Mindlin Plate Research Articles

Hydroelastic analysis of partially or totally immersed Mindlin plates with a hybrid isogeometric-based approach

Hydroelastic analysis of partially or totally immersed Mindlin plates with a hybrid isogeometric-based approach

Locking-Free HDG Methods for Reissner–Mindlin Plates Equations on Polygonal Meshes

Locking-Free HDG Methods for Reissner–Mindlin Plates Equations on Polygonal Meshes

Wave Transmission Analysis of Timoshenko Beam Junction with Mindlin Plates Connected in the Same Plane

Wave Transmission Analysis of Timoshenko Beam Junction with Mindlin Plates Connected in the Same Plane

Simulation of mixed-mode crack propagation in Mindlin plates by a hierarchical quadrature element method with minimal remeshing

Simulation of mixed-mode crack propagation in Mindlin plates by a hierarchical quadrature element method with minimal remeshing

The Hellan–Herrmann–Johnson and TDNNS methods for linear and nonlinear shells

In this paper we extend the recently introduced mixed Hellan–Herrmann–Johnson (HHJ) method for nonlinear Koiter shells to nonlinear Naghdi shells by means of a hierarchical approach. The additional shearing degrees of freedom are discretized by H(curl)-conforming Nédélec finite elements entailing a shear locking free method. By linearizing the models we obtain in the small strain regime linear Kirchhoff–Love and Reissner–Mindlin shell formulations, which reduce for plates to the originally proposed HHJ and TDNNS methods for Kirchhoff–Love and Reissner–Mindlin plates, respectively. By interpolating the membrane strains into the so-called Regge finite element space we obtain locking-free arbitrary order shell methods. Additionally, the methods can be directly applied to structures with kinks and branched shells. Several numerical examples and experiments are performed validating the excellent performance of the proposed shell elements.

Modeling and dynamic analysis of multi-layer plates supported by Pasternak foundation under thermo-mechanical loads using moving element method

This article aims to present a numerical paradigm for modeling and analyzing the dynamic responses of the multi-layer plate structure resting on the Pasternak foundation under thermo-mechanical loads by using the multi-layer moving plate method (MMPM), in which, two consecutive plates connected by a middle viscoelastic layer are simulated by the Mindlin plate theory. The governing equations of motion of the multi-layer plate supported by a Pasternak foundation under a moving load with the linear temperature distribution along the plate thickness are generally established by the virtual work principle. Then, a moving coordinate system attached to the moving load is employed to discretize the structural domain that surrounds loadings. This strategy not only helps to dramatically lessen the number of finite elements but also to completely curtail the updating procedure of the time-varying and spatially varying loads. Finally, the dynamic behavior of the structural system is obtained by the Newmark-beta approach. Outcomes gained from the proposed model are validated by comparing with previously published works. After that, the dynamic analysis of multi-layered connected plates is studied by changing the temperature surrounding the structure. Further, this study also considers the dynamic behavior of double-plate structure subjected to moving harmonic load and resting on an in-homogeneous foundation.

Novel quadrilateral strain-based finite element for static, free vibration, and buckling analysis of plates

This paper focuses on the static analysis, free vibration, and buckling validation of Reissner–Mindlin plates using a newly developed quadrilateral finite element, termed FNSBP, which is based on the strain approach. The proposed element is equipped with three essential external degrees of freedom at each of its four corner nodes. The displacement field is formulated using assumed strain functions that inherently satisfy the compatibility equations. This new element was developed to improve upon the previously introduced SBRP strain-based rectangular plate element. The effectiveness and robustness of the FNSBP element are evaluated through various numerical examples involving both thick and thin plates of different geometries. Its performance is benchmarked against displacement-based elements from the literature and analytical solutions, demonstrating superior accuracy and convergence in many cases. The results indicate that the FNSBP element provides significant improvements in computational efficiency and accuracy, especially for complex plate configurations under different loading and boundary conditions, validating its applicability in structural analysis.

Determination of material constants of piezoceramics using genetic algorithm

This study seeks to accurately determine the piezoelectric material constants of piezoceramic disks from the resonant frequencies of the disks using a genetic algorithm programmed according to the principles of plate theory. Mindlin's plate theory is judged to be most suitable for approximating the relationship between the in-plane and out-of-plane resonant frequencies and the material constants of a disk-shaped piezoceramic thick plate, which was programmed into a genetic algorithm in order to obtain all relevant piezoelectric material constants from measured resonant frequencies of sample piezoceramic disks through inverse calculation. To verify the accuracy of the material constants, finite element method was employed to derive the theoretical resonant frequencies along with the corresponding mode shapes, which were then compared with the actual resonant frequencies measured using amplitude-fluctuation electronic speckle pattern interferometry. The comparison shows that the genetic algorithm can successfully determine all desired material constants of piezoceramic disks from the measured resonant frequencies in a single operation, and that the resonant frequency values modeled using the constants more accurately correspond to the experimentally measured frequencies than those derived from material constants obtained using conventional methods.

An analytical method for the vibration characteristics of double panel structures with uniform elastic boundary supports

ABSTRACT In this paper, the methods for solving the free vibration and forced vibration of double panel structures under arbitrary boundaries are studied. Based on the Mindlin plate theory, the natural boundary between plates and plates is constrained by artificial springs, and the theoretical model of the structure is derived. The displacement function of in-plane vibration and bending vibration is expressed by a modified Fourier series expansion. The effectiveness and reliability of this method are verified by a series of numerical calculations and comparison with those of the finite element method. Through the parameter analysis, it can be seen that the vibration isolation effect can be increased by properly increasing the coupling height or making the connecting plate closer to the boundary of the bottom plate. The method presented in this paper can be used in vibration absorption and structural design of such structures.

An Energy Approach to the Modal Identification of a Variable Thickness Quartz Crystal Plate

The primary objective of modal identification for variable thickness quartz plates is to ascertain their dominant operating mode, which is essential for examining the vibration of beveled quartz resonators. These beveled resonators are plate structures with varying thicknesses. While the beveling process mitigates some spurious modes, it still presents challenges for modal identification. In this work, we introduce a modal identification technique based on the energy method. When a plate with variable thickness is in a resonant state of thickness–shear vibration, the proportions of strain energy and kinetic energy associated with the thickness–shear mode in the total energy reach their peak values. Near this frequency, their proportions are the highest, aiding in identifying the dominant mode. Our research was based on the Mindlin plate theory, and appropriate modal truncation were conducted by retaining three modes for the coupled vibration analysis. The governing equation of the coupled vibration was solved for eigenvalue problem, and the modal energy proportions were calculated based on the determined modal displacement and frequency. Finally, we computed the eigenvalue problems at different beveling time, as well as the modal energies associated with each mode. By calculating the energy proportions, we could clearly identify the dominant mode at each frequency. Our proposed method can effectively assist engineers in identifying vibration modes, facilitating the design and optimization of variable thickness quartz resonators for sensing applications.

Annular sector plates made of FG graphene origami-enabled auxetic metamaterial under bending and buckling loading: A VDQ-based approach

The behaviors of annular sector plates made of functionally graded graphene origami-enabled auxetic metamaterials (FG-GOEAMs) under the action of buckling and bending loads are investigated herein using a numerical approach. Multiple GOEAM layers with graphene origami (GOri) content are considered for the plate which changes in layer-wise patterns. Moreover, the material properties of the plate are calculated using novel genetic programming-assisted micromechanical models. In order to derive the governing equations, the Mindlin plate theory in conjunction with Hamilton’s principle is utilized within the framework of the variational differential quadrature (VDQ) method. The vector-matrix representation of the presented formulation can be beneficial from the viewpoint of implementing numerical approaches. The governing equations are then discretized and solved based on VDQ matrix differential and integral operators so as to obtain the critical buckling load and maximum lateral deflection of plates with different edge conditions. The effects of GOri content, folding degree and distribution pattern on the results are studied. It is revealed that increasing the GOri content leads to the negative Poisson’s ratio (NPR) effect which respectively reduces and increases the lateral deflection and the critical buckling load.

An isogeometric approach to dynamic structures for integrating topology optimization and optimal control at macro and micro scales

In the context of active vibration suppression, an integrated topology optimization framework is proposed for the optimal control of dynamic structures. This article formulates a new composite objective function by considering the external harmonic excitations with different excitation frequencies and the complex-valued displacement field. Within the proposed optimal control performance function, the symmetric weighting matrix, the magnitudes of which are related to the importance of the control forces, is integrated into dynamic compliance to serve the state displacement. Utilizing the non-uniform rational B-splines (NURBS) basis functions, the material interpolation model defined at control points is described by the NURBS surface, which is incorporated into the optimization formulation for optimal structural and vibration control design. The sensitivity results are deduced in terms of the pseudo-densities and the corresponding weights at control points from both macro and micro perspectives of structural dynamics. To demonstrate the effectiveness of the integrated topology optimization of dynamic structures at macro and micro scales, several numerical examples, including the topology optimization of solid beams in plane stress and Mindlin plates with 3D microstructures, are provided and discussed in terms of structural shape and topology, frequency response curve, and modal displacement.

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.

Bandgap characteristics of periodic mindlin plates under arbitrary boundary conditions via the Spectral Element Method

This paper employs the Spectral Element Method (SEM) to investigate the bandgap characteristics in periodic plates under arbitrary boundary conditions. The spectral element model of Mindlin plate is derived via force-displacement relations. The convergence of the SEM results for a single plate structure is validated by comparing them with the Finite Element Method (FEM) results at distinct mesh resolutions, demonstrating that SEM maintains high computational accuracy even with fewer elements. Like FEM, the spectral element matrix is assembled to establish the spectral element model of the periodic plate. The influences of boundary conditions, substructures, material properties, and geometric dimensions on the bandgap characteristics in the periodic plate are explored.

Vectfem: a generalized MATLAB-based vectorized algorithm for the computation of global matrix/force for finite elements of any type and approximation order in linear elasticity

In this paper, we introduce a new vectorized MATLAB-based algorithm for efficient serial computation of global matrix/force arising from finite element method (FEM) for meshes of any type and approximation order in linear elasticity. Because for-loops in MATLAB are very slow, we propose a modified process that takes advantage of vectorization and sparse assembly to achieve good performance while using the same memory as the standard algorithm. For this purpose, by using good programming practices, the implementation of this scheme is succinctly described and can be integrated into any MATLAB package dealing with FEM. Specifically, attention is paid to the calculation of the triplet (row index, column index, matrix components) as well as the assembly of the global stiffness matrix, mass matrix and force vector. Additionally, an extension of the proposed approach for Mindlin plate theory and functionally graded materials is outlined. Finally, the accuracy of this strategy is verified on selected numerical tests after comparing the obtained results with those of ABAQUS. In terms of performance, the study conducted on a set of meshes considering the standard algorithm and two other well-known MATLAB vectorized algorithms revealed that: (i) for a 2D beam problem meshed with P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{1}$$\\end{document}-triangle elements, a speedup of about 8 and 15 is achieved with sparse and fsparse, respectively. (ii) for a 3D plate problem meshed with P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{1}$$\\end{document}-tetrahedral elements, a speedup of about 4 and 8 is achieved with sparse and fsparse, respectively. When compared to ABAQUS performance, the proposed scheme results in a computational time that is about five times smaller.

A quadrilateral non-classical microplate element considering the voltage effect

A quadrilateral non-classical Mindlin element that incorporates voltage effects was introduced for examining the bending and free vibration of piezoelectric multilayer plates. The model was developed to examine multilayer plates in the presence of voltage effects. Hamilton’s principle was initially employed to formulate the equation of motion for a multilayer Mindlin plate, considering both size and voltage effects. The equations of motion were resolved through the Galerkin’s method. The suggested element is a rectangular element featuring four-nodes, each possessing 15 degrees of freedom, accounting for both bending and stretching deformations. This element meets the requirements for C0 continuity and C1 weak continuity, and incorporates size and voltage effects. The results were examined with both experimental and analytical data. Upon investigating the voltage effect, it was found that the stiffness matrix depends on both the magnitude and sign of the voltage. Furthermore, it has been demonstrated that the natural frequencies of higher modes are less affected by voltage variations compared to lower modes. In the end, the model was compared to experimental results obtained from a multilayer microcantilever.

Maximizing band gaps of single-phase phononic plates: Isogeometric optimal approach and 3D printing experimental validation

This work presents an effective isogeometric shape optimization approach for finding and widening the phononic band gaps of single-phase Mindlin plate structures. As the single-phase material phononic plate is easy to be manufactured by additive manufacturing, it has been investigated here by optimized its thickness profile to find and widen the band gaps. The proposed method utilizes a coarse B-spline surface to model the thickness profile of periodic plate and a fine B-spline surface to model the mid-surface of plate structure for simulation. The optimal design variables are the thickness variables, i.e., the z-components of the control points of the coarse B-spline surface. To avoid specifying the initial control point locations manually, the constrained dynamic maximization problem is solved by a particle swarm optimization (PSO) algorithm here. Various numerical examples demonstrate the effectiveness and reliability of the proposed method in finding optimal phononic band gaps of periodic plates. And the numerical results show that there is no band gap for hmax = 3 hmin, and the band gaps can be found and widen for hmax ≥ 5 hmin. The obtained band range is 638.9–823.6 Hz with a decreased central frequency of 731.25 Hz for hmax = 5 hmin and the width and the number of band gaps are increased as the maximum allowable thickness increases. Finally, one optimized design is fabricated through additive manufacturing, and the experimental frequency response is consistent with the results based on isogeometric analysis.

Exact Eigensolutions of Two-Dimensional Laminated Panel Flutter Based on Mindlin Plate Theory and the First-Order Piston Theory

The panel flutter analysis considering the shear deformation is performed with a particular focus on the exact eigensolutions and the energy transfer. According to Mindlin plate theory and the first-order piston theory, this work obtains the exact eigensolutions of two-dimensional (2D) panel flutter at supersonic speeds. The boundary conditions (BCs) considered include: two edges are simply supported (SS), two edges are clamped (CC), and two edges are clamped and free (CF) respectively. Using the obtained exact eigensolutions, the relations of the kinetic energy, the potential energy and the work done by aerodynamic force are examined for the states of divergent vibration and periodic vibration. Besides, the coupling effect of shear deformation and aerodynamic damping on flutter frequencies is also revealed, and the present results are compared with those of the Galerkin method and the 2D Kirchhoff panel.

A meshless computational framework for a modified dynamic system of vehicle coupled with plate structure

In previous simulations of train–bridge interaction systems (TBIS), the supporting system for the train are commonly treated as beam structures, leading to less accurate results, particularly for small-span cases. To address this limitation, a modified vertical TBIS is proposed. In the presented TBIS, the supporting system is considered as a Reissner–Mindlin plate, and the displacement field is described by first-order shear deformation theory (FSDT). To establish the model, radial point interpolation method (RPIM), a meshless method, is employed. Finally, a coupled dynamic equation is established to calculate various responses of the system. Several numerical examples are presented to illustrate the disparities between the system based on plate model and traditional beam model. The results indicate that the beam model yields higher estimates of the mid-span vertical displacement of the bridge, while the peak of the mid-span vertical acceleration is smaller compared to the plate model; additionally, it is observed that the carbody is primarily influenced by rail irregularities. Consequently, the proposed plate model offers distinct advantages over the beam model in providing comprehensive structural response information, thereby offering novel insights into bridge design and analysis. Additionally, this marks the inaugural application of the meshless method in the field of TBIS, which further extends the application scope of meshless methods.

An SPH formulation for general plate and shell structures with finite deformation and large rotation

In this paper, we propose a reduced-dimensional smoothed particle hydrodynamics (SPH) formulation for quasi-static and dynamic analyses of plate and shell structures undergoing finite deformation and large rotation. By exploiting Uflyand–Mindlin plate theory, the present surface-particle formulation is able to resolve the thin structures by using only one layer of particles at the mid-surface. To resolve the geometric non-linearity and capture finite deformation and large rotation, two reduced-dimensional linear-reproducing correction matrices are introduced, and weighted non-singularity conversions between the rotation angle and pseudo normal are formulated. A new non-isotropic Kelvin-Voigt damping is proposed especially for the both thin and moderately thick plate and shell structures to increase the numerical stability. In addition, a shear-scaled momentum-conserving hourglass control algorithm with an adaptive limiter is introduced to suppress the mismatches between the particle position and pseudo normal and those estimated with the deformation gradient. A comprehensive set of test problems, for which the analytical or numerical results from literature or those of the volume-particle SPH model are available for quantitative and qualitative comparison, are examined to demonstrate the accuracy and stability of the present method.