Mindlin First-order Shear Deformation Research Articles

A modified first-order plate theory of laminated piezoelectric plate actuators

Piezoelectric actuators with large output and high reliability are widely applied in electron device. In this paper, a modified first-order plate theory of laminated piezoelectric plates with upper and lower electrodes is presented. Then, based on the Mindlin first-order shear deformation theory, a quadratic function of electric potential across the thickness is used to describe the inverse piezoelectric effect. The laminated plate governing equations are derived in terms of the resultant force, moment and electric displacement. Further, supplementary equations are introduced through the continuity of electrical potential and the normal electric displacement across interface of each lamina, which are applied to solve the unknown electric functions in each lamina. The numerical results of displacement, deflection and electric potential are consistent with those obtained by the three-dimensional finite element method. Finally, with such a modified plate theory, regulation mechanisms of polarization direction and layer numbers are investigated on actuation properties of laminated piezoelectric actuators. It is shown that the polarization direction can change the driving displacement by an order of magnitude. It is expected that the modified theory can provide a more efficient and accurate approach for analysis of multilayer piezoelectric composite actuators.

Equilibrium and displacement elements for the design of plates and shells

This paper reconsiders the finite-element modelling of the linear elastic behaviour of plates and shells as governed by the Reissner–Mindlin first-order shear deformation theory. Particular attention is given to the problems associated with the locking of thin forms of structure when modelled with isoparametric conforming elements. As a means of ameliorating or removing these problems, three recent alternative types of elements are studied. Two are displacement elements which include different approaches to the definition of assumed strains, and the third is based on a hybrid equilibrium formulation of a flat shell element. The purpose of the paper is to compare and explain their performances and outputs in the context of two benchmark problems: a trapezoidal plate and the Scordelis−Lo cylindrical shell. Numerical examples are used to illustrate the convergence of stress-resultant contours as well as global quantities such as strain energy. The main conclusion is that while all three alternative types of element overcome locking with regard to displacements, the hybrid models are generally more efficient at providing good quality stress resultants. This is particularly so for those which contribute little to the total strain energy but yet may be significant in design.

Non-Newtonian fluid–structure interactions: Static response of a microchannel due to internal flow of a power-law fluid

We study fluid–structure interactions (FSIs) in a long and shallow microchannel, conveying a non-Newtonian fluid, at steady state. The microchannel has a linearly elastic and compliant top wall, while its three other walls are rigid. The fluid flowing inside the microchannel has a shear-dependent viscosity described by the power-law rheological model. We employ lubrication theory to solve for the flow problem inside the long and shallow microchannel. For the structural problem, we employ two plate theories, namely Kirchhoff–Love theory of thin plates and Reissner–Mindlin first-order shear deformation theory. The hydrodynamic pressure couples the flow and deformation problem by acting as a distributed load onto the soft top wall. Within our perturbative (lubrication theory) approach, we determine the relationship between the flow rate and the pressure gradient, which is a nonlinear first-order ordinary differential equation for the pressure. From the solution of this differential equation, all other quantities of interest in non-Newtonian microchannel FSIs follow. Through illustrative examples, we show the effect of FSI coupling strength and the plate thickness on the pressure drop across the microchannel. Through direct numerical simulation of non-Newtonian microchannel FSIs using commercial computational engineering tools, we benchmark the prediction from our mathematical theory for the flow rate–pressure drop relation and the structural deformation profile of the top wall. In doing so, we also establish the limits of applicability of our perturbative theory.

A continuum theory of through–the–thickness jacketed shells for the elasto-plastic analysis of confined composite structures: Theory and numerical assessment

Abstract The paper proposes a generalized shell formulation devised for the triaxial stress analysis of Through-the-Thickness (TT) confining mechanisms induced by TT Jacketing (TTJ) devices in laminated composite structures, such as masonry walls retrofitted by stirrups-tied FRP sheets and TT jacketed concrete sandwich panels. Assuming a smeared description of TT reinforcements, the proposed shell formulation is constructed as an enhancement of the classical laminated shell formulation based on the Equivalent Single Layer Mindlin First-order Shear Deformation Theory (ESL-FSDT). This enhancement captures TT stretching by adding the TT displacement field among the kinematic variables and permits to describe the smeared TTJ interaction between transverse uniaxial reinforcements and confined layers in terms of continuum equilibrium and compatibility equations. Statics and kinematics of the shell are developed by following standard work-association arguments and encompassing both TT-laminated and TT-functionally graded structures. A nonlinear elasto-plastic constitutive behavior of the core material and of the TT reinforcements is considered and explicit representations of the elasto-plastic tangent operator are derived. The TTJ formulation is combined with a MITC finite element formulation and implemented in the research FE code Opensees. Results of nonlinear structural analyses of walls subject to in-plane and out-of-plane bending show that the proposed TTJ approach provides physically meaningful predictions of the structural response and is capable to efficiently track a complex triaxial confining interaction which ultimately results into marked global structural effects of increased stiffness, strength and ductility.

Geometrically nonlinear static FE-simulation of multilayered magneto-electro-elastic composite structures

A fully geometrically nonlinear finite rotation shell element for static analysis of layered magneto-electro-elastic (MEE) coupled composite structures is proposed. Reissner–Mindlin first-order shear deformation (FOSD) theory with full geometrically nonlinear strain–displacement relations and finite rotations is used to obtain the variational formulation. The scalar electric and magnetic fields are assumed along with the quasi-static behavior of MEE layers. Electric and magnetic potentials are assumed to vary quadratically in the transverse direction. Three linear constitutive relations are used describe the magneto-electro-elastic coupling. The magneto-electro-elastic composite four node shell element behavior is refined by embodying an assumed natural strain (ANS) method for transverse shear strains, and an enhanced assumed strain (EAS) method for in-plane bending strains. The developed finite element model is deployed for static analysis of layered MEE structures in sensor and actuator configurations, and results are compared with the available references. Additionally, several numerical examples are simulated to show the potentiality and predictive capabilities of the proposed finite element method.

Finite rotation FE-simulation and active vibration control of smart composite laminated structures

The present article focuses on the nonlinear finite element simulation and control of large amplitude vibrations of smart piezolaminated composite structures. Full geometrically nonlinear finite rotation strain---displacement relations and Reissner---Mindlin first-order shear deformation hypothesis to include the transverse shear effects are considered to derive the variational formulation. A quadratic variation of electric potential is assumed in transverse direction. An assumed natural strain method for the shear strains, an enhanced assumed strain method for the membrane strains and an enhanced assumed gradient method for the electric field is incorporated to improve the behavior of a four-node shell element. Numerical simulations presented in this article show the accurate prediction capabilities of the proposed method, especially for structures undergoing finite deformations and rotations, in comparison to the results obtained by simplified nonlinear models available in references and also with those obtained by using the C3D20RE solid element for piezoelectric layers in the Abaqus code.

Static and dynamic finite rotation FE-analysis of thin-walled structures with piezoelectric sensor and actuator patches or layers

This paper deals with static and dynamic analysis of thin-walled structures with integrated piezoelectric layers as sensors and actuators in the geometrically nonlinear range of deformations. A variational formulation is derived by using the Reissner–Mindlin first-order shear deformation (FOSD) hypothesis and full geometrically nonlinear strain-displacement relations accounting for finite rotations. The finite rotations are treated by Rodriguez parameterization. In order to enhance the accuracy of a four-node shell element, a combination of an assumed natural strain (ANS) method for the shear strains, an enhanced assumed strain (EAS) method for the membrane strains and an enhanced assumed gradient (EAG) method for the electric field are employed. The present shell element has five mechanical degrees of freedom (DOFs) and three electrical DOFs per node. The Newton–Raphson method for static analysis and the Newmark method for dynamic analysis are used to perform linear and nonlinear simulations. In comparison to the results obtained by simplified nonlinear models reported in the existing literature, the finite-element simulations performed in this paper show the importance of the present model, precisely for structures undergoing finite deformations and rotations.

Accurate free vibration analysis of thick laminated circular plates with attached rigid core

This paper deals with the free vibration behavior of laminated transversely isotropic circular plates with axisymmetric rigid core attached at the center. The governing equations of motion are obtained based on Mindlin's first-order shear deformation plate theory. Two possible categories of vibration modes related to up-down translation of the core and wobbly rotation of the core about a diameter are studied. Accurate natural frequencies hitherto not reported in the literature are presented for a wide range of thickness-to-radius ratio, inner-to-outer radius ratio, mass and moment of inertia ratios of the core and various boundary conditions at the outer edge of the plate. Numerical results are compared with those of a three-dimensional finite element method (3-D FEM) to demonstrate the high accuracy and reliability of the current analysis.

Nonlinear free vibration of functionally graded shear deformable sector plates by a curved triangular p-element

Abstract A p -version of the finite element method based on a curved triangular p -element is developed and applied to nonlinear free vibration analysis of functionally graded sector plates. The material is assumed to be temperature dependent and graded in the thickness direction according to the power-law distribution in terms of volume fractions of the constituents. In the geometrically nonlinear formulation, the Von Karman assumptions with Mindlin first-order shear deformation theory are used. The shape functions are constructed from the shifted Legendre orthogonal polynomials. The curved edge of the sector plate is represented accurately using the blending function method. The nonlinear equation of motion is obtained using the harmonic balance method and solved iteratively using the linearized updated mode technique. The linear and nonlinear frequencies are calculated for a functionally graded SUS304/Si 3 N 4 clamped circular plate. The accuracy of the proposed method is demonstrated through convergence and comparison studies. Sector plates made out of three types of functionally graded materials (SUS304/Si 3 N 4 , AL/AL 2 O 3 , AL/ZrO 2 ) are considered. The effects of sector angle, thickness, and volume fraction exponent on the hardening behavior of a clamped sector plate are also investigated. It is shown that the increase or decrease of the hardening behavior depends upon these parameters.

Aeroelastic behavior of cantilevered rotating rectangular plates

Aeroelastic behavior of a supersonic rotating rectangular plate in the air medium is studied. For simulating the plate structure, the Mindlin first-order shear deformation plate theory along with Von Karman nonlinear terms is employed. Air dynamic pressure is modeled using first-order piston theory. The plate is placed inside a rigid baffle to remove shock waves generated by plate rotation and provide a uniform flow passing over the plate. Nonlinear dimensionless generalized equations of motion are presented based on the Kane dynamic method. After linearization of the nonlinear equations of motion, effect of different parameters including plate aspect ratio, thickness ratio, hub radius ratio and dimensionless rotation speed on aeroelastic behavior of the system are investigated. Frequency locking and high frequency flutter phenomenon are observed in the numerical results.

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.

DSC-element method for free vibration analysis of rectangular Mindlin plates

A novel DSC-element method is proposed to investigate the free vibration of moderately thick plates based on the well-known Mindlin first-order shear deformation plate theory. The development of the present approach not only employs the concept of finite element method, but also implements the discrete singular convolution (DSC) delta type wavelet kernel for the transverse vibration analysis. This numerical algorithm is allowed dividing the domain of Mindlin plates into a number of small discrete rectangular elements. As compared with the global numerical techniques i.e. the DSC–Ritz method, the flexibility is increased to treat complex boundary constraints. For validation, a series of numerical experiments for different meshes of Mindlin plates with assorted combinations of edge supports, plate thickness and aspect ratios is carried out. The established natural frequencies are directly compared and discussed with those reported by using the finite element and other numerical and analytical methods from the open literature.

NUMERICAL SOLUTIONS TO THE FREE VIBRATION PROBLEM OF MINDLIN SECTOR PLATES USING THE DISCRETE SINGULAR CONVOLUTION METHOD

This paper presents the free vibration analysis of annular sector plates based on Mindlin's first-order shear deformation theory using the discrete singular convolution (DSC) method. In the solution process, the governing equations of the motions and boundary conditions of the plate are discretized by the method. The effect of some geometric parameters on frequencies is investigated. Comparisons are made with existing numerical and analytical solutions in the literature. It is found that the DSC method yields accurate results for the plate problems under investigation.

Accurate natural frequencies of transversely isotropic moderately thick annular sector plates

This article introduces an analytical method for free vibration analysis of transversely isotropic moderately thick annular sector plates. The plate has simply supported radial edges and arbitrary conditions along the circular edges. The displacement field of Mindlin's first-order shear deformation plate theory is considered and, based on this, the governing equations of motion are obtained. By using a function, which will be referred to as the boundary layer function, the three coupled governing equations of motion have been converted into two uncoupled equations. The general solutions of these equations contain integer and non-integer order Bessel and/or modified Bessel functions of the first and second kinds. By applying the boundary conditions at the inner and outer radii, an eigenvalue problem for finding the natural frequencies is obtained. The nine distinct cases considered involve all possible combinations of boundary conditions along the circular edges. An accurate non-dimensional frequency parameter is presented for a wide range of sector angles, some thickness—radius ratios and different inner-to-outer radius ratios. Three mode shapes of an annular sector plate with a sector angle of 60° are also presented. Finally, the effects of sector angle, thickness—radius ratio, boundary conditions, and inner-to-outer radius ratio on the frequency parameter are discussed.

Dynamic responses of Reissner–Mindlin plates with free edges resting on tensionless elastic foundations

Dynamic response analysis is presented for a Reissner–Mindlin plate with four free edges resting on a tensionless elastic foundation of the Winkler-type and Pasternak-type. The mechanical loads consist of transverse partially distributed impulsive loads and in-plane static edge loads while the temperature field is assumed to exhibit a linear variation through the thickness of the plate. The material properties are assumed to be independent of temperature. The two cases of initially compressed plates and of initially heated plates are considered. The formulations are based on Reissner–Mindlin first-order shear deformation plate theory and include the plate–foundation interaction and thermal effects. A set of admissible functions is developed for the dynamic response analysis of moderately thick plates with four free edges. The Galerkin method, the Gauss–Legendre quadrature procedure and the Runge–Kutta technique are employed in conjunction with this set of admissible functions to determine the deflection-time and bending moment–time curves, as well as shape mode curves. An iterative scheme is developed to obtain numerical results without using any assumption on the shape of the contact region. The numerical illustrations concern moderately thick plates with four free edges resting on tensionless elastic foundations of the Winkler-type and Pasternak-type, from which results for conventional elastic foundations are obtained as comparators. The results confirm that the plate will have stronger dynamic behavior than its counterpart when it is supported by a tensionless elastic foundation.

Nonlinear dynamic analysis of composite laminated plates containing spatially oriented short fibers

The paper presents an analytical approach to examine the nonlinear dynamic responses of a laminated composite plate composed of spatially oriented short fibers in each layer of the composite. Using Mori–Tanaka mean field theory, the effective elastic moduli of each lamina are obtained explicitly as a function of the properties of the constituents, volume fraction, orientation angles, and fiber shape. The resulting moduli are further applied to analyze the nonlinear transient response of the laminated plate. The formulation is based on Mindlin first-order shear deformation theory and von-Karman nonlinear kinematics, and the methodology of the solution utilizes the fast converging finite double Chebyshev series. Houbolt time marching scheme and quadratic extrapolation technique are used for the temporal discretization and linearization, respectively. Numerical results are presented for laminated plates made of E-glass/Epoxy fiber reinforced composites.

DIFFERENTIAL QUADRATURE METHOD FOR VIBRATION ANALYSIS OF SHEAR DEFORMABLE ANNULAR SECTOR PLATES

This paper presents differential quadrature solutions for free vibration analysis of moderately thick annular sector plates based on the Mindlin first-order shear deformation theory. Numerical characteristics of the differential quadrature method are illustrated through solving selected annular sector plates with different boundary conditions, relative thickness ratios, inner-to-outer radius ratios and various sector angles. Parametric studies in terms of the vibration frequency parameters are thoroughly investigated.

Free vibration analysis of Mindlin sector plates: Numerical solutions by differential quadrature method

In this paper, the free vibration analysis of moderately thick sector plates based on Mindlin's first-order shear deformation theory is examined. The solutions to this problem are determined using the differential quadrature (DQ) method. For numerical evaluation, the DQ formulations for free vibration of the Mindlin sector plates in two-dimensional polar co-ordinate system are derived. To demonstrate the accuracy of the DQ method, convergence and comparison studies are carried out. In parametric study, the first eight frequency parameters are computed for sector plates with different boundary conditions, relative thickness ratios, and sector angles (30° ⩽ α ⩽ 360°). The effects of sector angle and relative thickness ratio on the frequency parameters are discussed.

On a consistent first-order shear-deformation theory for laminated plates

Abstract This paper systematically states the consistent first-order shear-deformation theory for laminated plates recently proposed by Qi and Knight. It assumes that only in an average sense does a straight line originally normal to the midplane remain straight and rotate relative to the normal of the midplane, and in a local sense a slight displacement perturbation around the average rotated line is also permitted after deformation. Since the curved line is very shallow, the present theory still approximates linear in-plane and constant transverse displacements through the thickness just as Reissner and Mindlin's first-order shear-deformation theory does. Reissner and Mindlin's theory leads to uniform transverse shear strain distributions by employing pointwise strain displacement relationships, and satisfies the transverse shear constitutive relationships only in an average corrected form. In contrast, Qi and Knight's theory accounts for variable transverse shear strain distributions by enforcing pointwise constitutive relationships, and relates transverse shear strains to kinematic unknowns only in a weighted-average form. Through-the-thickness transverse shear strains are thus consistent with the stress counterparts and their transverse-shear-stress-weighted-average values are just the nominal-uniform transverse shear strains which correspond to the average rotations. The new theory combines the advantages of several prevailing 2D laminated plate theories while overcoming their drawbacks. Numerical results for the cylindrical bending problem of orthotropic laminated plates exhibit excellent agreement between Qi and Knight's theory and Pagano's 3D exact elasticity results.

Benchmark vibration solutions for regular polygonal Mindlin plates

Benchmark vibration solutions for thick polygonal plates of arbitrary boundary conditions are presented. The Mindlin first-order shear deformation theory is used to derive the energy functional while the pb-2 Ritz method is employed for solution. By minimizing the difference between the maximum strain and kinetic energies with respect to unknown coefficients in the pb-2 admissible functions, the governing eigenvalue equation is derived. The vibration frequencies are obtained by solving this governing eigenvalue equation. A convergence study has been carried out to verify the accuracy of these results. In the absence of existing results for thick polygonal plates, the present results where possible are compared with available thin plate solutions by setting the thickness to apothem ratio to be very small (say, t/r=0.001). Sets of first known vibration frequencies for moderately thick regular polygonal plates are presented as an enhancement to existing literature.