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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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:
where α = mπ/a, β = nπ/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:
where
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
Nanoplate-2
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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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DOI: https://doi.org/10.1007/s40430-015-0370-0