Classical Plate Model Research Articles

Static and dynamic stabilities of modified gradient elastic Kirchhoff–Love plates

The static and dynamic stabilities of modified gradient elastic Kirchhoff–Love plates (MGEKLPs), which incorporate two length-scale parameters related to strain gradient and rotation gradient effects, are comprehensively analyzed under various load forms and boundary conditions (BCs). The study of static stability employs static balance method and an improved energy method by introducing higher-order deformation gradients and corresponding energy terms. Utilizing the variational method, a sixth-order fundamental buckling differential equation for MGEKLPs under both transverse and in-plane loads is derived, serving as the foundation for the static balance method. The static stability analysis of MGEKLPs examines the combined effects of strain and rotation gradients on size-dependent critical buckling loads. Building on generalized strain energy with higher-order deformation energy, the energy method of classical elastic thin plate model is enhanced and applied to the static stability analysis of MGEKLPs. This approach enables the investigation of static stability without being constrained by the need to solve complex differential equations, making it applicable to various BCs and load scenarios. While static stability provides a description of stable state of an elastic system, dynamic stability offers a more scientific and rigorous analysis. The dynamic stability of simplified gradient elastic Kirchhoff–Love plates (SGEKLPs) with curved edges and different BCs is further investigated by combining the generalized strain energy with Lyapunov’s second stability method, presenting the dynamic stability criterion in the form of norms. A strict description of the dynamic stability of a SGEKLP over the entire time domain is provided for different supporting conditions, including case where all edges are supported and case with free edges. The analysis of size-dependent static and dynamic stabilities offers theoretical guidance for designing elastic thin plates with microstructures.

A deviatoric couple stress Mindlin plate model and its degeneration

Based on the classical couple stress theory, the deviatoric couple stress theory (DCST), where both couple stress and curvature tensors are traceless, is developed. The deviatoric couple stress Mindlin plate is established based on a new hypothesis of the vanishing curvature of the transverse normal, and the governing equations and associated boundary conditions of the simplified size-dependent Mindlin plate are respectively derived by the principle of virtual work. In the absence of couple stresses, a simplified Mindlin plate model between the classical Kirchhoff plate model and classical Mindlin plate model is derived. To illustrate the simplified DCST-based Mindlin plate model, the static bending and free vibration of rectangular plates are studied. The dependence of the length scale parameter and curvature ratio on the deflection, rotation and natural frequency is analyzed. The numerical results show that the consideration of couple stresses and curvatures hardens the stiffness of the microstructure, resulting in a decrease in its deflection and rotation compared with that of the macroscopic structure under the same conditions, while the natural frequency increases, which is consistent with the size effect observed in experiments.

Asymptotic homogenization of flexoelectric composite plates with periodically varying thickness

A micromechanical model for the analysis of structurally periodic flexoelectric plates with periodically varying thickness is developed on the basis of asymptotic homogenization. The period of thickness variation is small and comparable to the plate thickness; accordingly, the plate is usually referred to as having a “rapidly varying thickness.” The stipulation for rapidly varying thickness is important because it is envisioned that the developed model can be applied to a broad range of thin plates, both stratified as well as plates endowed with an arbitrary distribution of reinforcements attached to the surface or embedded within the plate. The microscopic problem is implemented in two steps pertaining, respectively, to first- and second-gradient asymptotic homogenization. Each level of homogenization culminates in its own set of unit cell problems from which the effective coefficients of the homogenized structure can eventually be obtained. These effective coefficients couple the force and moment resultants as well as the averaged electric displacement with the first and second gradients of the macroscopic displacement and electric potential. Once the effective coefficients are obtained, the macroscopic problem is invoked, which provides a set of four differential equations from which the macroscopic variables of mechanical displacement and electric potential can be obtained. The model is illustrated by means of laminated flexoelectric composites as well as simple rib-reinforced plates. It is shown that in the limiting case of a thin, purely elastic plate, the derived model converges to the familiar classical plate model.

Weak-form differential quadrature element analysis of plate on a tensionless and frictional foundation using a higher-order kinematics

Plate-foundation systems, as a structure common in engineering practices, have received extensive scientific research. A novel technical scheme is proposed to improve the computational efficiency in predicting the nonlinear responses of a plate on the foundation. The core work of this technical scheme lies in two aspects. The first aspect is to create a novel plate-foundation system, where a higher order kinematical assumption common in Carrera Unified Formulation is used to model the plate, and a plate-foundation interaction model taking into account the tensionless and frictional characteristics is created. The adjustable kinematical parameters of the higher-order kinematical assumption endow the plate with flexible accuracy. The second aspect is to derive a two-dimensional (2D) differential quadrature (DQ) formula for 2D interpolation based on the one-dimensional (1D) DQ formula, and thus a weak-form differential quadrature element method (WDQEM) for nonlinear quasi-static analysis of the plate-foundation system is proposed by using the principle of virtual work and the 2D interpolation derived. The proposed WDQEM supports flexible nodal configuration in the element, making it degree-of-freedom-flexible and suitable to carry out both p- and h-version strategies in the numerical simulation. Several numerical examples are given to (i) verify the proposed technical scheme and the corresponding computer codes; (ii) to investigate the numerical performance of the WDQEM through comparison between the classical plate model and the 3D solid model, and (iii) to study some physical behaviors related to the tensionless and frictional characteristics of the plate-foundation system. The example results show that (i) the WDQEM converges much faster than the conventional finite element method due to the higher order approximation; (ii) the higher order kinematics, due to its capability of adjusting the accuracy of kinematical assumptions, can effectively reduce the computational scale; (iii) the smoothness problem in the post process of stresses analysis is greatly alleviated, (iv) the tensionless and frictional features of the plate-foundation system should be considered when evaluating both the vertical and horizontal bearing capacities of the plate.

On the hybrid atomistic-continuum model for vibrational analysis of α-, β-, and γ-graphyne circular nano-plates

The natural frequency of graphyne structures for all phases is calculated by molecular dynamics simulation. Tersoff potential function was used in molecular dynamics (MD) simulations. In this paper, the vibration behavior of α-, β-, and γ-graphyne circular nano-plates was investigated using a hybrid atomistic-continuum Model. The MD frequencies are higher than those obtained from the classical local plate model, so the use of the Eringen-nonlocal elasticity model is practically unusable. The obtained equations based on the stress-driven elasticity, modified couple stress theory, and strain gradient theory were solved by generalized differential quadrature rule (GDQR), which has a faster convergence than generalized differential quadrature method (GDQ). The value of the size parameter for each diameter was calculated using the molecular dynamics results. By changing the phase from alpha to beta and beta to gamma, value of stiffness is increased. Therefore, for a smaller size parameter, continuum mechanics results match the molecular dynamics results. The results show that the size effect parameter is not a fixed value and has different values in different conditions. For all three phases and all diameters, the calculated frequencies are greater than those from classical theory. This result shows that the graphyne structure is fantastic and peculiar.

A Variable Kinematic Multifield Model for the Lamb Wave Propagation Analysis in Smart Panels.

The present paper assessed the use of variable kinematic two-dimensional elements in the dynamic analysis of Lamb waves propagation in an isotropic plate with piezo-patches. The multi-field finite element model used in this work was based on the Carrera Unified Formulation which offers a versatile application enabling the model to apply the desired order theory. The used variable kinematic model allowed for the kinematic model to vary in space, thereby providing the possibility to implement a classical plate model in collaboration with a refined kinematic model in selected areas where higher order kinematics are needed. The propagation of the symmetric () and the antisymmetric () fundamental lamb waves in an isotropic strip was considered in both mechanical and piezo-elastic plate models. The convergence of the models was discussed for different kinematics approaches, under different mesh refinement, and under different time steps. The results were compared to the exact solution proposed in the literature in order to assess and further determine the effects of the different parameters used when dynamically modeling a Lamb wave propagating in such material. It was shown that the higher order kinematic models delivered a higher accuracy of the propagating wave evaluated using the corresponding Time Of Flight (TOF). Upon using the appropriate mesh refinement of 2000 elements and sufficient time steps of 4000 steps, the error between the TOF obtained analytically and numerically using a high order kinematics was found to be less than 1% for both types of fundamental Lamb waves and . Node-dependent kinematics models were also exploited in wave propagation to decrease the computational cost and to study their effect on the accuracy of the obtained results. The obtained results show, in both the mechanical and the piezo-electric models, that a reduction in the computational cost of up to 50% can be easily attained using such models while maintaining an error inferior to 1%.

Vibration analysis of a strain gradient plate model via a mesh-free moving Kriging Interpolation Method

The strain gradient elasticity theory is adopted to capture the small-scale effect of micro/nanostructures to derive the governing equation by using Hamilton's principle. A mesh-free method on the basis of moving Kriging interpolation, of which shape function possesses C2 continuum, is developed to investigate the vibration of the strain gradient plate. Owing to the inherent Kronecker delta function property of the moving Kriging interpolation, essential boundary conditions can be directly implemented. One superiority of the moving Kriging interpolation over the traditional moving least square approximation is that it can ensure the continuity and stability at the third order derivatives, which is confirmed by comparison with analytic solutions. Parametric studies in terms of small-scale parameters, boundary conditions and side length on the frequencies and mode shapes are carried out. Natural frequencies evaluated by strain gradient elasticity theory are found to be lower than those of classical plate model. Some lower natural frequencies corresponding to ultrahigh-order mode shapes appear. This demonstrates that the mesh-free moving Kriging interpolation method has high precision and good stability in the construction of the C2 continuum shape functions.

Triangular Functionally Graded Porous Moderately Thick Plates—Deformations and Free Vibrations

Since the finite element analysis of deformations for plates made of functionally graded materials cannot be carried out with the use of commercial FE packages, aconsistent method of analytical analysis is proposed in the paper. The method of the analysis is based on the application of the weighted residuals and the Bubnov–Galerkin method. The 2D formulation of moderately thick plates is adopted herein for classical and transverse shear deformation plate models (first and third order). Plate deformations subjected to uniform normalpressure and free vibrations are considered. The validity of the analytical model was verified by the comparison of results with FE analysis for isotropic plates. Two correction multipliers were proposed in order to take into account the unsymmetric composition of functionally graded porous plate walls.

Optimum design of two-material bending plate compliant devices

PurposeThe purpose of this study is to explore the optimum design of bending plate compliant mechanisms subjected to pure mechanical excitations using topological-derivative-based topology optimization. The main objective is to design the reinforcement in a plate of base material.Design/methodology/approachThe optimum design is performed by means of a level-set representation method guided by topological derivatives. Kirchhoff and Reissner–Mindlin models are used to solve the linear bending plate problem. A qualitative comparison has been carried out between the optimal obtained topologies for each model.FindingsThe proposed methodology was able to design reinforcement in a plate of the base material. The obtained reinforcements notably improve the device’s behavior. The shape and topology of the reinforcements vary depending on the mechanical plate model considered. In fact, in the Reissner–Mindlin solutions, very thin flexo-torsional hinges connecting big zones of the reinforcement material are designed.Originality/valueUp to date, the synthesis of ortho-planar mechanisms by means of continuum topology optimization was only boarded within a multi-physics context. In this work, the optimal design of pure ortho-planar compliance actuators is addressed. The best performance is found by analyzing the results for two classical mechanical plate models.

Computing wrinkling and restabilization of stretched sheets based on a consistent finite-strain plate theory

It was reported both in experiments and computations using some classical plate theories that wrinkles can appear in a uniaxially stretched rectangular hyperelastic film with clamped–clamped boundaries and can be suppressed upon further tension. Here, based on a recently-available consistent finite-strain plate theory, we investigate this complex instability problem with isola-center bifurcation (the nontrivial solution curve begins and ends at two distinct points on the trivial line) in more depth and present an efficient numerical algorithm. An advantage of this plate theory, beside its applicability to finite-strain problems, is its asymptotic consistency with the 3D field equations and surface traction conditions in a pointwise manner. Efficiency of the numerical algorithm and this consistency advantage of this plate model are examined for both Saint-Venant–Kirchhoff and incompressible neo-Hookean materials by comparing results using classical plate models. Moreover, a quantitative explanation of restabilization response is provided by exploring the difference of uniformly stretched sheets and ones with clamped–clamped boundaries. Effects of geometric and material parameters on the applicability of those classical plate models are carefully examined, and it is found that some of them may produce quantitative deviations in some situations.

Analysis of Boundary Layer Effects Due to Usual Boundary Conditions or Geometrical Defects in Elastic Plates Under Bending: An Improvement of the Love-Kirchhoff Model

We propose a model of flexural elastic plates accounting for boundary layer effects due to the most usual boundary conditions or to geometrical defects, constructed via matched asymptotic expansions. In particular, considering a rectangular plate clamped at two opposite edges while the other two are free, we derive the effective boundary conditions or effective transmission conditions that the two first terms of the outer expansion must satisfy. The new boundary value problems thus obtained are studied and compared with the classical Love-Kirchhoff plate model. Two examples of application illustrate the results.

Development of a track dynamics model using Mindlin plate theory and its application to coupled vehicle-floating slab track systems

Unlike track slabs in high speed railways, floating slabs in metro lines are usually relatively thicker and their thickness can often reach 0.5 m or more. When the thickness of the track slab is relatively large compared to the length and width, the influence of shear effect and rotatory inertia will become significant and it is more reliable to regard the track slab as an elastic thick plate than adopt the classical thin plate model in the simulation. This paper presents a three-dimensional dynamic model for coupled vehicle-floating slab track (CVFST) systems on the basis of Mindlin plate theory for the first time. The floating slab is described as an elastic rectangle Mindlin plate with free boundary conditions resting on steel springs, the mode shapes of the Mindlin plate are approximated by series of products of Timoshenko beam functions, and the corresponding vibration equations are solved by Rayleigh-Ritz method in time domain. The vehicle subsystem and track subsystem are coupled via wheel-rail nonlinear interactions. The effectiveness of the proposed model is validated by comparing with the measured data from an impact test and numerical results in the previous literature. Influence of size effect on the slab natural frequencies and dynamic responses of the CVFST system are investigated, and discrepancies of the calculation results between the developed model and the traditional thin plate model are simultaneously discussed. Some practical conclusions are drawn and the developed model may serve as a potent tool for more accurate assessment of train-induced vibrations in metro lines.

Vibration of functionally graded Mindlin plate based on a modified strain gradient elasticity theory

Vibrational behavior of functionally graded (FG) microplates is investigated by a new modified strain gradient Mindlin plate (MSGMP) model. With the help of Hamilton’s principle, the dynamic equation is easily obtained. Furthermore, the general forms of boundary conditions are gotten by using coordinate transformation. The MSGMP model can be degenerated to a couple stress elastic Mindlin plate model or the classical Mindlin plate (CMP) model. Analytical solutions of vibrational problem of a rectangular microplate with four simply supported edges are gotten. Numerical results reveal significant effects of the dimensionless nonlocal parameters, the power law index and vibration mode on the free vibration behavior of FG plate.

Higher-order structural theories for transient analysis of multi-mode Lamb waves with applications to damage detection

Addressing the need for more computationally efficient and accurate simulation tools for Lamb wave-based structural health monitoring (SHM) systems, this paper proposes the use of refined theories for the study of the wave propagation phenomena in thin-walled structures. When dealing with complex structures, classical plate models can be insufficiently accurate whereas 3D finite elements soon reach their limits due to the high computational requirements in ultrasonic wave analysis. As a solution to these issues, a finite element framework based on the Carrera's unified formulation (CUF) is presented for the simulation of Lamb waves in continua. Different structural theories are discussed and tested, showing that the Lamb modes can be well represented using beam elements with enriched kinematics. The validity and convergence rates of the proposed models are assessed through benchmark solutions. An example of damage detection in a typical reinforced panel is included to show the potential of the method.

Two-scale micropolar plate model for web-core sandwich panels

A 2-D micropolar equivalent single-layer (ESL), first-order shear deformation (FSDT) plate model for 3-D web-core sandwich panels is developed. First, a 3-D web-core unit cell is modeled by classical shell finite elements. A discrete-to-continuum transformation is applied to the microscale unit cell and its strain and kinetic energy densities are expressed in terms of the macroscale 2-D plate kinematics. The hyperelastic constitutive relations and the equations of motion (via Hamilton’s principle) for the plate are derived by assuming energy equivalence between the 3-D unit cell and the 2-D plate. The Navier solution is developed for the 2-D micropolar ESL-FSDT plate model to study the bending, buckling, and free vibration of simply-supported web-core sandwich panels. In a line load bending problem, a 2-D classical ESL-FSDT plate model yields displacement errors of 34–175% for face sheet thicknesses of 2–10 mm compared to a 3-D FE solution, whereas the 2-D micropolar model gives only small errors of 2.7–3.4% as it can emulate the 3-D deformations better through non-classical antisymmetric shear behavior and local bending and twisting.

A cell lines derived microfluidic liver model for investigation of hepatotoxicity induced by drug-drug interaction.

The poor metabolic ability of cell lines fails to meet the requirements of an in vitro model for drug interaction testing which is crucial for the development or clinical application of drugs. Herein, we describe a liver sinusoid-on-a-chip device composed of four kinds of transformed cell lines (HepG2 cells, LX-2 cells, EAhy926 cells, and U937 cells) that were ordered in a physiological distribution with artificial liver blood flow and biliary efflux flowing in the opposite direction. This microfluidic device applied three-dimensional culturing of HepG2 cells with high density (107 ml-1), forming a tightly connected monolayer of EAhy926 cells and achieving the active transport of drugs in HepG2 cells. Results showed that the device maintained synthetic and secretory functions, preserved cytochrome P450 1A1/2 and uridine diphosphate glucuronyltransferase enzymatic activities, as well as sensitivity of drug metabolism. The cell lines derived device enables the investigation of a drug-drug interaction study. We used it to test the hepatotoxicity of acetaminophen and the following combinations: "acetaminophen + rifampicin," "acetaminophen + omeprazole," and "acetaminophen + ciprofloxacin." The variations in hepatotoxicity of the combinations compared to acetaminophen alone, which is not found in a 96-well plate model, in the device were -17.15%, 14.88%, and -19.74%. In addition, this result was similar to the one tested by the classical primary hepatocyte plate model (-13.22%, 13.51%, and -15.81%). Thus, this cell lines derived liver model provides an alternative to investigate drug hepatotoxicity, drug-drug interaction.

On the local and non-local plate models of single layer graphene

In the application of continuum plate theories for modeling the mechanics of single layer graphene, significant inconsistencies exist in the literature on how to deal with the discrepancy between the continuum assumptions and discrete nature of the graphene lattice, whether using the classical plate theories or the more recent non-local elasticity based plate models. Physically vague properties such as graphene thickness and non-local parameter are often used as fitting parameters for calibrating the numerical results against those obtained from atomistic models or experimental data. However, very often a set of parameters that work well for a particular case can lead to large errors in other cases. This work attempts to answer the following question: can the mechanical behavior of single layer graphene sheets be described accurately and consistently using a continuum plate model? By using the results from atomic lattice mechanics calculations as reference for error analysis, we demonstrate that, when the sources of error in the classical plate model are properly identified, mechanics of single layer graphene can be accurately described by a modified non-local plate model with an interatomic potential based constitutive relation and a corrected boundary configuration.

Assessment of binary PSP technique for film cooling effectiveness measurement on nozzle vane cascade with cutback trailing edge

The present paper shows the results of an experimental investigation aiming to assess advantages and disadvantages of binary Pressure Sensitive Paints (PSP) technique for the measurement of film cooling effectiveness distribution. For this purpose, two test cases pertaining to gas turbine high pressure nozzle vane cooling systems have been used: a classical flat plate model with coolant injection through a single large scale shaped hole and a high loaded nozzle vane cascade with a pressure side cooling system that features two rows of cylindrical holes followed by a pressure side cutback with film cooling slots. Both models have been extensively tested: in particular, film cooling effectiveness data measured by means of well established thermal measurement techniques (Infrared thermocamera for the flat plate and wide banded Thermocromic Liquid Crystals for the vane cascade) are available from the literature. Pressure Sensitive Paints technique was able to provide high quality data all over the investigated test cases and operating conditions. Moreover, the binary version allows to fully eliminate any spurious temperature influence, thus making this technique well suited for the analysis of complex flows and geometries.

Derivation and verification of a lattice model for bending vibration of a plate

AbstractLattice models are successfully used in modelling of fracture of brittle materials. To date, most of the lattice multi‐dimensional (2D and 3D) models known to the authors describe either in‐plane or three‐dimensional mechanics of the materials. Only a few lattice models are available in the literature for the description of the out‐of‐plane mechanics of plates. However, the parameters of those lattice models have not been linked to those of the classical plate models, such as Mindlin‐Reissner plate theory, based on the classical continuum theory. Nor has the dynamic behaviour of the out‐of‐plane lattice models been compared to that of the classical plates. This gap is closed in this paper by means of developing a lattice model that reproduces the out‐of‐plane dynamics of a shear‐deformable plate in the low frequency band. The developed model can be applied in various fields of engineering. In this paper, however, it is discussed taking example of an ice sheet. The linear dynamics of the model is focused upon in this paper in order to show its consistency with a continuum plate theory in the low frequency band and underline the differences emerging at higher frequencies. The developed model is composed of masses and springs whose morphology and properties were derived to match the out‐of‐plane deformations of thick plates as described by the Mindlin‐Reissner theory. Bending, shear and torsion are taken into account. The eigenfrequencies and the steady‐state response of the model to a sinusoidal in time point load are computed and compared to those of a corresponding continuum plate. It is proven that the developed lattice predicts the same dynamic behaviour as the corresponding continuum plate at relatively low frequencies. At higher frequencies deviations occur. These are discussed in this paper in terms of the dispersion, anisotropy and specific boundary effects of the lattice model.

A posteriori analysis of classical plate elements

We outline the results of our recent article on the a posteriori error analysis of C1 finite elements for the classical Kirchhoff plate model with general boundary conditions. Numerical examples are given.