Mindlin Plate Theory Research Articles

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.

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.

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.

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.

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.

Theoretical investigation on scattering of ultrasonic Lamb waves at damaged area in quasi-isotropic CFRP laminates based on Mindlin plate theory

ABSTRACT With the wide application of carbon fibre reinforced plastic (CFRP) structure, a structural health monitoring (SHM) system that can easily detect damages in CFRP structures is urgently needed. Since it is possible to detect damages effectively in a large area in CFRP by using the scattering of Lamb waves, in this research, we investigated the scattering of Lamb waves at an impact damaged area in quasi-isotropic (QI) CFRP laminate. By modelling the impact damage as a cylinder area with a uniform stiffness degradation, the scattering waves are calculated based on Mindlin plate theory, which is more precise than Kirchhoff plate theory. It is shown that in the low-frequency region, the calculation results based on Kirchhoff and Mindlin plate theory was similar, but in a higher frequency range, the results differed. The calculation results were compared with the simulation results, suggesting the validity of Mindlin plate theory within a practical frequency range. Moreover, we validated the analytical and simulation results experimentally using a QI CFRP laminate with a hole. We also conducted some parameter study based on the analytical calculation to clarify influential stiffness components. Since scattering behaviour of Lamb waves can be calculated in a fast and easy way using this method, this work is useful for the successful development of a scattering-based SHM system.

Spectral collocation method for free vibration of sandwich plates containing a viscoelastic core

Viscoelastic constrained layer damping is a simple and efficient method for reducing noise, vibration, and fatigue in metal structures. In this work, a spectral collocation method utilizing a layer-wise plate theory was established to inspect the vibration characteristics of a sandwich plate with a viscoelastic core. The displacements of each layer satisfied the Mindlin plate theory. The core's transverse shear stress remained constant. The three layers’ transverse displacements were assumed to be identical. The displacement fields were reduced to nine variables by utilizing the interfaces’ continuity of the displacements. The equations of motion were obtained by utilizing Hamilton’s principle. The viscoelastic core’s material properties were considered as frequency-dependent. To address the complex eigenvalue problem, an iterative algorithm was used. The theoretical results were compared with published theoretical and experimental data for fully clamped rectangular sandwich plates to validate the proposed method. The modified Oberst beam method was applied by fitting the impulse response function to obtain the adhesive’s frequency-dependent storage modulus and loss factor. The natural frequencies and corresponding loss factors of three sandwich plates with different adhesives were measured by conducting impact tests. The numerical results from the present theoretical method agreed well with the measured results.

Free vibration of cracked FGM Mindlin plate in fluid

The free vibration of cracked functionally graded material (FGM) plate in fluid is investigated in this paper. The material components are supposed following the power law and formulated by the Voigt model. The Mindlin plate theory (MPT) including the shear deformation effect is established and the physical neutral plane is adopted to simplify the derivation due to the in-plane displacements of this plane assumed as zero. A massless linear rotational spring model (LRSM) is established to simulate the through-width surface crack. The hydrodynamic pressure at the fluid-plate interface is derived by the potential flow theory, and is regarded as the added mass of cracked FGM plate during the analyses of fluid-plate coupled vibration. The derivation and discretization of vibrational equations for cracked FGM plates in fluid are respectively achieved though the Hamilton’s principle and differential quadrature (DQ) method. The frequencies and mode shapes are iteratively calculated with these values of the vacuum case. The influences of immersed depth, fluid density, gradient index, crack depth, crack location, length-to-width ratio, length-to-thickness ratio, and boundaries on vibration characteristics are conducted in the section of numerical results. The present vibration analyses of cracked structures can be applicable to avoid structural resonance and failure.

Nonlinear modeling of nanoscale interaction forces between atomic force microscope and carbon nanotubes

This paper introduces a non-linear non-classical model of nanoscale interaction forces between atomic force microscope and carbon nanotubes in a manipulation process. To accurately model the 3D manipulation process, Prandtl-Tomlinson friction model and the Hertz, JKR, and Lundberg contact mechanics theories were employed, taking into account the CNT geometry and manipulation conditions. The applied forces exerted by both the substrate and the probe on the CNT were extracted. The dynamic characteristics of the nanorods and nanotubes were investigated by employing the non-linear modified couple stress Timoshenko beam. Additionally, the cantilever was modeled using the Mindlin plate theory. The findings showed a good agreement between the experimental data of multi-walled carbon nanotube with a diameter of 100 nm, and the theoretical simulations. In addition to critical force and time, CNT bending results were presented using various Timoshenko theories. In order to validate the results of the maximum bending, the outcomes derived from the computational modeling of a polystyrene nanorod using the Euler-Bernoulli and Timoshenko theories were juxtaposed with the existing classical bending outcomes. Finally, the effects of the size parameter were investigated in non-classical polystyrene models. The findings indicated that considering the size effects is important; For instance, the variance between classical and non-classical models can exceed 57% for l/D = 0.5.

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.

Large-amplitude vibration analysis of FG rectangular plates under rapid cooling

In the present research, the large-amplitude vibration behavior of rectangular plates subjected to cooling shock is investigated based on a numerical solution approach. It is considered that the plates are made of functionally graded materials as a mixture of stainless steel and low-carbon steel. The thermoelastic properties are also considered temperature-dependent which are estimated based on available experimental data. According to the Mindlin plate theory, the nonlinear governing equations of motion together with corresponding boundary conditions are derived. The temperature profile is also obtained based on a one-dimensional Fourier-type transient heat conduction equation. Moreover, two time-dependent thermal loading scenarios for cooling shock on the plate’s top face are considered. To solve the problem numerically, the well-known generalized differential quadrature technique is used for discretization considering the Chebyshev–Gauss–Lobatto grid. In addition, the governing equations are traced in time by Newmark’s time integration scheme. After showing the validity of developed approach, a parametric study is presented to investigate the stress changes along the thickness and the effects of thermal load rapidity time, geometry, magnitude of thermal load, and material properties on the nonlinear vibrations of plates with various boundary conditions subjected to sudden decrease of temperature. It is concluded that the thermal load rapidity time has a significant role in the vibrational behavior and the stress distribution of the plate.

Modelling of bidirectional functionally graded plates with geometric nonlinearity: A comparative dynamic study using whole domain and finite element method

Present work attempts to study the nonlinear dynamic behaviour of bidirectional functionally graded plates (BFG) along with unidirectional functionally graded plates (UFG). For BFG plates the material is graded along the thickness and length simultaneously and two types of UFG plates are considered namely thickness-wise (TFG) and length-wise (AFG) plates. The dynamic problem is formulated using two different methods namely, whole domain method and finite element method. The constitutive relation for the former is generated using classical plate theory whereas Mindlin plate theory is used in the later. Geometric nonlinearity is addressed here by using Von Karman type strain-displacement relation. Both clamped and simply supported boundary conditions are considered and the plate is assumed to be under action of uniformly distributed external load. The results of free and forced vibration analyses are presented as backbone curves and frequency response curves respectively. The results show that all three plates exhibit hardening type nonlinearity. It is also seen that the effect of material gradation parameters is more profound in simply supported plates compared to clamped plates. Specifically, these parameters do not seem to have much effect on BFG clamped plates in free vibration. The outcome of the analysis is compared with those in existing literature and fair agreement is observed between them.

The influence of surface effects on axisymmetric vibration of graded porous Mindlin circular nanoplate in a temperature field

AbstractThe surface effect significantly affects the mechanical response of nanoscale materials. In this work, we introduce Gurtin‐Murdoch surface elasticity theory into Mindlin plate theory to study the axisymmetric vibration characteristic of graded porous circular nanoplate in a temperature field. The main structure is a porous graded material with uniform or non‐uniform porosity distribution in its thickness. The numerical shooting technique was applied to solve the governing differential equations derived from Hamilton's principle. Responses for the vibration characteristic of the graded porous nanoplate for both clamped and simply supported boundary conditions were obtained. The numerical results show that when the surface effect is removed, the classical results of Mindlin circular plate will be obtained. Natural frequencies and mode shapes varying with thermal and porosity coefficients were present graphically. And then the effects of surface material properties on the axisymmetric vibration characteristic of the nanoplates were discussed in detail. The parametric study examined the influence of porosity coefficient, porosity distribution mode, surface elastic parameters, and residual surface stress on the vibration characteristics of porous circular nanoplates.

Coupled five-parameter dynamics of Mindlin and third-order shear deformable FG graphene-platelets reinforced viscoelastic plates with geometric and material imperfections

The present work investigates the coupled five-parameter dynamics, in particular, the coupled five-parameter structural resonance frequency, of functionally graded (FG) graphene-platelets reinforced viscoelastic plates in the presence of geometric and material (porosity) imperfections, based on the Mindlin theory and the third-order shear deformation plate theory (TSDPT). Various imperfections are often found in structures as a result of the challenges in manufacturing and operations over time. Geometric imperfection is considered in the FG reinforced plate structure by modelling an initial curvature in the out-of-plane direction. Material imperfections, in the form of closed-cell voids, are formulated as uniform and non-uniform FG porosity distributions. Uniform and thickness-wise FG gradation effects of graphene-platelets are accounted for in the determination of the effective material properties using a two-dimensional Halpin-Tsai micromechanics model, in conjunction with the rule of mixtures. Viscoelasticity of the FG imperfect plate structure is modelled by employing the Kelvin-Voigt model to consider energy dissipation. Using the Mindlin plate theory and the TSDPT, five equations of motion, which are coupled through the five parameters u, v, w, ϕ1, and ϕ2, are derived using Hamilton's variational principle. An assumed mode method is then employed to obtain the real and the imaginary components of resonance frequencies. The combined influences of the different effects on the structural resonance are examined. The results obtained offer a comprehensive understanding of how the FG graphene-platelets, FG porosity, plate thickness, geometric imperfection, and the utilisation of two different shear deformation plate theories collectively influence the coupled five-parameter dynamics of the FG graphene-platelets reinforced viscoelastic plate structure with imperfections. It was found that the difference in frequency resulted from the use of the two plate theories is most dominant with the FG GX and/or FG PX graphene-platelets reinforced porous plates, and that geometric imperfection was found to alter a graphene-platelets reinforced plate's sensitivity towards material imperfection.

An Analytical Solution on Vibration Reduction and Shear Force Mitigation of Cantilever Mindlin Plates Using Orthogonal Ribs

Vibration of aircraft wings and the dynamic stress concentration at the clamped edge are important research topics due to concerns on the safety of aircrafts. To have a better understanding of the problem, the free and forced vibration response of a ribbed rectangular cantilever plate representing a section of an aircraft wing is investigated in this study. A new analytical solution is developed for the vibration analysis of rib stiffened cantilever plates using Mindlin plate and Timoshenko beam theories alongside the finite integral transform technique. The one- and two-dimensional integral transforms are applied to the governing equations of beams and plates, respectively, where the coupling force components at the interface between the base plate and the beam(s) can be automatically defined during the integral transform. Eventually, the partial differential equations are transformed into a system of linear algebraic equations in which its derivation is rigorous and easily implemented. Good agreements are found between the results of analytical solution, finite element analysis (FEA) and related literature. The solution is then employed to study the vibration suppression of cantilever plates and the shear force at the clamped edge. It is found that the insertion of a pair of orthogonal ribs in the plate can effectively reduce its vibration. An optimum orthogonal ribbing pattern is obtained using multi-objective particle swarm optimization (MOPSO) algorithm, taking into consideration both the vibration suppression of the plate and the maximum induced shear force at the corners of the clamped edge.

Analysis of a Steel-Concrete Composite Plate resting on Axial Bars using the Finite Element Method

This study applied finite element analysis to a steel-concrete composite plate resting on the axial bar, using a four-node quadrilateral finite element for a steel-concrete composite plate combined with a truss element. Plate displacements and deformations were used to formulate the finite element method based on the first-order plate theory, also known as the Mindlin plate theory. The finite element analysis to combine steel-concrete composite plates and axial bars was implemented in MATLAB. Numerical examples were used in detail, showing a very small difference between the current study and the SAP2000 results.

Analysis of Concrete Pavement Slab resting on Non-uniform Elastic Foundation using the Finite Element Method

During the rigid pavement design, the concrete slab working on the base/subbase and subgrade is usually modeled as the slab on the elastic foundation. However, the non-uniform distribution of materials in the base or subgrade layers naturally exists in real conditions and should be considered. In this paper, a finite element method for calculating concrete pavement slabs on an elastic foundation with non-uniform stiffness distribution is developed. This study applies 4-node finite elements and Mindlin plate theory to formulate the finite element equations. The results predicted by the proposed approach are verified analytically. Calculation examples are conducted with the practical settings to investigate the influence of slab and foundation stiffness parameters on the slab displacement.

Constrained layer damping for mitigating vibration of a rotating disk-drum coupled structure

Rotating disk-drum coupled structures are commonly employed in large rotating machines. They are prone to have vibration issues caused by their decreasing thickness to meet lightweight demands. Constrained layer damping (CLD) is a widely used technique to reduce vibrations, but no studies have yet investigated its application on rotating coupled structures. In this paper, a universal model is proposed and experiments are conducted to investigate the influence of CLD on a rotating cylindrical shell ended by an annular disk. This approach considers the rotating effects, and formulates the equations of motion using the Rayleigh-Ritz method with the help of the Sanders shell theory and Mindlin plate theory. Different sets of artificial springs are applied to model the coupling and boundary conditions. The proposed approach is validated by comparing the numerical results with those obtained from modal and rotor experiments. Then the modal parameters and vibration responses are evaluated for different rotational speeds, and the influences of the location and thickness ratio of the CLD patch and the coupling stiffness on the dynamic characteristics of the system are discussed.

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.