scholarly journals Influence of Variable Nonlocal Parameter and Porosity on the Free Vibration Behavior of Functionally Graded Nanoplates

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Pham Van Vinh ◽  
Le Quang Huy
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Form Solution ◽  
Functionally Graded ◽  
Governing Equations ◽  
Vibration Behavior ◽  
Special Cases

This paper studies the influence of the variable nonlocal parameter and porosity on the free vibration behavior of the functionally graded nanoplates with porosity. Four patterns of distribution of the porosity through the thickness direction are considered. The classical nonlocal elasticity theory is modified to take into account the variation of the nonlocal parameter through the thickness of the nanoplates. The governing equations of motion are established using simple first-order shear deformation theory and Hamilton’s principle. The closed-form solution based on Navier’s technique is employed to solve the governing equations of motion of fully simply supported nanoplates. The accuracy of the present algorithm is proved via some comparison studies in some special cases. Then, the effects of the porosity, the variation of the nonlocal parameter, the power-law index, aspect ratio, and the side-to-thickness ratio on the free vibration of nanoscale porous plates are investigated carefully. The numerical results show that the porosity and nonlocal parameter have strong effects on the free vibration behavior of the nanoplates.

Size-dependent free vibration analysis of three-layered exponentially graded nanoplate with piezomagnetic face-sheets resting on Pasternak’s foundation

2017 ◽  
Vol 29 (5) ◽  
pp. 774-786 ◽  
Author(s):  
M Arefi ◽  
MH Zamani ◽  
M Kiani
Keyword(s):  
Free Vibration ◽  
Equations Of Motion ◽  
Nonlocal Parameter ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Governing Equations ◽  
First Order ◽  
Size Dependent ◽  
Inhomogeneity Parameter ◽  
Exponentially Graded

This work is devoted to the free vibration nonlocal analysis of an elastic three-layered nanoplate with exponentially graded graphene sheet core and piezomagnetic face-sheets. The rectangular elastic three-layered nanoplate is resting on Pasternak’s foundation. Material properties of the core are supposed to vary along the thickness direction based on the exponential function. The governing equations of motion are derived from Hamilton’s principle based on first-order shear deformation theory. In addition, Eringen’s nonlocal piezo-magneto-elasticity theory is used to consider size effects. The analytical solution is presented to solve seven governing equations of motion using Navier’s solution. Eventually, the natural frequency is scrutinized for different side length ratio, nonlocal parameter, inhomogeneity parameter, and parameters of foundation numerically. The comparison with various references is performed for validation of our analytical results.

Influence of Symmetrical Boundary Conditions on Free Vibration of Functionally Graded Cylindrical Shells With Ring Support

2009 ◽  
Author(s):  
M. R. Isvandzibaei ◽  
M. M. Najafizadeh ◽  
P. Khazaeinejad
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Governing Equations ◽  
Third Order ◽  
Graded Materials ◽  
Ring Support

In the present work, the free vibration of thin cylindrical shells with ring support made of functionally graded materials under various symmetrical boundary conditions is presented. Temperature and position dependent material properties are varied linearly through the thickness of the shell. The functionally graded cylindrical shell has ring support which is arbitrarily placed along the shell and imposed a zero lateral deflection. The third order shear deformation theory is employed to formulate the problem. The governing equations of motion are derived using the Hamilton’s principle. Results are presented on the frequency characteristics and influence of the boundary conditions and the locations of the ring support on the natural frequencies. The present analysis is validated by comparing the results with those available in the literature.

Study on free vibration behavior of rotating bidirectional functionally graded nano-beams based on Eringen’s nonlocal theory

2020 ◽  
Vol 234 (9) ◽  
pp. 1203-1217 ◽  
Author(s):  
Manash Malik ◽  
Debabrata Das
Keyword(s):  
Free Vibration ◽  
Nonlocal Parameter ◽  
Nonlocal Theory ◽  
Functionally Graded ◽  
Numerical Results ◽  
Mode Switching ◽  
Mode Shapes ◽  
Governing Equations ◽  
Vibration Behavior ◽  
Gradient Parameter

Free vibration behavior of rotating bidirectional functionally graded nano-beams is studied based on Eringen’s nonlocal theory. The beam material consists of ceramic and metal constituents, and the material is graded across the length and thickness directions. The mathematical formulation is framed on Euler–Bernoulli beam theory, and the system of governing equations is derived in variational form using Hamilton’s principle. The governing equations are discretized and transformed to an eigen value problem using Ritz method. The model is formulated to study the flapping and lead-lag motions due to free vibration. The model is verified with the available numerical results. The numerical results are presented in non-dimensional frequency-speed plane to study the influence of normalized nonlocal parameter, length gradient parameter, thickness gradient parameter, root radius parameter, and section aspect ratio. Some normalized mode shapes are presented to illustrate the mode switching phenomenon. The mathematical model of a nonlocal rotating bidirectional functionally graded material nano-beam is presented for the first time through this work and the reported results are new of its kind.

Free Vibration Analysis of Composite Material Plates "Case of a Typical Functionally Graded FG Plates Ceramic/Metal" with Porosities

2019 ◽  
Vol 25 ◽  
pp. 69-83 ◽  
Author(s):  
Slimane Merdaci
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Porous Plate ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Simply Supported ◽  
Fg Plates

This article presents the free vibration analysis of simply supported plate FG porous using a high order shear deformation theory. In is work the material properties of the porous plate FG vary across the thickness. The proposed theory contains four unknowns unlike the other theories which contain five unknowns. This theory has a parabolic shear deformation distribution across the thickness. So it is useless to use the shear correction factors. The Hamilton's principle will be used herein to determine the equations of motion. Since, the plate are simply supported the Navier procedure will be retained. To show the precision of this model, several comparisons have been made between the present results and those of existing theories in the literature for non-porous plates. Effects of the exponent graded and porosity factors are investigated.

Well-posed two-phase nonlocal integral models for free vibration of nanobeams in context with higher-order refined shear deformation theory

2021 ◽  
pp. 107754632110399
Author(s):  
Pei Zhang ◽  
Hai Qing
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Shear Deformation ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Higher Order ◽  
Well Posedness ◽  
Governing Equations ◽  
Two Phase

In this article, the well-posedness of several common nonlocal models for higher-order refined shear deformation beams is studied. Unlike the case of classic beams models, both strain-driven and stress-driven purely nonlocal theories lead to an ill-posed issue (i.e., there are excessive mandatory boundary conditions) when considering higher-order shear deformation assumption. As an effective remedy, the well-posedness of strain-driven and stress-driven two-phase nonlocal (StrainDTPN and StressDTPN) models is pertinently evidenced by studying the free vibration problem of nanobeams. The governing equations of motion and standard boundary conditions are derived from Hamilton’s principle. The integral constitutive relation is transformed equivalently to a differential form equipped with two constitutive boundary conditions. Using the generalized differential quadrature method (GDQM), the governing equations in terms of displacements are solved numerically. Numerical results show that both the StrainDTPN and StressDTPN models can predict consistent size-effects of beams with different boundary conditions.

scholarly journals Nonlocal Elasticity Response of Doubly-Curved Nanoshells

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 466 ◽  
Author(s):  
Mohammad Hassan Dindarloo ◽  
Li Li ◽  
Rossana Dimitri ◽  
Francesco Tornabene
Keyword(s):  
Nonlocal Parameter ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Theory Of Elasticity ◽  
Form Solution ◽  
Small Scale ◽  
Unified Framework ◽  
Geometrical Properties ◽  
Small Scale Effect ◽  
Doubly Curved

In this paper, we focus on the bending behavior of isotropic doubly-curved nanoshells based on a high-order shear deformation theory, whose shape functions are selected as an accurate combination of exponential and trigonometric functions instead of the classical polynomial functions. The small-scale effect of the nanostructure is modeled according to the differential law consequent, but is not equivalent to the strain-driven nonlocal integral theory of elasticity equipped with Helmholtz’s averaging kernel. The governing equations of the problem are obtained from the Hamilton’s principle, whereas the Navier’s series are proposed for a closed form solution of the structural problem involving simply-supported nanostructures. The work provides a unified framework for the bending study of both thin and thick symmetric doubly-curved shallow and deep nanoshells, while investigating spherical and cylindrical panels subjected to a point or a sinusoidal loading condition. The effect of several parameters, such as the nonlocal parameter, as well as the mechanical and geometrical properties, is investigated on the bending deflection of isotropic doubly-curved shallow and deep nanoshells. The numerical results from our investigation could be considered as valid benchmarks in the literature for possible further analyses of doubly-curved applications in nanotechnology.

Thermoelastic Stability of Imperfect Functionally Graded Plates Based on the Third Order Shear Deformation Theory

2006 ◽  
Author(s):  
B. Samsam Shariat ◽  
M. R. Eslami ◽  
A. Bagri
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Form Solution ◽  
Functionally Graded ◽  
Functionally Graded Plates ◽  
Equilibrium Equations ◽  
Third Order ◽  
The Third

Thermal buckling analysis of rectangular functionally graded plates with initial geometric imperfections is presented in this paper. It is assumed that the non-homogeneous mechanical properties vary linearly through the thickness of the plate. The plate is assumed to be under various types of thermal loadings, such as the uniform temperature rise and nonlinear temperature gradient through the thickness. A double-sine function for the geometric imperfection along the x and y-directions is considered. The equilibrium equations are derived using the third order shear deformation plate theory. Using a suitable method, equilibrium equations are reduced from 5 to 2 equations. The corresponding stability equations are established. Using these equations accompanied by the compatibility equation yield to the buckling loads in a closed form solution for each loading case. The results are compared with the known data in the literature.

Mechanical Buckling of Functionally Graded Cylindrical Shells Based on the First Order Shear Deformation Theory

2007 ◽  
Author(s):  
Ramin Narimani ◽  
Mehdi Karami Khorramabadi ◽  
Payam Khazaeinejad
Keyword(s):  
Shear Deformation ◽  
Deformation Theory ◽  
Cylindrical Shells ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Form Solution ◽  
Functionally Graded ◽  
Buckling Load ◽  
Critical Buckling Load ◽  
First Order

Buckling analysis of simply supported functionally graded cylindrical shells under mechanical loads is presented in this paper. The Young’s modulus of the shell is assumed to vary as a power form of the thickness coordinate variable. The shell is assumed to be under three types of mechanical loadings, namely, axial compression, uniform external lateral pressure, and hydrostatic pressure loading. The equilibrium and stability equations are derived based on the first order shear deformation theory. Resulting equations are employed to obtain the closed-form solution for the critical buckling load. The influences of dimension ratio, relative thickness and the functionally graded index on the critical buckling load are studied. The results are compared with the known data in the literature.

Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects

2020 ◽  
Vol 234 (18) ◽  
pp. 3667-3688
Author(s):  
Ismail Bensaid ◽  
Ahmed Amine Daikh ◽  
Ahmed Drai
Keyword(s):  
Free Vibration ◽  
Power Law ◽  
Functionally Graded Material ◽  
Volume Fraction ◽  
Nonlocal Parameter ◽  
Shear Deformation Theory ◽  
Nonlocal Elasticity Theory ◽  
Functionally Graded ◽  
Size Dependent ◽  
Graded Material

The investigation conducted in this paper aims to study free vibration and buckling behaviors of size-dependent functionally graded sandwich nanobeams. In order to take into account the small size effects, nonlocal elasticity theory of Eringen's is incorporated. Material properties of the functionally graded sandwich beams are supposed to change continuously through the thickness direction according to two forms of the volume fraction of constituents by power law functionally graded material and sigmoid law functionally graded material. These rules are modified to consider the effect of porosity, which covers four kinds of porosity distributions. Two types of sandwich nanobeams were provided: (a) homogeneous core and functionally graded skins and (b) functionally graded core and homogeneous skins. Third-order shear deformation theory without any shear correction factor in conjunction with Hamilton's principle is used to extract the governing equations of motions of porous functionally graded sandwich nanobeams and then solved analytically for two hinged ends. The effects of nonlocal parameter, length to thickness ratios, material graduation index, amount of porosity, porosity distribution shape, on the nondimensional frequency and critical buckling load of the functionally graded sandwich nanobeams made of porous materials are exhibited by a parametric study.

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