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Nonlinear static bending of single-crystalline circular nanoplates with cubic material anisotropy

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Abstract

In this work, a model is presented for nonlinear static bending of single circular nanoplates with cubic material anisotropy. The material anisotropy is modeled based on Zener ratio for which some new functions are presented to get closer formulations. The Ritz and Galerkin weighted residual methods are used to solve the derived nonlinear differential equation using different deflection surfaces defining two integral functions for material anisotropy in polar coordinate. The results show that the deflection surface remains approximately axisymmetric even at extreme material anisotropy levels similar to orthotropic materials. This is verified by FEM too. The comprehensive results are presented and discussed for the mostly used materials at MEMS/NEMS technology including Ag, Au, Al, Si, Ni, Mo, Cu and W single-crystalline nanoplates. It is observed that the size scale effects related to material surface stresses are greater for BCC nanoplates in comparison with FCC ones. Moreover, the Poisson’s ratio in [100] direction plays an essential role in the problem which is evaluated carefully. Finally, the effective Young’s modulus of the nanoplates with 10–80 nm of thickness is compared by the experiments for Ag nanowires using a correction factor due to lack of direct experiments for nanoplates.

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Abbreviations

(\(r, \theta , z\)):

Cylindrical coordinates

[100], [110] and [010]:

Standard crystallographic directions

(001) and (111):

Crystallographic planes

\(E_{\theta } \) :

Young’s modulus

\(\nu _{\theta } \) :

Poisson’s ratio

\(G_{\theta } \) :

Shear modulus

\(E_{[100]}\) and \(E_{[ 110 ]}\) :

Young’s modulus in [100] and [110] directions

\(\nu _{[100]} \) :

Poisson’s ratio in [100] direction

\(G_{[100]} \) and \(G_{[110]} \) :

Shear modulus in [100] and [110] directions

\(E^\mathrm{eff}\) :

Effective Young’s modulus

\(\tau ^{\mathrm{s}}, \tau _{0}\) :

Surface residual stress

\(E^{\mathrm{s}}\) :

Material surface elasticity

\(\sigma _{ij}^{\mathrm{Res}}\) :

Bulk residual stresses

\(R_{E}\) :

Anisotropy ratio

\(R_{\mathrm{Zen}} \) :

Zener ratio

\({[ S_{[ {100}]} ]}\) :

Compliance or flexibility matrix

\({[ T_{\theta } ]}\) :

Transformation matrix

[R]:

Engineering-tensor interchange matrix

\(\varepsilon _{r\theta }^{\mathrm{Res}} \) :

Residual strain components

\(N_{ii}^{\mathrm{Res}} \) :

Normal residual loads

\(D_{[100]}^\mathrm{eff} \) :

Effective flexural rigidity in [100 direction]

\(\,B_{[100]}^\mathrm{eff} \) :

Effective extensional rigidity in [100 direction]

\(\psi _{\theta }\) and \(\varphi _{\theta }\) :

Flexural rigidity orientation functions

\(\oint _1 \) :

1st integral function

\(\oint _2 \) :

2nd integral function

\({\bar{W}}={\tilde{w}}/h\) :

Normalized deflection

\(X=r/R\) :

Normalized radius

\(\eta =R/h\) :

Nanoplates’ aspect ratio

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Authors and Affiliations

  1. Faculty of Mechanics, Malek Ashtar University of Technology, Shahin Shahr, Iran

    Abbas Assadi

  2. Micro and Nano-Engineering, Faculty of Mechanics, Malek Ashtar University of Technology, Shahin Shahr, Iran

    Hossein Najaf

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  1. Abbas Assadi
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Correspondence to Abbas Assadi.

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Assadi, A., Najaf, H. Nonlinear static bending of single-crystalline circular nanoplates with cubic material anisotropy. Arch Appl Mech 90, 847–868 (2020). https://doi.org/10.1007/s00419-019-01643-9

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