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A novel meshless particle method for nonlocal analysis of two-directional functionally graded nanobeams

  • M. Rezaiee-PajandEmail author
  • M. Mokhtari
Technical Paper
  • 41 Downloads

Abstract

Based on the Reddy–Bickford beam theory (RBBT), a comprehensive study on high-order bending, buckling and free vibration of two-directional functionally graded (FG) nanobeams is presented. The variation of the material properties is assumed to obey arbitrary power-law form in both axial and thickness directions. The equations of motion are derived using Hamilton’s principle, and small-scale effects are captured by nonlocal elasticity theory of Eringen. The RBBT formulation leads to complicated governing equations, and 2D FGM introduces additional stiffness terms. Accordingly, the governing equations cannot be solved by classical analytical schemes, and thus, symmetric smoothed particle hydrodynamics (SSPH) meshless method is adopted as an efficient numerical solution approach. The revised super-Gauss function is used as the kernel function. To validate the developed SSPH code, benchmark problems are studied and the results are compared to the analytical solutions. In this context, excellent agreement is observed. Several numerical examples are included to illustrate the effects of gradient indexes, boundary conditions, size scale parameters, aspect and elastic modulus ratios on static and dynamic responses of two-directional FG nanobeams.

Keywords

Symmetric smoothed particle hydrodynamics (SSPH) Two-directional FG nanobeams, meshless method Static bending, buckling Free vibration 

List of symbols

K

Nonlocal modulus

\( \mu \)

Nonlocal parameter

E

Elastic modulus

G

Shear modulus

ρ

Density

V

Volume fraction

\( \sigma_{xx} \)

Axial stress

L

Beam length

d

Radius of the CSD

W

Kernel function

A

Cross sectional area

\( k_{z} ,k_{x} \)

Gradient indexes

e

Error norm

\( \varepsilon_{xx} \)

Axial strain

\( \sigma_{xz} \)

Shear stress

\( \gamma_{xz} \)

Shear strain

U

Potential energy

Ve

External work

T

Kinetic energy

\( I_{i} ,J_{i} \)

Inertia coefficients

h

Beam height

\( \rho_{0} \)

Scaling factor

p

Dimensionality of space

M

Number of nodes (particles)

k

Mode number

\( \lambda \)

Normalized critical buckling load

u

Axial displacement

w

Transverse deflection

\( \phi \)

Bending rotation

\( N_{0}^{cr} \)

Critical buckling load

\( \omega_{n} \)

Natural frequency

\( A_{i} ,B_{i} \)

Stiffness coefficients

I

Moment of inertia

b

Beam width

\( h_{0} \)

Smoothing length

\( N_{0} \)

Compressive load

q

Transverse load

\( \nabla \)

Laplacian operator

v

Poisson’s ratio

Notes

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Civil EngineeringFerdowsi University of MashhadMashhadIran

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