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Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method

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Abstract

In this article, buckling analysis of double-orthotropic nanoplates (DONP) embedded in elastic media under biaxial, uniaxial and shear loading is numerically studied. The analysis is based on non-local theory. Both two-variable refined plate theory (TVRPT) and first-order shear deformation plate theory (FSDT) are used to derive the governing equations. Generalized differential quadrature method (GDQM) is utilized to solve the governing equations. In buckling analysis, both in-phase and out-of-phase modes are studied. A graphene sheet is selected as the case study to investigate the numerical results. GDQM results are validated by comparing with the Navier’s solutions. After validating the formulation and method of solution, the effect of non-local parameter, geometrical parameters and boundary conditions on the critical buckling load of the double-orthotropic nanoplate are investigated and discussed in detail. It is shown that the effects of non-local parameter for shear buckling are more noticeable than that of biaxial buckling. Moreover, for higher values of non-local parameter, the shear buckling is not dependent on the van der Waals and Winkler moduli.

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Abbreviations

DONP:

Double-orthotropic nanoplate

DQM:

Differential quadrature method

GDQM:

Generalized differential quadrature method

TVRPT:

Two-variable refined plate theory

FSDT:

First-order shear deformation theory

HSDT:

Higher-order shear deformation theories

CPT:

Classical plate theory

NDCBL:

Non-dimensional critical buckling load

a, b, h :

Length, width and thickness of nanoplate, respectively

E ij :

Modulus of elasticity

v ij :

Poisson’s ratio

u b, v b, w b :

Bending displacements in the x, y and z directions, respectively

u s, v s, w s :

Shear displacements in the x, y and z directions, respectively

σ ij :

Stress components

ε ij :

Strain components

M ij :

Bending moment components

D ij :

Bending stiffness

A ij :

Extensional stiffness

G ij :

Shear modulus

x, y, z :

Cartesian coordinates

b:

Bending components

s:

Shear components

(),x , (),y :

Partial derivatives with respect to x and y coordinates, respectively

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Correspondence to Morteza Karimi.

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Technical Editor: Francisco Ricardo Cunha.

Appendices

Appendix A. Exact solution for two-variable refined plate theory

Based on the Navier’s method, the following displacement functions w b and w s are chosen to automatically satisfy the simply supported boundary conditions of the DONP:

$$\begin{aligned} w^{\text{b}} = \sum\limits_{m = 1}^{\infty } {\sum\limits_{n = 1}^{\infty } {w_{mn}^{\text{b}} } } \sin {\kern 1pt} (\alpha {\kern 1pt} x)\sin \left( {\beta y} \right), \hfill \\ w^{\text{s}} = \sum\limits_{m = 1}^{\infty } {\sum\limits_{n = 1}^{\infty } {w_{mn}^{\text{s}} } } \sin {\kern 1pt} (\alpha {\kern 1pt} x)\sin \left( {\beta y} \right), \hfill \\ \end{aligned}$$
(31)

where α = /a, β = /b and m, n, \(w_{mn}^{\text{b}}\) and \(w_{mn}^{\text{s}}\) are the half wave numbers and coefficients, respectively. Substituting Eq. (31) into Eqs. (19) and (20), the following system of equations are obtained:

$$\left[ {\begin{array}{*{20}c} {S_{11} } & {S_{12} } & {S_{13} } & {S_{14} } \\ {S_{21} } & {S_{22} } & {S_{23} } & {S_{24} } \\ {S_{31} } & {S_{32} } & {S_{33} } & {S_{34} } \\ {S_{41} } & {S_{42} } & {S_{43} } & {S_{44} } \\ \end{array} } \right]{\kern 1pt} {\kern 1pt} \,\left\{ {\begin{array}{*{20}c} {w_{1mn}^{b} } \\ {w_{1mn}^{s} } \\ {w_{2mn}^{b} } \\ {w_{2mn}^{s} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right\},$$
(32)

where

$$\begin{aligned} S_{11} = \, & D_{11} \alpha^{4} + 2\left( {D_{12} + 2D_{66} } \right)\alpha^{2} \beta^{2} + D_{22} \beta^{4} + K + g^{2} K\left( {\alpha^{2} + \beta^{2} } \right) + K_{\text{w}} + g^{2} K_{\text{w}} \left( {\alpha^{2} + \beta^{2} } \right) \\ & \quad + K_{\text{p}} \left( {\alpha^{2} + \beta^{2} } \right) + g^{2} K_{\text{p}} \left( {\alpha^{4} + 2{\kern 1pt} \alpha^{2} \beta^{2} + \beta^{4} } \right) - N_{xx} \alpha^{2} - N_{xx} g^{2} \left( {\alpha^{4} + \alpha^{2} \beta^{2} } \right) \\ & \quad - N_{yy} \beta^{2} - N_{yy} g^{2} \left( {\beta^{4} + \alpha^{2} \beta^{2} } \right) \\ S_{12} = \, & K + g^{2} K\left( {\alpha^{2} + \beta^{2} } \right) + K_{\text{w}} + g^{2} K_{\text{w}} \left( {\alpha^{2} + \beta^{2} } \right) + K_{\text{p}} \left( {\alpha^{2} + \beta^{2} } \right) + g^{2} K_{\text{p}} \left( {\alpha^{4} + 2\alpha^{2} \beta^{2} + \beta^{4} } \right) \\ & \quad - N_{xx} \alpha^{2} - N_{xx} g^{2} \left( {\alpha^{4} + \alpha^{2} \beta^{2} } \right) - N_{yy} \beta^{2} - N_{yy} g^{2} \left( {\beta^{4} + \alpha^{2} \beta^{2} } \right) \\ S_{21} = \, & S_{12} \\ S_{22} = & \left( {{1 \mathord{\left/ {\vphantom {1 {84}}} \right. \kern-0pt} {84}}} \right)D_{11} \alpha^{4} + 2\left( {D_{12} + 2D_{66} } \right)\alpha^{2} \beta^{2} + D_{22} \beta^{4} + A_{55} \alpha^{2} + A_{44} \beta^{2} + K + g^{2} K\left( {\alpha^{2} + \beta^{2} } \right) \\ & \quad + K_{\text{w}} + g^{2} K_{\text{w}} \left( {\alpha^{2} + \beta^{2} } \right) + K_{\text{p}} \left( {\alpha^{2} + \beta^{2} } \right) + g^{2} K_{\text{p}} \left( {\alpha^{4} + 2\alpha^{2} \beta^{2} + \beta^{4} } \right) \\ & \quad - N_{xx} \alpha^{2} - N_{xx} g^{2} \left( {\alpha^{4} + \alpha^{2} \beta^{2} } \right) - N_{yy} \beta^{2} - N_{yy} g^{2} \left( {\beta^{4} + \alpha^{2} \beta^{2} } \right) \\ S_{23} = \, & - K - g^{2} K\left( {\alpha^{2} + \beta^{2} } \right),\quad S_{24} = \, - K - g^{2} K\left( {\alpha^{2} + \beta^{2} } \right) \\ S_{31} = \, & S_{13} ,\quad S_{32} =\, S_{14} ,\quad S_{33} = S_{11} ,\quad S_{34} = S_{12} \\ S_{41} = \, & S_{23} ,\quad S_{42} = S_{24} ,\quad S_{43} = \, S_{21} ,\quad S_{44} = S_{22} . \\ \end{aligned}$$
(33)

The determinant of the coefficient matrix in Eq. (32) must be zero. Therefore, the buckling load for DONP is easily obtained.

Appendix B. First-order shear deformation plate theory

Based on the first-order shear deformation plate theory (FSDT), the equations of motion for a DONP can be obtained as follows:

Nanoplate-1

$$\begin{aligned} D_{11} \psi_{x,xx}^{1} + \left( {D_{12} + D_{66} } \right)\psi_{y,xy}^{1} + D_{66} \psi_{x,yy}^{1} - K_{s} Q_{55} h\left( {\psi_{x}^{1} + w_{,x}^{1} } \right) = 0 \hfill \\ D_{22} \psi_{y,yy}^{1} + \left( {D_{12} + D_{66} } \right)\psi_{x,xy}^{1} + D_{66} \psi_{y,xx}^{1} - k_{s} Q_{44} h\left( {\psi_{y}^{1} + w_{,y}^{1} } \right) = 0 \hfill \\ \hfill \\ A_{44} \left( {\psi_{x,x}^{1} + w_{,xx}^{1} } \right) + A_{55} \left( {\psi_{y,y}^{1} + w_{,yy}^{1} } \right) + K_{\text{w}} w^{1} - g^{2} K_{\text{w}} \left( {\frac{{\partial^{2} w^{1} }}{{\partial x^{2} }} + \frac{{\partial^{2} w^{1} }}{{\partial y^{2} }}} \right) \hfill \\ \quad \quad \quad \quad \quad \quad - K_{\text{p}} \left( {\frac{{\partial^{2} w^{1} }}{{\partial x^{2} }} + \frac{{\partial^{2} w^{1} }}{{\partial y^{2} }}} \right) + g^{2} K_{\text{p}} \left( {\frac{{\partial^{4} w^{1} }}{{\partial x^{4} }} + 2\frac{{\partial^{4} w^{1} }}{{\partial x^{2} \partial y^{2} }} + \frac{{\partial^{4} w^{1} }}{{\partial y^{4} }}} \right) + K\left( {w^{1} - w^{2} } \right) \hfill \\ \quad \quad \quad \quad \quad \quad - g^{2} K\left( {\frac{{\partial^{2} w^{1} }}{{\partial x^{2} }} + \frac{{\partial^{2} w^{1} }}{{\partial y^{2} }} - \frac{{\partial^{2} w^{2} }}{{\partial x^{2} }} - \frac{{\partial^{2} w^{2} }}{{\partial y^{2} }}} \right) + P\left( {w^{1} } \right) = 0 \hfill \\ \hfill \\ P\left( {w^{1} } \right) = N_{xx} \left( {\frac{{\partial^{2} w^{1} }}{{\partial x^{2} }}} \right) + N_{yy} \left( {\frac{{\partial^{2} w^{1} }}{{\partial y^{2} }}} \right) + 2N_{xy} \left( {\frac{{\partial^{2} w^{1} }}{\partial xy}} \right) - N_{xx} g^{2} \left( {\frac{{\partial^{4} w^{1} }}{{\partial x^{4} }} + \frac{{\partial^{4} w^{1} }}{{\partial x^{2} y^{2} }}} \right) \hfill \\ \quad \quad \quad \quad \quad \quad - N_{yy} g^{2} \left( {\frac{{\partial^{4} w^{1} }}{{\partial y^{4} }} + \frac{{\partial^{4} w^{1} }}{{\partial x^{2} y^{2} }}} \right) - 2N_{xy} g^{2} \left( {\frac{{\partial^{4} w^{1} }}{{\partial x^{3} \partial y}} + \frac{{\partial^{4} w^{1} }}{{\partial x\partial y^{3} }}} \right). \hfill \\ \end{aligned}$$
(34)

Nanoplate-2

$$\begin{aligned} D_{11} \psi_{x,xx}^{2} + \left( {D_{12} + D_{66} } \right)\psi_{y,xy}^{2} + D_{66} \psi_{x,yy}^{2} - k_{\text{s}} Q_{55} h\left( {\psi_{x}^{2} + w_{,x}^{2} } \right) = 0 \hfill \\ D_{22} \psi_{y,yy}^{2} + \left( {D_{12} + D_{66} } \right)\psi_{x,xy}^{2} + D_{66} \psi_{y,xx}^{2} - k_{\text{s}} Q_{44} h\left( {\psi_{y}^{2} + w_{,y}^{2} } \right) = 0 \hfill \\ \hfill \\ A_{44} \left( {\psi_{x,x}^{2} + w_{,xx}^{2} } \right) + A_{55} \left( {\psi_{y,y}^{2} + w_{,yy}^{2} } \right) + K_{\text{w}} w^{2} - g^{2} K_{\text{w}} \left( {\frac{{\partial^{2} w^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} w^{2} }}{{\partial y^{2} }}} \right) \hfill \\ \quad \quad \quad \quad \quad \quad - K_{\text{p}} \left( {\frac{{\partial^{2} w^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} w^{2} }}{{\partial y^{2} }}} \right) + g^{2} K_{\text{p}} \left( {\frac{{\partial^{4} w^{2} }}{{\partial x^{4} }} + 2\frac{{\partial^{4} w^{2} }}{{\partial x^{2} \partial y^{2} }} + \frac{{\partial^{4} w^{2} }}{{\partial y^{4} }}} \right) + K\left( {w^{2} - w^{1} } \right) \hfill \\ \quad \quad \quad \quad \quad \quad - g^{2} K\left( {\frac{{\partial^{2} w^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} w^{2} }}{{\partial y^{2} }} - \frac{{\partial^{2} w^{1} }}{{\partial x^{2} }} - \frac{{\partial^{2} w^{1} }}{{\partial y^{2} }}} \right) + P\left( {w^{2} } \right) = 0 \hfill \\ \hfill \\ P\left( {w^{2} } \right) = N_{xx} \left( {\frac{{\partial^{2} w^{2} }}{{\partial x^{2} }}} \right) + N_{yy} \left( {\frac{{\partial^{2} w^{2} }}{{\partial y^{2} }}} \right) + 2N_{xy} \left( {\frac{{\partial^{2} w^{2} }}{\partial xy}} \right) - N_{xx} g^{2} \left( {\frac{{\partial^{4} w^{2} }}{{\partial x^{4} }} + \frac{{\partial^{4} w^{2} }}{{\partial x^{2} y^{2} }}} \right) \hfill \\ \quad \quad \quad \quad \quad \quad - N_{yy} g^{2} \left( {\frac{{\partial^{4} w^{2} }}{{\partial y^{4} }} + \frac{{\partial^{4} w^{2} }}{{\partial x^{2} y^{2} }}} \right) - 2N_{xy} g^{2} \left( {\frac{{\partial^{4} w^{2} }}{{\partial x^{3} \partial y}} + \frac{{\partial^{4} w^{2} }}{{\partial x\partial y^{3} }}} \right). \hfill \\ \end{aligned}$$
(35)

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Shokrani, M.H., Karimi, M., Tehrani, M.S. et al. Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method. J Braz. Soc. Mech. Sci. Eng. 38, 2589–2606 (2016). https://doi.org/10.1007/s40430-015-0370-0

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