conditions are derived. Afterward, the generalized differential quadrature (GDQ) method is employed to discretize the governing differential equations along with various types of edge supports. Selected numerical results are given to indicate the influences of dimensionless length scale parameter, material index and radius-to-thickness ratio on the deflection, critical buckling load and natural frequency of FG circular/annular microplates.

Surface stress effect on the pull-in instability of circular nanoplates

The prime aim of the current study is to investigate the pull-in instability characteristics of hydrostatically and electrostatically actuated circular nanoplates including surface stress effect. Gurtin–Murdoch elasticity theory is incorporated into the classical plate theory to develop a non-classical plate model considering surface stress effect. The governing size-dependent differential equations of a hydrostatically and electrostatically actuated circular nanoplate are discretized along with simply-supported and clamped boundary conditions using the generalized differential quadrature (GDQ) method. Selected numerical results are given to indicate the capability of the present size-dependent plate model for predicting the normalized pull-in voltage and pull-in hydrostatic pressure of nanoplates with inclusion surface stress effect. It is found that the pull-in phenomenon of actuated nanoplate is strongly size-dependent. The results indicate that surface stress effect plays a more significant role in the pull-in phenomenon of nanoplates of lower thicknesses. This investigation might be helpful to evaluate the mechanical characteristics of electrostatical actuators.

A non-classical third-order shear deformation plate model based on a modified couple stress theory

A non-classical third-order shear deformation plate model is developed using a modified couple stress theory and Hamilton’s principle. The equations of motion and boundary conditions are simultaneously obtained through a variational formulation. This newly developed plate model contains one material length scale parameter and can capture both the size effect and the quadratic variation of shear strains and shear stresses along the plate thickness direction. It is shown that the new third-order shear deformation plate model recovers the non-classical Reddy-Levinson beam model and Mindlin plate model based on the modified couple stress theory as special cases. Also, the current non-classical plate model reduces to the classical elasticity-based third-order shear deformation plate model when the material length scale parameter is taken to be zero. To illustrate the new model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new plate model are smaller than those predicted by its classical elasticity-based counterpart, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significant when the plate thickness is small, but they are diminishing with increasing plate thickness.

Size-dependent dynamic pull-in instability of hydrostatically and electrostatically actuated circular microplates

In the present study, the dynamic pull-in instability and free vibration of circular microplates subjected to combined hydrostatic and electrostatic forces are investigated. To take size effects into account, the strain gradient elasticity theory is incorporated into the Kirchhoff plate theory to develop a nonclassical plate model including three internal material length scale parameters. By using Hamilton’s principle, the higher-order governing equation and the corresponding boundary conditions are obtained. Afterward, a generalized differential quadrature (GDQ) method is employed to discritize the governing differential equations along with simply supported and clamped edge supports. To evaluate the pull-in voltage and vibration frequencies of actuated microplates, the hydrostatic-electrostatic actuation is assumed to be calculated by neglecting the fringing field effects and utilizing the parallel plate approximation. Also, a comparison between the pull-in voltages predicted by the strain gradient theory and the degenerated ones is presented. It is revealed that increasing the dimensionless internal length scale parameter or decreasing the applied hydrostatic pressures leads to higher values of the pull-in voltage. Moreover, it is found that the value of pull-in hydrostatic pressure decreases corresponding to higher dimensionless internal length scale parameters and applied voltages.

Free vibration of size-dependent Mindlin microplates based on the modified couple stress theory

This paper develops a Mindlin microplate model based on the modified couple stress theory for the free vibration analysis of microplates. This non-classical plate model contains an internal material length scale parameter related to the material microstructures and is capable of interpreting the size effect that the classical Mindlin plate model is unable to describe. The higher-order governing equations of motion and boundary conditions are derived using the Hamilton principle. The p-version Ritz method is employed to determine the natural frequencies of the microplate with different boundary conditions. A detailed parametric study is conducted to study the influences of the length scale parameter, side-to-thickness ratio and aspect ratio on the free vibration characteristics of the microplate. It is found that the size effect is significant when the thickness of microplate is close to the material length scale parameter.

A non-classical Mindlin plate model based on a modified couple stress theory

A non-classical Mindlin plate model is developed using a modified couple stress theory. The equations of motion and boundary conditions are obtained simultaneously through a variational formulation based on Hamilton’s principle. The new model contains a material length scale parameter and can capture the size effect, unlike the classical Mindlin plate theory. In addition, the current model considers both stretching and bending of the plate, which differs from the classical Mindlin plate model. It is shown that the newly developed Mindlin plate model recovers the non-classical Timoshenko beam model based on the modified couple stress theory as a special case. Also, the current non-classical plate model reduces to the Mindlin plate model based on classical elasticity when the material length scale parameter is set to be zero. To illustrate the new Mindlin plate model, analytical solutions for the static bending and free vibration problems of a simply supported plate are obtained by directly applying the general forms of the governing equations and boundary conditions of the model. The numerical results show that the deflection and rotations predicted by the new model are smaller than those predicted by the classical Mindlin plate model, while the natural frequency of the plate predicted by the former is higher than that by the latter. It is further seen that the differences between the two sets of predicted values are significantly large when the plate thickness is small, but they are diminishing with increasing plate thickness.