On the Stress Singularities and Boundary Layer in Moderately Thick Functionally Graded Sectorial Plates

2010 ◽  
Author(s):  
A. R. Saidi ◽  
F. Hejripour ◽  
E. Jomehzadeh
Keyword(s):  
Boundary Layer ◽  
Plate Theory ◽  
Functionally Graded ◽  
Stress Singularities ◽  
Equilibrium Equations ◽  
Arbitrary Boundary ◽  
Edge Zone ◽  
Layer Function ◽  
Sector Plate ◽  
Boundary Layer Function

In this paper, the stress analysis of moderately thick functionally graded (FG) sector plate is developed for studying the singularities in vicinity of the vertex. Based on the first-order shear deformation plate theory, the governing partial differential equations are obtained. Using an analytical method and defining some new functions, the stretching and bending equilibrium equations are decoupled. Also, introducing a function, called boundary layer function, the three bending equations are converted into two decoupled equations called edge-zone and interior equations. These equations are solved analytically for the sector plate with the simply supported radial edges and arbitrary boundary condition along the circular edge. The singularities of shear force and moment resultants are discussed for both salient and re-entrant sectorial plates. Also, the effects of power of the FGM, thickness to length ratio on the stress singularities of the FG sector plates are investigated.

Elastic Foundation Analysis of Uniformly Loaded Functionally Graded Viscoelastic Sandwich Plates

2012 ◽  
Vol 28 (3) ◽  
pp. 439-452 ◽  
Author(s):  
A. M. Zenkour ◽  
M. Sobhy
Keyword(s):  
Power Law ◽  
Sandwich Plate ◽  
Plate Theory ◽  
Functionally Graded ◽  
Sandwich Plates ◽  
Equilibrium Equations ◽  
Graded Material ◽  
Plate Aspect Ratio ◽  
Type V ◽  
Power Law Index

AbstractThis paper deals with the static response of simply supported functionally graded material (FGM) viscoelastic sandwich plates subjected to transverse uniform loads. The FG sandwich plates are considered to be resting on Pasternak's elastic foundations. The sandwich plate is assumed to consist of a fully elastic core sandwiched by elastic-viscoelastic FGM layers. Material properties are graded according to a power-law variation from the interfaces to the faces of the plate. The equilibrium equations of the FG sandwich plate are given based on a trigonometric shear deformation plate theory. Using Illyushin's method, the governing equations of the viscoelastic sandwich plate can be solved. Parametric study on the bending analysis of FG sandwich plates is being investigated. These parameters include (i) power-law index, (ii) plate aspect ratio, (iii) side-to-thickness ratio, (iv) loading type, (v) foundation stiffnesses, and (vi) time parameter.

A closed form solution for bending/stretching analysis of functionally graded circular plates under asymmetric loading using the Green function

2010 ◽  
Vol 224 (6) ◽  
pp. 1153-1163 ◽  
Author(s):  
A R Saidi ◽  
A Naderi ◽  
E Jomehzadeh
Keyword(s):  
Green Function ◽  
Closed Form ◽  
Volume Fraction ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Form Solution ◽  
Functionally Graded ◽  
Transverse Displacement ◽  
Circular Plates ◽  
Equilibrium Equations

In this article, a closed-form solution for bending/stretching analysis of functionally graded (FG) circular plates under asymmetric loads is presented. It is assumed that the material properties of the FG plate are described by a power function of the thickness variable. The equilibrium equations are derived according to the classical plate theory using the principle of total potential energy. Two new functions are introduced to decouple the governing equilibrium equations. The three highly coupled partial differential equations are then converted into an independent equation in terms of transverse displacement. A closed-form solution for deflection of FG circular plates under arbitrary lateral eccentric concentrated force is obtained by defining a new coordinate system. This solution can be used as a Green function to obtain the closed-form solution of the FG plate under arbitrary loadings. Also, the solution is employed to solve some different asymmetric problems. Finally, the stress and displacement components are obtained exactly for each problem and the effect of volume fraction is also studied.

Maximization of Natural Frequencies for Functionally Graded Rectangular Plates

2021 ◽  
Vol 891 ◽  
pp. 116-121
Author(s):  
Aleksander Muc
Keyword(s):  
Analytical Methods ◽  
Natural Frequencies ◽  
Plate Theory ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Optimization Techniques ◽  
Point Of View ◽  
Rectangular Plates ◽  
Classical Plate Theory ◽  
Arbitrary Boundary

In this paper optimal design of free vibrations for functionally graded plates is studied using the analytical methods. The analytical methods can be employed for the solution of six of 21 arbitrary boundary conditions (the combinations of clamped, simply supported and free). The influence of various models of porosity and forms of different reinforcements with nanoplatelets and carbon nanotubes are investigated, including variations of stiffness/density along the thickness of a plate. The analysis is carried out for the classical plate theory. Parametric studies illustrate the possibility of increasing natural frequencies and the necessity of implementing the optimization techniques to find the best solutions from the engineering point of view.

Effective Approximation for Elliptic Partial Differential Equation with Periodical Boundary Value Problem

2010 ◽  
Vol 29-32 ◽  
pp. 1294-1300
Author(s):  
Xin Cai
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Partial Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Elliptic Partial Differential Equation ◽  
Partial Differential ◽  
Layer Function ◽  
Smooth Component ◽  
Boundary Layer Function

Elliptic partial differential equation with periodical boundary value problem was considered. The equation would degenerate to parabolic partial differential equation when small parameter tends to zero. This is a multi-scale problem. Firstly, the property of boundary layer was discussed. Secondly, the boundary layer function was presented. The smooth component was constructed according to the boundary layer function. Thirdly, finite difference scheme for the smooth component is proposed according to transition point in time direction. Finally, experiment was proposed to illustrate that our presented method is an effective computational method.

scholarly journals Singularly perturbed parabolic problem with oscillating initial condition

Filomat ◽  
2019 ◽  
Vol 33 (5) ◽  
pp. 1323-1327
Author(s):  
Asan Omuraliev ◽  
Ella Abylaeva
Keyword(s):  
Boundary Layer ◽  
Parabolic Problem ◽  
Singularly Perturbed ◽  
Initial Condition ◽  
Parabolic Boundary ◽  
Layer Function ◽  
Angular Boundary ◽  
Singularly Perturbed Parabolic Problem ◽  
Boundary Layer Function ◽  
Regularized Asymptotics

The aim of this paper is to construct regularized asymptotics of the solution of a singularly perturbed parabolic problem with an oscillating initial condition. The presence of a rapidly oscillating function in the initial condition has led to the appearance of a boundary layer function in the solution, which has the rapidly oscillating character of the change. In addition, it is shown that the asymptotics of the solution contains exponential, parabolic boundary layer functions and their products describing the angular boundary layers. Continuing the ideas of works [1, 3] a complete regularized asymptotics of the solution of the problem is constructed.

Dynamic Behavior Study of Functionally Graded Porous Nanoplates

2021 ◽  
Vol 33 ◽  
pp. 83-92
Author(s):  
Hadj Mostefa Adda ◽  
Bouchafa Ali ◽  
Merdaci Slimane
Keyword(s):  
Natural Frequencies ◽  
Plate Theory ◽  
Functionally Graded ◽  
Small Scale ◽  
Dimensionless Frequency ◽  
Equilibrium Equations ◽  
Local Principle ◽  
Behavior Study ◽  
Small Scale Effect ◽  
The Impact

This paper introduces the analytical solutions of complex behavior analysis utilizing high-order shear deformation plate theory of functionally graded FGM nano-plate content consisting of a mixture of metal and ceramics with porosity. To incorporate the small-scale effect, the non-local principle of elasticity is used. The impact of variance of material properties such as thickness-length ratio, aspect ratio, power-law exponent and porosity factor on natural frequencies of FG nano-plate is examined. Compared to those achieved from other researchers, the latest solutions are. Using the simulated displacements theory, equilibrium equations are obtained. Current solutions of the dimensionless frequency are compared with those of the finite element method. The effect of geometry, material variations of nonlocal FG nano-plates and the porosity factor on their natural frequencies are investigated in this review. The results are in good agreement with those of the literature.

Bending Analysis of Thick Laminated Rectangular Plates Using a Boundary Layer Function

2010 ◽  
Vol 224 (10) ◽  
pp. 2073-2081 ◽  
Author(s):  
A R Saidi ◽  
S R Atashipour ◽  
H Keshavarzi
Keyword(s):  
Boundary Layer ◽  
Shear Deformation Theory ◽  
Transversely Isotropic ◽  
Rectangular Plates ◽  
Governing Equations ◽  
Third Order ◽  
Transversely Isotropic Plate ◽  
Reissner Theory ◽  
Layer Function ◽  
Boundary Layer Function

In this article, the governing bending equations of thick laminated transversely isotropic rectangular plates are derived based on third-order shear deformation theory (TSDT). Using a new function, called the boundary layer function, the three coupled governing equations are converted to two decoupled equations. These equations are in terms of the deflection of the plate and the mentioned boundary layer function, which are written in invariant form. By solving the decoupled equations, a Levy-type analytical solution is presented for bending of a transversely isotropic plate. Finally, numerical results are presented for boundary layer phenomenon and its effects in TSDT. It is shown that all of the boundary layer effects in Mindlin—Reissner theory appear in this theory. However, it is shown that the intensity of the boundary layer effects in TSDT exceeds that of the Mindlin—Reissner theory.

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