Reissner-Mindlin Plate Theory Research Articles

Free vibration of functionally graded graphene platelets reinforced porous composite plate with a diamond-shaped hole and two cracks

This paper investigates the free vibration of the functionally graded graphene platelets reinforced porous composite (FG-GPLRPC) plate with a diamond-shaped hole and two cracks. Based on the Mindlin-Reissner plate theory and Hamilton principle, the governing equations and boundary conditions are derived. These equations are discretized using the generalized differential quadrature finite element method (GDQFEM), and the natural frequency is obtained by numerical solutions. The effects of the hole size, crack length and crack angle on the natural frequency are discussed. The results show that the natural frequency decreases with increasing crack angle and crack length.

Forced vibration characteristics of viscoelastic variable stiffness laminated composite plates using time and frequency domain approaches

The linear forced vibration characteristics of viscoelastic variable stiffness laminated composite plates (VVSLC) are studied using a generalized Maxwell model and finite element method. The integral form of the viscoelastic constitutive relation is converted to the incremental form for finite element formulation based on the Reissner–Mindlin plate theory. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response directly in the time domain is obtained using shooting technique coupled with Newmark’s time integration method. The implementation of shooting technique for Boltzmann integral-based viscoelasticity for curvilinear fibre composite plates is done for the first time in this study. For the comparison purpose, the response/resonance frequency/modal loss factor is also obtained using equivalent complex modulus based viscoelastic correspondence principle. It is observed that the variation in fibre orientation and boundary conditions leads to significant variations in response, stress/moment resultant amplitude and the damping factor of the VVSLC plates. Further, the present time domain based approach is capable of predicting damping factor at all forcing frequencies whereas complex eigenvalue analysis can predict damping factor only at discrete resonance frequency. Based on the detailed studies, it is found that the curvilinear fibre composite plates depict a significant reduction in response/moment resultants compared to straight fibre composite plates.

Coupled effect of light incidence angle and deformation on the light-induced bending of liquid crystal elastomers plates

In this paper, the light-induced bending of a liquid crystal elastomers (LCEs) plate with the coupled effect of light incidence angle and deformation is studied for the first time. Based on the Reissner-Mindlin plate theory, a finite element model incorporating the coupled effect is developed to investigate the bending behavior of the LCEs plate. Deflections of the LCEs plate subjected to the light illumination with different light incidence angles are given. It is found that the coupled effect will significantly affect the bending deformation of the LCEs plate such as the location of the peak and the bending shape. Comparisons of the spontaneous bending of the LCEs plate with and without the coupled effect demonstrate the necessity of this study, especially for the LCEs plate subjected to the non-normal incidence light.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

The effects of shear deformation and rotary inertia on the electrical analogs of beams and plates for multimodal piezoelectric damping

AbstractAnalogous electrical networks were previously derived from the Euler–Bernoulli and Kirchhoff–Love theories to represent beams and plates, respectively, for use in multimodal structural vibration damping. However, these networks do not account for shear deformations or rotary inertia, which can result in suboptimal vibration damping performance when used on moderately thick beams and plates. In this paper, we investigate the incorporation of shear deformation and rotary inertia using Timoshenko–Ehrenfest beam theory and Mindlin–Reissner plate theory to develop improved electrical networks that can more accurately represent thick beams and plates. Our findings suggest that the inclusion of shear deformation and rotary inertia can significantly improve the frequency coherence of the electrical networks and multimodal vibration damping for thicker structures. The electrical analogs presented here are of use for various applications, especially to conveniently design complex circuit topologies in fields spanning from vibration attenuation to energy harvesting.

Free vibration of functionally graded graphene platelets reinforced composite porous L-shaped folded plate

This paper investigates the free vibration of a functionally graded graphene platelets reinforced composite (FG-GPLRC) porous L-shaped folded plate. According to the folded position, the L-shaped folded plate is divided into three rectangular plates. Based on Mindlin-Reissner plate theory and Hamilton’s principle, the governing equations for free vibration of each subelement are deduced. Utilizing the generalized differential quadrature element (GDQE) method, the governing equations for each subelement are discretized and numerically solved to obtain the natural frequency. The study focuses on discussing the influence of several factors, including porosity distribution, porosity coefficient, plate size, crank angle, boundary conditions, GPL weight fraction, and GPL pattern on the natural frequency. The findings reveal that the incorporation of GPL enhances the mechanical performance of the porous folded plate compared to a pure porous plate. Dispersion of more GPL near the upper and lower layers enhances the stiffness of the FG-GPLRC porous folded plate, leading to higher natural frequency. Additionally, variations in the crank angle have a significant impact on the natural frequency.

Shear locking free polygonal elements for the analysis of functionally graded plates using (n + 1) integration scheme and Reissner-Mindlin theory

A new shear-locking free polygonal elements is proposed for static bending and free vibration analysis of thin/thick functionally graded material plates. The domain is discretized with arbitrary polygons and Wachspress interpolants are employed for approximating the unknowns. The plate kinematics is based on Reissner-Mindlin plate theory and shear locking syndrome is suppressed by employing the novel n + 1 integration scheme based on Richardson extrapolation technique. Within this, a two stage sampling technique is introduced with the Weighted Richardson scheme to compute the system matrices. From the systematic study, it is inferred that the proposed scheme passes patch tests and yields accurate results.

Buckling analysis of functionally graded plates in thermal environment by using a meshfree method

In material, the pre-stress state caused by temperature variation of the working environment is one of the main reasons which cause buckling in structures. This paper aims to analyse the critical temperature rise to cause buckling in functionally graded material (FGM) under different boundary conditions. Reissner-Mindlin plate theory is employed in this paper with the radial point interpolation method (RPIM). The RPIM has the advantage compared to other meshfree methods in the satisfaction of the Kronecker delta property which allows directly imposing boundary conditions on scattered nodes. Factors affecting to thermal buckling phenomenon are investigated. The accuracy of solutions will be shown by comparing them to reliable results in previous studies

Damping optimization of viscoelastic thin structures, application and analysis

In this work, damping properties of bending viscoelastic thin structures are enhanced by topology optimization. Homogeneous linear viscoelastic plates are optimized and compared when modeled by either the Kirchhoff–Love or Reissner–Mindlin plate theories as well as by the bulk 3D viscoelastic constitutive equations. Mechanical equations are numerically solved by the finite element method and designs are represented by the level-set approach. High performance computing techniques allow to solve the transient viscoelastic problem for very thin 3D meshes, enabling a wider range of applications. The considered isotropic material is characterized by a generalized Maxwell model accounting for the viscoelasticity of both Young modulus and Poisson’s ratio. Numerical results show considerable design differences according to the chosen mechanical model, and highlights a counter-intuitive section shrinking phenomenon discussed at length. The final numerical example extends the problem to an actual shoe sole application, performing its damping optimization in an industrial context.

Analytical and numerical solution for multiple shape memory effect of smart corrugated-core sandwich panels with different patterns

Shape memory polymers (SMPS) are a class of smart materials used in various industries due to their unique properties. On the other hand, sandwich structures have attracted the attention of many researchers and industries due to their low weight and high strength. Therefore, this study it is aimed to present a semi-analytical solution to describe the behavior of a sandwich plate made of temperature-sensitive SMPs with a corrugated structure, based on the Reissner-Mindlin plate theory together with the viscoelastic theory. The solution is implemented for both dual shape memory (shape and force recovery) and triple shape memory scenarios. Three types of structure, including rectangular, triangular and trapezoidal, have been studied. In addition, in order to examine the impact of the support type, simply supported and clamped boundary conditions have also been investigated for three different geometries. Also, this work examines the impact of thermal stress caused by expansion. In addition to the semi-analytical solution, finite element modeling has also been carried out in all problems. Comparing the results in different geometries and supports shows the proposed semi-analytical solution's ability to predict the material's behavior with great accuracy. Moreover, by examining different cores and supports, one may observe a significant difference in the amount and distribution of the force and deformation, which are key to the design. Therefore, the semi-analytical solution can be considered reliable and accurate for various problems. In addition, compared to the finite element solution, the analytical solution enjoys a much higher speed; thus, it can be used for optimization and design problems that require a huge number of simulations.

Dynamic Analysis of Mindlin Plate Subjected to a Moving Mass-Spring-Damper System Using the Moving Element Method

The moving element method (MEM) is applied in this paper to study the dynamic behavior of the Mindlin plate subjected to a moving mass-spring-damper system. Reissner- Mindlin plate theory is utilized to study the plate structure. The plate is modeled by the Isoparametric quadrilateral nine-node element. The governing equation of motion is built into a coordinate system which moves with the moving mass-spring-damper system and is based on the principle of virtual work. The MEM is proposed to solve the governing equation of motion of the plate resting on a Pasternak foundation under the moving mass-spring-damper system including roughness of plate surface which is a dynamic source. Numerical results are investigated and verified by comparing with the published results. The effects of different parameters, such as spring stiffness, damping coefficient and velocity of the moving load system, plate thickness, foundation parameters and roughness of the plate surface on the dynamic response of the Mindlin plate are investigated.

Convergence studies for static analysis of thin plates on Pasternak Foundations

Convergence studies for the static analysis of thin plates resting on Pasternak foundations is performed. The plates are discretized using two different finite elements, the formulations of which are based on the Kirchhoff and Reissner-Mindlin plate theories. The shear locking problem which arises when full integration is used in the finite element implementation of Reissner-Mindlin plate theory is eliminated with selective integration. The Pasternak foundation is accounted for by adding the parameter matrices of an existing soil finite element to the stiffness matrix terms of the plate finite elements corresponding to deflections. Convergence rates for different boundary conditions, plate thicknesses and soil parameters are obtained and given comparatively through numerical examples.

Improving Felippa Bergan Triangular element by using UI approach for analysis of isotropic and FGM sandwich plates

This work presents the enhancement of Felippa Bergan Triangular (FBT) element from thin (Kirchhoff-Love plate theory) to thick plate (Reissner-Mindlin plate theory) using the unified and integrated approach. In FBT, the total transverse displacement w is used as a single independent variable; meanwhile, in the new triangular plate UI-T3 element, the single independent variable is the bending displacement (wb). The difference between the two approaches lies in the transverse displacement w. In the UI approach, the transverse displacement w is expressed as the sum of the transverse bending displacement wb and transverse shear displacements ws. The first and second derivatives of the bending displacement wb are the rotations and curvatures. The free formulation concept allows for non-conforming displacement functions that cover the lower and higher modes and are linearly independent. Rigid body mode and constant strain mode should be obtained by the lower mode displacement function. These lower and higher modes must create orthogonal bending energies to pass the patch test. The proposed element called UI-T3 is locking-free, has a proper rank, and gives satisfying results in the standard benchmark problems for isotropic plates and Functional Graded Material (FGM) sandwich plates.

Wave attenuation in viscoelastic hierarchical plates

Phononic crystals (PCs) are periodic structures obtained by the spatial arrangement of materials with contrasting properties, which can be designed to efficiently manipulate mechanical waves. Plate structures can be modeled using the Mindlin–Reissner plate theory and have been extensively used to analyze the dispersion relations of PCs. Although the analysis of the propagating characteristics of PCs may be sufficient for simple elastic structures, analyzing the evanescent wave behavior becomes fundamental if the PC contains viscoelastic components. Another complication is that increasingly intricate material distributions in the unit cell of PCs with hierarchical configuration may render the calculation of the complex band structure (i.e., considering both propagating and evanescent waves) prohibitive due to excessive computational workload. In this work, we propose a new extended plane wave expansion formulation to compute the complex band structure of thick PC plates with arbitrary material distribution using the Mindlin–Reissner plate theory containing constituents with a viscoelastic behavior approximated by a Kelvin–Voigt model. We apply the method to investigate the evanescent behavior of periodic hierarchically structured plates for either (i) a hard purely elastic matrix with soft viscoelastic inclusions or (ii) a soft viscoelastic matrix with hard purely elastic inclusions. Our results show that for (i), an increase in the hierarchical order leads to a weight reduction with relatively preserved attenuation characteristics, including attenuation peaks due to locally resonant modes that present a decrease in attenuation upon increasing viscosity levels. For (ii), changing the hierarchical order implies in opening band gaps in distinct frequency ranges, with an overall attenuation improved by an increase in the viscosity levels.

On the static, vibration, and transient responses of micro-plates made of materials with different microstructures

In the present paper, for the first time, theoretical analysis for static, dynamic, and transient responses of micro-plates made of different materials were comprehensively performed. The constitutive relations of analyzed materials satisfy assumptions of the modified couple stress theory. Additionally, values of length scale parameters of studied materials are taken from well-established experimental results presented in the literature. A generalized model of micro-plates made from Cu, Ni, Ti, and Epoxy satisfies assumptions of Reissner–Mindlin plate theory. The formulated generalized boundary value problem is solved by analytical and the radial basis function finite difference (RBF-FD) methods. A convergence study is performed to show the advantages and correctness of the obtained numerical solution. The meshless numerical method is easy to implement and it provides accurate results that are in excellent agreement with analytical solutions. Effects of material parameters, boundary conditions, and assumptions of classical and the modified couple stress theory on the static bending, free vibration and transient responses of micro-plates are presented and discussed.

Locking-free 6-noded triangular shell elements based on hierarchic optimisation

Conforming lower-order shell elements based on Reissner-Mindlin plate theory generally exhibit an over-stiff response under loading, typically manifested through various forms of locking. A recently developed hierarchic optimisation approach addresses locking by enriching the conforming strains with hierarchic strain terms towards an objective ‘smoother’ strain distribution afforded by the element, which has proven to be effective in relieving shear, membrane and distortion locking in 9-noded quadrilateral shell elements. Nevertheless, in some practical structural problems that involve complex geometry, triangular shell elements are required to avoid a highly distorted mesh of quadrilateral elements. This paper presents a family of 6-noded Reissner-Mindlin triangular shell elements based on the hierarchic optimisation approach. The proposed curved triangular shell elements not only effectively alleviate inaccuracies arising from locking, but also embrace the desirable characteristics of spatial isotropy and insensitivity to element distortion. The family of 6-noded triangular elements have been incorporated within a co-rotational framework to allow large displacement analysis of thin to moderately thick plates and shells. Several numerical examples are finally presented to demonstrate the effectiveness and accuracy of the proposed 6-noded shell element formulation as well as its superior locking-free performance compared to existing shell elements. • Novel family of 6-noded locking-free curved shell elements. • Mitigation of shear, membrane and distortion locking via hierarchic optimisation. • Objective and corrective assumed-strain variants including spatial isotropy. • Incorporation within co-rotational framework for large displacement analysis. • Element variant H3O6 offers superior performance compared to MITC6 element.

A Node-Based Strain Smoothing Technique for Free Vibration Analysis of Textile-Like Sheet Materials

This paper presents an implementation of the node-based smoothed finite element method and Reissner-Mindlin plate theory for a four node isoparametric shell element to improve the numerical precision and computational efficiency subjected to free vibration analysis of textile-like sheet materials. A one smoothing cell integration scheme in the strain smoothing technique is implemented to contrast the shear locking phenomenon that may exists in the analysis for moderately-thick and thick shell models. Various numerical results of free vibration analysis for a multi-layer nonwoven fabric sample are compared with other existing analytical solutions and numerical solutions in literatures to demonstrate the effectiveness of the present method. An advantage of the present formulation is that it can improve the numerical precision without decreasing the computational efficiency.

Nonlocal strain gradient IGA numerical solution for static bending, free vibration and buckling of sigmoid FG sandwich nanoplate

For the first time, a numerical isogeometric numerical solution based on the nonlocal strain gradient elasticity theory for static bending, free vibration, and buckling of sigmoid functionally graded (S-FG) nanoplate is presented. Two configurations of S-FG sandwich material, including isotropic core and FG core, are considered. A parameter is proposed to define the location of the neutral axis through the cross-section. The simple Reissner-Mindlin plate theory, the nonlocal strain gradient theory, and Hamilton's principle are employed to establish the general equilibrium of S-FG nanoplate that contains two small size coefficients, including nonlocal and strain gradient parameters. The static bending, free vibration, and buckling responses of S-FG nanoplate are explored using isogeometric analysis with a NURBS basic function. The accuracy of the presented model is verified through the comparison with other solutions for nanoplate. As a result of numerical investigation studies, the static bending, free vibration, and buckling responses of S-FG are significantly affected by the material variation along the thickness direction, the neutral axis location, nonlocal parameter, strain gradient parameter, and material index.

A novel stiffness prediction method with constructed microscopic displacement field for periodic composite plates

A novel method for predicting effective stiffnesses of composite plates with in-plane periodicity is proposed according to two-scale strain energy equivalence. The macroscale energy is reached with the Reissner-Mindlin plate theory, while the microscale energy is obtained through the newly constructed displacement functions of a unit cell within the framework of the three-dimensional elasticity theory. The new displacement functions consist of homogenized displacements, two decoupled expansion terms, and a higher-order perturbed displacement term. The elaborate finite element formulations are presented for realizing the present method in Comsol Multiphysics. Several numerical experiments demonstrate the effectiveness of the new stiffness prediction method.

Computational analysis of the nonlinear vibrational behavior of perforated plates with initial imperfection using NURBS-based isogeometric approach

Abstract In this article, an isogeometric analysis through NURBS basis functions is presented to study the nonlinear vibrational behavior of perforated plates with initial imperfection. In this regard, the governing equations of plate dynamics, as well as the displacement–strain relations, are derived using the Mindlin–Reissner plate theory by considering von Karman nonlinearity. The geometry of the structure is formed by selecting the order of NURBS basis functions and the number of control points according to the physics of the problem. Since similar basis functions are utilized to estimate the accurate geometry and displacement field of the domain, the order of the basic functions and the number of control points are optimized for the proper approximation of the unknown field variables. By utilizing the energy approach and Hamilton principle and discretizing the equations of motion, the vibrational response of the perforated imperfect plate is extracted through an eigenvalue problem. The results of linear vibrations, geometrically nonlinear vibrations, and nonlinear vibrations of imperfect plates are separately validated by considering the previously reported findings, which shows a satisfactory agreement. Thereafter, a coefficient of the first mode shape is considered as the initial imperfection and the vibrational analysis is reexamined. Furthermore, the nonlinear vibrations of the perforated plate with initial imperfection are analysed using an iterative approach. The effects of the perforated hole, initial imperfection, and geometric nonlinearity are also addressed and discussed.