Shear Deformation in Rectangular Auxetic Plates

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Teik-Cheng Lim
Keyword(s):  
Shear Deformation ◽  
Poisson’S Ratio ◽  
Plate Thickness ◽  
Poisson's Ratio ◽  
Kirchhoff Plate ◽  
Thin Plates ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Bending Deformation ◽  
Plate Deflection

Solids that exhibit negative Poisson's ratio are called auxetic materials. This paper examines the extent of transverse shear deformation with reference to bending deformation in simply supported auxetic plates as a ratio of Mindlin-to-Kirchhoff plate deflection for polygonal plates in general, with special emphasis on rectangular plates. Results for square plates show that the Mindlin plate deflection approximates the Kirchhoff plate deflection not only when the plate thickness is negligible, as is obviously known, but also when (a) the Poisson's ratio of the plate is very negative under all load distributions, as well as (b) at the central portion of the plate when the load is uniformly distributed. Hence geometrically thick plates are mechanically equivalent to thin plates if the plate Poisson's ratio is sufficiently negative. The high suppression of shear deformation in favor of bending deformation in auxetic plates suggests its usefulness for bending-based plate sensors that require larger difference in the in-plane strains between the opposing plate surfaces with minimal transverse deflection.

Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

2020 ◽  
Vol 71 (7) ◽  
pp. 853-867
Author(s):  
Phuc Pham Minh
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Phase Field ◽  
Deformation Theory ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Rectangular Plates ◽  
Equilibrium Equations ◽  
Third Order ◽  
Free Vibration Frequency

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions

Optimal location of ring support for heavy Mindlin plates under axisymmetric loading

2017 ◽  
Vol 232 (7) ◽  
pp. 1270-1279
Author(s):  
CN Xie ◽  
XF Li
Keyword(s):  
Poisson’S Ratio ◽  
Plate Theory ◽  
Outer Radius ◽  
Optimal Location ◽  
Poisson's Ratio ◽  
Mindlin Plate ◽  
Axisymmetric Loading ◽  
Applied Loading ◽  
Ring Support ◽  
Governing Differential Equation

Bending of an annular thick plate resting on a ring support is analyzed under the action of power-law axisymmetric loading. A single governing differential equation for the Mindlin plate theory is derived. By solving associated boundary value problem, the optimal support location is determined to achieve minimizing the maximum deflection of a moderately thick circular or annular plate. The minimum sag of a heavy solid circular plate with or without the center support under self-weight is also analyzed. In addition to applied loading and the restraint of plate's rims, the optimal location of the ring support is also related to Poisson's ratio and the ratio of inner-to-outer radius. Auxetic plates with negative Poisson's ratio require larger ring support's radius, and conventional plates require smaller ring support's radius. Usually, the optimal support location is closer to the outer rim and far to the inner rim for a plate of self-weight. The obtained results are useful in safety design of circular or annular plates under complicated loading.

Flexural Vibrations of Rectangular Plates

1956 ◽  
Vol 23 (3) ◽  
pp. 430-436
Author(s):  
R. D. Mindlin ◽  
A. Schacknow ◽  
H. Deresiewicz
Keyword(s):  
Shear Deformation ◽  
Classical Theory ◽  
The Other ◽  
Thin Plates ◽  
Rectangular Plates ◽  
Rotatory Inertia ◽  
Simply Supported ◽  
Independent Families ◽  
Flexural Vibrations ◽  
Elastically Supported

Abstract The influence of rotatory inertia and shear deformation on the flexural vibrations of isotropic, rectangular plates is investigated. Three independent families of modes are possible when the edges are simply supported. Coupling of the modes is studied for the case of one pair of parallel edges free and the other pair simply supported. The development of the coupling is traced by means of a solution for elastically supported edges. Special attention is given to the higher modes and frequencies of vibration which are beyond the range of applicability of the classical theory of thin plates.

Studies Regarding the Shear Correction Factor in the Mindlin Plate Theory for Sandwich Plates

2017 ◽  
Vol 21 ◽  
pp. 301-308 ◽  
Author(s):  
Mihai Vrabie ◽  
Radu Chiriac ◽  
Sergiu Andrei Băetu
Keyword(s):  
Shear Deformation ◽  
Correction Factor ◽  
Plate Theory ◽  
Plate Thickness ◽  
Shear Stiffness ◽  
Laminated Plates ◽  
Mindlin Plate ◽  
Sandwich Plates ◽  
Shear Correction Factor ◽  
Mindlin Plate Theory

The displacement field from the first-order shear deformation plate theory (FSDT) extents the cinematic aspect of the classical theory of laminated plates (CLPT), including a transverse shear deformation, considered constant on the plate thickness. In order to correct this aspect, in FSDT (named also the Mindlin plate theory) a coefficient Ks was inserted, named shear correction factor, used as a multiplier in the shear stiffness equation of the plate. In this paper are presented the most popular methods for determination of the shear correction factor, identifying the differences between them. To emphasize the influence of the shear correction factor on the stress response, a numerical parametric study was done on some sandwich plates filled with polyurethane foam. The processing of the obtained results allow drawing some conclusions useful in the designing of this type of sandwich plates.

The review of the bond-based peridynamics modeling

2019 ◽  
Vol 04 (01) ◽  
pp. 1830001 ◽  
Author(s):  
Duanfeng Han ◽  
Yiheng Zhang ◽  
Qing Wang ◽  
Wei Lu ◽  
Bin Jia
Keyword(s):  
Meshless Method ◽  
Poisson’S Ratio ◽  
Review Paper ◽  
Three Dimensional ◽  
Poisson's Ratio ◽  
Thin Plates ◽  
Advantages And Disadvantages ◽  
Starting Point ◽  
Material Points ◽  
Elastic Spring

Peridynamics theory is a nonlocal meshless method that replaces differential equations with spatial integral equations, and has shown good applicability and reliability in the analysis of discontinuities. Further, with characteristics of clear physical meaning and simple and reliable numerical calculation, the bond-based peridynamics method has been widely applied in the field. However, this method describes the interaction between material points simply using a single elastic “spring”, and thus leads to a fixed Poisson’s ratio, relatively low computational efficiency and other inherent problems. As such, the goal of this review paper is to provide a summary of the various methods of bond-based peridynamics modeling, particularly those that have overcome the limitations of the Poisson’s ratio, considered the shear deformation and modeling of two-dimensional thin plates for bending and three-dimensional anisotropic composites, as well as explored coupling with finite element methods. This review will determine the advantages and disadvantages of such methods and serve as a starting point for researchers in the development of peridynamics theory.

scholarly journals Static Bending Analysis of Auxetic Plate by FEM and a New Third-Order Shear Deformation Plate Theory

2020 ◽  
Vol 36 (1) ◽  
Author(s):  
Pham Hong Cong ◽  
Pham Minh Phuc ◽  
Hoang Thi Thiem ◽  
Duong Tuan Manh ◽  
Nguyen Dinh Duc
Keyword(s):  
Finite Element ◽  
Shear Deformation ◽  
Poisson’S Ratio ◽  
Plate Theory ◽  
Poisson's Ratio ◽  
Element Formulation ◽  
Negative Poisson’S Ratio ◽  
Third Order ◽  
Shear Deformation Plate Theory ◽  
Negative Poisson's Ratio

In this paper, a finite element method (FEM) and a new third-order shear deformation plate theory are proposed to investigate a static bending model of auxetic plates with negative Poisson’s ratio. The three – layer sandwich plate is consisted of auxetic honeycombs core layer with negative Poisson’s ratio integrated, isotropic homogeneous materials at the top and bottom of surfaces. A displacement-based finite element formulation associated with a novel third-order shear deformation plate theory without any requirement of shear correction factors is thus developed. The results show the effects of geometrical parameters, boundary conditions, uniform transverse pressure on the static bending of auxetic plates with negative Poisson’s ratio. Numerical examples are solved, then compared with the published literatures to validate the feasibility and accuracy of proposed analysis method. Keywords: Static bending; New third-order shear deformation plate theory; Auxetic material.

scholarly journals Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

2020 ◽  
Vol 71 (7) ◽  
pp. 853-867
Author(s):  
Phuc Pham Minh
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Phase Field ◽  
Deformation Theory ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Rectangular Plates ◽  
Equilibrium Equations ◽  
Third Order ◽  
Free Vibration Frequency

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions

ANALYSIS OF AUXETIC BEAMS AS RESONANT FREQUENCY BIOSENSORS

2012 ◽  
Vol 12 (05) ◽  
pp. 1240027 ◽  
Author(s):  
TEIK-CHENG LIM
Keyword(s):  
Shear Deformation ◽  
Cross Sections ◽  
Timoshenko Beam ◽  
Poisson’S Ratio ◽  
Beam Theory ◽  
Timoshenko Beam Theory ◽  
Poisson's Ratio ◽  
Bernoulli Beam ◽  
Bernoulli Beam Theory ◽  
Euler Bernoulli Beam

The mechanics of beam vibration is of fundamental importance in understanding the shift of resonant frequency of microcantilever and nanocantilever sensors. Unlike the simpler Euler–Bernoulli beam theory, the Timoshenko beam theory takes into consideration rotational inertia and shear deformation. For the case of microcantilevers and nanocantilevers, the minute size, and hence low mass, means that the topmost deviation from the Euler–Bernoulli beam theory to be expected is shear deformation. This paper considers the extent of shear deformation for varying Poisson's ratio of the beam material, with special emphasis on solids with negative Poisson's ratio, which are also known as auxetic materials. Here, it is shown that the Timoshenko beam theory approaches the Euler–Bernoulli beam theory if the beams are of solid cross-sections and the beam material possess high auxeticity. However, the Timoshenko beam theory is significantly different from the Euler–Bernoulli beam theory for beams in the form of thin-walled tubes regardless of the beam material's Poisson's ratio. It is herein proposed that calculations on beam vibration can be greatly simplified for highly auxetic beams with solid cross-sections due to the small shear correction term in the Timoshenko beam deflection equation.

Nonlinear Vibration of Plates and Higher Order Theory for Different Boundary Conditions

2010 ◽  
Author(s):  
Marco Amabili ◽  
Kostas Karagiozis ◽  
Sirwan Farhadi ◽  
Korosh Khorshidi
Keyword(s):  
Boundary Conditions ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Laminate Composite ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Rectangular Plates ◽  
Different Boundary Conditions ◽  
Simply Supported

There are numerous applications of plate structures found in structural, aerospace and marine engineering. The present study extends the previous work by Amabili and Sirwan [1] investigating the performance of isotropic and laminate composite rectangular plates with different boundary conditions subjected to an external point force with an excitation frequency that lies in the neighbourhood of the fundamental mode of the plate. The analysis is performed using three different nonlinear plate theories, namely: i) the classical Von Ka´rman theory, ii) first-order shear deformation theory, and iii) third-order shear deformation theory. Three different boundary conditions are considered in the investigation: a) classical clamped boundary conditions, b) simply-supported ends with immovable edges, and c) simply-supported ends with movable boundaries. In addition, the effect of thickness was also considered in the analysis and different values for the plate thickness were assumed. The results investigate the accuracy of lower order theories versus higher order shear deformation theories, the effect of boundary conditions and highlight the differences in the responses obtained from isotropic and laminate composite rectangular plates.

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