Classical Plate Model Research Articles

A non-classical model for an orthotropic Kirchhoff plate embedded in a viscoelastic medium

A non-classical model for an orthotropic Kirchhoff plate embedded in a viscoelastic medium is developed by using an extended version of the modified couple stress theory and a three-parameter foundation model. The equations of motion and the boundary conditions are simultaneously obtained through a variational formulation based on Hamilton’s principle. The new plate model contains three material length scale parameters to capture the microstructure effect, one damping coefficient to account for the viscoelastic damping effect, and two foundation moduli to represent the foundation effect. The current non-classical orthotropic plate model includes the isotropic plate model incorporating the microstructure effect and the classical elasticity-based orthotropic plate model as special cases. To illustrate the new model, the static bending and free vibration problems of a simply supported orthotropic 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 predicted by the current model is smaller than that predicted by the classical model, and the difference is large when the plate thickness is small but diminishes as the thickness becomes large. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. In addition, the damping ratio predicted by the new plate model is lower than that predicted by the classical plate model, and the difference is diminishing as the plate thickness increases. These predicted trends of the size effect at the micron scale agree with those observed experimentally. Furthermore, it is quantitatively shown that the presence of the foundation enlarges the plate natural frequency.

Dynamic analysis of a transversely isotropic non-classical thin plate

This study investigates the dynamic analysis of a transversely isotropic thin plate. The plate is made of hyperelastic John\'s material and its constitutive law is obtained by taken the Frechect derivative of the highlighted energy function with respect to the geometry of deformation. The three-dimensional equation governing the motion of the plate is expressed in terms of first Piola-Kirchhoff\'s stress tensor. In the reduction to an equivalent two-dimensional plate equation, the obtained model generalizes the classical plate equation of motion. It is obtained that the plate under consideration exhibits harmonic force within its planes whereas this force varnishes in the classical plate model. The presence of harmonic forces within the planes of the considered plate increases the natural and resonance frequencies of the plate in free and forced vibrations respectively. Further, the parameter characterizing the transversely isotropic structure of the plate is observed to increase the plate flexural rigidity which in turn increases both the natural and resonance frequencies. Finally, this study reinforces the view that non-classical models of problems in elasticity provide ample opportunity to reveal important phenomena which classical models often fail to apprehend.

Micromechanical analysis of piezo-magneto-thermo-elastic T-ribbed and Π-ribbed plates

ABSTRACTThis article determines the closed-form expressions for the effective properties of fully coupled piezo-magneto-thermo-elastic T-ribbed and Π-ribbed plates consisting of arbitrary orthotropic constituents. To evaluate these properties, the authors invoke a general, previously developed asymptotic homogenization model. Of particular interest and practical importance are the product properties, i.e., the pyroelectric, pyromagnetic, and magnetoelectric coefficients; these properties are manifested in the macroscopic composite but are usually absent from the micromechanical behavior of the individual constituents; they are a consequence of the coupling between mechanical, electrical, magnetic, and thermal fields and are realized through strain transfer between the various constituents. The practical importance of the work lies in the fact that it can be successfully employed to tailor the effective properties of a smart composite T- or Π-ribbed plate to the requirements of a particular engineering application by changing certain geometric or material parameters of interest. The validity of the results is illustrated via comparison with other models. Finally, it is shown that in the limiting case of a thin elastic plate of uniform thickness, the derived model converges to the familiar classical plate model.

A nonlinear thick plate formulation based on the modified strain gradient theory

ABSTRACTA geometric nonlinear first-order shear deformation theory-based formulation is presented to analyze microplates. The formulations derived herein are based on a modified strain gradient theory and the von Karman nonlinear strains. The modified strain gradient theory includes five material length scale parameters capable to capture the size effects in small scales. The governing equations of motion and the most general form of boundary conditions of an arbitrary-shaped plate are derived using the principle of virtual displacements. The analysis is general and can be reduced to the modified couple stress plate model or the classical plate model.

Micromechanical modeling of thin composite and reinforced magnetoelectric plates – Effective elastic, piezoelectric and piezomagnetic coefficients

A comprehensive asymptotic homogenization model for the analysis of composite and reinforced magnetoelectric plates is developed. The model culminates in a set of unit cell problems via which the relevant expressions for the effective coefficients are derived. The micromechanical and geometrical platform pertinent to the desired model is set up in this paper which also obtains the effective elastic, piezoelectric and piezomagnetic coefficients; the remaining properties, including product properties, are calculated in another work. Examples of structures that can be examined include diagonally- and triangularly-reinforced plates, hexagonal honeycomb sandwich plates and others. The developed model can be used to customize the effective properties of a reinforced plate to the requirements of a particular engineering application by changing some geometric, structural or material parameter of interest. It is shown that in the limiting case of a thin composite plate of uniform thickness whereby electrical conductivity is ignored and all pertinent quantities are time-averaged the presented model converges to the familiar classical plate model. Overall, this paper represents an important addition to the existing literature in terms of the complex geometries that can be designed and analyzed and at the same time constitutes an important refinement over previously published work.

A Non-Classical Model for Circular Mindlin Plates Incorporating Microstructure and Surface Energy Effects

A new non-classical model for circular Mindlin plates 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, and the microstructure and surface energy effects are treated in a unified manner. The new plate model contains a material length scale parameter to account for the microstructure effect and three surface elasticity constants to describe the surface energy effect. The current non-classical model includes the plate models considering the microstructure influence only and the surface energy effect alone as special cases, and it recovers the classical elasticity-based circular Mindlin plate model when both the microstructure and surface energy effects are suppressed. To illustrate the new model, the static bending problem of a clamped circular Mindlin plate under a uniform normal load is analytically solved by directly applying the general formulas derived.

Variational formulations and general boundary conditions for sixth-order boundary value problems of gradient-elastic Kirchhoff plates

Abstract Sixth-order boundary value problems for gradient-elastic Kirchhoff plate bending models are formulated in a variational form within an H 3 Sobolev space setting. Existence and uniqueness of the weak solutions are then established by proving the continuity and coercivity of the associated symmetric bilinear forms. Complete sets of boundary conditions, including both the essential and the natural conditions, are derived accordingly. In particular, the gradient-elastic Kirchhoff plate models feature two different types of clamped and simply supported boundary conditions in contrast to the classical Kirchhoff plate model. These new types of boundary conditions are given additional attributes singly and doubly ; referring to free and prescribed normal curvature, respectively. The formulations and results of the analyzed strain gradient plate models are compared to two other generalized Kirchhoff plate models which follow a modified strain gradient elasticity theory and a modified couple stress theory. It is clarified that the results are extendable to these model variants as well. Finally, the relationship of the natural boundary conditions to external loadings is analyzed.

Active sloshing control in a smart flexible cylindrical floating roof tank

An exact three dimensional fully-coupled hydro-elastic analysis for transient liquid sloshing in a partially-filled vertically-standing flexible circular cylindrical shell container fitted with a freely floating smart piezo-sandwich thin elastic circular plate is presented. The problem formulation is based on the linear water wave theory, the classical (Kirchhoff/Sanders) thin plate and shell models, Maxwell's equations of electrodynamics, Stokes’ transformation, and eigen-function expansions in cylindrical coordinates. The control action is achieved by combined volume displacement and volume velocity feedbacks (VDF, VVF) implemented in a second order active damping (AD) compensator via two competent evolutionary heuristic optimization techniques that systematically tune the controller gain parameters while constraining the floating panel displacement and control voltage. The uncontrolled and controlled transient responses of the coupled hydro-elastic system under various external disturbances (i.e., a harmonic base excitation, a real seismic event, a severe launch vehicle liftoff event, and a distributed impulsive transverse load on the floating panel) are calculated by means of Durbin's numerical inverse Laplace transform scheme. Moreover, the free vibration characteristics of the coupled fluid/structure interaction (FSI) system are briefly studied. The superior performance of the proposed active floating roof control configuration in effective suppression of the key hydro-elastic parameters (panel displacement, and shell displacements/stresses) is demonstrated. It is also found that, in the current FSI control problem, the Multi-objective Particle Swarm Optimization (MOPSO)-based ADC outperforms the Non-dominated Sorting Genetic Algorithm (NSGA-II)-based method, in terms of convergence rate and computational effort. Limiting cases are examined and the precision of results is verified by comparisons with the existing data as well as with the results produced by a commercial finite element package.

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.

Heat as a tracer for understanding transport processes in fractured media: Theory and field assessment from multiscale thermal push‐pull tracer tests

AbstractThe characterization and modeling of heat transfer in fractured media is particularly challenging as the existence of fractures at multiple scales induces highly localized flow patterns. From a theoretical and numerical analysis of heat transfer in simple conceptual models of fractured media, we show that flow channeling has a significant effect on the scaling of heat recovery in both space and time. The late time tailing of heat recovery under channeled flow is shown to diverge from the behavior expected for the classical parallel plate model and follow the scaling for a simple channel modeled as a tube. This scaling, which differs significantly from known scalings in mobile‐immobile systems, is of purely geometrical origin: late time heat transfer from the matrix to a channel corresponds dimensionally to a radial diffusion process, while heat transfer from the matrix to a plate may be considered as a one‐dimensional process. This phenomenon is also manifested on the spatial scaling of heat recovery as flow channeling affects the decay of the thermal breakthrough peak amplitude and the increase of the peak time with scale. These findings are supported by the results of a field experimental campaign performed on the fractured rock site of Ploemeur. The scaling of heat recovery in time and space, measured from thermal breakthrough curves measured through a series of push‐pull tests at different scales, shows a clear signature of flow channeling. The whole data set can thus be successfully represented by a multichannel model parametrized by the mean channel density and aperture. These findings, which bring new insights on the effect of flow channeling on heat transfer in fractured rocks, show how heat recovery in geothermal tests may be controlled by fracture geometry. In addition, this highlights the interest of thermal push‐pull tests as a complement to solute tracers tests to infer fracture aperture and geometry.

An application of a size-dependent model on microplate with elastic medium based on strain gradient elasticity theory

A size-dependent Kirchhoff micro-plate model resting on elastic medium is developed based on the strain gradient elasticity theory. Three material length scale parameters are introduced in the model, and those parameters may effectively capture the size effect. The model can degenerate into the modified couple stress plate model or the classical plate model by setting two (l 0 and l 1) or all (l 0, l 1 and l 2) of the material length scale parameters to be zero. Analytical solutions for the static bending, buckling and free vibration problems of a rectangular micro-plate with all edges simply supported are obtained. The results predicted by the present model are compared with those predicted by the degraded models. Influences of the elastic medium on the static bending, buckling, and free vibration are discussed. The results show that the present model can predict prominent size-dependent normalized stiffness, buckling load, and natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter. The study may be helpful to guide the design of microplate-based devices resting on elastic medium for a wide range of potential applications.

Size-dependent bending and vibration behaviors of piezoelectric circular nanoplates

The size-dependent bending and vibration behaviors of a clamped piezoelectric circular nanoplate are investigated by using a modified Kirchhoff plate model. The flexoelectricity, the surface effect and the non-local elastic effect are taken into account in the modified model by decomposing the electric Gibbs free energy into the bulk and surface parts and including the strain gradient and the electric field gradient terms into the bulk energy density function. Different from the results predicted by the classical plate model, the proposed model predicts size-dependent behaviors of the piezoelectric thin plate with nanoscale thickness. Comparisons among the models considering the flexoelectricity, the surface effect and the non-local elastic effect individually, the current model and the classical model are also given in this study. Simulation results indicate that the electromechanical coupling properties, the transverse displacements and the resonant frequencies of the plate are significantly influenced by each individual effect as well as their combined effects. It is also indicated that such effects are affected by the external applied electric potential and the plate geometries. Neglecting any individual effect may induce inaccurate characterization of the electromechanical coupling of the piezoelectric nanoplate. Therefore, the current plate model is expected to provide more accurate predictions of the electromechanical coupling and the mechanical behaviors of piezoelectric circular nanoplate-based devices in the nanoelectromechanical systems.

Buckling and vibration of flexoelectric nanofilms subjected to mechanical loads

Piezoelectric nanofilms (PNFs) are widely used in microelectromechanical systems, buckling commonly occurs when subjected to compressive mechanical loads in their applications. In this paper we comprehensively study the flexoelectric effect on the buckling and vibrational behaviors of PNFs. The results from the analytical solutions indicate the significance of the flexoelectricity. The critical buckling loads and natural frequency are enhanced by the flexoelectricity. Analytical results indicate that the critical buckling load is not only influenced by the thickness of the PNFs, but also by the in-plane aspect ratio. When the thickness of the PNFs is several micrometers, the critical buckling load predicted by the present model is much higher than the prediction by the classical piezoelectric plate model. And the natural frequency calculated by the current model is much higher than that obtained by the classical piezoelectricity plate theory when the thickness is several tens of nanometers.

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.

Vibration analysis of nano-structure multilayered graphene sheets using modified strain gradient theory

In this paper, for the first time, the modified strain gradient theory is used as a new size-dependent Kirchhoff micro-plate model to study the effect of interlayer van der Waals (vdW) force for the vibration analysis of multilayered graphene sheets (MLGSs). The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. After obtaining the governing equations based on modified strain gradient theory via principle of minimum potential energy, as only infinitesimal vibration is considered, the net pressure due to the vdW interaction is assumed to be linearly proportional to the deflection between two layers. To solve the governing equation subjected to the boundary conditions, the Fourier series is assumed for w = w(x, y). To show the accuracy of the formulations, present results in specific cases are compared with available results in literature and a good agreement can be seen. The results indicate that the present model can predict prominent natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.

Nitsche method for isogeometric analysis of Reissner–Mindlin plate with non-conforming multi-patches

Nitsche method application in non-conforming plate is presented in the context of isogeometric analysis. Reissner–Mindlin plate theory is employed to build governing equation and stiffness matrix. We use this theory to solve the elasticity problems of various classical plate models, and compare the obtained results to those from single-patch models and the exact solutions in Kirchhoff theory. The solutions of problem involving the use of complex model are as well obtained using the same Reissner–Mindlin theory and compared to the results from finite element method. All models are built with NURBS (non-uniform rational B-spline) patches with non-conforming mesh along the common boundaries. The algorithms of knot insertion and order elevation are applied to enrich the basis functions of NURBS patches. The results of numerical examples show the accuracy, robustness and high convergence rate of this method.

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.

An equivalent classical plate model of corrugated structures

An equivalent classical plate model of corrugated structures is derived using the variational asymptotic method. Starting from a thin shell theory, we carry out an asymptotic analysis of the strain energy in terms of the smallness of a single corrugation with respect to the characteristic length of macroscopic deformation of the corrugated structure. We obtained the complete set of analytical formulas for effective plate stiffnesses valid for both shallow and deep corrugations. These formulas can reproduce the well-known classical plate stiffnesses when the corrugated structure is degenerated to a flat plate. The extension–bending coupling stiffnesses are obtained the first time. The complete set of relations are also derived for recovering the local fields of corrugated structures.

Decoupling the nonlocal elasticity equations for thermo-mechanical vibration of circular graphene sheets including surface effects

This article deals with the axisymmetric vibration properties of circular single-layered graphene sheet embedded in a polymer matrix under thermo-mechanical loading. Both surface and nonlocal effects are taken into account. To this end, Gurtin–Murdoch continuum elasticity in conjunction with the nonlocal elasticity theory is used to develop a modified continuum plate model for free vibration analysis of the nanoplates. The governing equations are derived by decoupling the nonlocal constitutive equations of Eringen theory in the polar coordinate. Galerkin׳s method is used to obtain the vibration frequencies. To verify the accuracy of the Galerkin results, a differential quadrature (DQ) solution is also developed. Galerkin results are successfully verified with those of the DQ method. A good agreement is also found between the present results and experimental data. Further, in comparison to the available molecular dynamics simulation results, the present formulation with appropriate values of surface and nonlocal parameters provides more accurate results than those by the classical plate model.

A Nonclassical Model for Circular Mindlin Plates Based on a Modified Couple Stress Theory

A nonclassical model for circular Mindlin plates subjected to axisymmetric loading is developed using a modified couple stress theory. The equations of motion and boundary conditions are simultaneously obtained 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 existing circular Mindlin plate models based on classical elasticity. In addition, both the stretching and bending of the plate are considered in the formulation. The current plate model reduces to the classical elasticity-based Mindlin plate model when the material length scale parameter is set to be zero. Additionally, the new circular Mindlin plate model recovers the circular Kirchhoff plate model as a special case. To illustrate the new model, the static bending problem of a clamped solid circular Mindlin plate subjected to an axisymmetrically distributed normal pressure is analytically solved by directly applying the new model and using the Fourier–Bessel series. The numerical results show that the deflection and rotation angle predicted by the new model are smaller than those predicted by the classical Mindlin plate model. 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 the increase of the plate thickness.