Eigen-frequencies in thin elastic 3-D domains and Reissner-Mindlin plate models

2001 ◽  
Vol 25 (1) ◽  
pp. 21-48 ◽  
Author(s):  
Monique Dauge ◽  
Zohar Yosibash
Keyword(s):  
Mindlin Plate ◽  
Plate Models ◽  
Eigen Frequencies

Nonlocal Approaches for the Vibration of Lattice Plates Including Both Shear and Bending Interactions

2018 ◽  
Vol 18 (07) ◽  
pp. 1850094 ◽  
Author(s):  
F. Hache ◽  
N. Challamel ◽  
I. Elishakoff
Keyword(s):  
Natural Frequencies ◽  
Scale Effects ◽  
Dynamical Behavior ◽  
Mindlin Plate ◽  
Length Scale ◽  
Small Length ◽  
Simply Supported ◽  
Small Length Scale ◽  
Plate Models ◽  
Scale Coefficient

The present study investigates the dynamical behavior of lattice plates, including both bending and shear interactions. The exact natural frequencies of this lattice plate are calculated for simply supported boundary conditions. These exact solutions are compared with some continuous nonlocal plate solutions that account for some scale effects due to the lattice spacing. Two continualized and one phenomenological nonlocal UflyandMindlin plate models that take into account both the rotary inertia and the shear effects are developed for capturing the small length scale effect of microstructured (or lattice) thick plates by associating the small length scale coefficient introduced in the nonlocal approach to some length scale coefficients given in a Taylor or a rational series expansion. The nonlocal phenomenological model constitutes the stress gradient Eringen’s model applied at the plate scale. The continualization process constructs continuous equation from the one of the discrete lattice models. The governing partial differential equations are solved in displacement for each nonlocal plate model. An exact analytical vibration solution is obtained for the natural frequencies of the simply supported rectangular nonlocal plate. As expected, it is found that the continualized models lead to a constant small length scale coefficient, whereas for the phenomenological nonlocal approaches, the coefficient, calibrated with respect to the element size of the microstructured plate, is structure-dependent. Moreover, comparing the natural frequencies of the continuous models with the exact discrete one, it is concluded that the continualized models provide much more accurate results than the nonlocal Uflyand–Mindlin plate models.

THE REISSNER–MINDLIN PLATE IS THE Γ-LIMIT OF COSSERAT ELASTICITY

2010 ◽  
Vol 20 (09) ◽  
pp. 1553-1590 ◽  
Author(s):  
PATRIZIO NEFF ◽  
KWON-IL HONG ◽  
JENA JEONG
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Methods ◽  
Mindlin Plate ◽  
Linear Scaling ◽  
Plate Model ◽  
Cosserat Elasticity ◽  
Plate Models ◽  
Membrane Bending ◽  
Bulk Model

The linear Reissner–Mindlin membrane-bending plate model is the rigourous Γ-limit for zero thickness of a linear isotropic Cosserat bulk model with symmetric curvature. For this result we use the natural nonlinear scaling for the displacements u and the linear scaling for the infinitesimal microrotations Ā ∈ 𝔰𝔬(3). We also provide formal calculations for other combinations of scalings by retrieving other plate models previously proposed in the literature by formal asymptotic methods as corresponding Γ-limits. No boundary conditions on the microrotations are prescribed.

scholarly journals Polyharmonic Multiquadric Particular Solutions for Reissner/Mindlin Plate

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Chia-Cheng Tsai
Keyword(s):  
Present Method ◽  
Thick Plate ◽  
Mindlin Plate ◽  
Shear Forces ◽  
Helmholtz Operator ◽  
Particular Solutions ◽  
Product Operator ◽  
Plate Models ◽  
Operator Decomposition ◽  
Helmholtz Operators

Analytical particular solutions of the polyharmonic multiquadrics are derived for both the Reissner and Mindlin thick-plate models in a unified formulation. In the derivation, the three coupled second-order partial differential equations are converted into a product operator of biharmonic and Helmholtz operators using the Hörmander operator decomposition technique. Then a method is introduced to eliminate the Helmholtz operator, which enables the utilization of the polyharmonic multiquadrics. Then, the analytical particular solutions of displacements, shear forces, and bending or twisting moments corresponding to the polyharmonic multiquadrics are all explicitly derived. Numerical examples are carried out to validate these particular solutions. The results obtained by the present method are more accurate than those by the traditional multiquadrics and splines.

Vibration Analysis of Coupled Multilayer Structures with Discrete Connections for Noise Prediction

2020 ◽  
Vol 20 (04) ◽  
pp. 2050051
Author(s):  
Zheng Lu ◽  
Junzuo Li ◽  
Qi Li
Keyword(s):  
Finite Element ◽  
Infinite Plate ◽  
Urban Rail Transit ◽  
Multilayer Structures ◽  
Mindlin Plate ◽  
Finite Element Models ◽  
Urban Rail ◽  
Plate Models ◽  
Similar Accuracy ◽  
Track Bridge

It is often necessary to calculate the vibration of noise from multilayer structures comprising several substructures coupled with discrete connections. A dynamic flexibility method (DFM) is adopted to decouple the multilayer substructures, which allows the interface forces among the substructures to be directly solved using a linear equation of deformation compatibility. The structural vibrations and power flows into each substructure can then be calculated. To illustrate the use of the DFM, a coupled train–track–bridge system for urban rail transit traffic is investigated as a case study. Two infinite plate models are used to model the U-shaped bridge substructure to improve the computing efficiency compared with the finite element models in calculating high-frequency vibration. The applicability of the infinite plate models is discussed in terms of various rail positions on the bridge, the thickness of the rail support blocks, and multiple wheels that interface with the rail. The results show that the Mindlin plate model has similar accuracy but much greater computing efficiency than the finite element models. With the vibration results from the DFM, the associated wheel–rail noise and structure-borne noise from the bridge are then calculated together with a 2D acoustic model. Good agreement is observed between the predicted noises and the measured data.

PLATE BUCKLING ANALYSIS USING A GENERAL HIGHER-ORDER SHEAR DEFORMATION THEORY

2013 ◽  
Vol 13 (05) ◽  
pp. 1350028 ◽  
Author(s):  
NOËL CHALLAMEL ◽  
GJERMUND KOLVIK ◽  
JOSTEIN HELLESLAND
Keyword(s):  
Plate Theory ◽  
Shear Deformation Theory ◽  
Gradient Elasticity ◽  
Composite Plates ◽  
Sensitivity Study ◽  
Higher Order ◽  
Constitutive Law ◽  
Mindlin Plate ◽  
Shear Plate ◽  
Plate Models

The buckling of higher-order shear plates is studied in this paper with a unified formalism. It is shown that usual higher-order shear plate models can be classified as gradient elasticity Mindlin plate models, by augmenting the constitutive law with the shear strain gradient. These equivalences are useful for a hierarchical classification of usual plate theories comprising Kirchhoff plate theory, Mindlin plate theory and third-order shear plate theories. The same conclusions were derived by Challamel [Mech. Res. Commun.38 (2011) 388] for higher-order shear beam models. A consistent variational presentation is derived for all generic plate theories, leading to meaningful buckling solutions. In particular, the variationally-based boundary conditions are obtained for general loading configurations. The buckling of the isotropic or orthotropic composite plates is then investigated analytically for simply supported plates under uniaxial or hydrostatic in-plane loading. An analytical buckling formula is derived that is common to all higher-order shear plate models. It is shown that cubic-based interpolation models for the displacement field are kinematically equivalent, and lead to the same buckling load results. This conclusion concerns for instance the plate models of Reddy [J. Appl. Mech.51 (1984) 745] or the one of Shi [Int. J. Solids Struct.44 (2007) 4299] even though these models are statically distinct (leading to different stress calculations along the cross-section). Finally, a numerical sensitivity study is made.

scholarly journals Natural Frequencies of the Moderately Thick Plate Models with Uniformly Distrubuted Mass

2019 ◽  
Vol 21 (1) ◽  
pp. 13-30
Author(s):  
Marin Grbac ◽  
Dragan Ribarić
Keyword(s):  
Dynamic Analysis ◽  
Shear Strain ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Mass Matrix ◽  
Mindlin Plate ◽  
Benchmark Problems ◽  
Transverse Displacement ◽  
Assumed Strain ◽  
Plate Models

A four-node finite element is developed for modeling plates according to the Mindlin plate theory and it is constructed with the assumed shear strain approach. The element is previously verified in a static analysis on the benchmark problems of moderately thick and extremely thin plate models and compared to the other elements known from the literature. As starting interpolations, a complete cubic polynomial for the transverse displacement field and quadratic polynomials for the two rotation fields are used, and they are problem dependent at the same time. Some unfavorable terms are excluded from the derived shear strain expression to avoid locking phenomena in the thin geometry conditions. In this paper, the proposed element is tested for the dynamic analysis calculating the natural frequencies of plate vibrations with the uniformly distributed mass. The influence of the element consistent mass matrix is analyzed on the first 12 vibration modes. The results are verified on the circular plate model and compared to the existing analytical solutions as well as the results of other four-node elements from the literature. The goal of this paper is to demonstrate the efficiency of the proposed assumed strain element also in the dynamic analysis of plane structures.

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