Mindlin Plate Theory Research Articles

On the formulation and evaluation of old and new efficient low order triangular plate bending elements with shear effects

This paper deals with the presentation of a general, variational principle as theoretical support for the development of simple and efficient triangular elements having only one displacement and two rotations at the corner nodes to model thin to thick plates based on the first order Reissner- Mindlin plate theory. The functional is a modified Hellinger–Reissner mixed expression in terms of the kinematic variables (transverse displacement w and rotations βx and βy), and independent transverse shear strains (γx and γy). The approximations of the five independent variables of the mixed formulation take into account the accumulated knowledge on existing performing 2D Timoshenko beams and triangular elements such as T3γs, MITC3, DKT, DST, DKMT. The present mixed variational support is useful, not only to give a unique theoretical support to the above existing elements, usually based on assumed natural strain formulations, but it allows also to propose new simple and efficient elements, here called BAK1, BAK2 and BAK3. The paper includes a detailed presentation of results of patch tests for very thin and very thick plates, for convergence of displacements, bending moments and shear forces, for clamped circular and simply supported square plates and for s-norm convergence tests considering regular and irregular meshes. Shear force distribution is also considered for situations with boundary layer effects.

Theoretical analysis of transverse and planar vibrations for the piezoceramic disk based on Mindlin plate theory

In this paper, the out-of-plane and in-plane displacements coupled behavior of the transverse and planar vibration characteristics for piezoceramic disks under fully-clamped and traction-free boundary conditions are investigated theoretically base on the Mindlin's and Kirchhoff's plate models. By varying the radius-to-thickness ratios, the piezoceramic disks with moderate and thin thickness are considered in the analysis. It is verified that Mindlin's plate theory with first-order shear deformation hypothesis provides sufficient accuracy for transverse vibration analysis in comparison with Kirchhoff's model; however, this circumstance does not occur for planar vibration analysis. With the aid of theoretical analysis, both the resonant frequency and the corresponding mode shape are obtained for piezoceramic disks with various radius-to-thickness ratios. In addition, maximum normalized displacement components in three directions are also explored to identify the coupled vibration modes. Numerical calculations using the finite element method (FEM) are performed and the results are compared with the theoretical analysis. Excellent consistence between the theoretical and numerical results are found for the three-dimensional coupled vibration characteristics of piezoceramic disks.

Prediction of local resonance band gaps in 2D elastic metamaterials via Bloch mode identification

This paper investigates methods of band gap formation in two-dimensional locally resonant elastic metamaterials and proposes a technique for computing such gaps as algebraic functions of integrated Bloch modes and metamaterial parameters. Mode shapes associated with the deformation of unit cells consisting of a hard matrix, soft filler, and hard resonator are investigated near the lower and upper bounds of the first local resonance band gap. Similarities reveal that Bloch modes at these bounds have the same non-dimensional geometries and locations within the irreducible Brillouin Zone irrespective of material properties, matrix and resonator thicknesses, and unit cell size. A theoretical model for the unit cell incorporating Mindlin plate theory is employed to compute the frequency of these modes in transverse vibration using conservation of energy. A least squares system identification algorithm is then described to generate surface equations for non-dimensional Bloch mode shapes computed by finite element analysis, employing a computation-saving Kronecker factorization and yielding continuous surface equations for the non-dimensional Bloch modes when rectangular cell components are utilized. Nine sample metamaterials are analyzed with this approach generating reliable predictions of the first band gap bounds. The presented framework offers insights into band gap formation, opens avenues in inverse unit cell design, and provides versatile algebraic relationships between band gap frequencies and unit cell parameters in such resonant metamaterials.

Development of an analytical framework for viscoelastic corrugated-core sandwich plates and validation against FEM

In this study, an analytical procedure for the bending problem of a viscoelastic sandwich plate with a corrugated core is presented. Reissner–Mindlin plate theory and N-termed Prony series are employed to define the elastic and time-dependent contributions of the governing equations, respectively. Three different corrugation patterns, i.e., rectangular, trapezoidal, and triangular, are examined. Moreover, the structure is analyzed under both simply support and clamp boundary conditions. The calibrated material parameters of polymethyl methacrylate (PMMA) for the Generalized Maxwell rheological model are employed to show the viscoelastic response of the structure. A 3D finite element simulation of the problem is also conducted to confirm the accuracy of the analytical formulation. The two well-known creep and stress relaxation phenomena of the viscoelastic materials are examined for the mentioned corrugation cores and both boundary conditions analytically and numerically. The time-dependent dimensionless deflection and resultant von Mises stress distributions are provided. Besides, the variation of the results with various rise-times and applied load are studied in detail. The von Mises stress contours of the upper surface of the structure at the end of the creep test are also presented. The finite element method outcomes verify the analytical results with excellent compatibility. The proposed analytical procedure can be used as an efficient tool to study the effects of various parameters such as material, geometrical constants, and corrugation pattern on bending of viscoelastic sandwich plates with corrugated core problems for design and optimization, which involves a high number of simulations.

Transverse Free Vibration of Axisymmetric Functionally Graded Circular Nanoplates with Radial Loads

The functionally graded circular nanoplate is a commonly seen component in the nano-electromechanical system. It is indispensable to examine free vibration behaviors of such an axisymmetric nanostructure subjected to uniformly distributed loads in the radial direction. Although the vibration engineering and technology have been fully studied at the macro-scale, there are still many unsolved problems at the micro-scale. The present research aims to promote the theoretical characterization of vibration behaviors at micro/nano-scale and further provide a basis for the development of vibration testing technologies. Using the nonlocal strain gradient approach and Mindlin plate theory, we develop the theoretical model describing the transverse free vibration of the axisymmetric functionally graded circular nanoplate. First, by considering the nonlocal strain gradient constitutive relation, we derive the equation of motion via Hamilton’s principle in polar coordinate system. Subsequently, the differential quadrature method is employed to solve the equation of motion numerically. The natural frequencies of circular nanoplates decrease with an increase of a radial compression while increase with an increase of a radial tension. The first-mode natural frequency reduces to zero under a certain radial compression, bringing about the dynamical instability. The natural frequencies are sensitive to the radial compression, and the clamped boundary constraint is more resistant to external loads than the simply supported one. An increase in the nonlocal parameter results in lower natural frequencies, while an increase in the strain gradient characteristic parameter results in higher ones. It is demonstrated that the strain gradient characteristic parameter has a threshold in the present model for functionally graded circular nanoplates. In the circumstance of a lower nonlocal parameter than the strain gradient characteristic parameter, the circular nanoplate shows hardening behaviors. In the circumstance of a greater nonlocal parameter, the circular nanoplate shows softening behaviors. When the two internal characteristic parameters are equal, the stiffness of nanoplates remains unchanged and degenerates into its classical counterpart.

Multiple scattering of flexural waves on Mindlin plates with circular scatterers

AbstractThe multiple scattering of flexural waves on an elastic plate with circular scatterers is analyzed in the frequency domain based on the Mindlin plate theory accounting for the rotary inertia and shear deformation of the plate. To this purpose, a semi‐analytical numerical method is formulated as an extension of the previous study based on the Kirchhoff plate theory. It consists of expressing the flexural wave field in terms of the superposition of the wave function expansion, and determining the expansion coefficients by a collocation technique. As demonstrative examples, the transmission of a plane flexural wave across a square array of circular through‐thickness holes or thin‐plate inclusions is analyzed using the proposed method. The comparison between the results based on the Mindlin and Kirchhoff theories is shown for the case of multiple holes. The analysis shows that the transmission amplitude of the flexural wave is reduced at certain frequencies due to the Bragg reflection by the inclusions. In the case of thin‐plate inclusions, the resonance of the inclusions also brings about a sharp decrease of the transmission amplitude.

An advanced train-slab track spatially coupled dynamics model: Theoretical methodologies and numerical applications

Train-slab track spatial interactions generally deal with system vibrations in vertical, lateral and longitudinal directions. In the majority of conventional dynamics models, vertical-lateral coupling effects are of main concern, and great progress has also been achieved in the field of train longitudinal dynamics in recent years. However, for modelling strategies of track structures, longitudinal coupling effects are always neglected and lateral vibration of track slab is assumed as rigid behavior. To this end, the classical vehicle-track coupled dynamics theory is extended in this paper by thoroughly implementing the track flexibility and advanced coupling relations into a three-dimensional train-slab track system. Rail axial, torsional and bending vibrations, as well as in-plane and flexural vibrations of track slab based on Kirchhoff or Mindlin plate theory, are simultaneously taken into account. Eigenvalues and forced vibrations of the track slab for both in-plane and flexural vibrations are tackled by a set of powerful trigonometric series serving as the unified trial functions. Moreover, wheel-rail, rail-track slab and track slab-base interactions are partially re-deduced due to the participation of foregoing new degrees of freedom (DOFs). The reliability of proposed dynamics model has been verified by the previous literature, ANSYS software and co-simulation technique. Comparisons between the presented model (PM) and conventional model (CM) indicate that dynamic responses are sensitive to the longitudinal coupling effects and slab in-plane vibration which should be paid adequate attention for more accurate evaluations on train-slab track interactions.

Size-dependent vibration of laminated composite nanoplate with piezo-magnetic face sheets

Laminated composites are extensively employed in many aerospace structures due to their excellent mechanical properties. In this paper, a size-dependent model on the basis of the nonlocal strain gradient theory is adopted to reveal the vibration behavior of laminated nanoplate with piezo-magnetic face sheets in its upper and lower surfaces. The governing equations are derived by employing the Hamilton’s principle and Mindlin plate theory. The validation of the present study is carried out by comparison with two previous works and good agreements are achieved. By comparing the vibrational frequencies of composite laminated core sandwich nanoplate with and without the piezo-magnetic face sheets, it is demonstrated that the upper and lower piezo-magnetic face sheets will extensively enhance the vibrational frequencies of laminated core sandwich nanoplates. Furthermore, a comprehensive numerical investigation is performed to examine the influence of the cross-ply laminated type, external electric and magnetic potentials, thickness ratio, size scale parameters, as well as aspect and width-to-thickness ratios on the vibration of the laminated core piezo-magnetic sandwich nanoplate. It is expected that the current work can provide some helpful guidelines for employing the piezo-magnetic surfaces as sensors and actuators to control their vibration behaviors of composite laminated nanostructures.

Nonlinear finite element analysis within strain gradient elasticity: Reissner-Mindlin plate theory versus three-dimensional theory

Nonlinear plate bending within Mindlin's strain gradient elasticity theory (SGT) is investigated by employing somewhat non-standard finite element methods. The main goal is to compare the bending results provided by the geometrically nonlinear three-dimensional (3D) theory and the geometrically nonlinear Reissner–Mindlin plate theory, i.e., the first-order shear deformation plate theory (FSDT), within the SGT. For the 3D theory, the nonlinear Green–Lagrange strain relations are adopted, while the von Kármán nonlinear strains are employed for the FSDT. The matrix-vector forms of the energy functionals are derived for both models. In order to perform the corresponding finite element discretizations, a quasi-C1-continuous 4-node tetrahedral solid element and a quasi-C1-continuous 6-node triangular plate element are employed for the 3D model and plate model, respectively. The first-order derivatives of the primal problem quantities are employed as additional nodal values to respond to the continuity requirements of class C1. A variety of computational results highlighting the differences between the 3D and FSDT models are given for two different plate geometries: a rectangular plate with a circular hole and an elliptical plate.

The effect of buckling and post-buckling behavior of laminated composite plates with rotationally restrained and Pasternak foundation on stacking sequence optimization

This paper presents a stacking sequence optimization for maximizing the buckling load of rotationally restrained laminated composite rectangular plates with different boundary conditions resting on an elastic Pasternak foundation subjected to uniaxial and biaxial in-plane static loads. The Mindlin Plate Theory (MPT), which considers the first-order shear deformation effect, is used to extract the characteristic equations of the plates under in-plane loading, including plate-foundation interaction. The buckling problem of the laminated plates is analyzed by the Rayleigh–Ritz method. The aim of optimization is to maximize the buckling load and post-buckling load capacity by using the Genetic Algorithm (GA) method, and the design variable is the ply orientation. The results showed that the optimal orientation, θ, of the laminated square plate under biaxial in-plane loading with various conditions is 〖45〗^∘ approximately. The existence of a foundation, clamped boundary conditions, and high aspect ratio lead to increase the optimal orientation.

Flexural wave band gaps in a ternary periodic metamaterial plate using the plane wave expansion method

Low- and mid-frequency range vibrations usually present issues in their insulation. The development of periodic metamaterials (artificially engineered structures) has been thoroughly studied since their periodicity and geometric properties may yield good attenuation properties in such frequency ranges. Several works investigated periodic structures presenting binary or ternary material configurations using the Kirchhoff plate theory or binary material configurations using the Mindlin plate theory. In this paper, a ternary periodic metamaterials made of concentric circular inclusions of a rigid material coated with a soft material embedded in a matrix made of a low-cost recyclable polymer is investigated using the Mindlin plate theory combined with the plane wave expansion method. Computed band diagrams are compared to those obtained using the Kirchhoff plate theory and three-dimensional models. Additionally, the effect of different boundary conditions is investigated on finite structures using the finite element method. Our results assess the validity of the Mindlin plate model in such case and indicate its use in practical applications.

Isogeometric Collocation Method to solve the strong form equation of UI-RM Plate Theory

This work presents the formulation of the isogeometric collocation method to solve the strong form equation of a unified and integrated approach of Reissner Mindlin plate theory (UI-RM). In this plate theory model, the total displacement is expressed in terms of bending and shear displacements. Rotations, curvatures, and shear strains are represented as the first, the second, and the third derivatives of the bending displacement, respectively. The proposed formulation is free from shear locking in the Kirchhoff limit and is equally applicable to thin and thick plates. The displacement field is approximated using the B-splines functions, and the strong form equation of the fourth-order is solved using the collocation approach. The convergence properties and accuracy are demonstrated with square plate problems of thin and thick plates with different boundary conditions. Two approaches are used for convergence tests, e.g., increasing the polynomial degree (NELT = 1✕1 with p = 4, 5, 6, 7) and increasing the number of element (NELT = 1✕1, 2✕2, 3✕3, 4✕4 with p = 4) with the number of control variable (NCV) is used as a comparable equivalent variable. Compared with DKMQ element of a 64✕64 mesh as the reference for all L/h, the problem analysis with isogeometric collocation on UI-RM plate theory exhibits satisfying results.

Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

Based on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. Finally, the effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.

Optimal core design of MR based smart panels subject of flutter instability

This study deals with aeroelastic and flutter analyses of the partially treated MR sandwich plates in the supersonic airflow. It also identifies the optimal locations of the MR segments in the core layer, which results in maximum flutter boundaries of the MR sandwich plate, in response to the airflow. Classical and Mindlin plate theories, Hamilton’s principle and first order linear piston theory are employed to derive the governing equations of motion of the structure. The assumed mode technique is used to solve the equations and specify the instability bounds of the sandwich plate. An experiment on the dynamic characteristics of the partially treated sandwich plate incorporating MR fluid as the core layer is conducted to demonstrate validity of the developed model. Then, the effect of number and locations of the MR pockets on the flutter suppression of the sandwich plates is comprehensively investigated. Also, an optimization problem is formulated to identify the optimal locations of the MR treatments, which maximize flutter boundaries of the structure in different levels of the magnetic flux density.

Identification of Vibration Modes of Quartz Crystal Plates with Proportion of Strain and Kinetic Energies

For the design of quartz crystal resonators, finding and determining the vibration modes have always been very important and cumbersome. Vibration modes are usually identified through plotting displacement patterns of each coupled modes and making comparisons. Over the years, there is not much improvement in the identification procedure while tremendous efforts have been made in refining the equations of the Mindlin plate theory to obtain more accurate results, such as the adoption of the Finite Element Method (FEM) by implementing the high-order Mindlin plate equations for efficient analysis. However, due to the old fashioned mode identification method, the FEM application is still inadequate and cannot be fully automated for this purpose. To have this situation improved, a method using the proportions of strain and kinetic energies to characterize the energy level of each vibration mode is proposed. With solutions of displacements, the energy distribution of each vibration mode is calculated and the mode with the highest energy concentration at a specific frequency is designated as the dominant mode. The results have been validated with the traditional approach by plotting mode shapes at each frequency. Clearly, this energy approach will be advantageous with the FEM analysis for vibration mode identification automatically.

ANLYTICAL SOLUTION OF THERMOELASTIC DAMPING IN RECTANGULAR MINDLIN MICRO PLATES

Analytical solution of thermoelastic damping for rectangular Mindlin micro plate with the four edges simply supported is presented for the first time. Based on the Mindlin plate theory and the one-way coupled heat conduct theory, governing differential equations for thermo-elastically coupled free vibration of the micro plate are formulated. Ignoring the in-plane variation of the temperature gradient, analytical solution of temperature field in terms of the kinematic parameters is obtained under the adiabatic boundary conditions at the top and bottom surfaces. Furthermore, the equations of structural vibration including the thermal bending moment are transformed into a fourth-order partial differential equation only in terms of the deflection. By using the mathematical similarity between the eigenvalue problems under the simply supported boundary conditions, analytical solution of the complex natural frequency of the Mindlin plate is expressed in terms of the frequency of isothermal Kirchhoff plate. Then the inverse quality factor which represents the level of the thermoelastic damping is obtained. Finally, the effects of the shear deformation, the material and the geometry parameters on the thermoelastic damping are examined in detail by the numerical results. The numerical results show that thermoelastic damping estimated by Mindlin plate theory is less than that by Kirchhoff plate theory. The difference between the values evaluated by the two plate theories becomes very significant near the critical thickness. Moreover, along with the increase of the side-to-thickness aspect ratio,the maximum of the thermoelastic damping in micro Mindlin plate increase monotonically, however, that of micro Kirchhoff plate keeps constant.

A peridynamic Reissner‐Mindlin shell theory

SummaryIn this work, we have developed a state‐based peridynamics theory for nonlinear Reissener‐Mindlin shells to model and predict large deformation of shell structures with thick wall. The nonlocal peridynamic theory of solids offers an integral formulation that is an alternative to traditional local continuum mechanics models based on partial differential equations. This formulation is applicable for solving the material failure problems involved in discontinuous displacement fields. The governing equations of the state‐based peridynamic shell theory are derived based on the nonlocal balance laws by adopting the kinematic assumption of the Reissner and Mindlin plate and shell theories. In the numerical calculations, the stress points are employed to ensure the numerical stability. Several numerical examples are conducted to validate the nonlocal structure mechanics model and to verify the accuracy as well as the convergence of the proposed shell theory.

Analysis of Thermoelastic Damping of Functionally Graded Material Micro Plate Based on the Mindlin Plate Theory

In this paper, thermoelastic damping (TED) in a simply supported rectangular functionally graded material (FGM) micro plate with continuous variation of the material properties along the thickness direction is performed. The equations of motion and the heat conduction equation coupled with the thermal effects are derived based on the Mindlin plate theory and the one-way coupled heat conduction theory, respectively. The heat conduction equation with variable coefficients is solved by using the layer-wise homogenization approach. Analytical solution of TED is obtained by complex frequency method. Numerical results of TED are presented for the rectangular FGM micro plate made of ceramic-metal constituents with the power-law gradient profile. The effects of the shear deformation, the material gradient index, the plate thickness on the TED of the FGM micro plate are studied.

Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates.

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.

Dynamic solution for vehicle–track interaction considering the elastoplasticity of track slabs

In this work, an entire multi-rigid-body system with two-stage suspension systems is assumed to run on a slab track system, where the elastoplasticity of the track slab is modeled by Mindlin plate theory considered in a framework of three-dimensional vehicle–track dynamic system. Considering the elastoplasticity of track slabs, the nonlinearity and time- and stress-dependent track slab stiffness matrices must be accounted for in the time-domain dynamic solutions; therefore, a two-step iterative algorithm separating the vehicle–track system into vehicle–rail subsystem and track slab system is developed to solve the dynamic equations of motion for vehicle–track interactions. The numerical studies have shown the accuracy and efficiency of this method in calibrating the vehicle–track dynamic behavior in conditions of track slab elastoplastic deformation. Besides, it has shown that the longitudinal unevenness of the track support stiffness has significant influence on the plasticity of track slabs.