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Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory

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Abstract

A continuum model based on the nonlocal theory of elasticity is developed for buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates under uniform in-plane compression. The nanoplate is considered to be rested on two-parameter Winkler-Pasternak elastic foundation. The principle of virtual work is used to derive the governing vibration and stability equations. The weighted residual statements of the equations of motion are performed and the well-known Galerkin method is employed to obtain the nonlocal “Quadratic Functional” for embedded micro/nano-plates. The Ritz functions are taken to form an expression for transverse displacement which satisfies the kinematic boundary conditions. In this way, the entire nanoplate is considered as a single super-continuum element. Employing the Ritz functions eliminates the need for mesh generation and thus large number of degrees of freedom arising in discretization methods such as finite element (FE). The results show obvious dependency of critical buckling loads on the non-locality of the micro/nano elliptical plate, especially, at very small dimensions.

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Abbreviations

a,b :

semi-major axis and semi-minor axis of elliptical nanoplate

[B]:

buckling matrix of the nanoplate

C i :

unknown coefficients in the assumed expression for w

D ij :

components of bending rigidity tensor

E x , E y :

Young’s moduli of the material of nanoplate in x and y directions

G xy :

shear modulus of the material of nanoplate

h :

thickness of elliptical nanoplate

k w :

Winkler coefficient of two-parameter elastic foundation

k p :

Pasternak (shear) coefficient of two-parameter elastic foundation

[K]:

stiffness matrix of the nanoplate

m 0 :

mass per unit of area of the nanoplate

m 2 :

mass moment of inertia of the nanoplate

M xx ,M yy ,M xy :

moment resultants

[M]:

mass matrix of the nanoplate

N xx ,N yy ,N xy :

in-plane stress resultants

q 0 :

transverse distributed load

u,v :

displacement of the nanoplate particles in x and y directions

w :

displacement of the nanoplate particles in z direction

χ :

weight function

ε ij :

components of strain tensor

Φ i :

Ritz functions

λ b :

buckling parameter

μ :

nonlocal parameter

ν x ,ν y :

Poisson’s ratios of nanoplate in x and y directions

\(\varPi_{\mathrm{tot}}^{(nl)}\) :

nonlocal quadratic functional for nanoplates

ρ :

mass per unit of volume of the nanoplate

ω :

vibration frequency of the nanoplate

\(\sigma_{ij}^{(l)}\) :

components of local stress tensor

\(\sigma_{ij}^{(nl)}\) :

components of nonlocal stress tensor

∇2 :

Laplacian operator in three-dimensional Cartesian coordinate system

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Authors and Affiliations

  1. Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Amin Anjomshoa

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  1. Amin Anjomshoa
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Correspondence to Amin Anjomshoa.

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Anjomshoa, A. Application of Ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory. Meccanica 48, 1337–1353 (2013). https://doi.org/10.1007/s11012-012-9670-y

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  • DOI: https://doi.org/10.1007/s11012-012-9670-y

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