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Surface and Nonlocal Effects on Coupled In-Plane Shear Buckling and Vibration of Single-Layered Graphene Sheets Resting on Elastic Media and Thermal Environments using DQM

Published online by Cambridge University Press:  10 May 2018

F. Abdollahi  and
A. Ghassemi
F. Abdollahi
Affiliation:
Department of Mechanical EngineeringNajafabad BranchIslamic Azad UniversityNajafabad, Iran
A. Ghassemi*
Affiliation:
Modern Manufacturing Technologies Research CenterNajafabad BranchIslamic Azad UniversityNajafabad, Iran
*
*Corresponding author ([email protected])
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Abstract

In this article, surface and nonlocal effects are explored in the analysis of buckling and vibration in rectangular single-layered graphene sheets embedded in elastic media and subjected to coupled in-plane loadings and thermal conditions. The small-scale and surface effects are taken into account using the Eringen's nonlocal elasticity and Gurtin-Murdoch's theory, respectively. Using the principle of virtual work, the governing equations considering small-scale are derived for the nanoplate bulk and surface. The differential quadrature method (DQM) is utilized for the solution of the relevant problems and the results are validated against Navier's solutions. The impacts of the nonlocal parameter, Winkler and shear elastic moduli, temperature rise, boundary conditions, and the in-plane biaxial, uniaxial, and shear loadings on the surface effects of buckling and vibration are investigated. Numerical results show that increasing nonlocal parameter leads to enhanced surface effects on both buckling and vibration. This is in contrast to those reported elsewhere. Moreover, increasing in-plane loads are observed to enhance surface effects on vibration. On the other hand, the nonlocal parameter is observed to have more pronounced effects on shear buckling and vibration of plates subjected to coupled in-plane shear loads than those subjected to biaxial and uniaxial loads. This is while surface effects have greater impacts on biaxial buckling and vibration of nanoplates than on shear buckling and vibration.

Keywords

Surface effectShear buckling/vibrationNonlocal elasticitySingle-layered graphene sheets
Type
Research Article
Information
Journal of Mechanics , Volume 34 , Issue 6 , December 2018 , pp. 847 - 862
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2018 

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References

1. Ball, P., “Roll up for the Revolution,” Nature, 414, pp. 142–144 (2001).Google Scholar
2. Li, C. and Chou, T. W., “A Structural Mechanics Approach for the Analysis of Carbon Nanotubes,” International Journal of Solids and Structures, 40, pp. 2487–2499 (2003).Google Scholar
3. Govindjee, S. and Sackman, J. L., “On the Use of Continuum Mechanics to Estimate the Properties of Nanotubes,” Solid State Communications, 110, pp. 227–230 (1999).Google Scholar
4. Gurtin, M. E. and Murdoch, A. I., “Surface Stress in Solids,” International Journal of Solids and Structures, 14, pp. 431–440 (1978).Google Scholar
5. Ansari, R. and Sahmani, S., “Bending Behavior and Buckling of Nanobeams Including Surface Stress Effects Corresponding to Different Beam Theories,” International Journal of Engineering Science, 49, pp. 1244–1255 (2011).Google Scholar
6. Lu, T. Q., Zhang, W. X. and Wang, T. J., “The Surface Effect on the Strain Energy Release Rate of Buckling Delamination in Thin Film – Substrate Systems,” International Journal of Engineering Science, 49, pp. 967–975 (2011).Google Scholar
7. Yan, Z. and Jiang, L. Y., “Surface Effects on the Electroelastic Responses of a Thin Piezoelectric Plate with Nanoscale Thickness,” Journal of Physics D: Applied Physics, 45, pp. 255401–255409 (2012).Google Scholar
8. Yan, Z. and Jiang, L. Y., “Vibration and Buckling Analysis of a Piezoelectric Nanoplate Considering Surface Effects and in-Plane Constraints,” Proceedings of the Royal Society of London A, 468, pp. 3458–3475 (2012).Google Scholar
9. Zhang, J., Wang, C. and Chen, W., “Surface and Piezoelectric Effects on the Buckling of Piezoelectric Nanofilms Due to Mechanical Loads,” Meccanica, 49, pp. 181–189 (2014).Google Scholar
10. Asemi, S. R. and Farajpour, A., “Decoupling the Nonlocal Elasticity Equations for Thermo-Mechanical Vibration of Circular Graphene Sheets Including Surface Effects,” Physica E: Low-Dimensional Systems and Nanostructures, 60, pp. 80–90 (2014).Google Scholar
11. Mahmoud, F. F., Eltaher, M. A., Alshorbagy, A. E. and Meletis, E. I., “Static Analysis of Nanobeams Including Surface Effects by Nonlocal Finite Element,” Journal of Mechanical Science and Technology, 26, pp. 3555–3563 (2012).Google Scholar
12. Eltaher, M. A., Mahmoud, F. F., Assie, E. I. and Meletis, E. I., “Coupling Effects of Nonlocal and Surface Energy on Vibration Analysis of Nanobeams,” Applied Mathematics and Computation, 224, pp. 760–774 (2013).Google Scholar
13. Mohammadi, M., Goodarzi, M., Ghayour, M. and Alivand, S., “Small Scale Effect on the Vibration of Orthotropic Plates Embedded in an Elastic Medium and under Biaxial in-Plane Pre-Load via Nonlocal Elasticity Theory,” Journal of Solid Mechanics, 4, pp. 128–143 (2012).Google Scholar
14. Mohammadi, M., Farajpour, A., Goodarzi, M. and Shehni Nezhad Pour, H., “Numerical Study of the Effect of Shear in-Plane Load on the Vibration Analysis of Graphene Sheet Embedded in an Elastic Medium,” Computational Materials Science, 82, pp. 510–520 (2014).Google Scholar
15. Murmu, T. and Pradhan, S. C., “Vibration Analysis of Nanoplates under Uniaxial Prestressed Conditions via Nonlocal Elasticity,” Journal of Applied Physics, 106, pp. 104301–104309 (2009).Google Scholar
16. Mohammadi, M., Farajpour, A., Moradi, A. and Ghayour, M., “Shear Buckling of Orthotropic Rectangular Graphene Sheet Rmbedded in an Elastic Medium in Thermal Environment,” Composites Part B: Engineering, 56, pp. 629–637 (2014).Google Scholar
17. Sobhy, M., “Natural Frequency and Buckling of Orthotropic Nanoplates Resting on Two-Parameter Elastic Foundations with Various Boundary Conditions,” Journal of Mechanics, 30, pp. 443–453 (2014).Google Scholar
18. Zenkour, A. M. and Sobhy, M., “Nonlocal Elasticity Theory for Thermal Buckling of Nanoplates Lying on Winkler–Pasternak Elastic Substrate Medium,” Physica E: Low-Dimensional Systems and Nanostructures, 53, pp. 251–259 (2013).Google Scholar
19. Sobhy, M., “Thermomechanical Bending and Free Vibration of Single-Layered Graphene Sheets Embedded in an Elastic Medium,” Physica E: Low-Dimensional Systems and Nanostructures, 56, pp. 400–409 (2014).Google Scholar
20. Sobhy, M., “Generalized Two-Variable Plate Theory for Multi-Layered Graphene Sheets with Arbitrary Boundary Conditions,” Acta Mechanica, 225, pp. 2521–2538 (2014).Google Scholar
21. Sobhy, M., “Levy-Type Solution for Bending of Single-Layered Graphene Sheets in Thermal Environment Using the Two-Variable Plate Theory,” International Journal of Mechanical Sciences, 90, pp. 171–178 (2015).Google Scholar
22. Sobhy, M., “Hygrothermal Vibration of Orthotropic Double-Layered Graphene Sheets Embedded in an Elastic Medium Using the Two-Variable Plate Theory,” Applied Mathematical Modelling, 40, pp. 85–99 (2016).Google Scholar
23. Sobhy, M. and Radwan, A. F., “A New Quasi 3D Nonlocal Plate Theory for Vibration and Buckling of FGM Nanoplates,” International Journal of Applied Mechanics, 9, pp. 1750008 (2017).Google Scholar
24. Sobhy, M., “A Comprehensive Study on FGM Nanoplates Embedded in an Elastic Medium,” Composite Structure, 134, pp. 966–980 (2015).Google Scholar
25. Alzahrani, E. O., Zenkour, A. M. and Sobhy, M., “Small Scale Effect on Hygro-Thermo-Mechanical Bending of Nanoplates Embedded in an Elastic Medium,” Composite Structure, 105, pp. 163–172 (2013).Google Scholar
26. Sobhy, M., “Hygrothermal Deformation of Orthotropic Nanoplates Based on the State-Space Concept,” Composites Part B: Engineering, 79, pp. 224–235 (2015).Google Scholar
27. Yang, K. et al., “Buckling Behavior of Substrate Supported Graphene Sheets,” Materials, 9, pp. 32 (2016).Google Scholar
28. Karličić, D., Kozić, P. and Pavlović, R., “Free Transverse Vibration of Nonlocal Viscoelastic Orthotropic Multi-Nanoplate System (MNPS) Embedded in a Viscoelastic Medium,” Composite Structures, 115, pp. 89–99 (2014).Google Scholar
29. Karličić, D., Adhikari, S., Murmu, T. and Cajić, M., “Exact Closed-Form Solution for Non-Local Vibration and Biaxial Buckling of Bonded Multi-Nanoplate System,” Composites Part B: Engineering, 66, pp. 328–339 (2014).Google Scholar
30. Karličić, D., Cajić, M., Kozić, P. and Pavlović, I., “Temperature Effects on the Vibration and Stability Behavior of Multi-Layered Graphene Sheets Embedded in an Elastic Medium,” Composite Structures, 131, pp. 672–681 (2015).Google Scholar
31. Karličić, D., Cajić, M., Adhikari, S., Kozić, P. and Murmu, T., “Vibrating Nonlocal Multi-Nanoplate System under Inplane Magnetic Field,” European Journal of Mechanics-A/Solids, 64, pp. 29–45 (2017).Google Scholar
32. Sarrami-Foroushani, S. and Azhari, M., “Nonlocal Vibration and Buckling Analysis of Single and Multi-Layered Graphene Sheets Using Finite Strip Method Including Van Der Waals Effects,” Physica E: Low-Dimensional Systems and Nanostructures, 57, pp. 83–95 (2014).Google Scholar
33. Anjomshoa, A., Shahidi, A. R., Hassani, B. and Jomehzadeh, E., “Finite Element Buckling Analysis of Multi-Layered Graphene Sheets on Elastic Substrate Based on Nonlocal Elasticity Theory,” Applied Mathematical Modelling, 38, pp. 5934–5955 (2014).Google Scholar
34. Nazemnezhad, R. and Hosseini-Hashemi, S., “Free Vibration Analysis of Multi-Layer Graphene Nanoribbons Incorporating Interlayer Shear Effect via Molecular Dynamics Simulations and Nonlocal Elasticity,” Physics Letters A, 378, pp. 3225–3232 (2014).Google Scholar
35. Hosseini, M., Bahreman, M. and Jamalpoor, A., “Thermomechanical Vibration Analysis of FGM Viscoelastic Multi-Nanoplate System Incorporating the Surface Effects via Nonlocal Elasticity Theory,” Microsystem Technologies, DOI: 10.1007/s00542-016-3133-7 (2016).Google Scholar
36. Wang, K. F. and Wang, B. L., “Combining Effects of Surface Energy and Non-Local Elasticity on the Buckling of Nanoplates,” Micro and Nano Letters, 6, pp. 941–943 (2011).Google Scholar
37. Wang, K. F. and Wang, B. L., “Vibration of Nanoscale Plates with Surface Energy via Nonlocal Elasticity,” Physica E: Low-Dimensional Systems and Nanostructures, 44, pp. 448–453 (2011).Google Scholar
38. Malekzadeh, P. and Shojaee, M., “A Two-Variable First-Order Shear Deformation Theory Coupled with Surface and Nonlocal Effects for Free Vibration of Nanoplates,” Journal of Vibration and Control, 21, pp. 2755–2772 (2013).Google Scholar
39. Eringen, A. C. and Edelen, D. G. B., “On Nonlocal Elasticity,” International Journal of Engineering Science, 10, pp. 233–248 (1972).Google Scholar
40. Karimi, M. and Shahidi, A. R., “Thermo-Mechanical Vibration, Buckling, and Bending of Orthotropic Graphene Sheets Based on Nonlocal Two-Variable Refined Plate Theory Using Finite Difference Method Considering Surface Energy Effects,” Proceedings of the Institution of Mechanical Engineers, Part N: Journal of Nanomaterials, Nanoengineering and Nanosystems, 231, pp. 111–130 (2017).Google Scholar
41. Shokrani, M. H., Karimi, M., Tehrani, M. S. and Mirdamadi, H. R., “Buckling Analysis of Double-Orthotropic Nanoplates Embedded in Elastic Media Based on Non-Local Two-Variable Refined Plate Theory Using the GDQ Method,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, 38, pp. 2589–2606 (2016).Google Scholar
42. Karimi, M. and Shahidi, A. R., “Nonlocal, Refined Plate, and Surface Effects Theories Used to Analyze Free Vibration of Magnetoelectroelastic Nanoplates under Thermo-Mechanical and Shear Loadings,” Applied Physics A, DOI: 10.1007/s00339-017-0828-2 (2017)Google Scholar
43. Karimi, M., Shahidi, A. R. and Ziaei-Rad, S., “Surface Layer and Nonlocal Parameter Effects on the in-Phase and out-of-Phase Natural Frequencies of a Double-Layer Piezoelectric Nanoplate under Thermo-Electro-Mechanical Loadings,” Microsystem Technologies, DOI: 10.1007/s00542-017-3395-8 (2017).Google Scholar
44. Karimi, M., Shokrani, M. H. and Shahidi, A. R., “Size-Dependent Free Vibration Analysis of Rectangular Nanoplates with the Consideration of Surface Effects Using Finite Difference Method,” Journal of Applied and Computational Mechanics, 1, pp. 122–133 (2015).Google Scholar
45. Karimi, M., Mirdamadi, H. R. and Shahidi, A. R., “Positive and Negative Surface Effects on the Buckling and Vibration of Rectangular Nanoplates under Biaxial and Shear in-Plane Loadings Based on Non-Local Elasticity Theory,” Journal of the Brazilian Society of Mechanical Sciences and Engineering, 39, pp. 1391–1404 (2017).Google Scholar
46. Karimi, M., Mirdamadi, H. R. and Shahidi, A. R., “Shear Vibration and Buckling of Double-Layer Orthotropic Nanoplates Based on RPT Resting on Elastic Foundations by DQM Including Surface Effects,” Microsystem Technologies, 23, pp. 765–797 (2017).Google Scholar
47. Karimi, M., Haddad, H. A. and Shahidi, A. R., “Combining Surface Effects and Non-Local Two Variable Refined Plate Theories on the Shear/Biaxial Buckling and Vibration of Silver Nanoplates,” Micro and Nano Letters, 10, pp. 276–281 (2015).Google Scholar
48. Karimi, M. and Shahidi, A. R., “Finite Difference Method for Sixth Order Derivatives of Differential Equations in Buckling of Nanoplates Due to Coupled Surface Energy and Non-Local Elasticity Theories,” International Journal of Nano Dimension, 6, pp. 525–538 (2015).Google Scholar
49. Karimi, M. and Shahidi, A. R., “Finite Difference Method for Biaxial and Uniaxial Buckling of Rectangular Silver Nanoplates Resting on Elastic Foundations in Thermal Environments Based on Surface Stress and Nonlocal Elasticity Theories,” Journal of Solid Mechanics, 8, pp. 719–733 (2016).Google Scholar
50. Shu, C., “Differential Quadrature and Its Application in Engineering,” Springer, Berlin (2000).Google Scholar
51. Radic, N., Jeremic, D., Trifkovic, S. and Milutinovic, M., “Buckling Analysis of Double-Orthotropic Nanoplates Embedded in Pasternak Elastic Medium Using Nonlocal Elasticity Theory,” Composites Part B: Engineering, 61, pp. 162–171 (2014).Google Scholar
52. Bassily, S. F. and Dickinson, S. M., “Buckling and Lateral Vibration of Rectangular Plates Subject to in-Plane Loads-a Ritz Approach,” Journal of Sound and Vibration, 24, pp. 219–239 (1972).Google Scholar