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.

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.

Failure behavior of hierarchical corrugated sandwich structures with second-order core based on Mindlin plate theory

2017 ◽  
Vol 21 (2) ◽  
pp. 552-579 ◽  
Author(s):  
Gang Li ◽  
Zhaokai Li ◽  
Peng Hao ◽  
Yutian Wang ◽  
Yaochu Fang
Keyword(s):  
Thick Plate ◽  
Failure Modes ◽  
Sandwich Structures ◽  
Plate Theory ◽  
Second Order ◽  
Mindlin Plate ◽  
Failure Behavior ◽  
Plate Model ◽  
Moderately Thick Plate ◽  
Mindlin Plate Theory

For hierarchical corrugated sandwich structures with second-order core, the prediction error of failure behavior by existing methods becomes unacceptable with the increase of structure thickness. In this study, a novel analytical model called moderately thick plate model is developed based on Mindlin plate theory, which can be used to analyze the failure behavior of hierarchical corrugated structures with second-order core under compression or shear loads. Then, the analytical expressions of nominal stress for six competing failure modes are derived based on the moderately thick plate model. The results of six different unit structures based on the moderately thick plate model agree quite well the ones by finite element methods. Furthermore, the influence of different structure thicknesses is investigated to validate the applicability of the moderately thick plate model. According to the comparative results with the thin plate model, the proposed moderately thick plate model has a better precision with the increase of the ratio of thickness to width for failure components.

scholarly journals Bending analysis of magnetoelectroelastic nanoplates resting on Pasternak elastic foundation based on nonlocal theory

2020 ◽  
Vol 41 (12) ◽  
pp. 1769-1786
Author(s):  
Wenjie Feng ◽  
Zhen Yan ◽  
Ji Lin ◽  
C. Z. Zhang
Keyword(s):  
Elastic Foundation ◽  
Plate Theory ◽  
Nonlocal Theory ◽  
Mindlin Plate ◽  
Governing Equations ◽  
General Applicability ◽  
Particular Solutions ◽  
Pasternak Elastic Foundation ◽  
Mindlin Plate Theory ◽  
Selection Of

AbstractBased on the nonlocal theory and Mindlin plate theory, the governing equations (i.e., a system of partial differential equations (PDEs) for bending problem) of magnetoelectroelastic (MEE) nanoplates resting on the Pasternak elastic foundation are first derived by the variational principle. The polynomial particular solutions corresponding to the established model are then obtained and further employed as basis functions with the method of particular solutions (MPS) to solve the governing equations numerically. It is confirmed that for the present bending model, the new solution strategy possesses more general applicability and superior flexibility in the selection of collocation points. Finally, the effects of different boundary conditions, applied loads, and geometrical shapes on the bending properties of MEE nanoplates are evaluated by using the developed method. Some important conclusions are drawn, which should be helpful for the design and applications of electromagnetic nanoplate structures.

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.

Closed-Form Solutions for Eigenbuckling of Rectangular Mindlin Plate

2016 ◽  
Vol 16 (08) ◽  
pp. 1550079 ◽  
Author(s):  
Yufeng Xing ◽  
Wei Xiang
Keyword(s):  
Closed Form ◽  
Present Method ◽  
Separation Of Variables ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Major Development ◽  
First Time ◽  
Good Agreement

This paper studies the eigenbuckling of Mindlin plate with two adjacent edges clamped and the remaining edges simply supported or clamped by using the separation of variables method, and the concise and explicit closed-form solutions are obtained for the first time. The cases involving free edges can also be dealt with if there are two opposite edges simply supported. The closed-form solutions are in good agreement with the existing solutions, thus the validity of present method and accuracy of the obtained solutions are verified. This paper proves to be a major development of analytical method since it has long been acknowledged that the eigenbuckling of rectangular plates without two parallel edges simply supported are not amenable to analytical solutions.

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.

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